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Mathematicians Bridge Finite-Infinite Divide

quantamagazine.org

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Re: Mathematicians Bridge Finite-Infinite Divide

#41

Post by GregBuchholz »

tnecniv wrote: post #9
Smaug123 wrote: post #6

Earlier quoted context omitted.

Not quite, I think, although I haven't read the original paper. There are some mathematicians who doubt that there is an infinite object (we call such mathematicians "finitist"). For such mathematicians, there is a large chunk of the mathematical literature they just can't use, because it relies inherently on the existence of an infinite set. Ramsey's theorem for pairs looks like it relies on the existence of an infi…

How do these finitists handle things like the real numbers? Do they just not consider questions that require the notion of infinity?

>How do these finitists handle things like the real numbers?

You may enjoy "Meta Math" by Gregory Chaitin (jump to Chapter 5 for the impatient):

http://arxiv.org/abs/math/0404335

"Finally, and perhaps even more devastatingly, it turns out that the set of all reals that can be individually named or specified or even defined or referred to—constructively or not—within a formal language or within an individual FAS, has probability zero. Summary: reals are unnameable with probability one.

So the set of real numbers, while natural—indeed, immediately given—geometrically, nevertheless remains quite elusive:

Why should I believe in a real number if I can’t calculate it, if I can’t prove what its bits are, and if I can’t even refer to it? And each of these things happens with probability one!"

You might also want to take a look at some of Norman Wildbergers work, like:

"Set Theory: Should You Believe?"

web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf

Re: Mathematicians Bridge Finite-Infinite Divide

#42

Post by Double_Cast »

mcherm wrote: post #19
tnecniv wrote: post #9

Earlier quoted context omitted.

How do these finitists handle things like the real numbers? Do they just not consider questions that require the notion of infinity?

In high-school calculus (if you've taken that), you apply things like "d/dx" which LOOKS like it is just a fraction, dividing "d" by "d times x". The notation was first dreamed up by mathematicians who were thinking "but if we keep making the d-slices really REALLY small we would move from the discrete approximation to the formula for the correct continuous answer". Unfortunately, while SOME such formulas worked (the…

Question: Isn't there an axiom that says "for any real number, there's always a bigger number"? What stopped Patey and Yokoyama from proving Ramsey's Theorem For Pairs/Triples by saying "for any pair which satisfies some relation X, there exists another pair which also satisfies relation X"?

Re: Mathematicians Bridge Finite-Infinite Divide

#43

Post by ZeroFries »

auggierose wrote: post #17

Great article, very well written and actually understandable without hunting down mathematical definitions. I think it is quite cool that people are still hunting down the various different incarnations of infinity and are able to prove deep results like this. On the other hand, I don't think that these questions are as essential as they are made out to be. How do we know that everything finite is on unshakable found…

I think you're on to something here. We often assume a playing field of finite, seperate, individual objects interacting with other finite, seperate, individual objects, but I feel this is more a relic of how our brains break down reality than what reality actually is.

Re: Mathematicians Bridge Finite-Infinite Divide

#44

Post by xviia »

bradleyjg wrote: post #40
antognini wrote: post #39

Earlier quoted context omitted.

In addition, James Simons was a mathematician prior to his work in the finance industry, hence his partiality toward funding a magazine that often writes about pure mathematics.

I believe he is now back at Stony Brook in the mathematics department. Not full time, he has a lot going on, but some kind of affiliation.

I bet being an academic is a lot better when you don't have to worry about being a post-doc searching for a faculty position :/

Re: Mathematicians Bridge Finite-Infinite Divide

#45

Post by auggierose »

Smaug123 wrote: post #34
auggierose wrote: post #31

Earlier quoted context omitted.

I am aware of that. That's why analogies are usually not very useful.

I have no idea how you learn, but analogies are basically the only way I use of absorbing new concepts.

Interesting. I learn by playing around with the concept until I gain some intuition about it. No analogies required.

Re: Mathematicians Bridge Finite-Infinite Divide

#46

Post by tikhonj »

xviia wrote: post #44
bradleyjg wrote: post #40

Earlier quoted context omitted.

I believe he is now back at Stony Brook in the mathematics department. Not full time, he has a lot going on, but some kind of affiliation.

I bet being an academic is a lot better when you don't have to worry about being a post-doc searching for a faculty position :/

To be fair, he was a successful researcher who became professor and later chair of the Mathematics department at Stony Brook before starting Renaissance, so he got past all that the normal way.

Re: Mathematicians Bridge Finite-Infinite Divide

#47

Post by mathattack »

defen wrote: post #35
xviia wrote: post #33

Earlier quoted context omitted.

It makes me curious how Quanta is funded. The journal is very high quality and in a niche market: pure science journalism. I can't imagine they make enough subscription revenue; maybe it's funded mostly through the Simons Foundation? Either way, glad it exists!

I believe it's funded by James Simons

Yes. He also funds things like Mathematics museums. [0] He's an interesting individual who also wants to stay current in Math.

[0] http://www.bloomberg.com/news/articles/2011-11-01/harvard-gr...

Re: Mathematicians Bridge Finite-Infinite Divide

#48

Post by Manishearth »

ZeroFries wrote: post #43
auggierose wrote: post #17

Great article, very well written and actually understandable without hunting down mathematical definitions. I think it is quite cool that people are still hunting down the various different incarnations of infinity and are able to prove deep results like this. On the other hand, I don't think that these questions are as essential as they are made out to be. How do we know that everything finite is on unshakable found…

I think you're on to something here. We often assume a playing field of finite, seperate, individual objects interacting with other finite, seperate, individual objects, but I feel this is more a relic of how our brains break down reality than what reality actually is.

Your recasting of the problem reminds me of the Axiom of Choice, which leads to similar questions if not included.

Re: Mathematicians Bridge Finite-Infinite Divide

#49

Post by SilasX »

n4r9 wrote: post #20
scribu wrote: post #18

As a noob, I'm having a hard time grasping why the Ramsey pairing is interesting: If you pair up every member of an infinite set with every member of that same infinite set, of course you'll get an infinite subset for almost every predicate about a pairing, just by virtue of starting from an infinite superset. It seems like the theorem would be interesting if it said something about the "magnitude" of the subset, not…

It's hard to know what you're finding uninteresting/confusing from your description. Perhaps it would help to note that every possible pairing is given a colour, and in your infinite subset every possible pairing within that subset has the same colour. To me this is quite counter-intuitive. There is a somewhat enlightening comparison to be made between Ramsey on pairs and the result that every real sequence x_n has a…

I have the same confusion as the GP, so let me try to ask a clarifying question:

Is the challenge to partition the pairs (of natural numbers) such that both partitions are infinite and monochromatic? Is that the hard thing that was proved here?

If so, how about "color red if a = b-1, blue otherwise"? Then the infinite subset (0, 1), (1, 2), (2, 3), ... is monochromatic.

What criterion did my partition there fail to satisfy?

Re: Mathematicians Bridge Finite-Infinite Divide

#50

Post by jsprogrammer »

Manishearth wrote: post #48
ZeroFries wrote: post #43

Earlier quoted context omitted.

I think you're on to something here. We often assume a playing field of finite, seperate, individual objects interacting with other finite, seperate, individual objects, but I feel this is more a relic of how our brains break down reality than what reality actually is.

Your recasting of the problem reminds me of the Axiom of Choice, which leads to similar questions if not included.

Axiom of choice is just an ad hoc theorem taken as axiom. Check out HoTT.
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