Peirce on Mathematics
Posted by David Corfield
By the winter of 1897-8, the jobless philosopher Charles Peirce was financially crippled, continuing his studies despite the cold, unable to heat his house. Regretting this situation, William James, at Harvard, organised for Peirce to give a series of lectures for much needed remuneration. Peirce wanted to talk about formal logic, but James persuaded him that this would severely reduce his audience (“Now be a good boy and think a more popular plan out. I don’t want the audience to dwindle to 3 or 4…”), and that he would be better off discussing “topics of a vitally important character”.
He did adjust the content somewhat, as you can see in the published lectures Reasoning and the Logic of Things, (Kenneth Ketner ed., Harvard University Press, 1992), where you can read some of the correspondence between Peirce and James.
In the first lecture, Peirce outlines his understanding of the architectonic structure of the sciences, dividing them into the psychical and the physical. He notes that on both of these sides, the sciences are growing increasing nomological (law-like), tending towards the state of general psychics and general physics. These in turn are “surely developing into parts of metaphysics,” and
Metaphysics in its turn is gradually and surely taking on the character of a logic. And finally logic seems destined to become more and more converted to mathematics.
Naturally, Peirce then asks “And now whither is mathematics tending?” He notes that “Mathematics is based wholly upon hypotheses, which would seem to be entirely arbritrary.” And continues by observing that nobody now can cover the whole field, and that most important developments are made by specialists. Then,
For that reason you would expect the arbritrary hypotheses of the different mathematicians to shoot out in every direction into the boundless void of arbritrariness. But you do not find any such thing. On the contrary, what you find is that men working in fields as remote from one another as the African Fields are from the Klondike, reproduce the same forms of novel hypotheses. Riemann had apparently never heard of his contemporary Listing. The latter was a naturalistic Geometer, occupied with the shapes of leaves and birds’ nests, while the former was working upon analytical functions. And yet that which seems the most arbitrary in the ideas created by the two men, are one and the same form. This phenomenon is not an isolated one; it characterizes the mathematics of our times, as is, indeed, well-known. All this crowd of creators of forms for which the real world affords no parallel, each man arbitrarily following his own sweet will, are, as we now begin to discern, gradually uncovering one great Cosmos of Forms, a world of potential being. The pure mathematician himself feels that this is so. He is not indeed in the habit of publishing any of his sentiments nor even his generalizations. The fashion in mathematics is to print nothing but demonstrations, and the reader is left to divine the workings of the man’s mind from the sequence of those demonstrations. But if you enjoy the good fortune of talking with a number of mathematicians of a high order, you will find that the typical Pure Mathematician is a sort of Platonist. Only, he is Platonist who corrects the Heraclitan error that the Eternal is not Continuous. The Eternal is for him a world, a cosmos, in which the universe of actual existence is nothing but an arbitrary locus. The end that Pure Mathematics is pursuing is to discover that real potential world. (pp. 120-1)
This allows him a dig at James – “Once you become inflated with that idea vital importance seems to be a very low kind of importance, indeed.”
Re: Peirce on Mathematics
A tragic life indeed. Apparently Peirce had enemies in high places, who blocked his ability to win the academic appointments he so richly deserved.
Peirce also suffered from a facial neuralgia which is believed to have been trigeminal neuralgia, one of the most painful medical conditions known. (When triggered by, e.g., a kiss to the cheek or a light breeze, an attack is typically likened to a violent electric shock lasting several seconds – and there may be dozens or even hundreds of such episodes in the course of a day. Before effective treatments became available, it was known as the “suicide disease”, for evident reasons.) In the Wikipedia article on Peirce, it is averred that this condition may have substantially contributed to his growing isolation in his later years.