Chapter 3 Algebraic Grammar, Geometric Grammar: Vailati and Peirce
Susanna Marietti
During the last years of his life, Giovanni Vailati devoted himself more to the strenuous defense of pragmatism than to scientific research. He spoke of prag- matism as the philosophical current he adhered to. The twentieth century was beginning and we can hear in Vailati’s words some echoes that belong between Charles Sanders Peirce’s pragmatism and an anticipation of those verification theories of meaning that would be successful in Europe and then in the United States a couple of decades later. Vailati spoke of the advantages of applying the analysis in terms of expectations to the propositions we deal with. They consist:
first of all in the possibility of obtaining some kinds of expression of our or others’ beliefs apt more than every other to clarify to which operations or researches we should resort in order to confirm or refute them; sec- ondly in the fact that it is easier to distinguish, among our statements, those that can indeed be confirmed or refuted from those that escape every sort of evidence or refutation in the strict sense of the word, since they only refer to states of consciousness, whose presence can only and without appeal be evaluated by the single individual, or since they only look like assertions, actually being just sentences without meaning.1
And further:
one of the inclinations most distinctive of the new philosophical doctrine called “pragmatism”—and its very definition, if we accept the one given by the American mathematician Ch. S. Peirce who introduced it— consists in seeing in the process of deduction something more than an instrument directed to establish the truth of theories or of scientific or philosophical hypothesis. According to the pragmatist, examining or
1 “Il Pragmatismo e i vari modi di non dir niente” (1909), in Vailati 2003: 285. All translations from Vailati’s papers are mine.
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determining the concrete consequences of a theory or abstract doctrine is not only an instrument to make sure of its truth, but also an instrument to make sure of its having a meaning, of its having a sense and which one.2
Vailati was well acquainted with Peirce’s pragmatist essays published on The Popular Science Monthly in 1877–78, and certainly also with the anti-Cartesian essays came up about a decade before on the Journal of Speculative Philosophy. Less clear is what else he knew of Peirce’s thought, which he always recalls rather fragmentarily. He certainly had no considerable information about Peircean works upon the method of deductive sciences, which were for the most part collected in manuscripts and inevitably intertwined with Peirce’s semiotic notions. Vailati shares some general assumptions with Peircean phi- losophy. As a consequence of that, he shows some connection also with Peirce’s philosophy of mathematics, one of the theoretical fields in which the Ameri- can philosopher was most engaged. And, as a matter of fact, Vailati tightly con- nects his pragmatism to mathematics, a science which is at the core of his cul- tural background. However, whoever is familiar with Peirce’s philosophy of mathematics can strongly perceive that something is missing in Vailati’s papers. They lack the awareness of the semiotic character of the signs employed within the mathe- matical processes. Consequently, they lack a specific attention to the level of the sign in itself. Vailati’s philosophical-mathematical statements, even if they are in accordance with pragmatism, always deal with the relation between the linguistic tools employed by mathematics and the possible extra-linguistic ob- ject denoted by them, entirely leaving out the relation—a true Peircean novelty—between such linguistic tools and that interpretant that makes the sign employed significant. In fact, if we take into account the notion of the interpretant, the mathe- matical sign is no longer a transparent notation connected without mediation with the object. Rather, it becomes an articulate material structure, an instru- ment of work that is helpfully cumbersome in order to reach new information about the supposed objects dealt with. This is at the heart of Peircean notion of mathematical processes: deduction proceeds upon concrete signs embody- ing it, and it is only thanks to these signs that the mathematician becomes aware of those relations among the elements of the system at hand in whose explicitation deductive inference consists. Without a graphical sign which can cooperate in this process mathematical thought cannot be carried on. The choice of the best notation is not decided by the pursuit of transparency and
2 “Su alcuni aspetti del movimento filosofico contemporaneo in Italia” (1907), in Vailati 1980: 276.