Comments on: Finding Cantor’s proof that there are transcendental numbers

By: Why Read the Heroes? « Pink Iguana

Why Read the Heroes? « Pink Iguana — Fri, 06 Apr 2012 20:54:55 +0000

[...] – Rouse Ball chair, Cambridge U, Fields Medal 1998, see ICM 2010 or Finding Cantor’s proof that there are transcendental numbers, and he was piqued to comment Re: Steig Larsson, or perhaps the translator Reg Keeling in Wiles [...]

By: Aidan Rocke

Aidan Rocke — Sat, 28 May 2011 08:28:39 +0000

The difference between a finite-state machine(computer) and a human is the capacity to make mistakes. This capacity underlies human creativity. It’s different from the ability of genetic algorithms to evolve because there’s no way at present to pre-determine human mistakes(good or bad) perhaps because humans exist in a dynamic equilibrium. On the other hand, from the inception of a computer each procedure is defined recursively and so every procedure must be predictable.

By: K. P. Hart

K. P. Hart — Tue, 12 Apr 2011 14:54:05 +0000

From Dauben’s biography and other sources I got the impression that Cantor got to the (un)countability question through his work on the uniqueness of trigonometric series and that the application to transcendentals was there to make the real result, the uncountability of R, more palatable to the mathematical community.

By: Lorin

Lorin — Wed, 06 Apr 2011 22:58:11 +0000

Hallo!I want to thank you for this kind of articles. I’ve always thought is quite useful this work you often do to break an idea in many simple insincts, but i never had a really concrete example on myself of why it can help. Today i’ve one:some months ago i read on Ben Green page for student this problem(“for enthusiasts” ):are there 2 real invertible matrices 2×2,that generates the free group on 2 generators?…after some frustrating attempt to construct the two matrices(that is the analogue of find an explicit trascendental number),i realized the potential analogy with your discussion:we want to find something free of relation,let’s look how much do we loose every step of relation;and i was very happy to realize that the situation is exactly in analogy with Cantor one(where here cardinality can be substituted with measure);you can describe every relation as subset of zeros of some real noncostant polinomial in 8 variables, then at every step you loose a set of measure 0, and at the end you have just a countable number of steps(the words),then you loose just a set of measure 0(all this part you can also do with Baire theorem)….sorry if maybe is a little bit OT but i found the analogy very nice, and also the fact that the proof can came up to mind very easier after having seen some decompositions of ideas such that you have done here…Bye

By: observer

observer — Thu, 20 Jan 2011 12:00:16 +0000

Hi Prof. Gowers,
Also, there is a result on Fibonacci integers (the multiplicative group generated by the Fibonacci numbers) in Journal of Number Theory 131 (2011) 440-457: a near-asymptotic formula for their counting function, the proof based on both combinatorial and analytic arguments.

By: observer

observer — Tue, 18 Jan 2011 16:53:05 +0000

Hi Prof. Gowers,
Speaking about numbers, you might find interesting the following result:
The set of all prime numbers is locally connected in Furstenberg’s topology, not locally connected in Golomb’s and Kirch’s topologies (Demonstratio Mathematica Vol. XLIII No 4 2010).

By: Uwe Stroinski

Uwe Stroinski — Tue, 28 Dec 2010 08:10:04 +0000

The computer has to come up with the idea to apply the correct argument to the correct set. Brute-force would be a possible strategy however the huge amount of available information could make this impractible. Hence we search for rules to reduce the size of the search space. At one point we must commit on these rules (like in a chess machine). Is there a general rule or pattern reproducing Russel’s insight? Or does our code contain some a-posteriori insight like …. ApplyTo(DiagonalArgument, UniversalSet)?

By: John Sidles

John Sidles — Mon, 27 Dec 2010 15:42:30 +0000

There’s a terrific Chauvenet Lecture by Saunders Mac Lane, Hamiltonian mechanics and geometry (1970), in which Mac Lane argues that a mathematician’s notion of “naturality” and a physicists notion of “physicality” are really the the same notion, and that this identity arises because both mathematicians and physicists require that their understanding be coordinate-independent.

I posted about Mac Lane’s point-of-view on Dick Lipton’s weblog, under the topic Unexpected connections In mathematics … but if you think about it, Mac Lane’s observation is relevant to Tim’s question too.

Namely, for computers to derive insights into mathematical naturality from the direct experience of physicality—as mathematicians so often have done—then a necessary element of programming a mathematical computer is to provide them with hands, eyes, ears, mobility … and curiosity.

Example: for a computer to conceive of links, knots, and braids … it might be very helpful if the computer first learned to weave fabrics, smith chains, and tie shoes.

As Bill Thurston puts it:

Mathematical concepts are abstract, so it ends up that there are many different ways that they can sit in our brains. A given mathematical concept might be primarily a symbolic equation, a picture, a rhythmic pattern, a short movie—or best of all, an integrated combination of several different representations.

These non-symbolic mental models for mathematical concepts are extremely important, but unfortunately, many of them are hard to share.

Mathematics sings when we feel it in our whole brain. People are generally inhibited about even trying to share their personal mental models. People like music, but they are afraid to sing.

You only learn to sing by singing.

We have a considerable ways to go, before mathematical computer programs “learn to sing by singing.”

By: observer

observer — Mon, 27 Dec 2010 14:36:52 +0000

Apologies for an off-topic comment. In base 12 let 10=a and 11=b. Then 131=ab.

By: Aatu Koskensilta

Aatu Koskensilta — Thu, 23 Dec 2010 15:04:13 +0000

The story has it that Russell discovered his paradox when pondering Cantor’s diagonal argument, as applied to the universal set. If we consider the identity mapping the set of all sets that are not members of themselves automatically pops out. So, given the diagonal argument, there’s a simple and plausible step-by-step story explaining how to get to Russell’s paradox.

By: J. L.

J. L. — Wed, 15 Dec 2010 03:57:31 +0000

I think another important part of creativity is letting yourself make mistakes and unjustified assumptions. Since I only (partially) understand my own creative process, I’ll have to give an example from there. I was trying to prove a certain result one time. I had already proven a partial result on it using a certain equation, which I imagined as a diamond shape in my mind. However, it seemed that a new idea would be needed to solve it completely. After working on it for a few months, I finally had an idea that eventually allowed me to prove the complete result. I imagined that there was another equation that was a larger diamond. At this point, I had no idea how this larger diamond would help prove my result. It turned out that it didn’t exist, but a similar equation did exist, as long as I made a certain assumption which I really had no justification for making. Luckily, that assumption turned out to be true, and the equation ended up working to prove the full result.

The point is, I think that for a computer to be able to make a similar “creative” leap in a reasonable amount of time, it would have to be allowed to make random or unjustifiable assumptions, and “play” with unproven things. The trick would be figuring out how to keep these assumptions sufficiently “creative” while sufficiently relevant to the problem at hand.

By: Uwe Stroinski

Uwe Stroinski — Mon, 13 Dec 2010 17:31:59 +0000

Reading Frege’s work and coming up with Russel’s antinomy might be hard for a computer (although not exactly within the scope of your challenge).

However, as long as you are allowed to change the program it is not really a Turing machine we are up against. You might consider to commit on the axioms you feed into your machine. If mathematics is not like chess it should be easy for us to come up with proofs your machine will never be able to produce. On the other hand, if mathematics is just like chess, you will eventually win by presenting us better and better programs.

By: gowers

gowers — Mon, 13 Dec 2010 15:32:33 +0000

It was sloppy of me to use the word “indirect”, I agree. I haven’t looked at Cantor’s paper, but I would hazard a guess that his thought process was quite similar to the one outlined above: that he started out thinking about why there were transcendental numbers, realized that his argument was very general, and then gave the general argument with the existence of transcendental numbers as an “application”. That is, perhaps the application was his main concern to start with, but not by the time he had finished the paper. I could be completely wrong though (and in particular have not read the article by Robert Gray mentioned in the next comment).

By: gowers

gowers — Mon, 13 Dec 2010 15:27:08 +0000

I knew I shouldn’t have commented on Dick Lipton’s Wikileaks post …

By: Anonymous

Anonymous — Mon, 13 Dec 2010 12:51:04 +0000

Update: It has just disappeared, fortunately for me.

By: Anonymous

Anonymous — Mon, 13 Dec 2010 12:48:06 +0000

Dear Prof. Gowers,

I am seeing a naked woman’s back right at the end of your post, on the frontpage and in this page, with the legend “Ads by Google”.

Sincerely,

By: Pepe

Pepe — Mon, 13 Dec 2010 10:17:03 +0000

Robert Gray, “Georg Cantor and transcendental numbers”, American Mathematical Monthly, vol. 101 (1994), pages 819-832

By: Martin Goldstern

Martin Goldstern — Sun, 12 Dec 2010 23:20:55 +0000

I would not call Cantor’s proof “indirect”. In section 1 of his 1874 paper “Ueber eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen” (On a property of the set of all real algebraic numbers) he first gives an explicit enumeration of the set of all algebraic (real) numbers, and in section 2 he gives an explicit (albeit impractical) construction of a real number that cannot appear in this enumeration. Thus, he shows that there are transcendental numbers by constructing a sequence of algebraic numbers that must converge to a transcendental number.

It is not clear to me what Cantor’s motivation was. The title of the article suggests he was more interested in the yet unnamed concept of countability (or some kind of effective countability); he calls his proof in section 2 an “application” of this concept. It does not sound as if this application was his main concern.

By: roy

roy — Sun, 12 Dec 2010 21:06:03 +0000

Jonathan, thanks; I have the dimmest notion of how categories work, but it’s something I’d like to learn more about.

By: Jonathan Vos Post

Jonathan Vos Post — Sun, 12 Dec 2010 01:06:10 +0000

“On Analogy”. Aristotle’s word was “paradeigma”, also known as “reasoning by example”. He analyzed analogical inference as a mixed syllogism that combined inductive and deductive components. Peirce later analyzed it as involving all three modes of inference: abductive, deductive, and inductive. Jon Awbrey collected a few fragments of source materials here:
http://mywikibiz.com/Talk:Inquiry

By: John Sidles

John Sidles — Sat, 11 Dec 2010 23:50:38 +0000

With an admixture of serious and tongue-in-cheek humor ... an easy-to-implement strategy for identifying hard-to-compute mathematical results is to search MathSciNet for articles whose titles contain the phrase "What is ...". This query returns a list of 838 articles. Terry Tao's "What is good mathematics?" asserts that "the concept of mathematical quality is a high-dimensional one, and lacks an obvious canonical total ordering" ... such sets are generically hard to search. Of course, Notices of the AMS regularly publishes "What is ...?" articles, and in aggregate these span most of the range of modern mathematical inquiry. My favorite titles are plaintive: "If multi-agent learning is the answer, what is the question?" or humorous: "What is the meaning of these constant interruptions?" or self-referential: "What is a question?" These "What is ...?" articles (informally) sample a class of mathematical questions that are very difficult for computers to answer ... and still more difficult for computers to ask ... and isn't the same broadly true of humans? Thank you, Tim, for this fine weblog ... and thank you too, for your many fine comments on many other forums. Happy holidays to all!

By: Jonathan Vos Post

Jonathan Vos Post — Sat, 11 Dec 2010 21:35:41 +0000

In partial answer to Roy: Peter J. Freyd and Andrej Scedrov gave a graphical calculus for allegories. On n-Catgory Cafe the question was asked: how is that related to string diagram calculus for predicate logic? See, for example:
[book] Categories, Allegories, by Peter J. Freyd and Andrej Scedrov, Elsevier, 1990.

and

“Functorial polymorphism”,
Bainbridge, E S | Freyd, P J | Scedrov, A | Scott, P J
THEOR. COMP. SCI. Vol. 70, no. 1, pp. 35-64. 1990
“In the past several years types have become an important component of programming language design. They provide a logical framework to ensure that programs meet given specifications, support a partial correctness or verification mechanism, enhance software maintenance, and encourage the systematic building of complex modules from simpler ones. These features are crucial in large-scale programming projects requiring coordination among many teams of programmers. “

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Fri, 10 Dec 2010 11:27:25 +0000

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By: Roy

Roy — Fri, 10 Dec 2010 02:36:50 +0000

At the level of possibly unedifying generality, I would be interested in how analogies might be translated into computational routines. In your own account of Liouville’s proof, you realized that it can plausibly be described as a diagonal argument. It reminds me a bit of the Poincare quote about mathematics consisting of calling different things by the same name.

Another, possibly overused example, would be Lie looking for continuous symmetry groups as an analogue to Galois Theory. It seems like a fundamental thought process.