All We Need Is a Paradigm
By Phin Upham
I worked on “All We Need is a Paradigm,” a collection of essays I edited in 2009 from The Harvard Review of Philosophy. One of the brilliant minds who contributed to the book is Richard G. Heck, Jr., a contemporary philosopher whose fields of interest include the philosophy of language and logic, and the work of Gottlob Frege.
In “Frege’s Theorem: An Introduction” Richard Heck examines the epistemic status of our understanding of arithmetic—is arithmetic an outcome of reason and are its principles analytically true, or do they depend on human intuition and cognition? Frege tries to resolve this tension by showing that arithmetic truths can be drawn directly from premises that are fundamental truths of pure logic without extraneous assumptions. Since existing systems of logic were insufficient to this task, Frege tried to develop a system he calls Begriffsschrift which is a full second-order logic which accommodates quantification over ‘concepts’. This system of logic restricted inferential steps to syntactic criteria revealing all assumptions being used in all proofs.
Frege transforms the epistemological problems of arithmetic with his system and mathematical argumentation making proofs possible. Frege’s system contributes to axiomizing arithmetic as Euclid axiomized geometry. His formalization brings the resources of mathematical logic to bear now on philosophical problems as well and from this flows questions of what can be solved or proven from what assumptions. The analytic philosophy of Russell and Wittgenstein draws deeply on Frege’s Begriffsschrift. Heck also challenges the certainty of Hume’s principle in logic upon which Frege depends, suggesting ways in which it could be challenged and showing its internal contradictions.