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The Paradox of the Proof

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Re: The Paradox of the Proof

#31

Post by stephencanon »

freefrag wrote: post #17
pfortuny wrote: post #14

Earlier quoted context omitted.

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…

A (somewhat tongue-in-cheek) question to ponder: If Hardy were alive today, would he turn his back on Number Theory as too useful, and study something more theoretical like the existence of long-time solutions to Navier-Stokes?

Re: The Paradox of the Proof

#32

Post by pseut »

codeulike wrote: post #10

If a programmer locked himself away for 14 years and then emerged and announced he'd written a completely bug free OS, there would be skepticism. Code needs to be battle tested by other people to find the bugs. Mathematics is the same, to an extent; one guy working alone for 14 years is likely to have missed ideas and perspectives that could illuminate flaws in his reasoning. Maths bugs. If he's produced hundreds of…

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 the puzzle to figure out the scope of the error.

Re: The Paradox of the Proof

#33

Post by calinet6 »

robinhouston wrote: post #5

This is a great article. The writer has obviously spent a lot of time speaking to Mochizuki’s colleagues, and has explained the whole strange situation in a way that is layman-friendly without being wrong. It’s interesting that it’s part of an experiment in donation-funded journalism: I was sufficiently impressed that I donated a few dollars, but I fear that is not likely to be a common enough response to be a viable…

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. With 10,000 people all following this same path, the author would "only" get $1600. Readership is likely to drop off exponentially after the initial publicity. Publishing two articles of similar quality per month would get the author a nice living wage; not too bad, but only if they can reach an audience of 10,000 interested readers who are in on this compensation system. Slim chance.

Perhaps it works as a supplementary reward system. Another question: does this incentivize the right things? Will such metrics lead to longer articles which aren't necessarily interesting to read, but just take a long time? Does it disincentivize quality in any way?

More questions: how much would people be willing to contribute—$10 per month? $30 per month? How is it distributed—manually or automatically, by time or by rating? Is the small bit better than nothing? Does this incentivize people to a) produce good content, or b) pay for its consumption? Would people voluntarily buy into this system, or does it need to be a restrictive thing where only contributors can read articles?

Just thinking.

Re: The Paradox of the Proof

#34

Post by pseut »

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 counterpoint is that the guy has a serious track record before now (according the the article). If he didn't, no one would pay any attention to these claims. Since he does, you have to think that he might have proven it.

Re: The Paradox of the Proof

#36

Post by sanderjd »

robinhouston wrote: post #5

This is a great article. The writer has obviously spent a lot of time speaking to Mochizuki’s colleagues, and has explained the whole strange situation in a way that is layman-friendly without being wrong. It’s interesting that it’s part of an experiment in donation-funded journalism: I was sufficiently impressed that I donated a few dollars, but I fear that is not likely to be a common enough response to be a viable…

This is exactly the sort of thing that I wish I could use something like bitcoin for. I groaned when I clicked "Pay Now" and saw PayPal. It doubles my time-investment in the article and significantly drags down my average enjoyment per minute.

Re: The Paradox of the Proof

#37

Post by brazzy »

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…

> What other conclusions can be drawn if one were to find this explanation appealing?

That even if we find a Grand Unified Theory of Everything that fully explains the physical world, we'll never run out of things to discover.

> Are there other explanations of the relationship between math and our reality, that you've found appealing?

None that makes sense, given what I know.

> Is there a consensus among mathematicians as to what higher-order math, essentially is in pursuit of or should be in pursuit of?

The potential for more math. Mathematicians look for theorems and constructs that are "interesting", which basically means that they are neither simple nor random. Prime numbers would be uninteresting if they were regular, but also if they were random. In fact they are neither, there's a multitude of patterns - the Wikipedia category "Classes of prime numbers" has 69 pages in it!

> Is it an exercise of random shooting of darts hoping that some "mathematical truth" sticks and corresponds to some observable phenomenon?

Yes, but some people don't care where the darts go, while others care about observable phenomena and notice any darts lying near them and start asking around who shot those and what they were doing...

Re: The Paradox of the Proof

#38

Post by btilly »

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…

The book that you want to read is The Mathematical Experience. Trust me on this. There is no book that I know which is as good at conveying the feeling or breadth of doing mathematics, with real examples, at a level that is both largely accessible to high school students, and enlightening for math professors.

Re: The Paradox of the Proof

#39

Post by raverbashing »

claudius wrote: post #24
raverbashing wrote: post #20

Earlier quoted context omitted.

" 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 after you spent ten years reformulating basic maths in your fancy new logic, people will look at your papers and won’t understand a word, which appears to be more or less what happened to our poor protagonist in the OP. Furthermore, I have to admit I don’t see the immediate advantage such a reconstruction would bring with it.

Well, the OPs reinvention looks like something more high level

Well, there may not be immediate advantages, but in math you never know. There are several hard problems in one domain that are trivial in another domain, for example.

An example from physics: http://en.wikipedia.org/wiki/Hamiltonian_mechanics

Re: The Paradox of the Proof

#40

Post by brazzy »

pseut wrote: post #30
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…

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's shirking his responsibility by not explaining how to understand the result, regardless of whether the proof is correct or incorrect.

The main problem caused by his "shirking" is that it leaves the correctness of the proof uncertain and trying to fully understand it is a lot of work that could turn out to be wasted. People want to confirm the proof, and Mochizuki could make it much easier and less costly for them, but for some reason declines to do so.

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