## Pragmatism and Mathematical Logic, 1900-1914

Abstract:

In a paper first published in Leonardo (February 1906) and reprinted in The Monist next to Peirce’s Prolegomena (October 1906), Vailati identified several “pragmatic characteristics” of mathematical logic. He argued that the fact that the inventor of pragmatism, Peirce, was himself an original mathematical logician provides “a significant indication of the intimate connection” between mathematical logic and pragmatism. The aim of this paper is to chart the early encounters between pragmatism (understood as a philosophical method) and mathematical logic (understood as a scientific theory). Vailati’s enthusiasm made him list a wide range of pragmatic features that characterise mathematical logic, including “repugnance to the vague, indefinite, generic”, “interest shown for historical researches in the development of scientific theories”, and “conciseness and rapidity of expression”. A more persuasive characteristic is “the arbitrary character” of postulates and definitions, depending on the ends and applications one has in mind. As such, both pragmatism and mathematical logic promote an “instrumental” view of theories, where the “value, and even meaning” of an assertion is related to the use made or meant to be made of it. It is not clear, however, what Vailati makes of his argument: Do mathematical logicians in general tend to proceed along with pragmatistic methods, even without consciously claiming to be pragmatists?
Peirce was a champion of both mathematical logic and pragmatism. But he also took the two deeply connected. He argued that the maxim of pragmati(ci)sm is really a consequence of certain key logical and semeiotic principles. But those arguments were hardly published at the time and the wedlock of pragmatism and mathematical logic failed to materialise. Several contemporary mathematical logicians began to oppose pragmatism. Couturat claimed in his 1905 lecture that pragmatism was contrary to the “true conception and right appreciation” of logic. Objection was that pragmatism subordinates truth to utility, or worse defines truth by means of utility. As such, it stands (and falls) together with psychologism, sociologism and moralism; all of logic’s enemies. Schiller (1912) fired from an opposite direction: we should propagate pragmatism but demolish formal logic, as formal rules are meaningless without actual use. Interestingly, not only Couturat and Schiller adhered only to either mathematical logic or pragmatism, but they pitted one against the other. We explain the conflict of arguments by observing that, while Couturat, Schiller and many others mistook pragmatism for its Jamesian ‘one-world’ variant and not as the Peirce-Vailati pragmaticism advocating methodological pluralism, Schiller and Peirce had a common enemy in Russell’s ‘peanosation’ of formal logic. We identify the key details of the debates that took place e.g. in the 1908 International Congress of Philosophy where a spectrum of views concerning pragmatism and mathematisation of logic clashed (Peirce, James, Dewey, Royce, Baldwin, Ladd-Franklin, Calderoni, Vailati, Russell, Peano), and also point out the key lessons to be drawn from these debates to contemporary philosophy of logic and science that has begun to re-highlight the importance of practices.