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The Paradox of the Proof

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Re: The Paradox of the Proof

#11

Post by crntaylor »

brazzy wrote: post #4

> For centuries, mathematicians have strived towards a single goal: to understand how the universe works, and describe it. Um, no. Mathematics itself has absolutely nothing to do with "describing the universe". Mathematics is purely abstract. It has no inherent relationship whatsoever with reality. That certain mathematical constructs can be used to model certain aspects of the real world is basically a lucky coincid…

I think there's a reasonable argument to be made that mathematical constructs are part of the universe.

David Deutsch has a method for ascertaining whether something can can be said to exist or not - ask whether it "kicks back" when you interact with it, in the sense that simulating the response of the thing you're considering in a totally convincing way would involve an effort as large as building a new universe for that thing to exist in.

Rocks "exist" because, if you wanted to build a totally convincing simulation of a rock, you'd need to include all of modern physics as we currently understand it.

Other minds "exist" because simulating them convincingly would basically require you to construct a true articificial intelligence (strong AI).

Mathematics "exists" because giving someone the genuine experience of doing mathematics when they really weren't would involve a simulation of almost unfathomable complexity.

There is a very real truth in the statement that '1 + 1 = 2', or the statement that there are arbitrarily long arithmetic progressions of prime numbers, or any number of other mathematical results. The world of mathematics is a very real part of the universe, that "kicks back" by constantly surprising us when we think about it.

So I don't think the article, which is very well written and researched, deserves the middlebrow "Um, no" scorn that you treated it to.

Re: The Paradox of the Proof

#12

Post by claudius »

krajzeg wrote: post #8

Don't get me wrong, but from what I've seen of the world of science, the only people who exhibit this type of behaviour are frauds and delusional pseudoscientists. All the warning signs of pseudoscience are there: "I couldn't possibly explain it" as a response to lecture invitations, working on something for extended periods of time without sharing results, using lots of obscure terminology which is not standard for…

Well, the ‘warning signs’ are definitely there, but then it doesn’t appear entirely unlikely that he simply invested so much time to build up an entirely new field that it will be impossible to explain it in a few lectures. And given that Perelmann apparently still lives with his parents and refused this fancy price, I don’t know exactly what to expect from a successful mathematician.

So, yes, something is fishy, but so was the weird idea of curved spacetime.

Re: The Paradox of the Proof

#13

Post by claudius »

raverbashing wrote: post #6

(ok, rant ahead) I think mathematicians have a weird way of thinking about problems. First-order logic for example: http://en.wikipedia.org/wiki/First-order_logic It's quirky to think, for example, on the natural numbers that 'exists an operation + and a null element under that operation 0' This is "very understandable" by humans, but very difficult to compute. As such as this conjecture is stated in a way that looks…

If you want general results, you need to abstract away from standard preschool algebra and build up models that then let you get said general results. That might appear ‘weird’, but I don’t see how else you could get even to calculus, not to mention, for example, the algebra driving a sensible description of quantum mechanics or differential geometry.

You called it a rant, but could you maybe still make a suggestion on how to better think about problems?

Re: The Paradox of the Proof

#14

Post by pfortuny »

brazzy wrote: post #4

> For centuries, mathematicians have strived towards a single goal: to understand how the universe works, and describe it. Um, no. Mathematics itself has absolutely nothing to do with "describing the universe". Mathematics is purely abstract. It has no inherent relationship whatsoever with reality. That certain mathematical constructs can be used to model certain aspects of the real world is basically a lucky coincid…

You are wrong. It is reality what drives mathematicians to work, not a pure abstraction. It is because of reality that maths is interesting. It is because of Geo-metry that algebra is so relevant.

Not to speak of Calculus...

Re: The Paradox of the Proof

#15

Post by shocks »

raverbashing wrote: post #6

(ok, rant ahead) I think mathematicians have a weird way of thinking about problems. First-order logic for example: http://en.wikipedia.org/wiki/First-order_logic It's quirky to think, for example, on the natural numbers that 'exists an operation + and a null element under that operation 0' This is "very understandable" by humans, but very difficult to compute. As such as this conjecture is stated in a way that looks…

First-order Logic is a fascinatingly deep and interesting topic. The level of abstraction is a requirement - it makes it far easier to work with, prove, and understand.

Source: 4th year mathematics student.

Re: The Paradox of the Proof

#16

Post by vilda »

It's a good tradition in math that you as the author have to convince others that your theorem is correct. If you cannot convince other fellow colleagues then you can't claim you have a proof.

That's why a lot of mathematicians are sceptical against computationally constructed proofs such as state space exploration. Or, at least, they don't like the taste of it :)

Re: The Paradox of the Proof

#17

Post by freefrag »

pfortuny wrote: post #14
brazzy wrote: post #4

> For centuries, mathematicians have strived towards a single goal: to understand how the universe works, and describe it. Um, no. Mathematics itself has absolutely nothing to do with "describing the universe". Mathematics is purely abstract. It has no inherent relationship whatsoever with reality. That certain mathematical constructs can be used to model certain aspects of the real world is basically a lucky coincid…

You are wrong. It is reality what drives mathematicians to work, not a pure abstraction. It is because of reality that maths is interesting. It is because of Geo-metry that algebra is so relevant. Not to speak of Calculus...

He's not wrong.

To many mathematicians reality is largely irrelevant. For example G. H. Hardy, an extremely prominent British Mathematician, believed that true Mathematics is an art form and is not useful. He dismissed applications of mathematics as dull and boring.

And I can relate to him. It's incredible how complex, beautiful structures arise from a couple of simple axioms. It doesn't matter if what you study will be relevant or not, what matters is that it's fun and stimulating to explore.

Re: The Paradox of the Proof

#18

Post by sigmavirus24 »

krajzeg wrote: post #8

Don't get me wrong, but from what I've seen of the world of science, the only people who exhibit this type of behaviour are frauds and delusional pseudoscientists. All the warning signs of pseudoscience are there: "I couldn't possibly explain it" as a response to lecture invitations, working on something for extended periods of time without sharing results, using lots of obscure terminology which is not standard for…

Also making claims about typical mathematicians and attempting to apply them to those already determined to be atypical is a bit dodgy in and of itself. Perelman was not your typical mathematician and he behaved atypically as well. I'm not certain but I don't think he travelled and lectured on his proof of the Poincaré Conjecture. He even turned down a sizable award size. Just because an atypical scientist behaves atypical doesn't make their work any less valid.

I agree it seems odd but even in my limited exploration into Pure Mathematics I have seen alternative proofs made with ideas not native to the field. Why must that necessarily make this proof invalid?

Re: The Paradox of the Proof

#19

Post by krajzeg »

sigmavirus24 wrote: post #18
krajzeg wrote: post #8

Don't get me wrong, but from what I've seen of the world of science, the only people who exhibit this type of behaviour are frauds and delusional pseudoscientists. All the warning signs of pseudoscience are there: "I couldn't possibly explain it" as a response to lecture invitations, working on something for extended periods of time without sharing results, using lots of obscure terminology which is not standard for…

Also making claims about typical mathematicians and attempting to apply them to those already determined to be atypical is a bit dodgy in and of itself. Perelman was not your typical mathematician and he behaved atypically as well. I'm not certain but I don't think he travelled and lectured on his proof of the Poincaré Conjecture. He even turned down a sizable award size. Just because an atypical scientist behaves at…

I'm far from claiming that the proof is invalid based on my feelings on the subject.

My only issue is that in the article, and in other writing on the subject, nobody is even contemplating the possibility that Mochizuki's work might be unreadable for reasons other than it being too brilliant to grasp.

I wanted to present an alternative possibility which seems to be disregarded at the moment in favor of the attractive "eccentric genius" narrative.

On the topic of Perelman - he did reject the Fields Medal. However, he did give a series of talks at MIT, Princeton and other places a year after publishing his proof.

Re: The Paradox of the Proof

#20

Post by raverbashing »

claudius wrote: post #13
raverbashing wrote: post #6

(ok, rant ahead) I think mathematicians have a weird way of thinking about problems. First-order logic for example: http://en.wikipedia.org/wiki/First-order_logic It's quirky to think, for example, on the natural numbers that 'exists an operation + and a null element under that operation 0' This is "very understandable" by humans, but very difficult to compute. As such as this conjecture is stated in a way that looks…

If you want general results, you need to abstract away from standard preschool algebra and build up models that then let you get said general results. That might appear ‘weird’, but I don’t see how else you could get even to calculus, not to mention, for example, the algebra driving a sensible description of quantum mechanics or differential geometry. You called it a rant, but could you maybe still make a suggestion…

" you need to abstract away from standard preschool algebra and build up models that then let you get said general results"

Yes, of course.

" but could you maybe still make a suggestion on how to better think about problems?"

And that's what I meant. Thinking about problems in a different way (but still provable, and still working in a similar way)

For example, for Peano arithmetic you have that equality is symmetric (and transitive)

Now, there are several ways to explain that, and it's usually explained more or less by "for all X and all Y, if X = Y then Y = X"

Now, it would maybe be interesting to have a 'different explanation' that is as powerful as first order logic but works differently (and maybe easier to compute)

For example, it may be possible to write Peano arithmetic as a grammar (so zero would be ' ', one would be I, two would be II, etc)

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