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Nereid
2006-Mar-30, 09:32 PM
In the why math? thread (http://www.bautforum.com/showthread.php?t=39653) in the ATM section, I said I'd start a thread (http://www.bautforum.com/showpost.php?p=711731&postcount=56), here in General Science, on infinities in your daily life. This is it.

We all know that F(ahrenheit) and C(elsius) are quite arbitrary - both for the zero point and the scale. Along came K(elvin), which at least has a non-arbitrary zero point, even if its scale is still arbitrary.

We've no doubt read about how close to 0K researchers have been able to get, and maybe we've read about a way to 'show' how unattainable 0K is, by taking the log of the temperature (0K then becomes -∞, which is, 'of course', unattainable).

But why do we use a linear scale for temperature? Why don't we use a logarithmic one? After all, such a scale would 'show' the unattainability of '0K' much more clearly, wouldn't it? And anyway, we already have several logarithmic thingies, don't we? pH, and decibels, and earthquakes (think Richter), for example.

It turns out that there is an interesting aspect to temperature and logs ... and it has to do with entropy. In one definition of temperature, it is just dE/dS, where E is the total energy of the system, and S the entropy; and S is just the logarithm of the number of microstates of the system.

And this definition leads to the possibility of negative temperatures (http://en.wikipedia.org/wiki/Negative_temperature), which, despite what you might think, are not colder than 0K, but hotter than +∞!

So, where might you encounter an infinity (temperature in this case) in your daily life? The CD/DVD player in your PC likely has a component that goes through a state which has a temperature of +∞ (and one with a temperature of -∞) during its normal operation. So does the barcode scanner in your local supermarket. And a great deal of other circumstances too.

But this is just a mathematician's (or physicist's) trick, right? Nothing is really infinite in nature, right?

Well, I guess that's up to you to decide - how does 'infinity', a mathematical concept, relate to 'reality'? If you concede, perhaps ever so reluctantly, that the theories of physics have at least some relationship to 'reality', then I guess you've no choice but to conclude that 'infinities' do, indeed, occur in nature (and, as the above illustrates, in your everyday life).

Another example - much more open and shut (perhaps).

'Heat capacity (http://en.wikipedia.org/wiki/Heat_capacity)' is a pretty darn concrete thing, right? I mean, for goodness sake, you can measure it! And, since you can't measure 'infinity', there's no way that could possibly be infinitite, in 'reality', right? Wrong (http://en.wikipedia.org/wiki/Phase_(matter)).

snarkophilus
2006-Mar-31, 02:06 AM
I can think of a few, though they're maybe not what you are looking for.

The first occurs every time I walk toward a door. First I get halfway there. Then I get halfway from that. Then halfway from that. That happens an infinite number of times, until I touch the door at the infinity'th time division (though a finite time later).

In the game of go, depending upon the rule set, you can reach situations where the game would go on for an infinite number of moves, though typically the game is halted long before that.

In terms of measurement, I have two eggs in my hand. You have no eggs in your hand. We can measure that. But I have some infinity times as many eggs as you do.

Those infinite heat capacities are curious things, but I'd think that they're more of a mathematical anomaly than a real-world situation. Transitions across those boundaries do occur, typically through some intermediate stage. I think that there are even continuous parameter spaces that are non-analytic (using the terminology of that article), but don't quote me on that.

Grey
2006-Mar-31, 03:23 AM
And to be pedantic, those states of a laser can only be considered to have a temperature of negative infinity if you include only the temperature contribution from the energy of the state change used in the stimulated emission itself, where you have a population inversion.laser. There are other contributions (random motions of the atoms, other atomic state changes, and so forth), and if you include all of these microstates, you'll find that the total temperature for any macroscopic configuration is always a finite positive number.

I will agree, however, that it's perfectly possible in principle for things in nature to be infinite. The universe itself might be such a thing.

Ken G
2006-Mar-31, 08:16 AM
Even more pedantic, since it is easy to make things zero, and physical expressions allow ratios, it is pretty easy to also make infinity. For example, is not the number of trips I've taken to Chicago, per trip to Miami, a physical entity? But I've been to Chicago many times, and I've never been to Miami...

czeslaw
2006-Mar-31, 08:54 AM
It is an excellent problem.
Zero temperature means zero oscillations of the atom and higher density, higher temperature more oscillations and lower density.
To create a singularity in a Black Hole the oscillations should stops and the temperature reaches an absolute zero. Is it possible to separate baryons from bosons in such a closed system as the Black Hole is ?
A gravitational energy is a potential energy which could be transmitted in kinetic energy (oscillation).
The star loses its energy in supernova outburst – The emission of the photons and neutrinos.
When a neutron star collapses into a Black Hole the gravitational pressure is like in relativistic collision creating matter-antimatter pairs. It will cause an energetic sound wave. In a Black Hole closed system this sound wave will move till event horizon and back into the centre and drives the rotations as a whirling top. This balanced energy will prevent to create a superdense cold singularity.
The energy of gravitational pressure = GM^2/R^4 (pressure) x 4piR^3/3 (volume)
is balanced by kinetic oscillations
E=0,5 Mv^2
Very small Black Hole emits more energy then absorbs but Hawking proved – this BH will evaporate (explode).
May be you know a way how to seperate an energy from baryons in the Black Hole ?

tusenfem
2006-Mar-31, 10:14 AM
It is an excellent problem.
Zero temperature means zero oscillations of the atom and higher density, higher temperature more oscillations and lower density.
To create a singularity in a Black Hole the oscillations should stops and the temperature reaches an absolute zero. Is it possible to separate baryons from bosons in such a closed system as the Black Hole is ?
A gravitational energy is a potential energy which could be transmitted in kinetic energy (oscillation).
The star loses its energy in supernova outburst – The emission of the photons and neutrinos.
When a neutron star collapses into a Black Hole the gravitational pressure is like in relativistic collision creating matter-antimatter pairs. It will cause an energetic sound wave. In a Black Hole closed system this sound wave will move till event horizon and back into the centre and drives the rotations as a whirling top. This balanced energy will prevent to create a superdense cold singularity.
The energy of gravitational pressure = GM^2/R^4 (pressure) x 4piR^3/3 (volume)
is balanced by kinetic oscillations
E=0,5 Mv^2
Very small Black Hole emits more energy then absorbs but Hawking proved – this BH will evaporate (explode).
May be you know a way how to seperate an energy from baryons in the Black Hole ?

Whatever does this dr*vel have to do with the original post by Nereid? Not everything revolves around the inside of a black hole and Hawking radiation.

czeslaw
2006-Mar-31, 12:01 PM
Whatever does this dr*vel have to do with the original post by Nereid? Not everything revolves around the inside of a black hole and Hawking radiation.
Nereid wrote about extremaly low remperature.
I am interesting how to create such a cold singularity inside a Black Hole.
It is good example of the infinity, I think.

Nereid
2006-Mar-31, 12:36 PM
Nereid wrote about extremaly low remperature.
I am interesting how to create such a cold singularity inside a Black Hole.
It is good example of the infinity, I think.Maybe, but they are certainly not the sorts of things that you encounter 'every day', are they!

Thanks to Grey and Ken G and snarkophilus - extending the discussion to just what 'infinity' is, and how it may be part of your physical reality (in an 'every day' sense).

The relationship is, indeed, quite nuanced, with many different perspectives.

There are infinities galore, in every atom of your body, by one reading of a powerful technique in quantum theory (http://en.wikipedia.org/wiki/Renormalization).

If I may take this a little further ... to what extent does the existence of an 'infinity' in nature depend upon a (physical/physics) theory? Can there be an 'infinity' without a theory? Note that the theory may be implied, rather than explicit.

tusenfem
2006-Mar-31, 01:42 PM
Well, there is ofcourse the well known fact that I am infinitely more intelligent than the other people here on the board ;-)

As for infinity in nature. I am mostly aware of plasma waves in my daily life (well, in the office that is). Naturally, there you get infinities in the equations, because of the setup of the Maxwell equations, the equotion of motion and harmonic waves. No matter how you turn it, there is always a "root", which is basically a nice name for an infinity, e.g. divide by (w - w0). Now, the interesting thing here is, that we like this root, because that is where the wave mode is (where w0 can be substituted for more difficult stuff).
So, is the infinitiy natural? I would say yes, but not in the matematical sense that it goes to infty but in that something special happens there. It is just the mathematical word for "something interesting happens", which can, unfortunately also be misunderstood, because sometimes infinity means "you are wrong."

Relmuis
2006-Mar-31, 01:54 PM
A perfect insulator has infinite resistance.

(Yes, you can probably cook it, or something by applying enough voltage. But for low voltages the slope of voltage as a function of current is vertical.)

dvb
2006-Mar-31, 05:16 PM
The first occurs every time I walk toward a door. First I get halfway there. Then I get halfway from that. Then halfway from that. That happens an infinite number of times, until I touch the door at the infinity'th time division (though a finite time later). I've heard this one before. Just curious though, if you measured the distance in atoms between you, and the door, then halved that distance, wouldn't you reach a point where you end up with a single indivisible atom? I'm not saying that atoms are indivisible, but perhaps the unit of measurement is what's causing this infinity. An atom represents a point in space, whereas an arbitrary unit of measurement does not, thus can be divided to infinity. Just a thought.

Nereid
2006-Mar-31, 05:28 PM
Well, there is ofcourse the well known fact that I am infinitely more intelligent than the other people here on the board ;-)

As for infinity in nature. I am mostly aware of plasma waves in my daily life (well, in the office that is). Naturally, there you get infinities in the equations, because of the setup of the Maxwell equations, the equotion of motion and harmonic waves. No matter how you turn it, there is always a "root", which is basically a nice name for an infinity, e.g. divide by (w - w0). Now, the interesting thing here is, that we like this root, because that is where the wave mode is (where w0 can be substituted for more difficult stuff).
So, is the infinitiy natural? I would say yes, but not in the matematical sense that it goes to infty but in that something special happens there. It is just the mathematical word for "something interesting happens", which can, unfortunately also be misunderstood, because sometimes infinity means "you are wrong."And where would any one of us encounter these plasma waves, in our daily lives, tusenfem? For example, if you're lucky enough to live at a high northern or southern latitude, would those beautiful aurorae have these kinds of waves?

zebo-the-fat
2006-Mar-31, 06:01 PM
The amount of logical thought used by most politicians is infinately small!:D

hhEb09'1
2006-Mar-31, 06:08 PM
If I may take this a little further ... to what extent does the existence of an 'infinity' in nature depend upon a (physical/physics) theory? Can there be an 'infinity' without a theory? Note that the theory may be implied, rather than explicit.I'd like a little clarification on this question. Is the question "Can there be an infinity without a theory, implied or explicit?" Or is it "Can there be an infnity without an explicit theory?" The last would allow an implicit theory.

And the next question would be what is an example of an implicit theory.

gzhpcu
2006-Mar-31, 06:42 PM
IMHO, infinity is just a mathematical construct. Sure, the universe might be infinite, but this is just a supposition. The symbol of infinity is needed, for example, in math to cover division by zero, or to set bounds for an integral, etc. Also (again just IMHO), o dimensional particles, 1 dimensional strings are mathematical constructs and do not correspond to reality.

gzhpcu
2006-Mar-31, 06:43 PM
P.S. Remember the endless discussion of whether 0.9999999..... equals 1.000000? In math, yes, in reality no.

Nereid
2006-Mar-31, 07:27 PM
The first occurs every time I walk toward a door. First I get halfway there. Then I get halfway from that. Then halfway from that. That happens an infinite number of times, until I touch the door at the infinity'th time division (though a finite time later).I've heard this one before. Just curious though, if you measured the distance in atoms between you, and the door, then halved that distance, wouldn't you reach a point where you end up with a single indivisible atom? I'm not saying that atoms are indivisible, but perhaps the unit of measurement is what's causing this infinity. An atom represents a point in space, whereas an arbitrary unit of measurement does not, thus can be divided to infinity. Just a thought.
I'd like a little clarification on this question. Is the question "Can there be an infinity without a theory, implied or explicit?" Or is it "Can there be an infnity without an explicit theory?" The last would allow an implicit theory.

And the next question would be what is an example of an implicit theory.Perfect timing! :)

There's an implicit theory - that space (and time?) are continuous.

Disinfo Agent
2006-Mar-31, 07:28 PM
So, where might you encounter an infinity (temperature in this case) in your daily life? The CD/DVD player in your PC likely has a component that goes through a state which has a temperature of +infinity (and one with a temperature of -infinity K) during its normal operation.Measured in which scale? Surely you don't mean +infinity K and -infinity K... Or do you?

But this is just a mathematician's (or physicist's) trick, right? Nothing is really infinite in nature, right?Discounting pedantic (and debatable) examples like the ones in Zeno's paradox, that's what we've observed so far. I don't, however, discount the possibility that there may be physical infinities in the universe which we haven't found yet.

Disinfo Agent
2006-Mar-31, 07:31 PM
P.S. Remember the endless discussion of whether 0.9999999..... equals 1.000000? In math, yes, in reality no.I'd hate to resurrect that discussion, but I have to say you still don't get it. They are the same, in math and in reality.

Grey
2006-Mar-31, 07:36 PM
A perfect insulator has infinite resistance.Well, of course there's no such thing as a perfect insulator, as far as we know. Apply enough voltage across anything, and it will conduct electricity.

IMHO, infinity is just a mathematical construct. Sure, the universe might be infinite, but this is just a supposition.But if the universe is infinite, then it's certainly not just a mathematical concept. Are you prepared to claim that the universe cannot possibly be infinite, because you're absolutely certain that nothing can be infinite?

The symbol of infinity is needed, for example, in math to cover division by zero, or to set bounds for an integral, etc.But division by zero is something that can certainly happen, right? Ken's simple example of the ratio of trips he's taken to Chicago and Miami shows that. If you'd prefer something more physical, how about the ratio of the rest masses of an electron and a photon. Hmm, or we could take the ratio of the rest masses of a graviton and a photon, and we'd get something undefined.

Disinfo Agent
2006-Mar-31, 07:41 PM
Another example - much more open and shut (perhaps).

'Heat capacity (http://en.wikipedia.org/wiki/Heat_capacity)' is a pretty darn concrete thing, right? I mean, for goodness sake, you can measure it! And, since you can't measure 'infinity', there's no way that could possibly be infinitite, in 'reality', right? Wrong (http://en.wikipedia.org/wiki/Phase_(matter)).Yes, heat capacity can be measured in 'reality', and it can become infinite -- in theory. But can you show me an infinite measurement of heat capacity?

gzhpcu
2006-Mar-31, 09:02 PM
But if the universe is infinite, then it's certainly not just a mathematical concept. Are you prepared to claim that the universe cannot possibly be infinite, because you're absolutely certain that nothing can be infinite?

But division by zero is something that can certainly happen, right? Ken's simple example of the ratio of trips he's taken to Chicago and Miami shows that. If you'd prefer something more physical, how about the ratio of the rest masses of an electron and a photon. Hmm, or we could take the ratio of the rest masses of a graviton and a photon, and we'd get something undefined.

To the first question, no I am not certain, but it can't be proven either way.
To the second question, it seems to me you are basing yourself on statements made by mathematical theories.

gzhpcu
2006-Mar-31, 09:03 PM
I'd hate to resurrect that discussion, but I have to say you still don't get it. They are the same, in math and in reality.

Maybe I am obtuse, but prove it in reality and not mathematical terms...

Grey
2006-Mar-31, 09:23 PM
To the first question, no I am not certain, but it can't be proven either way.So that means that you'd be willing to accept that something could be infinite in reality. Sounds good.

To the second question, it seems to me you are basing yourself on statements made by mathematical theories.I guess I don't understand what you mean. Certainly whenever I talk about measuring something, I'm going to have to use math in some way. Unless we're dead wrong about the mass of a photon, isn't it true that the ratio of the mass of an electron to the mass of a photon is infinite? We often talk about the ratio of the masses of a proton and an electron, or a muon and an electron; why shouldn't we allowed to talk about the ratio these masses?

Maybe I am obtuse, but prove it in reality and not mathematical terms...How can you even define 0.9999... and 1, except in mathematical terms?

Disinfo Agent
2006-Mar-31, 09:27 PM
Maybe I am obtuse, but prove it in reality and not mathematical terms...But numbers are mathematical! That's like asking me to prove evolution with math, and not in biological terms...

I think what you haven't understood yet is that both '0.999...' and '1', in this context, are just names. It seems you are thinking that there is a one, true physical '1' and a one, true physical '0.999...' out there in the universe, or perhaps physical processes of some kind, which may disprove the equality. But '0.999...' is just a mathematical name, a label. That's what it's always been.

Nereid
2006-Mar-31, 09:49 PM
Yes, heat capacity can be measured in 'reality', and it can become infinite -- in theory. But can you show me an infinite measurement of heat capacity?Excellent question! :clap:

Now let me see, ... to 'measure' something I need some kind of an instrument, and to be sure that is a faithful servant, telling me the whole measurement and nothing but the measurement, I need to understand how that instrument goes about turning an input of some sort into an output (of the numerical kind, of course).

Hmm, don't I need to have confidence in what the instrument does to have confidence in what it tells me? (assume for now that its input is OK)

And how do I tell that it really, truly is giving me a 'heat capacity' (or a 'temperature', or ...)? (Hint: you need a theory, implicit or explicit).

snarkophilus
2006-Mar-31, 09:57 PM
I've heard this one before. Just curious though, if you measured the distance in atoms between you, and the door, then halved that distance, wouldn't you reach a point where you end up with a single indivisible atom? I'm not saying that atoms are indivisible, but perhaps the unit of measurement is what's causing this infinity. An atom represents a point in space, whereas an arbitrary unit of measurement does not, thus can be divided to infinity. Just a thought.
I think that this is an open problem, though one that no one worries about. Does space without particles in it really exist? Can two particles be an arbitrary (in terms of real numbers) distance apart? Or is it that the universe consists only of particles and their interactions, and the distance between them is illusory?

P.S. Remember the endless discussion of whether 0.9999999..... equals 1.000000? In math, yes, in reality no.
Oh dear... and I'm one of those people who says, "in reality, yes." How embarrassing. :D

Here's another infinity: if I fly around the Earth in a straight line, I can go infinite distance without hitting anything (provided I don't crash, of course).

In my pencil, there are atoms. The number of atoms, plus the number of groups of atoms, plus the number of groups of groups of atoms, plus... et cetera... is infinity. (This is a real world example of why we use the notation 2^Z = R)

Finally, an infinite number of monkeys on an infinite number of typewriters... only might produce some Shakespeare. After all, there's an infinite amount of gibberish out there, and not all infinities are created equal. Not that I've ever seen an infinite number of monkeys....

As an aside, all infinities we come up with are going to be explainable as "just mathematical constructs." That's just the nature of how we describe things. How does it go? "God created the integers; everything else is the work of man." -- Kroenecker. I'll take that one further and say that man created the integers, too. You can't talk about infinity without bringing math into it in some way. I think the point of this thread is to think up different ways of doing so, and relate that to the real world.

gzhpcu
2006-Apr-01, 04:23 AM
"I guess I don't understand what you mean. Certainly whenever I talk about measuring something, I'm going to have to use math in some way. Unless we're dead wrong about the mass of a photon, isn't it true that the ratio of the mass of an electron to the mass of a photon is infinite? We often talk about the ratio of the masses of a proton and an electron, or a muon and an electron; why shouldn't we allowed to talk about the ratio these masses?"

Because we are making assumptions about the photon, among other things that it has a zero rest mass. I know of no experiment which proves this.

gzhpcu
2006-Apr-01, 04:27 AM
But numbers are mathematical! That's like asking me to prove evolution with math, and not in biological terms...

I think what you haven't understood yet is that both '0.999...' and '1', in this context, are just names. It seems you are thinking that there is a one, true physical '1' and a one, true physical '0.999...' out there in the universe, or perhaps physical processes of some kind, which may disprove the equality. But '0.999...' is just a mathematical name, a label. That's what it's always been.

Actually, I do understand that they are just names. Math is totally abstract. I do not think there is a physical "1" out there, but it is a way to count isn't it?

gzhpcu
2006-Apr-01, 04:33 AM
I think that this is an open problem, though one that no one worries about. Does space without particles in it really exist? Can two particles be an arbitrary (in terms of real numbers) distance apart? Or is it that the universe consists only of particles and their interactions, and the distance between them is illusory?

Oh dear... and I'm one of those people who says, "in reality, yes." How embarrassing. :D

Here's another infinity: if I fly around the Earth in a straight line, I can go infinite distance without hitting anything (provided I don't crash, of course).

In my pencil, there are atoms. The number of atoms, plus the number of groups of atoms, plus the number of groups of groups of atoms, plus... et cetera... is infinity. (This is a real world example of why we use the notation 2^Z = R)

Finally, an infinite number of monkeys on an infinite number of typewriters... only might produce some Shakespeare. After all, there's an infinite amount of gibberish out there, and not all infinities are created equal. Not that I've ever seen an infinite number of monkeys....

As an aside, all infinities we come up with are going to be explainable as "just mathematical constructs." That's just the nature of how we describe things. How does it go? "God created the integers; everything else is the work of man." -- Kroenecker. I'll take that one further and say that man created the integers, too. You can't talk about infinity without bringing math into it in some way. I think the point of this thread is to think up different ways of doing so, and relate that to the real world.

You said it yourself here: just mathematical constructs. First: To go an infinite distance you need infinite time (and it is not yet proven that there is infinite time). Second: your pencil example - certainly a high number, but not infinite (though it might seem so, "for all practical purposes"). Third: The monkeys example has nothing to do with reality.

snarkophilus
2006-Apr-01, 07:53 AM
Second: your pencil example - certainly a high number, but not infinite (though it might seem so, "for all practical purposes").

It certainly is infinite. Pick an integer. Any integer. This number is bigger. That is what infinity is.

Here's a concrete idea of what I mean. Suppose I have but one atom. We'll denote the collection containing that atom {a}, and the collection containing nothing {}. Now, I want the collection containing all possible collections of atoms. There are two such collections, one containing that atom, and one not containing it. So the first collection is { {}, {a} }. Now, the collection of collections looks like { {}, {{a}}, {{}}, {{a},{}} }. Then the collection of collections of collections...

You can clearly see how this goes on. If you pick an arbitrary integer n, then the log_2(n)'th collection has that many elements in it. Since you can choose whichever collection you want, you can always pick one that has more elements than n. And that means there are infinity (a countable infinity) such elements. There are also infinity collections, which can be proven by noting that each collection has more elements than the last.

gzhpcu
2006-Apr-01, 10:58 AM
It certainly is infinite. Pick an integer. Any integer. This number is bigger. That is what infinity is.

Here's a concrete idea of what I mean. Suppose I have but one atom. We'll denote the collection containing that atom {a}, and the collection containing nothing {}. Now, I want the collection containing all possible collections of atoms. There are two such collections, one containing that atom, and one not containing it. So the first collection is { {}, {a} }. Now, the collection of collections looks like { {}, {{a}}, {{}}, {{a},{}} }. Then the collection of collections of collections...

You can clearly see how this goes on. If you pick an arbitrary integer n, then the log_2(n)'th collection has that many elements in it. Since you can choose whichever collection you want, you can always pick one that has more elements than n. And that means there are infinity (a countable infinity) such elements. There are also infinity collections, which can be proven by noting that each collection has more elements than the last.

Theoretically yes. Mathematically yes. However, I thought this thread was discussed the occurence of an infinite number of something in reality. IMHO: in reality, no. (well, let's say perhaps - if the universe is infinite - but certainly not proven)

Ken G
2006-Apr-01, 03:29 PM
Here's perhaps a more everyday example of an infinite mathematical contribution to a physical quantity. The two electrons in the ground state of the helium atom have an interaction energy distribution with an infinity in it when you consider contributions from the two particles being at the same point. The infinity is integrated away to get a physical energy, but it is there in the contribution function.

snarkophilus
2006-Apr-01, 03:35 PM
Theoretically yes. Mathematically yes. However, I thought this thread was discussed the occurence of an infinite number of something in reality. IMHO: in reality, no. (well, let's say perhaps - if the universe is infinite - but certainly not proven)

Mathematically yes, but no? You mean to tell me that if you count ten apples, using mathematics, that you mathematically have ten apples, but you don't really have ten?

A collection of objects is as close to reality as you can come, I think. What would you accept as infinity if not that? One apple after another, all in a big line, going on forever (and beyond)? Well, the number of collections (and collections of collections, and so on) of anything is the same number as that many apples. If you had such a line of apples, you could match them up one for one with those collections. I suppose we could argue the existence of collections....

On a side note, we're not necessarily looking for an infinite number of something, although this example does describe an infinite number of things. We're just looking for examples of infinity that occur in real life.

snarkophilus
2006-Apr-01, 03:43 PM
Here's perhaps a more everyday example of an infinite mathematical contribution to a physical quantity. The two electrons in the ground state of the helium atom have an interaction energy distribution with an infinity in it when you consider contributions from the two particles being at the same point. The infinity is integrated away to get a physical energy, but it is there in the contribution function.

Oh ho! I hadn't thought of that, but of course it's true for all sorts of systems. In chemical reaction dynamics, the energy limit as two atomic nuclei approach is considered to be infinity.

The funny thing is that although we treat that energy as infinite in calculations, it might not be. Presumably, if you could overlap two electrons like that, some funky physics would take place and something with a finite energy would be the result.

There are all those nasty infinities in QED that Tomonaga, Schwinger, and Feynman got the Nobel for. Since that theory basically runs the universe... :D

Ken G
2006-Apr-01, 03:46 PM
Yes, I think if we allow infinity to mean "larger than anything our instruments can measure", then it does indeed show up in a real and quasi-measurable sense.

gzhpcu
2006-Apr-01, 03:51 PM
OK, maybe I have misunderstood the entire gist of this thread. I thought the idea was to give a real example in real life of infinity. In this vein: show me an infinite number of apples in real life. Pardon me if I still don't seem to get it, but if we are not looking for an infinite number of something, what do you understand by "examples of infinity that occur in real life"?

There might be mathematical models for everyday life which use the symbol infinity for integration, for example, but IMHO this is just a mathematical idealization to achieve a result.

Disinfo Agent
2006-Apr-01, 06:24 PM
Actually, I do understand that they are just names. Math is totally abstract.Then you understand why it doesn't make sense to say the following?

Remember the endless discussion of whether 0.9999999..... equals 1.000000? In math, yes, in reality no.

I do not think there is a physical "1" out there, but it is a way to count isn't it?Counting is just one of the various uses of numbers. You only need integers to count, but we have more numbers than the integers.

Here's perhaps a more everyday example of an infinite mathematical contribution to a physical quantity. The two electrons in the ground state of the helium atom have an interaction energy distribution with an infinity in it when you consider contributions from the two particles being at the same point. The infinity is integrated away to get a physical energy, but it is there in the contribution function....Which is a theoretical entity, not a measurement.

gzhpcu
2006-Apr-01, 07:03 PM
Maybe I am not expressing myself clearly (I hope... :-)). When I said above in reality no, I meant to say that there is no physical example of an integer equaling infinity or 0.99999999...... in nature which we can measure and prove.

Ken G
2006-Apr-02, 12:42 AM
I'm not sure what is being said here. That only numbers that come up in measurements are real? Measurements don't produce numbers at all, people do. Measurements are movement of a needle, or of electrical charges or currents, or triggering something to flip. Numbers are mathematical constructs made from those measurements, and there are ways to construct infinity, and zero, and everything in between. Nature, on the other hand, must "know" what to do without the limitations of numbers, but we have no access to that process outside of nature itself.

Peptron
2006-Apr-02, 04:02 AM
I'd say just take the Mandelbrot fractal and count the sides.

gzhpcu
2006-Apr-02, 09:11 AM
I'd say just take the Mandelbrot fractal and count the sides.

This is a mathematical set. IMHO, an object with the shape of a Mandelbrot fractal and an infinite number of sides does not exist in nature. If I am wrong, please give an example.

Grey
2006-Apr-02, 01:48 PM
Because we are making assumptions about the photon, among other things that it has a zero rest mass. I know of no experiment which proves this.Perhaps not proves, but that's our best measurement at this point. We're never absolutely certain of anything. If you want to say we can't be sure, then that's true of any measurement we make, not just ones that give us results that are infinite or zero. So in that case, some of the things we think are finite might in fact be infinite instead.

When I said above in reality no, I meant to say that there is no physical example of an integer equaling infinity or 0.99999999......Well, perhaps not an example of an infinite integer, but there are plenty of examples of having 0.999... objects, since that's just equal to 1, by definition.

gzhpcu
2006-Apr-02, 04:08 PM
All I am saying is that while perhaps possible, we have no definite proof. The 0.999..... equal to 1 is a mathematical definition, IMHO it is not relevant in the real world. If, otherwise, please give an example in the real world.

hhEb09'1
2006-Apr-02, 04:20 PM
All I am saying is that while perhaps possible, we have no definite proof. The 0.999..... equal to 1 is a mathematical definition, IMHO it is not relevant in the real world. If, otherwise, please give an example in the real world.I think a lot of examples have already been given, in the real world. However, you seem to insist that all those examples are not real world, but mathematical in nature.

When you said
P.S. Remember the endless discussion of whether 0.9999999..... equals 1.000000? In math, yes, in reality no.you made a statement about reality. Since math and science are one way of understanding the real world, it is certainly reasonable that the conclusions of math and science are also about the real world. It is up to you to show that they actually do not apply to the real world, in this case. In other words, you have to justify your statement. You can't turn it around and say it is up to us to prove something without mathematics--especially since anyone here will admit that there never can be such a proof of anything.

So, either what you said is meaningless, or you can back it up. Can you?

gzhpcu
2006-Apr-02, 04:32 PM
I am not saying "prove something without math". Without math we would be nowheres. I am just saying that infinity is a mathematical symbol and does not exist in the real world. This, however, is an idealization. Very useful, but still a replacement for "practically zero" or "practically infinite" Zero for example, has already given us problems with QM and its 0 dimensional particles. This breaks down at the Planck length, and M-theory has gotten around that with strings and branes.

hhEb09'1
2006-Apr-02, 04:39 PM
I am not saying "prove something without math". Without math we would be nowheres. I am just saying that infinity is a mathematical symbol and does not exist in the real world. This, however, is an idealization. Very useful, but still a replacement for "practically zero" or "practically infinite" Zero for example, has already given us problems with QM and its 0 dimensional particles. This breaks down at the Planck length, and M-theory has gotten around that with strings and branes.OK, but what do you mean by 0.999... does not equal 1 in the real world? Or did you mean something else by that?

Do you mean that 0.999... (and 1) don't exist in the real world?

Grey
2006-Apr-02, 04:41 PM
All I am saying is that while perhaps possible, we have no definite proof. The 0.999..... equal to 1 is a mathematical definition, IMHO it is not relevant in the real world. If, otherwise, please give an example in the real world.If I were to give you a collection of 1 object (or 2, or 3, or...), you would presumably acknowledge that to do so, I would be using our agreed upon definitions of 1, 2, 2, and so forth, right? If I give you half an apple, we'd have to use the standard definition of "half" (and we'd probably agree that I was likely to have given you not quite half an apple, or a tiny bit more, because my measuring isn't that precise). In some sense, those definitions of numbers could be called "just mathematics", but they certainly apply to the real world (unless, as snarkophilus suggested above, you deny that, in which case if I count ten apples, I have ten apples "mathematically" but not "in reality").

So, if I want to give you 0.9999... apples, I'd similarly have to use the agreed on definition of 0.9999.... And in spite of claims to the contrary, that number that we represent with those symbols is indeed well-defined and unambiguous, just as are the numbers 1, 2, 3, and so forth, and it's precisely equal to 1. If using the definition of 0.9999... is "just mathematics" and not "reality", then so is using the definition of 1, or 2, or 3, and we can't talk about any measurement existing in "reality".

gzhpcu
2006-Apr-02, 05:06 PM
Please remember, this is just my opinion and nothing else: When I count objects, I will say "one apple", "two apples". This corresponds to reality. We really have "one" apple, or "two" apples. I will not say "0.99999... apple", however, even if mathematically this makes sense, because it does not exist. Coming from the other direction, say 0.9 apple, I do not have an infinite number of 9's which I can add behind the decimal point, because at some point I will reach a fundamental particle which can no longer be broken down.

hhEb09'1
2006-Apr-02, 05:54 PM
Please remember, this is just my opinion and nothing else: When I count objects, I will say "one apple", "two apples". This corresponds to reality. We really have "one" apple, or "two" apples. I will not say "0.99999... apple", however, even if mathematically this makes sense, because it does not exist. Coming from the other direction, say 0.9 apple, I do not have an infinite number of 9's which I can add behind the decimal point, because at some point I will reach a fundamental particle which can no longer be broken down.So, 1, 2, 3, ... exist, but other numbers do not?

Do 2/3 or 7/13 exist?

I'm not being facetious. A hundred years ago, this was a topic of hot debate even among mathematicians. I think it was Kronecker that said, "God created the integers, all the rest is the work of Man."

Is that what you mean?

gzhpcu
2006-Apr-02, 06:07 PM
So, 1, 2, 3, ... exist, but other numbers do not?

Do 2/3 or 7/13 exist?

I'm not being facetious. A hundred years ago, this was a topic of hot debate even among mathematicians. I think it was Kronecker that said, "God created the integers, all the rest is the work of Man."

Is that what you mean?

I would think that in the case of the apple, 2/3's might exist for some apples. Say, for example, if for a particular apple, the total amount of elementary particles is exactly divisible by 3, yes it would exist for this particular example. If for another apple, the amount of elementary particles making up the apple are not divisible by 3, then no. Of course, for all practical purposes we will speak of 2/3's of an apple, but imply an approximation.

hhEb09'1
2006-Apr-02, 06:14 PM
I would think that in the case of the apple, 2/3's might exist for some apples. Say, for example, if for a particular apple, the total amount of elementary particles is exactly divisible by 3, yes it would exist for this particular example. If for another apple, the amount of elementary particles making up the apple are not divisible by 3, then no. Of course, for all practical purposes we will speak of 2/3's of an apple, but imply an approximation.OK, so that's not the same objection that Kronecker made.

What about 1/7? If 1/7 exists, does the infinite decimal .142857142857... ?

Are you saying that not only does .999... not exist, but neither does .333...?

Grey
2006-Apr-02, 06:52 PM
Fractions need not apply only to subdividing a single object. Surely if I have 63 apples, it's meaningful to talk about 2/3 or the apples. Or is that also just mathematics that don't actually apply to reality?

Why are the definitions for integers acceptable as "reality" when decimals are not? Is it just that the definitions are more complex, and only very simple mathematics can be considered to be real? Where does the line of complexity get drawn? Is anything with an infinite decimal expansion not real? As Pace points out, that eliminates a whole category of simple rational fractions. And why should those fractions be singled out, anyway? The only reason that 1/2 has a finite decimal expansion and 1/3 does not is because we use base 10 and not base 12 or 6.

gzhpcu
2006-Apr-02, 07:28 PM
Of course not. Sure if I have 63 apples, I can talk about having 2/3rds of the apples, because the number 63 is divisible by 3. So, in this case, the mathematical result applies to reality. I am just saying, that in some cases, it does not.
IMHO, an infinite decimal expansion is not real in the case I mentioned, that is to say, when I speak of 1/3rd of an apple and the number of elementary particles of the apple is not exactly divisible by 3.

gzhpcu
2006-Apr-02, 07:40 PM
OK, so that's not the same objection that Kronecker made.

What about 1/7? If 1/7 exists, does the infinite decimal .142857142857... ?

Are you saying that not only does .999... not exist, but neither does .333...?

If the amount of fundamental particles of an apple can be divided by 7, then 1/7 exists in reality for this apple, then also the infinite decimal will exist for the apple.

gzhpcu
2006-Apr-02, 07:41 PM
P.S. I also take back the statement about 0.9999.... not being equal to 1 in reality... :-)

hhEb09'1
2006-Apr-02, 07:54 PM
P.S. I also take back the statement about 0.9999.... not being equal to 1 in reality... :-)...choke... I love you man :)

Disinfo Agent
2006-Apr-03, 12:39 PM
Measurements are movement of a needle, or of electrical charges or currents, or triggering something to flip. Numbers are mathematical constructs made from those measurements, and there are ways to construct infinity, and zero, and everything in between.From physical measurements? Please exemplify.

Ken G
2006-Apr-03, 01:23 PM
What are you asking me to do, get infinity from measurements? Trivial. I will divide the number of hands I have by the number of tails. My ratio of hands to tails is infinity. What do you mean by a measurement, that I am not allowed to divide? All measurement involves division, we divide a real result by the conventional unit result. That's what I mean when I say that measurements aren't numbers until the human mind gets into the game, and when that happens, we have mathematics. This distinction between "real" numbers and "mentally constructed" numbers simply does not exist.

Disinfo Agent
2006-Apr-03, 01:32 PM
What are you asking me to do, get infinity from measurements? Trivial. I will divide the number of hands I have by the number of tails. My ratio of hands to tails is infinity.That's not a value you got from a measurement, though. It's the ratio of two measurements (well, two counts, to be more exact). You can get any kind of number you choose by applying arbitrary operations to measurements. For example, if I multiply the number of fingers on my right hand by i, I get 5i. Oh, look! Imaginary numbers exist in nature!

What do you mean by a measurement, that I am not allowed to divide? All measurement involves division, we divide a real result by the conventional unit result.You are stretching the semantics beyond recognition, there.

Ken G
2006-Apr-03, 01:48 PM
It's not semantics, it is the definition of a number. I would like you to define measurement, and define number, and then explain how you tell a "real" connection between these from a concocted one. And by the way, imaginary numbers do exist in nature, to the same extent that any mathematical concept does (the motion of nature's elementary constituents follow the algegra of imaginary numbers).

Disinfo Agent
2006-Apr-03, 02:13 PM
It's not semantics, it is the definition of a number.No, it isn't. The mathematical definitions of number do not assume the existence of fundamental physical units like the metre or the second. And, even if they did, I don't need to divide 5 metres by 1 metre to know that it equals 5 units of length. I can just count the units.

I would like you to define measurement, and define number, and then explain how you tell a "real" connection between these from a concocted one.Well, I would like you and Nereid to show me an infinite physical measurement. You still haven't.

And by the way, imaginary numbers do exist in nature, to the same extent that any mathematical concept does (the motion of nature's elementary constituents follow the algegra of imaginary numbers).Question: what's the physical meaning of 5i, where 5 is the number of fingers in my right hand?

korjik
2006-Apr-03, 07:24 PM
And where would any one of us encounter these plasma waves, in our daily lives, tusenfem? For example, if you're lucky enough to live at a high northern or southern latitude, would those beautiful aurorae have these kinds of waves?

Prolly in a plasma TV, but I dont know for sure. Plasma heating in a tokomac or the VASIMR is based on frequency resonances also. VASIMR at least I do know for sure.

MRI is based on nuclear magnetic resonance, which has a 1/0.

Any resonance in physics is based around a 1/(w-w0) as w -> w0 term. These 1/0 terms are where the term singularity comes from.

The best example I can think of tho is conductivity in a superconductor.

hhEb09'1
2006-Apr-03, 07:31 PM
Well, I would like you and Nereid to show me an infinite physical measurement. You still haven't.Well, we're finite beings :)

But what about the tangent of an angle? Isn't that OK? It corresponds to the length from x-axis to point of tangency of the line segment tangent to the unit circul at the angle. It doesn't have to be sine divided by cosine.

Fixed definition, as below

Disinfo Agent
2006-Apr-03, 08:23 PM
We usually say that the tangent is not defined for some angles...

P.S. Graph here. (http://mathworld.wolfram.com/Tangent.html)

hhEb09'1
2006-Apr-03, 08:35 PM
We usually say that the tangent is not defined for some angles... Yes, because it is infinite. :)

That's sorta the way that these things are dealt with. Sometimes, anyway.

PS: I just noticed that I messed up the graphical definition of tangent. Using the sine/cosine definition, it's clear why it is undefined, because you divide by zero. But the (corrected) graphical definition doesn't have that problem--the tangent line just ends up being infinite.

snarkophilus
2006-Apr-03, 08:49 PM
No, it isn't. The mathematical definitions of number do not assume the existence of fundamental physical units like the metre or the second. And, even if they did, I don't need to divide 5 metres by 1 metre to know that it equals 5 units of length. I can just count the units.
Except that the concept of counting itself is a mathematical thing. 5 = 4 + 1 + 0 = 3 + 1 + 1 + 0 = 2 + 1 + 1 + 1 + 0 = 1 + 1 + 1 + 1 + 1 + 0. That's the definition of 5. It's a purely mathematical (recursive) definition, and it's more or less arbitrary that you use that counting scheme. You could equally well count by multiplying by successively larger numbers, and then to compare two numbers, divide the first by the second, rather than subtracting (for example). In terms of the physical world, it's the same end result, but in terms of the mathematical concept, addition and subtraction is easier.

When you say five meters, what you are really saying is that you have a preset piece of reality that measures one meter, and you are multiplying that by five. That's math as well.

Counting 5i fingers has the same meaning as counting 5 fingers. The choice of the natural numbers as the basis for counting is arbitrary, because you count in terms of units, and you divide the units out when comparing ratios. You could equally well call the natural numbers 0, 1s, 2s, 3s, 4s..., and the imaginary numbers 0, 1i, 2i, 3i, 4i.... The mathematical rules for dealing with s numbers is a little bit different than those for dealing with i numbers, but then the rules for dealing with vectors and matrices are different again. It's not a problem -- you just need to adjust your algorithms to make the results consistent.

Question: what's the physical meaning of 5i, where 5 is the number of fingers in my right hand?

That's a curious question. I'll pose a related one with the same answer: what's the meaning of 10 kg m / s, where 10 is the number of toes on my feet?

We usually say that the tangent is not defined for some angles.

Implicit in that is that you mean it is undefined in the real numbers. That does not mean that it does not have a value in some other number system. 5/7 is undefined in the integers.

Disinfo Agent
2006-Apr-03, 09:04 PM
Yes, because it is infinite. :)

That's sorta the way that these things are dealt with. Sometimes, anyway.

PS: I just noticed that I messed up the graphical definition of tangent. Using the sine/cosine definition, it's clear why it is undefined, because you divide by zero. But the (corrected) graphical definition doesn't have that problem--the tangent line just ends up being infinite.There's also a geometrical -- dare I say physical? ;) -- argument against defining the tangent of 90&#186;. In the traditional definition of the trigonometric functions (http://pasture.ecn.purdue.edu/~agen215/trnglprt.gif), we start from a right triangle, and focus on one of its non-right angles, theta. The tangent is then the ratio between the length of the side of the triangle opposite theta, and the length of the side adjacent to theta. But if we let theta increase to 90&#186;, the triangle degenerates into a line segment. It won't be a real triangle anymore.
Mathematically, it's convenient to overlook that and extend most trigonometric definitions to angles of 90&#186;, but physically a 'triangle' with two right angles is a very dubious entity (http://britton.disted.camosun.bc.ca/escher/waterfall.jpg). (Well, perhaps in non-Euclidean geometries it isn't...)

Disinfo Agent
2006-Apr-03, 10:04 PM
Except that the concept of counting itself is a mathematical thing. 5 = 4 + 1 + 0 = 3 + 1 + 1 + 0 = 2 + 1 + 1 + 1 + 0 = 1 + 1 + 1 + 1 + 1 + 0. That's the definition of 5. It's a purely mathematical (recursive) definition, and it's more or less arbitrary that you use that counting scheme. You could equally well count by multiplying by successively larger numbers, and then to compare two numbers, divide the first by the second, rather than subtracting (for example). In terms of the physical world, it's the same end result, but in terms of the mathematical concept, addition and subtraction is easier.

When you say five meters, what you are really saying is that you have a preset piece of reality that measures one meter, and you are multiplying that by five. That's math as well.

Counting 5i fingers has the same meaning as counting 5 fingers. The choice of the natural numbers as the basis for counting is arbitrary, because you count in terms of units, and you divide the units out when comparing ratios. You could equally well call the natural numbers 0, 1s, 2s, 3s, 4s..., and the imaginary numbers 0, 1i, 2i, 3i, 4i.... The mathematical rules for dealing with s numbers is a little bit different than those for dealing with i numbers, but then the rules for dealing with vectors and matrices are different again. It's not a problem -- you just need to adjust your algorithms to make the results consistent.What would be the physical interpretation of 2i x 2i = -4, in that counting system?

For example, we can say that 2 x 2 = 4 means that if I fill two baskets with two pumpkins, then I have a total of 4 pumpkins in the two baskets. What would be the equivalent interpretation for 2i x 2i = -4?

That's a curious question. I'll pose a related one with the same answer: what's the meaning of 10 kg m / s, where 10 is the number of toes on my feet?Bad dimensional analysis.

Implicit in that is that you mean it is undefined in the real numbers.Not only that, but in the usual extended real line, which has a positive infinity and a negative infinity, the tangent of 90º is also undefined.

That does not mean that it does not have a value in some other number system. 5/7 is undefined in the integers.Of course it can have a value in some other number system, but if you're going to use some other number system in physics you should first explain what you need it for.

Nereid
2006-Apr-03, 11:56 PM
Yes, heat capacity can be measured in 'reality', and it can become infinite -- in theory. But can you show me an infinite measurement of heat capacity?
Now let me see, ... to 'measure' something I need some kind of an instrument, and to be sure that is a faithful servant, telling me the whole measurement and nothing but the measurement, I need to understand how that instrument goes about turning an input of some sort into an output (of the numerical kind, of course).

Hmm, don't I need to have confidence in what the instrument does to have confidence in what it tells me? (assume for now that its input is OK)
A perfect insulator has infinite resistance.
Well, I would like you [Ken G] and Nereid to show me an infinite physical measurement. You still haven't.Let's take Relmius' example.

I go to my friendly local Dick Smith's electronics/hobbyist shop, and I buy a digital multimeter. I put the battery in, and turn it on.

I turn the dial to resistance, and put the probes across the "Relmius' insulator". My multimeter reads "∞".

Now, let's look at what's happening inside the multimeter (I'll make some assumptions; what matters is the general principle, not the details).

Some sensor inside the multimeter detects a current flowing through the probes; some electronics converts that detected current into some code (several bytes long, expressed as voltages on some control lines), which ends up as something on the LCD display.

In the case of the Relmius' insulator, the sensor detects zero current. The logic circuits converts this to the symbol "∞" on the LCD panel.

Well, of course there's no such thing as a perfect insulator, as far as we know. Apply enough voltage across anything, and it will conduct electricity.How do we reconcile the fact that we have, per Disinfo Agent's demand, clearly demonstrated "an infinite physical measurement", yet we also know, per Grey, "there's no such thing as a perfect insulator"!

So is my Dick Smith multimeter lying to me?

hhEb09'1
2006-Apr-04, 02:55 AM
There's also a geometrical -- dare I say physical? ;) -- argument against defining the tangent of 90º. In the traditional definition of the trigonometric functions (http://pasture.ecn.purdue.edu/~agen215/trnglprt.gif), we start from a right triangle, and focus on one of its non-right angles, theta. Hold up, that may have been the way that you and I were introduced to the trig functions, but that is not the traditional way. The diagram that I was talking about is more traditional, where the measure of the line along the tangent line is the tangent of the angle, and the measure of the secant line is the secant of the angle. So, tangent does go to infinity at 90 degrees.

Why do you think they call them "tangent" and "secant"? :)

snarkophilus
2006-Apr-04, 06:32 AM
So is my Dick Smith multimeter lying to me?

In a word, yes. Air has a finite resistance. Ask any lightning bolt. :)

Heck, even empty space has a finite resistance.

http://en.wikipedia.org/wiki/Permittivity

Actually, if one were to read that, he would find that it is an example of a physical quantity that has a measurable yet complex value.

Ken G
2006-Apr-04, 07:49 AM
What would be the physical interpretation of 2i x 2i = -4, in that counting system?

I was going to address your earlier questions, but snarkophilus did it beautifully for me, I have nothing to add to that except my complete agreement. As for 2i X 2i = -4, this is no different from 2 meters by 2 meters = 4 square meters. A square meter is a totally different thing from a meter, just as -1 is totally different from i. Yet multiplications can connect them. That's also math, but it is math that connects to the real world. I think the subtext of all this is the mystery of why does math, a human concept, connect with the real world in the first place. Apparently our minds have made sense of the real world using math. That is the place of infinity in math, and in the real world, in a nutshell. From this perspective, all numbers have "reality" to whatever extent they have a place in helping us conceptualize and quantify our reality. Ratios are a key part of that, and so are zeros, and ratios involving zeros can give infinity. On that basis, infinity has a perfectly valid claim to reality.

Nereid
2006-Apr-04, 08:51 AM
In a word, yes. Air has a finite resistance. Ask any lightning bolt. :)

Heck, even empty space has a finite resistance.

http://en.wikipedia.org/wiki/Permittivity

Actually, if one were to read that, he would find that it is an example of a physical quantity that has a measurable yet complex value.That may be so, but Disinfo Agent (DA) asked for "an infinite physical measurement", which I provided.

That the multi-meter's 'measurement' was not physically infinite requires that you accord your view of reality primacy over the output of a measuring device. If you go down that route, then you would probably need to reply to DA with something like "we know our theories of reality are correct; here is a 'physically real' situation where infinity occurs, and here is an instrument which will produce '∞' on its dial as it 'measures' that physical reality".

Disinfo Agent
2006-Apr-04, 06:03 PM
Hold up, that may have been the way that you and I were introduced to the trig functions, but that is not the traditional way. The diagram that I was talking about is more traditional, where the measure of the line along the tangent line is the tangent of the angle, and the measure of the secant line is the secant of the angle. So, tangent does go to infinity at 90 degrees.I never said it didn't go to infinity...

I go to my friendly local Dick Smith's electronics/hobbyist shop, and I buy a digital multimeter. I put the battery in, and turn it on.

I turn the dial to resistance, and put the probes across the "Relmius' insulator". My multimeter reads "infinity".

Now, let's look at what's happening inside the multimeter (I'll make some assumptions; what matters is the general principle, not the details).

Some sensor inside the multimeter detects a current flowing through the probes; some electronics converts that detected current into some code (several bytes long, expressed as voltages on some control lines), which ends up as something on the LCD display.

In the case of the Relmius' insulator, the sensor detects zero current. The logic circuits converts this to the symbol "infinity" on the LCD panel.So, what the sensor actually measured was a zero (or a near zero), not an infinity. The infinite (or near infinite) value is deduced from a theoretical relation.

Now what about Grey's comment?How do we reconcile the fact that we have, per Disinfo Agent's demand, clearly demonstrated "an infinite physical measurement", yet we also know, per Grey, "there's no such thing as a perfect insulator"!See above. In any event, though, there are no perfectly exact physical measurements, either.

Disinfo Agent
2006-Apr-04, 06:21 PM
In a word, yes. Air has a finite resistance. Ask any lightning bolt. :)

Heck, even empty space has a finite resistance.

http://en.wikipedia.org/wiki/Permittivity

Actually, if one were to read that, he would find that it is an example of a physical quantity that has a measurable yet complex value.Only if one were not very careful in such reading.

I was going to address your earlier questions, but snarkophilus did it beautifully for me, I have nothing to add to that except my complete agreement.He never said what 5i means, when 5 is the number of fingers I have in my right hand. Try again.

As for 2i X 2i = -4, this is no different from 2 meters by 2 meters = 4 square meters. A square meter is a totally different thing from a meter, just as -1 is totally different from i. Yet multiplications can connect them. That's also math, but it is math that connects to the real world. I think the subtext of all this is the mystery of why does math, a human concept, connect with the real world in the first place. Apparently our minds have made sense of the real world using math. That is the place of infinity in math, and in the real world, in a nutshell. From this perspective, all numbers have "reality" to whatever extent they have a place in helping us conceptualize and quantify our reality. Ratios are a key part of that, and so are zeros, and ratios involving zeros can give infinity. On that basis, infinity has a perfectly valid claim to reality.Handwaving. There is a simple, physical interpretation of 2 metres x 2 metres = 4 square metres, or 8 x 8 = 64. The latter, for example, tells us that inside a chess table there 64 squares.
What meaning of this kind would 2i x 2i = -4 have, in your "parallel integer" number system?

Nereid
2006-Apr-04, 07:10 PM
[snip]
I go to my friendly local Dick Smith's electronics/hobbyist shop, and I buy a digital multimeter. I put the battery in, and turn it on.

I turn the dial to resistance, and put the probes across the "Relmius' insulator". My multimeter reads "infinity".

Now, let's look at what's happening inside the multimeter (I'll make some assumptions; what matters is the general principle, not the details).

Some sensor inside the multimeter detects a current flowing through the probes; some electronics converts that detected current into some code (several bytes long, expressed as voltages on some control lines), which ends up as something on the LCD display.

In the case of the Relmius' insulator, the sensor detects zero current. The logic circuits converts this to the symbol "infinity" on the LCD panel.So, what the sensor actually measured was a zero (or a near zero), not an infinity. The infinite (or near infinite) value is deduced from a theoretical relation.We're making good progress.

It's your turn: tell us what kind of multimeter you'd like us to examine, in terms of what the sensor (or sensors) at its heart is.

Separately, define what you mean by 'measure', first with specific reference to the sensor at the heart of the multimeter, and then generalise that to 'measurement' in the broad.

(to telegraph what my reply will likely contain: 'zero' is not what any sensor can 'measure'. Further, whatever it is that a sensor detects (that's the word I used), it has meaning only in terms of a theory.)

Now what about Grey's comment?How do we reconcile the fact that we have, per Disinfo Agent's demand, clearly demonstrated "an infinite physical measurement", yet we also know, per Grey, "there's no such thing as a perfect insulator"!See above. In any event, though, there are no perfectly exact physical measurements, either.See above - without a definition of what 'measurement' is, I (for one) have no way to address whether any can (or can't) be 'perfectly exact' (must less 'physical').

Disinfo Agent
2006-Apr-04, 08:10 PM
I don't feel that the concept of measurement is important in this context, but, if you think it requires further clarification, then tell me your definition of 'measurement', and we'll take it from there, as long as I manage to understand your definition.

snarkophilus
2006-Apr-04, 08:30 PM
Only if one were not very careful in such reading.
Read the article. Permittivity takes on complex values. You can measure them.

He never said what 5i means, when 5 is the number of fingers I have in my right hand. Try again.

5i means exactly what it says. You have five groups of i, just like if you wrote 5 a in algebra, or 5 kg in physics. That was the point I was trying to make with 10 kg m / s. All of those units are just algebraic symbols. What is a kilogram? You might say it's what so many moles of carbon-12 mass. That's fine, but it still doesn't tell you what it really is. You might say it's how much a particular object warps space-time. Again, you can use that, but it still doesn't tell you what it is.

How do you compare two objects' masses? I take an apple and put it on a scale, getting 0.5 kg. Then I take a watermelon and get 2.0 kg. How do I compare those? I have to use math. I say 2.0 kg / 0.5 kg = 4.0. Note that the kilograms must be included for me to get the proper multiple. That's because the kg parameter cancels out. And it really does; it's not just a dimensional analysis thing or a placeholder or a trick to help you remember. It doesn't matter what a kg is, exactly, as long as it's a useful tool for us to use with the math.

So 10 kg is 10 groups of kg, where 10 takes on its mathematical definition. 5i is 5 groups of i. And you can do the same things with i as with kg. That i^2 happens to be an integer is useful, sure, but that's because we've invented it to be that way. Treat i as a unit, but a special one, one better defined than those SI units, because it has a direct connection to the number of fingers on your hand. Maybe we'll one day discover that kg^50 is -pi. Not likely, since we don't know what kg really means, but if it turns out that that result is convenient for our math, then we'll redefine kg to make it that way.

(5 fingers * i)^2 = -25 fingers^2. It has as much meaning as 30 J, if given the right context. Or take the kinematics equation v^2 - v0^2 = 2ad. That could just as nicely be written v^2 + (v0 * i)^2 = 2ad. What's the meaning of 6i m/s? If you look at that equation and let v0 = 6 m/s, you can see a possible meaning. In fact, you can actually tell what the i means in that equation, but you still don't know what a meter really (really) is.

Furthermore, as I said before, there's no reason to assume that the real numbers hold a special place in the world. That we use the notation 5*i is only because we wanted to represent that imaginary numbers have a simple relation to real numbers. We could equally well use imaginary numbers and call them 1,2,3,4,5, and invent a number n such that n = 1^2. Then 1n, 2n, 3n would be the natural numbers, and you'd be arguing that we couldn't measure those in real life, but only the more usual imaginary numbers.

Nereid
2006-Apr-04, 09:14 PM
I don't feel that the concept of measurement is important in this context, but, if you think it requires further clarification, then tell me your definition of 'measurement', and we'll take it from there, as long as I manage to understand your definition.OK.

'Measurement' is a uniquely human activity. It refers to a process/concept which is mostly arbitrary. Though the details vary somewhat (or considerably, your mileage may vary), it involves the specification of a (theory-based) thing ('resistance', for example), and an (almost) entirely arbitrary zero point and scale (e.g. the freezing point of water, at 1 atm, is 32, the boiling point 212), which usually (but not always) can be mapped one-to-one onto the reals.

The specifics of how a particular 'measurement' can (in principle) be obtained are highly context-dependent; the description of those contexts is permeated by a large number of (invented by humans) theories.

Some examples: the measurement of resistance, by a 'Dick Smith' multimeter, is valid only in an utterly trivially tiny proportion of space-time environments/contexts (try getting a measurement of resistance, using that device, in the IPM or ISM, for example; or in the Sun) the measurement of temperature, by a 'mercury thermometer', is valid in an utterly trivially tiny proportion ... (you get the idea)Now, since it was your challenge, DA, without knowing what you consider to be a valid 'measurement', I don't see how anyone can provide you with samples that show 'an infinite physical measurement'.

Worse, without your a priori specification of just what you mean by these words, any such attempt to provide you with such samples is easy to foil (by you) ... you simply need say 'that's not what I meant by {X}' (or words to that effect).

Or did I misunderstand something vital?

Disinfo Agent
2006-Apr-04, 09:21 PM
Read the article. Permittivity takes on complex values. You can measure them.Where does the article say that you can measure complex permittivity, exactly? As far as I've been able to gather, what they measure are real valued quantities which are related to the real and imaginary parts of permittivity (which do have physical interpretations). But the physical interpretation of complex permittivity itself elludes me.

5i means exactly what it says. You have five groups of i, just like if you wrote 5 a in algebra, or 5 kg in physics. That was the point I was trying to make with 10 kg m / s. All of those units are just algebraic symbols. What is a kilogram? You might say it's what so many moles of carbon-12 mass. That's fine, but it still doesn't tell you what it really is. You might say it's how much a particular object warps space-time. Again, you can use that, but it still doesn't tell you what it is.

How do you compare two objects' masses? I take an apple and put it on a scale, getting 0.5 kg. Then I take a watermelon and get 2.0 kg. How do I compare those? I have to use math. I say 2.0 kg / 0.5 kg = 4.0. Note that the kilograms must be included for me to get the proper multiple. That's because the kg parameter cancels out. And it really does; it's not just a dimensional analysis thing or a placeholder or a trick to help you remember. It doesn't matter what a kg is, exactly, as long as it's a useful tool for us to use with the math.

So 10 kg is 10 groups of kg, where 10 takes on its mathematical definition. 5i is 5 groups of i. And you can do the same things with i as with kg. That i^2 happens to be an integer is useful, sure, but that's because we've invented it to be that way. Treat i as a unit, but a special one, one better defined than those SI units, because it has a direct connection to the number of fingers on your hand. Maybe we'll one day discover that kg^50 is -pi. Not likely, since we don't know what kg really means, but if it turns out that that result is convenient for our math, then we'll redefine kg to make it that way.How does any of that relate to my fingers?

(5 fingers * i)^2 = -25 fingers^2. It has as much meaning as 30 J, if given the right context.And which 'right context' is that, exactly?

Or take the kinematics equation v^2 - v0^2 = 2ad. That could just as nicely be written v^2 + (v0 * i)^2 = 2ad. What's the meaning of 6i m/s? If you look at that equation and let v0 = 6 m/s, you can see a possible meaning. In fact, you can actually tell what the i means in that equation [...]No, I can't. You tell me what it is I'm supposed to be 'seeing'.

[...], but you still don't know what a meter really (really) is.

Furthermore, as I said before, there's no reason to assume that the real numbers hold a special place in the world.I disagree. There's a reason why we call them 'the real numbers', and it's precisely the opposite of what you've just said. They are special.

That we use the notation 5*i is only because we wanted to represent that imaginary numbers have a simple relation to real numbers. We could equally well use imaginary numbers and call them 1,2,3,4,5, and invent a number n such that n = 1^2. Then 1n, 2n, 3n would be the natural numbers, and you'd be arguing that we couldn't measure those in real life, but only the more usual imaginary numbers.What would be the point of that, physically speaking?

Disinfo Agent
2006-Apr-04, 09:30 PM
'Measurement' is a uniquely human activity. It refers to a process/concept which is mostly arbitrary. Though the details vary somewhat (or considerably, your mileage may vary), it involves the specification of a (theory-based) thing ('resistance', for example), and an (almost) entirely arbitrary zero point and scale (e.g. the freezing point of water, at 1 atm, is 32, the boiling point 212), which usually (but not always) can be mapped one-to-one onto the reals.I don't think I agree that zero metres, or zero metres per second, or zero kilogrammes, are '(almost) entirely arbitrary zero' points. What makes you say that?

Now, since it was your challenge, DA, without knowing what you consider to be a valid 'measurement', I don't see how anyone can provide you with samples that show 'an infinite physical measurement'.I don't accept the phrasing "without knowing what you consider to be a valid 'measurement'". I know a measurement when I see it. You were the one who asked for definitions.

Worse, without your a priori specification of just what you mean by these words, any such attempt to provide you with such samples is easy to foil (by you) ... you simply need say 'that's not what I meant by {X}' (or words to that effect).

Or did I misunderstand something vital?Why this defensive assumption that I'm here to 'foil' your samples by playing semantic games? When did I give you reason to think that of me?

Nereid
2006-Apr-04, 09:52 PM
I don't think I agree that zero metres, or zero metres per second, or zero kilogrammes, are '(almost) entirely arbitrary zero' points. What makes you say that?OK, so what do you mean by 'metre', 'second', 'kilogram'? What do you mean by 'zero'?
I don't accept the phrasing "without knowing what you consider to be a valid 'measurement'". I know a measurement when I see it. You were the one who asked for definitions.Aye, and there's the rub.

I (and presumably every other BAUT member, except you) have no way of knowing what you 'know', or what you don't 'know' (or 'see') ... other than by you telling us.
Why this defensive assumption that I'm here to 'foil' your samples by playing semantic games? When did I give you reason to think that of me?See above - if you can't define what you mean by the terms you consider critical to the case you (seem to be trying to) make*, then on what basis can we continue having a discussion?

*to date, the method seems to have been: X attempts a reply, DA negates it by claiming (in effect) that (key term) means (to DA) something quite different than what X used (in the reply). If this isn't a clear demonstration of the importance of unambiguous definitions (not semantics), ...

Disinfo Agent
2006-Apr-04, 10:24 PM
OK, so what do you mean by 'metre', 'second', 'kilogram'? What do you mean by 'zero'?Aye, and there's the rub.I see no reason not to use the definitions of the SI.

I (and presumably every other BAUT member, except you) have no way of knowing what you 'know', or what you don't 'know' (or 'see') ... other than by you telling us.I think I misunderstood that part of your post, actually. Sorry about that. It's been a long afternoon...

See above - if you can't define what you mean by the terms you consider critical to the case you (seem to be trying to) make*, then on what basis can we continue having a discussion?

*to date, the method seems to have been: X attempts a reply, DA negates it by claiming (in effect) that (key term) means (to DA) something quite different than what X used (in the reply). If this isn't a clear demonstration of the importance of unambiguous definitions (not semantics), ...I don't see it that way at all. Here's my interpretation of the events so far:

1 - DA asks for examples of infinite physical measurements (or 'infinities in physical reality', as you first worded it).
2 - Other posters give what they think are good examples.
3 - DA points out that those are actually not valid examples, usually because they are not the direct result of a measurement, but rather involve some theoretical extrapolation as an intermediate step.

Now, this may simply mean that we place the boundaries between 'theory' and 'observations' at different places. I'm willing to admit that. And maybe you'll tell me that observation is inseparable from theory, with which I think you know I agree. Nevertheless, most scientists do think of 'observation' and 'theory' as at least two separate stages in the scientific method. My main counterargument to most of the examples that have been presented so far is that they fall more under the 'theory' stage than under the 'observation' stage of the process. As such, I think their physical reality is at least arguable, and I would even say doubtful. For me, they remain theory-based 'extreme ideal cases', not observed realities.

spoonman
2006-Apr-05, 01:55 AM
When I saw this thread I immediately thought of tangents and my computer science O Level, then I saw tangents had already been mentioned.

The reason it made me think of my O Level, which I took in 1984 when I was 18, was that I decided for one of my coursework projects I'd do a 3D program. This was in the days when our high powered computer network consisted of 4 Research Machines 480Zs, with Z80A processors and I think 48k RAM. Problem was I had no idea how to calculate 3D points on a 2D projection on the screen and had to make it up. I remember many lessons when I was supposed to be programming just desperately drawing triangles on bits of paper trying to figure out what to program.

In the end, I came up with this weird system with an origin some arbitrary distance "beyond the screen" rather than at the viewer's position, or something, and a lot of trigonometry that unfortunately required tangents. And when a point was on an axis, the damned thing went to infinity and threw up an error.

In the end I gave up trying to get better math into it, and just had the program watch for 90 degrees to appear, and at that point kludge in an arbitrarily high (but less than infinity!) value.

Hey, it worked, and I got an "A" :)

John Dlugosz
2006-Apr-07, 08:25 PM
I turn the dial to resistance, and put the probes across the "Relmius' insulator". My multimeter reads "∞".

On first reading, I thought there really was something called a "Relmius' insulator", just as there are "black bodies" for testing light sensors. It could work like a regulator, exactly compensating for the probe's potential with a reverse potential of its own, so it always reads 0 current.

Think of the way a superconductor will exclude magnetic fields. It's not really infinitely tough; it sets up a current that exactly cancels out the input.

--John

Jacotus
2006-Apr-11, 03:00 AM
I am going to jump in with my own interpretation of the original topic. I firmly believe that nothing physical can be infinite. Not merely a human idea or abstraction (such as a ratio...), but something that actually exists. Nereid's example of the ideal resistor is really what made this click in my head. In this example, there can be a voltage difference across the resistor. This is fine and physical. No current flows, but a digital multimeter can read infinity. But there is no physical infinity. What is physical is that there is zero current (zero net charge flowing through the resistor on any timescale).

Well, ok, what else? Someone mentioned counting groups and then groups of groups, etc. This is clearly not physically meaningful: I could have one apple, and count that one apple forever, but there are not an infinite number of apples.

Furthermore, I can't think of any situation where a physical infinity would not result in infinite energy, which in itself is not physical.

Nereid
2006-Apr-11, 06:21 AM
Welcome to BAUT, Jacotus!

What about my initial example (temperature, spin temperature)? Heat capacity for certain phase transitions?

More fundamentally, how about the fact that 'physical meaning' becomes 'a dense web of theories, glued together with lots of math' when you look at it more closely?

Or perhaps you have some Platonic ideal universe, which can (ultimately) be grasped by 'minds [perhaps] immeasurably superior to ours*'? In which there is nothing that has the 'essence' infinity?

*from which famous scifi novel, referring to what? (Hint: Richard Burton)

Disinfo Agent
2006-Apr-20, 01:27 PM
Here's someone else (http://www.bautforum.com/showthread.php?t=40149) who doesn't take imaginary numbers in physics literally.

Ken G
2006-Apr-22, 03:09 AM
Imaginary numbers are like all numbers-- human constructs. There is no number that is "real", not even real numbers. Nature has no idea what a number is. You say, wait, what about the five fingers on my hand? Perfect example. Nature does not say you have five fingers, it says you have a thumb, and index finger, etc (or really, the particles that compose it, and all the uncertainties inherent). What on Earth does nature need with the number 5? I'm asking this, seriously. Some on this thread are arguing that some numbers are real, and others are inventions of the human mind, but I see no evidence for this position whatsoever. They are all inventions of the human mind.

Digix
2006-Apr-22, 03:47 PM
infinity = something /0 that the only way to get it, unless we talk about universe size or maximum possible numbers.

Disinfo Agent
2006-Apr-22, 08:39 PM
Imaginary numbers are like all numbers-- human constructs. There is no number that is "real", not even real numbers. Nature has no idea what a number is. You say, wait, what about the five fingers on my hand? Perfect example. Nature does not say you have five fingers, it says you have a thumb, and index finger, etc (or really, the particles that compose it, and all the uncertainties inherent). What on Earth does nature need with the number 5? I'm asking this, seriously. Some on this thread are arguing that some numbers are real, and others are inventions of the human mind, but I see no evidence for this position whatsoever. They are all inventions of the human mind.I take it, then, that your reply to the title of this thread would be 'Never'...

Ken G
2006-Apr-22, 10:13 PM
My answer would depend on a more careful explanation of the meaning of the word "encounter". If we include, in encounter, the constructs of our minds, then I would agree with those who have suggested ways that infinities are encountered. If we include only infinities that exist independently of the human mind, that would be here even if we weren't, then I would say no-- but the same would hold for the number 5. Even the number zero is a construction, because due to quantum mechanics, we understand that emptyness doesn't really happen either.

Nereid
2006-Apr-22, 10:38 PM
I take it, then, that your reply to the title of this thread would be 'Never'...Thanks, this bring us very neatly back to the OP.
My answer would depend on a more careful explanation of the meaning of the word "encounter". If we include, in encounter, the constructs of our minds, then I would agree with those who have suggested ways that infinities are encountered. If we include only infinities that exist independently of the human mind, that would be here even if we weren't, then I would say no-- but the same would hold for the number 5. Even the number zero is a construction, because due to quantum mechanics, we understand that emptyness doesn't really happen either.Thanks too to KenG! :clap:

In other words, if you accept 'five' (as having a physical reality), then you must accept 'infinity' as having exactly the same kind of physical reality.

(and so you encounter infinities of the 'physical reality' kind all day, every day, of your life).

Disinfo Agent
2006-Apr-24, 12:19 PM
Even the number zero is a construction, because due to quantum mechanics, we understand that emptyness doesn't really happen either.Who said the number zero is just a representation of the vacuum?

In other words, if you accept 'five' (as having a physical reality), then you must accept 'infinity' as having exactly the same kind of physical reality.I don't know if Ken G agrees with that, but I certainly don't.

Sure, numbers are abstractions. Counting and measuring are mental acts. But not all measurements are created equal, because some involve a lot more of theorizing and simplifying than others.

What I've been arguing all along is that some measurement acts produce results with a more convincingly physical plausibility than others; 2.5 amperes is a physically plausible measurement; infinite amperes is not.

By the way, if all types of measurements are equivalent as you claim, then why is it that intelligent and well informed people like the Celestial Mechanic make statements like the following?

CM: "I think that the use of complex numbers here really hurt the general understanding of special relativity in the early decades by making it seem almost 'mystical'. There's a place for complex numbers in physics, lots of places in fact, and I use them all the time. But changing a real observable coordinate into a pure imaginary number is not a good idea."I'm not sure I would agree with him that the use of complex numbers hurt the general understanding of special relativity, but I do know this: complex numbers, as used in electromagnetism and general relativity, are just computational shortcuts. They are not physically necessary. Anything you say with complex numbers can be said equally well with real numbers only (though that might make some calculations more cumbersome).

snarkophilus
2006-Apr-24, 06:47 PM
I'm not sure I would agree with him that the use of complex numbers hurt the general understanding of special relativity, but I do know this: complex numbers, as used in electromagnetism and general relativity, are just computational shortcuts. They are not physically necessary. Anything you say with complex numbers can be said equally well with real numbers only (though that might make some calculations more cumbersome).

Ah, but one could make the same argument for the real numbers, saying that it's sufficient to use rational numbers and slightly more complex calculations. And from that, you could say that integers are all that is necessary. Where do you draw the line?

You can measure the space group of a crystal (so I claim). Well, really, you're measuring some x-ray diffraction data, counting clicks on a few detectors and a timing circuit. Then you convert that to real and complex numbers (sort of -- within machine precision), fit it, then convert it into a space group. So where in that process does it stop being a measurement and become something more? When do your numbers stop having a physical representation?

Is it when you apply a theory to your data to get another value? Well, it's a theory that your ruler remains the same length between observations, and that if it measures two lines and both are 5 cm long, then they are the same length. So that doesn't quite work. So the question again: when does your interpretation and converstion of numbers stop having physical representation?

I say that it doesn't. Anything you derive from physical data is a measurement. There's a reason for this, too. With even a simple measurement, like counting apples or putting a ruler to a piece of paper to see how long a line is, there's a lot of approximation and computation going on. Your eye is a complicated detector, you're doing a huge amount of work to relate your interaction with the environment to your abstract concept of numbers, and you're not even certain that you're seeing everything. You make all sorts of assumptions and generalizations just looking at an apple.

Even if you disagree with that, most people will agree that anything that is sufficiently easy can be considered a measurement. A gas chromatograph peak constitutes a measurement, even though the underlying process which obtains that peak is very complicated, because you don't need to rebuild the machine every time. You just punch some buttons and out comes your measurement. This is the same as putting a ruler to paper and seeing that your line is 3.2 cm, or counting five apples. You don't need to re-learn how to see every time you do that.

In this way, you can measure complex values. So long as the process to get those values is relatively simple, most people will call it a measurement, unless they have a hang-up about complex numbers (now I've said it! ;) ). People used to have trouble accepting irrational numbers, too.

Ken G
2006-Apr-24, 06:58 PM
I agree completely with snarkophilus. To some people, simple addition is mind-boggling, while others prove theorems on mulitidmensional manifolds in their sleep. Just because one type of abstract thinking is generally viewed as more difficult than another does not weaken its connection to reality (and I interpret Celestial Mechanics' statement that complex numbers are unnecessarily confusing, not that they are less real. After all, if you use a technique that does not invoke complex numbers, you are doing all the same steps in your calculation, you are just using labels that some might call more or less confusing.) So one can go snarkophilus' route, and say that all abstract thinking that helps us make a prediction or understand an experiment is as much a part of "reality" as any other, you can try to recognize levels of connectedness, as Disinfo Agent does, or you can go to the other extreme and say that none of it has anything at all to do with reality, only with thought about reality. I'll leave it to the philosophers to debate, but my personal inclination is the last of these. When you lay a ruler and compare lengths, the concept of "length" is yours, the concept of "compare" is yours, and the concept of "ruler" as a standard for comparing lengths is yours. What does nature need with any of these concepts? Reality dances to a drummer that we can scarcely imagine, yet we do make progress by trying our best to grasp it conceptually. Infinity plays a role there as much as any other number.

Nereid
2006-Apr-24, 07:58 PM
Thanks (once again) to KenG (and to snarkophilus) :clap:

... for triggering another idea about 'physical reality'.

If '5' is a mental construct, then we can ask what it's representation is, in terms of processes and states in the human brain.

While some progress has been made towards working out such details, a detailed answer is not possible today. However, many folk would accept that it is possible, in some sense.

So, '5' can have a 'physical reality', in terms of processes and states in your brain (or mine, or ...).

And so can '∞'.

We can go further ... '5' could be a state in a (silicon-based) circuit. Sure it's a representation of '5', but it has no less of a physical reality for that. And if '5' can have such a physical reality, then so can '∞'.

(Perhaps Van Rijn's invisible elf also has a physical reality then?)

John Dlugosz
2006-Apr-24, 07:59 PM
Ah, but one could make the same argument for the real numbers, saying that it's sufficient to use rational numbers and slightly more complex calculations. And from that, you could say that integers are all that is necessary. Where do you draw the line?

With logic only: a very few logical productions and one initial fact will give you a system that can be interpreted as representing non-negative integers in a unary base and defining addition.

Everything else is meerly a mathematical convenience, and you could do without it by having somewhat more complex calculations.

references: Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter
Playing with Infinity by Rozsa Peter

While looking those up, I happened upon From Five Fingers to Infinity: A Journey Through the History of Mathematics which sounds interesting relative to this thread!

--John

snarkophilus
2006-Apr-24, 10:12 PM
While some progress has been made towards working out such details, a detailed answer is not possible today. However, many folk would accept that it is possible, in some sense.

So, '5' can have a 'physical reality', in terms of processes and states in your brain (or mine, or ...).

And so can '∞'.

Hmm... the difficulty with symbols in the brain is that they are highly context-sensitive. That is, there's nothing to say that your interpretation of '5' is in any way the same as my interpretation of '5' unless there is a mathematical centre hard-wired in there, one which is the same in all people. Some psychological studies seem to suggest that maybe for '5' that is true, but it is certainly not true for, say, '754.'

The representation in a circuit is a good idea, though. You set up an isomorphism between some aspect of reality and another aspect of reality. That constitutes a measurement.

(Perhaps Van Rijn's invisible elf also has a physical reality then?)

I think we've shown in another thread that she might. ;)

snarkophilus
2006-Apr-24, 10:19 PM
With logic only: a very few logical productions and one initial fact will give you a system that can be interpreted as representing non-negative integers in a unary base and defining addition.

Everything else is meerly a mathematical convenience, and you could do without it by having somewhat more complex calculations.

The Peano axioms produce the integers, yes (I assume that's what you mean). But that's still an abstract concept. It's only related to reality because we've invented it to correspond to reality. In fact, logic is purely abstract as well. There's no guarantee that it works in all real-world situations. Why choose that axiom schema over any other?

As an aside, it occurs to me that these ideas are created in an evolutionary manner. That is, if you could study the history of thought, you'd see that it resembles evolution among organisms. Something is observed (an environment is created). A number of theories are formed (organisms enters the environment). Something that contradicts that theory is discovered (the environment changes). Those theories which don't correspond to the new information are discarded (unfit organisms die, new ones become more prominent and reproduce). And so on.

At least, in theory. Sadly, there are a number of unfit theories out there. Consider that analogy a reflexive relation. :)

Disinfo Agent
2006-Apr-24, 10:23 PM
Ah, but one could make the same argument for the real numbers, saying that it's sufficient to use rational numbers and slightly more complex calculations.Conceptually, it isn't enough. Remember the Pythagorians and the square root of 2?

You can measure the space group of a crystal (so I claim). Well, really, you're measuring some x-ray diffraction data, counting clicks on a few detectors and a timing circuit. Then you convert that to real and complex numbers (sort of -- within machine precision), fit it, then convert it into a space group. So where in that process does it stop being a measurement and become something more? When do your numbers stop having a physical representation?I wish I knew enough about space groups of crystals to comment on your example, but unfortunately I do not. :(

Is it when you apply a theory to your data to get another value? Well, it's a theory that your ruler remains the same length between observations, and that if it measures two lines and both are 5 cm long, then they are the same length.Is that a theory, though? It could be argued that it's a basic principle without which science simply becomes impossible.

Even if you disagree with that, most people will agree that anything that is sufficiently easy can be considered a measurement.If you're going to appeal to popularity, I think I'll start a poll on how many people think there are real physical infinities in the universe. :p

Disinfo Agent
2006-Apr-24, 10:51 PM
I agree completely with snarkophilus. To some people, simple addition is mind-boggling, while others prove theorems on mulitidmensional manifolds in their sleep. Just because one type of abstract thinking is generally viewed as more difficult than another does not weaken its connection to realityFine, but you are assuming there is a connection to reality. That's arguable. I would even say doubtful, in the case of infinite numbers and complex numbers.

So one can go snarkophilus' route, and say that all abstract thinking that helps us make a prediction or understand an experiment is as much a part of "reality" as any other, you can try to recognize levels of connectedness, as Disinfo Agent does, or you can go to the other extreme and say that none of it has anything at all to do with reality, only with thought about reality. I'll leave it to the philosophers to debate, but my personal inclination is the last of these. When you lay a ruler and compare lengths, the concept of "length" is yours, the concept of "compare" is yours, and the concept of "ruler" as a standard for comparing lengths is yours. What does nature need with any of these concepts? Reality dances to a drummer that we can scarcely imagine, yet we do make progress by trying our best to grasp it conceptually. Infinity plays a role there as much as any other number.Here's the problem I see with treating measurements as a theoretical construct. Whenever someone arrives in this forum with an "against the mainstream claim", such as that UFOs are proof of ETI, or that evolution is just a theory as good as ID, or that the Twin Towers were blown up with explosives, the standard reply is "But the evidence...!" There's even a conversation in the same vein going on next door (http://www.bautforum.com/showthread.php?t=39322&page=7).

But if your position is that measurements are ultimately theoretical constructs, then doesn't that turn any science into "just a theory", and make the ID proponents' case for them?

It also seems to me that your position in this thread is inconsistent with your rejection, elsewhere (http://www.bautforum.com/showthread.php?p=693470#post693470), of the scientificity of interpretations of quantum mechanics. You have complex numbers in the formulas of relativity (http://www.bautforum.com/showthread.php?t=40149), and they give end results that physicists use. You get infinites in Maxwell's equations (http://www.bautforum.com/showpost.php?p=715353&postcount=9), and the end result is physically useful... And you can also sum over all histories (http://www.bautforum.com/showthread.php?t=39582) in a quantum mechanical problem, and get the right answer. How can you accept the former two (http://www.bautforum.com/showthread.php?p=716782#post716782), but reject the latter?

snarkophilus
2006-Apr-25, 12:02 AM
But if your position is that measurements are ultimately theoretical constructs, then doesn't that turn any science into "just a theory", and make the ID proponents' case for them?

But that's the essence of science, isn't it? We recognize that everything is "just a theory." We just say that the only useful information is that to which we can determine a decent probability of being true. ID = not science because no probability can be assigned. Peano axioms and derived mathematics = highly probable to be true in most cases, due to a massive number of observations.

Take the act of measuring with a ruler. You put the ruler next to object A and read 4 cm. You put the ruler next to object B and read 4 cm. Then you put the two objects together and note that they are the same length. Repeat a number of times to verify that it's consistent. That's science. Now you can extrapolate and say that it's very likely that your use of the ruler gives a valid measurement. Of course, the same can be applied to measurement of spectral lines or time or anything else.

Far from making a case for ID proponents, this view strengthens science. The whole idea is to keep your mind open, no? If that means entertaining ID ideas until you decide whether or not they are supported by facts or whether or not additional facts can be determined which would help in the decision, then that's for the best. That's responsible science.

As an example of a situation where "just a theory" was useful, I'll give the Einstein-Bohr debates and the EPR paper. That whole thing hinged heavily upon what the nature of measurement is, but from a slightly different view point than what we are discussing (though I'm sure we could tie them together if we felt it necessary). Bohr believed firmly in the predictions of QM, but Einstein clung firmly to more classical ideas. Their arguments led to the EPR paper, which ultimately opened up the whole investigation into entanglement and all sorts of other neat effects. Einstein was wrong, but even that was beneficial. Opposing views are the best thing for the advancement of knowledge. Who knows? Maybe one day some ID proponent or opponent will come up with a way to test some aspect of it or another. That'll be an interesting day!

snarkophilus
2006-Apr-25, 12:08 AM
Quote:
Originally Posted by snarkophilus
Ah, but one could make the same argument for the real numbers, saying that it's sufficient to use rational numbers and slightly more complex calculations.
Conceptually, it isn't enough. Remember the Pythagorians and the square root of 2?

It is enough. Formally, the real numbers are just sets of rational numbers. (Possibly infinite sets, sure, but the decimal representation of a real number may also be infinite.)

As to the poll about common meaning, I thought hard about bringing it up. :) However, I think we're better just to discuss what we mean by a measurement and then try to relate that to the OP.

Disinfo Agent
2006-Apr-26, 09:11 PM
But that's the essence of science, isn't it? We recognize that everything is "just a theory." We just say that the only useful information is that to which we can determine a decent probability of being true. ID = not science because no probability can be assigned. Peano axioms and derived mathematics = highly probable to be true in most cases, due to a massive number of observations.But where does that probability come from, if all you have is theory upon theory?...

Take the act of measuring with a ruler. You put the ruler next to object A and read 4 cm. You put the ruler next to object B and read 4 cm. Then you put the two objects together and note that they are the same length. Repeat a number of times to verify that it's consistent. That's science. Now you can extrapolate and say that it's very likely that your use of the ruler gives a valid measurement. Of course, the same can be applied to measurement of spectral lines or time or anything else.According to Ken G and Nereid, the measurements you got are entirely theoretical constructs based on assumptions you have made about the measuring process itself. What if some of those assumptions are wrong?

Far from making a case for ID proponents, this view strengthens science. The whole idea is to keep your mind open, no?...but not so much that your brains fall off. :p ;)

If that means entertaining ID ideas until you decide whether or not they are supported by facts or whether or not additional facts can be determined which would help in the decision, then that's for the best. That's responsible science.Wait; how do you decide that they're 'supported by facts', if the facts themselves are just deductions made from other theories?

As an example of a situation where "just a theory" was useful, I'll give the Einstein-Bohr debates and the EPR paper. That whole thing hinged heavily upon what the nature of measurement is, but from a slightly different view point than what we are discussing (though I'm sure we could tie them together if we felt it necessary). Bohr believed firmly in the predictions of QM, but Einstein clung firmly to more classical ideas. Their arguments led to the EPR paper, which ultimately opened up the whole investigation into entanglement and all sorts of other neat effects. Einstein was wrong, but even that was beneficial.
Looking here (http://math.ucr.edu/home/baez/physics/Quantum/bells_inequality.html), it seems that the matter is still somewhat open...

It is enough. Formally, the real numbers are just sets of rational numbers. (Possibly infinite sets, sure, but the decimal representation of a real number may also be infinite.)But you need those sets of rational numbers to describe all possible physical lengths. You can't do without them, or you'll have 'holes' in your rulers.

snarkophilus
2006-Apr-26, 09:48 PM
But where does that probability come from, if all you have is theory upon theory?...

It really is all just theory upon theory. What you do is at some arbitrary point, you say, "I'm assuming that X has a 100% probability of being true." Then X is an axiom. Often we'll take the principles of logic to be axiomatic, along with elements of math proven using them (despite the incompleteness of mathematics, by the way).

Then, you have experiment-dependent axioms. These are things like, "I measured that there's a spectral line at 729 Hz. I am going to declare that certainly I saw it, and there is no chance whatsoever that my ten detectors, for all five thousand samples, were just glitchy."

Of course, if you later find out that there really was no spectral line at 729 Hz, you may need to know that you made those assumptions. Maybe you got a bad batch of detectors. Maybe you were measuring in a black hole, and causality was violated, so your logic doesn't apply any more. Maybe the tooth fairy was playing with the apparatus. (The assumption that the tooth fairy does not interfere with your experiments is axiomatic, too.) It's important to recognize that no matter what you assume, there's always a chance that you are in error.

Heck, even probability theory is just a theory. It's just that it happens to work out rather often and has good predictive capability.

According to Ken G and Nereid, the measurements you got are entirely theoretical constructs based on assumptions you have made about the measuring process itself. What if some of those assumptions are wrong?

Then your measurements are probably wrong. This happens all the time, especially in theoretical science, and especially in the QM world.

...but not so much that your brains fall off. :p ;)

Ha ha, well... yes. :)

As a slight aside, I think that it's very important, from the point of view of the scientific method, to accept that the scientific method itself might indeed be completely wrong. It could be that we've just tricked ourselves a great deal, and this idea of testable hypotheses and experimentation actually doesn't work. It's just not very likely (at all), so there's no point in dwelling on it until we hit a case where it doesn't work. We'll take the correctness of the scientific method as an axiom until we discover that it causes more problems than it solves.

clop
2006-Apr-27, 03:45 AM
I can think of an infinity in everyday life.

It's the number of points you need to list to define the circumference of a circle without using a construct.

clop

edit: oh I've just noticed it has to be of a physical reality kind, and there are no real circles.

edit: mind you, it doesn't have to be a perfect circle, it can be any curve - how about I say "the number of discrete points you need to list to define the last orbit of the earth around the sun" - that's even better because it isn't a perfect circle and all constructs are approximations.

edit: how about the number of terms needed in a fourier series to accurately represent a non-sinusoid.

edit: or the number of terms needed in a binomial expansion to accurately represent a square root.

edit: this is a real one - infinity is the number of different possible orientations the seeds in an apple will have if you cut lots of apples in half - no two apples will ever be exactly the same.

clop
2006-Apr-27, 04:54 AM
Hmmm in terms of "infinity in physical reality" we can define infinity as any number that we are unable to handle in reality. It doesn't actually have to be infinity. Some numbers are so big that they might just as well be infinity, because they are literally too big for us to deal with. So big that we can't solve very simple finite problems that have these numbers in them.

One famous puzzle comes to mind - The Travelling Salesman.

A salesman is planning to travel to a number of different cities to sell his goods, starting out from, and ending at, his home city. We have to select a route for him which will minimise the distance he has to travel. Assuming we know the distance between each pair of cities on his tour, we have all the data necessary to plan the route.

It's not that easy. The problem becomes exponentially more difficult as the number of cities increases, and approximation algorithms do not give the optimal route.

12 cities doesn't sound a lot does it? You might think you could work the answer out for 12 cities with a piece of paper and a pencil, but with 20,000,000 possible routes to check, even if you could do them at a rate of one every 2 minutes, it would take you 76 years to complete the task.

And 16 cities would have 653,000,000,000 possible solutions. That's two and a half million years with your pencil.

In 1954 the world record solution was for just 49 cities.

In 1992 the world record was up to 3038 cities.

In 1994 the world record was 7397 cities.

In 1998 it was 13509 cities.

In 2004 the world record was up to 24978 cities.

The current record, set in 2005, is for 33810 cities. The computation took 15.7 CPU years.

So to define infinity to you, I just have to ask you to solve the travelling salesman problem for 80,000 cities. It may be that the human race will never develop a method to find the solution to this.

It's a finite problem with real world applications - but it's intractable - i.e. infinite - in our reality.

clop

Ken G
2006-Apr-27, 05:53 AM
Here's an even simpler one. Essentially any arbitrarily chosen length will be in an irrational ratio to the length standard, so will require an infinite number of decimal places to specify exactly in terms of that length standard. Granted, you'll never get an infinite number of decimal places in the actual measurement because you'll have to settle for some level of precision, but that's a measurement convention, not "reality". If one is to argue that measurements never yield infinite results because they are not allowed to by convention, that is a rather different (and self-fulfilling) argument than that reality contains no infinities.

clop
2006-Apr-27, 12:40 PM
Here's an even simpler one. Essentially any arbitrarily chosen length will be in an irrational ratio to the length standard, so will require an infinite number of decimal places to specify exactly in terms of that length standard. Granted, you'll never get an infinite number of decimal places in the actual measurement because you'll have to settle for some level of precision, but that's a measurement convention, not "reality". If one is to argue that measurements never yield infinite results because they are not allowed to by convention, that is a rather different (and self-fulfilling) argument than that reality contains no infinities.

Well yes but like you say, as our measurement technique is limited to the resolution of our measuring standard, in reality all measurements can be expressed using non-infinite values. Anyone can define infinity as just halving the gap forever, but it doesn't correspond to our reality if we can't resolve the gap. At least the salesman problem (and others) gives an essentially-infinite value from a simple and very much finite construct. No gap halving required.

clop

hhEb09'1
2006-Apr-28, 02:11 AM
It's a finite problem with real world applications - but it's intractable - i.e. infinite - in our reality.i.e. infinite? I don't think intractable means infinite. :)

And I am as impressed as you are with the difficulty of the traveling salesperson problem. Building a building with 30 septillion kilogram bricks is intractable on earth, but the number is not infinite.

clop
2006-Apr-28, 02:51 AM
i.e. infinite? I don't think intractable means infinite. :)

And I am as impressed as you are with the difficulty of the traveling salesperson problem. Building a building with 30 septillion kilogram bricks is intractable on earth, but the number is not infinite.

All I'm saying is that the number is so large that it might as well be infinite, practically speaking.

clop

snarkophilus
2006-Apr-28, 06:31 AM
All I'm saying is that the number is so large that it might as well be infinite, practically speaking.

clop

Well, the classic definition of infinity is "arbitrarily large." But if you can find an exact real (or complex, or whatever) value for a problem, no matter how big, then it's not infinite.

clop
2006-Apr-28, 12:13 PM
Well, the classic definition of infinity is "arbitrarily large." But if you can find an exact real (or complex, or whatever) value for a problem, no matter how big, then it's not infinite.

But you can't find a value for the travelling salesman problem with 80,000 cities, even if you wanted to. And we may never be able to either. No it's not infinite but it might as well be from our point of view. It's more infinite than anything we can measure and halve.

The original post was asking about infinities we encounter in physical reality. I haven't seen any other example of a number so large that we can't work with it. Accurately measuring the length of a piece of string is no good because we can only measure it to resolution of our measurement technique, then it becomes finite. All number constructs are conceptual and do not have basis in "physical reality". There are no true infinities in nature that are not made non-infinite by our measurement limits. So I'm giving you a number that is so big in physical reality (the solutions to 80,000 cities) that the combined computing power of every computer on earth cannot solve the problem in our lifetimes. That's big enough for me.

clop

Ken G
2006-Apr-28, 01:01 PM
Accurately measuring the length of a piece of string is no good because we can only measure it to resolution of our measurement technique, then it becomes finite.

My point was, does "reality" include a concept of the exact length of something, even if we can't measure it? If it does, then there's a kind of infinity-- infinite decimal places to express an exact length. If it does not, and length is inseparable from measurement so is a purely human ideal, then you can't distinguish "real" things from "ideal" things-- all our words are ideals. Snarkophilus and I have argued for the latter, but either way, you still encounter "real" infinities.

hhEb09'1
2006-Apr-28, 04:18 PM
But you can't find a value for the travelling salesman problem with 80,000 cities, even if you wanted to. And we may never be able to either. No it's not infinite but it might as well be from our point of view. It's more infinite than anything we can measure and halve.I may not be able to find an exact value, but I can easily write down a number that is larger: 80,000! (factorial)

As everyone knows, that's nowhere close to infinity :)

clop
2006-Apr-28, 04:46 PM
I may not be able to find an exact value, but I can easily write down a number that is larger: 80,000! (factorial)

As everyone knows, that's nowhere close to infinity :)

No, 80,000! is not a real thing in our physical reality. It's a construct of a concept. The travelling salesman problem is a real problem in our physical reality. You can't just say a number and keep adding one to it and say that's infinity because it has no use to anyone.

clop

snarkophilus
2006-Apr-28, 08:34 PM
But you can't find a value for the travelling salesman problem with 80,000 cities, even if you wanted to. And we may never be able to either. No it's not infinite but it might as well be from our point of view. It's more infinite than anything we can measure and halve.

Well, you can find out how many comparisons you would need to make. You can set an upper limit on the time it would take to solve the problem. So that's not really infinity.

And actually, TSP is only difficult on current computer architectures. It's just a matter of time until quantum computing makes that problem relatively easy (though there are other problems that are still hard on quantum computers).

There are no true infinities in nature that are not made non-infinite by our measurement limits.

This was the point of the discussion. :) I contend that there are, indeed, infinities in nature, and I've given a few examples (as have some others) to support that. If you are willing to accept TSP as a real-world example, then my "sets of sets" example (way back near the start of the thread) may convince you that there are infinities in reality.

Disinfo Agent
2006-Apr-28, 09:59 PM
Here's an even simpler one. Essentially any arbitrarily chosen length will be in an irrational ratio to the length standard, so will require an infinite number of decimal places to specify exactly in terms of that length standard. Granted, you'll never get an infinite number of decimal places in the actual measurement because you'll have to settle for some level of precision, but that's a measurement convention, not "reality".We really (no pun intended) have very different perspectives about this. I would say the opposite: it's the fact that you have to stop at some decimal place in practice that is "reality". The exact infinite decimal expansion is the abstraction.

No, 80,000! is not a real thing in our physical reality. It's a construct of a concept. The travelling salesman problem is a real problem in our physical reality. You can't just say a number and keep adding one to it and say that's infinity because it has no use to anyone.

clopIf a problem is real, can we be sure that its solution is real, too? :)

Ken G
2006-Apr-28, 10:31 PM
We really (no pun intended) have very different perspectives about this. I would say the opposite: it's the fact that you have to stop at some decimal place in practice that is "reality". The exact infinite decimal expansion is the abstraction.

I suppose it is inevitable that this discussion must hinge on an operational definition of "reality". I tend to think of reality as that which is, that which all of our measurements derive from, that which we only approximate whenever we attempt to quantify it. Thus for me, any measurement that yields a quantity of some kind is a replacement of reality with a conceptualization of reality. Such a measurement is the intersection of reality with the human act of trying to understand it. Your approach is rather to think of the reality as the measurement, but then, what will you call that from which the measurement derives? Also, a measurement has an arbitrary character, which is how precise to do you want to be. Two people can "measure" the same event with differing precisions, and then by your definition of reality, there are two separate realities going on there. If so, then what will you call whatever aspect is the same for both? Or do you contend, like some postmodernists, that reality is purely subjective?

hhEb09'1
2006-Apr-29, 02:32 PM
I may not be able to find an exact value, but I can easily write down a number that is larger: 80,000! (factorial)

As everyone knows, that's nowhere close to infinity :)

No, 80,000! is not a real thing in our physical reality. It's a construct of a concept. The travelling salesman problem is a real problem in our physical reality. You can't just say a number and keep adding one to it and say that's infinity because it has no use to anyone.No?

When you said "no", what was it that you were disagreeing with?

clop
2006-Apr-29, 02:53 PM
No?

When you said "no", what was it that you were disagreeing with?

He said he could write 80,000! and I was saying no, he couldn't write it. Sure, you can write "80,000!" but what's that? It's not in our physical reality; it's a construct in our head. To make it physical reality you'd have to make 80,000! dots on a piece of paper or something.

Now as far as I can gather from a bit of googling, there are around 2x10E77 atoms in the visible universe.

That's only about 58! So we can't even do 58! dots.

I think you see the problem with 80,000!

clop

gzhpcu
2006-Apr-29, 03:20 PM
Here's an even simpler one. Essentially any arbitrarily chosen length will be in an irrational ratio to the length standard, so will require an infinite number of decimal places to specify exactly in terms of that length standard. Granted, you'll never get an infinite number of decimal places in the actual measurement because you'll have to settle for some level of precision, but that's a measurement convention, not "reality". If one is to argue that measurements never yield infinite results because they are not allowed to by convention, that is a rather different (and self-fulfilling) argument than that reality contains no infinities.

IMHO, you will never get an infinite number of decimal points, because it seems that we do not have an infinitely small building block in nature (reality). Disregarding QM, because it breaks down at the Planck length, we can not assume 0 dimensional points as being the basic building blocks. The next theory, M-theory, postulates strings the size of the Planck length, so this would set a lower limit to the granularity of the universe. If correct, this would exclude an infinite number of decimal points.

gzhpcu
2006-Apr-29, 03:26 PM
I can think of an infinity in everyday life.

It's the number of points you need to list to define the circumference of a circle without using a construct.

clop

edit: oh I've just noticed it has to be of a physical reality kind, and there are no real circles.

edit: mind you, it doesn't have to be a perfect circle, it can be any curve - how about I say "the number of discrete points you need to list to define the last orbit of the earth around the sun" - that's even better because it isn't a perfect circle and all constructs are approximations.

edit: how about the number of terms needed in a fourier series to accurately represent a non-sinusoid.

edit: or the number of terms needed in a binomial expansion to accurately represent a square root.

edit: this is a real one - infinity is the number of different possible orientations the seeds in an apple will have if you cut lots of apples in half - no two apples will ever be exactly the same.

IMHO, none of these examples correspond to everyday life. A discrete point (I assume you mean 0 dimensional) is a mathematical construct. Does not exist in reality. The applies example isn't either: an infinite number of apples to cut in the universe does not exist.

clop
2006-Apr-29, 03:35 PM
IMHO, none of these examples correspond to everyday life. A discrete point (I assume you mean 0 dimensional) is a mathematical construct. Does not exist in reality. The applies example isn't either: an infinite number of apples to cut in the universe does not exist.

Yes you're right. The first examples are just numbers in equations. No basis in reality. The orbit of the earth is physical but still depends on finite resolution of measurement. The apples do have a finite number of combinations using our current measurement techniques.

clop

Disinfo Agent
2006-Apr-29, 03:37 PM
I suppose it is inevitable that this discussion must hinge on an operational definition of "reality". I tend to think of reality as that which is, that which all of our measurements derive from, that which we only approximate whenever we attempt to quantify it. Thus for me, any measurement that yields a quantity of some kind is a replacement of reality with a conceptualization of reality. Such a measurement is the intersection of reality with the human act of trying to understand it. Your approach is rather to think of the reality as the measurement, but then, what will you call that from which the measurement derives? Also, a measurement has an arbitrary character, which is how precise to do you want to be. Two people can "measure" the same event with differing precisions, and then by your definition of reality, there are two separate realities going on there. If so, then what will you call whatever aspect is the same for both? Or do you contend, like some postmodernists, that reality is purely subjective?You've reminded me of this discussion (http://www.bautforum.com/showthread.php?t=38524). :)

Granted, I was using the word 'reality' in a different sense from the one we gave it in that earlier discussion. But the title of this thread has the phrase 'physical reality'. For me, this is composed of what we perceive of reality, and how we perceive it. Which may just be a very incomplete and perhaps distorted glimpse of what truly exists, but it's all we have to base science on (apart from reason).

There may be something else out there, of course -- such as infinite curvatures of space, or protons and photons, or exact infinite decimal expansions -- but we have no way of knowing that for sure. What we know, in science, comes from what we can count or measure.

snarkophilus
2006-Apr-29, 05:34 PM
He said he could write 80,000! and I was saying no, he couldn't write it. Sure, you can write "80,000!" but what's that? It's not in our physical reality; it's a construct in our head. To make it physical reality you'd have to make 80,000! dots on a piece of paper or something.

Now as far as I can gather from a bit of googling, there are around 2x10E77 atoms in the visible universe.

That's only about 58! So we can't even do 58! dots.

I think you see the problem with 80,000!

clop

Hmm... but the number 6 is just a construct in your head, too. After all, what do you mean by 6? If you say, "I have 6 apples," you really mean that you've identified a correspondence between some measurement you've made of the outside world (via your eyes, for instance) and some internal representation of numbers.

That aside, say there are only 2e77 atoms in the universe. What if you represent reality as the links between those atoms? That is to say, what if you consider the connection between each atom as one unit? Then you can count much higher indeed. In fact, you will be to count to 2e77! in that way. Then, what if you want to count the number of groups of links? Now you're getting big, and that's still physical reality, no? (It's 2e77!!/1! + 2e77!!/2! + 2e77!!/3! +... combinations. I'm too stupid right now to think of what that adds up to.) If not, then your dots are also not physical reality, as they are just groups of atoms (unless, of course, you deny that atomic interactions are part of reality).

You can extend that scheme, take any integer (really, any integer), and in a finite and relatively small number of steps, figure out where in the scheme that integer belongs. That's infinity.

clop
2006-Apr-29, 06:18 PM
Hmm... but the number 6 is just a construct in your head, too. After all, what do you mean by 6? If you say, "I have 6 apples," you really mean that you've identified a correspondence between some measurement you've made of the outside world (via your eyes, for instance) and some internal representation of numbers.

That aside, say there are only 2e77 atoms in the universe. What if you represent reality as the links between those atoms? That is to say, what if you consider the connection between each atom as one unit? Then you can count much higher indeed. In fact, you will be to count to 2e77! in that way. Then, what if you want to count the number of groups of links? Now you're getting big, and that's still physical reality, no? (It's 2e77!!/1! + 2e77!!/2! + 2e77!!/3! +... combinations. I'm too stupid right now to think of what that adds up to.) If not, then your dots are also not physical reality, as they are just groups of atoms (unless, of course, you deny that atomic interactions are part of reality).

You can extend that scheme, take any integer (really, any integer), and in a finite and relatively small number of steps, figure out where in the scheme that integer belongs. That's infinity.

Yes I think you are right. But I got the gist from the original post that the infinity in question has to relate to some kind of everyday physical thing. The travelling salesman problem (TSP) does relate to several real world applications. One of the more recent solutions of the TSP was specifically to find the quickest way for a robot to drill all the holes in a printed circuit board, and so minimise the time (and cost) required to make the board. This is an everyday application.

Your idea of linking every atom in the universe with every other atom in the universe does not have an everyday real world application (and by the way it's nowhere near as big as 2e77!, it's a summation series of n=1 to (2e77-1) which is around 1e77*2e77 which is only about 98!, a tiny tiny number compared to 80,000! - your 2e77! is the number of possible solutions for doing a universal tour of each atom in the universe, not the number of links). However, I can see an application for the number of links in theoretical physics. It would be a comprehensive index of every single gravitational force in the universe. Now wouldn't that be handy.

clop

neilzero
2006-Apr-29, 06:32 PM
Worse some people have poor comprehension of practical applications and projects that have hope of being engineered sucessfully. Some of these are quite influential and are mudding the waters. Neil

snarkophilus
2006-Apr-29, 07:59 PM
Your idea of linking every atom in the universe with every other atom in the universe does not have an everyday real world application (and by the way it's nowhere near as big as 2e77!, it's a summation series of n=1 to (2e77-1) which is around 1e77*2e77 which is only about 98!, a tiny tiny number compared to 80,000! - your 2e77! is the number of possible solutions for doing a universal tour of each atom in the universe, not the number of links). However, I can see an application for the number of links in theoretical physics. It would be a comprehensive index of every single gravitational force in the universe. Now wouldn't that be handy.

Ah yes... but I was thinking not just 1-1 links, but also links between three atoms, links between four, et cetera.

Anyway, yes. An index of all gravitational forces, and EM forces, and everything else... it'd be useful if we could use it! :D

clop
2006-Apr-29, 08:37 PM
Ah yes... but I was thinking not just 1-1 links, but also links between three atoms, links between four, et cetera.

Anyway, yes. An index of all gravitational forces, and EM forces, and everything else... it'd be useful if we could use it! :D

Even then the positions and distances would be approximations and the entire index would need to be recompiled every moment.

I wonder if time is a continuous or discrete coordinate. I've often thought that time could be a discrete variable, but the quantum interaction of all the particles in the universe, that define time, makes it appear continuous to us.

clop

PS I'm sitting in Chicago airport bored out of my head.

Disinfo Agent
2006-Apr-29, 08:42 PM
Even then the positions and distances would be approximations and the entire index would need to be recompiled every moment.It seems like a paradoxical concept. You should need more particles to store all that information than there are in the universe itself. Perhaps God can do it. ;)

Ken G
2006-Apr-30, 04:26 PM
. But the title of this thread has the phrase 'physical reality'. For me, this is composed of what we perceive of reality, and how we perceive it. Which may just be a very incomplete and perhaps distorted glimpse of what truly exists, but it's all we have to base science on (apart from reason).
So physical reality, as you apply the term, is more than just the measurements themselves, it requires analysis by some inteligence. A scale is just a spring that flexes. So your interpretation of physical reality includes the application of numbers as constructs of our minds. That's also where infinity comes from.

gzhpcu
2006-Apr-30, 06:32 PM
When we measure something, we need to agree on a unit of measurement, and then measure in multiples of that unit of measurement. The multiples of the unit measurement is represented in our minds by numbers. Numbers allow us to count and measure. I still do not see where infinity comes in. It is just an abstract mathematical symbol.

Ken G
2006-Apr-30, 07:38 PM
Infinity comes in as soon as you are allowed to mentally manipulate numbers that come from "reality". and that's inherent in the concept of number. For example, not only can we use numbers to count relative to some unit standard, we can form ratios of any such counts (like speed, for example, which is a physical quantity that is not counted relative to a standard unit, but instead is a ratio of such counts). It's true that we don't see infinite speeds, but we can just as easily define inverse speed. Inverse speed is the time you take to travel a certain distance. Or we can use inverse acceleration as a more common example, like a car that goes from zero to 50 mph in 6 seconds. That's as physical a measure as actual acceleration (and means more to car enthusiasts), yet it becomes infinite for objects that don't move at all. I don't see any fundamental difference here-- systems that allow zeroes will also allow infinities, and they will be just as "physical".

Nereid
2006-Apr-30, 08:34 PM
When we measure something, we need to agree on a unit of measurement, and then measure in multiples of that unit of measurement [snip]And that's fine, but it's also only a subset of what we do.

"Measurement" also implies "theory"; neither are necessarily fully axiomatised, internally consistent, etc; the 'fundamentalness' (if there is such a word) of "measurement" is just a story we tell ourselves, there's nothing 'in reality' which demands that "measurement according to an agreed set of units" be in place first.

So, you either accept the 'reality' of the (implicit) theory (or theories) that "measurement" entails (and so 'infinity, of the physical reality kind' is but a mere consequence), or you don't (in which case you have to establish - for us all, if your 'reality' is anything other than a purely personal one - that your measurement is 'theory-free', or that your theory cannot admit 'infinity', that ...). The latter is going to be a very hard sell, given how powerful the interlocking, logically consistent results from numerous work in/on mathematics is.

gzhpcu
2006-May-01, 06:23 AM
Infinity comes in as soon as you are allowed to mentally manipulate numbers that come from "reality". and that's inherent in the concept of number. For example, not only can we use numbers to count relative to some unit standard, we can form ratios of any such counts (like speed, for example, which is a physical quantity that is not counted relative to a standard unit, but instead is a ratio of such counts). It's true that we don't see infinite speeds, but we can just as easily define inverse speed. Inverse speed is the time you take to travel a certain distance. Or we can use inverse acceleration as a more common example, like a car that goes from zero to 50 mph in 6 seconds. That's as physical a measure as actual acceleration (and means more to car enthusiasts), yet it becomes infinite for objects that don't move at all. I don't see any fundamental difference here-- systems that allow zeroes will also allow infinities, and they will be just as "physical".

IMHO, for the example for the object standing still: the time is infinite only mathematically. In reality, not so, for a number of reasons:

1) it is not proven the universe will last for infinity,
2) no object really "stands still" in reality,
3) even less will an object stand still for eternity
4) should the universe at any time implode, the object will no longer be standing still

All I am trying to say, is that the examples are valid mathematically, but do not correspond to what happens in the real world.

gzhpcu
2006-May-01, 06:31 AM
And that's fine, but it's also only a subset of what we do.

"Measurement" also implies "theory"; neither are necessarily fully axiomatised, internally consistent, etc; the 'fundamentalness' (if there is such a word) of "measurement" is just a story we tell ourselves, there's nothing 'in reality' which demands that "measurement according to an agreed set of units" be in place first.

So, you either accept the 'reality' of the (implicit) theory (or theories) that "measurement" entails (and so 'infinity, of the physical reality kind' is but a mere consequence), or you don't (in which case you have to establish - for us all, if your 'reality' is anything other than a purely personal one - that your measurement is 'theory-free', or that your theory cannot admit 'infinity', that ...). The latter is going to be a very hard sell, given how powerful the interlocking, logically consistent results from numerous work in/on mathematics is.

Not quite sure I follow you, will attempt to answer, if I misunderstood you, please correct me:

- All theories need mathematics to describe them.
- The universe is immensely complex
- In order to attempt to understand it, we have iteratively come up with progressively more refined theories
- Mathematics is an invention of the human mind to try to cope with the complexity of the universe, to try to understand it
- All theories are approximative, because at some point an idealized assumption is made (e.g. 0 dimensional points, 1 dimensional strings, etc.)
- The concept of infinity comes up in mathematics

It is not proven that infinity comes up in the real world.

snarkophilus
2006-May-01, 07:19 AM
IMHO, for the example for the object standing still: the time is infinite only mathematically. In reality, not so, for a number of reasons:

1) it is not proven the universe will last for infinity,
2) no object really "stands still" in reality,
3) even less will an object stand still for eternity
4) should the universe at any time implode, the object will no longer be standing still

All I am trying to say, is that the examples are valid mathematically, but do not correspond to what happens in the real world.

Here's the difficulty with that interpretation: every value you measure is (perhaps) valid mathematically, but none of them correspond to what happens in the real world, exactly.

What is momentum? What is it, really? How do you measure it? You can't really answer those except in terms of other things. You can say, "it's mass times velocity," but what do you mean when you say "times"? That's a theoretical concept. For that matter, what is mass, really? Again, purely a thing of theory, a construct of the mind.

All of measurement is defined in terms of other measurements, and the whole mess comes from theory. And in those theories, you can have infinite numbers, complex numbers, et cetera. Furthermore, those theories might describe the world very closely. But without a theory to tell you what mass (for example) is, there's no measurement at all, so without infinite numbers, there is also no measurement.

The universe doesn't need to last for infinity time in order to measure an infinite quantity. The quantum number of a free electron is infinity, for instance. That means that just by looking at the sodium street lamps outside my house, I am measuring an infinite quantity, because I can see from that colour that there is ionized sodium in there. Sure, I'm making an approximation (that it's just me and the lamp), but everything I measure requires the exact same assumptions.

If I measure the distance between me and you, I'm assuming that there's no warp in space between us. I'm assuming that the light is not being sent an unreasonable distance, then refocussed and amplified. I'm assuming that my measuring apparatus is behaving as it does under some other conditions (again, measurement is relative). It's all relative, and whenever you can compare things like that, you can get infinities, and they're just as valid mathematically as are the real numbers you get. That means that they have the exact same correspondence to reality that the real numbers get.

Actually, I think that summary is better than any specific example I can give. Infinite numbers can have the same degree of correspondence to reality that real numbers can have, because of the way in which quantities are measured: through comparison.

gzhpcu
2006-May-01, 12:22 PM
We attempt to understand the universe with the help of theories. These theories are described using mathematics. Infinities occur in mathematical equations. Ergo they will occur in theories. The discussion of the free electron is theory-base/math-based. In this context, infinities may occur. But not in nature.

I still do not see where infinities occur in the real world (nature). Please give me an example where such infinities actually occur, and not artificial, math-based ones.

hhEb09'1
2006-May-01, 01:49 PM
I still do not see where infinities occur in the real world (nature). Please give me an example where such infinities actually occur, and not artificial, math-based ones.That's kinda like asking for an example of where we use language in the real world, without using words :)

We can express the coordinates of a finite point using an infinite decimal representation, or we can talk about the angle whose tangent is infinite but is actually a very finite pi over two. We've found ways to deal with the infinities, whether it is through another attribute or renormalization or a reparametrization. I suppose that makes them less real, but I'm not so sure.

What is wrong with having a line with infinite slope? You can look at it from another coordinate system, and it'll have a finite slope, but does that make its slope in the first system less real? Or is the idea of slope not real?

Ken G
2006-May-01, 02:38 PM
Well put, hh. It strikes me that this thread is an interesting way to look at all kinds of quantitative concepts, and underneath it all, we are probably not all disagreeing as much as it might appear.

gzhpcu
2006-May-01, 03:25 PM
That's kinda like asking for an example of where we use language in the real world, without using words :)

We can express the coordinates of a finite point using an infinite decimal representation, or we can talk about the angle whose tangent is infinite but is actually a very finite pi over two. We've found ways to deal with the infinities, whether it is through another attribute or renormalization or a reparametrization. I suppose that makes them less real, but I'm not so sure.

What is wrong with having a line with infinite slope? You can look at it from another coordinate system, and it'll have a finite slope, but does that make its slope in the first system less real? Or is the idea of slope not real?

All I am saying is that a line with infinite slope does not exist in the real world. By real world, I mean in nature. The physical world out there.

hhEb09'1
2006-May-01, 03:31 PM
All I am saying is that a line with infinite slope does not exist in the real world. By real world, I mean in nature. The physical world out there.Still, what does that mean? Do you mean, no line has a slope? Or do you mean, we could never guarantee that a particular line was perfectly vertical? Or do you mean, there is no such thing as lines, in the real world?

gzhpcu
2006-May-01, 03:46 PM
Well, first of all, you are equating a 90 degree angle with what you define as being "an infinite slope". This is a mathematical definition. Also, the concept of a line is mathematical. What I am looking for is an example where infinity can apply in nature. Actually in nature, not in a mathematical model of nature.

hhEb09'1
2006-May-01, 04:17 PM
Well, first of all, you are equating a 90 degree angle with what you define as being "an infinite slope". This is a mathematical definition. Also, the concept of a line is mathematical. What I am looking for is an example where infinity can apply in nature. Actually in nature, not in a mathematical model of nature.Infinity is also a mathematical concept. Your question is meaningless, if you do not allow mathematical terms.

So far as we know, time is infinite. It may not be, but it certainly could be. How about that?

gzhpcu
2006-May-01, 04:54 PM
Sure time might be infinite, but the jury is still out. We do not know for sure either way.

The mathematical symbol for infinity is what I do not particularly care for (in the context of nature).

hhEb09'1
2006-May-01, 05:24 PM
Sure time might be infinite, but the jury is still out. We do not know for sure either way.That's another way to deal with infinities. :) We just don't know anything for sure.

What's the beef with the symbol? or did you explain that earlier?

gzhpcu
2006-May-01, 05:51 PM
My problem with the symbol? Simply, that IMHO, it represents a number which does not exist in nature.

Nereid
2006-May-01, 06:11 PM
My problem with the symbol? Simply, that IMHO, it represents a number which does not exist in nature.And this is in contrast with other 'numbers', which do 'exist in nature'?

And the basis for your belief for the existence of (any) number(s) in nature is ...?

gzhpcu
2006-May-01, 07:42 PM
No, just infinity. Remember all other numbers are represented using the digits 0 - 9. Infinity is just represented by a special symbol.

Nereid
2006-May-01, 10:48 PM
No, just infinity. Remember all other numbers are represented using the digits 0 - 9. Infinity is just represented by a special symbol.Let's see now ... there are, what, ~10^80 baryons in the observable universe; so does the number 100^100 'exist in nature'? How about 9999^9999^9999 ( (99999999)9999 )?

I can represent the 'number' 0.5 as 1/2, or 3/6, or ... is √2 a 'number'? Afterall, I can't represent it 'using the digits 0 - 9'.

Suppose, instead of ∞, I write '1/0', have I thus 'represented' infinity 'using the digits 0 - 9'?

And which particular infinity do you have a problem with - the infinity of the integers, or the reals, or ... (or perhaps all of them)?

gzhpcu
2006-May-02, 03:46 AM
Let's see now ... there are, what, ~10^80 baryons in the observable universe; so does the number 100^100 'exist in nature'? How about 9999^9999^9999 ( (99999999)9999 )?

I can represent the 'number' 0.5 as 1/2, or 3/6, or ... is √2 a 'number'? Afterall, I can't represent it 'using the digits 0 - 9'.

Suppose, instead of ∞, I write '1/0', have I thus 'represented' infinity 'using the digits 0 - 9'?

And which particular infinity do you have a problem with - the infinity of the integers, or the reals, or ... (or perhaps all of them)?

The numbers you represent above are large but not infinite. The equations can be resolved. You can write out the digits. ∞ is a special symbol, a special case. IMHO, it does not appear in nature.

I have a problem with the symbol ∞. Your example of 1/0: you can not solve this equation other than supplying ∞.

clop
2006-May-02, 07:16 AM
The numbers you represent above are large but not infinite. The equations can be resolved. You can write out the digits.

And once again, my entire point, how exactly is it possible to write out the digits of a number larger than the number of baryons in the universe.

clop

snarkophilus
2006-May-02, 07:33 AM
The numbers you represent above are large but not infinite. The equations can be resolved. You can write out the digits. ∞ is a special symbol, a special case. IMHO, it does not appear in nature.

I have a problem with the symbol ∞. Your example of 1/0: you can not solve this equation other than supplying ∞.

How is the square root sign not a special symbol? And what does the symbol have to do with a number, anyway?

Suppose you tell me that since sqrt(2) can be written using an infinite number of the digits 0-9, it is a real number. Well, 1000000... (with infinity 0's) can also be written using only those digits, so it's just as much a number as sqrt(2). Or do you perhaps deny the existence of transcendental numbers (like pi and e)?

The symbol you use for a number is unimportant. It's just a convenience, a construct. You could equally well use an arbitrary system of dots and lines and banana cream pies to represent numbers (though performing arithmetic would be more difficult and delicious). The number itself is two things:
1) A theoretical object created from a set of rules (axioms) which can be used with respect to operators.
2) A theoretical object whose properties have a correspondence to some real-world phenomenon, either directly or via operations on the class of all numbers.

gzhpcu
2006-May-02, 08:09 AM
The square root is a symbol for a mathematical operation, and not a number. The numbers you mention are just that: numbers. A product of our imagination. Math is a product of our imagination.

What it all boils down to, is that there is no infinite anything in the physical universe. (unless, of course, the universe is infinite in size and will exist for infinite time - which is not proven)

Nereid
2006-May-02, 08:27 AM
The square root is a symbol for a mathematical operation, and not a number. The numbers you mention are just that: numbers. A product of our imagination. Math is a product of our imagination.

What it all boils down to, is that there is no infinite anything in the physical universe. (unless, of course, the universe is infinite in size and will exist for infinite time - which is not proven)By the same logic, there are no numbers in the physical universe, ... certainly no number such as 100100, nor any negative number, nor zero, ...

(and if you try to make a case for, say, 'one', then whatever method you use can be employed to show, by the same logic, that 'infinity' has the same status ... unless your method for 'one' is, in effect, nothing more than "I declare it to be so").

gzhpcu
2006-May-02, 08:37 AM
I am probably thick, but I do not really follow. What I am trying to say, is that the symbol ∞ does not seem to me to be applicable in nature. For example: I count exactly one apple. I can not count ∞ apples.

I am not saying that all the results of mathematics are figments of our imagination. Many of the numbers which result can be interpreted meaningfully. But I do not see how ∞ can.

Disinfo Agent
2006-May-02, 11:04 AM
But the title of this thread has the phrase 'physical reality'. For me, this is composed of what we perceive of reality, and how we perceive it. Which may just be a very incomplete and perhaps distorted glimpse of what truly exists, but it's all we have to base science on (apart from reason).
So physical reality, as you apply the term, is more than just the measurements themselves, it requires analysis by some inteligence. A scale is just a spring that flexes. So your interpretation of physical reality includes the application of numbers as constructs of our minds.I did not come up with the phrase 'physical reality', Nereid did. I can't be sure about what he meant by it, but, from the context, it seemed clear that he was referring to observational data, as opposed to theory or interpretation. Was I wrong?

My problem with the symbol? Simply, that IMHO, it represents a number which does not exist in nature.How do you know it doesn't exist in nature?

gzhpcu
2006-May-02, 12:08 PM
How do I know it doesn't exist in nature? I am not so arrogant as to say I know anything at all with 100% certainty. It just seems to me that it does not, because I have yet to see an example which convinces me. Why do I tend to think it does not exist in nature? Well, because it would imply an infinite amount of something: energy, time, dimension, particles. If the universe were proven to be timeless and infinite, then that would be another matter.

Disinfo Agent
2006-May-02, 12:24 PM
Why do I tend to think it does not exist in nature? Well, because it would imply an infinite amount of something: energy, time, dimension, particles.And what's the problem with that?

gzhpcu
2006-May-02, 12:35 PM
The problem is that IMHO those infinities most probably do not exist except mathematically. If our universe did really begin with a BB (be it due to some unknown fluctuation or brane collision, or whatever), then it is finite.

gzhpcu
2006-May-02, 12:37 PM
This discussion is somewhat akin to the one I had with Grey on 0 dimensional particles. I just don't believe in 0 dimensional particles. 0 dimensional for me is nothing. The concept of infinity has the same reaction on me.

Grey
2006-May-02, 01:00 PM
The problem is that IMHO those infinities most probably do not exist except mathematically. If our universe did really begin with a BB (be it due to some unknown fluctuation or brane collision, or whatever), then it is finite.Well, no. A big bang universe could still be infinite. As far as I can tell, an infinite universe could never be proven to your satisfaction. It's clear that however long you wait around to see if time is infinite, it will always be a finite length of time. And however far you look, anything you see will always be a finite distance away. It's possible that we'll discover that models of the universe that are infinite fit the observations better than models that are finite right now the issue is still open, but with better observations, it might be clearly settled one way or the other). For most cosmologists, that would be the same as showing that the universe is infinite, but I expect you would simply dismiss that as theory and not "reality", and insist that the universe might still be finite, and we just haven't figured out the right model.

gzhpcu
2006-May-02, 01:28 PM
Well, no. A big bang universe could still be infinite. As far as I can tell, an infinite universe could never be proven to your satisfaction. It's clear that however long you wait around to see if time is infinite, it will always be a finite length of time. And however far you look, anything you see will always be a finite distance away. It's possible that we'll discover that models of the universe that are infinite fit the observations better than models that are finite right now the issue is still open, but with better observations, it might be clearly settled one way or the other). For most cosmologists, that would be the same as showing that the universe is infinite, but I expect you would simply dismiss that as theory and not "reality", and insist that the universe might still be finite, and we just haven't figured out the right model.

Say, looks like you are getting to know me Grey... ;-)

Grey
2006-May-02, 02:24 PM
Say, looks like you are getting to know me Grey... ;-);) That seems to be an attitude that dismisses scientific understanding, though. That is, if the theory that best fits the data clashes with a preconceived idea of yours (like "nothing can be infinite" or "everything that exists has to occupy volume"), you'll hold onto your original idea, and assume that it's "just a theory" and doesn't really correspond to "reality". Do you think we can ever understand "reality" at all?

gzhpcu
2006-May-02, 03:34 PM
;) That seems to be an attitude that dismisses scientific understanding, though. That is, if the theory that best fits the data clashes with a preconceived idea of yours (like "nothing can be infinite" or "everything that exists has to occupy volume"), you'll hold onto your original idea, and assume that it's "just a theory" and doesn't really correspond to "reality". Do you think we can ever understand "reality" at all?

Do I think we can ever really understand "reality"? Probably not. Certainly not with mathematical models. (before anybody jumps up: math is great to explain how things function, and without it, scientific progress would be inconceivable. It just IMHO does not correspond to what is "really" out there.)

As far as "nothing can be infinite" and "everything that exists has to occupy volume", are concerned, I (just me), personally, have seen nothing to convince me to the contrary. For example: QM breaks down due to being based on 0 dimensional points. Many of the developers of QM considered a more complex structure, because they also did not care for a 0 dimensional point, but they saw that the equations would be much too complex. Ergo, this is a mathematical simplification, and the price that is paid is at the Planck length, where another theory (string theory) is needed to coexist with GR. This most probably will not be the final theory either.

Disinfo Agent
2006-May-02, 06:07 PM
As far as "nothing can be infinite" and "everything that exists has to occupy volume", are concerned, I (just me), personally, have seen nothing to convince me to the contrary.Google found me this:

High energy scattering from electrons shows no "size" of the electron down to a resolution of about 10-3 fermis, and at that size a preposterously high spin rate of some 1032 radian/s would be required to match the observed angular momentum.

source (http://230nsc1.phy-astr.gsu.edu/hbase/spin.html)

Or how about the zero mass of the photon?

A non-zero rest mass would lead to a change in the inverse square Coulomb law of electrostatic forces. There would be a small damping factor making it weaker over very large distances.

A limit on the photon mass can be obtained through satellite measurements of planetary magnetic fields. The Charge Composition Explorer spacecraft was used to derive a limit of 6x10-16 eV with high certainty. This was slightly improved in 1998 by Roderic Lakes in a laborartory experiment which looked for anomalous forces on a Cavendish balance. The new limit is 7x10-17 eV.

Its mass is smaller than the smallest we can currently measure. I'd say that's compelling evidence...

Nereid
2006-May-02, 06:57 PM
By the same logic, there are no numbers in the physical universe, ... certainly no number such as 100100, nor any negative number, nor zero, ...

(and if you try to make a case for, say, 'one', then whatever method you use can be employed to show, by the same logic, that 'infinity' has the same status ... unless your method for 'one' is, in effect, nothing more than "I declare it to be so").
I am probably thick, but I do not really follow. What I am trying to say, is that the symbol ∞ does not seem to me to be applicable in nature.So, it's just the symbol? Or is it the concept which the symbol represents?

Perhaps you could explain what you mean by "applicable in nature"?

Also, "seem to me" allows the possibility that you and "nature" may be out of synch - yes? no? maybe?
For example: I count exactly one apple. I can not count ∞ apples.OK, so can you count 0 apples? -1 apples? 100100 apples? 1/2 apples? 1/3 apples? 1/100 apples? 1/(1035) apples? pi apples?

What other tests are permitted, in the gzhpcu view, to establish "applicab[ility] in nature"?
I am not saying that all the results of mathematics are figments of our imagination. Many of the numbers which result can be interpreted meaningfully. But I do not see how ∞ can.OK, so now we have left "applicable in nature" and arrived at "meaningfully".

What, according to gzhpcu, constitutes the hallmarks of "meaningfulness"?

gzhpcu
2006-May-02, 07:03 PM
According to M-theory, the size of the electron is 10^-35 m long. Very small, yet not 0 dimensional.

Just because a mass is smaller than we can measure does not mean it is zero dimensional by any means.

Disinfo Agent
2006-May-02, 07:08 PM
There is far more evidence for Quantum Mechanics than there is independent evidence for M-theory (I'm being kind, here). The scientific thing to do is to let the scales tip towards zero mass photons and pointlike electrons, at least for the time being.

gzhpcu
2006-May-02, 07:14 PM
So, it's just the symbol? Or is it the concept which the symbol represents?

Perhaps you could explain what you mean by "applicable in nature"?

Also, "seem to me" allows the possibility that you and "nature" may be out of synch - yes? no? maybe?OK, so can you count 0 apples? -1 apples? 100100 apples? 1/2 apples? 1/3 apples? 1/100 apples? 1/(1035) apples? pi apples?

What other tests are permitted, in the gzhpcu view, to establish "applicab[ility] in nature"?OK, so now we have left "applicable in nature" and arrived at "meaningfully".

What, according to gzhpcu, constitutes the hallmarks of "meaningfulness"?

By "applicable in nature", I mean if the result exists in the physical world. It seems to me that the concept of infinity is not applicable to the real world.

In your apples examples, any number or fraction of an apple can exist in nature, but an infinite amount of apples does not actually exist. Your examples are all "meaningful", except for an infinite amount of apples which is not, since this will never occur in nature.

The test, simply put, is: can the result be relevant (actually, demonstrably exist) in the real world.

gzhpcu
2006-May-02, 07:16 PM
There is far more evidence for Quantum Mechanics than there is independent evidence for M-theory (I'm being kind, here). The scientific thing to do is to let the scales tip towards zero mass photons and pointlike electrons, at least for the time being.

As far as prediction of how things work, yes. As far as what reality really is, not IMHO.

snarkophilus
2006-May-02, 08:05 PM
I am probably thick, but I do not really follow. What I am trying to say, is that the symbol ∞ does not seem to me to be applicable in nature. For example: I count exactly one apple. I can not count ∞ apples.

I am not saying that all the results of mathematics are figments of our imagination. Many of the numbers which result can be interpreted meaningfully. But I do not see how ∞ can.

But you agreed that all numbers are theoretical constructs. I find it hard to see why you would accept one type of number (integers) but not another (cardinal numbers) as being valid. Perhaps it is just because you've learned that there can be a 1-1 correspondence between certain types of number and certain aspects of reality... but even then, that still doesn't explain why you would accept irrational numbers.

How about pi? There's an interesting number indeed. Ratio of the circumference to the diameter of a circle. But since no perfect circle exists in reality (after all, any physical object probably consists of a finite number of particles), is pi actually found in reality?

Nereid
2006-May-02, 08:10 PM
By "applicable in nature", I mean if the result exists in the physical world.How could you tell if, in fact, "the result exists in the physical world"?

How could I tell?
It seems to me that the concept of infinity is not applicable to the real world.Then neither is the concept of 100100, surely? By your definition, "the result" - 100100 apples (or anything) - most certainly does not "exist in the physical world".

Why, then, is 100100 (apples) acceptable to you, but ∞ (apples) is not?
In your apples examples, any number or fraction of an apple can exist in nature,Are you sure? Can you tell me how I (or you) can make 1/(100100) apples? 100100 apples?
but an infinite amount of apples does not actually exist. Your examples are all "meaningful", except for an infinite amount of apples which is not, since this will never occur in nature.There are an infinite number of 'fraction of an apple' that likewise "will never occur in nature"!*
The test, simply put, is: can the result be relevant (actually, demonstrably exist) in the real world.Good.

Then, as I have shown, only a trivially tiny number of the numbers you claim to be 'meaningful' (a.k.a. "relevant (actually, demonstrably exist)") actually are. The overwhelming majority (in fact, an infinite number of) the numbers you had claimed as "meaningful", fail to meet your more detailed criterion ("relevant (actually, demonstrably exist)").

Oh, and you didn't answer my questions about -1, 0, and pi apples.

Does -1 apples "actually, demonstrably exist"?

*The rationals are (countably) infinite.

hhEb09'1
2006-May-02, 08:10 PM
But since no perfect circle exists in reality (after all, any physical object probably consists of a finite number of particles), is pi actually found in reality?He would almost certainly have to say it does not exist (in reality). And it has its own symbol too! :)

Disinfo Agent
2006-May-02, 08:44 PM
There is far more evidence for Quantum Mechanics than there is independent evidence for M-theory (I'm being kind, here). The scientific thing to do is to let the scales tip towards zero mass photons and pointlike electrons, at least for the time being.As far as prediction of how things work, yes. As far as what reality really is, not IMHO.How can you tell the difference between a good prediction of how things work, and what reality really is?

gzhpcu
2006-May-03, 04:35 AM
But you agreed that all numbers are theoretical constructs. I find it hard to see why you would accept one type of number (integers) but not another (cardinal numbers) as being valid. Perhaps it is just because you've learned that there can be a 1-1 correspondence between certain types of number and certain aspects of reality... but even then, that still doesn't explain why you would accept irrational numbers.

How about pi? There's an interesting number indeed. Ratio of the circumference to the diameter of a circle. But since no perfect circle exists in reality (after all, any physical object probably consists of a finite number of particles), is pi actually found in reality?

Actually, I was just concentrating on infinity (have my hands full with that as it is.... :-), but now that you mention it, the same objection applies to pi. Just a mathematical abstractionl

gzhpcu
2006-May-03, 04:38 AM
How could you tell if, in fact, "the result exists in the physical world"?

How could I tell?Then neither is the concept of 100100, surely? By your definition, "the result" - 100100 apples (or anything) - most certainly does not "exist in the physical world".

Why, then, is 100100 (apples) acceptable to you, but ∞ (apples) is not?Are you sure? Can you tell me how I (or you) can make 1/(100100) apples? 100100 apples?There are an infinite number of 'fraction of an apple' that likewise "will never occur in nature"!*Good.

Then, as I have shown, only a trivially tiny number of the numbers you claim to be 'meaningful' (a.k.a. "relevant (actually, demonstrably exist)") actually are. The overwhelming majority (in fact, an infinite number of) the numbers you had claimed as "meaningful", fail to meet your more detailed criterion ("relevant (actually, demonstrably exist)").

Oh, and you didn't answer my questions about -1, 0, and pi apples.

Does -1 apples "actually, demonstrably exist"?

*The rationals are (countably) infinite.

You are right. There are a lot more numbers which will never occur in nature. I just did not want to expand the discussion, since this thread is on infinities.

Quite clearly -1 apples and pi apples do not exist in nature. 0 apples could, when you are counting apples and there aren't any.

gzhpcu
2006-May-03, 04:39 AM
He would almost certainly have to say it does not exist (in reality). And it has its own symbol too! :)

Quite right!

gzhpcu
2006-May-03, 04:47 AM
How can you tell the difference between a good prediction of how things work, and what reality really is?

I can predict how something works, without really knowing what it is (black box models).

GR says that gravity is not a force, just a geometric distortion of space. Maybe that corresponds to reality, but maybe not, despite the excellent results the theory exhibits. Meantime, the search for the graviton continues....

QM says that the fundamental particles are 0 dimensional. However, QM and GR are incompatible at the Planck length.

So, to answer your question, we can't with certainty. Might even be impossible, since the dimension we have to investigate are far beyond the capacities of our accelerators to investigate them.

Sure, I know I might be totally wrong, but two things which I can't swallow so easily are 0 dimensional particles and an infinite amount of anything in our universe.

Ken G
2006-May-03, 12:51 PM
I agree that an infinite amount is hard to imagine, but ratios (like velocity) are also physical quantities, and other types of infinity have been mentioned. There is even the infinity of counting that is possible if you generalize what you are counting, as has been mentioned, so it's not just an infinite amount that we are discussing as a "real" infinity.

Nereid
2006-May-03, 01:02 PM
You are right. There are a lot more numbers which will never occur in nature. I just did not want to expand the discussion, since this thread is on infinities.Ah, but that's the point ... they're all connected.

Once you allow "one", and "two" (as having some 'physical reality'), you must allow infinity!

Your only alternative is to abandon logic.

(Wait, logic is not "relevant (actually, demonstrably exist)", does it? If that's the case, then "one" and "two" may well exist, but you'd never know!)
Quite clearly -1 apples and pi apples do not exist in nature. 0 apples could, when you are counting apples and there aren't any.OK, so you have 1 apple, and I have 2 apples. The ratio of gzhpcu apples to Nereid apples is 0.5 (1/2); the ratio of Nereid apples to gzhpcu apples is 2.

You eat your apple. You now have 0 apples.

What is the ratio of gzhpcu to Nereid apples? It is 0.

What is the ratio of Nereid apples to gzhpcu apples?

Disinfo Agent
2006-May-03, 01:29 PM
Once you allow "one", and "two" (as having some 'physical reality'), you must allow infinity!

Your only alternative is to abandon logic.[Mr. Spock's voice]That's illogical.[/Mr. Spock's voice]

OK, so you have 1 apple, and I have 2 apples. The ratio of gzhpcu apples to Nereid apples is 0.5 (1/2); the ratio of Nereid apples to gzhpcu apples is 2.

You eat your apple. You now have 0 apples.

What is the ratio of gzhpcu to Nereid apples? It is 0.

What is the ratio of Nereid apples to gzhpcu apples?You are assuming that any and all ratios made from physical quantities must be themselves physically meaningful. As I argued before, that's not necessarily the case.

Nereid
2006-May-03, 01:45 PM
[snip]

You are assuming that any and all ratios made from physical quantities must be themselves physically meaningful. As I argued before, that's not necessarily the case.And what distinguishes the 'physical meaningfullness' of some (such) ratios from the others?

What tests - hopefully objective, physically meaningful ones - can be applied to make such determinations?

Disinfo Agent
2006-May-03, 01:56 PM
Ken G mentioned above that velocities are physically meaningful ratios.* I agree, and we do have instruments that measure velocities.
We don't have instruments that measure 'the ratio of gzhpcu to Nereid apples'; that's a pure abstraction.

I am not making this notion up, by the way. The OECD glossary of statistical terms distinguishes primay units (http://stats.oecd.org/glossary/detail.asp?ID=3793) from derived statistics (http://stats.oecd.org/glossary/detail.asp?ID=3744).

*Technically, only average speeds are ratios; velocities in general are derivatives, which is a different mathematical operation. But let's not worry with this.

Ken G
2006-May-03, 02:08 PM
Ken G mentioned above that velocities are physically meaningful ratios.* I agree, and we do have instruments that measure velocities.

But is the ratio of distance over time not an arbitrary way to measure motion? If I choose to ratio the time to the distance, is that not equally good (like the acceleration analogy I gave, where we talk about the time it takes a car to go from zero to 60mph)? If we can measure velocity, and a speed of zero is one possibility, then we can also measure inverse speed, and infinity.

Disinfo Agent
2006-May-03, 02:13 PM
But is the ratio of distance over time not an arbitrary way to measure motion?I can't think of a better one.

If I choose to ratio the time to the distance, is that not equally good (like the acceleration analogy I gave, where we talk about the time it takes a car to go from zero to 60mph)?But you are still using 60 mph as a reference.

Relmuis
2006-May-03, 02:21 PM
Quantum effects might (spoil this / come to the rescue). After all, if one measures a certain property, and it is infinite, its inverse must be exactly zero. And if a measurement yields exactly zero, there is some other measurement which will no longer yield results with any precision at all. (The Heisenberg uncertainty relation.) For example, if a particle's speed is (known to be) exactly zero, the particle might be located just anywhere in the universe, which means that we cannot measure the speed of a particle on Earth and find it to be exactly zero. Which in turn means that the inverse speed can be known only to be larger than some large number of seconds per meter, but not to be infinite.

Ken G
2006-May-03, 03:10 PM
But you are still using 60 mph as a reference.
That's irrelevant, I could equally well measure the time it takes a car to travel a standard distance, I just chose this example because it is already in use.

Ken G
2006-May-03, 03:12 PM
Quantum effects might (spoil this / come to the rescue).
That's why I earlier said that infinity was as "real" as zero. But if one wants to argue that neither infinity nor zero are "real", it follows pretty quickly that no numbers are "real", which is the alternative stance.

Grey
2006-May-03, 03:30 PM
I can predict how something works, without really knowing what it is (black box models).But surely the converse is not true. That is, I cannot adequately describe what something is without including, as part of the description, how it behaves and interacts with other somethings. Any description of what an electron really is will have to involve the mathematical treatment of it, including things as simple as its inherent properties (mass, charge, spin), as well as an explanation of how those properties affect its interactions. How would you propose that I avoid using math in such a description?

Disinfo Agent
2006-May-03, 04:15 PM
I can predict how something works, without really knowing what it is (black box models).Yes, but your prediction may be wrong.

I could equally well measure the time it takes a car to travel a standard distanceI guess the problem with that is that in practice we're not satisfied with fixing a certain distance and measuring all speeds as times across that distance. We want to be able to measure speeds over different distances and time lengths, and compare them.

P.S. By the way, if you decide to measure speed as the necessary time to cross a standard distance, then you're not making a ratio.

Nereid
2006-May-03, 04:18 PM
Ken G mentioned above that velocities are physically meaningful ratios.* I agree, and we do have instruments that measure velocities.
We don't have instruments that measure 'the ratio of gzhpcu to Nereid apples'; that's a pure abstraction.

I am not making this notion up, by the way. The OECD glossary of statistical terms distinguishes primay units (http://stats.oecd.org/glossary/detail.asp?ID=3793) from derived statistics (http://stats.oecd.org/glossary/detail.asp?ID=3744).

*Technically, only average speeds are ratios; velocities in general are derivatives, which is a different mathematical operation. But let's not worry with this.So the critical distinction, re what is, and what is not, a "physically meaningful ratio", is convenience? history? convention?

Whatever it is, the Disinfo Agent criterion doesn't (seem to) meet the gzhpcu criterion ("relevant (actually, demonstrably exist)").

If I can invent an instrument that 'measures' 'the ratio of gzhpcu to Nereid apples' (or anything else that, purely by chance you understand, is capable of showing the dreaded "∞" symbol on its output LCD (or any other dreaded symbol - such as "i", or "π", or "e", or "-1", ...), then I have met the Disinfo Agent criterion for demonstrating the pure reality of 'infinity' (or whatever is the antonym of "a pure abstraction")?

Disinfo Agent
2006-May-03, 05:07 PM
So the critical distinction, re what is, and what is not, a "physically meaningful ratio", is convenience? history? convention?How do you figure that?

Whatever it is, the Disinfo Agent criterion doesn't (seem to) meet the gzhpcu criterion ("relevant (actually, demonstrably exist)").Whether or not my criterion meets Gzhpcu's criterion, your lack of criteria doesn't meet logic.

If I can invent an instrument that 'measures' 'the ratio of gzhpcu to Nereid apples' (or anything else that, purely by chance you understand, is capable of showing the dreaded "infinity" symbol on its output LCD (or any other dreaded symbol - such as "i", or "pi", or "e", or "-1", ...), then I have met the Disinfo Agent criterion for demonstrating the pure reality of 'infinity' (or whatever is the antonym of "a pure abstraction")?

[symbols rewritten by me]Didn't you bring up the LCD example just a few pages ago? What was the reply you got then?

snarkophilus
2006-May-03, 08:54 PM
We don't have instruments that measure 'the ratio of gzhpcu to Nereid apples'

We could easily build one!

snarkophilus
2006-May-03, 08:58 PM
I am not making this notion up, by the way. The OECD glossary of statistical terms distinguishes primay units (http://stats.oecd.org/glossary/detail.asp?ID=3793) from derived statistics (http://stats.oecd.org/glossary/detail.asp?ID=3744).[/SIZE]

The first concerns a statistical unit of record which is basic in the sense that it does not depend upon any derived calculations, for example: persons, miles, tons, gallons, thousands of an article

Everything we measure is dependent upon a derived calculation, although it is often true that those calculations are handled implicitly in our brains. Nevertheless, it requires a massive amount of processing power to measure two people (for instance). Ask anyone who's ever tried to program an AI for this....

For that reason, I take issue with that definition. They're just setting up an axiom schema, declaring that certain things are basic elements of reality, but that's just for convenience (because you have to start somewhere).

Disinfo Agent
2006-May-03, 09:17 PM
We could easily build one!Tell me more.

gzhpcu
2006-May-04, 05:07 AM
But surely the converse is not true. That is, I cannot adequately describe what something is without including, as part of the description, how it behaves and interacts with other somethings. Any description of what an electron really is will have to involve the mathematical treatment of it, including things as simple as its inherent properties (mass, charge, spin), as well as an explanation of how those properties affect its interactions. How would you propose that I avoid using math in such a description?

I am not proposing to avoid math. I am just saying IMHO math will always result in an approximation. The correct theory probably needs a mathematical model of infinite complexity... ;-)

gzhpcu
2006-May-04, 05:15 AM
So the critical distinction, re what is, and what is not, a "physically meaningful ratio", is convenience? history? convention?

Whatever it is, the Disinfo Agent criterion doesn't (seem to) meet the gzhpcu criterion ("relevant (actually, demonstrably exist)").

If I can invent an instrument that 'measures' 'the ratio of gzhpcu to Nereid apples' (or anything else that, purely by chance you understand, is capable of showing the dreaded "∞" symbol on its output LCD (or any other dreaded symbol - such as "i", or "π", or "e", or "-1", ...), then I have met the Disinfo Agent criterion for demonstrating the pure reality of 'infinity' (or whatever is the antonym of "a pure abstraction")?

The ∞ will show up in mathematical equations when one attempts to do something nonsensical like division by zero. In reality, ratios do not exist. Have you ever seen a ratio in nature? Whereas things like size of an object, number of objects, etc. do exist.

snarkophilus
2006-May-04, 05:23 AM
Tell me more.

Here is my scheme for a machine that measures the ratio of gzhpcu apples to Nereid apples, for finite numbers of apples.

First, we get ourselves a slave. I am thinking of Powers' famous sculpture, though not made of marble, and perhaps more appropriately dressed (though that is not strictly necessary). Her eyes should work, and she should be versed in counting. Actually, a computer would do, too, but would be much less pleasing to watch as it ran.

Next, we have our slave put out her hand, holding it vertically and dividing the world with the plane of her hand. The division on her left is her "left side," and the other division is her "right side."

Third, we put Nereid's apples on her left side, and gzhpcu's apples on her right side. (If we have more apples than will fit on the Earth, we can extend our experiment into space. If we can not move the apples, then we mark them as belonging to one or the other and have her move in order to count them.)

That completes the apparatus setup. If we expect that the experiment will take longer than a single lifetime to complete, we can always acquire/produce more slaves. After all, we'll have lots of food for them.

To run the machine, we say (probably in Greek), "Mademoiselle Slave, if you would be so kind, please run the following experiment: if there are no apples on your left, write down the symbol <infinity>. If there not zero apples on your left, write down the number of apples on your right, and beneath that a horizontal line, and beneath that the number of apples on your left.

:D

01101001
2006-May-04, 06:10 AM
The ∞ will show up in mathematical equations when one attempts to do something nonsensical like division by zero. In reality, ratios do not exist. Have you ever seen a ratio in nature? Whereas things like size of an object, number of objects, etc. do exist.

Yeah, right. When babies are born, the parents only want to know how many and what their lengths and weights are.

Wait. And, then they check the number of fingers and toes per baby, and the number of fingers and toes per hand and foot.

Ken G
2006-May-04, 10:18 AM
P.S. By the way, if you decide to measure speed as the necessary time to cross a standard distance, then you're not making a ratio.
Yes you are, because you would not actually need to use the standard distance. You can use any distance, and scale it up using a ratio. That's the normal approach, although not used for cars I admit. The car analogy is of limited value.

Ken G
2006-May-04, 10:23 AM
Whereas things like size of an object, number of objects, etc. do exist.
Correction, objects exist. That's all. Numbers of objects are an abstraction of an intelligent mind, and may be said to "exist" only in the same sense as all abstractions, like the concept of ratios, or infinity.

Nereid
2006-May-04, 01:26 PM
Both DA and gzhpcu have indicated problems with ratios (because they easily show the 'physical reality' of an infinity?).

KenG used velocity as an example of how a standard definition/measurement, via convention, convenience, or history, may avoid a ratio that could yield an infinity, in the real world.

Earlier, I introduced a Dick Smith electronics multimeter, and used it measure the resistance of something - let's say the sensor works by detecting a current (the rest is signal processing, A/D conversion, display chips and logic, etc). If the sensor does not detect a current, then the resistance is infinite. No if's, no but's, no maybe's - the instrument works as advertised, the material has an infinite resistance. And as it's in the real world - the lump of whatever is about as physical as it gets - we have a "result exists in the physical world" (to quote gzhpcu).

(Electrical) conductance can illustrate the other side of ratios, reality, infinities etc.

Of course we all know that 'electrical conductance' is defined as inverse resistance (resistivity?), so that the stuff we found had infinite resistance (with our Dick Smith multimeter) has zero conductance.

But what about superconductors? They have zero resistance. No problems for DA or gzhpcu - not an infinity, a result which exists in the physical world, etc.

But wait! Superconductors will have infinite conductance!

(must be a pure abstraction, quite unlike resistance)

gzhpcu
2006-May-04, 05:00 PM
Correction, objects exist. That's all. Numbers of objects are an abstraction of an intelligent mind, and may be said to "exist" only in the same sense as all abstractions, like the concept of ratios, or infinity.

Isn't any kind of thinking an abstraction? I am just saying that some mathematical abstractions, such as infinity, have no relevance in the real world.

Nereid
2006-May-04, 05:11 PM
Isn't any kind of thinking an abstraction? I am just saying that some mathematical abstractions, such as infinity, have no relevance in the real world.At the risk of going over ground we've already covered ...

Numbers are 'mathematical abstractions', whether they are 'one', 'two', 'zero', 'minus one', 'pi', 'e', or 'infinity'.

Once you admit that any number has 'relevance in the real world', you have admitted that they all have such relevance.

You may, as DA seems to be trying to do, limit that 'relevance', by choosing (arbitrary) definitions of units, and (arbitrary) limits on the physical properties you choose to apply numbers to (e.g. resistance, but not conductance) - which amounts to (arbitrary) choices about the applicability of (today's) theories to 'the real world'.

However you wish to avoid 'infinity', it will involve considerable numbers of arbitrary choices, which choices bear no connection with 'the real world' other than your personal preference.

But perhaps I missed some subtle aspect ...

Disinfo Agent
2006-May-04, 06:12 PM
Oooh, all that suspense and then this:

To run the machine, we say (probably in Greek), "Mademoiselle Slave, if you would be so kind, please run the following experiment: if there are no apples on your left, write down the symbol <infinity>. If there not zero apples on your left, write down the number of apples on your right, and beneath that a horizontal line, and beneath that the number of apples on your left.

:DYou're devious! :evil: :naughty: :lol: :D

Disinfo Agent
2006-May-04, 06:14 PM
Yes you are, because you would not actually need to use the standard distance. You can use any distance, and scale it up using a ratio.You can, but you don't have to.

That's the normal approach, although not used for cars I admit. The car analogy is of limited value.I agree. :p

hhEb09'1
2006-May-04, 06:39 PM
I am just saying that some mathematical abstractions, such as infinity, have no relevance in the real world.Yabbut, didn't you say the same thing about pi? Boy, do I disagree with that :)

John Dlugosz
2006-May-04, 08:22 PM
The current record, set in 2005, is for 33810 cities. The computation took 15.7 CPU years.

So to define infinity to you, I just have to ask you to solve the travelling salesman problem for 80,000 cities. It may be that the human race will never develop a method to find the solution to this.

It's a finite problem with real world applications - but it's intractable - i.e. infinite - in our reality.

clop

So use a quantum computer. Or, the next best thing: an analog analogue. Create a system where the shortest path gives the least energy, and let it do its thing.

gzhpcu
2006-May-04, 09:29 PM
At the risk of going over ground we've already covered ...

Numbers are 'mathematical abstractions', whether they are 'one', 'two', 'zero', 'minus one', 'pi', 'e', or 'infinity'.

Once you admit that any number has 'relevance in the real world', you have admitted that they all have such relevance.

You may, as DA seems to be trying to do, limit that 'relevance', by choosing (arbitrary) definitions of units, and (arbitrary) limits on the physical properties you choose to apply numbers to (e.g. resistance, but not conductance) - which amounts to (arbitrary) choices about the applicability of (today's) theories to 'the real world'.

However you wish to avoid 'infinity', it will involve considerable numbers of arbitrary choices, which choices bear no connection with 'the real world' other than your personal preference.

But perhaps I missed some subtle aspect ...
I do not understand why saying that any number has relevance in the real world one can conclude that all numbers, including infinity have relevance.

However, I will concede that your point about infinite resistance (an open circuit, for example) is making me rethink my resistance to infinity... Let me sleep over that one, getting late at night over here... might have to admit you have a point there... :-)

Disinfo Agent
2006-May-04, 10:18 PM
Both DA and gzhpcu have indicated problems with ratios (because they easily show the 'physical reality' of an infinity?).That is the topic of this thread, isn't it?

Earlier, I introduced a Dick Smith electronics multimeter, and used it measure the resistance of something - let's say the sensor works by detecting a current (the rest is signal processing, A/D conversion, display chips and logic, etc). If the sensor does not detect a current, then the resistance is infinite. No if's, no but's, no maybe's - the instrument works as advertised, the material has an infinite resistance.That deduction is based on Ohm's Law (http://www.physics.uoguelph.ca/tutorials/ohm/Q.ohm.intro.html), right?

V = R I

The sensor measures I=0, and you know that V has a positive value, therefore R must be infinite. Pretty straightforward.

But let's look more closely at what's happening...

Let's take Relmius' example.

I go to my friendly local Dick Smith's electronics/hobbyist shop, and I buy a digital multimeter. I put the battery in, and turn it on.

I turn the dial to resistance, and put the probes across the "Relmius' insulator". My multimeter reads "infinity".

Now, let's look at what's happening inside the multimeter (I'll make some assumptions; what matters is the general principle, not the details).

Some sensor inside the multimeter detects a current flowing through the probes; some electronics converts that detected current into some code (several bytes long, expressed as voltages on some control lines), which ends up as something on the LCD display.

In the case of the Relmius' insulator, the sensor detects zero current. The logic circuits converts this to the symbol "infinity" on the LCD panel.

Now what about Grey's comment?How do we reconcile the fact that we have, per Disinfo Agent's demand, clearly demonstrated "an infinite physical measurement", yet we also know, per Grey, "there's no such thing as a perfect insulator"!

So is my Dick Smith multimeter lying to me?You are lying to yourself. Ohm's Law is an ideal simplification which only describes accurately what happens in "ohmic resistors". That is to say, it is only applicable in conditions where voltage and current are proportional.

But in the example you have given, they are not proportional (http://mathworld.wolfram.com/DirectlyProportional.html). If they were, V would have to be zero when I is zero.

And as it's in the real world - the lump of whatever is about as physical as it gets - we have a "result exists in the physical world" (to quote gzhpcu).

(Electrical) conductance can illustrate the other side of ratios, reality, infinities etc.

Of course we all know that 'electrical conductance' is defined as inverse resistance (resistivity?), so that the stuff we found had infinite resistance (with our Dick Smith multimeter) has zero conductance.

But what about superconductors? They have zero resistance. No problems for DA or gzhpcu - not an infinity, a result which exists in the physical world, etc.

But wait! Superconductors will have infinite conductance!

(must be a pure abstraction, quite unlike resistance)You wrote before "there's no such thing as a perfect insulator". Well, Nereid, I don't think there's any such thing as a perfect superconductor, either.

You may, as DA seems to be trying to do, limit that 'relevance', by choosing (arbitrary) definitions of units, and (arbitrary) limits on the physical properties you choose to apply numbers to (e.g. resistance, but not conductance) - which amounts to (arbitrary) choices about the applicability of (today's) theories to 'the real world'.

However you wish to avoid 'infinity', it will involve considerable numbers of arbitrary choices, which choices bear no connection with 'the real world' other than your personal preference.

But perhaps I missed some subtle aspect ...I believe you did. Above, Ken G proposed measuring speed as time/distance rather than distance/time, but I pointed out several problems with that approach. I also said I could not think of a better way to measure a speed than as distance/time. So far, no one here has come up with one. This seems "significant" -- not "arbitrary".

gzhpcu
2006-May-05, 04:56 AM
Having had a good night's sleep, waking up and reading Disinfo Agent's last post, has put the world back in place.

Getting back to the numbers discussion: Numbers such as one, two, three IMHO are quite different from pi or infinity, for example. Why? Because one, two, etc. I can easily represent by ticking off fingers on my hand. I just happen to give them an arbitrary name. This does not seem like an abstraction to me. Pi and infinity, I can't. These are really mathematical abstractions. So, one, two, etc. are different classes and have a relevance in the real world. Infinity doesn't, unless you want to have it as a place holder for a huge number.

gzhpcu
2006-May-05, 05:14 AM
I found this in the Wikipedia:

http://en.wikipedia.org/wiki/Infinite

In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e. counting). It is therefore assumed by physicists that no measurable quantity could have an infinite value, for instance by taking an infinite value in an extended real number system (see also: hyperreal number), or by requiring the counting of an infinite number of events. It is for example presumed impossible for any body to have infinite mass or infinite energy. There exists the concept of infinite entities (such as an infinite plane wave) but there are no means to generate such things. Likewise, perpetual motion machines theoretically generate infinite energy by attaining 100% efficiency or greater, and emulate every conceivable open system; the impossible problem follows of knowing that the output is actually infinite when the source or mechanism exceeds any known and understood system.

It should be pointed out that this practice of refusing infinite values for measurable quantities does not come from a priori or ideological motivations, but rather from more methodological and pragmatic motivations. One of the needs of any physical and scientific therory is to give usable formulas that correspond to or at least approximate reality. As an example if any object of infinite gravitational mass were to exist, any usage of the formula to calculate the gravitational force would lead to an infinite result, which would be of no benefit since the result would be always the same regardless of the position and the mass of the other object. The formula would be useful neither to compute the force between two objects of finite mass nor to compute their motions. If an infinite mass object were to exist, any object of finite mass would be attracted with infinite force (and hence acceleration) by the infinite mass object, which is not what we can observe in reality.

Ken G
2006-May-05, 06:47 AM
Above, Ken G proposed measuring speed as time/distance rather than distance/time, but I pointed out several problems with that approach. I also said I could not think of a better way to measure a speed than as distance/time. So far, no one here has come up with one.
That's the definition of speed. But there are no "problems" with the use of inverse speed, it's just a convention. Speed is going to be more convenient because it is very close to being additive, but again, it's pure convention. All we need is for (1) zero speed to exist and (2) for inverse speed to be "real" ("convenient" is not a requirement) and we're done here.

Nereid
2006-May-05, 08:28 AM
That is the topic of this thread, isn't it?

That deduction is based on Ohm's Law (http://www.physics.uoguelph.ca/tutorials/ohm/Q.ohm.intro.html), right?

V = R I

The sensor measures I=0, and you know that V has a positive value, therefore R must be infinite. Pretty straightforward.

But let's look more closely at what's happening...

You are lying to yourself. Ohm's Law is an ideal simplification which only describes accurately what happens in "ohmic resistors". That is to say, it is only applicable in conditions where voltage and current are proportional.

But in the example you have given, they are not proportional (http://mathworld.wolfram.com/DirectlyProportional.html). If they were, V would have to be zero when I is zero.

You wrote before "there's no such thing as a perfect insulator". Well, Nereid, I don't think there's any such thing as a perfect superconductor, either.

I believe you did. Above, Ken G proposed measuring speed as time/distance rather than distance/time, but I pointed out several problems with that approach. I also said I could not think of a better way to measure a speed than as distance/time. So far, no one here has come up with one. This seems "significant" -- not "arbitrary".Some time back (around post #170?), gzhpcu provided us with his criterion for acceptability (http://www.bautforum.com/showpost.php?p=736309&postcount=173) (of infinity, or anything else, presumably): "By "applicable in nature", I mean if the result exists in the physical world." and "The test, simply put, is: can the result be relevant (actually, demonstrably exist) in the real world."

In other words, an operational definition, one that abandons Plato and ideals (gzhpcu later confirmed that his reality is operational, in post #182 (http://www.bautforum.com/showpost.php?p=736593&postcount=182)).

Using this criterion, reality is nothing more than what you can 'see'; instruments such as the Dick Smith multimeter are merely extensions of your 'seeing' (assuming, of course, that by 'result' gzhpcu means something like 'what you can see/touch/feel/hear/smell/taste'). In this case, the 'result' is easy to demonstrate - the LCD displays "∞" (and, we assume, the multimeter works, as advertised).

Does 'electrical resistance' or 'electrical conductance' "actually, demonstrably exist"? I guess we'd have to ask gzhpcu - if the answer is 'yes', then it seems to me the only loophole is that instruments which measure either are not accurate (but that takes us back to Plato, does it not?).

Nereid
2006-May-05, 08:33 AM
Having had a good night's sleep, waking up and reading Disinfo Agent's last post, has put the world back in place.

Getting back to the numbers discussion: Numbers such as one, two, three IMHO are quite different from pi or infinity, for example. Why? Because one, two, etc. I can easily represent by ticking off fingers on my hand. I just happen to give them an arbitrary name. This does not seem like an abstraction to me. Pi and infinity, I can't. These are really mathematical abstractions. So, one, two, etc. are different classes and have a relevance in the real world. Infinity doesn't, unless you want to have it as a place holder for a huge number.OK, so what about "zero"? "half"? "minus one"?

And for what numbers (positive integers) does "easily represent" fail? "five", "six", "hundred", "wan" (the Chinese word for what is called ten thousand in English), "trillion", ...?

snarkophilus
2006-May-05, 09:25 AM
OK, so what about "zero"? "half"? "minus one"?

And for what numbers (positive integers) does "easily represent" fail? "five", "six", "hundred", "wan" (the Chinese word for what is called ten thousand in English), "trillion", ...?

Googol? Googolplex? Skewes' number? The Moser? Or how about the biggest one with practical application, Graham's number?

All are very big; all are finite. All are easy to represent (I just wrote them down, after all). Only one is even within conceivable orders of magnitude of anything countable in the universe.

How about plugging a couple of those into the Ackermann function? (Big numbers are cool. :D )

Disinfo Agent
2006-May-05, 11:04 AM
That's the definition of speed. But there are no "problems" with the use of inverse speed, it's just a convention. Speed is going to be more convenient because it is very close to being additive, but again, it's pure convention.You're contradicting yourself, there. If speed is "more convenient" than inverse speed, then I'd say that's a bit of a "problem" for inverse speed.

Not to mention the other objections I made, which you seem to be have forgotten about.

Some time back (around post #170?), gzhpcu provided us with his criterion for acceptability (http://www.bautforum.com/showpost.php?p=736309&postcount=173) (of infinity, or anything else, presumably): "By "applicable in nature", I mean if the result exists in the physical world." and "The test, simply put, is: can the result be relevant (actually, demonstrably exist) in the real world."

In other words, an operational definition, one that abandons Plato and ideals (gzhpcu later confirmed that his reality is operational, in post #182 (http://www.bautforum.com/showpost.php?p=736593&postcount=182)).

Using this criterion, reality is nothing more than what you can 'see'; instruments such as the Dick Smith multimeter are merely extensions of your 'seeing' (assuming, of course, that by 'result' gzhpcu means something like 'what you can see/touch/feel/hear/smell/taste'). In this case, the 'result' is easy to demonstrate - the LCD displays "infinity" (and, we assume, the multimeter works, as advertised).Whoa, there!

:exclaim: First of all, I never said or implied that I agree with all of Gzhpcu's ideas. In fact, I have disagreed with things he wrote, several times in this thread!

But let's leave that aside for now. :question: Gzhpcu, do you agree with Nereid that "seeing is believing"; meaning, if you saw the infinity symbol plastered on an LCD screen, would that prove to you that infinities exist in 'physical reality'?

Ken G
2006-May-05, 02:49 PM
You're contradicting yourself, there. If speed is "more convenient" than inverse speed, then I'd say that's a bit of a "problem" for inverse speed.

I see no connection between the words "convenient" and "problem", and no "contradiction" in my position. Einstein's theory of gravity is so inconvenient that it has only been solved in a number of situations you can count on one hand. Is this a "problem" with the theory? Remember, this thread is about reality. Who says it has to be convenient?

Not to mention the other objections I made, which you seem to be have forgotten about.

Pretty snide comment, DA. I commented on everything I thought was germaine, if I missed something important feel free to repeat it. I certainly cannot promise I will make specific mention of everything you say, however.

Disinfo Agent
2006-May-05, 03:22 PM
When you write things like:

[...] there are no "problems" with the use of inverse speed, it's just a convention....it's clear that you've rejected the objections I made to your "inverse speed" concept in the previous page. Yet you haven't refuted them. I can only conclude that you didn't understand my objections, but dismiss what you don't understand as "not germaine".

I see no connection between the words "convenient" and "problem", and no "contradiction" in my position. Einstein's theory of gravity is so inconvenient that it has only been solved in a number of situations you can count on one hand. Is this a "problem" with the theory? Remember, this thread is about reality. Who says it has to be convenient?I have presented objections to what I was able to make out of the "inverse speed" counterargument. Not all of them have been refuted (if any), as far as I'm able to make out. I remain unconvinced by that counterargument.

Nereid
2006-May-05, 03:47 PM
[snip]
Whoa, there!

:exclaim: First of all, I never said or implied that I agree with all of Gzhpcu's ideas. In fact, I have disagreed with things he wrote, several times in this thread!

[snip]Yes, quite right.

Can you remind me please: what is the Disinfo Agent definition of 'physical reality'?

How can one go about determining whether one encounters any particular number (e.g. 1, 2, 1/2, pi, 0, -1, √2) in terms of it being 'physically real'?

gzhpcu
2006-May-05, 03:56 PM
OK, so what about "zero"? "half"? "minus one"?

And for what numbers (positive integers) does "easily represent" fail? "five", "six", "hundred", "wan" (the Chinese word for what is called ten thousand in English), "trillion", ...?

Gosh, I will give it try (always IMHO!!): zero (no fingers), half (half a finger) both OK. five, six, hundred, wan, trillion all OK. "minus one" by itself, problematic, I would think, only if combined in a mathematical operation "five" "minus one" -> "4".

gzhpcu
2006-May-05, 04:04 PM
But let's leave that aside for now. :question: Gzhpcu, do you agree with Nereid that "seeing is believing"; meaning, if you saw the infinity symbol plastered on an LCD screen, would that prove to you that infinities exist in 'physical reality'?

No, I would not.

gzhpcu
2006-May-05, 04:18 PM
Some time back (around post #170?), gzhpcu provided us with his criterion for acceptability (http://www.bautforum.com/showpost.php?p=736309&postcount=173) (of infinity, or anything else, presumably): "By "applicable in nature", I mean if the result exists in the physical world." and "The test, simply put, is: can the result be relevant (actually, demonstrably exist) in the real world."

In other words, an operational definition, one that abandons Plato and ideals (gzhpcu later confirmed that his reality is operational, in post #182 (http://www.bautforum.com/showpost.php?p=736593&postcount=182)).

Using this criterion, reality is nothing more than what you can 'see'; instruments such as the Dick Smith multimeter are merely extensions of your 'seeing' (assuming, of course, that by 'result' gzhpcu means something like 'what you can see/touch/feel/hear/smell/taste'). In this case, the 'result' is easy to demonstrate - the LCD displays "∞" (and, we assume, the multimeter works, as advertised).

Does 'electrical resistance' or 'electrical conductance' "actually, demonstrably exist"? I guess we'd have to ask gzhpcu - if the answer is 'yes', then it seems to me the only loophole is that instruments which measure either are not accurate (but that takes us back to Plato, does it not?).

It would appear to me that electrical resistance exists. In fact, I really think you have made a point, and the concept of infinite resistance is valid.

Disinfo Agent
2006-May-05, 04:52 PM
Yes, quite right.

Can you remind me please: what is the Disinfo Agent definition of 'physical reality'?

How can one go about determining whether one encounters any particular number (e.g. 1, 2, 1/2, pi, 0, -1, ?2) in terms of it being 'physically real'?When you started this thread, and named it "How often, every day, do you encounter infinities of the 'physical reality' kind?", you didn't give us any definition of 'physically real', either. Nor have you defined it since then, as far as I remember. Why should I have to do what you don't?

Let me be quite clear: I am not saying you should have defined 'physically real'. You probably assumed it was clear enough for any reasonable, serious person, and that any minor semantic differences could be cleared as they appeared in the conversation -- as do I.

But let's leave that aside for now. Gzhpcu, do you agree with Nereid that "seeing is believing"; meaning, if you saw the infinity symbol plastered on an LCD screen, would that prove to you that infinities exist in 'physical reality'?
No, I would not.Thank you. But now it seems that you are finding Nereid's LCD argument persuasive. I think it would be interesting if you could explain why...

Nereid
2006-May-05, 04:57 PM
But let's leave that aside for now. Gzhpcu, do you agree with Nereid that "seeing is believing"; meaning, if you saw the infinity symbol plastered on an LCD screen, would that prove to you that infinities exist in 'physical reality'?No, I would not.There are some rather important aspects rather too quickly skipped, in DA's summary.

Here's what I said (bold added):
Using this criterion, reality is nothing more than what you can 'see'; instruments such as the Dick Smith multimeter are merely extensions of your 'seeing' (assuming, of course, that by 'result' gzhpcu means something like 'what you can see/touch/feel/hear/smell/taste'). In this case, the 'result' is easy to demonstrate - the LCD displays "∞" (and, we assume, the multimeter works, as advertised).The key part, in gzhpcu's criterion, IMHO, is "demonstrably"*

If you cannot demonstrate (the result), then it does not have relevance, it does not exist (in the physical world).

The corollary (if you can demonstrate something, it has relevance, it exists) - this is DA's summary - is not obviously true.

In the case of our 'seeing', we have various checks that we apply, to convince ourselves that what we 'see' is indeed a demonstration (in gzhpcu's sense) - independent observers, multiple observations, use of different methods of 'seeing' that we know from previous experience are reliable substitutes for our 'seeing', make sure no magicians are around, and so on.**

Then there's "the multimeter works, as advertised". This is, of course, essential. And it encapsulates an efficiency that is breath-taking to contemplate ... that \$20 device effectively and efficiently (if it is working properly) distills the labour and inspiration of thousands and thousands of physicist scientists, engineers, mathematicians ... labour and inspiration that was spent over centuries and millenia. To create, from first principles, from the raw earth, fire, water and air, such a device is surely the work of more than one lifetime! Yet, with the device, one can demonstrate (so that someone else can 'see') that there is an infinity, of the physical reality kind, right there on the desk ... all in less than one minute.

So, this thread is about infinities of the 'physical reality' kind. Here's what gzhpcu just said:
It would appear to me that electrical resistance exists.But as DA said earlier (http://www.bautforum.com/showpost.php?p=737990&postcount=214), 'electrical resistance' is an abstraction, something of a deduction, an ideal simplification*** ... and I'm sure we'd all agree that it is based in some theory (or theories) of the physics kind.

Perhaps, then, this pushes the question of 'physical reality' back a bit? To the question of the relationship between (physics) theories and (physical) reality?

*"By "applicable in nature", I mean if the result exists in the physical world." and "The test, simply put, is: can the result be relevant (actually, demonstrably exist) in the real world."
**Since BAUT is a science-based discussion forum, I'm sure readers will recognise this as a shorthand for much of the standard protocol for the doing of science.
***I'm somewhat distorting what DA actually said, but not too much I hope, and have retained the essence.

Disinfo Agent
2006-May-05, 05:26 PM
So, this thread is about infinities of the 'physical reality' kind. Here's what gzhpcu just said:But as DA said earlier (http://www.bautforum.com/showpost.php?p=737990&postcount=214), 'electrical resistance' is an abstraction, something of a deduction, an ideal simplification*** ... and I'm sure we'd all agree that it is based in some theory (or theories) of the physics kind.

***I'm somewhat distorting what DA actually said, but not too much I hope, and have retained the essence.I wouldn't mind having my words reinterpreted a little bit, so to speak, but I'm afraid what you wrote is almost the opposite of my position. Here's what I've obviously failed to get across:

- Electrical resistence is not (just) an abstraction. It's a real, physical property, as well.
- What is an abstraction and a simplification is Ohm's Law. Because it's not exactly true (real-life resistors are not 100% like ideal ones), and it doesn't apply well under all circumstances (even if 'misapplying' it in borderline cases can still give informative, approximate insights into those cases).

I hope you see better now why I can't agree that this is all a matter of theory.

By the way, you've just noted:

In the case of our 'seeing', we have various checks that we apply, to convince ourselves that what we 'see' is indeed a demonstration (in gzhpcu's sense) - independent observers, multiple observations, use of different methods of 'seeing' that we know from previous experience are reliable substitutes for our 'seeing', make sure no magicians are around, and so on.**Would you describe all those checks as based on theory?

snarkophilus
2006-May-05, 07:52 PM
Gosh, I will give it try (always IMHO!!): zero (no fingers), half (half a finger) both OK. five, six, hundred, wan, trillion all OK. "minus one" by itself, problematic, I would think, only if combined in a mathematical operation "five" "minus one" -> "4".

What differentiates half a finger from a whole finger? What if my finger is smaller than yours? What if I cut part of my finger off, but it's not exactly in half?

Better yet, what if I remove a spherical volume (say, a hydrogen atom) from my finger? Do I have 1 - 4.3 * pi * a0^3 of a finger? Is that a physical instance of pi?

snarkophilus
2006-May-05, 07:56 PM
It would appear to me that electrical resistance exists. In fact, I really think you have made a point, and the concept of infinite resistance is valid.

Curiously enough, I disagree with this example. :) Apply enough voltage, and you can get a current running through anything I can think of. QM predicts this. (I was thinking about black holes as possible perfect resistors, but then I realised they can hold a charge, which means they can conduct -- I guess through e-/e+ production at the horizon.)

Nereid
2006-May-05, 08:58 PM
Curiously enough, I disagree with this example. :) Apply enough voltage, and you can get a current running through anything I can think of. QM predicts this. (I was thinking about black holes as possible perfect resistors, but then I realised they can hold a charge, which means they can conduct -- I guess through e-/e+ production at the horizon.)Which is not in line with gzhpcu operational definition!

Operationally, a demonstration comes with its complete context and environment ... the multimeter tests resistance by applying not 'enough voltage' but ~9v (or whatever the Dick Smith multimeter uses).

The resistance measured is also on {date} at {time} at {place}; such operational definitions ("demonstrations") carry no (necessary) requirements re timelessness ("this {lump of stuff} has ∞ resistance, has had so since it was created {x billion years ago}, and will continue to have such ∞ resistance until it is destroyed when the Sun goes red giant").

gzhpcu
2006-May-06, 05:16 AM
What differentiates half a finger from a whole finger? What if my finger is smaller than yours? What if I cut part of my finger off, but it's not exactly in half?
It is just a visual image of a concept ("half") which we give a tag to and apply it for other comparisons.

Better yet, what if I remove a spherical volume (say, a hydrogen atom) from my finger? Do I have 1 - 4.3 * pi * a0^3 of a finger? Is that a physical instance of pi?

pi is a much more abstract concept, which we have derived mathematically. We only work with an approximation of pi, not with the actual pi, which has an infinite number of decimal points. We do not work with the actual physical instance of pi anymore than we work with the actual physical instance of infinity.

gzhpcu
2006-May-06, 05:17 AM
Curiously enough, I disagree with this example. :) Apply enough voltage, and you can get a current running through anything I can think of. QM predicts this. (I was thinking about black holes as possible perfect resistors, but then I realised they can hold a charge, which means they can conduct -- I guess through e-/e+ production at the horizon.)

True, guess you can getting a tunneling effect, can't you?

clop
2006-May-06, 11:08 AM
Right I have finally thought of an infinity of an everyday physical kind that does have basis in actual objects, not numbers, and does not depend on limited resolution of measurement of anything. It's a purely physical thing.

It's the amount of energy required to accelerate a ball bearing to the speed of light.

There we go. Easy.

clop

Disinfo Agent
2006-May-06, 11:54 AM
I asked you before (http://www.bautforum.com/showpost.php?p=733944&postcount=120), with a little irony, whether all real problems had real solutions. That's because I don't think they necessarily do. The example you've just given is one such real problem without any real solution (as far as we know).

In other words, you may need infinite energy, but the amount of energy you need is just not there.

clop
2006-May-06, 01:48 PM
In other words, you may need infinite energy, but the amount of energy you need is just not there.

If it were there it wouldn't be infinite would it?

Like, duh.

Disinfo Agent
2006-May-06, 02:08 PM
If it's not there, then you don't get to "encounter" it, "every day", in "physical reality". Disqualified, Mr. Duh.

clop
2006-May-06, 02:27 PM
If it's not there, then you don't get to "encounter" it, "every day", in "physical reality". Disqualified, Mr. Duh.

By definition, infinity can't exist Mr Duhinfo Agent.

At least my version is physical and relates to objects, not conceptual numbers.

Like, meh.

Disinfo Agent
2006-May-06, 03:07 PM
By definition, infinity can't exist [...]Really? Which definition are you talking about?

By the way, you might want to tone down your ad hominems. They're against the forum rules (http://www.bautforum.com/showthread.php?p=564845).

gzhpcu
2006-May-06, 03:33 PM
Right I have finally thought of an infinity of an everyday physical kind that does have basis in actual objects, not numbers, and does not depend on limited resolution of measurement of anything. It's a purely physical thing.

It's the amount of energy required to accelerate a ball bearing to the speed of light.

There we go. Easy.

clop

Nope. Infinite energy does not exist to accelerate the ball bearing to the speed of light in reality.

gzhpcu
2006-May-06, 03:35 PM
By definition, infinity can't exist Mr Duhinfo Agent.

At least my version is physical and relates to objects, not conceptual numbers.

Like, meh.

Your version does not exist in reality.

Ken G
2006-May-07, 08:44 AM
I found this in the Wikipedia:

http://en.wikipedia.org/wiki/Infinite
A perfect example of why you can't rely too much on Wikipedia. This is not an authoritative statement by any means, and I will point out several specific objections:

It is therefore assumed by physicists that no measurable quantity could have an infinite value, for instance by taking an infinite value in an extended real number system (see also: hyperreal number), or by requiring the counting of an infinite number of events.
OK, so now it is merely an "assumption" that we have no infinites? Later they will try to argue that it would be unphysical, but that is circular reasoning if it is an initial assumption. And exactly why is this assumption at all necessary?

It should be pointed out that this practice of refusing infinite values for measurable quantities does not come from a priori or ideological motivations, but rather from more methodological and pragmatic motivations.

What "practice" are they talking about, exactly? Is there some axiom of physics I don't know about that requires me to "refuse" infinite values? What branch of physics does this play any role in?

One of the needs of any physical and scientific therory is to give usable formulas that correspond to or at least approximate reality.

I added the bold face. Note the "correspond to" part of this sentence has no separate meaning from the bold part. Then they go on to point out that an infinite plane wave is an abstraction that can't be created in reality, but they fail to recognize that so is any finite description of a wave. They have failed to draw any meaningful distinction between an infinite wave and a finite one! Furthermore, a large part of actual physics is done by expanding on infinite plane wave basis functions, so their place in physics is at least as important , if not more so, as finitely bound wave modes.

As an example if any object of infinite gravitational mass were to exist, any usage of the formula to calculate the gravitational force would lead to an infinite result, which would be of no benefit since the result would be always the same regardless of the position and the mass of the other object.

Now they are merely pointing out the vagaries of the algebra of infinities. It is a purely philosophical objection, and merely serves to limit what you can use infinities for. If you don't use them that way, you're fine. Indeed, the entire field of renormalization was developed to handle the awkward algebra of infinities. I find it rather shocking that Wikipedia appears to be oblivious to this.

Nice try, Wiki.

Ken G
2006-May-07, 09:02 AM
...it's clear that you've rejected the objections I made to your "inverse speed" concept in the previous page. Yet you haven't refuted them. I can only conclude that you didn't understand my objections, but dismiss what you don't understand as "not germaine".

You're right that I rejected your objections, and wrong that I didn't understand them. I presume you are either talking about the fact that velocity is "better" than inverse velocity, a point I did not comment on because it is totally irrelevant to a discussion about "reality" (it would be relevant to a discussion about expedience), or you are talking about this statement you made:

I guess the problem with that is that in practice we're not satisfied with fixing a certain distance and measuring all speeds as times across that distance. We want to be able to measure speeds over different distances and time lengths, and compare them.

P.S. By the way, if you decide to measure speed as the necessary time to cross a standard distance, then you're not making a ratio.

Again your statements are about expedience, not reality, but they are easily dismissed anyway. When we quote a speed of 50 mph, are we "satisfying" ourselves with only considering distances that accrue over an hour of time? No, the necessary rescaling to any actual time period is implied. The exact same situation applies to inverse speed, I have no idea why you see a fundamental difference between these concepts. They differ entirely by convention, are equally applicable to "reality", and one is more popular entirely for reasons of expedience. Or am I still failing to grasp some deep subtlety in your position?

I have presented objections to what I was able to make out of the "inverse speed" counterargument. Not all of them have been refuted (if any), as far as I'm able to make out.
Yes, they have all be refuted.

Disinfo Agent
2006-May-09, 12:52 PM
FLASHBACK:

In this post (http://www.bautforum.com/showpost.php?p=736818&postcount=188), Ken G came up with his idea of measuring velocity as a ratio of time over distance. I told him right away (http://www.bautforum.com/showpost.php?p=736826&postcount=189) why that was nonsense:

Ken G
But is the ratio of distance over time not an arbitrary way to measure motion?

Disinfo Agent (http://www.bautforum.com/showpost.php?p=736826&postcount=189) OBJECTION NR. 1
I can't think of a better one.

Ken G
Silence

Ken G
If I choose to ratio the time to the distance, is that not equally good (like the acceleration analogy I gave, where we talk about the time it takes a car to go from zero to 60mph)?

Disinfo Agent (http://www.bautforum.com/showpost.php?p=736923&postcount=194) OBJECTION NR. 2
I guess the problem with that is that in practice we're not satisfied with fixing a certain distance and measuring all speeds as times across that distance. We want to be able to measure speeds over different distances and time lengths, and compare them.

Ken G
Silence

Disinfo Agent (http://www.bautforum.com/showpost.php?p=736923&postcount=194) OBJECTION NR. 3
By the way, if you decide to measure speed as the necessary time to cross a standard distance, then you're not making a ratio.

Ken G
Yes you are, because you would not actually need to use the standard distance. You can use any distance, and scale it up using a ratio. [His reply didn't even make much sense here. I suppose he got this objection mixed up with the previous one.]

Disinfo Agent (http://www.bautforum.com/showpost.php?p=737831&postcount=210)
You can, but you don't have to.

Ken G
Crickets

have presented objections to what I was able to make out of the "inverse speed" counterargument. Not all of them have been refuted (if any), as far as I'm able to make out.Yes, they have all be refuted.Just who do you think you're fooling, Ken?

snarkophilus
2006-May-09, 07:52 PM
Ken G
But is the ratio of distance over time not an arbitrary way to measure motion?

Disinfo Agent (http://www.bautforum.com/showpost.php?p=736826&postcount=189) OBJECTION NR. 1
I can't think of a better one.

I can't think of a solution to the Riemann Hypothesis, but that doesn't make it invalid. Your objection doesn't really say why inverse speed isn't a good unit of measurement. I can think of a few situations in which inverse speed would be an appropriate measurement. For instance, it may be that we are performing a measurement where we observe a difference in position, but no difference in time. That would give an infinite speed, which makes calculations difficult. Much better sense to say that such a measurement yield 0 inverse speed.

As to your other objections, please tell me if I understand what you are saying correctly. You claim that in the real world, for each case where we perform a measurement of speed, we select the distance over which we wish to measure that speed, and then measure a time. Then we can know the speed of that object without doing any scaling, and this is sufficient for all such measurements. You assume that we measure non-zero times, and therefore the only division performed is the distance/time operation, which is finite.

Ken G, on the other hand, is saying that we can't really do that, and that there is always an element of scaling required, and therefore division is a valid operation in terms of interpreting reality.

I'm inclined to agree with Ken G on this aspect. I do all my measuring in m/s (or cm/s or A/s or nm/s, or something like that). There is always scaling involved in these calculations, because I simply don't have time to wait for my molecules to travel a whole meter, and it's completely impractical to measure how far I run in angstroms (although it's impressive to say that I ran about 350000000000000 A yesterday). When we get to astronomical distances, it's impossible to make those measurements directly, anyway, so comparison would be impossible without some scaling.

It's also completely impractical to measure two things over the same distance all the time. If I want to know how fast I was driving my car yesterday in relation to how fast I am driving my brother's car today, I can't go back in time and measure over the same distance -- odds are I don't even know what distance I drove. Therefore, I need to scale to a common unit (say, the kilometer per hour) in order to get any meaningful information. It's not just that I can, but that I must.

You sort of agreed with this in your objection #2. I'm not sure why, given what you said there, that you feel this is a problem.

The only disagreement I have with Ken G is that this isn't a great example, because it's rare that we'd scale to orders of infinity, which is kind of the point of the thread. On the other hand, hopefully it now convinces you that ratios are intrinsic to any robust measurement system.

Ken G
2006-May-10, 09:41 AM
Yes, each of Disinfo Agent's "objections" to inverse speed is easily disposed of, and I already did so, as there is really no fundamental difference between that choice of measure and the normal inverse one. snarkophilus seems to be on that page as well, though I'm not sure what is meant by scaling to orders of infinity. The way I see it, any measure that admits zero as an allowable result also brings in potential infinities via ratios involving zeroes. I have not heard any objections from Disinfo Agent that hold any traction for me, but probably this is not such an important issue at the end of the day.

gzhpcu
2006-May-10, 04:14 PM
Yes, each of Disinfo Agent's "objections" to inverse speed is easily disposed of, and I already did so, as there is really no fundamental difference between that choice of measure and the normal inverse one. snarkophilus seems to be on that page as well, though I'm not sure what is meant by scaling to orders of infinity. The way I see it, any measure that admits zero as an allowable result also brings in potential infinities via ratios involving zeroes. I have not heard any objections from Disinfo Agent that hold any traction for me, but probably this is not such an important issue at the end of the day.
To me, division by zero does not make sense, and is of mathematical interest only. In physical reality, I still have not seen an example of infinity occuring.

Ken G
2006-May-11, 03:40 PM
To me, division by zero does not make sense, and is of mathematical interest only. In physical reality, I still have not seen an example of infinity occuring.
This is merely a statement of incredulity-- "I don't think division by zero makes sense, so I exclude measurements of inverse velocity from the outset". Yes, you can rule out infinities if you so choose, but that is a circular argument when applied to whether or not they have a place in the description of reality. The application to reality is in the answer to these two questions: 1)How many miles does an immobile object move in an hour? 2) How long do you have to wait for an immobile object to move a mile?