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The Paradox of the Proof

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Re: The Paradox of the Proof

#101

Post by gwern »

ics wrote: post #84
gwern wrote: post #60

Earlier quoted context omitted.

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?

> Is there any way of even roughly estimating how much Flattr might bring in? Of course: you try it, write up a massive future post about the results, and voila! But that probably wasn't the answer you wanted. I have no idea how many people are using it, but the idea sounds pretty great and if there's a good crowd to try it out on then HN readers wouldn't be a bad bet. I don't think anyone would mind another small bu…

> Of course: you try it, write up a massive future post about the results, and voila!

~-~

> I don't think anyone would mind another small button between PayPal and BTC.

Yes, fair enough... I think I may just be making excuses at this point to not try it out.

Re: The Paradox of the Proof

#102

Post by krakensden »

ajuc wrote: post #45
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).

To be honest, I read "Inter-universal Geometer", and I immediately wondered if it was an Anathem reference.

Re: The Paradox of the Proof

#103

Post by ics »

cookingrobot wrote: post #88

Mochizuki has this 1 page pdf posted on his website that tries to explain Inter-universal Teichmuller Theory through analogy to a Japanese animation: http://www.kurims.kyoto-u.ac.jp/~motizuki/sokkuri-hausu-link... I wish I could find the animation itself, but that link is broken.

Song from the album: http://youtu.be/feOLbipVGEU

If you find a copy of his explanation in Japanese I'll try to see if he's referring to a literal animation or not. I couldn't find one and he doesn't really refer to anything beyond the theme/concept and the girl's theta-like eyes.

Edit: Something else I found (https://www.jstage.jst.go.jp/article/essfr/6/3/6_160/_pdf).

Re: The Paradox of the Proof

#104

Post by ricardobeat »

brazzy wrote: post #27
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...

> It is reality what drives mathematicians to work, not a pure abstraction. In general, no. > It is because of reality that maths is interesting. It is because of Geo-metry that algebra is so relevant. Perhaps to you, but not to most mathematicians, certainly not those working in academical settings. Have you ever looked at group theory or topology? I don't think you've ever done the kind of math that mathematicians…

> invent constructs that have no basis in reality

Regardless of how abstract you get, how far you go, math is always tied to our physical reality. The basic operations are reflections on properties of our universe. The "kind of stuff" mathematicians do" allows us to model and reason about our world in ways that wouldn't be possible any other way (that we know of).

Re: The Paradox of the Proof

#105

Post by xtrumanx »

jerf wrote: post #98
_j5l3 wrote: post #96

Earlier quoted context omitted.

As someone newly entering the workforce, this is a really depressing view into how bureaucracy can squash innovation.

Do not be depressed, at least because of this. Few people will ever write anything that is correct, can not be meaningfully simplified, yet incomprehensible and essentially unreviewable. The only other thing that comes to mind are the top-grade encryption algorithms. (They are reviewed, but even after extensive, extensive review by the very smartest people in the field, there's still an irreducible part when selectin…

> Do not be depressed, at least because of this. Few people will ever write anything that is correct, can not be meaningfully simplified, yet incomprehensible and essentially unreviewable.

You don't have to write something that is essentially unreviewable to end up in the scenario described earlier. You just have to work with a team that isn't prepared to learn new things and rejects thing that they don't understand.

I've been studying functional programming lately and can easily imagine being told my code is "incomprehensible" because I wrote it in a functional style instead using loops and variables.

Re: The Paradox of the Proof

#106

Post by brazzy »

ricardobeat wrote: post #104
brazzy wrote: post #27

Earlier quoted context omitted.

> It is reality what drives mathematicians to work, not a pure abstraction. In general, no. > It is because of reality that maths is interesting. It is because of Geo-metry that algebra is so relevant. Perhaps to you, but not to most mathematicians, certainly not those working in academical settings. Have you ever looked at group theory or topology? I don't think you've ever done the kind of math that mathematicians…

> invent constructs that have no basis in reality Regardless of how abstract you get, how far you go, math is always tied to our physical reality. The basic operations are reflections on properties of our universe. The "kind of stuff" mathematicians do" allows us to model and reason about our world in ways that wouldn't be possible any other way (that we know of).

> The basic operations are reflections on properties of our universe.

No, this is most definitely not true for all branches of mathematics.

Re: The Paradox of the Proof

#107

Post by bbq »

crntaylor wrote: post #11
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…

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

What if there are 0 universes? Does 0 exist without any universes?

Re: The Paradox of the Proof

#108

Post by lmm »

csense wrote: post #95
lmm wrote: post #59

Earlier quoted context omitted.

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 c…

> Hardy rejoiced in the idea that his work on primes would never be a weapon of war What about RSA encryption, and the former US export controls thereon? Also, what about more exotic mathematical objects, like finite fields for example, that have a lot of structural properties but don't seem to model much of anything in the real world, but are still useful theoretically?

>What about RSA encryption, and the former US export controls thereon?

That's exactly the point.

>Also, what about more exotic mathematical objects, like finite fields for example, that have a lot of structural properties but don't seem to model much of anything in the real world, but are still useful theoretically?

If that theoretical use applies to physics then to my mind the theoretical objects are as real as anything else. I have a friend who likes to say that electrons don't "really" exist (because they can't be directly observed), they're just a theoretical object that makes our calculations easier. I guess if you pressed me on philosophy I'd say that the only "real" things are sensory observations - we posit objects such as "that table" because they make the explanation of our visual perceptions easier. If positing a finite field makes some part of physics simpler, that seems to be the same thing.

Now for areas of mathematics that are isolated from the rest it's much easier to say they don't represent anything real. But one of the things I was trying to say in the grandparent post is that there's an uncanny tendency for these areas to not stay isolated for long.

Re: The Paradox of the Proof

#109

Post by stianan »

ics wrote: post #103
cookingrobot wrote: post #88

Mochizuki has this 1 page pdf posted on his website that tries to explain Inter-universal Teichmuller Theory through analogy to a Japanese animation: http://www.kurims.kyoto-u.ac.jp/~motizuki/sokkuri-hausu-link... I wish I could find the animation itself, but that link is broken.

Song from the album: http://youtu.be/feOLbipVGEU If you find a copy of his explanation in Japanese I'll try to see if he's referring to a literal animation or not. I couldn't find one and he doesn't really refer to anything beyond the theme/concept and the girl's theta-like eyes. Edit: Something else I found ( https://www.jstage.jst.go.jp/article/essfr/6/3/6_160/_pdf ).

Is that article written by Mochizuki? Can you make sense of it?

Re: The Paradox of the Proof

#110

Post by ag412 »

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 is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of laws of nature and of the human mind's capacity to divine them." -Eugene Wigner

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

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