Richard Lawrence

In this article, which is based on my dissertation work, I examine the connection between inquiry and meaning by taking problems in elementary algebra as a case study. In elementary algebra, an equation like \(x^2 - 6x + 9 = 0\) gives a problem: what is the value of \(x\)? We answer this question with another equation that gives the value of \(x\): \(x = 3\). Since answers to many other kinds of questions can be conceived as giving the values of variables, the process of answering an algebraic question serves as a model for inquiries more generally. But a puzzle arises about what these equations mean. These two equations are truth-conditionally equivalent; each is true if, and only if, \(x\) takes the number \(3\) as its value. So in what sense could the second equation be an informative answer? Why does it count as solving the problem that the first equation poses? I argue that four different features are required for an equation to give the value of a variable and thereby answer an algebraic question: the variable must be in the scope of the problem statement; the values given must be in the range of the variable; the statement giving the values must represent a complete solution; and it must be in a canonical form. These features are a guide to the structure of inquiries more generally: to answer a question is to find a statement that has these four features.

Since completing my dissertation, my work has focused on historical research about Frege’s distinction between objects and concepts. In the Foundations of Arithmetic, Frege explicitly introduces this distinction in order to offer a critique of formalism in mathematics. In this article, I examine Frege’s early engagement with formalism in the Foundations, where his main formalist interlocutor is Hermann Hankel. Hankel’s text is not well-known among Frege scholars, but I argue that it had an important influence on Frege. Hankel, like Frege, wants to show against Kant that arithmetic is analytic, and he does so by arguing that basic arithmetic truths like \(7 + 5 = 12\) can be deduced from purely analytic axioms. Hankel’s formalism thus anticipates Frege’s logicism to a significant extent. This undercuts Frege’s claim that his logicism is “completely different” from Hankel’s formalism, and raises the question of where the differences really lie. I argue that Hankel and Frege both recognize concepts as a kind of content which may be freely postulated or defined. But Frege differs from Hankel in recognizing that arithmetical terms have a different kind of content which we cannot merely postulate; before we can use arithmetical terms, we first need to prove that there are objects which serve as their contents. Frege thus aligns the distinction between concepts and objects with a practical distinction in mathematical inquiry, between what can be postulated, and what must be proven.

There is additional historical work to be done on Frege’s engagement with formalism. While Frege’s opposition to formalism is well known, the views of the formalist figures he engages with, and the details of Frege’s arguments against them, are under-explored. Moreover, his criticisms of formalism are important textual sources for his thinking about objects, functions, and the category of Bedeutung in general. Frege continues to stress in his later work that formalism fails because it postulates, rather than demonstrates, the existence of mathematical objects, which results from a failure to distinguish objects from concepts. He also sometimes charges formalism with failing to distinguish signs from their Bedeutung entirely, or with assigning them a kind of Bedeutung that makes them unsuitable for use in science. To make sense of these claims and the consequences they have for Frege’s overall theory, we need a better picture of his other formalist interlocutors, including Heine and Thomae (whose views stem from Weierstrass), Hilbert, and Wittgenstein. My plan for the next three years is to write a book on this topic. The book will investigate the views of these interlocutors on their own terms, in order to bring unknown details to light, to describe their impact on Frege’s views, and to explain the influence that they had—both through Frege and in spite of him—on the later development of analytic philosophy. I have developed this research into a grant proposal; the proposal is currently under review at the Deutsche Forschungsgemeinschaft.