The Paradox of the Proof
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The Paradox of the Proof
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Re: The Paradox of the Proof
#2I actually read the whole article word-to-word. May be this personality of Shinichi Mochizuki appeals to me or may be i find maths more interesting than i admit. But i dont know which.
Also, i find it quite surprising that the proof for problems in domains as elementary as number theory, should have to be so complex, sort of baffles me. I hope i can rise up to the level to begin to understand this lingo or that someone brings it down to the level where i can find it interesting to read, like this article :D
Re: The Paradox of the Proof
#3I think this points out the necessity to develop better proof assistant systems [1], in particular for automated proof checking [2]. However, I have never interacted with such systems and thus don't know whether it will be possible to just feed Mochizuki's formidable constructions into it.
Re: The Paradox of the Proof
#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 coincidence, but not very interesting to mathematicians - that's what physicists do.
Re: The Paradox of the Proof
#5This 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 way of funding this sort of time-consuming journalism. I would love to be wrong about that.
There are interesting parallels and contrasts with Thomas Hales’s proof of the Kepler conjecture. In that case, as far as I know Hales did everything possible to help his colleagues understand the proof, but even so it was so long and involved that the referees declared themselves unable to be certain it was correct. Since then, he has been working on a formal machine-verifiable version of the proof under the banner of the Flyspeck Project: http://code.google.com/p/flyspeck/
Re: The Paradox of the Proof
#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 qualitative (but has a good definition), still, usually it seems that proofs are even harder for theorems defined like that
Re: The Paradox of the Proof
#7I think this points out the necessity to develop better proof assistant systems [1], in particular for automated proof checking [2]. However, I have never interacted with such systems and thus don't know whether it will be possible to just feed Mochizuki's formidable constructions into it. [1] http://en.wikipedia.org/wiki/Proof_assistant [2] http://en.wikipedia.org/wiki/Automated_proof_checking
I would not expect so. Mechanized proof systems tend to require a lot more detail than one would put into a proof meant for humans to read. There's been a lot of work in automating part of the generation of a proof, but that still requires a human to look at what the automation came up with and intervene to guide it in the right direction.
Re: The Paradox of the Proof
#8Don'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 field, one long and obfuscated paper instead of building toward the result incrementally, etc. Using the title "Inter-universal geometer" instead of calling yourself a mathematician is also strange.
The natural way for a mathematician to behave after seemingly solving an important problem is exactly the opposite of what this guy is doing. And the only rational reason to do so is if there is actually something wrong about the work.
I lack the know-how to arrive at my own opinion by reading the paper, but this situation is definitely fishy. And it's not like a bona-fide scientist losing it and becoming an pseudo-scientist obsessed about a topic is completely unheard of.
Re: The Paradox of the Proof
#9[deleted]
Re: The Paradox of the Proof
#10If 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 pages of complex reasoning, on his own, however smart he is I'd say there's a high chance he's missed something.
Humans need to collaborate in areas of high complexity. With a single brain, there's too high a chance of bias hiding the problems.
(Repost of my previous comment https://news.ycombinator.com/item?id=4829806)