Carnap, Rudolf. 1963: "Intellectual Autobiography", inand consists in Carnap's reminiscences of Frege,The Philosophy of Rudolf Carnap(ed., P. Schilpp), pp. 3-84.

In 1909 we moved to Jena. From 1910 to 1914 I studied at the Universities of Jena and Freiburg/i.B. First I concentrated on philosophy and mathematics; later, physics and philosophy were my major fields. ... Within the field of philosophy, I was mainly interested in the theory of knowledge and in the philosophy of science. On the other hand, in the field of logic, lecture courses and books by philosophers appeared to me dull and entirely obsolete after I had become acquainted with a genuine logic through Frege's lectures. I studied Kant's philosophy with Bruno Bauch in Jena. ... (pp. 3-4)

... But the most fruitful inspiration I received from university lectures did not come from those in the fields of philosophy proper or mathematics proper, but rather from the lectures of Frege on the borderlands between those fields, namely symbolic logic and the foundations of mathematics.

Gottlob Frege (1848-1925) was at that time, although past 60, only a Professor Extraordinarius (Associate Professor) of mathematics in Jena. His work was practically unknown in Germany; neither mathematicians nor philosophers paid any attention to it. It was obvious that Frege was deeply disappointed and sometimes bitter about this dead silence. No publishing house was willing to bring out his main work, the two volumes ofGrundgesetze der Arithmetik; he had it printed at his own expense. In addition, there was the disappointment over Russell's discovery of the famous antinomy which occurs both in Frege's system and in Cantor's set theory. I do not remember that he ever discussed in his lectures the problem of this antinomy and the question of possible modifications of his system in order to eliminate it. But from the Appendix of the second volume it is clear that he was confident that a satisfactory way for overcoming the difficulty could be found. He did not share the pessimism with respect to the "foundations crisis" of mathematics sometimes expressed by other authors.

In the fall of 1910, I attended Frege's course "Begriffsschrift" (conceptual notation, ideography), out of curiosity, not knowing anything either of the man or the subject except for a friend's remark that somebody had found it interesting. We found a small number of other students, there. Frege looked old beyond his years. He was of small stature, rather shy, extremely introverted. He seldom looked at the audience. Ordinarily we saw only his back, while he drew the strange diagrams of his symbolism on the blackboard and explained them. Never did a student ask a question or make a remark, whether during the lecture or afterwards. The possibility of a discussion seemed out of the question.

Towards the end of the semester Frege indicated that the new logic to which he introduced us, could serve for the construction of the whole of mathematics. This remark aroused our curiosity. In the summer semester of 1913, my friend and I decided to attend Frege's course "Begriffsschrift II". This time the entire class consisted of the two of us and a retired major of the army who studied some of the new ideas in mathematics as a hobby. It was from the major that I first heard of Cantor's set theory, which no professor had ever mentioned. In this small group Frege felt more at ease and thawed out a bit more. There were still no questions or discussions. But Frege occasionally made critical remarks about other conceptions, sometimes with irony and even sarcasm. In particular he attacked the formalists, those who declared that numbers were mere symbols. Although his main works do not show much of his witty irony, there exists a delightful little satireUeber die Zahlen des Herrn H. Schubert. In this pamphlet he ridicules the definition which H. Schubert had given in an article on the foundations of mathematics, which was published as the first article in the first volume of the first edition of the largeEnzyklopädie der mathematischen Wissenschaften. (Schubert's article was fortunately replaced in the second edition by an excellent contribution by Hermes and Scholz.) Frege points out that Schubert discovered a new principle, which Frege proposed to call the principle of the non-distinction of the distinct, and he showed further that his principle could be used in a most fruitful way in order to reach the most amazing conclusions.

In the second advanced course onBegriffsschrift, Frege explained various applications, among them some which are not contained in his publications, e.g., a definition of the continuity of a function, and of the limit of a function, the distinction between ordinary and uniform convergence. All of these concepts were expressible with the help of the quantifiers, which appear in his system of logic for the first time. He gave also a demonstration of the logical mistake in the ontological proof for the existence of God.

Although Frege gave quite a number of examples of interesting applications of his symbolism in mathematics, he usually did not discuss general philosophical problems. It is evident from his works that he saw the great philosophical importance of the new instrument which he had created, but he did not convey a clear impression of this to his students. Thus, although I was intensely interested in his system of logic, I was not aware at the time of its great significance. Only much later, after the first world war, when I read Frege and Russell's books with greater attention, did I recognize the value of Frege's work not only for the foundations of mathematics, but for philosophy in general.

In the summer semester of 1914 I attended Frege's course,Logik in der Mathematik. Here he examined critically some of the customary conceptions and formulations in mathematics. He deplored the fact that mathematicians did not even seem to aim at the construction of a unified, well-founded system of mathematics, and therefore showed a lack of interest in foundations. He pointed out a certain looseness in the customary formulation of axioms, definitions and proofs, even in works of the more prominent mathematicians. As an example, he quoted Weierstrass' definition: "A number is a series of things of the same kind" ("...eine Reihe gleichartiger Dinge"). He criticized in particular the lack of attention to certain fundamental distinctions, e.g., the distinction between the symbol and the symbolised, that between a logical concept and a mental image or act, and that between a function and the value of a function. Unfortunately, his admonitions go mostly unheeded even today. (pp. 4-6)

What a great passage! For the record, though, as I know you know, Jeff, Cantor's set theory was informal but not inconsistent -- he didn't accept anything like a naive comprehension principle and early on made a sort of rudimentary distinction between sets and proper classes.

ReplyDeleteThanks, Chris.

ReplyDeleteYes, you're right, of course. My guess is maybe Carnap didn't look at Cantor's original writings, and so knew only second-hand about how Cantor thought about these things.

What I'm fascinated by are these interesting coincidences - Carnap studying with Frege at Jena; Frege sending LW off to study with Russell. Ramsey being at Cambridge, and befriending LW. Then Carnap, LW, Popper and Godel all in Vienna, circa 1926-1936.

Jeff