Tag Archives: philosophy of mathematics

Frege, Thomae, and Formalism / Review of work on Wittgenstein and Russell

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Volume 11.2 of The Journal for the History of Analytical Philosophy (JHAP) has now been published online, with full open-access:

https://jhaponline.org/jhap/issue/view/494

It features an article by Richard Lawrence, entitled “Frege, Thomae, and Formalism: Shifting Perspectives” Here is the abstract:

Mathematical formalism is the the view that numbers are “signs” and that arithmetic is like a game played with such signs. Frege’s colleague Thomae defended formalism using an analogy with chess, and Frege’s critique of this analogy has had a major influence on discussions in analytic philosophy about signs, rules, meaning, and mathematics. Here I offer a new interpretation of formalism as defended by Thomae and his predecessors, paying close attention to the mathematical details and historical context. I argue that for Thomae, the formal standpoint is an algebraic perspective on a domain of objects, and a “sign” is not a linguistic expression or mark, but a representation of an object within that perspective. Thomae exploits a shift into this perspective to give a purely algebraic construction of the real numbers from the rational numbers. I suggest that Thomae’s chess analogy is intended to provide a model for such shifts in perspective.

The volume also contains a review of Wittgenstein’s Critique of Russell’s Multiple Relation Theory of Judgement, by James Connelly (Anthem Press 2021), written by Samuel Lebens.

JHAP is a free, open-access, peer-reviewed journal. It is available at https://jhaponline.org/. Submissions welcome!

Denoting Concepts and Ontology in Russell’s Principles of Mathematics

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Volume 10.7 of The Journal for the History of Analytical Philosophy (JHAP) has now been published online, with full open-access:

https://jhaponline.org/jhap/issue/view/480

It features an article by Wouter Adriaan Cohen, entitled “Denoting Concepts and Ontology in Russell’s Principles of Mathematics.” Here is the abstract:

Bertrand Russell’s Principles of Mathematics (1903) gives rise to several interpretational challenges, especially concerning the theory of denoting concepts. Only relatively recently, for instance, has it been properly realised that Russell accepted denoting concepts that do not denote anything. Such empty denoting concepts are sometimes thought to enable Russell, whether he was aware of it or not, to avoid commitment to some of the problematic non-existent entities he seems to accept, such as the Homeric gods and chimeras. In this paper, I argue first that the theory of denoting concepts in Principles of Mathematics has been generally misunderstood. According to the interpretation I defend, if a denoting concept shifts what a proposition is about, then the aggregate of the denoted terms will also be a constituent of the proposition. I then show that Russell therefore could not have avoided commitment to the Homeric gods and chimeras by appealing to empty denoting concepts. Finally, I develop what I think is the best understanding of the ontology of Principles of Mathematics by interpreting some difficult passages.

JHAP is a free, open-access, peer-reviewed journal. It is available at https://jhaponline.org/. Submissions welcome!