Warren Goldfarb '- The Philosophy of Mathematics in Early Positivism It is commonly held that the philosophy of logic and foundations of mathematics were of central moment in the development of logical pos- itivism. This common wisdom is, of course, correct. It is also commonly held that there was a single logical-positivist doctrine on the nature of logic and mathematics. Here the common wisdom oversimplifies. For although the major positivist writers were in agreement on the general shape of such a doctrine, differences in their views emerge on closer inspection. A look at the evolution of their views and the differences among them can shed light on the positions at which mature positivism arrived. It is natural to divide developments into three periods. The first lies prior to the formulation of classical logical positivism, that is, before 1928 or so. Although the roots of many positivist concerns can be seen in the writings of both Schlick and Carnap in this period, it is surpris- ing to note how small a role is played by considerations in philosophy of mathematics. The second period centers on 1930: the main event here is the assimilation and appropriation of the views of Wittgenstein's Tractatus. Despite Schlick, Hahn, and Carnap's unanimity (against Witt- genstein) that mathematics as well as logic are tautologous, there are subtle disparities — obscured by the common terminology — in their positions. Finally, the third period is marked by the emergence of Car- nap's distinctive position given in The Logical Syntax of Language. To my mind, that position marks a profoundly original shift in the concep- tion of the philosophy of mathematics. The earlier views can serve as a contrast that helps to bring out the nature of Carnap's contribution. 1. Protopositivism Schlick, the founder of the Vienna Circle, sounded many characteristic themes of positivism well before he could properly be called a positivist. 213 214 Warren Goldfarb However, his first writing that touches on the philosophy of mathemat- ics displays a rather different bent. In "The Nature of Truth in Modern Logic" (1910), Schlick mentions that the work of Hilbert, Russell, and Louis Couturat on the principles of mathematics could be viewed as casting in doubt the Kantian notion that the certainty of mathematical propositions rests on intuition; but then he winds up taking Poincare's side against the logicists, with respect to both the ineliminability of mathematical induction by means of a reduction to logic and the jus- tification for induction as lying in the affirmation of powers of mind. "Here too then, we find the 'eternal verities' established through inner or perfect experience" (Schlick 1979a [1910], 85). Over the next few years, though, Schlick's thinking develops away from such Kantianism. His fundamental theme becomes the limited role that intuition or perception plays in knowledge. Knowledge, for him, is conceptual and not grounded in the intuitive. Although there remain some Kantian aspects in Schlick's general system, the "structural real- ism" he espouses in this period, he explicitly rejects Kant's doctrine of pure intuition. Consequently, of course, he shall have to deny the exis- tence of synthetic a priori truths, and this will raise the question of how mathematics is to be accounted for. Schlick's answer, in the Allgemeine Erkenntnislehre (1918), invokes Hilbert's notion of implicit definition. He calls Hilbert's method "a path that is of the greatest significance for epistemology." That method is "simply to stipulate that the basic or primitive concepts are to be de- fined just by the fact that they satisfied the axioms" (1918, 32; 1975 [1925], 33). Hilbert, of course, devised this notion to apply to geom- etry. Schlick generalizes it to cover other branches of mathematics — he explicitly mentions number theory and presumably wishes to include analysis — as well as the empirical sciences. He admits that "a system of truths created with the aid of implicit definitions does not at any point rest on the ground of reality" (1918, 35; 1975 [1925], 37); "reality" here means empirical reality. He adds that, for geometry and the empirical sciences, in the end we apply the implicitly defined concepts to the in- tuitive, although "the moment we carry over a conceptual relation to intuitive examples, we are no longer assured of complete rigor." Schlick is extremely vague, in this book, on the nature of the links between the conceptual and the intuitive or empirical; he talks of the concepts coming in contact with the intuitive, but never says what this comes to.1 Schlick's distinction between free-floating, implicitly defined, ab- stract concepts and a less rigorous application of those concepts to empirical reality seems particularly unsuitable as an account of num- ber theory and analysis. Surprisingly enough, Schlick seems to claim THE PHILOSOPHY OF MATHEMATICS IN EARLY POSITIVISM 215 that in "such an abstract science as number theory" there is never any contact with the intuitive, with the real (1918, 35; 1975 [1925], 37). And then, of course, the Kantian question is never answered; the appli- cability of the a priori sciences of number theory and analysis to the empirical world is never explained; and the need for pure intuition is not eliminated. A more general difficulty looms here, in any attempt to use im- plicit definition to replace pure intuition as a grounding for mathematics. Schlick recognizes that we can take an axiom system to provide an ac- ceptable implicit definition only if the system is consistent (1918, 36; 1975 [1925], 38). Now Poincare had argued over a decade earlier that precisely this need for a consistency proof shows that implicit defini- tions alone cannot provide the basis for mathematics; if the mathematics is to be legitimized, more must be assumed, namely, whatever principles are used in demonstrating consistency. Thus the recourse to implicit def- initions does not, by itself, show that pure intuition can be eliminated (Poincare 1905, 820). Schlick, despite raising the issue of consistency, simply ignores such an objection, calling the provision of a consistency proof "an internal affair of the theory in question" (1918, 37; 1975 [1925], 39). In any case, Schlick winds up claiming that mathematics is entirely analytic (1918, 96; 1975 [1925], 115) and for that reason does not lead to new knowledge. Here he means analytic in just Kant's sense (1918, 97; 1975 [1925], 75). This is also odd, one might think. Even if we ac- cept Schlick's employment of implicit definitions, we still need logic to draw the consequences of the axioms, and presumably such logic would go beyond the subject-predicate containment that figures in the Kantian definition. (True enough, in his Grundlagen der Arithmetik [1884] Frege called logic "analytic," but in this he was using the word in a redefined sense, not the original Kantian one.) In fact, at this time Schlick does not believe that logic outstrips the Kantian analytic: he believes that the only inferences needed are syllogistic. As he puts it: All truths that have precise logical interconnection (that is, that ad- mit of being deduced from one another) can be represented as far as their mutual linkage is concerned by means of syllogisms, specifically in the mood Barbara. The Aristotelian theory of inference needs no modification or extension in order to be applicable to modern science. What is necessary is only that the theory of concepts be deepened. (1918, 89; 1975 [1925], 107) This is astonishing. Thirty-four years after Frege's Grundlagen and eight years after volume 1 of Principia Mathematica, Schlick thinks 216 Warren Goldfarb himself to have provided an alternative to the Kantian account of math- ematics by a breezy invocation of implicit definition and a fallback on syllogisms, and without any attention to details of the logic of the sit- uation. Nor do matters change for ten years. All of the remarks cited above are retained in the 1925 second edition of the Allgemeine Er- kenntnislehre. Moreover, implicit definition retains its central place in "Erleben, Erkennen, und Metaphysik" (1926). (By this point, though, Schlick no longer uses the "analytic" character of mathematics, as obtained via implicit definitions and syllogisms, as an independent ar- gument against Kant. He exploits a far more direct argument available by then, namely, that general relativity had simply shown Kant to be wrong about geometry and hence about pure intuition.) There is no further consideration of philosophy of logic or foun- dations of mathematics in Schlick until he fell under the influence of Wittgenstein's Tractatus. This was a period in which, in contrast, his general view of physical science develops considerably. In the Allge- meine Erkenntnislehre, the conceptual realm and the empirical world seem to be taken as independent, although with some points of contact. In the 1920s, Schlick starts formulating a different picture, spurred by his understanding of general relativity. In this picture, the conceptual is an overlay on an empirical world already given. In Schlick's account of relativity, there are event-coincidences, and then the conceptual (the im- plicitly defined, the conventional) includes things like a coordinization and a metric, which simply give us different ways of talking about the same underlying reality. Schlick's use (or overuse) of implicit definitions is the unstated tar- get of Carnap's earliest writing on philosophy of mathematics and logic, the 1927 paper "Eigentliche und Uneigentliche Begriffe." For Carnap, a proper concept is one built up by explicit definitions from basic con- cepts; Carnap includes here both real (that is, empirical) concepts and formal (that is, logico-mathematical) ones. An improper concept is one introduced through an axiom system, that is, by implicit definition.