Gödel, Hilbert and Brouwer

Is mathematics a consistent theory? Or, rather, is there a danger of finding a correct mathematical proof for a false statement like “0 = 1″?  These questions became quite relevant at the end of the nineteenth century, when some mathematical truths dating back many centuries were shattered and mathematicians started to feel the need for completely rigorous and solid foundations for their discipline.  Continue reading

About Conjectures: Shmuel Weinberger

The following paragraph is taken from the original “too personal for publication draft” of an article entitled ” ‘Final values’ of functors” by Shmuel Weinberger for a volume in honor of Guido Mislin’s retirement from ETH. (L’enseignement Mathematique 54(2008), 180-182.) Shmuel’s remarks about making conjectures and the different types of conjectures appear here for the first time. 

 

“Final Values” of FunctorsProf. G. Mislin

 

Shmuel Weinberger

 

A personal letter for, and to, Guido Mislin, a mathematician who always epitomized for me elegance, taste, and precision.

 

 Guido Mislin      

I have, on very rare occasions, conjectured things I really believe with evidence by analogy and the religious belief in the essential simplicity of the mathematical universe (looked at correctly)[1].  Besides beauty, such a conjecture[2] pragmatically serves as a guide through unknown landscape.  And even an unreliable guide helps to point in the right direction. In fact, an unreliable guide known to be unreliable can be useful indeed.  Sometimes one makes conjectures knowing them to be false, but feeling that their falsity is a deep phenomenon and most of the predictions made with the conjecture as guide will be true[3].  On other times, I have conjectured to lay down the gauntlet:  “See, you can’t even disprove this ridiculous idea.” [4] On yet other times, the conjectures come from daydreams:  it would be so nice if this were true[5].  And, yet on others one makes a conjecture hoping to probe the landscape that other conjectures have already illuminated[6].

The conjecture (and speculations) that I would like to present to you, Guido, is motivated by ideas I learnt from Goodwillie and Weiss, but I think it is also much in the spirit of the way you sometimes approach mathematics and it is somewhere between the last two kinds…

 


[1] These include the package of conjectures about homology manifolds made in [BFMW], which incidentally flatly contradict other standard conjectures like the Bing-Borsuk conjecture.

[2] Here I have in mind things like the Riemann hypothesis, indeed huge swaths of number theory; in topology and geometry, Thurston’s geometricization conjecture, Baum-Connes conjecture and Novikov conjectures, and so on are examples.

[3] Here I am thinking of the equivariant version of the Borel conjecture, or the stratified version. See [W, Chapter 13].

[4] I think this is what Lott had in mind (although he was usually careful to call it a question) — but I have less concern for my reputation, and am willing to conjecture the opposite of what I believe -when he formulated the zero-in-the-spectrum problem for all complete manifolds.  And it was disproved in [FW].  On the other hand, my dear friend and mentor Donald Newman made a conjecture of this same sort once in rational approximation, only to have it proven by V.A.Popov.

[5] This must be the case for, say, Gromov’s questions about large Riemannian manifolds [G] or the Weinberger conjecture discussed first in print in [D]

[6] The zero in the spectrum problem for universal covers of aspherical manifolds or even uniformly contractible manifolds, also discussed by Lott, is of this sort.  The conjecture of rationality of the difference of twisted APS invariants for homotopy equivalent manifolds was made after realizing that it would follow from the Borel conjecture for torsion free groups.  By now, it’s been proven three times.  [FLW][K][GHW] and [HR]. 

 

As further afterthoughts Shmuel writes: Continue reading

About Mathematics

This post collects some brief philosophical thoughts about mathematics that appeared as part of my paper “Combinatorics with a geometric flavor: some examples,” from the proceedings of the conference “Vision in Mathematics, towards 2000.” I added two small items (the first and fifth).  

1. Mathematical truths – theorems.

“There are infinitely many primes;” “The three angles of a triangle add up to 180 degrees;”  “A continuous real function defined in a closed interval attains there its maximum;” “A non-constant polynomial over the complex numbers has a solution;” If you substitute a matrix A in its characteristic polynomial you get zero;” “A simply connected closed 3-dimensional manifold is homeomorphic to a sphere.”   

These truths appear very different from truths in other areas of life. This sharp difference is the secret to some of the successes of mathematics and explains also its limitation.

What makes a mathematical theorem important, deep, or central? 

 

 

2. Proofs, more proofs, “proofs from the book” and computer proofs

Science has a dual role: exploring and explaining. In mathematics, unlike other sciences, mathematical proofs are used as the basic tool for both tasks: to explore mathematical facts and to explain them.

The meaning of a mathematical proof is quite stable. It seems unharmed by the “foundation crisis” and the incompleteness results in the beginning of the 20th century, and unaffected by the recent notions of randomized and interactive proofs in theoretical computer science. Still, long and complicated proofs,
as well as computerized proofs, raise questions about the nature of mathematical explanations.

Proofs are gradually becoming intolerably difficult. This may suggest that soon our days of successfully tackling a large percentage of the problems we pose are over. Also, this may reflect the small incentives to simplify.

Be that as it may, we cannot be satisfied without repeatedly finding new connections and new proofs, and we should not give up hope to find simple and illuminating proofs that can be presented in the classroom. Continue reading

The Golden Room and the Golden Mountain

 

Christine Björner’s words at the Stockholm Festive Combinatorics are now available to all our readers. What makes this moving and interesting, beyond the intimate context of the conference, is our (mathematician’s) struggle (and usually repeated failures) to explain to others what we are doing and why we are doing it.

 

 

 

 

The Golden Room and the Golden Mountain

 

 

Christine Björner

 

 

 

I want to tell you a story about Anders. Actually two stories. The story of The Golden Room. And the story of The Golden Mountain.

 

I’ll begin with the Golden Room.

 

Once when I had seen Anders, night after night, week after week, working at his desk, totally immersed in a world of his own, I asked him this question: Can you explain to me what you are doing? And he answered: Christine, I have found the most beautiful room. The whole room is a mosaic of gold, dazzling in its splendour. What I am trying to do is make this room visible to others.

 

I am not a mathematician, but this metaphor gave me an insight into the magical world that all of you who have come to the Festive Combinatorics conference share. I want to honour today, in Anders, and all of you, not only your beautiful minds, but your intuition, your persistence, and your passion for truth.

 

   

 

 

 

A papyrus showing Queen Ankhes-Tut, wife of King Tut, with her husband in the golden room. The Egyptian Museum, Cairo.

Now I will tell you the second story. The story of The Golden Mountain.

Continue reading

Is Mathematics a Science?

Many people do not regard mathematics as a science since it does not directly probe our physical reality; some mathematicians even like to think about mathematics as being closer to art, music or literature. But is there really a big difference between exploring the physical reality and exploring the logical/mathematical reality?

It is perhaps too early to have an open discussion thread on this blog but let me try anyway. What do you think? Is mathematics a science?

(This post was influenced by a recent post on Peter Woit’s blog where he referred to an article by Robert Matthews entitled: “Do we need to change the definition of Science?”)

 Updates: (June 18; June 30 ) A related very successful discussion titled “What is Purity” is taking place Continue reading