I mentioned that Mark Colyvan, by way of epilogue to his Introduction to the Philosophy of Mathematics has a list of choice “mathematical results that have some philosophical interest, or in some cases are just very cool pieces of mathematics”. Some might be interested in knowing about his list. It is divided into three main parts (leaving aside the epilogue to the epilogue on “Some interesting numbers”). So we have, with his dates,
1. Philosophers’s Favourites: The Tarski-Banach Theorem (1924), Löwenheim-Skolem Theorem (1922), Godel’s Incompleteness Theorems (1931) , Cantor’s Theorem (1891), Independence of Continuum Hypothesis (1963) , Four-Colour Theorem (1976), Fermat’s Last Theorem (1995) Bayes’ Theorem (1763), The Irrationality of Square root of 2 (ca. 500 BC), The Infinitude of the Primes (ca. 300 BC).
2. The Under-Appreciated Classics: The Borsuk-Ulam Theorem (1933), Riemann Rearrangement Theorem (1854), Gauss’s Theorema Egregium (1828), Residue Theorem (1831), Poincare Conjecture (2002), Prime Number Theorem (1849). The Fundamental Theorems of Calculus (ca. 1675), Lindemann’s Theorem (1882), Fundamental Theorem of Algebra (1816), Fundamental Theorem of Arithmetic (ca. 300 BC).
3. Some Famous Open Problems: The Riemann Hypothesis, The Twin Prime Conjecture, Goldbach’s Conjecture, Infinitude of the Mersenne Primes.
Well, that strikes me as a fairly random list. And Colyvan’s comments are a pretty mixed bag too. For example, I’d have thought that one of the things of interest about the Prime Number Theorem is that, though it originally looked ‘deep’, something that required some serious apparatus to prove, it has latterly been shown to have an elementary proof (indeed can be proved in IDelta_0 + exp). Now questions about ‘depth’ of proof, and what can be revealed by proofs of different kinds, are surely of some philosophical interest, but Colyvan misses the chance to hint at them. Again, what’s the philosophical interest of the proof of the Poincaré conjecture? Colyvan gives us some human interest gossip about Grigori Perelman turning down the Fields medal and so on: but what’s the philosophical point? Perhaps Colyvan has the Poincaré conjecture on the list because it is “cool”: but then why is it especially cool (apart from the fact that it resisted proof for a long time?).
However, I’m certainly not at all averse to Colyvan’s project of giving a list of conceptually interesting mathematical problems and proofs: it could be a fun and illuminating project. What would you put on the list and why? [For more initial thoughts, see the comments on the previous blog post.]
Its surprising the Halting Problem isn’t here. Also some results concerning P vs NP like maybe 3-SAT is NP-complete or things like relativization and natural proofs.
Yes, and perhaps also results in descriptive complexity, such as Fagin’s theorem relating NP and existential second-order logic.
Considerable philosophical interest should attach to results that tell us that some proposition holds almost surely. Exactly how should an epistemologist slide a cigarette paper in between something happening with probability 1, and its always happening? A nice example is the fact that a countably infinite random graph, generated with any fixed probability other than 0 or 1, is almost surely the Rado Graph.
I belive that (most) representation theorems have philosophical morals. Which ones exactly? Well, that is a matter of taste.
I think the theorem of Steiner has some interest. It states: Given a projective transformation f, between the pencil of lines passing through a point X and the pencil of lines passing through a point Y, the set C of intersection points between a line x and its image f(x) forms a conic.
One sees how can appear structures from the relations between different objects.
Urysohn’s metrization theorem gives sufficient conditions, that are strictly topological, for a space to have a distance function defined on it. It answers a question of direct philosophical interest: What does a space have to be like if real numbers are to measure its relations?
Notably, the relationship between metrical properties of space and non-metrical (viz., projective) properties was of great interest to Russell dating to the 1890s. After Principia, Russell was interested in Hausdorff’s fundamental work on the separation properties at the core of metrizablity. Russell mentions Urysohn’s theorem in The Analysis of Matter and the topological properties of physical space are treated as fundamental in Human Knowledge.