Ars Mathematica

Here is a page with individual GIFs/PNGs for each math symbol. This is useful for the occasional inclusion of math formulas on a web page, for example. It also has directions on how to build more complex formulas using just HTML. The page is part of the Metamath project.

The Economist has an article on the question of the hardest language to learn. They suggest that a language called Tuyuca is the answer. What makes Tuyuca unusual is that verbs carry an ending that indicates whether the statement is thought to be true or known to be true with certainty. Imagine a language with one tense for conjectures, and another for theorems.

This post at Geomblog is a nice survey of the different approaches to computational topology, which includes both computational approaches to topology and applications of topology to particular areas within computation.

Famously, Netflix offered a million dollar prize for a group that could beat Netflix’ own algorithm for predicting customer movie rankings. The contest, whose main beneficiary was Netflix itself, garnered considerable participation.

The Analytics X Competition is a similar contest, but aimed at benefiting the public directly. The contest is to predict crime rates by zip code in the city of Philadelphia. (The contest is privately funded, though, so the prizes are much smaller. Still it’s an interesting idea.)

The New York Times recently featured this defense of a recent controversial article:

[Hoyt replied to this criticism by saying]—with emphasis in the original— that “The story says O’Keefe dressed up as a pimp and trained his hidden camera on Acorn counselors. It does not say he did those two things at the same time.”

Apparently, the New York Times has adopted as part of its style-book the truth-functional and beloved of first-order logicians everywhere. (In most forms of formal logic the statement “A and B” is true if and only if each of A and B are true. Outside of logic, and rarely works that way. Even programming languages frequently deviate from a pure truth-functional and.)

I’m now enamored by the idea that the New York Times style guide is being taken over by logicians. The next step will be when they start publishing an edition in Loglan.

(Joke swiped from Neddy Merrill at Edge of the American West.)

I just found an interesting website: Connexions. It is a website that hosts a large variety of textbooks and academic monographs, all available under an open-content license. A significant fraction of the collection are applied mathematics books that range from the high school level to the graduate level.

One reason math is hard to teach is that you have someone who found something easy to understand trying to each it to someone who finds it hard to understand. This post at Language Log is an interesting example of someone in a different field running into the same problem.

Linguists use the idea that language constructs are recursive. The example that Mark at Language Log gives is that of “stone traffic barrier”. Here, “stone” and “traffic” are both adjectives. Adjectives can modify not only single-world nouns, but more complex phrases that stand in for nouns. Here, “stone” modifies “traffic barrier”. If you parenthesize it like it’s a piece of mathematics, you would write “(stone (traffic barrier)”. Mark comments that he’s regularly surprised how difficult people find the concept.

It also makes me wonder if mathematics is hard to teach at a basic syntactic level in a way that we would never appreciate. Syntactically, mathematics is defined entirely in terms of recursive syntactic constructions, as the Wikipedia page for well-formed formula illustrates. Getting across the idea of order of operations of arithmetic would be pretty hard if you’re talking to somebody who hasn’t learned the idea of recursive syntax in the first place. It makes me wonder if learning how to diagram English sentences would make it easier to learn algebra.

What scientists say and what journalists here are frequently two different things. Tim Lambert at Deltoid has decided to take the fight to the enemy by calling journalists out by name. For example, here he targets a sensationalist story by Jonathan Leake on how researchers have shown that Facebook causes students to do worse in school. The actual research was much more preliminary and equivocal.

Also, I had no idea that deltoid was a geometrical term. (I think of the muscle.)

I thought I would give a series of examples in increasing exoticness, starting with rings.

A ring has a signature that consists of two binary operations, + and ., one unary operation (-), and two constants (or nullary operations), 0 and 1. The axioms for a ring consist of the associative law for both + and ., the commutative law for +, and the appropriate axioms for -, 0, and 1. Since these all consist of equations, these are all Horn clauses.

Commutative rings are also axiomitizable by adding the commutative law for multiplication. Commutativity is just one of a large family of additional equations that can imposed on the theory of a ring. The whole topic is well-studied, and goes by the name of polynomial identity rings.

In the usual axiomatization of rings, unary minus is not explicitly included. Instead, the existence of an additive inverse is postulated, and uniqueness is proven. This axiom is not a Horn clause, but since the element is unique, you can reformulate it by introducing a new function for the additive inverse.

This trick is not universally applicable. The axioms for division rings, where every nonzero element has a multiplicative inverse, cannot be given in terms of Horn clauses, even by introducing a new function symbol for inverses. One immediate problem is that the inverse is only a partial function, since zero does not have an inverse. This problem can be finessed by extending the notion of Horn clauses to partially-defined functions (something that I will address in a later post). A more serious problem is that there is no way to express the notion that an element is either zero, or has an inverse.

Integral domains also cannot be axiomatized by Horn clauses: the notion that ab = 0 implies that a = 0 or b = 0 is inexpressible. Horn clauses can express the idea that a ring has a zero nilradical, by introducing infinitely many axioms of the form aⁿ = 0 implies a = 0, one for each n.