Hilbert’s Foundations of Geometry

Project Gutenberg has David Hilbert’s Foundations of Geometry available. It is a translation of Hilbert’s Grundlagen der Geometrie, which is famous as the first modern axiomization of Euclidean geometry. The difference between Hilbert’s approach and that of Euclid is that Hilbert fills in all of the fiddly little details required to meet modern standards of rigor.

The book is elementary, and (as translated by Townsend) is a pleasant read. Much of the book centers around constructing the field of real numbers in terms of the axiomized geometrical constructions. This in turn allows Hilbert to show that the set of axioms is complete. The topic leads naturally to one of the main themes of research in plane geometry in the early part of the last century, which is to consider different algebraic objects and how they can serve as coordinates for different notions of affine or projective planes. The reals can be replaced with an arbitrary division ring, for example. For a projective plane, the most general object is a planar ternary ring, with has a ternary operation that serves as a hybrid of addition and multiplication. Determining the projective planes with a finite number of points is still an open question.

Best Possible Rejection Letter

Andrew Gelman quotes from the best possible rejection letter from a journal (sent to Charles Babbage):

It is no inconsiderable degree of reluctance that I decline the offer of any Paper from you. I think, however, you will upon reconsideration of the subject be of the opinion that I have no other alternative. The subjects you propose for a series of Mathematical and Metaphysical Essays are so profound, that there is perhaps not a single subscriber to our Journal who could follow them.

I encourage all journals to adopt this as the standard form letter for rejection.

Scientific Swindler

I came across this story about a 19th century swindler who targeted geologists. He would pose (convincingly) as an expert, and exploit the collegiality of geologists to “borrow” money, or valuable specimens, equipment, and books. You would never hear a similar story about mathematicians because they have nothing worth stealing.

Functor of Points Versus Locally Ringed Spaces

A year ago, the Secret Blogging Seminar had a long thread on how to teach algebraic geometry, one that I never managed to read in its entirety before now. The thread was very interesting. The original post started with the premise that the best way to introduce the idea of an affine scheme is to use the set of maximal ideals, rather than the set of prime ideals as is standard. This is sufficient for classical algebraic geometry.

The thread then wanders off into a different question, of whether the best way to define schemes is as locally ringed space, or rather in terms of a scheme’s functor of points. In the functor of points view, you think of a scheme S as being given by the functor Hom(Spec A, S), as A varies of all rings. You can think of “Spec A” as representing a generalized point — true points correspond to fields. The idea of a generalized point never appealed to me, but still the functor of points view seems more natural to me. What do we know about a general scheme? We know how to map affine schemes into it, and how these pieces are glued together. I remember when I first tried to read Hartshorne, and I learned about sheaves, and then ringed spaces, and then when I got to the requirement that the induced map between stalks had to be local homomorphisms, I got frustrated that after all that machinery, you still needed a weird extra condition to get the right definition. With the functor of points, we start with affine schemes as the dual of the category of rings, and then say what we mean by “gluing” affine pieces together. The Secret Blogging Seminar thread, though, has several practicing algebraic geometers arguing that some notions, such as proper morphism, are much clearer from the locally-ringed space point of view.

Tai’s Method

I just ran across a slightly odd story. In an old post, the (now-defunct) physics blog Flip Tomato points to an abstract for a paper in a diabetes journal that makes a curious claim. The author claims that she has found a new method (named after herself) for calculating the area under a curve by subdividing it into rectangles and triangles and adding it up. This sounds almost exactly like the trapezoid rule from first-year calculus, reported as a new discovery in 1974.

If you’re curious, you can see the abstract of the paper here.

nLab link

The nLab — a math wiki devoted to n-categories and related topics — is having temporary DNS troubles, so if you go to the usual address you’ll get one of those landing pages that domain squatters love so much. There’s a secret alternative URL, but since I love you guys so much, I’m going let you in on the secret.