Overview and Context
Philosophische Grundlegung der Mathematik (Philosophical Foundations of Mathematics) is a book by the German-born philosopher Rudolf Carnap, the central figure of the logical empiricist (Vienna Circle and later logical positivist) movement. Carnap wrote it in the 1920s and early 1930s, and it is the most sustained treatment he gave to the foundations of mathematics in the period when he was moving from an earlier, broadly constructivist and neo-Kantian standpoint toward the logicist and then the syntactic conception of philosophy that dominated his later work. The book belongs to the same period as his Der logische Aufbau der Welt (The Logical Structure of the World, 1928) and Logische Syntax der Sprache (The Logical Syntax of Language, 1934), and it is best read as a bridge between the project of reconstructing knowledge in a constructed logical system and the later program of logical syntax.
The central problem the book addresses is the old question of what mathematical knowledge is and how it can be secure. Carnap was writing against the background of a crisis in the foundations of mathematics that had been produced by the paradoxes of set theory and by the competing programs of Frege, Russell, Hilbert, Brouwer, and Poincaré. Each of these programs offered a different answer to the question of whether mathematics rests on logic, on intuition, on formal manipulation of symbols, or on some combination of these. Carnap’s aim was to clarify the relations among these positions, to argue for a particular resolution, and to show how the resolution fits into a general philosophy of science and language.
Main Thesis and the Argument’s Structure
The thesis that organizes the book is that mathematics is, at bottom, a body of analytic, logical truths that can be reconstructed within a formal language, and that the apparent difference between mathematical and empirical knowledge is a difference of role within a language rather than a difference of subject matter or source. Mathematical statements do not describe a special realm of abstract objects that we apprehend by intuition. Rather, they are consequences of the rules that govern the symbols we use. This view is a version of logicism, the program associated with Frege and Russell that holds that arithmetic and analysis can be derived from purely logical principles plus definitions. Carnap accepts the broad logicist aim but reshapes it in several important ways.
The argument proceeds in roughly three stages. The first stage is a critical survey of the existing foundational schools, examining what each takes mathematical objects and mathematical proof to be and where each runs into difficulty. The second stage develops the logicist reconstruction, showing how number concepts and the basic structures of arithmetic can be defined in terms of logical notions and how the principles of arithmetic can be derived from them. The third stage addresses the philosophical status of the resulting system, arguing that the question of whether mathematical statements are true is to be answered by reference to the rules of the language in which they are formulated, and not by appeal to an independent faculty of intuition.
The Critique of Rival Foundational Programs
A substantial part of the book is devoted to examining the rival programs. Carnap considers the intuitionist position, associated with Brouwer, which holds that mathematics is a free creation of the mind and that existence in mathematics means constructibility. He acknowledges the force of the intuitionist critique of nonconstructive existence proofs and of the unrestricted use of the law of excluded middle when applied to infinite totalities. But he argues that the intuitionist’s appeal to a primordial intuition of time as the basis of the natural numbers is not needed for the reconstruction of arithmetic and that it introduces an unnecessary epistemological commitment. The intuition that Brouwer treats as the source of mathematical certainty can, on Carnap’s account, be replaced by the explicit stipulation of the rules of a formal system, which is a more transparent and checkable basis.
He treats the formalist program of Hilbert with more sympathy but identifies a basic limitation. Hilbert’s idea of treating mathematics as the manipulation of uninterpreted signs according to explicit rules is, in Carnap’s reading, the right direction, because it makes the rules themselves the object of study. But Hilbert wanted to secure the consistency of mathematics by finitary means, and Carnap regards the restriction to finitary reasoning as an arbitrary limitation on what can count as an acceptable proof about the formal system. The formalist insight that the rules of a system determine what follows within it is, for Carnap, the decisive point, and it leads him toward a syntactic conception of logic that does not require any privileged finitary standpoint.
The logicism of Frege and Russell receives the most detailed attention. Carnap accepts the central idea that arithmetic can be derived from logic and definitions. He discusses how the concept of cardinal number can be defined through the relation of equinumerosity, how the natural numbers can be characterized through the notion of inductive sequence, and how the axiom of infinity, which Russell needed in order to secure the existence of an infinite domain, presents a special difficulty for the logicist program. Carnap is candid that the logicist reduction does not come for free: it depends on assumptions about the existence of individuals and of infinite classes that cannot themselves be established by pure logic alone. He treats this not as a refutation of logicism but as a reason to be precise about what logic must include.
The Logicist Reconstruction
The constructive core of the book sets out a system in which the basic concepts of arithmetic are defined in terms of logical primitives. Carnap proceeds by introducing a hierarchy of types of objects and propositional functions, in the manner of the theory of types developed by Russell and Whitehead, and then showing how the familiar objects of arithmetic can be defined within this hierarchy. Natural numbers are defined as the classes that are equinumerous with the classes of finite sequences, and the successor relation is defined so as to yield the familiar structure of the natural numbers. Addition, multiplication, and the basic laws governing them are then derived as theorems from these definitions together with the logical axioms of the system.
The reconstruction is intended to show that the axioms of arithmetic, which Peano had given as independent postulates, are not independent truths about a special domain but follow from the definitions of the concepts involved. This is the sense in which Carnap takes mathematics to be analytic: its theorems are true in virtue of the meanings the system assigns to its terms and the rules by which those terms are combined, and no further empirical or intuitive input is required. The derivation is designed to be explicit enough that each step can be checked against the rules, which is part of Carnap’s concern with rigor and with making the foundations of mathematics open to public verification.
The Status of Mathematical Truth
The philosophical payoff of the reconstruction is a theory of mathematical truth that Carnap develops in the later sections. He argues that the question of whether a mathematical statement is true cannot be separated from the question of which language it belongs to and what rules govern that language. A statement is true within a system when it can be derived from the system’s rules, and to ask whether it is true in some stronger, absolute sense is, on his view, to ask a question that has no determinate answer unless one specifies the language in which the question is posed. This relativization of truth to a language is one of the seeds of the later principle of tolerance, which Carnap made the centerpiece of his philosophy of logic. The principle holds that there is no fixed set of correct logical rules, and that each language may be constructed according to its own conventions, provided those conventions are stated clearly.
Carnap connects this view to the distinction between analytic and synthetic statements. Analytic truths, including the truths of logic and mathematics, are true in virtue of the rules of the language. Synthetic truths, including the truths of empirical science, depend on how the world is. Mathematics falls on the analytic side, which means that it makes no assertion about the world and therefore cannot be refuted by experience. This does not make mathematics trivial, since the consequences of a rule system can be very rich and surprising. It means that the source of mathematical knowledge lies in the rules we adopt and in our capacity to reason correctly from them, not in a special access to abstract objects.
Relation to the Broader Philosophy of Science
The book’s conclusions are closely tied to Carnap’s general program in the philosophy of science. Carnap wanted to show that a unified account of knowledge could be given in which logic and mathematics provide the structure, empirical concepts are constructed from basic experiential elements, and the truths of science are those that stand in the right relations to these structures. The foundations of mathematics thus serve as a model case for the analysis of how concepts are constructed from more basic ones, and how their truth conditions depend on the rules of the framework in which they are stated. The reconstruction of arithmetic in the book is therefore both an end in itself and an illustration of the method that Carnap applied to the reconstruction of empirical knowledge in the Aufbau.
Reception and Significance
The book occupies an important place in the history of the logical empiricist movement and in the development of the philosophy of logic in the twentieth century. It shows Carnap working through the logicist tradition in detail and preparing the ground for his later conventionalist and syntactic views. Its treatment of the rival foundational programs is often cited as a clear and fair-minded survey of the options available in the early twentieth century, and its discussion of the relations among logic, mathematics, and language anticipates several themes that became central in Carnap’s later work. Readers interested in how the foundational debates of the 1920s and early 1930s were understood by a leading participant will find the book valuable both for its arguments and for its account of the dialectic among the main schools.
