erikp responds to me:
quote:
Look, we can go on and on and about Gdel, and insist that nobody understands Gdel, except for you
No, we can't. In order to discuss who understands Godel, you need to know enough about set theory to be able to understand that you don't understand it. It's the practical aspect of the general claim that the more you know, the more you realize you don't know much at all.
I'm not saying I understand Godel. I'm saying I understand more than you. Hint: You need to stop reading Wikipedia and start reading the actual source. There's a reason I mentioned the Continuum Hypothesis. It's pretty much the foundational reason why we have the Incompleteness Theorems in the first place. Godel didn't come up with these things out of the blue. It is because of the work of the set theorists such as Russell who were working to do what Hilbert was envisioning: Create a mechanical mathematics. If you could define your symbology and rules, you could conceivably create a machine that would spit out proofs of theorems.
But the work of Russell and his paradox found that there were some issues with that claim. Self-reference is problematic for a mechanical process. The question of the size of the continuum has long been a puzzle and Godel's work showing that its assumption as true doesn't lead to a contradiction seemed to be a good thing...until Cohen came along and showed that its assumption as false doesn't lead to a contradiction, either. Godel had developed the Incompleteness Theorems before this, but it had been thought that such undecidable questions would be esoteric ones. That something as basic and simple as "How big is the size of the Real numbers?" would be undecidable was a big blow.
This is what I'm saying when I know more about it than you. As a trained mathematician, I know the history. I've had to do some of the foundational aspects regarding the problem. I don't claim to be the world's greatest mathematician. But I do claim to be a better mathematician than you.
Once again, I ask you the question you refused to answer back in
Message 30:
What does "incomplete" mean?
quote:
(Wiki):
You need to stop reading Wikipedia and go to original sources. Stanley Jaki is a physicist, not a mathematician. I do not expect him to understand what incompleteness means. Incompleteness is a trait of axiomatic set theories. At what point does physics become an axiomatic set theory?
Jaki isn't here to answer that question. You are. If you have writings of Jaki that show him demonstrating how physics is an axiomatic set theory, then by all means bring them forward.
Do you?
Since you seem to love Wikipedia, let's see what it has to say about Jaki:
Since any 'theory of everything' will certainly be consistent, it must be either incomplete or unable to prove basic facts about the integers.
That's great. When would a physical theory become an axiomatic set theory? The universe is inherently mathematical in nature and physics is applied mathematics, but that doesn't mean that physics and mathematics are interchangeable.
And sure enough, going beyond Wikipedia and reading the actual sources, you find that Jaki and Hawking weren't actually saying incompleteness applies directly. They were using it as a metaphor:
What we need, is a formulation of M theory, that takes account of the black hole information limit, but then our experience with supergravity and string theory, and the analogy of Goedels theorem, suggest that even this formulation, will be imcomplete.
-- Stephen Hawking, "Gdel and the end of physics"
Hmm..."Analogy." What might that mean? From reading the rest of it, the claim is that Hawking thinks that physics cannot be reduced to a finite series of statements. This is analogous to the Incompleteness Theorems, but it is not the same. If Hawking is right and physics is not reducible to a finite description, it is not directly because of incompleteness. It is, instead, analogous to incompleteness.
Now, Jaki writes:
Therefore any theory of physics, which contained more than a trivial form of mathematics, was subject to the restriction of Gdel's theorem.
-- Stanley Jaki, "A Late Awakening to Gdel in Physics"
Now this, unfortunately, is an error. Physics doesn't "contain" mathematics. Physics uses mathematics and is applied mathematics, but it is not simply mathematics. That is, the objects of physics behave mathematically, but they are not the objects of mathematics. Therefore, we cannot apply the traits of mathematics to physics out of hand.
Trivial example: Numbers are infinitely divisible. Objects in physics are not. One of the wonderful tricks of math is that you can take a sphere and by dividing it into parts, you can reassemble them into two complete spheres of the exact same size as the first, no holes, no gaps. But this requires you to be able to divide it down into infinitely many infinitesimal pieces.
You can't do that with an actual object.
So given that physics is a limited subset of mathematics, why would we expect it to have all of the problems of mathematics?
Physics is not axiomatic set theory. Therefore, why should we expect incompleteness to apply? Why do we expect physics to depend upon those statements that in current mathematics are undecidable? The size of the continuum is undecidable. Is there something in physics that requires the size of the continuum to be known? From what we can tell, reality isn't continuous.
Jaki continues:
Then there is a book with the title
The End of Science by John Horgan, a senior member on the staff of Scientific American. The book begins with the declaration that "Gdel's theorem denies us the possibility of constructing a complete, consistent description of physical reality."11 This, is, of course, not what the theorem denies.
-- Stanley Jaki, "A Late Awakening to Gdel in Physics"
But the thing is, that's exactly what you're claiming: That the Incompleteness Theorems deny the possibility of a complete, consistent description of physical reality.
But wait, there's more:
Gdel's theorem does not mean that physicists cannot come up with a theory of everything or TOE in short.
-- Stanley Jaki, "A Late Awakening to Gdel in Physics"
So if your own source denies your claim, where does that leave your claim?
Of course, Jaki then screws up his own statement:
But in terms of Gdel's theorem such a theory cannot be taken for something which is necessarily true.
-- Stanley Jaki, "A Late Awakening to Gdel in Physics"
Why not? The condensed version seems to be that because physics uses simple arithmetic, that necessarily means that all the incompleteness of describing simple arithmetic applies to physics.
But that isn't true on its face. If all you need out of simple arithmetic is to be able to say that 1 + 1 = 2, then what is the problem? Since you seem to love the argument from authority so much:
It’s indeed the case that if the laws of physics are formulated in a formal system S which includes the concepts and axioms of arithmetic as well as physical notions such as time, space, mass, charge, velocity, etc., and if S is consistent then there are propositions of higher arithmetic which are undecidable by S. But this tells us nothing about the specifically physical laws encapsulated in S, which could conceivably be complete as such.
-- Solomon Feferman, "The nature and significance of Gdel’s incompleteness theorems"
And then there's this:
Nothing in the incompleteness theorem excludes the possibility of our producing a complete theory of stars, ghosts and cats, all rolled into one, as long as what we say about stars, ghosts and cats can’t be interpreted as statements about the natural numbers.
-- Torkel Franzén,
Gdel's Theorem: An Incomplete Guide to its Use and Abuse
Hint: Franzén takes on Hawking to show that his attempt to model numbers in physics as blocks of wood is inappropriate.
quote:
The fact that you need to introduce a "perfect theory",
Incorrect. Do you not understand how indirect proof works? You start by assuming to be true that which you are trying to show to be false. By leading yourself to a contradiction, you show that what you assumed to be true is actually false.
Classic example: Is there a largest prime number? The answer is no and the common way to do so is by assuming that there is one. It doesn't actually exist, but you assume it to exist in order to show that its existence leads to a contradiction. That's the entire point.
Assume there is a perfect theory. Your "measure of falsifiability" would declare it to be false. But that's a contradiction. Therefore, your "measure of falsifiability" is false.
quote:
which inevitably takes all possible factors and influences into account to explain phenomena, and therefore amounts to the "theory of everything"
Incorrect. Presburger arithmetic is a perfect theory. It is complete, consistent, and decidable. It takes into account all possible factors and influences to explain phenomena.
But it is hardly a "theory of everything."
Note, your "measure of falsifiability" would declare Presburger arithmetic to be false. But it is true. Therefore, your "measure of falsifiability" is false.
quote:
There is no need to distinguish between the "perfect theory" and the "other theory", because your perfect theory is an impossibility.
Presburger arithmetic doesn't exist? Then how did we manage to define it?
quote:
Furthermore, the idea that Gdel does not apply to gravity, is refuted in "The Relevance of Physics".
Where? Chapter, page, and full quote in complete context, please.
Remember, your own source denies you:
Gdel's theorem does not mean that physicists cannot come up with a theory of everything or TOE in short.
-- Stanley Jaki, "A Late Awakening to Gdel in Physics"
What happens to your claim now?
quote:
Now first explain why in addition to Stephen Hawking being wrong, Stanley Jaki is also wrong
I already did: They are physicists, not mathematicians. We cannot expect them to understand the details of set theory. Jaki makes that exact comment, himself:
The paper ["Formally undecidable propositions of Principia Mathematica and related Systems I."] could not be easy reading for most physicists, or even for most mathematicians for that matter.
As for Hawking, he doesn't say what you claim he says. He simply uses the incompleteness theorems as a metaphor for the problems facing physics.
At any rate, they are not here. You are. It would help if you would stop using Wikipedia and started using original source material.
I'm still waiting for you to answer the question I asked back in
Message 30:
What does "incomplete" mean?
Edited by Rrhain, : Added references of responses to Jaki's claim and Hawking's misunderstanding.
Rrhain
Thank you for your submission to
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