Let’s start with your first choice, The Foundations of Arithmetic by Gottlob Frege. This is cited as one of the most influential books of the 20th century.

Yes, although it wasn’t written in the 20th century, it became very influential in the 20th century. Frege wrote it in 1884 in the middle of a large project in the philosophy of mathematics. The first part of the project was to develop a new system of logic capable of formalising the notion of proof in mathematics. The result was a system that replaced the classical Aristotelian logic of the syllogism, and represented the most important advance in logic in 2,000 years.

The second part of his project was to demonstrate that the axioms of arithmetic can be derived from his system of logic plus his logical definitions of all arithmetical concepts. He outlines the strategy for doing this in The Foundations of Arithmetic. Simply put, the idea is to show first that arithmetic is at bottom nothing but an elaboration of pure logic, and second that higher mathematics is at bottom nothing more than an elaboration of arithmetic. So the goal, as explained in the book, is to show how all of mathematics can be established with the unchallengeable a priori certainty of pure logic.

And why does this book continue to be so successful?

It is the best, most accessible work ever in the philosophy of mathematics. It is also beautifully conceived and executed. For those who want to know what philosophical analysis is, this is among the best example ever produced. Its vision, though complicated in details, is simple and compelling. In the end, Frege didn’t accomplish everything he hoped for. But he did succeed in laying the foundation for the stunning advances in mathematical logic in the 20th century that themselves provided frameworks for modern theories both of computation and of linguistically encoded information. 

Your next choice, Naming and Necessity by Saul Kripke, is also widely seen as one of the most important philosophical works of the 20th century. How does he fit in with Frege?

This work is a kind of bookend, if you like, to Frege, who initiated a tradition which came to be known as analytic philosophy. In 1970, Saul Kripke, who is also an analytic philosopher, pointed out that despite all the progress made in following Frege’s emphasis on logic, language and meaning, there are certain limitations of that project that we must transcend.

Kripke’s central message emerges from a discussion of three distinctions. One is the distinction between necessary and contingent truth. A necessary truth is one that is true, and would have remained so no matter what possible state the world was in. A contingent truth is one that is true, but could have been false. For example, it is true that we are talking today, but we could have decided otherwise, in which case the claim that we are talking would have been false. This distinction between necessary and contingent truths is traditionally illustrated by saying that the truths of logic and mathematics are necessary, whereas those of natural and social science are contingent.

The second distinction is between those truths we can know a priori, just by thinking about them, and other truths, knowledge of which requires empirical observation and experiment for confirmation. As before, this distinction – between a priori and a posteriori truths – is traditionally illustrated by saying that our logical and mathematical knowledge is a priori, whereas our empirical, scientific knowledge is a posteriori. In short, the metaphysical distinction between what is and couldn’t have been otherwise vs what is but could have been otherwise was thought to coincide with the epistemological distinction between what we can know without empirical confirmation vs what we can know only though empirical confirmation.

Why should that be so? Well, that brings us to the third distinction – between analytic and synthetic truths. An analytic truth is a sentence made true simply by what the words mean – bachelors being unmarried, for example. By contrast, a synthetic truth is made true by corresponding to facts in the world. Throughout most of the 20th century this distinction between the meanings of two classes of sentences was assumed to explain the coincidence of the necessary with the a priori, and the contingent with the a posteriori. If a statement is made true by its meaning alone, then of course it would have remained true even if the facts of the world had been different, and of course it can be known without empirical confirmation, since understanding what it means is enough to know that it is true.

What Kripke shows in Naming and Necessity is that the difference between the analytic and the synthetic can’t explain this coincidence because there is, in fact, no coincidence to explain. Contrary to what had been assumed, there are necessary truths knowable only by experience – including many important scientific truths – and there are contingent truths that can be known a priori.