The Foundations of Arithmetic (Gottlob Frege)

Though Gottlob Frege might not be a popular or mainstream thinker, he laid down the foundation for one of the most important philosophical  movements in history, namely, analytic philosophy. While Aristotle’s logic couldn’t even represent trivial inferences in Euclidean geometry, Frege’s work lead to the formalization of Russel’s Principia Mathematica, Gödel’s incompleteness theorems, and Tarski’s theory of thruth.

Frege wrote Die Grundlagen der Arithmetik in 1884 with the intent of laying down an a priori analytic-based treatise on what we refer to as numbers. We now know in retrospect that logicism as a singular way to reduce all of mathematics to pure logic did not survive the test of time. However, its rigorous logical formalism did.

Using words to describe mathematics certainly is difficult. Saying that the number 1 is a thing is like defining a definite article in terms of its indefinite article. In Grundlagen, Frege defines numbers as objects that assert something about a concept. What does this mean? In daily speech, we usually use numbers much in the same way we use adjectives: “I see five ants” or “I see red ants”. While both properties may be regarded as objective, the difference is that every ant is red, but not every ant is five. Numbers are not properties of the ants themselves. This becomes more apparent when we convert the sentence to, “The number of ants I see is five”, where the word five is used as a singular term rather than adjectivally. The number five then belongs to a concept of ant that I see. So what is a natural number?

Frege defines ‘the number that belongs to a concept Φ’ as the extension of ‘equal to the concept Ψ’. Equality is defined as: a mutual univocal correlation of the extensions of concept terms. Zero, in turn, would be the concept of being non self-identical. Another way of expressing zero would be as the extension of a concept that has no objects falling under it. I’m not entirely certain if Frege was aware of it or if we all simply have misunderstood his ideas, but to understand the notion of univocal (one-to-one) correlation, we must first define the natural number 1. He’s given us a circular argument! A bit disappointing but not fatal to his legacy.

While Grundlagen leaves out definitions for complex numbers, imaginary numbers, infinitesimals, and more, it sets the bar on other important matters. I particularly enjoyed Frege’s long digressions psychologism which he strongly disliked. The inductive proofs that fancy calculus mathematicians pulled out of their hats seemed to irritate him especially. I would recommend this book to someone that’s interested in mathematical philosophy but doesn’t actually want to learn too much mathematics.