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parallaxicality
2007-Sep-05, 09:46 PM
someone posted this on another site, and I have no idea how it works. I've been stretching my knowledge of algebra but I can't seem to find the flaw in it.

a = b

Multiply both sides by a

a^2 = ab

Subtract b^2 from each side

a^2 - b^2 = ab - b^2

Factor

(a+b)(a-b) = b(a-b)

Divide by a-b

a+b = b

But a = b, so

b + b = b

So

b = 2b

Divide by b

1 = 2

01101001
2007-Sep-05, 09:49 PM
someone posted this on another site, and I have no idea how it works. I've been stretching my knowledge of algebra but I can't seem to find the flaw in it.

Divide by a-b

STOP!

Since a = b, this is division by zero.

publius
2007-Sep-05, 09:57 PM
The flaw there is diving by zero. You start out with the statement a = b, which means (a - b) = 0. All the rest is just "smoke and mirrors" to make it look like you're doing something complicated, like many puzzles depend on.

When an equation provides a nonsense result, like 1 = 2, that means that some step was invalid. A more complex expression was really the same nonsense, you just reduced it to something more obvious. The nonsense here is dividing by zero.

-Richard

BioSci
2007-Sep-05, 10:13 PM
a^2 = ab ...........................OK

Subtract b^2 from each side
a^2 - b^2 = ab - b^2 ...........Red flag warning: [0=0]

Factor
(a+b)(a-b) = b(a-b) .............Danger! heading for a cliff: (a+b)x(0)= (b)x(0)

Divide by a-b
a+b = b .............................Over the edge!

:)

Ufonaut99
2007-Sep-06, 04:03 AM
You've also got to be careful how you handle i (Imaginary numbers) :)

1) -1 = -1
2) 1/-1 = -1/1
3) sqrt(1/-1) = sqrt(-1/1)
4) sqrt(1)/sqrt(-1) = sqrt(-1)/sqrt(1)
5) 1/i = i/1
6) 1/i = i
7) 1 = i * i
8) 1 = -1

Maksutov
2007-Sep-06, 04:10 AM
1=2. How does this work?Simple. Just add 1. Stir and serve.

hhEb09'1
2007-Sep-06, 08:49 AM
You've also got to be careful how you handle i (Imaginary numbers) Not so much i as sqrt, as sqrt((2)2)=sqrt((-2)2) has a related problem when "cancelling" the square. The complex numbers do introduce more possibilities.

astromark
2007-Sep-06, 10:11 AM
And this my friends is what is our downfall. As long as there are people whom are willing to argue that one is two. We are doomed. Doomed...
I am certain that at no time has or does one ever equal two. What ever value you attribute to one. The two is twice of it. Do not throw the now famous line at me that one zero is equal to two zero's, I may become hysterical :)
As has been clearly stated here.. No amount of fiddling with the very impressive numerics of the highest levels of algebra will or can altar this truth. 1 does not equal 2.
Eight posts and we are off on a tangent to the OP. -- So the answer is no. Its nonsense.

hhEb09'1
2007-Sep-06, 10:18 AM
Eight posts and we are off on a tangent to the OP. -- So the answer is no. Its nonsense.What was the question again? :)

parallaxicality
2007-Sep-06, 10:48 AM
As has been clearly stated here.. No amount of fiddling with the very impressive numerics of the highest levels of algebra will or can altar this truth. 1 does not equal 2.

Sometimes, Winston. Sometimes it equals five. Sometimes it equals three. Sometimes it equals all of them at once. You must try harder. It is not easy to become sane.

antoniseb
2007-Sep-06, 12:09 PM
I recall back in the old days of non-precision cosmology we sued to say that 1=10. The offered proof was this:

11! = 11x10!

11!/! = 11x10!/!

11/11 = 11x10/11

1 = 10

John Mendenhall
2007-Sep-06, 04:48 PM
The math (?) examples above demonstrate nicely my objections to 'thought experiments'. Some well meaning person (for UT, probably an ATM'er) proposes a complicated thought experiment, usually about SR or GR, and the mainstreamers get to spend boodles of time finding the divide by zero, or other flaw.

Publius ought to receive the UT Medal of Honor for patience.

01101001
2007-Sep-06, 05:50 PM
11!/!

Unrecognized. What does 11!/! mean? Or even 1/!?

Or is the step just: "divide both sides by an exclamation mark"?

pghnative
2007-Sep-06, 05:54 PM
yes, but does 0.999... = 2?

<ducks and runs>

Ken G
2007-Sep-06, 06:36 PM
Unrecognized. What does 11!/! mean? Or even 1/!?

Or is the step just: "divide both sides by an exclamation mark"?
Yes, I think that's what it means-- it's a joke about taking algebra rules too dogmatically. There's no substitute for the old maxim, "know thy function"-- but modern graphical routines sure make that a lot easier to do!

antoniseb
2007-Sep-06, 06:39 PM
Unrecognized. What does 11!/! mean? Or even 1/!?
Or is the step just: "divide both sides by an exclamation mark"?
This was never taken seriously. The exclamation mark is the Factorial symbol, which I presume you knew. The proof was very tongue-in-cheek, and presented to astronomy students to make the point that 1=10, but for some reason 1<>100.

astromark
2007-Sep-06, 07:38 PM
Falls of chair... bangs head on floor, see's stars,. Oh yes astronomy.:)

01101001
2007-Sep-06, 09:16 PM
This was never taken seriously. The exclamation mark is the Factorial symbol, which I presume you knew. The proof was very tongue-in-cheek, and presented to astronomy students to make the point that 1=10, but for some reason 1<>100.

Yes, I realize it is not serious and ! is factorial

Please explain the notation. What does 11!/! mean? What does 1/! mean? Is this division by punctuation? I don't get it.

What was a student supposed to think was happening here? This use of symbols is unfamiliar to me. I am a stubborn student.

Did Ken G guess right?

peter eldergill
2007-Sep-06, 09:49 PM
Whe I teach mathematical induction, I offer up a "proof" that n = n + 1, by intentionally leaving out a crucial step in showing that we can find a "starting point" (usually by letting n = 1)

I do this to see if a student will question the teacher as to the utter nonsense of the statement and to try to instill the importance of a first step in the method of induction.

Reminds me also of a funny joke which was circulating a while back:

Q: Expand (x+y)^10
A: ( x + y )^10
( x + y )^10
( x + y ) ^ 10

Heh, math teachers are funniest people on the planet (even funnier than proctologists)

Pete

Decayed Orbit
2007-Sep-06, 09:50 PM
someone posted this on another site, and I have no idea how it works. I've been stretching my knowledge of algebra but I can't seem to find the flaw in it.

Other people have pointed out the flaw, but if you get stuck on something like this, you can just try plugging in actual numbers. Pick a=b=3, and see where the equalities stop being true. You find it pretty quickly that way.

Decayed Orbit
2007-Sep-06, 09:51 PM
Whe I teach mathematical induction, I offer up a "proof" that n = n + 1, by intentionally leaving out a crucial step in showing that we can find a "starting point" (usually by letting n = 1)

Can you clarify? Do you mean you use the normal rules of algebra to show that if n = n+1, then n+1 = n+2?

Nowhere Man
2007-Sep-06, 10:14 PM
Please explain the notation. What does 11!/! mean? What does 1/! mean? Is this division by punctuation? I don't get it.

What was a student supposed to think was happening here? This use of symbols is unfamiliar to me. I am a stubborn student.

The technical term for this line of reasoning is joke. (http://en.wikipedia.org/wiki/Joke)

Fred

hhEb09'1
2007-Sep-06, 10:24 PM
The technical term for this line of reasoning is joke. (http://en.wikipedia.org/wiki/Joke)Kinda like cancelling sixes and getting the right answer:

16/64

01101001
2007-Sep-06, 10:27 PM
The technical term for this line of reasoning is joke. (http://en.wikipedia.org/wiki/Joke)

Fred

Well, duh. Yes, it's a "funny" non-proof that 1 = 10.

If you understand the joke, you tell me what 11!/! means in English. I don't speak nonsense. Is it too much to ask for an explanation? Is it division by punctuation? What kind of students would tolerate a moment of that? I reallly want to know.

If you like that joke, you'll howl at:

2-#%

19=+

4/\$?22

Ha! Get it?

antoniseb
2007-Sep-06, 10:54 PM
If you understand the joke, you tell me what 11!/! means in English.

Sorry, I forgot that some readers don't know what factorial means. Factorial is the product of all integers greater than one and less than or equal to the number given. The symbol '!' indicates the factorial operation. Thus:

2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 7 * 6! = 5040
etc.

The numbers go up very quickly.
Factorials are very important in expressing series solutions to complicated math problems.

Nowhere Man
2007-Sep-06, 11:16 PM
OK, you know that the ! operator means factorial.

By the basic rules of algebra, 11x/x = 11, because the xes cancel out.

If you're symbol-minded, you might get carried away and claim that 11!/! = 11 by the same rule. The absurdity of this statement is what makes it funny.

If that doesn't make you laugh, than maybe I should fall into an open sewer and die*.

Fred

*Mel Brooks reference

Lord Jubjub
2007-Sep-06, 11:52 PM
Another simpler way of expressing that joke would be:

1*2/*=12

Ken G
2007-Sep-07, 12:09 AM
A joke my high school math teacher always told made fun of a student, who when asked what 5/5 was, said "the fives cancel, so it's nothing".

Dragon Star
2007-Sep-07, 12:16 AM
yes, but does 0.999... = 2?

<ducks and runs>

*chases pghnative around with a sickle*

01101001
2007-Sep-07, 01:20 AM
OK, you know that the ! operator means factorial.

Just as I said.

By the basic rules of algebra, 11x/x = 11, because the xes cancel out.

Because x/x is 1, when x nonzero. But, colloquially called cancelling out. OK. That's semi-justifiable.

If you're symbol-minded, you might get carried away and claim that 11!/! = 11 by the same rule. The absurdity of this statement is what makes it funny.

Yeah, funny to some. I recognized and appreciated the "cancelling" humor.

However, I'm still wondering about how the punctuation got into the denominator. The step before sticks in my craw. I take it, that it's some handwaving like: "Divide both sides by exclamation mark," that I and Ken G guess at, and I really haven't seen confirmed. I'll assume it.

Fine. OK. I don't know about others' cultures, but in mine, students were expected to interrupt a teacher's exposition if they didn't understand something. This is where I would have asked what "divide by exclamation mark" means. What do the comedic teachers do then?

I really want to know. I am serious. What? Does the teacher just try to wink at this point, and escape on charm? Is there anything to say? Hope no one asks? If one does, just give up and not bother with the conclusion? Maybe, with smart students, everyone always sees what's up and goes along for the ride.

Maybe I'm curious because Antoniseb only provided the text, not the banter that goes with it, at a minimum, the justifications for each step, at least the justification for the erroneous step.

And, again, please, please, I do get that this is a bogus proof, using unreasonable steps to reach an unsupportable answer. I've enjoyed many such in my life.

I have advanced degrees in symbol manipulation. In college, I've taught it, and mathematical algorithms, and generic problem solving. I don't recall, but I've probably done bogus proofs in class. And, I've been paid to produce written humor. I know how to laugh. I know what's up.

I'm sorry, but this isn't one of the good ones, because, unless I'm missing something, the bogus step lacks any subtlety, has no essential surprise factor, is simply symbolic gibberish, contains no ironic twist. Frankly, I'd sooner laugh at someone falling into an open sewer and dying.

One might as well employ, more directly, with more subtlety and less bumbling:

given: 1 = 1
since x + 0 = x; add 0 to the righthand side: 1 = 1 0
therefore: 1 = 10

Now that's funny. And less boring.

Look, I'm not trying to over-over-overanalyze this. It's not thesis-defense time, here. I wanted to find the humor. Perhaps I might want to use this some day. There are always children with vulnerable young minds about to challenge.

I respect Antoniseb, and the wisdom and intelligence inherent therein. And I know Antoniseb can't hit every one out of the park, too. I was just curious about the missing plot details, and I felt like my questions were being intentionally evaded or frustratingly misunderstood.

I appreciate someone finally beginning to address my question as asked. I knew I shouldn't relax my standards. Thanks. Really.

Ken G
2007-Sep-07, 02:23 AM
I really want to know. I am serious. What? Does the teacher just try to wink at this point, and escape on charm? Is there anything to say? Hope no one asks? If one does, just give up and not bother with the conclusion?

I think a way to do it might be to eliminate the step where ! is explicitly on the bottom, and write it
1 x 11! = 10! x 11
and just say "cancel the ! on both sides, and the 11 too, yielding 1=10."
Then if a student says "but why can you cancel the ! on both sides", you simply say "exactly the question you should be asking. This is not really a proof that 1=10, it's a proof that you have to understand what you're doing and not just symbol manipulate."

cran
2007-Sep-07, 02:38 AM
What does 11!/! mean? What does 1/! mean? Is this division by punctuation? I don't get it.

I think it means "divide by 0" ...

Factorial is the product of all integers greater than one and less than or equal to the number given.
if no number is given, then there can be no product ... 0*n = 0

so 11!/! = 11!/0! = 11!/0 = naughty!

... yes?

01101001
2007-Sep-07, 02:43 AM
I think it means "divide by 0" ...

I'm sure it doesn't.

if no number is given, then there can be no product ... 0*n = 0

That would be an undefined operator sans operand. Meaning less. Meaningless.

If it did mean 0!, that happens to be defined to be 1 (Mathworld (http://mathworld.wolfram.com/Factorial.html)), so don't run too far with that ball.

Ken G
2007-Sep-07, 04:15 AM
Yes, 0! is defined as 1, because the recursive defining relation of n! is
n! = n*(n-1)!
so if you start the induction with n=1, you need to define 0! = 1 to make the swan take flight. You could always just start at n=2 instead, but I think the idea is to extend it to as low an n as you can before it just doesn't work any more (like n = 0 in the above).

peter eldergill
2007-Sep-07, 12:44 PM
n! can also be thought of as the number of arrangements of n distinct objects in a row. So how many ways can you arrange 0 objects? One way, so 0! can be defined as 1

Gotta go, functions won't teach themselves!

Pete

hhEb09'1
2007-Sep-07, 03:30 PM
n! can also be thought of as the number of arrangements of n distinct objects in a row. So how many ways can you arrange 0 objects? One way, so 0! can be defined as 1Even better, I think, is that n!/m! represents the number of ways that n distinct objects can be placed in a row of (n-m) objects. In other words, if n=5, and m=2, then 5!/2! is the number of ways to place the 5 objects in rows of 3. It's just 5 x 4 x 3.

Of course, the number of ways to place n objects in rows of n is n!, but that formula would say it's n!/(n-n)!, so (n-n)! must be 1.

peter eldergill
2007-Sep-07, 04:14 PM
To answer an earlier question about "proving" that n = n + 1

The idea is using induction (but i'm skipping the first step, which is the crucial flaw in the proof)

Assume the statement is true for n = k, that is k = k+1

We now prove the next statement true for n = k + 1

That is, prove k+1 = k+2

R.S. = k+2 = (k+1) +1
but k+1 = k (by assumtion)

so (k+1) +1 = (k) +1 + L.S

so the statement is true by induction

The false step is not finding an original statement for this to work. If we let
n=1, the statement is 1=2, false. For n=2, we get 2=3, false...etc...

There is nowhere to "start", so the proof i've given is not valid

Pete

Decayed Orbit
2007-Sep-07, 05:12 PM
To answer an earlier question about "proving" that n = n + 1

The idea is using induction (but i'm skipping the first step, which is the crucial flaw in the proof)

Assume the statement is true for n = k, that is k = k+1

We now prove the next statement true for n = k + 1

That is, prove k+1 = k+2

R.S. = k+2 = (k+1) +1
but k+1 = k (by assumtion)

so (k+1) +1 = (k) +1 + L.S

so the statement is true by induction

The false step is not finding an original statement for this to work. If we let
n=1, the statement is 1=2, false. For n=2, we get 2=3, false...etc...

There is nowhere to "start", so the proof i've given is not valid

Pete

OK, that's what I thought, I just wanted to confirm.

I've seen people make basically this same mistake, although disguised more subtly....

Warren Platts
2007-Sep-07, 05:28 PM
Quote:
a = b

Multiply both sides by a

a^2 = ab

Subtract b^2 from each side

a^2 - b^2 = ab - b^2

Factor

(a+b)(a-b) = b(a-b)

Divide by a-b

a+b = b

But a = b, so

b + b = b

So

b = 2b

Divide by b

1 = 2 Dividing by the (a-b) actually yeilds:

∞(a+b)=b∞

which eventually yeilds:

∞=2∞

Anywhere else, that's called a reductio ad absurdum. But Mathematics gets a pass. We just call it incomplete, rather than inconsistent! :D

parallaxicality
2007-Sep-07, 05:39 PM
Wait though, isn't infinity like zero, in that two infinities are still infinity? If so than that statement isn't actually false. It does raise the question though of whether it's possible to cancel infinities out and be left with 1=2.

Disinfo Agent
2007-Sep-07, 05:58 PM
Yes. The problem isn't actually with the division by zero, which you can define appropriately in some extensions of the real numbers. But then you lose an important property of algebra, called the cancellation law, and the equations cease to be equivalent.

I recall back in the old days of non-precision cosmology we sued to say that 1=10. The offered proof was this:

11! = 11x10!

11!/! = 11x10!/!

11/11 = 11x10/11

1 = 10
I think a way to do it might be to eliminate the step where ! is explicitly on the bottom, and write it
1 x 11! = 10! x 11
and just say "cancel the ! on both sides, and the 11 too, yielding 1=10."

Cheshire cat math. :D

Ken G
2007-Sep-07, 09:29 PM
Of course, the number of ways to place n objects in rows of n is n!, but that formula would say it's n!/(n-n)!, so (n-n)! must be 1.Yeah, that's a good one, it's basically connecting 0!=1 to the fact that the nth position in the row of n objects "didn't count anyway" because it was completely determined by the rest, but you can count it if you like without changing anything. But that does come down quite similarly to saying what Peter did, that when you have none left to place, you can only do one thing.

Ken G
2007-Sep-07, 09:43 PM
There is nowhere to "start", so the proof i've given is not valid

True, but note the induction piece is kind of window dressing, because if you are allowed to start with an incorrect premise, you can actually prove anything. If I say k=k+1 for some k, I can prove n=n+1 for any n, even noninteger n, without using induction-- I can use algebra and subtract k-n from both sides. That reductio ad absurdum actually does prove something-- it proves k cannot equal k+1 for any k that obeys that algebra. So there are basically two types of erroneous proofs that 1=2, one type simply uses incorrect methodology, while the other inserts an incorrect assumption whose truth might be unknown-- and the result proves the falsity of the assumption, rather than that 1=2. So this type of reasoning actually is useful, but in the backwards sense that we're playing with. Perhaps that's the part that was bothering astromark, he couldn't put in words his frustration that the logic was being inverted for fun. I guess for some people that's like playing bullfighter with your country's flag.

Nowhere Man
2007-Sep-07, 10:53 PM
However, I'm still wondering about how the punctuation got into the denominator. The step before sticks in my craw. I take it, that it's some handwaving like: "Divide both sides by exclamation mark," that I and Ken G guess at, and I really haven't seen confirmed. I'll assume it.
11! = 11x10!

11!/! = 11x10!/! <--

11/11 = 11x10/11

1 = 10

That's right, in step two, both sides were divided by 1/!. Algebra is partly symbol manipulation , but if you manipulate the wrong symbols, you get weird stuff.

Fine. OK. I don't know about others' cultures, but in mine, students were expected to interrupt a teacher's exposition if they didn't understand something. This is where I would have asked what "divide by exclamation mark" means. What do the comedic teachers do then?
It's a useful tool to see if your students are awake and paying attention. If no one speaks up, then you've lost your audience in one way or another.

And now I'm done.

Fred

Warren Platts
2007-Sep-07, 10:55 PM
So there are basically two types of erroneous proofs that 1=2, one type simply uses incorrect methodology, while the other inserts an incorrect assumption whose truth might be unknown-- and the result proves the falsity of the assumption, rather than that 1=2. There's also the 3rd sense: that mathematics is intrinsically contradictory. It might be the case that there is a contradiction that would imply that 1=2, that is so well hidden, it hasn't been discovered yet.

After all, the bottom line of Gödel's proof was in the logical form of a conditional: if wid(c) then 17genR. I'm not qualified to say what the inscription '17genR' really means. But the 'wid(c)' was German that might be translated as 'the mathematical system axiomatized by Russell and Whitehead in their Principia Mathematica is logically consistent'.

This is the same system in use today--correct me if I'm wrong. By two philosophers no less. But there's an ambiguity somewhere. How are we interpret Gödel's result? Roger Penrose flat out said in his Emperor's New Mind that Gödel's proof implied that math is inconsistent. Wittgenstein said that as long as the contradiction is well hidden--what difference does it make? Turing said that it's not a problem until a bridge collapses.

Disinfo Agent
2007-Sep-08, 01:36 AM
Russell and Whitehead were mathematicians, as well as philosophers.
As for Gödel's Theorem, Gödel himself noted that it was a hurdle for 'logic', but not for 'mathematics'. (http://www.bautforum.com/against-mainstream/7216-blind-helpless-confined-within-sphere.html#post128187)

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. -- Kurt Gödel

Warren Platts
2007-Sep-08, 01:53 AM
Russell and Whitehead were mathematicians, as well as philosophers.
As for Gödel's Theorem, Gödel himself noted that it was a hurdle for 'logic', but not for 'mathematics'. (http://www.bautforum.com/against-mainstream/7216-blind-helpless-confined-within-sphere.html#post128187)
Of course it's not a problem for mathematics. Mathematics is as flexible as Silly Putty®. Turing himself proposed a system whereby "oracle machines" would decide which of an infinite series of Gödel sentences should be added to an infinite list of axioms that would render math both complete and consistent.

hhEb09'1
2007-Sep-08, 03:41 AM
How are we interpret Gödel's result? Roger Penrose flat out said in his Emperor's New Mind that Gödel's proof implied that math is inconsistent. I think I still have that book around somewhere. What page? I'm pretty sure that that is wrong, I'd like to see how he justifies that.

Ken G
2007-Sep-08, 07:15 AM
There's also the 3rd sense: that mathematics is intrinsically contradictory. It might be the case that there is a contradiction that would imply that 1=2, that is so well hidden, it hasn't been discovered yet. You are quite right, and spawned a new thread in the process, but note that so far there is zero evidence that there really is a valid argument that 1=2, i.e., that math is inconsistent. Godel tells us there's no way to prove that our suspicion is true, but we hold it, based on experience, nevertheless. What else can a brain do than to hold a suspicion based on experience? Math proofs are really no different, this should not surprise us, because math has to be linked to experience to have any meaning.

How are we interpret Gödel's result? Roger Penrose flat out said in his Emperor's New Mind that Gödel's proof implied that math is inconsistent. Wittgenstein said that as long as the contradiction is well hidden--what difference does it make? Turing said that it's not a problem until a bridge collapses.

I'm with hhEb09'1-- no one knows if math is inconsistent or incomplete, but it certainly seems far more likely to me that it is incomplete than that it contains an explicit "poison pill" somewhere hard to find that could allow us to prove that false things are true, or make a bridge collapse. It certainly could be in there, and very hard to find, and if we never find it then its presence is not terribly significant. The important thing is that we always check what we are proving against experience, in math as much as in science. Godel's theorem: math is the same thing as science.

Ken G
2007-Sep-08, 07:25 AM
Of course it's not a problem for mathematics. Mathematics is as flexible as Silly Putty®.

Methinks you are rather forgetting the real purpose of mathematics, which is not a logic game that gives us the pleasing sensation of certainty, but rather a useful way of determining truisms that connect to reality and can be checked there. On the latter score, math does rather better than Silly Putty.

Warren Platts
2007-Sep-08, 12:27 PM
I think I still have that book around somewhere. What page? I'm pretty sure that that is wrong, I'd like to see how he justifies that.IIRC it was just a passing mention in a footnote. Now I've got motivation to unpack my own books--I'll try to find it.

I'm with hhEb09'1-- no one knows if math is inconsistent or incomplete, but it certainly seems far more likely to me that it is incomplete than that it contains an explicit "poison pill" somewhere hard to find that could allow us to prove that false things are true, or make a bridge collapse. It certainly could be in there, and very hard to find, and if we never find it then its presence is not terribly significant. The important thing is that we always check what we are proving against experience, in math as much as in science. Godel's theorem: math is the same thing as science. If you found a contradiction, couldn't you just add one or the other sentences to your list of axioms?

cran
2007-Sep-09, 12:09 AM
Yes, 0! is defined as 1, because the recursive defining relation of n! is
n! = n*(n-1)!
so if you start the induction with n=1, you need to define 0! = 1 to make the swan take flight. You could always just start at n=2 instead, but I think the idea is to extend it to as low an n as you can before it just doesn't work any more (like n = 0 in the above).
OK ... so 0(0-1) = 02 - 0*1 = 1?

so it's a proof that 0 = 1? ... :shifty:

grant hutchison
2007-Sep-09, 12:35 AM
How are we interpret Gödel's result? Roger Penrose flat out said in his Emperor's New Mind that Gödel's proof implied that math is inconsistent.I think I still have that book around somewhere. What page? I'm pretty sure that that is wrong, I'd like to see how he justifies that.It would certainly be surprising for Penrose to make such a claim, since his thesis in The Emperor's New Mind hinges on the "other limb" of Gödel's theorem. Penrose uses Gödel's result that we can construct true statements which are not mathematically derivable within a given mathematical formalism to suggest that our minds are not purely algorithmic. If he thought mathematics had been proven to be inconsistent, he'd have trouble building his case.
A scan through the references to "Gödel's theorem" given in the index doesn't seem to turn up anything that contradict the above.

Grant Hutchison

01101001
2007-Sep-09, 01:03 AM
OK ... so 0(0-1) = 02 - 0*1 = 1?

No. In the recursive version of the definition, you simply define 0! to be 1. That is the starting point.

That's what 0! means, 1.

Then n! for integer n > 0 is n*(n-1)!

So 1! is 1*(0!) or 1*1 or 1.
So 2! is 2*(1!) or 2*1 or 2.
Etc.

Ken G
2007-Sep-09, 01:36 AM
If you found a contradiction, couldn't you just add one or the other sentences to your list of axioms?

I suppose you could trade the inconsistency for incompleteness. But as soon as you knew that the old math was inconsistent, it would be hard to believe the new one wasn't. But probably, what we really mean here is "effective consistency"-- if no human proof would ever be likely to use an inconsistent part because they are so profoundly buried. In that case, the difference between inconsistency and incompleteness is pretty "academic"-- truisms that couldn't be proven, or falsnesses that could, would likely never actually be encountered. It's unsettling not to know, though, if one puts too much faith in mathematics over experience.

Ken G
2007-Sep-09, 01:39 AM
OK ... so 0(0-1) = 02 - 0*1 = 1?

so it's a proof that 0 = 1? ... :shifty:

Yes, that's another "fake proof", but as 01101001 said, the reason it's not a real proof is that the statement you are applying is recursive, so it's not meant to find the value of the first member of the sequence (0!), it is meant to find the value of the succeeding members (n>0). The first member always has to be either found, or simply defined (0! = 1), separately from the recursive relation-- that's how induction works!

hhEb09'1
2007-Sep-09, 01:55 AM
But probably, what we really mean here is "effective consistency"-- if no human proof would ever be likely to use an inconsistent part because they are so profoundly buried. In that case, the difference between inconsistency and incompleteness is pretty "academic"-- truisms that couldn't be proven, or falsnesses that could, would likely never actually be encountered. It's unsettling not to know, though, if one puts too much faith in mathematics over experience.At that point though there wouldn't be an inconsistent "part". Once you have a statement p such that p and its opposite (p and ~p) are true, then everything's opposite is true, in that system. (p and ~p) implies q for all q, including ~q. In other words (q and ~q) is true for every statement q. That may look like purely academic wordplay, but it's not.

Just because all proofs suddenly become trivial doesn't mean that more robust, or non-obvious, proofs don't exist. I'd hate to see the day though when proofs were valued just for how much sweat went into them. :)

publius
2007-Sep-09, 02:04 AM
At that point though there wouldn't be an inconsistent "part".

That is a very important point to realize. When a "formal system" (I wish I could define that) is found to contradict itself somewhere, then it contradicts itself it *everywhere*. It can prove and disprove everything. So, if we were to find that mathematics contradicted itself, then the whole thing collapses. Every theorem that has been proven would be meaningless as everyone of those theorems can then be disproven. The whole thing falls apart.

-Richard

astromark
2007-Sep-09, 02:20 AM
Will you please stop this... you are doing my feeble head in. Left is right and right is wrong., this I can explain but wont until asked to... cos its silly and boring:) I am taking on board the foolishness of mathematics and how what is correct in math., may be manipulated to be wrong by the fundamentalist mathematician. Of whom I know none and wish to meat , never.
This for me is likened to that little story you all do not know... Where the philosopher proves by complex use of logarithmic calculations that black is in fact unlit white and consequentially is run down by a 'mac' truck on a pedestrian crossing... I might be on my own here with this view but, fiddling with what we know is akin to fixing that which is not broken... :) and more coffee...
This subject has two, ( Thats half of four ) and ( twice of one ) Threads running and, both are coming quickly to the same conclusion; ie, that this is just interesting enough but still very wrong. With special thanks to Ken for providing the clarity I was looking for.

Ken G
2007-Sep-09, 02:41 AM
At that point though there wouldn't be an inconsistent "part". Once you have a statement p such that p and its opposite (p and ~p) are true, then everything's opposite is true, in that system. (p and ~p) implies q for all q, including ~q. In other words (q and ~q) is true for every statement q. No, this is just my point. If you have consistent systems, you can get away with confusing what is "true" for what is "provable", but it's important to recognize that what is "true" is external to the logical system, and in math, it is specified by experience. This becomes a very important distinction when the system is not consistent, because then you can actually prove things that are not true. So you can prove p and ~p, but that does not mean that p and ~p are true when they appear in the predicate of another proposition. It means you have to restore consistency by finding out which is true, p or ~p, and excise the offending proof that led to the false result. Once you do that, you have no problem with being able to prove q and ~q, and it was certainly never true in any event that both q and ~q were true. The point is, the statement (p and ~p) impiles q is a statement about the truth of p or ~p, not a statement about the provability of p or ~p. We simply assume that provability implies truth, but you cannot make that assumption in inconsistent systems. That's why math is science-- you can't assume or define truth, you have to find it from experience.

Ken G
2007-Sep-09, 02:45 AM
That is a very important point to realize. When a "formal system" (I wish I could define that) is found to contradict itself somewhere, then it contradicts itself it *everywhere*. It can prove and disprove everything. So, if we were to find that mathematics contradicted itself, then the whole thing collapses. Every theorem that has been proven would be meaningless as everyone of those theorems can then be disproven. The whole thing falls apart.
Again, I think this statement confuses provability with truth. Math has value, not by virtue of it giving us ways of proving things, but by virtue of its tendency to lead us to things that are true. So far the two are interchangeable, but if math is inconsistent, then we must recognize the real goal is the latter, not the former, and nothing would fall apart if we can maintain the latter goal. In other words, which is more fundamentally the goal of math and formal logic: to give us a warm feeling of confidence that we have "proven" something, or to lead us to discovering things that our experience validates are true? And as you answer that, note that every proof at some point has to connect to experience, that's the point where we are "convinced the proof has reduced to a truism". Typically, the proof is just a time-saving placekeeper, it says "don't bother checking experience any more, it will always work", but if math has the potential for inconsistency, it just means we can't save ourselves that work, we just have to always keep checking with experience. It becomes science, so that's what it is, by Godel.

mugaliens
2007-Sep-09, 06:58 AM
Divide by a-b

STOP!

Since a = b, this is division by zero.

Bingo, 01101001!

I'm glad someone who knows the "secret" to this puzzle posted first. I'm sick and tired of seeing it going around the internet. It's high time this "trick" was trashed.

hhEb09'1
2007-Sep-09, 10:28 AM
No, this is just my point. If you have consistent systems, you can get away with confusing what is "true" for what is "provable", but it's important to recognize that what is "true" is external to the logical system, and in math, it is specified by experience. I'm surprised to see you say that, I'd thought you were an advocate of "we can never really know what is true". Have I been misreading you?
Again, I think this statement confuses provability with truth.You have quoted publius's entire comment, but he has no mention of "truth" so I don't see how you can say the statement confuses provability and truth. In my post, I do talk about true statements, but I'm not confusing provability and truth either--in the context, true or false have meaning differently than the "real" world. If, within the formal system, our model, we come to the conclusion somehow that it is raining outside that doesn't mean that it is actually raining outside.

Typically, the proof is just a time-saving placekeeper, it says "don't bother checking experience any more, it will always work", but if math has the potential for inconsistency, it just means we can't save ourselves that work, we just have to always keep checking with experience. It becomes science, so that's what it is, by Godel.Within a system, there are many true statements that are never proven, the axioms. But they are not reflections of the "true" world, necessarily. Non-euclidean geometry takes as one of its axioms the opposite of what we assume in Euclidean geometry. Which one is the "true" one? Can both be true in the real world? I'm thinking yes, they can, in the right context.

I wish it were so easy as "It means you have to restore consistency by finding out which is true, p or ~p, and excise the offending proof that led to the false result." :)

cran
2007-Sep-09, 12:01 PM
No. In the recursive version of the definition, you simply define 0! to be 1. That is the starting point.

That's what 0! means, 1.

Then n! for integer n > 0 is n*(n-1)!

So 1! is 1*(0!) or 1*1 or 1.
So 2! is 2*(1!) or 2*1 or 2.
Etc.
This is where my old math teacher would have looked at me ...
and groaned ...
and seen a terrier latching on ...

it comes down to "it is so because (I/we/someone) said so!" ... ?
"we simply define 0! to be 1" ...

we can't apply the formula - n*(n-1)!, because it is wouldn't work ...
(because both 0-1 and 0 are less than 1)
so we simply say ... it is ... ??

actually, I don't see how it works for 1 either,
except by virtue of this arbitrary determination ...
so 1! is also a "special case" ...?

we get: Then n! for integer n > 0 is n*(n-1)!
instead of n! is valid for n > 1 ...
eg, like prime numbers - the smallest value is 2 ...
we don't seem to have any problems excluding 1
from being a prime number
(any number divisible only by itself and 1)

doesn't the definition (as per the link provided earlier)
require that n be a positive integer ?
and that n! is the product of n and all positive integers from 1 to n-1?

Q. When did 0 become a positive integer?

The logic provided for the "special case" of 0! -
that there is exactly 1 way to arrange 0 objects
is not correct as far as I can see -
there are exactly 0 ways to arrange 0 objects in a sequence ...
and exactly 1 way to arrange 1 object in a sequence ...

So, instead of saying 0! is meaningless (like n/0 is meaningless),
"we"(?) define it as a "special case" and assign the value 1?

Yes, that's another "fake proof", but as 01101001 said, the reason it's not a real proof is that the statement you are applying is recursive, so it's not meant to find the value of the first member of the sequence (0!), it is meant to find the value of the succeeding members (n>0). The first member always has to be either found, or simply defined (0! = 1), separately from the recursive relation-- that's how induction works!
mea culpa time: I don't understand the terms recursive or induction in this context ...
though I imagine that recursive has something to do with a repeating pattern of action within a larger action ...

and I did find this (the bold is mine):

Mathematics A two-part method of proving a theorem involving an integral parameter. First the theorem is verified for the smallest admissible value of the integer. Then it is proven that if the theorem is true for any value of the integer, it is true for the next greater value. The final proof contains the two parts. - thefreedictionary

is there a reason why the smallest admissable value is not 2?
:shifty:
forgive me if I'm being dense ...
I was a highschool dropout years before I went to university ...

01101001
2007-Sep-09, 01:35 PM
So, instead of saying 0! is meaningless (like n/0 is meaningless),
"we"(?) define it as a "special case" and assign the value 1?

In a recursive definition, there is always a special case.

Wikipedia: Recursion (http://en.wikipedia.org/wiki/Recursion):

In mathematics and computer science, recursion specifies (or constructs) a class of objects or methods (or an object from a certain class) by defining a few very simple base cases or methods (often just one), and then defining rules to break down complex cases into simpler cases.

We who will employ the factorial function can define it however we wish, to make it useful.

We could define another function denoted by ? and say 0? = 42, and integer n > 0 n? = n*(n-1)?, but that is a less useful function.

We could declare 0! is undefined and special case 1! is 1 and integer n > 1 n! = n*(n-1)!, but that is a less useful function.

What have you got against making factorial useful? If 0! is not defined as 1, then the arithmetic of usage becomes far more special-casey than the definition.

By the way, there are several other definitions of factorial, non-recursive ones, and they all yield 0! = 1.

You should embrace the definition that makes 0!=1, if only for convenience. If you don't, you are welcome to do it the hard way. It's no skin off my nose.

parallaxicality
2007-Sep-09, 01:39 PM
Well, I can officially say that I'm completely lost. This topic escaped the meagre gravity of Planet Parallaxicality about 40 posts ago. Pretty much sums up why I'm a science buff and not a scientist I suppose.

But thanks for answering my question. :)

Ken G
2007-Sep-09, 03:11 PM
I'm surprised to see you say that, I'd thought you were an advocate of "we can never really know what is true". Have I been misreading you?Apparently. What I've said is that we determine what is true when we choose the means we will apply to decide that. The word "truth" requires an implicit assumption about what we mean by it. But in this case, I'm contrasting two approaches to truth: a formal proof and a confrontation with experience. If neither is relevant to what one means by truth in some situation, then I'm simply not talking about that. I'm restricting to the issue at hand: a system is consistent, in the useful (not formal) sense, when certain things that are considered true based on our experience with them, can be connected via a reasoning process that has proven reliable when confronted with past experience, to "proven" things that may be expected to also be true when confronted with experience. The formal sense is that it is consistent if you can never prove both p and ~p, but note we'd still have to replace our mathematics if we could only prove p but p was shown by experience to be wrong!

You have quoted publius's entire comment, but he has no mention of "truth" so I don't see how you can say the statement confuses provability and truth.Then I need to explain. The issue mentioned was whether or not mathematics "collapses" if shown to be inconsistent. I'm saying that's only true if you think the purpose of mathematics is to achieve a sense of certainty by following rules of proof, like a kind of mental recreation. If you instead think the purpose of mathematics is a way to generate new things that will be true based on old ones that are (by your truth convention, of course), then it does not collapse, but you would have to excise or avoid mathematical techniques or reasoning processes that prove false things. (Based on Godel's proof, it seems likely said techniques would be so obscure anyway that they would not likely be encountered.) My overarching point is that when you say p implies q, in the second form of the goals of math, you want the truth of p to imply the truth of q, simply the provability of p implying the provability of q is of no independent interest, it's like doing a logic puzzle for fun.

In my post, I do talk about true statements, but I'm not confusing provability and truth either--in the context, true or false have meaning differently than the "real" world. If, within the formal system, our model, we come to the conclusion somehow that it is raining outside that doesn't mean that it is actually raining outside.
Then you are interested in the recreational elements. I assumed you were talking about meaning, and meaning comes from connection to experience-- i.e., truth in the application of mathematics.

Within a system, there are many true statements that are never proven, the axioms. Exactly, that's where we connect to experience. We build a useful system by starting with axioms that we will simply assume are true, and see where they lead us. If they lead to a bunch of other things that are true, we have confidence our axioms are "good" and our reasoning is "good", but only on a provisional basis. The use of axioms only underscores my point that math is really science, when it has meaning and is not seen as merely a recreational game played with symbols.

Non-euclidean geometry takes as one of its axioms the opposite of what we assume in Euclidean geometry. Which one is the "true" one? Can both be true in the real world? I'm thinking yes, they can, in the right context.Of course, because the very concept of "truth" inherently involves a chosen level of precision. In applications where both axioms are suitably precise, they are both "true", and in applications where one violates the desired precision, then that one is no longer true. That's all part of the options we invoke when we choose our truth-determining mechanisms, a crucial part of science that also appears in the meaning-generating step in mathematics.

I wish it were so easy as "It means you have to restore consistency by finding out which is true, p or ~p, and excise the offending proof that led to the false result."
Do you doubt that this is precisely what we would actually do if faced with that problem? Notice how completely routine that problem is for science.

publius
2007-Sep-09, 07:35 PM
This discussion needs to be moved into the Godel thread, because that's where it belongs. The wind-baggy academese sounding sub-title above is my quirky sense of humor. :lol: These types of questions can get big shot "high priest" types to waxing rather full of themselves sometimes, and that's the type of pompous sounding titles they tend to write with this stuff.

And so in that spirit, with our tongue firmly in our cheek making light of such pomposity, we'll call our favorite "finite formal system" that includes what call math and logic, what we do when we derive and prove things, "Principia Mathematica". Now, Principia Mathematica is a very important tool. We want it to be sane. When we prove something with Principia Mathematica (PM), we want that to mean something.

I am no confusing PM-provable with "true". That's only as good as the axioms. And yes, Ken, experience, when we use PM in physics and science in general to look at the real world. The fact that PM's proofs seem to agree with with what the real world does is evidence that PM is not just an academic exercise, some toy creation of the mind, but something that works.

For example, we formulate the laws of nature in PM's language. We then use PM to derive results, exploring the logical (the meaning of which is PM itself) consequences of those first principles.

It was hoped that PM could be shown, proved, to be both consistent and complete. That was sort of a holy grail. That is, we could use PM to show that PM is consistent (sane), doesn't contradict itself, and that it could (in principle) answer any question about its domain.

That itself was not even a quest to establish "truth" (again, it's only as good as its axioms and rules), just that PM, within itself, was consistent and complete.

I stress again, this was not about truth, put about the meaning of PM-provable.

Godel came along and just let the wind out of those sails completely. No matter how big and grand PM becomes, so long as it is "not infinite", there will always be a question it cannot decide. And I actually think that infinity is the biggest, baddest infinity of them all, Absolute Infinity, the infinity of infinities, that which cannot be named (any property you can conceive of is held by a "lesser infinity") -- see Rucker's "Infinity and the Mind" where he discusses this.

And not only that, but PM cannot prove its own consistency. And if it does every prove its own consistency, then it has proven it is inconsistent. PM is thus insane. And it is insane within its own context. Again, this is not about truth.

And that is a disaster. I don't think the import of PM insanity is appreciated. If PM is found to contradict itself, that is, both A and NotA are PM-provable for some A, then that is true for *EVERY A.

Everything is provable, everything is disprovable. Every conceivable statement about "math" can be both proven and disproven. That means every proof, every PM-derived theorem is called into question. PM proof no longer means anything, because everything that has been proven can be disproven by the same system.

And that's what I mean by insanity. If that happens PM is insane. Every result, every theorem, that has been used in physics and all of science is then called into question. Think of all that "existence and uniqueness", etc stuff that has important things to say about solutions to equations of physical laws. All of that is meaningless, because the opposite can be proven as well!

If PM contradicts itself on one thing, it contradicts itself on everything. And this is not about truth, but about PM as a tool.

It is not a simple matter of "excising" the proof of that first discovered contradiction. It calls into question the whole thing. It may not have anything to do with that particular proof of itself, but the very axioms, and even the rules of how you build theorem from axiom!

Yes, we would have to come up with a new mathematics (NM). And in that new mathematics, well established theorems might not be NM-provable or might be NM-disprovable. Everything would be out the door with NM.

-Richard

Ken G
2007-Sep-09, 09:49 PM
I am no confusing PM-provable with "true". That's only as good as the axioms. And yes, Ken, experience, when we use PM in physics and science in general to look at the real world. The fact that PM's proofs seem to agree with with what the real world does is evidence that PM is not just an academic exercise, some toy creation of the mind, but something that works.But I would go even farther-- it's not just when we apply PM to physics that experience comes into play, it's the very choice of axioms that PM uses. Every proof reaches completion when it has been reduced to the axioms, but note, mathematicians do not say "and we see we have reduced to an axiom, so we win the game", they say "and this is obviously true-- we have reduced the complex theorem to something obvious." That's a very important difference, and it is only if PM lost that property would it collapse. If we knew PM was inherently incomplete at some deep point that no one had ever penetrated to, yet everything that it reduced to an axiom continued to be as true as the axioms when confronted with experience, it would not be a big problem for any but the "high priests" who probably have too high opinion of the standard they should be shooting for.

For example, we formulate the laws of nature in PM's language. We then use PM to derive results, exploring the logical (the meaning of which is PM itself) consequences of those first principles.
Right, and the value of the whole enterprise is judged only by the confrontation with experience, i.e., observation.

That itself was not even a quest to establish "truth" (again, it's only as good as its axioms and rules), just that PM, within itself, was consistent and complete.
That's completely true, but its significance was overblown, in my opinion. I'm saying the real value of PM comes from its ability to connect "true" axioms with "true" conclusions (where we use some other method to establish truth, generally observations), and truth is never going to be a complete or consistent concept (it's inconsistent because truth depends on the level of precision, and it's incomplete because we only have so much experience to draw on, and only so much intelligence to subject it to). Thus, what difference does it make if PM itself is consistent and complete, when truth isn't? The proof is in the pudding-- does it achieve the desired result. So the high priests are disappointed that PM is the same as science, but I ask, on what basis could we have expected anything else? I'm amazed it even comes remotely close, that's the real puzzle.

And not only that, but PM cannot prove its own consistency. And if it does every prove its own consistency, then it has proven it is inconsistent. PM is thus insane. And it is insane within its own context. I would say that is an insane context to have used-- it's not the actual context where PM gets its value or meaning, again because all value and meaning interfaces with experience. The rest is a recreational game like chess.

And that is a disaster. I don't think the import of PM insanity is appreciated. If PM is found to contradict itself, that is, both A and NotA are PM-provable for some A, then that is true for *EVERY A.
Right-- but is that a disaster? What it would do is disconnect the meaning of "provable" from the meaning of "truth"-- everything is now provable. But will there still not be a subset of proofs that lead to something true, where truth comes from this other measure? And if we merely restrict to proofs with a "good track record", will we have a problem? You might say, "just because a particular approach has a good track record doesn't guarantee it will work the next time", but I will just say, "comes with the territory-- science does fine despite this very same problem". There are no guarantees in life, why does PM need them? Is it not a tool of life?

Every result, every theorem, that has been used in physics and all of science is then called into question. But that was already true, when you talk about the confrontation with experience-- because of the reliance on axioms! All we are doing is introducing a new but much less likely way that we could trip up, when we already had a very easy one-- the idealizations and axioms we have to assume to make progress.

Think of all that "existence and uniqueness", etc stuff that has important things to say about solutions to equations of physical laws. All of that is meaningless, because the opposite can be proven as well! But we don't really care if existence and uniqueness can be proven, we only care if it's true! If we had assumed a proof necessarily implies truth, we now must accept that truth is judged in more practical terms-- but science already knew that, and PM should have as well-- they are both human constructions. Philosophically, I agree the impact is great-- it means PM is not magic. In hindsight, maybe that isn't so surprising-- maybe what was insane was expecting that we were magically hooking into a path to absolute truth.

If PM contradicts itself on one thing, it contradicts itself on everything. And this is not about truth, but about PM as a tool. When PM is thought of as a tool, that does bring in the concept of truth. The only way to leave that out is to think of it as something entirely self-contained, like solving an upside-down jigsaw puzzle. That was never its intent though-- we're supposed to "flip it over" and see some truth written on it, that we are piecing together, and only then is it a tool.

It is not a simple matter of "excising" the proof of that first discovered contradiction. It calls into question the whole thing. It may not have anything to do with that particular proof of itself, but the very axioms, and even the rules of how you build theorem from axiom!
Let's consider all the "truths" that have been discovered using PM and later verified to be indeed true, to some level of precision and given some mechanism for deciding truth. Will they all vanish if PM is inconsistent? It reminds me of Wile E. Coyote not falling until he notices there's no ground below him!

Yes, we would have to come up with a new mathematics (NM). And in that new mathematics, well established theorems might not be NM-provable or might be NM-disprovable. Everything would be out the door with NM.Not everything-- in fact very little. NM would be built on the back of PM-- we'd just stick to the approaches with a good track record. But that's always what we do in a search for truth! I really don't think it would make that much difference, I think that's what Wittgenstein must have meant. As for the "bridge collapsing" of Turing, that's exactly what would not happen-- that's Wile E. Coyote again.

cran
2007-Sep-10, 02:31 AM
In a recursive definition, there is always a special case. if you say so ...
because the link you provided doesn't -
it mentions simple cases ...

We who will employ the factorial function can define it however we wish, to make it useful. Of course you can ...
but if you choose to accept and defend an arbitrary definition,
why are you surprised when someone else doesn't understand?

What have you got against making factorial useful? If 0! is not defined as 1, then the arithmetic of usage becomes far more special-casey than the definition.useful? - I have nothing against being useful ...
though I'd prefer sensible, and consistent ...
but hey, if multiplying by 1 (twice) is something you find useful
(rather than a tad redundant) then knock yourself out ...

By the way, there are several other definitions of factorial, non-recursive ones, and they all yield 0! = 1.if you say so ...

You should embrace the definition that makes 0!=1, if only for convenience. If you don't, you are welcome to do it the hard way. It's no skin off my nose. I "should embrace" something I don't understand, that is arbitrarily defined purely for convenience,
and to avoid upsetting those who have embraced it?
:confused:
I should simply take it on ... faith? :shifty:

My, hasn't the language of science come a long way?
:(

Warren Platts
2007-Sep-10, 03:00 AM
PM cannot prove its own consistency. And if it does every prove its own consistency, then it has proven it is inconsistent. PM is thus insane. And it is insane within its own context. Again, this is not about truth.

And that is a disaster. I don't think the import of PM insanity is appreciated. If PM is found to contradict itself, that is, both A and NotA are PM-provable for some A, then that is true for *EVERY A.

Everything is provable, everything is disprovable. Every conceivable statement about "math" can be both proven and disproven. That means every proof, every PM-derived theorem is called into question. PM proof no longer means anything, because everything that has been proven can be disproven by the same system.

And that's what I mean by insanity. If that happens PM is insane. Every result, every theorem, that has been used in physics and all of science is then called into question. Think of all that "existence and uniqueness", etc stuff that has important things to say about solutions to equations of physical laws. All of that is meaningless, because the opposite can be proven as well! Check out this dialogue between Turing and Wittgenstein. Who do you think makes the better point? (From www.turing.org.uk (http://www.turing.org.uk/turing/index.html))

Wittgenstein:... Think of the case of the Liar. It is very queer in a way that this should have puzzled anyone — much more extraordinary than you might think... Because the thing works like this: if a man says 'I am lying' we say that it follows that he is not lying, from which it follows that he is lying and so on. Well, so what? You can go on like that until you are black in the face. Why not? It doesn't matter. ...it is just a useless language-game, and why should anyone be excited?
Turing: What puzzles one is that one usually uses a contradiction as a criterion for having done something wrong. But in this case one cannot find anything done wrong.
W: Yes — and more: nothing has been done wrong, ... where will the harm come?
T: The real harm will not come in unless there is an application, in which a bridge may fall down or something of that sort.
W: ... The question is: Why are people afraid of contradictions? It is easy to understand why they should be afraid of contradictions, etc., outside mathematics. The question is: Why should they be afraid of contradictions inside mathematics? Turing says, 'Because something may go wrong with the application.' But nothing need go wrong. And if something does go wrong — if the bridge breaks down — then your mistake was of the kind of using a wrong natural law. ...
T: You cannot be confident about applying your calculus until you know that there are no hidden contradictions in it.
W: There seems to me an enormous mistake there. ... Suppose I convince Rhees of the paradox of the Liar, and he says, 'I lie, therefore I do not lie, therefore I lie and I do not lie, therefore we have a contradiction, therefore 2 x 2 = 369.' Well, we should not call this 'multiplication,' that is all...
T: Although you do not know that the bridge will fall if there are no contradictions, yet it is almost certain that if there are contradictions it will go wrong somewhere.
W: But nothing has ever gone wrong that way yet...
I kind of have to go with Wittgenstein on this one. We've got bridges collapsing all over the place, but I doubt it's because of inherent contradictions within PM! :D

If PM contradicts itself on one thing, it contradicts itself on everything. And this is not about truth, but about PM as a tool.

It is not a simple matter of "excising" the proof of that first discovered contradiction. It calls into question the whole thing. It may not have anything to do with that particular proof of itself, but the very axioms, and even the rules of how you build theorem from axiom!

Yes, we would have to come up with a new mathematics (NM). And in that new mathematics, well established theorems might not be NM-provable or might be NM-disprovable. Everything would be out the door with NM.Actually, Alan Turing did his Ph.D. thesis on this very problem. His solution was to propose an infinite mathematics based on "ordinal logics". He supplemented his computing machine with "oracle machines". Turing's system goes around computing theorems, and then when it finds a pair of those unprovable sentences, the oracle-machine applies human-style, non-computable intution to the problem and assigns truth-values to the sentences, that then become part of the axioms. Thus, each iteration of this process generates a new NM, bigger than the NM that came before, and so on ad infinitum. The infinite ordinal logic would be complete IIRC.

01101001
2007-Sep-10, 04:29 AM
if you say so ...
because the link you provided doesn't -
it mentions simple cases ...

Nitpicking. My cite agrees with my language. "A case" can mean at least one; it doesn't have to mean "only one". If it matters: There is always at least one special case, possibly more, in a recursive definition. Surely that doesn't make you happier. Don't you want no special cases?

Of course you can ...
but if you choose to accept and defend an arbitrary definition,
why are you surprised when someone else doesn't understand?

Look, this isn't about being right, in the true/false sense. It's about being right in the quality sense. When we define a function, we are building a tool to make our computation, our symbol manipulation, our notation, our thinking, easier. It's anything but arbitrary. It's useful.

0! = 1 makes things simpler. Understand?

If you want a bad tool, go create your own. Define it any way you want. Mathematics is egalitarian. Build your tools the way you want to use them.

useful? - I have nothing against being useful ...
though I'd prefer sensible, and consistent ...
but hey, if multiplying by 1 (twice) is something you find useful
(rather than a tad redundant) then knock yourself out ...

I suggest you stop focusing on the 1x2x3x...xn definition of factorial and use one of the several others. Maybe then you'll be recognize it's not about what you like. It's not about aesthetics. It's about functionality. It is about being sensible. It is about being consistent. That's why 0! = 1.

I "should embrace" something I don't understand, that is arbitrarily defined purely for convenience,
and to avoid upsetting those who have embraced it? :confused:
I should simply take it on ... faith? :shifty:
My, hasn't the language of science come a long way? :(

So? It's up to you take take my embracing advice or not. Use the tool. Don't use the tool. Take it on faith if that's all you got; you've got my reasons, but you don't seem to want to hear them.

I'm not claiming that the definition mathematicians use is somehow correct; it's only a definition of a tool. I only claim the standard factorial function, with 0! = 1, is effective for those who use it.

If you wish to be some sort of contrarian, define your own factorial tool, one that is undefined for 0, and if it proves to be more effective, the mathematicians, and even I, will start using it. If it's clumsier, we won't. Do let us know.

Ken G
2007-Sep-10, 06:31 AM
I kind of have to go with Wittgenstein on this one. We've got bridges collapsing all over the place, but I doubt it's because of inherent contradictions within PM! Yes, Wittgenstein's position on this sounds very much like what I've been advocating as well. It's a tempest in a teacup unless you, for some reason, take mathematics more seriously than we have any particular reason or need to, when really what we care about is simply the "school of hard knocks".

He supplemented his computing machine with "oracle machines". Turing's system goes around computing theorems, and then when it finds a pair of those unprovable sentences, the oracle-machine applies human-style, non-computable intution to the problem and assigns truth-values to the sentences, that then become part of the axioms.
Now that sounds to me like someone who takes their mathematics more seriously than they have any reason or need to! Or maybe one can think of what he's doing as a kind of contingency plan, a hypothetical outline of what we would really need to do if a specific contradiction was actually encountered. It's always useful to have thought out the contingencies, but it seems too hypothetical to be of much use...

cran
2007-Sep-10, 06:41 AM
Nitpicking. My cite agrees with my language. "A case" can mean at least one; it doesn't have to mean "only one". If it matters: There is always at least one special case, possibly more, in a recursive definition. Surely that doesn't make you happier. Don't you want no special cases?

Look, this isn't about being right, in the true/false sense. It's about being right in the quality sense. When we define a function, we are building a tool to make our computation, our symbol manipulation, our notation, our thinking, easier. It's anything but arbitrary. It's useful.

0! = 1 makes things simpler. Understand? No ...
and that's exactly what I want to do - understand ...
because if I understand the "why",
then I can agree with you about the "what" ...
I understand that you find it "useful" ... and now "simpler" ...
but not why it is either ...
you consider me a nitpicker, and a contrarian ...
perhaps ...
but I consider me a student who is struggling with a concept,
who feels like he is being scorned by a teacher
who can't be bothered to explain, merely dictate ...
or is it because I'm not brilliant or intuitive enough
to see what you see?

If you want a bad tool, go create your own. Define it any way you want. Mathematics is egalitarian. Build your tools the way you want to use them.

I suggest you stop focusing on the 1x2x3x...xn definition of factorial and use one of the several others. Maybe then you'll be recognize it's not about what you like. It's not about aesthetics. It's about functionality. It is about being sensible. It is about being consistent. That's why 0! = 1.well, I don't know any others (I don't really know that one, apparently) ...
only that they must exist ...
I never mentioned aesthetics, but whatever ...
OK, but why is it sensible, or necessary, to multiply by 1 (twice over)?
and, apart from being self-consistent, how is it consistent?
is it consistent with (n*0) ?

So? It's up to you take take my embracing advice or not. Use the tool. Don't use the tool. Take it on faith if that's all you got; you've got my reasons, but you don't seem to want to hear them.

I'm not claiming that the definition mathematicians use is somehow correct; it's only a definition of a tool. I only claim the standard factorial function, with 0! = 1, is effective for those who use it.

If you wish to be some sort of contrarian, define your own factorial tool, one that is undefined for 0, and if it proves to be more effective, the mathematicians, and even I, will start using it. If it's clumsier, we won't. Do let us know.not your reasons, merely your position and experience (and derision) ... reasons would include why it is so ...

unlike computer programmers and mathematicians,
I've had to work on construction sites, and in factories ...
my experience is that people who use tools they don't understand -
get hurt ...
and when I started out, I had to learn how and why a particular tool ...
later, I had to do my best to explain it to the new guy ...
and if he said he didn't understand, or asked questions like "why?",
I would consider him a willing student,
rather than a nitpicker or contrarian, or someone not worth the bother ...

Ken G
2007-Sep-10, 07:02 AM
I've had to work on construction sites, and in factories ...
my experience is that people who use tools they don't understand -
get hurt ...

This is indeed a useful analogy, so let's use it to achieve the understanding you seek. Let's say you have a tool, and one supervisor tells you "you can use this tool on any floor of a building, except in the basement, you cannot use it in a basement", and another supervisor later tells you "actually, you can also use that tool in a basement, but you have to treat it special down there, it's trickier down there so you have to take the following special precautions". Which supervisor has told you something more useful? That's like what happens when you extend the factorial to include 0!=1.

Disinfo Agent
2007-Sep-11, 10:48 AM
This is where my old math teacher would have looked at me ...
and groaned ...
and seen a terrier latching on ...

it comes down to "it is so because (I/we/someone) said so!" ... ?
"we simply define 0! to be 1" ...

we can't apply the formula - n*(n-1)!, because it is wouldn't work ...
(because both 0-1 and 0 are less than 1)
so we simply say ... it is ... ??Alternatively, you can do it this way (which is how it's usually taught in schools, I think):

For n>1, define n! as n * (n-1) * ... * 1. Note that this definition is equivalent to n! = n * (n-1)! for n>2.

For other integers, define n! as the number that verifies n! = n * (n-1)! This works for n=2, iff 1!=1, and for n=1 iff 0!=1. For non-positive integers, there is no solution to the equation in bold, so leave (-n)! undefined.
In other words, the factorial of n, f(n)=n!, can be defined as the solution of the equation n! = n * (n-1)! which verifies f(1)=1.

cran
2007-Sep-12, 02:29 AM
Disinfo Agent, Ken G ...

thanks for trying ... I'm just too dense
(or my brain is hardwired to primary school arithmetic) ...

I can understand that there is a need for 0! ...
but I can't see what it is ...

I understand that both 1! and 0! equal 1 because that makes it simpler/more useful ...
whatever "it" is ...

I'm quite used to "exceptions to the rule" or "treat x differently under y circumstances" in english ...
and yes, even in labouring jobs (filling and bonding agents spring to mind) ...
but, I guess to learn that something like this even exists in maths ...
well, it's kind of like finding out that Santa Claus is
really uncle Charlie with a fake beard ... :sad:

Ken G
2007-Sep-12, 02:46 AM
Uncle Charlie is a mathematician...

hhEb09'1
2007-Sep-12, 03:13 AM
I'm surprised to see you say that, I'd thought you were an advocate of "we can never really know what is true". Have I been misreading you?Apparently. What I've said is that we determine what is true when we choose the means we will apply to decide that. The word "truth" requires an implicit assumption about what we mean by it.To me, that sounds like "we can never really know what is true". I'll have to go back and look at some of the contexts.

If you instead think the purpose of mathematics is a way to generate new things that will be true based on old ones that are (by your truth convention, of course), then it does not collapse, but you would have to excise or avoid mathematical techniques or reasoning processes that prove false things. (Based on Godel's proof, it seems likely said techniques would be so obscure anyway that they would not likely be encountered.)
A mathematical formal system is built upon a set of axioms and definitions. If an inconsistency is found, it is not just located in a farflung corner of our wild numerical wilderness that can be avoided by only driving on the autobahns--an inconsistency that crops up means that there is an inconsistency in the axioms. As you allude to, those axioms have been vetted and compared to our experiences with the real world, ad nauseam. An inconsistency, any inconsistency, would point to a fundamental disconnect between not only our formal system, but also our understanding of how the world works.

Then you are interested in the recreational elements. I assumed you were talking about meaning, and meaning comes from connection to experience-- i.e., truth in the application of mathematics.But, that is what mathematics is all about. :)

Calling that "recreational" is like calling quantum mechanics "basically bunk (http://science.reddit.com/info/1j6iz/comments)". It's a form of disparagement.
Do you doubt that this is precisely what we would actually do if faced with that problem? Notice how completely routine that problem is for science.I think I've explained why probably it could not be what we would actually do.

YNow that sounds to me like someone who takes their mathematics more seriously than they have any reason or need to!Maybe this will help: wikiHow (http://www.wikihow.com/Get-Through-High-School-With-Minimal-Work) :)
Or maybe one can think of what he's doing as a kind of contingency plan, a hypothetical outline of what we would really need to do if a specific contradiction was actually encountered. It's always useful to have thought out the contingencies, but it seems too hypothetical to be of much use...It looks to me (I've never read his Ph.D. thesis) that he's addressing the issue of completeness, not inconsistency. When he finds an unprovable theorem, it and its opposite are both potential axioms to be added to the system--like Euclid's fifth.

01101001
2007-Sep-12, 03:21 AM
Wikipedia: Empty product (http://en.wikipedia.org/wiki/Empty_product)

In mathematics, an empty product, or nullary product, is the result of multiplying no numbers. Its numerical value is 1, the multiplicative identity, just as the empty sum—the result of adding no numbers—is zero, or the additive identity.
[...]
Two often-seen instances are a0 = 1 (any number raised to the zeroth power is one) and 0! = 1 (the factorial of zero is one).

hhEb09'1
2007-Sep-12, 03:41 AM
Another good example, 01101001. If a to the zeroth power were something else, say zero, what would you get when you multiplied it by a? :)

'Course, n! means "multiply together all the numbers from n to 1" to a lot of people, so there is the confusion--in that definition, is n in there are not when you compute 0!

Ken G
2007-Sep-12, 04:39 AM
To me, that sounds like "we can never really know what is true". I'll have to go back and look at some of the contexts.
Are you talking about some kind of "absolute truth independent from human thought" here? I think you'll have a tough time defining such a concept. In the real world, all we have to go on to interact with a concept of truth is human thinking, and that requires we make some choices. If we say that 1+1=2 is true, then of course we can know that it is true that 1+1=2, so why would "we choose our truths" sound like "we can never know what is true"? The point is, \$ + % = @ is symbolic games, but 1+1=2 says something quite a bit more-- it has a truth that connects with experience. It has a truth that is objectively reproducible-- it has a truth that is science. Ergo, the meaning of math comes from the same place as the meaning of science, which is why I say math is science when it is not a recreational game. Don't worry, the mathematicians can say science is math-- but they have to have the meaning of math that makes this work.

A mathematical formal system is built upon a set of axioms and definitions. Which axioms and definitions? Randomly chosen, like the rules of chess? Chess is a formal system-- mathematics as we know it is not, it is a system based in experience that is easily mistaken for a formal system if you forget why it exists in the first place.

If an inconsistency is found, it is not just located in a farflung corner of our wild numerical wilderness that can be avoided by only driving on the autobahns--an inconsistency that crops up means that there is an inconsistency in the axioms. If you include "first order logic" as one of the axioms, yes (that is not the usual meaning of axiom). But so what? Of course the axioms are inconsistent-- we chose them! They were not delivered to us in a holy chariot, we picked them because we liked them. And we'll still like them if they turn out to be inconsistent, just as we'll still like the autobahn if it has a few potholes. Drive around 'em-- math will still serve the purpose for which it was invented and why it is taught in school, like driving itself.

As you allude to, those axioms have been vetted and compared to our experiences with the real world, ad nauseam. An inconsistency, any inconsistency, would point to a fundamental disconnect between not only our formal system, but also our understanding of how the world works.And that's a problem how? Does that not happen every time we turn a corner? What's the big surprise, I'm mystified. General relativity is inconsistent with quantum mechanics, yet we use them both without the slightest flinch, we just avoid the Planck scale. Why should we be surprised math itself could be like that? The surprise is that this is such a rare and esoteric problem in math, that's the real mystery.

Calling that "recreational" is like calling quantum mechanics "basically bunk (http://science.reddit.com/info/1j6iz/comments)". It's a form of disparagement. The point is, you have to choose your position, you can't have it both ways. If math is purely formal, then we are free to pick its rules to be as consistent as we like-- but it has no applications so it's a recreational game. If the goal is to have meaning, i.e., applications, then we must at some point connect to experience, and then we are not free to pick our rules as we like-- we are as stuck with potential inconsistencies as our understanding of how the world works is stuck with potential inconsistencies. Why did anyone think we would be privy to a formal system that would also connect to experience? Even logic has this problem-- formal logic is not applicable to experience, experience-based logic is not formal (it's fuzzier, it includes things like self-referential statements, or trying to guess a poker player's hand from his betting). What's the surprise here, exactly?

I think I've explained why probably it could not be what we would actually do.
Then I missed it, because I still have no doubt at all that this is just what we'd do, assuming the inconsistency was esoteric and rarely encountered, which we already know from experience with math as it is used. Math is a tool-- and it can slip and cut your hand, and you're just that much more careful next time. It seriously needs to get off its high horse, and just accept that it has a proven track record for being a great tool indeed.

It looks to me (I've never read his Ph.D. thesis) that he's addressing the issue of completeness, not inconsistency. When he finds an unprovable theorem, it and its opposite are both potential axioms to be added to the system--like Euclid's fifth.
Yes, you're right, but it can be used as a contingency plan if an inconsistency is encountered, just in reverse-- the oracle machines trace back to whatever are the axioms leading to the inconcistency, and excise them. That will no doubt introduce incompleteness problems, so run the machines forward again and pick new axioms that don't introduce new inconsistencies. If you make a mistake, go back and do it again. I haven't read his thesis either, but this follows from his argument as it is summarized-- an oracle machine that could pick an axiom based on "what is true" could also trace an inconsistency to one that isn't. The point is, his oracle machines are doing what I mentioned above-- explicitly embedding in math a guess about "what is true", and iterating the process. That's science, not math-- it's just "science by oracle". Wouldn't that be nice.

But come to think of it, we already have such oracle machines at our disposal: it's called reality. Which raises the interesting question: is reality complete and consistent? I expect it is, or I'd like to believe it is, but there is still no complete and consistent projection of reality onto the instruments we have command of. No surprise there, the surprise is we come as close as we do.

hhEb09'1
2007-Sep-12, 05:29 AM
Are you talking about some kind of "absolute truth independent from human thought" here? I think you'll have a tough time defining such a concept.I was talking about whatever you were talking about, in the post I quoted. Looking back, I see that you used it in quotes, so maybe you were talking about something different.
If we say that 1+1=2 is true, then of course we can know that it is true that 1+1=2, so why would "we choose our truths" sound like "we can never know what is true"?Are you saying that we chose 1+1=2?

Besides, I didn't think we were talking about individual instances, but more an encompassing reality.
Chess is a formal system-- mathematics as we know it is not, it is a system based in experience that is easily mistaken for a formal system if you forget why it exists in the first place.In the context of Godel's Theorems, we are talking about formal systems, surely. They wouldn't be applied otherwise.
Of course the axioms are inconsistent-- we chose them!We don't know that they are inconsistent.
And we'll still like them if they turn out to be inconsistent, just as we'll still like the autobahn if it has a few potholes. Drive around 'em-- math will still serve the purpose for which it was invented and why it is taught in school, like driving itself.I doubt it will be shown to be inconsistent. The theorem says we can't show that it is consistent, not that it is inconsistent.

The autobahn of an inconsistent system would be a nuclear carnage, not a few potholes. :)
And that's a problem how? Does that not happen every time we turn a corner? What's the big surprise, I'm mystified.A fundamental disconnect in our understanding of how the world works? What happens everytime we turn a corner?

Then I missed it, because I still have no doubt at all that this is just what we'd do, assuming the inconsistency was esoteric and rarely encountered, which we already know from experience with math as it is used.That's not what the Theorems are about.
It seriously needs to get off its high horse, and just accept that it has a proven track record for being a great tool indeed.Math is on a high horse? I mean, other than for being a great tool indeed? What's going on here?

Did a mathematician steal your wife? :)

Ken G
2007-Sep-12, 06:13 AM
I was talking about whatever you were talking about, in the post I quoted. Looking back, I see that you used it in quotes, so maybe you were talking about something different.I probably meant that we choose our truths when we establish what constitutes a way of establishing truth. That's not an arbitrary choice, but it is still a choice, just as it is in mathematics. So we only know what we choose to be true, not what is "really true" in some absolute and unimpeachable sense.
Are you saying that we chose 1+1=2?
I'm saying that if those are just a series of symbols, then yes, we certainly do choose it to be true, but if each symbol is intended to have some meaning that is independent of mathematics, that meaning must come from experience and is therefore established by the rules of science. The point is, if we adopt the latter meaning of those symbols, then we do not choose the truth of the equation, but we also lose control over the consistency and completeness of the result-- and that should surprise us not in the least. It's all about not confusing what is provable and what is true, by some standard of truth, yet mathematics marries the two in an important and yet too often unrecognized way. That marriage is what I feel is at the heart of Godel-- our attempt to capture reality in a bottle, only to discover that what we can get in there is not both consistent and complete (I suspect neither).

In the context of Godel's Theorems, we are talking about formal systems, surely. No, because Godel's theorems don't apply to all formal systems, only to formal systems with certain properties that also happen to be the ones we associate with reality. Hence the "crisis" is not that some systems that have nothing to do with reality cannot be both consistent and complete, it is that the very system that we like to associate with our understanding of reality cannot be! Is that not the crisis here? Do you think this thread would be fretting this crisis if Godel had shown that quaternion algebra cannot be both complete and consistent? I don't think so, Godel got us where we live, and showed that there is a limit to what we can know about the way we conceptualize reality. Yet we face that "crisis" every morning when we get out of bed, Godel has added nothing except to demonstrate what we might have already suspected: we face fundamental limitations.

I doubt it will be shown to be inconsistent. The theorem says we can't show that it is consistent, not that it is inconsistent.I'm well aware of that, and I share your doubt that it will ever be shown to be inconsistent, yet I also doubt that it isn't inconsistent. That it will never be shown so is just my point-- it's not a practical problem, but then neither is its incompleteness. A tempest in a teacup, the whole business-- not to take anything away from Godel's genius, but I wish he had unified the four forces instead.

The autobahn of an inconsistent system would be a nuclear carnage, not a few potholes.So you keep claiming, with no evidence at all to support this! We have agreed we'll probably never even find the inconsistencies, you think because they're not there and I think because they're too deeply buried for our limited intellect to root out. But if we did find them, it would likely be down some incredibly obscure avenue that had no practical applications and no one had ever used in a derivation before anyway. Yes that avenue could then be used to "prove" anything, but so what? It would only teach us what we should already recognize: the need to distinguish what can be proven true from how to derive something that is actually true in our experience.

Look at what we're doing right now, we're using logic to connect assumptions to conclusions. How do we know the conclusions we reach are correct, simply because we are using logic to reach it from axioms we accept? We don't, we just know that we like the axioms, and we accept logical equivalence as a meaningful thing, all based on experience. But experience is an uncertain master. I accept that, it doesn't bother me at all-- what difference then does it make if there's also a potential for inconsistency in the formal system we're using that is embedded in a way we are very unlikely to encounter in an argument of this type? I happen to think that the potential for error based on erroneous axioms or inadequacies in basic logical equivalence in a real application are far more likely sources of problems, and we have no defense and no guarantees against either of those. Why has Godel made us worry, we already had plenty to be worried about!

Math is on a high horse? I mean, other than for being a great tool indeed? What's going on here?The "high horse" is to think that if we could just know it was consistent, then it would be an oracle for truth. That's wrong-- even a consistent system can't guarantee truth. Yes, math is on a great steed, a charger of unparalleled speed and stamina-- so why does it need to think that it is infallible too? Where comes this belief, this faith, that because something that is untrue from experience has never been shown logically equivalent to an axiom that is thought to be true from experience, that this somehow requires math to be infallible in order to be of use to us? Talk about a "what have you done for me lately" kind of attitude!

Did a mathematician steal your wife? You jest, but in all seriousness, I'm not dissing mathematicians, I'm their best friend. I'm saying that they don't have to be plying an infallible trade in order to be able to derive important connections between things we already think are true and things which we're led to want to check out the truth of. In short, the marriage between what is provable and what is true is not going to end in divorce over one isolated transgression that happened in Vegas.

hhEb09'1
2007-Sep-12, 07:12 AM
I probably meant that we choose our truths when we establish what constitutes a way of establishing truth. That's not an arbitrary choice, but it is still a choice, just as it is in mathematics. So we only know what we choose to be true, not what is "really true" in some absolute and unimpeachable sense.OK, thanks for the further explanation. But I'm surprised to see you say that, what you did.
No, because Godel's theorems don't apply to all formal systems, only to formal systems with certain properties that also happen to be the ones we associate with reality.But, since we're talking about Godel's theorems, we are talking about formal systems, in that context. You said "Chess is a formal system-- mathematics as we know it is not" but what you mean is "mathematics as Ken G uses it is not". In the context of Godel's theorems, it is. Simply put, that's why Godel's theorems are irrelevant to your mathematics. They're not a formal system. Not really mathematics, in other words.
I'm well aware of that, and I share your doubt that it will ever be shown to be inconsistent, yet I also doubt that it isn't inconsistent. That it will never be shown so is just my point-- it's not a practical problem, but then neither is its incompleteness.Godel did not prove that it could never be shown to be inconsistent.
So you keep claiming, with no evidence at all to support this!I've worked with provably inconsistent systems before. That's what they do.

I accept that, it doesn't bother me at all-- what difference then does it make if there's also a potential for inconsistency in the formal system we're using that is embedded in a way we are very unlikely to encounter in an argument of this type?Godel didn't show that there was potential for inconsistency. All he did was show that we couldn't prove consistency.

I happen to think that the potential for error based on erroneous axioms or inadequacies in basic logical equivalence in a real application are far more likely sources of problems, and we have no defense and no guarantees against either of those. Why has Godel made us worry, we already had plenty to be worried about!That's the very problem that he was working on. And a lot of other people too.

The "high horse" is to think that if we could just know it was consistent, then it would be an oracle for truth. That's wrong-- even a consistent system can't guarantee truth. Yes, math is on a great steed, a charger of unparalleled speed and stamina-- so why does it need to think that it is infallible too? Where comes this belief, this faith, that because something that is untrue from experience has never been shown logically equivalent to an axiom that is thought to be true from experience, that this somehow requires math to be infallible in order to be of use to us?You want to know where it comes from? It's a straw horse. :)

You've built it up from your misunderstanding.

Ken G
2007-Sep-12, 02:51 PM
You said "Chess is a formal system-- mathematics as we know it is not" but what you mean is "mathematics as Ken G uses it is not". No, what I meant, and what I said, is that mathematics the way that it is used that put it into school curricula and into this thread, is not. As yet that contention remains uncontested.

Simply put, that's why Godel's theorems are irrelevant to your mathematics. They're not a formal system.The consistency of a system is determined by its axioms, we've established that. The consistency/completeness limitations are determined by the structure of the axioms, Godel established that. What we are calling "the mathematics that we learn in school so that we can better understand reality" (all else being a recreational game in comparison, unless the results of the game later get implemented into the axioms we use to understand reality, as is often the case but is still part of my argument) involves both a choice of the axioms and a choice of their structure and what will constitute a valid logical equivalence to those axioms. Those are the choices we make, they are not formal because they are not made formally-- they are made by experience.

Once those informal choices have been made, not before, the result becomes a formal system that Godel's proofs apply to. Had Godel's proofs not applied to the particular choices we made, would there still be a "crisis"? Why or why not? Your answer there is crucial to understand what we are really talking about here.

I've worked with provably inconsistent systems before. That's what they do.I don't understand your point-- if you've worked with provably inconsistent, then this is just what I'm saying-- such systems can still be worked with, it's no "crisis".

Godel didn't show that there was potential for inconsistency. All he did was show that we couldn't prove consistency.Again, I'm well aware of that and nothing I've said suggests otherwise. We are talking about our suspicions-- you think it isn't inconsistent, I think it is, but what really matters is whether or not that distinction matters in practice. I don't think it matters at all in practice.

That's the very problem that he was working on. And a lot of other people too.Fine, then we agree: that's the real problem.

You want to know where it comes from? It's a straw horse. :)

You've built it up from your misunderstanding.
You have not made that point, you have merely assumed it. I asked a question, you did not answer it-- again, I asked why you think you have any right to expect a formal system that was expressly designed to approximate reality as closely as possible to be either complete or consistent. Hmmm?

Disinfo Agent
2007-Sep-12, 06:13 PM
I'm quite used to "exceptions to the rule" or "treat x differently under y circumstances" in english ...
and yes, even in labouring jobs (filling and bonding agents spring to mind) ...
but, I guess to learn that something like this even exists in maths ...
well, it's kind of like finding out that Santa Claus is
really uncle Charlie with a fake beard ... :sad:But, Cran, what we've been trying to show you is that they are not exceptions to the rule; quite the contrary, 1! = 0! = 1 are direct, necessary consequences of the rule that defines factorials. But you have to look at the right rule!

Let me try again. The factorial function was probably defined to you for the first time as

n! = n * (n-1) * ... * 1 (1),

that is the product of all positive integers smaller than or equal to n. For instance:

4! = 4 * 3 * 2 * 1 = 24
3! = 3 * 2 * 1 = 6
2! = 2 * 1 = 2

If you accept this definition, it seems clear that 1! should equal 1. Agreed?

Now, on to 0! To make this extension, we need to express the definition of the factorial function in a different way. The one above doesn't work any more, obviously. But (1) clearly says the same as the following:

n! = 1, for n=1
n! = n * (n-1)!, otherwise
Let's understand how this recursive definition works. It essentially says that:

1! = 1
2! = 2 * 1! = 2
3! = 3 * 2! = 3 * 2 = 6
4! = 4 * 3! = 4 * 6 = 24
etc.

The latter definition is clearly equivalent to the former, but the beauty of it is that it can be extended down to n=0. Indeed, to have:

1! = 1 * 0!

we must define 0! as 1. :)

hhEb09'1
2007-Sep-13, 07:03 AM
What we are calling "the mathematics that we learn in school so that we can better understand reality" (all else being a recreational game in comparison, unless the results of the game later get implemented into the axioms we use to understand reality, as is often the case but is still part of my argument)OK, stop that. As near as I can tell, "all else" is next to nothing. It is not a recreational game, although mathematicians like to have fun as much as physicists do.

Once those informal choices have been made, not before, the result becomes a formal system that Godel's proofs apply to. Had Godel's proofs not applied to the particular choices we made, would there still be a "crisis"? Why or why not? Your answer there is crucial to understand what we are really talking about here.Why is there a crisis?
I don't understand your point-- if you've worked with provably inconsistent, then this is just what I'm saying-- such systems can still be worked with, it's no "crisis".?? I've worked with papers I've graded F too :)

I meant that I'd worked with inconsistent systems, say, as exercises. A problem in class might be a formal system which had abstract rules like "All B are G" and "F and L imply M". Recreational stuff :)

Again, I'm well aware of that and nothing I've said suggests otherwise. We are talking about our suspicions-- you think it isn't inconsistent, I think it is, Not a very scientific attitude to have. You have zero evidence that it is inconsistent.
You have not made that point, you have merely assumed it. I asked a question, you did not answer it-- again, I asked why you think you have any right to expect a formal system that was expressly designed to approximate reality as closely as possible to be either complete or consistent. Hmmm?I DO NOT EXPECT IT TO BE COMPLETE! IT IS NOT COMPLETE!

OK, I'm better now.

One of the "shortcuts" to proving an abstract formal system to be consistent was to find a model. If we could translate the abstract terms into a physical model that satisfied all the axioms (B would be angles of an octogon and G would be 135º etc.), then the system was consistent--you could hold "the system" in your hands. You can't do that with an inconsistent system (because it would satisfy both "it exists" and "it doesn't exist").

Here's an easy inconsistent system that you can work with: "There is exactly two apples in my right hand" and "There is exactly three apples in my right hand". There is no way to make a model that satisfies both statements, whereas the apparently more complicated "There is exactly two apples in my right hand" and "There is exactly three apples in my left hand" and "There is exactly five apples in both my hands." can be modeled successfully. (note: we do not actually have to derive "it doesn't exist" in the first case, it's inconsistency should be apparent long before that)

As much as we try to model the formal system of mathematics into reality, it is too complicated, and there is no chance of proving consistency. But we have tried to map it into reality, and that is our attempt. Where could we possibly have failed? The apparent inconsistency between quantum mechanics and general relativity would seem to imply that if we were to try to add both as axioms to the formal system, we'd end up with an inconsistent system. Perhaps we've already pulled off such a trick, but we've tried to avoid that. If we've done so successfully (and there is absolutely no evidence that we have not) then we have a consistent system.

The reason I object to your disparaging remarks about recreational math is that the work is serious, and is just as much an attempt to understand reality, whatever that is.

But one doesn't have to be serious while doing serious work, TG.

Ken G
2007-Sep-13, 03:26 PM
OK, stop that. As near as I can tell, "all else" is next to nothing. It is not a recreational game, although mathematicians like to have fun as much as physicists do.So if a formal mathematical system has no demonstrated relevance to reality, you object to calling it a "recreational game"? All right, then we can call it a "formal system with no relevance to reality". Note that if any formal systems do have relevance, it is because they are linked to experience-- and hence are not formal in terms of their construction, only their application. This is my point.

I meant that I'd worked with inconsistent systems, say, as exercises. A problem in class might be a formal system which had abstract rules like "All B are G" and "F and L imply M". Recreational stuff Bingo, recreational because there's no relevance to reality. But if there is relevance-- mere coincidence you say? Does the meaning of a formal system rest on coincidence, or on an experience-guided heuristic approach to choosing its axioms (which is what I'm calling science because it is)?

Not a very scientific attitude to have. You have zero evidence that it is inconsistent. I DO NOT EXPECT IT TO BE COMPLETE! IT IS NOT COMPLETE!So you keep saying, but you have zero evidence to support that. And note, I never claimed it was inconsistent, I said "what if" it's consistent, would it be a "crisis" or just handled the way science handles inconsistencies?

One of the "shortcuts" to proving an abstract formal system to be consistent was to find a model. If we could translate the abstract terms into a physical model that satisfied all the axioms (B would be angles of an octogon and G would be 135º etc.), then the system was consistent--you could hold "the system" in your hands. This is precisely my point. By resting it on experience, you imagine you have established consistency. You have not-- your conclusion is actually an assumption. It is part of science, not math, because you cannot prove it, you just hope it.

You can't do that with an inconsistent system (because it would satisfy both "it exists" and "it doesn't exist").Unless reality is inconsistent, of course. And we have no reason to expect it isn't, I've shown that already. Recall that seeing 100 black squirrels is solid evidence that 97% of the squirrels in the set you sampled are black, and that's all you can say.

Here's an easy inconsistent system that you can work with: "There is exactly two apples in my right hand" and "There is exactly three apples in my right hand". Straw man. One inconsistent system does not have the properties of all inconsistent systems, you expressly chose one that experience rejects. For those with experience with quantum systems, a far better example could be chosen.
As much as we try to model the formal system of mathematics into reality, it is too complicated, and there is no chance of proving consistency. But we have tried to map it into reality, and that is our attempt. Where could we possibly have failed? We didn't fail, we did just fine, look at all it achieves. The sole mistake was thinking the attempt was possible, rather than just very useful.
If we've done so successfully (and there is absolutely no evidence that we have not) then we have a consistent system.I'm not sure what you mean, we know perfectly well we have not pulled it off-- the two systems are incompatible at the Planck scale, the axioms are contradictory.

The reason I object to your disparaging remarks about recreational math is that the work is serious, and is just as much an attempt to understand reality, whatever that is.It is purely your interpretation that makes them disparaging, I am merely pointing out that meaning comes from experience. That's just a fact.

cran
2007-Sep-17, 06:04 PM
Sorry Disinfo Agent, I've been sidetracked (or is that sidelined?) for a few days ...

But, Cran, what we've been trying to show you is that they are not exceptions to the rule; quite the contrary, 1! = 0! = 1 are direct, necessary consequences of the rule that defines factorials. But you have to look at the right rule!

Let me try again. The factorial function was probably defined to you for the first time as
n! = n * (n-1) * ... * 1 (1),that is the product of all positive integers smaller than or equal to n. For instance:
4! = 4 * 3 * 2 * 1 = 24
3! = 3 * 2 * 1 = 6
2! = 2 * 1 = 2
If you accept this definition, it seems clear that 1! should equal 1. Agreed? only if I can understand or ignore the bit which says 1*(1-1) = 1 ... if that's not an exception to a rule, I don't know what is ...

Now, on to 0! To make this extension, we need to express the definition of the factorial function in a different way. The one above doesn't work any more, obviously. But (1) clearly says the same as the following:
n! = 1, for n=1
n! = n * (n-1)!, otherwiseLet's understand how this recursive definition works. It essentially says that:
1! = 1
2! = 2 * 1! = 2
3! = 3 * 2! = 3 * 2 = 6
4! = 4 * 3! = 4 * 6 = 24
etc.
The latter definition is clearly equivalent to the former, but the beauty of it is that it can be extended down to n=0. Indeed, to have:

1! = 1 * 0!
we must define 0! as 1. :)But why do you/we need to?
why the need to extend it to n=0? ...
all I can see is that for any n > 1, the last two steps are redundant ...
just multiplying by 1, twice ...
the answer will not change from the point where (n - 1) = 1
which occurs quite naturally when n = 2 ...
so, I guess there's still something I'm not seeing ...

Disinfo Agent
2007-Sep-17, 06:57 PM
Sorry Disinfo Agent, I've been sidetracked (or is that sidelined?) for a few days ...
only if I can understand or ignore the bit which says 1*(1-1) = 1 ... if that's not an exception to a rule, I don't know what is ...
The factorial function was probably defined to you for the first time as

n! = n * (n-1) * ... * 1 (1),

that is the product of all positive integers smaller than or equal to n
I thought it was clear that with this definition the product must stop at 1. Everyone knows that, right?

But why do you/we need to?
why the need to extend it to n=0? ...Substitute n=1 in the second formula, and see what you get...

n! = n * (n-1)!

mugaliens
2007-Sep-18, 05:41 PM
yes, but does 0.999... = 2?

<ducks and runs>

On my computer it does.

<rounding error>

As for sanity, it has it's place in life. But it's much more fun to skirt the edge...

cran
2007-Sep-19, 04:11 PM
I thought it was clear that with this definition the product must stop at 1. Everyone knows that, right?

Substitute n=1 in the second formula, and see what you get...

n! = n * (n-1)!

I get 1! = 1*(1-1)! [= 1] ... or 1 = 0 [= 1] ...

0! = 0*(0-1)! [= 1] ... or 0 = 0 [= 1]

which leaves me right where I started ... :shifty:

but then, why stop there? ...
there's more fun to be had at < 0 ...
-1! = -1*(-1-1)! = 2
-2! = -2*(-2-1)! = 6

01101001
2007-Sep-19, 04:43 PM
-1! = -1*(-1-1)! = 2

Since you're exploring the domain of the factorial function, maybe this more grown-up version will intrigue you -- except the negative integers are still right out.

The gamma function (http://en.wikipedia.org/wiki/Gamma_function):

In mathematics, the Gamma function (represented by the capitalized Greek letter Γ) is an extension of the factorial function to real and complex numbers. For a complex number z with positive real part it is defined by

which can be extended to the rest of the complex plane, excepting the non-positive integers.

If z is a positive integer, then

Γ(z) = (z - 1)!

Note that the Gamma function is defined for not just natural numbers, but upon reals, positive and negative, and also upon complex numbers, yet excluding only those pesky non-positive integers.

That's a very sophisticated function, useful in a number of fields, and I do feel it would be a bit of a travesty to dumb it down, so that Γ(1) = 0! was not 1 but special-cased, in an otherwise perfectly smooth part of the function, to be undefined.

(And, if you haven't seen it, you might also not enjoy the older topic: Why is 0! = 1? (http://www.bautforum.com/off-topic-babbling/49231-why-0-1-a.html) Only went for about 50 articles.)

Disinfo Agent
2007-Sep-19, 04:49 PM
I get 1! = 1*(1-1)! [= 1] ... or 1 = 0 [= 1] ... Where did the factorial sign on the right go?