The Foundations of Arithmetic: A Logico-Mathematical Enquiry Into the Concept of Number

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Gueguen & Psillos (2017) fn6 write:

Abstraction principles were characterized as such…by Frege (1884, § 63-67) to define the concept of cardinal number. It is significant that the main idea was employed by Duhem.

About footnote #6 of Duhem’s Critical Analysis of Mechanicism and his Defense of a Formal Conception of Theoretical Physics {Gueguen & Psillos (2017)}:

Recall that Frege’s 1884 definition of number (in Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung über den Begriff der Zahl) is predicated upon the concept of function. His theory of number is a second-order functional calculus with individual variables (terms/objects/arguments denoting truth-values) and function variables (formulas denoting the “extension” or class of a given numerical object under a specific concept). That’s how he thinks of “abstraction” in his war against psychologism and subjectivity in mathematical theory.

For example, a binary function of two variables x and y , denoted F((x), (y)), will stand for a class of objects falling under a given concept F if F((x), (y)) is always a truth-value. In that kind of statements, the function operation replaces the classical predication.

As for Russell’s theory of types, it is predicated upon a set theoretical view of variables of “concatenated” higher-orders. That’s how his theory may be thought of as having to do with abstraction of several degrees. For example, a variable for individuals, another for sets of individuals, another yet for sets of sets of individuals, etc.

The parallel you further highlight by quoting Duhem from Prémices philosophiques (to clarify the Frege-Duhem connection footnoted in the Anti-Scepticism and Epistemic Humility in Pierre Duhem’s Philosophy of Science article, p. 57) does make a lot of sense when talking about Duhem’s view of the use of abstraction as essentially equivalent (as you point out) to “Oresme’s discovery of how to assign numbers (analogically) to qualities.”

With reference to Frege, I would recap it by saying that there seems to be a legitimate connection between the Fregean view of numbers in terms of “extensions” (--> relations of analogy between classes, a number [in the Fregean sense] being a class of similar classes) and the Duhemian/Oresmean view about the epistemological function of “abstraction” in relation to physical theories.

I’m not aware of any direct influence of Oresme’s own “triangle” on Pascal’s triangle arithmétique. I know not of any place Pascal may have mentioned he knew of Le Livre du ciel et du monde. But he very well might have, and even actually referred to it—without me knowing it, as I haven’t gone through a detailed reading of Pascal’s entire corpus, which also includes many letters (notably to Fermat in mid-1654, precisely on his triangle).

It is associated to some extent with “the so-called intuitionism and formalism debate in mathematics.” But, more specifically, the anti-psychologism stance Frege and others took in relation to foundational problems in mathematics hinges around the objectivity of logic (of logical truth), which Frege expressly emphasized demonstrably obtains apart from empiricism and the psychological determinisms influencing mental processing.

His discussion of “analogy” is not so clearly identified as being about “analogy.” But whether with respect to what he understands and defines bedeutung (reference) to mean in functional logic or to the definition of a number as a class of similar classes, the notion of analogous modes of representation (“extensions”) for a given referent is quite intrinsic to his semantic theory and its epistemological implications. In that regard, it may be said not to be without connection with (as I wrote before) the Duhemian/Oresmean view about the epistemological function of “abstraction” in relation to physical theories, even though Frege did not “discuss analogy directly, in the process of [indicatively] understanding how to assign numbers to qualities.”