Greetings, all, and thank you, Walt, for inviting me to post here. I hope you find my take on the mathematical arts interesting enough to read and respond to my posts!

Since I’m a believer in the importance of context, let me say that I’m posting this from Brussels, Belgium, where I happen to be for a meeting. This afternoon I caught a major exhibition of the arts of the first Russian Avant Garde, held at Bozar, the Center for Fine Arts in Brussels. Why am I reporting this on a site devoted to mathematics, I hear you cry? Well, I think a typical pure mathematician would be struck by the geometrical nature of cubist, futurist or constructivist art, and particularly that of the Russians who are the focus of this exhibit. The cubists sought to reveal an object from all perspectives simultaneously, the futurists to capture the dynanism of machines and the colours of metals, and the constructivists to distill visual art to its essential and abstract forms and colours.

Indeed, our typical mathematician would not be mistaken in seeing geometry in this art. In the last decades of the 19th century and the early years of the 20th, there was widespread public interest in the ideas which had recently revolutionized geometry — non-Euclidean geometry, David Hilbert’s axiomatization of geometry (1899), and ideas of “the fourth dimension”. Two of the leading artists of this period, Kazimir Malevich and Piet Mondrian, both sought to represent these new ideas from geometry in their art, and said so explicitly.

If this topic interests you, there is some further reading below.

I’ll have more to say on Hilbert and the intellectual trouble that his axiomatization of geometry caused the philosopher Gottlob Frege in a later post.

References:

M. Dabrowski [1992]: Malevich and Mondrian: nonobjective form as the expression of the “absolute'”, pp. 145-168, in: G. H. Roman and V. H. Marquardt (Editors): The Avant-Garde Frontier: Russia Meets the West, 1910-1930. University Press of Florida, Gainesville, FL, USA.

L. D. Henderson [1983]: The Fourth Dimension and Non-Euclidean Geometry in Modern Art. Princeton University Press, Princeton, NJ, USA.

D. Hilbert [1899]: Grundlagen der Geometrie. pp. 3-92, in: Festschrift zur Feier der Enthullung des Gauss-Weber-Denkmals in Gottingen. Teubner, Leipzig, Germany. (Translated by E. J. Townsend as, “Foundations of Geometry”, Open Court, Chicago, IL, USA. 1910.)

The latest issue of the Bulletin of the AMS of now available (though most of the articles have been available online for a while).

Eric Schechter’s Handbook of Analysis and Its Foundations has a cool quote from constructivist mathematician Errett Bishop:

Mathematics belongs to man, not to God. We are not interested in properties of the positive integers that have no descriptive meaning for finite man. When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself.

I’ve just run across an interesting thought experiment known as Newcomb’s paradox. Suppose there is a being, called the Predictor, that presents you with a choice. There are two boxes. The first box may or may not contain $1,000,000. The second box always contains $1,000. You can choose to open either one box or both boxes. While you are making your choice, the Predictor does not touch the boxes in any way — whether or not the first box contains money is already determined.

Many people have encountered the Predictor before, and have discovered that he seems to always predict what you are going to do. Anyone who has ever chosen to open just the first box receives the $1,000,000. Any who has ever chosen to open both boxes finds the first box empty, and only receives $1,000.

Which would you choose?

The December issue of the Notices of the AMS is now available.

Here’s an interesting site: Jeff Miller’s Earliest Known Uses of Some of the Words of Mathematics, which offers an account of the introduction of various mathematical and statistical terms.

I’m not sure why, but this comment by Easwaran reminded me that the book Toposes, Triples, and Theories, by Michael Barr and Charles Wells, is available for downloading, if your vices run in that particular direction. A topos is a category-theoretic analogue of a set theory. The category of sets for a topos, but there many others. A triple (now usually called a monad) is a category-theoretic analogue of an algebra (in the sense of universal algebra). I don’t remember what a theory is.

I was wondering what the spatial analogue of a stochastic process was. It turns out that they’re known as either spatial processes or random fields, and there’s a book available on the subject online, Random Fields and Geometry, by Robert Adler and Jonathan Taylor.