### Mathematical tidbits

I’ve been flat on my back with the flu for the last three days, without an original thought in my head. So no philosophy. Instead a few more sites on mathematics:

- Carrés multimagies
Christian Boyer. Carrés multimagiques (multimagic squares, multimagischen Quadrate) are magic squares such that if every number in the square is replaced its square (cube, etc.) the result is also a magic square.
8 1 6 3 5 7 4 9 2 *n*×*n*array of natural numbers (in a “normal” magic square, the consecutive numbers from 1 to n2) for which every row, column, and diagonal adds up to the same number. There are also magic*cubes*, for which every line of numbers between two faces or vertices (and, in a perfect magic cube, every line of numbers between two edges)—every “row, column, pillar, and space diagonal”—adds up to the same number. In 2003 Boyer and Walter Trump discovered a 5×5 perfect magic cube (the smallest possible). There is an animated version of it at Mathworld. Boyer’s site is a labor of love. It belongs to the Anneau des Mathématiques Francophones, which has 87, mostly pedagogical, sites. - Mathcurve
Robert Ferréol. Une “encyclopédie des formes mathématiques remarquables”. All mathematical objects are abstract, I suppose. But some are more abstract than others. (Is there such a thing as a particular abstractum?) The study of curves and surfaces is rich in what a mathematician would call “concrete” examples. The
*Mathcurve*site includes plane curves, space curves, surfaces, fractals, and polyhedra, including some in animation. There is also a fairly extensive bibliography. Philosophers have had almost nothing to say about what you might call the “natural history” side of mathematics: the gathering and classification of examples, some of which may be no more than curiosities (like the bimagic squares above), some of which turn out to be crucial (like Torricelli’s surface, which, though infinite in extent, is finite in volume). A good cheap book on plane curves is J. Dennis Lawrence,*A catalog of special plane curves*(Dover, 1972) · 0486602885 (find it at ABEBooks). - New Proof of the Four-Color Theorem
Neil Robertson, Daniel P. Sanders, Paul Seymour and Robin Thomas. Twenty years ago, the first proof of the Theorem aroused some interest among philosophers because its proof, part of which was an enormous proof by cases, was carried out on computers. (See Thomas Tymoczko, “The Four-Color Problem and Its Philosophical Significance”, in
*New directions in the philosophy of mathematics*, rev. ed., Princeton, 1998 · 0691034982.) Mathematicians accepted the result, but were, on the whole, disappointed that the proof of what would seem to be a simple property of the plane (“every map can be colored with no more than four colors”) did not have a conceptually more revealing proof. (I seem to remember Jean Dieudonné dismissing it as unfruitful.) You might call it an*unwanted*particular, a quirky fact that wasn’t supposed to be quirky. Robertson et al. have simplified the proof and produced a faster algorithm for proving that a configuration is four-colorable. They acknowledge that the proof is “not a proof in the traditional sense”, and cannot be made into one:We should mention that both our programs use only integer arithmetic, and so we need not be concerned with round-off errors and similar dangers of floating point arithmetic. However, an argument can be made that our ‘proof’ is not a proof in the traditional sense, because it contains steps that can never be verified by humans. In particular, we have not proved the correctness of the compiler we compiled our programs on, nor have we proved the infallibility of the hardware we ran our programs on. These have to be taken on faith, and are conceivably a source of error. However, from a practical point of view, the chance of a computer error that appears consistently in exactly the same way on all runs of our programs on all the compilers under all the operating systems that our programs run on is infinitesimally small compared to the chance of a human error during the same amount of case-checking. Apart from this hypothetical possibility of a computer consistently giving an incorrect answer, the rest of our proof can be verified in the same way as traditional mathematical proofs. We concede, however, that verifying a computer program is much more difficult than checking a mathematical proof of the same length.K. Appel and W. Haken, Every planar map is four colorable. Part I. Discharging,*Illinois J. Math.*21 (1977), 429–490.K. Appel, W. Haken and J. Koch, Every planar map is four colorable. Part II. Reducibility,*Illinois J. Math.*21 (1977), 491—567.K. Appel and W. Haken, Every planar map is four colorable,*Contemporary Math.*98 (1989).N. Robertson, D. P. Sanders, P. D. Seymour and R. Thomas, The four colour theorem,*J. Combin. Theory*Ser. B. 70 (1997), 2–44.N. Robertson, D. P. Sanders, P. D. Seymour and R. Thomas, A new proof of the four colour theorem,*Electron. Res. Announc. Amer. Math. Soc.*2 (1996), 17–25 (electronic).