Monday, April 23, 2012

Mathematical Nominalism vs. Mathematical Realism

From the comments to the posting here:
Of course we want to hear about mathematics!
You should do some posts on Thomism and mathematics. The literature out there on St. Thomas's conception of mathematics seems very sparse. I especially would like to see some studies of how a Thomist views calculus, infinitesimals, etc.
There are some things, for example: My quotes of St. Thomas on mathematics; ch. 6 of Mullahy's Thomism & Mathematical Physics (PDF pgs. 264 ff.) is entitled "The Nature of Mathematical Abstraction"; "A neglected Thomistic text on the foundation of mathematics" by Armand A Maurer; and St. Thomas on the object of geometry by Vincent Edward Smith. These are all online for free.
Also, if you don't know about Augustin-Louis Cauchy, read this Dictionary of Scientific Biography article on him. I quote it: "More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy)." In his Cours d'Analyse, he proved the Fundamental Theorems of Calculus and Algebra, described partial fraction decomposition, devised convergence tests, and was the first to devise the ε-δ formulation of the limit (cf. this article by a prominent Cauchy historian; I love how it begins! A proper understanding of calculus is absolutely vital for understanding the power and limits of modern mathematical physics.). Of course Cauchy is also famous for complex integrals. Since he was a devout Catholic, I've been wondering if he was well-versed in Thomism. He studied classics before becoming a mathematician, so it's possible. I'm not sure he wrote any treatise on the philosophy of mathematics, though.
My follow-up:
Definitely read St. Thomas's In Boethium De Trinitate, q. 5, a. 3: "Does Mathematics Treat, Without Motion and Matter, of What Exists in Matter?". (Questions 5 and 6 of In Boethium De Trinitate are also called St. Thomas's Division and Methods of the Sciences.) Cf. also St. Thomas's commentary In II Phys. lect. 3.
From what I have read, it seems that Aristotelian philosophy of mathematics does well making sense of the everyday mathematics your average person on the street uses, e.g. arithmetic, combinatorics, geometry, and other discrete disciplines.
Aristotle and St. Thomas certainly distinguished discrete and continuous quantity (cf. Summa I q. 3 a. 3: "punctum non est principium nisi quantitatis continuae, et unitas quantitatis discretae": "a point is the principle of continuous quantity alone; and unity, of discontinuous quantity").
However, like you say, it would be interesting to see if an account of infinitesimals, calculus, the larger infinities, etc. could be given, while remaining a realist (rather than nominalist) theory of mathematics.
St. Thomas, in his commentary on Aristotle's Physics, deals very well with the "mathematical nominalist objection" to actual infinities in nature (be they infinitely large {cardinalities of Cantor sets}, infinitely small {infinitesimals}, etc.): From In III Phys. lect. 7:
[...] [Aristotle] says that it is impossible for the infinite to be separated from sensible things, in such a way that the infinite should be something existing of itself, as the Platonists laid down. For if the infinite is laid down as something separated, either it has a certain quantity (namely, continuous, which is size [magnitudo], or discrete, which is number [multitudo]), or not. If it is a substance without either the accident of size or that of number, then the infinite must be indivisible—since whatever is divisible is either number or size. But if something is indivisible, it will not be infinite except in the first way, namely, as something is called “infinite” [viz., "not finite"] which is not by nature susceptible to being passed through [better translation: "which is not surpass-able"], in the same way that a sound is said to be “invisible” [as not being by nature susceptible to being seen], but this is not what is intended in the present inquiry concerning the infinite, nor by those who laid down the infinite. For they did not intend to lay down the infinite as something indivisible, but as something that could not be passed through, i.e., as being susceptible to such, but with the passage having no completion.
So, it appears the mathematical nominalists are correct when it comes to mathematicals indirectly "separated [abstracted] from sensible things" (e.g., ∞, א, irrationals, etc., which are only ens rationis, "beings of reason" or "mind-dependent beings"), but they are incorrect in claiming that mathematicals of discrete (e.g., the number 2) and continuous quantity (e.g., a regular dodecahedron) can only be ens rationis (or our names of those ens rationis). The issue boils down to whether or not actual infinities exist in nature (cf., e.g., the whole Physics lecture above or the Summa questions: "Whether an actually infinite magnitude [magnitudo] can exist?" and "Whether an infinite multitude [multitudo] can exist?").
Also, Pierre Duhem—an early 20th century French, Catholic, Thomist physicist, philosopher of physics, and founder of the discipline of the history of medieval physics—wrote, contrary to Poincaré, that mathematical induction can be reduced to a finite syllogistic process.
Happy Easter!

No comments:

Post a Comment