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Argos
2003-Sep-02, 03:06 PM
Roger Penrose, some years ago, approached the question of the limits of knowledge (http://www.edge.org/documents/ThirdCulture/v-Ch.14.html). He notes that there are mathematical problems that cannot be solved algorithmically [the Gödel´s theorem (“that you can produce statements that you can't prove using any system of rules you've laid down ahead of time”, from the page I linked)]. This means that our understanding of the universe is not only limited by the Principle of Uncertainty, but also by the defects of the main tool we use to achieve that understanding: mathematics itself.

This is pretty discouraging. It seems to indicate that all our efforts are in vain; that we are definitely confined within a “sphere of knowledge” that can´t be enlarged to encompass the “totum”. We live in a universe that refuses to open it´s secrets; we are submitted to a “Cosmic Censor”. I´d like to know what the fellow members, especially the mathematically gifted ones (what I´m not), have to say about how the Godel´s theorem affects cosmology and how could we get around it.

kilopi
2003-Sep-02, 04:04 PM
Mathematics may be the language of Science, but Science is not just mathematics. The distance to Mars cannot be proved by mathematics, it can only be observed.

Science is a series of observations, modeled with mathematics. There may well be true mathematical statements about the Cosmos that we cannot prove--but we can make observations that support the statements. That is, essentially, what we do now.

Argos
2003-Sep-02, 04:34 PM
Mathematics may be the language of Science, but Science is not just mathematics. The distance to Mars cannot be proved by mathematics, it can only be observed.

Science is a series of observations, modeled with mathematics. There may well be true mathematical statements about the Cosmos that we cannot prove--but we can make observations that support the statements. That is, essentially, what we do now.

Yes, but what about the subatomic scale? Given the observational impossibilities at this level we have to resort to maths. But how to compute quantum phenomena if there are non-computable things?

Cougar
2003-Sep-02, 04:38 PM
It seems to indicate that all our efforts are in vain....

Well, obviously not. Discoveries are being made in all fields of study. Knowledge of the cosmos is advancing.


...that we are definitely confined within a “sphere of knowledge” that can´t be enlarged to encompass the “totum”.

Yet we advance. It looks like we will never know everything, but it certainly seems like we can continue to get closer. I am reminded of a more optimistic view expressed by Rocky Kolb....


"It is rather ironic that the job of a scientist is to understand nature, and if the scientist completely succeeds, the reward is unemployment. But of the many things that concern me in the day-to-day existence of a scientist, waking up one morning and discovering that there are no problems to solve is rather low on the list."


We live in a universe that refuses to open its secrets....

Clearly Nature has shown us many of her secrets, and there is no ordinance disallowing further illumination.

Like science, mathematics and logic are not rigidly etched in stone. Who's to say our future views of these fields will not undergo some kind of phase transition that opens up the possibility of more complete understanding?

Another pertinent quote (sorry, I'm a quote collector) comes from Keith Devlin, author of the excellent book, Goodbye, Descartes, professor of mathematics, Dean of the School of Science at St. Mary's College, and senior researcher at Stanford University's Center for the Study of Language and Communication....


"The evidence continues to mount that the answers to the age-old questions concerning the nature of thought, communication, and action will not be found until we go beyond the boundaries imposed by the legacy of Plato, Aristotle, Descartes, and all the other great thinkers in that two-thousand-year intellectual tradition."

DJ
2003-Sep-02, 04:47 PM
I think it's even worse than it appears.

Some of my past posts often include the question: Can the cell know the body? (Translated: can a red blood cell know it's just a cell within some sort of organism? can it know all the things that an organism knows? or can it only know and react to it's environment, never realizing it's a piece of a very big picture.)

Even worse now, I find in my Astronomy magazine that Time itself is not quantized. This appears to be cosmologically significant, for if *some* light can travel a little slower than the planck time (10^-43 seconds), it means that our concept of red-shift observations could change, and may change the size and age of the universe.

Of course, non-quantized time feels better... Instead of life being like a Compact Disc or DVD -- mere samples of frames of reference dancing by like the pictures in a movies -- we suddenly find time to be different, to be linear without breaks or sampling.

NOW do you know where you are? <muffled chuckle>

DJ

kilopi
2003-Sep-02, 04:53 PM
Yes, but what about the subatomic scale? Given the observational impossibilities at this level we have to resort to maths. But how to compute quantum phenomena if there are non-computable things?
That's the Copenhagen Interpretation--that quantum mechanics is truly random. If so, we can use statistics. Our observations are statistical variables.

But that is a different problem from what Gödel was treating. He wasn't discussing things that are not observable--he was discussing theorems that are neither provable nor disprovable.
The infinity of prime pairs may not be provable (if so, we'll never be able to prove that!) But we seem to be able to observe new prime pairs, as long as we search.

DJ
2003-Sep-02, 04:54 PM
Another pertinent quote (sorry, I'm a quote collector) comes from Keith Devlin, author of the excellent book, Goodbye, Descartes, professor of mathematics, Dean of the School of Science at St. Mary's College, and senior researcher at Stanford University's Center for the Study of Language and Communication....


"The evidence continues to mount that the answers to the age-old questions concerning the nature of thought, communication, and action will not be found until we go beyond the boundaries imposed by the legacy of Plato, Aristotle, Descartes, and all the other great thinkers in that two-thousand-year intellectual tradition."


My advice is not for moving away from the amazing Thought Triad you've mentioned, but instead moving closer to them. Often, we must be reminded to always read between the lines.


DJ

Grey
2003-Sep-02, 06:50 PM
Yes, but what about the subatomic scale? Given the observational impossibilities at this level we have to resort to maths. But how to compute quantum phenomena if there are non-computable things?
I don't think this is really a problem. We do observe things on a subatomic scale, and while there may be uncertainties imposed by quantum mechanics, it seems as if those very uncertainties are part of what we want to observe. To the Copenhagen interpretation, it's not that there's something there, but hidden from us by the uncertainty principle, it's that Nature itself is uncertain.

As far as scale goes, we've been able to probe the structure of a proton directly and demonstrate that the charge within it is split into three components of +2/3, +2/3, and -1/3; that is, we've made direct observations of the quarks that compose it. I don't see any reason to think that if it turns out that the quarks have structure of their own that we won't eventually be able to observe that as well.

Cougar
2003-Sep-02, 08:38 PM
I particularly liked the comments by various physicists at the end Argos' cited article by Penrose, (http://www.edge.org/documents/ThirdCulture/v-Ch.14.html) especially the comments by Gell-Mann and Varela who thought Penrose was pretty far off base in his conjecture that human consciousness was somehow directly determined by "quantum" processes....

Kebsis
2003-Sep-02, 08:59 PM
Perhaps somewhere within our sphere of knowledge is the knowledge to make our sphere larger.

DJ
2003-Sep-02, 09:03 PM
I particularly liked the comments by various physicists at the end Argos' cited article by Penrose, (http://www.edge.org/documents/ThirdCulture/v-Ch.14.html) especially the comments by Gell-Mann and Varela who thought Penrose was pretty far off base in his conjecture that human consciousness was somehow directly determined by "quantum" processes....

Human consciousness is not directly determined by quantum processes. Indeed, it is the other way around. At least, I think so! :D

mike alexander
2003-Sep-02, 11:04 PM
As far as scale goes, we've been able to probe the structure of a proton directly and demonstrate that the charge within it is split into three components of +2/3, +2/3, and -1/3

Now that we have a more primitive measure of charge, shouldn't the values be replaced by +/- 1 and +/- 2?

Gsquare
2003-Sep-03, 01:25 AM
Perhaps somewhere within our sphere of knowledge is the knowledge to make our sphere larger.

What if it turns out to be a cube of knowledge? :D

G^2

Kebsis
2003-Sep-03, 02:05 AM
Well I guess we're all just outta luck then :(

cyrek1
2003-Sep-03, 02:43 PM
cyreks comment:

The problem with the BB Universe is that we cannot look back in time because of the redshift barrier.
Our deepest redshift observered today is 6+. How will we be able to see an object with a redshift of say, 10 or 20?
Our limit to see such an object will be by our ability to use the spectral analysis to discern such a redshift.

Cougar
2003-Sep-03, 03:43 PM
The problem with the BB Universe is that we cannot look back in time because of the redshift barrier.
Our deepest redshift observered today is 6+. How will we be able to see an object with a redshift of say, 10 or 20?

Gee, Cy, we can look back in time 11 or 12 billion years, and you're complaining that we can't look back 13 billion?

Actually, doesn't the microwave background have a redshift of about 1,000? There, we're seeing light from when the universe was only about 300,000 years old. That's pretty far back.

Unfortunately, it's doubtful we'll be seeing further back than that, since before then the universe was so hot and dense, it was opaque - light couldn't travel very far without running into some other particle.

Grey
2003-Sep-03, 04:31 PM
Now that we have a more primitive measure of charge, shouldn't the values be replaced by +/- 1 and +/- 2?
Probably, but think of all the books and papers that would have to be revised! :)

It probably would also have been easier if electrons had been defined as the positive charge carriers (when the labels were set, it was just an arbitrary choice, since nobody could detect the individual particles at that point), so that current flows in the direction that the particles are actually moving. I think we're just stuck with historical artifacts like that. Besides, we may someday discover that quarks have structure, and then we'd have to go through it all over again! :-?

cyrek1
2003-Sep-04, 06:41 PM
Cougar wrote:
Actually, doesn't the microwave background have a redshift of about 1,000? There, we're seeing light from when the universe was only about 300,000 years old. That's pretty far back.

cyreks reply:
The CMBR is purely hypothetical.
I will be posting a rebuttal to the source of the CMBR soon with an explanation of why it ain't so.

I am talking about observational evidence of objects in the above post that can be analyzed.

russ_watters
2003-Sep-04, 08:08 PM
cyreks reply:
The CMBR is purely hypothetical. Not according to the Nobel Prize comitte or the entire mainstream physics community...

Emspak
2003-Sep-04, 08:41 PM
If Cyrek1 is going to trot out Eric Lerner's The Big Bang Never Happened we should link it to the dozens of threads that have a coherent explanation of why BB is a pretty good theory and why it trumps, as far as most scientists are concerned, steady-state universes.

(Lerner said the CMBR and the cosmic ray background are due to a stedy-state universe. He does not address the fact that the sky is dark, nor does he seem to understand entropy well. Lerner also refs Prigogine's work on complex chemistry in a rather bizarre way).

But before we get too OT, I'll lay it out here:

1. The sky is dark. Therefore the universe is finite in some fashion since if it wasn't the sky would be bright as the surface of the sun.

2. Relativity theory imlpies finite, non-steady-state universes. Einstein knew this -- and he introduced a cosmological constant to maintain the steady state, but it didn't work. Basically, we know relativity works (atoms move real fast they get heavy. Clocks slow down when they are accelerated, etc.) so it would seem that the steady state universe loses out.

3. The CMBR is detectable by anyone with an antenna of even moderate power. It is not perfectly isotropic but it does cover the whole sky. Hypothetical? So are the stars, by that kid of reasoning.

Now there are all sorts of variations on BB. Inflation, string and not-string theory, coming from a black hole in another universe, et cetera. Lot of debate there too.

But the standard models seem to have served rather well thus far, and I don't imagine any radical reworkings any time soon. I could be wrong :) so could everyone else.

informant
2003-Sep-04, 08:59 PM
My knowledge of Gödel’s Theorem and its implications is scarce, but anyway:
As I understand it, Gödel’s Theorem poses some questions for deductive sciences, where you derive all your conclusions logically from a set of axioms. Roughly, the theorem is about “How many theorems can you prove, given a set of assumptions?”
But the natural sciences aren’t really about proving theorems, in the mathematical, logical-deductive sense of the word. As kilopi noted above so concisely, in the end it all comes down to the physical evidence.
“Proof” in the natural sciences is ultimately based on an accumulation of observational data to support a given theory, not on the derivation of consequences from assumptions.

Even in what concerns mathematics, Gödel himself writes:


It might seem at first that the set-theoretical paradoxes would doom to failure such an undertaking, but closer examination shows that they cause no trouble at all. They are a very serious problem, not for mathematics, however, but rather for logic and epistemology. As far as sets occur in mathematics (at least in the mathematics of today, including all of Cantor's set theory), they are sets of integers, or of rational numbers, (i.e., of pairs of integers), or of real numbers (i.e., sets of rational numbers), or of functions of real numbers (i.e., of sets of pairs of real numbers), etc. When theorems about all sets (or the existence of sets in general) are asserted, they can always be interpreted without any difficulty to mean that they hold for sets of integers as well as for sets of sets of integers, etc. (respectively, that there either exist sets of integers, or sets of sets of integers, or... etc., which have the asserted property). This concept of set, however, according to which a set is something obtainable from the integers (or some other well-defined objects) by iterated application of the operation 'set of', not something obtained by dividing the totality of all existing things into two categories, has never led to any antinomy whatsoever; that is, the perfectly 'naive' and uncritical working with this concept of set has so far proved completely self-consistent.

source (http://www.friesian.com/goedel/chap-1.htm)

cyrek1
2003-Sep-07, 04:16 PM
Notice to russ_waters and Emspak:

Cyreks reply:

See my new post about this issue that was posted today (9-7-03).

Gsquare
2003-Sep-08, 12:15 AM
.....This means that our understanding of the universe is not only limited by the Principle of Uncertainty, but also by the defects of the main tool we use to achieve that understanding: mathematics itself.

.

Not necessarily; your conclusion is based upon the assumption that Mathematics is a mere subjective construct.
Here's someone else's opinion about this issue:

"...without hesitation that there is, in my opinion, a right way, and that we are capable of finding it. Our experience hitherto justifies us in believing that nature is the realization of the simpliest conceivable mathematical ideas. I am convinced that we can discover by means of purely mathematical constructions the concepts and laws connecting them with each other, which furnish the key to understanding of natural phenomena. Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it. Experience remains, of course, the sole criterion of the physical utility of a mathematical construction. But creative principle resides in mathematics." -
-A. Einstein,
Zur Theorie der Browschen Bewegung, Annalen der Physik, ser. 4, vol. 19, pp.371 - 381

G^2 :D