I am trying to figure out whether the mathematician Giuseppe Peano (1858-1932), most notably known for his standard axiomatization of the natural numbers, held a view of Logicism or rather of Formalism. But I nevertheless get confused and hence my question: Was Peano's view closer to Russell's logicism or was it closer to Hilbert's formalism? What was Peano's view regarding foundations?
You can see Arithmetices principia: nova methodo exposita (1889), translated into:
- Jean van Heijenoort (editor), From Frege to Gödel: A Source Book in Mathematical Logic (1967), page 85-on:
Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution. The difficulty has its main source in the ambiguity of language. [...] My goal has been to undertake this examination, and in this paper I am presenting the results of my study, as well as some applications to arithmetic.
I have denoted by signs all ideas that occur in the principles of arithmetic, so that every proposition is stated only by means of these signs.
The signs belong either to logic or to arithmetic proper. The signs of logic that occur here are ten in number, although not all are necessary.
Thus, it seems that logic and arithmetic - contrary to the founding fathers of logicism: Frege and Russell - are distinct.
But the signs K for classes and ∈ [est] for membership are listed as logical signs, and thus the theory of classes is part of logic, and this is common to Frege and Russell.
For a detailed discussion, see:
- Hubert C. Kenendy, The Mathematical Philosophy of Giuseppe Peano (1963),
where it is argued for a "rejection of the logicist thesis of the reduction of mathematics to logic." [Also available here].