I had not realized the 4-dimensional case was so hard that the much more famous 3-dimensional case was "another" contender for the hardest. I had thought it was sort of like the 4 color map theorem on stereoids, where first it was proved in 5+ dimensions (Smale, 1961), then 4 dimensions (Freedman, 1982), and finally 3 (Perelman, 2006), getting harder and harder at each step.
New math book rescues landmark topology proof
111–120 of 171 posts
Re: New math book rescues landmark topology proof
#112Re: New math book rescues landmark topology proof
#113If 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 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_...
Others have already mentioned it, but Gödel's incompleteness theorem doesn't need a book-length treatment. My undergraduate mathematical logic class covered the proof in a little more than a week.
Re: New math book rescues landmark topology proof
#114What about this guy's book? https://math.uchicago.edu/~dannyc/books/4dpoincare/4dpoincar...
Re: New math book rescues landmark topology proof
#115I had not realized the 4-dimensional case was so hard that the much more famous 3-dimensional case was "another" contender for the hardest. I had thought it was sort of like the 4 color map theorem on stereoids, where first it was proved in 5+ dimensions (Smale, 1961), then 4 dimensions (Freedman, 1982), and finally 3 (Perelman, 2006), getting harder and harder at each step.
And the proofs of this one fact are all completely different depending on the dimension. I wonder if there are any prospects of a unified proof for all dimensions.
Re: New math book rescues landmark topology proof
#116If 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.
Kevin Hartnett's popularizations of mathematical research are amazing, I've never seen anything like them before. (Said as a complete amateur with sketchy coverage of undergraduate pure math).
Re: New math book rescues landmark topology proof
#117If 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 than your average lay person, but I've also hardly used most of it since college. I'm curious if it'd be a worthwhile read or just something that'd end up sitting on my shelf, mostly unread...
Re: New math book rescues landmark topology proof
#118If 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 will only cost you a little over a Benjamin to find out.
Re: New math book rescues landmark topology proof
#119Earlier quoted context omitted.
> The halting problem means that you cannot make a program (a static piece of code) that can determine in a finite amount of time if another program halts operating on some input. Interesting. My first thought was, couldn't we just create programs that are designed to keep growing? Isn't that both completely possible and more powerful than the model of computing where programs are static? I realized it may not be ent…
> Interesting. My first thought was, couldn't we just create programs that are designed to keep growing? Isn't that both completely possible and more powerful than the model of computing where programs are static? 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…
> 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 itself powered by a finite program?
A program which keeps growing can make use of all the code it has seen before, all the results it has seen from running code previously. Although any of its inputs are finite strings, there are no theoretical limitations regarding the source of the inputs it can receive. They come from outside.
The core idea of the undecidability of the halting problem is that we can construct a machine M that can ask the halting machine for a bit, flip the bit, give it back to the same static halting machine code, thus forcing it to produce the wrong bit.
Any human in that situation would change their understanding of the situation, and we would tend to afford them a chance to change their answer based on their new understanding. Being able to adjust execution strategy after seeing new information is core to human life. Why do we neglect to provide computers opportunities which are available to all humans?
A somewhat trivial reply to Turing's original negator machine M would be to emit a new version of the halting code which checks for the exact code given by M and then returns the opposite of what was returned when M calls out to H. Then, in the argument, we would need to address the possibility that the H machine called out in M to may provide a different answer when we pass it all back through H again. The argument wouldn't work as-is if we afforded H the ability to respond to the situation.
What would the statement be? "It is not possible, given the information the halting code just received, it could change its answer response to the contradiction of its previous answer." A child can figure out this simple trick - we all understand why it's a trick - why do we not let a computer the same opportunity in our theory?
> So long as you have enough memory, ...
True enough. My point was in fact that if the inputs our machine receives are all needed, we would run out of memory eventually.
Re: New math book rescues landmark topology proof
#120Earlier quoted context omitted.
The classification of finite simple groups [1] is apparently ~11-12 books long (when it's finished). Yes, several books for a single theorem. [1] https://en.wikipedia.org/wiki/Classification_of_finite_simpl...
I don't wanna know how much books the classification of finite complex groups is gonna take.
Not very much actually. Finite simple groups are kind of like prime numbers: any finite group can be broken down into simple subgroups similar to how any natural number can be broken down into a multiset of prime number factors.
That's why the classification of finite simple groups is so important, it's essentially a classification of all finite groups.
https://en.wikipedia.org/wiki/Composition_series#Uniqueness:...