Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Some Fundamental Theorems in Mathematics (2019) [pdf]

people.math.harvard.edu

Post reply on HN
Print view View original

81–90 of 96 posts

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#81

Post by guerrilla »

generationP wrote: post #78

How useful is this? Not much that these things have in common beyond "results cited a lot". Can't say that the subjects are chosen very precisely either -- the Fundamental Theorem of Algebra isn't actually a theorem of algebra; Tychonoff's theorem is fundamental only to the set-theoretical part of topology; the Fundamental Theorem of Counting is just a particular case of the "Fubini" interchange-of-summations formula…

> Can't say that the subjects are chosen very precisely either

The explanation of how they were chosen starts on page 80.

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#83

Post by canjobear »

ogogmad wrote: post #75
dan-robertson wrote: post #31

I recall learning the fundamental theorem of applied maths. The approximate phrasing is “in applied maths, if it looks right then it is” and a more precise phrasing is that “in applied maths, all reasonable series converge, all functions are continuous, differentiable and smooth almost everywhere, all limits (and integrals) exist, all sums or integrals commute, all Taylor series are good approximations, all of the si…

> in applied maths ... all functions are continuous I guess that's related to the fact that all computable functions from real numbers to real numbers are continuous. It seems reasonable that the laws of physics should be computable to a large extent otherwise we'd have been able to build a hypercomputer by now.

A step function is not computable?

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#84

Post by ngcc_hk »

hnarn wrote: post #36
ilyagr wrote: post #35

Earlier quoted context omitted.

> every equation has one solution So when you see the sun rising, cherish the moment, as it has never happened before and will never happen again.

In a philosophical manner of speaking, that's true. You never step into the same river twice.

Well there are 3 fragments related to this, one likely fake. It is a way to wake you up in his crazy paradox.

There are books and articles and may be one can start with the key phrase What is same river

And the hidden issue is you. If you think it is unique every time frame ... what is same river and same you. Can One same you even step into the River

And if step into it and swim in parallel is the molecules mostly the same ...

Back to What is river? If it is keep on changing as any river that is not changing and moving is not a river, the sameness if you accept then you can step into the same river as many time as you like. But that is wrong per the statement. Hence what is “same river”.

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#85

Post by ngcc_hk »

jointpdf wrote: post #79
ivansavz wrote: post #5

Wow I really like this document. As someone who really like mathematics but didn't get to do this as my major in undergrad, I'm missing out on so much, but also there is not enough time to start with undergrad books and do three years of basics... This document with all the clear statements at least allows me to see what is there to learn. Yes it's long, but compare that to dozens of the other books I would have to r…

You might enjoy Oliver Knill’s lecture notes on undergrad math topics, which are similarly concise: http://people.math.harvard.edu/~knill/teach/index.html

Thanks. Same as the poster. But some “horrible” memory did meet me as I did math stat. Within 1/2 hour the lecturer tried to do a proof of central limit theory using an assumption that one can draw sphere to fill up a 3D space. Reading this does frighten me to a great extent at the same time appreciate what humanity have reached for doing such apparent “useless” things. Public goods are hard as it might look useless but unlike private goods it can be consumed and reuse many times. And some may find use of matrix in AI, number theory and geometry in encryption ... all because it is a public good that can be shared.

The trick whoonearth waste their life to create the first and develop later such “useless” things.

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#86

Post by whatshisface »

canjobear wrote: post #83
ogogmad wrote: post #75

Earlier quoted context omitted.

> in applied maths ... all functions are continuous I guess that's related to the fact that all computable functions from real numbers to real numbers are continuous. It seems reasonable that the laws of physics should be computable to a large extent otherwise we'd have been able to build a hypercomputer by now.

A step function is not computable?

If I gave you 0.0000000... as input, you would have to scan forever to find out whether I put in a nonzero digit. Scanning to a partial depth would not result in partial accuracy, because the step function is discontinuous.

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#87

Post by hermod »

canjobear wrote: post #83
ogogmad wrote: post #75

Earlier quoted context omitted.

> in applied maths ... all functions are continuous I guess that's related to the fact that all computable functions from real numbers to real numbers are continuous. It seems reasonable that the laws of physics should be computable to a large extent otherwise we'd have been able to build a hypercomputer by now.

A step function is not computable?

In fact, yes! This is counterintuitive if you think of the floats as a model for the reals, but in fact the floats give the wrong intuition, being (a finite subset of) rationals. [1] details this and refers to two related articles (particularly [2], which I find a lighter read). This fact is also remarked on in [3] (by example of the signum function being uncomputable).

For the particular example of a step function, consider the function f which is 1 on the positive reals (>=0) and 0 otherwise. What representation of the real numbers would you choose for the computation of f(0) to terminate in a finite amount of time? It cannot be the binary representation, as those are infinite. Your algorithm, terminating in a finite amount of time, will have checked only a finite number N of binary digits of the argument, and so I can choose x = 0.0{..N..}01 and obtain f(0) = f(x) by this algorithm, which is incorrect. You can choose the Cauchy sequence representation, or the Dedekind cut one, but this problem will persist (and [1] proves this in general).

You can "cheat" by saying that the leading two bits will store the sign of the number: 00 for 0, 01 for positive numbers and 10 for negative ones. But then suddenly arithmetic operations are not computable (see comments on [2])!

[1] https://lukepalmer.wordpress.com/2008/08/11/all-functions-ar...

[2] http://math.andrej.com/2006/03/27/sometimes-all-functions-ar...

[3] Vereshchagin, N., Shen, A. Computable Functions. https://bookstore.ams.org/stml-19/

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#88

Post by ghj »

dan-robertson wrote: post #51
itcrowd wrote: post #39

Earlier quoted context omitted.

Do you remember where you read it? It's a nice and funny formulation but a quick attempt at googling didn't bring this up for me.

It’s basically just a joke. I got it from a professor of mine. But like all good jokes, there’s some element of truth in it: most problems in applied maths relate to something physical, and in (non-quantum) physics, something happens and only one thing happens. Therefore your problems are either bad models of reality or they have exactly one solution. It often turns out that even if your methods aren’t technically co…

> I think this example is well known but I can’t think what it is

Could be the basel problem though he probably used this trick many times in his prolific life:

https://en.wikipedia.org/wiki/Basel_problem#Euler's_approach

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#89

Post by hermod »

domnomnom wrote: post #28
jimhefferon wrote: post #22

Earlier quoted context omitted.

That's a fine book. But to answer that question surely calls for emperical data? Generations of teachers of mathematics-- surely not all incompetent-- have found that many people, perhaps most people, do not love math. There are very good teachers (I had a few), and very good ways to teach it (shout out to the Mr Barton Maths podcast), but experience shows that it is a hard go much of the time.

Good, keep all the maths for me.

This was downvoted, but I think it's an interesting argument, in this context and in general.

Should we be rooting for things that make supply of labor in our field more plentiful? Wouldn't it be in our self-interest to prevent as many people as possible from learning math and computer science? One could argue it wouldn't be in the interest of humans as a species, but I'm skeptical that the marginal loss to each of us is greater than the gain.

Post reply on HN