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The Paradox of the Proof

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Re: The Paradox of the Proof

#41

Post by spitx »

brazzy wrote: post #37
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 mathematicia…

>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.

That's odd. In my experience, these detours in these discussions come as a result of how vague this whole terrain is. There is not even a smidgen of agreement on even the basic of agreed things.

Can't mathematicians and theorists not agree on what areas of math are most helpful and what areas least helpful in arriving at a Unified Theory?

This leads up to the other question of consensus among mathematicians.

>The potential for more math. Mathematicians look for theorems and constructs that are "interesting", which basically means that they are neither simple nor random.

Isn't there a faction of math people who strive towards a defined, non-abstract direction as opposed to fostering a laissez-faire approach to mathematics scholarship that naturally creates more math so that their area of expertise gets recognition and not to mention substantial purses of money?

Is math so poorly funded that all mathematicians lead a hand to mouth existence and therefore collectively as some sort of cabal, have to resort to these self-preservation tactics?

Come on. Really?

Re: The Paradox of the Proof

#42

Post by pfortuny »

30thElement wrote: post #22
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...

As someone with a degree in applied math, the pure abstract is more interesting than the applied. Applied math is like building really amazing and intricate sand castles on the beach. Pure math is like building the same sand castle, but in the sky and it's kept aloft purely by how beautiful it is, freed from constraints like "touches the ground" and "can support itself under gravity". A lot of my friends feel the sam…

Look, I am an Algebraic Geometer and have done Schemes and whatnot. It is REALITY and this has nothing to do with "applied" or "pure". The fact that it is abstact has nothing to do with its being unreal.

Just to clarify: I am an expert too.

Re: The Paradox of the Proof

#43

Post by brazzy »

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 about the work.

Who says the reason has to be rational? Maybe he just has developed a bad case of stage fright?

> And it's not like a bona-fide scientist losing it and becoming an pseudo-scientist obsessed about a topic is completely unheard of.

Can you give some examples?

Re: The Paradox of the Proof

#44

Post by codeulike »

pseut wrote: post #32
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 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.

Re: The Paradox of the Proof

#45

Post by ajuc »

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…

There was great discussion of this in Anathem by Neal Stephenson. It's sci-fi book, has story etc, but I mostly liked it because of such discussions between characters presented with invented terminology (so you don't skip over them out of familiarity).

Re: The Paradox of the Proof

#46

Post by nijk »

calinet6 wrote: post #33
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…

Flattr, etc

Re: The Paradox of the Proof

#47

Post by nijk »

claudius wrote: post #12
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…

It doesn't matter where he lives. It matters what he writes.

Re: The Paradox of the Proof

#48

Post by brazzy »

spitx wrote: post #41
brazzy wrote: post #37

Earlier quoted context omitted.

> 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 mathematicia…

>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. That's odd. In my experience, these detours in these discussions come as a result of how vague this whole terrain is. There is not even a smidgen of agreement on even the basic of agreed things. Can't mathematicians and theorists not agree on what areas of math are most helpful…

> Can't mathematicians and theorists not agree on what areas of math are most helpful and what areas least helpful in arriving at a Unified Theory?

Not fully, and with good reason (see below).

But actually, my point was that the existence of mathematical constructs that do not correspond to any physical reality means that there will be math to do when (and if) all physics has been done.

> Isn't there a faction of math people who strive towards a defined, non-abstract direction as opposed to fostering a laissez-faire approach to mathematics scholarship that naturally creates more math so that their area of expertise gets recognition and not to mention substantial purses of money?

Yes and no. There is the branch of applied math, and I'm sure they get funding more easily.

But there are also mathematicians (cited several times in the comments here) who see math as art and want to do it for its own sake.

And it has happened quite often that these "pure" mathematicians came up with enirely new stuff that only afterwards (and without anyone foreseeing it) turned out to be useful in modelling physical processes. Even among mathematicians you sometimes find that you can prove something in one field by using constructs and theorems from an entirely different field that nobody thought was in any way related. I believe Wiles' proof of Fermat's Last Theorem was like that.

Re: The Paradox of the Proof

#49

Post by msilenus »

calinet6 wrote: post #33
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…

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

Re: The Paradox of the Proof

#50

Post by claudius »

raverbashing wrote: post #39
claudius wrote: post #24

Earlier quoted context omitted.

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

> There are several hard problems in one domain that are trivial in another domain, for example.

Certainly, and this is pretty much what OP did, invent a new domain to solve a problem – Hamilton aka Lord Kelvin merely reformulated the problem slightly, and while I personally love Hamiltonian mechanics, I don’t think it is comparable to ‘inter-universal geometry’ or replacing first order logic with something else.

So, yes, a different field may provide a different perspective and hence easier solution, but if you want to replace first order logic, you’re not looking at a different/new field in maths, you’re looking at rebuilding maths.

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