Mathematics: items of interest
A few items on mathematics.
- Mathematical structures
Peter Jipsen. A wiki-ish list of mathematical structures, mostly algebraic. Quite a menagerie. People who know about structures and have competence in LaTeX can contribute. Instances include tense algebras (pdf) (there don’t seem to be any relaxed algebras), modal algebras (pdf), and logic algebras (pdf). More algebraic structures are listed here than at Mathworld, but at the moment some entries are just stubs.
- Mathematics in Isabelle
Jeremy Avigad. “The project goal [is] to formalize, and develop tools to assist in the formalization of, portions of mathematics in Isabelle’s higher-order logic”. Last year a proof of the Prime Number theorem (a theorem on the number of primes less than a given natural number n) was formalized;Research on the Prime Number Theorem has yielded one of the largest numbers with a proper name: Skewes’ number: 10^(10^(10^34))). For some time mathematicians believed that π(n), the number of primes less than n, was strictly less than Li(n), a function devised by Gauss as an approximation of π(n). J. E. Littlewood proved in 1914 that Li(n) > π(n) for infinitely many sufficiently large n. S. Skewes showed that the first n for which Li(n) > π(n) is less than the number later named after him. This bound has since been reduced to 1.39822 · 10^316 by Bays and Hudson (Mathematical computing 69(2000):1285–1296). In 1955 Skewes showed that if the Riemann Hypothesis is false, then Li(n) ≥ π(n) for at least one n less than 10^(10^(10^(10^3))), which is much larger than Skewes’ number; this is called the second Skewes number. An even larger number is Graham’s number. I wonder what Kant would have thought of them (see the Critique of judgment §26, Ak. 254).Avigad has written an informative note on the proof, with some brief remarks on its philosophical significance. Other projects with similar aims include the Logosphere Project, Mathscheme (“an integrated framework for computer algebra and computer theorem proving”), and Mathweb, which includes a database of definitions and theorems, along with various projects for coding and storing mathematical information. Of these projects, the Logosphere project and Avigad’s seem to be the most active.
- Lewis Carroll puzzles and games
Cathy Dean. Carroll/Dodgson was both a logician and an entertainer of children. The result of putting the two together was puzzles of permanent interest. Only Richard Smullyan comes to mind as a rival in that department. Has any philosopher tried to explain how logic can be funny? Jokes involving the use of logic—e.g., the syllogism—seem to depend on incongruity between subject matter and method, or between method and results. But you won’t find me offering Yet Another Failed Theory of Humor. Not here.