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New math book rescues landmark topology proof

quantamagazine.org

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Re: New math book rescues landmark topology proof

#131

Post by eynsham »

srcreigh wrote: post #76
mindcrime wrote: post #62

Earlier quoted context omitted.

I wonder how many other results in maths require an entire book length treatment (or multiple book length treatments)? The one that jumps out to my mind is Gödel's Incompleteness Theorem(s)[1]. I know there are at least a couple of complete books dealing exclusively with this result. [1]: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...

I suspect someone could write a book (or an HN comment at least) about the popular confusion about Godel's 1st incompleteness theorem. :-) In fact we (my brother and I) agreed that it bears a striking resemblance to Turing's halting problem: They both seem to show what computers or math can't do, but in fact both theorems only reveal limitations of certain models of computers and/or math. We both have undergrad degre…

> reveal limitations of certain models of computers

Modulo weird stuff with black holes or whatever (not that I am qualified to understand whether they count), there are no known counterexamples to the Church-Turing thesis.

Re: New math book rescues landmark topology proof

#132

Post by est31 »

spekcular wrote: post #4

This has been a long time coming. There is an infamous MathOverflow thread about the proof (also linked in the article), with the following comment summarizing the state of affairs ~10 years ago: > There is no other evidence. In fact there is absolutely no evidence what so ever. I have never met a mathematician who could convince me that he or she understood Freedman`s proof. I attempted to read that monstrosity of a…

This is more due to the inability of promotion committees to capture the usefulness of scientific work than due to the uselessness of this for science. Sure, if you have nothing else to show for yourself, it might be bad for your career. But I feel like people who work in academia already forego many career opportunities. That being said, you still need to put food onto the table.

But posterity will remember you. Today, Bach is considered one of the most prolific musical composers of history. But back when he lived, he didn't receive that much fame. In fact, after his death, people forgot about him. And Bach lived in a time before copyright. He used existing themes by other people and rearranged them for the instruments he used. Many of those folks might have been famous back then, but now many their names are only known to historians (exceptions exist, e.g. Vivaldi).

Re: New math book rescues landmark topology proof

#133

Post by slim »

hermitdev wrote: post #117
leephillips wrote: post #2

If I understand this article, it’s about a 500-page book that is devoted to one proof of one theorem in topology. I find that amazing. And cheers to Quantum Magazine for regularly publishing popularizations of research mathematics. I know many, at times, take issue with their simplifications and framing, but they’re trying, where almost no one else with their reach covers these areas at all.

I hope they have a quality editor. Would hate to have a situation where they made a mistake on page 5, rendering the subsequent 495 pages worthless gibberish without a correction. (I know there's a bit from a TV show or movie like this, but I can't recall where. Maybe Good Will Hunting?) On a serious note, I'm wondering how approachable this book is to non-mathematicians. I mean, I've had more formal math education t…

It's peppa pig episode where mommy pig writes a book and george overflows the buffer of her computer by scoring very high in the chicken game

Re: New math book rescues landmark topology proof

#134

Post by dqv »

slim wrote: post #133
hermitdev wrote: post #117

Earlier quoted context omitted.

I hope they have a quality editor. Would hate to have a situation where they made a mistake on page 5, rendering the subsequent 495 pages worthless gibberish without a correction. (I know there's a bit from a TV show or movie like this, but I can't recall where. Maybe Good Will Hunting?) On a serious note, I'm wondering how approachable this book is to non-mathematicians. I mean, I've had more formal math education t…

It's peppa pig episode where mommy pig writes a book and george overflows the buffer of her computer by scoring very high in the chicken game

Oh my god that's hilarious

Here's the link: https://youtu.be/IMAav163SFQ?t=42

Re: New math book rescues landmark topology proof

#135

Post by [deleted] »

unanswered wrote: post #129
srcreigh wrote: post #119

Earlier quoted context omitted.

> What does it mean to "create programs that are designed to keep growing"? If they keep going by executing a finite program (if modified by a non-static program, what created that program?) on their code and resuming, that would not be more powerful (Turing machines can do that). To me, your question is like asking this: Do all the programs that we humans ask computers to compute come from a finite program? Is life…

Your "growing program" can (presumably, since we don't have a formal definition) be interpreted by a finite program; i.e. which can treat unlimited memory as instructions to execute. This means, in a very strict and formal sense, that it is not a more powerful model of computing. Therefore it is subject to the halting problem. But even if it were a more powerful model, it would still be subject to an analogous haltin…

[deleted]

Re: New math book rescues landmark topology proof

#136

Post by hvidgaard »

srcreigh wrote: post #76
mindcrime wrote: post #62

Earlier quoted context omitted.

I wonder how many other results in maths require an entire book length treatment (or multiple book length treatments)? The one that jumps out to my mind is Gödel's Incompleteness Theorem(s)[1]. I know there are at least a couple of complete books dealing exclusively with this result. [1]: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...

I suspect someone could write a book (or an HN comment at least) about the popular confusion about Godel's 1st incompleteness theorem. :-) In fact we (my brother and I) agreed that it bears a striking resemblance to Turing's halting problem: They both seem to show what computers or math can't do, but in fact both theorems only reveal limitations of certain models of computers and/or math. We both have undergrad degre…

> Would the proof completely fall apart if the algorithm could update itself to handle the new information? The only thing stopping it is that in the formulation of the problem, the algorithm is assumed to not be able to change with new information (ie it's not an "online" algorithm).

First an online algorithm is an algorithm that process a stream instead of the full input. It's not because it changes.

Regarding the halting problem, lets assume that you give me a program can change itself and you claim solves the halting problem. Use Turings proof on that and it proves that you have not solved the halting problem for all inputs, because it just gave you one that it produced the wrong answer for.

Re: New math book rescues landmark topology proof

#137

Post by [deleted] »

unanswered wrote: post #129
srcreigh wrote: post #119

Earlier quoted context omitted.

> What does it mean to "create programs that are designed to keep growing"? If they keep going by executing a finite program (if modified by a non-static program, what created that program?) on their code and resuming, that would not be more powerful (Turing machines can do that). To me, your question is like asking this: Do all the programs that we humans ask computers to compute come from a finite program? Is life…

Your "growing program" can (presumably, since we don't have a formal definition) be interpreted by a finite program; i.e. which can treat unlimited memory as instructions to execute. This means, in a very strict and formal sense, that it is not a more powerful model of computing. Therefore it is subject to the halting problem. But even if it were a more powerful model, it would still be subject to an analogous haltin…

[deleted]

Re: New math book rescues landmark topology proof

#138

Post by red_trumpet »

hermitdev wrote: post #117
leephillips wrote: post #2

If I understand this article, it’s about a 500-page book that is devoted to one proof of one theorem in topology. I find that amazing. And cheers to Quantum Magazine for regularly publishing popularizations of research mathematics. I know many, at times, take issue with their simplifications and framing, but they’re trying, where almost no one else with their reach covers these areas at all.

I hope they have a quality editor. Would hate to have a situation where they made a mistake on page 5, rendering the subsequent 495 pages worthless gibberish without a correction. (I know there's a bit from a TV show or movie like this, but I can't recall where. Maybe Good Will Hunting?) On a serious note, I'm wondering how approachable this book is to non-mathematicians. I mean, I've had more formal math education t…

As a mathematician myself, I highly doubt it would be readable for non-mathematicians. Typically reading math books is work, even for mathematicians, and not so much something you do to relax in the evening.

Re: New math book rescues landmark topology proof

#139

Post by moomin »

lou1306 wrote: post #72
mindcrime wrote: post #62

Earlier quoted context omitted.

I wonder how many other results in maths require an entire book length treatment (or multiple book length treatments)? The one that jumps out to my mind is Gödel's Incompleteness Theorem(s)[1]. I know there are at least a couple of complete books dealing exclusively with this result. [1]: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...

Godel's theorem is actually not that complex. Sure, you may think of "Godel, Escher, Bach", but that book is stuffed with sooo much material that is only tangentially related to the theorem. Of course the consequences of that theorem are legion, and they do warrant many books worth of discussion. But the proof itself can fit in a long-ish blog post, I think. Fermat's last theorem, on the other hand...

For those interested:

Assume a formal language that can model the natural numbers.

Consider the set of all provable statements. This is countable because it’s a subset of the list of strings of a finite number of characters, which is countable.

Consider the statement “Statement N is false”. This is both a valid statement and cannot appear in the list.

Godel proved significantly more than that, but that’s the basic result. It’s basically the same as the diagonal argument of real numbers being uncountable and it’s also basically the halting problem.

Re: New math book rescues landmark topology proof

#140

Post by hansvm »

EamonnMR wrote: post #82
srcreigh wrote: post #76

Earlier quoted context omitted.

I suspect someone could write a book (or an HN comment at least) about the popular confusion about Godel's 1st incompleteness theorem. :-) In fact we (my brother and I) agreed that it bears a striking resemblance to Turing's halting problem: They both seem to show what computers or math can't do, but in fact both theorems only reveal limitations of certain models of computers and/or math. We both have undergrad degre…

I like to think of the halting problem as similar to the inability to predict the future. In order to predict the future you'd need a model of the whole universe, including the model. Once you think of it like that you realize that you can't beat halting in the classical universe.

The halting problem has that same style of thinking, but the universe doesn't necessarily play by such rules. For one, it's still an open question whether the universe could properly contain a model of itself (hence, needing a model of the universe plus the model isn't a guaranteed absurdity).
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