Category Archives: What is Mathematics

Mathematics, Science, and Blogs

Michael Nielsen wrote a lovely essay entitled “Doing science online” about  mathematics, science,  and blogs. Michael’s primary example is a post over Terry Tao’s blog about the Navier-Stokes equation and he suggests blogs as a way of scaling up scientific conversation. Michael is writing a book called “The Future of Science.” He is a strong advocate of doing science in the open, and regard these changes as truly revolutionary.  (The term “Science 2.0” is mentioned in the remarks.)

Michael’s post triggered Tim Gowers to present his thoughts about massive collaboration in mathematics, and this post is also very interesting with interesting follow-up remarks.  Tim Gowers mentioned the n-category cafe as a place where a whole research programme is advanced on a blog. Terry Tao mentioned comments on posts on his open-problems series as having some value. He  mentioned, in particular, the post on Mahler’s conjecture. (Also I think some discussions over Scott Aaronson’s blog had the nature of discussing specific technical math (coming from CS) problems.)

Tim actually proposes an experiment: trying to solve collectively a specific math problem. This would be interesting!!! I suppose we need to give such an effort over a blog a longer than ususal life-span – a few months perhaps. (And maybe not to start with a terribly difficult problem.) (What can be an appropriate control experiment though?)

Ben Webster in “Secret blogging seminar”  mentioned, in this context, earlier interesting related posts about “Working in secret“.  

Christian Elsholtz mentioned on Gowers’s blog an intermediate problem (called “Moser’s cube problem”) where you look not for combinatorial lines (where the undetermined coordinates should be 1 in x 2 in y and 3 in z), and not for an affine line (where it should be 1,2,3 in x y and z in any order), but for a line: it can be 1 in x 2 in y and 3 in z or 3 in x 2 in y and 1 in z. 

Update: Things are moving fast regarding Gowers’s massive collaboration experiment. He peoposes to study together a new approach to the k=3 “density Hales -Jewett theorem”. A background post apears here. Hillel Furstenberg and Izzy Katznelson’s proof of the density Hales-Jewett theorem was a crowning achievement of the ergodic theory method towards Szemeredi’s theorem. Like the case for Furstenberg’s proof of Szemeredi’s theorem itself the case k=3 was considerably simpler and had appeared in an earlier paper by Hillel and Izzy. The recent extensions of Szemeredi regularity lemma that led to simpler combinatorial proof of Szemeredi’s theorem did not led so far to simpler proofs for the density Hales-Jewett case. If you look at Tim’s background post let me ask you this: What is the case k=2 of the density Hales-Jewett’s theorem? It is something familiar that we talked about! 

Here is a particularly silly problem that I suggested at some point along the discussions: How large can a family of subsets of an n-element set be without having two sets A and B such that the number of elements in A but not in B is twice the number of elements in B but not in A?

Update: This problem was completely reolved by Imre Leader and Eoin Long, their paper Tilted Sperner families contains also related results and conjectures.

Massive collaboration in art (click the picture for details)

Q: What is the case k=2of the density Hales-Jewett’s theorem? A: It is  Sperner’s theorem! (that we discussed in this post.)

I will keep updating about news from Tim’s project. [Last update is from October 21].  More updates: Tim’s project is getting quickly off the ground. A useful wiki was established. More update: It is probably successful

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Fundamental Impossibilities


An Understanding of our fundamental limitations is among the most important contributions of science and of mathematics. There are quite a few cases where things that seemed possible and had been pursued for centuries in fact turned out to be fundamentally impossible. Ancient geometers thought that any two geometric lengths are commensurable, namely, measurable by the same common unit. However, for a right triangle with equal legs, the leg and the hypotenuse are incommensurable. In modern language (based on the Pythagorean theorem), this is the assertion that the square root of two is not a rational number. This was a big surprise in 600 BCE in ancient Greece (the story is that this discovery, attributed to a Pythagorean named Hippasus, perplexed Pythagoras to such an extent that he let Hippasus drown). Two centuries later, Euclid devoted the tenth book of his work The Elements to irrational quantities. The irrationality of the square root of 2 is an important landmark in mathematics. Similarly, the starting point of modern algebra can be traced back to another impossibility result. Algebraists found formulas for solving equations of degrees two, three, and four. Abel and Galois proved Continue reading

Can Category Theory Serve as the Foundation of Mathematics?

Usually the foundation of mathematics is thought of as having two pillars: mathematical logic and set theory.  We briefly discussed mathematical logic and the foundation of mathematics in the story of Gödel, Brouwer, and Hilbert. The story of set theory is one of the most exciting in the history of mathematics, and its main hero was Georg Cantor, who discovered that there are many types of “infinity.”

Mathematical logic was always considered a very abstract part of mathematical activity, related to philosophy and quite separate from applications of mathematics. With the advent of computers, however, this perception completely changed. Logic was the first, and for many years, the main mathematical discipline used in the development of computers, and to this day large parts of computer science can be regarded as “applied logic.”

While mathematical logic and set theory indeed make up the language spanning all fields of mathematics, mathematicians rarely speak it. To borrow notions from computers, basic mathematical logic can be regarded as the “machine language” for mathematicians who usually use much higher languages and who do not worry about “compilation.” (Compilation is the process of translating a high programming language into machine language.)

The story of Category Theory is markedly different from that of mathematical logic and set theory. It was invented Continue reading

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.

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