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The Paradox of the Proof

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Re: The Paradox of the Proof

#51

Post by voyou »

spitx wrote: post #26

"For centuries, mathematicians have strived towards a single goal: to understand how the universe works, and describe it." I cannot resist engaging that loaded premise. It is a thought that I've wanted - for some time - for someone to skilfully dissect and lay bare for easy comprehension. The closest I've come to seeing it, is the following - a concise and accessible yet well-rounded explanation of the relevance (or…

It's cool that "Alex Knapp, a science writer at Forbes" has solved one of the major issues in the philosophy of mathematics by, erm, just asserting that Platonism is false. Of course, maths is a tool, but that doesn't mean that it is just a tool, and Knapp doesn't provide any arguments for that conclusion. I can use a rock as a tool, but that doesn't mean the rock has no existence independent of my mind. In fact, the only reason the rock works as a tool is because it has an independent existence (I can hit things with it); the same might be true of maths.

Now, there are arguments for positions like Knapp's, but they're more complicated and contested than Knapp's facile post suggests. See, for instance, this Stanford Encyclopedia of Philosophy article: http://plato.stanford.edu/entries/fictionalism-mathematics/

Re: The Paradox of the Proof

#52

Post by calinet6 »

msilenus wrote: post #49
calinet6 wrote: post #33

Earlier quoted context omitted.

I wonder what would happen if, say, I allocated $10 per month to compensating articles and journalists I enjoyed. And then it was distributed evenly by time I spent on the page, every single day. So if I spent 30 minutes reading this article, 10 minutes reading another, 2 minutes on 10 others, this author would get $0.16 ($0.33 per day, times (30 mins, out of 60 total mins of time = 0.5)). That's really not that much…

There is a micro-payment service called Flattr [1] which does something like this. It was founded by Peter Sunde, one of the pirate bay founders. It's actually quite successfull in the German web community especially for podcasters. There are German podcasters which earn about 1000 Euro per month via Flattr. [1] https://flattr.com

Cool, I was aware of it, I was just thinking about making it automatic. :) Easy extension to flattr though. Excellent, thanks.

Re: The Paradox of the Proof

#53

Post by Rexxar »

brazzy wrote: post #43
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…

> the only people who exhibit this type of behaviour are frauds and delusional pseudoscientists. But those usually produce work that is very obviously nonsensical, and want attention rather than avoiding it. Mochizuki has earned the privilege of having his work evaluated on its own merits, and so far nobody has found any obvious flaws. > And the only rational reason to do so is if there is actually something wrong ab…

> Can you give some examples?

Isaac Newton : http://en.wikipedia.org/wiki/Isaac_Newton%27s_occult_studies

Re: The Paradox of the Proof

#54

Post by pseut »

brazzy wrote: post #40
pseut wrote: post #30

Earlier quoted context omitted.

The statement's a little hyperbolic, but I think you're misunderstanding the author's point. The point is that math is aimed at trying to build understanding in contrast to producing logically airtight proofs. The proofs are important to make sure that our understanding is right, but there's more to it than that. I agree that's not the literal sentence you quoted, but it's definitely the theme of the article: accordi…

> The point is that math is aimed at trying to build understanding in contrast to producing logically airtight proofs. The proofs are important to make sure that our understanding is right, but there's more to it than that. Understanding is necessary to be certain the proof is correct. After that is ensured, many mathematicians will happily use the proof and base their work on it without understanding it. > Mochizuki…

From TFA: "Even an incorrect proof is better than no proof, because if the ideas are novel, they may still be useful for other problems, or inspire another mathematician to figure out the right answer."

edit: I can see taking issue with the word, "universe" in the original quote. I have trouble seeing how the word "understand" is debatable.

Re: The Paradox of the Proof

#55

Post by peterwoo »

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…

Here's an amusing counterargument:

http://pauli.uni-muenster.de/~munsteg/arnold.html

Re: The Paradox of the Proof

#56

Post by pseut »

codeulike wrote: post #44
pseut wrote: post #32

Earlier quoted context omitted.

It's a shame that you've disregarded all of the replies to your original comment. tl;dr there are degrees of bugs and many are easily fixed. edit: more constructively, Imagine that you're working on the NYT crossword and I come along and point out that 45 across is wrong and tell you what the answer should be. Do you then throw away the rest of your work? No, you fix the part that's wrong and then check the rest of t…

I don't really accept the notion that inconsistencies in a giant mathematical proof will always show themselves. That does happen sometimes, but if you're breaking new ground (as Mochizuki seems to be) its much more likely that things will seem consistent to you, but are actually inconsistent because you made a mistake somewhere.

Right, that's why other people are trying to verify the proof and aren't just taking it on faith. They're almost certainly going to find errors, the question is whether or not those errors are easily fixed, difficult to fix, or fundamentally impossible to fix.

Re: The Paradox of the Proof

#57

Post by brazzy »

peterwoo wrote: post #55
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…

Here's an amusing counterargument: http://pauli.uni-muenster.de/~munsteg/arnold.html

That's not an argument, that's an almost incoherent rant.

Re: The Paradox of the Proof

#58

Post by pjungwir »

This situation reminds me of one of my favorite stories about Plato [1]. He gave a lecture in Athens on The Good, and lots of people showed up, but the whole talk was about mathematics! I hope some folks somewhere will take the time to understand Mochizuki's world. The part about Yamashita studying with him privately is encouraging.

[1] described in this paper: http://www.jstor.org/discover/10.2307/4182081?uid=3739856&#3...

Re: The Paradox of the Proof

#59

Post by lmm »

spitx wrote: post #26

"For centuries, mathematicians have strived towards a single goal: to understand how the universe works, and describe it." I cannot resist engaging that loaded premise. It is a thought that I've wanted - for some time - for someone to skilfully dissect and lay bare for easy comprehension. The closest I've come to seeing it, is the following - a concise and accessible yet well-rounded explanation of the relevance (or…

I feel like mathematics applies too closely to physics for it to be just a tossing of darts. With a bit of familiarity complex numbers are the "obviously best" numbers, the simplest (sort of) and most mathematically interesting; that they are so central to quantum mechanics is too much for coincidence. Then there's the fact that mathematical truths really are true in models to which their axioms apply (as far as we can measure), unlike any other rule of inference, so the association between the symbols we manipulate is not arbitrary.

Many mathematical areas initially seem to be blind alleys - Hardy rejoiced in the idea that his work on primes would never be a weapon of war. But there's an uncanny tendency for connections to emerge, and theorems in one field imply something new in a different one. Again, it seems too much for mere coincidence.

Moving from what's widely supported to my own views, my suspicion is that reality itself is a mathematical phenomenon (at one time I was expecting to work on the Causal Sets hypothesis, which attempts to explain the universe as arising from certain partially ordered sets).

Re: The Paradox of the Proof

#60

Post by gwern »

msilenus wrote: post #49
calinet6 wrote: post #33

Earlier quoted context omitted.

I wonder what would happen if, say, I allocated $10 per month to compensating articles and journalists I enjoyed. And then it was distributed evenly by time I spent on the page, every single day. So if I spent 30 minutes reading this article, 10 minutes reading another, 2 minutes on 10 others, this author would get $0.16 ($0.33 per day, times (30 mins, out of 60 total mins of time = 0.5)). That's really not that much…

There is a micro-payment service called Flattr [1] which does something like this. It was founded by Peter Sunde, one of the pirate bay founders. It's actually quite successfull in the German web community especially for podcasters. There are German podcasters which earn about 1000 Euro per month via Flattr. [1] https://flattr.com

Someone actually mentioned Flattr to me today as a way to take donations for my own writings; I've always begged off because I've never wanted to take the time to set it up and clutter my site (more), but now here I see Flattr brought up again... Is there any way of even roughly estimating how much Flattr might bring in?
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