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Indescribable numbers: The theorem that made me fall in love with math

blog.ram.rachum.com

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Re: Indescribable numbers: The theorem that made me fall in love with math

#11

Post by tripzilch »

DanielBMarkham wrote: post #2

This is good. I really liked it. But just to continue the mind-games, yes there are an infinity more indescribable numbers than there are describable ones, but should there be ? Given the balloon example, he makes the case that as an imaginary god: "...stare imploringly into the siren call of the final ellipsis, for you know that no matter how often you expand it, I will always smile when giving you more and more dig…

What you describe has to do with mathematical constructivism[0], and also with the Axiom of Choice, and it's really complicated (to me, I'm not a mathematician, and I only sort-of get it).

You may have heard about the Banach-Tarski paradox[1], which tells you that if you assume "Real numbers" are actually reality, and the Axiom of Choice, you can divide a sphere into five pieces (one of which is just a point) and rearrange them into two spheres of the same volume.

Obviously you can't do that in reality. This is the point where it gets really complex and I won't fill in the details as I understand them, because they are probably wrong.

R. Buckminster Fuller also wrote some ideas on this constructivism (or something a lot like it), basically advocating against the set of "Real Numbers" as an accurate depiction of reality. For instance if you make a 1m x 1m square out of a real material. The sides of that square will have N atoms, and the number of atoms (and how far they are apart, which is a property of the material used) is really what describes the length. This is a whole number. But according to geometry math, the diagonal of this square has a length of sqrt(2) metres, or sqrt(2) * N atoms. That can't be because you can't have an irrational number of atoms. Then what is really going on suddenly depends on how these atoms are packed inside the square, and again, it gets really complex and that's where my understanding (of both atomic physics and mathematical constructivism) stops.

[0] http://en.wikipedia.org/wiki/Constructivism_(mathematics)

[1] http://en.wikipedia.org/wiki/Banach-Tarski_paradox

Re: Indescribable numbers: The theorem that made me fall in love with math

#14

Post by cool-RR »

ColinDabritz wrote: post #13

Excellent article. If you like this sort of thing, you may enjoy Busy Beaver numbers, my favorite treatment of which is the essay "Who Can Name the Bigger Number?" http://www.scottaaronson.com/writings/bignumbers.html

Yes! I read that article years ago and was absolutely captivated!

Re: Indescribable numbers: The theorem that made me fall in love with math

#15

Post by StevenXC »

DanielBMarkham wrote: post #2

This is good. I really liked it. But just to continue the mind-games, yes there are an infinity more indescribable numbers than there are describable ones, but should there be ? Given the balloon example, he makes the case that as an imaginary god: "...stare imploringly into the siren call of the final ellipsis, for you know that no matter how often you expand it, I will always smile when giving you more and more dig…

When I tell many people that I do my research in infinite combinatorial games, they immediately assume that my research is more impressive[0] than my officemate's research dealing with finite combinatorics. I always move quickly to correct them: dealing with infinity often has the effect of making mathematics easier, and unfairly "trivializes" distinctly difficult problems. After all, in my work, 2^100 is "just finite", even though that many milliseconds would compose over 40 quintillion years.

To your point, I don't know the answer of whether the universe is finitely describable or not. But I certainly know it's easier to abstract many small aspects of the universe with an approximation by an "infinite model" than to deal with the ridiculously large finite numbers involved.

[0] if more useless

Re: Indescribable numbers: The theorem that made me fall in love with math

#16

Post by xerula »

ColinWright wrote: post #5
memming wrote: post #3

Cantor's diagonal argument. I learned it all in high-school!

You learned it, but the author worked it out for himself. Good for him - kudos.

He had probably previously encountered Cantor's diagonalization technique for proving the uncountability of the reals. He just saw how to apply the same technique to the question of describability.

Re: Indescribable numbers: The theorem that made me fall in love with math

#17

Post by tome »

graycat wrote: post #6

Ah, he's just getting started on his journey into the set of real numbers! Eventually he will discover, "God made the integers. All else is man made.". In particular, man made the real numbers to be complete which means that every sequence that appears to converge, that is, meets, the Cauchy criterion, actually does converge. Really his discoveries are about the completeness property of the real numbers. So, in parti…

> In particular, man made the real numbers to be complete which means that every sequence that appears to converge, that is, meets, the Cauchy criterion, actually does converge.

Indeed, but by the same argument as the author's, there are more Cauchy sequences than can be described, so it looks like the real numbers are much bigger than necessary to do mathematics :)

Re: Indescribable numbers: The theorem that made me fall in love with math

#19

Post by graycat »

tome wrote: post #17
graycat wrote: post #6

Ah, he's just getting started on his journey into the set of real numbers! Eventually he will discover, "God made the integers. All else is man made.". In particular, man made the real numbers to be complete which means that every sequence that appears to converge, that is, meets, the Cauchy criterion, actually does converge. Really his discoveries are about the completeness property of the real numbers. So, in parti…

> In particular, man made the real numbers to be complete which means that every sequence that appears to converge, that is, meets, the Cauchy criterion, actually does converge. Indeed, but by the same argument as the author's, there are more Cauchy sequences than can be described, so it looks like the real numbers are much bigger than necessary to do mathematics :)

No, to "do mathematics", e.g., show that the Riemann integral exists, that e and pi exist, etc., we want completeness. Then we are done: The reals are the only complete Archemedean ordered field! So, we have no choice!

Re: Indescribable numbers: The theorem that made me fall in love with math

#20

Post by xyzzyz »

The interesting thing here is that it's much harder to put this problem properly into mathematical terms than it is to solve it. The whole insight here is that you a "description" of number is just some finite sequence of symbols from a finite alphabet. Now, if you understand why cardinality of continuum is greater than aleph null, it's totally straightforward to show that there are only countably many descriptions, but continuum many reals, it's the kind of problem you give freshmen students on their first encounter with cardinalities. Thus, the big achievement of this guy is to actually come up with idea of indescribable numbers by himself and interpreting this question mathematically. In other words, the questions are usually more important than answers.

There's one minor, but nevertheless important mistake made by the author. Author says :

>The infinity just a step bigger than [aleph null] is Aleph one, the infinity of real numbers: The infinity of an impossibly dense line of numbers

While cardinality of reals is certainly larger than cardinality of integers, it is not true that it's just one step larger. Funny thing is that it's not false either: it was proved by Cantor and Cohen that it's impossible to prove or disprove that aleph one is cardinality of continuum. This is the famous continuum hypothesis.

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