1. Dedekind, Cassirer, and the Conceptual Turn in 19 th -Century Mathematics Erich Reck University of California at Riverside San Francisco, March 31, 2016

2. More P ARTICULAR F OCUS : Richard Dedekind’s contributions to this conceptual turn. Putting his contributions into context in several ways, thus exploring their philosophical significance. Ernst Cassirer’s reception of Dedekind’s work, including Cassirer’s attempt to reconcile Dedekind and Kant. Remaining, or re-surfacing, questions about “intuition” and “logic”. 2 O UR T HEME : P OST -K ANTIAN T HEORIES OF C ONCEPTS My take on this theme: New philosophical and scientific views, in the 19th century and later, on the nature and function of concepts, especially ones that can be seen as reactions to Kant. This includes corresponding developments in nineteenth century mathematics, as well as philosophical reflections on them. Specifically: The “conceptual turn” in 19 th -century mathematics.

3. Main works for our purposes: 1)Continuity and Irrational Numbers [Stetigkeit und irrationale Zahlen], 1872 2)The Nature and Meaning of Numbers [Was sind und was sollen die Zahlen?], 1888 3)But also his contributions to algebraic number theory and to algebra more generally, 1850s-1890s Re 1): The concepts of field, line-completeness (continuity) via cuts (Dedekind cuts), the concept of complete ordered field (containing Q). Re 2): The concepts of set/class, infinity (Dedekind-infinite), simple infinity, chain (Dedekind-Peano axioms) (cf. Ferreirós, Sieg, Schlimm). Re 3): The concepts of field (subfield of C), algebraic number, group, module, lattice (Avigad, Corry, Schlimm; cf. Reck on Dedekind in SEP). 3 D EDEKIND ’ S C ONTRIBUTIONS

4. 4 Derive the Calculus, then also the more basic theory of the natural numbers, from these concepts (cf. Klev, Sieg & Morris). Foil: Appeal to “intuitive” evidence (e.g., for the Mean Value Theorem); taking Euclidean geometry (straight edge & compass etc.), an intuitive notion of magnitude, etc., as basic, also for N,…, R, C. Methodological: New proofs (rules for calculating with square roots, Mean Value Theorem, mathematical induction); more precision (denseness versus completeness), increased generality (fields etc.). Foundational: Show that arithmetic, in the narrower and broader sense, is a “part of logic”, i.e., based on the “laws of thought” alone. Overall background, especially for defining the relevant concepts: a general theory of sets [Systeme] and functions [Abbildungen]. D EDEKIND ’ S G OALS

5. 5 1)Gauss: Prove mathematical results based on “concepts” (“notions”) rather than formalisms (“notations”). 2)Riemann: Isolate “the right”, “fruitful” concepts, e.g., in complex function theory; base proofs on them; notion of manifold. 3)Dirichlet: Avoid “mind-less computations”, replace them by more “intelligible” proofs from concepts; generalized notion of function. Re 1): Dedekind was Gauss’ last doctoral student in Göttingen. Re 2): Riemann was Dedekind’s friend, and intellectual model, when they were both writing their “habilitation” (second dissertation) there. Re 3): After Gauss’ death, Dirichlet became Dedekind’s senior mentor in Göttingen – long conversations after Dirichlet’s lectures on number theory, edited and later published by Dedekind. Cf. Howard Stein: “Logos, Logic, Logistiké: …” (1988) (also Laugwitz and Ferreirós on Riemann, Tappenden on Riemann and Frege, etc.). I NFLUENCES ON D EDEKIND

6. Anti-“geometricist”: Appeals to geometric evidence are not precise enough; claims about the different status of geometry and arithmetic. Anti-“formalist”: Mathematics is not just manipulation of symbols; results independent from particular symbolic representations. Anti-“computationalist”: Avoid lengthy computations, get to the conceptual core; possibilities for generalization, abstraction. Completion of the “arithmetization of analysis” (Cauchy, Bolzano, Weierstrass, Cantor), also the “arithmetization of algebra”. The birth of “pure mathematics”: arithmetic, from N through Z, Q, R, to C; new parts of algebra (“abstract algebra”) (cf. Ferreirós). Towards a general “axiomatic” approach, in a novel, non-traditional sense (Hilbert’s “formal axiomatics”, “implicit definitions”) (cf. Sieg). Into the 20 th and 21 st Centuries: Set-theoretic foundations; Bourbaki’s architectonic; category theory (Lawvere: “conceptual mathematics”); even HoTT (partly). 6 T HE C ONCEPTUAL T URN

7. Anti-Kantian thrust: From previous appeals to “intuition”, “space and time” (cf. Hamilton’s Kantianism) to “concepts”, “logic” alone. Towards Logicism: The use of concepts formulated within a general theory of sets and functions; purely logical deductions from them. Towards structuralism: As the result of using such concepts and logical deductions, we characterize basic “structural features”. But: What about the underlying logical principles, especially for existential claims (infinite sets, system of cuts, etc.)? Left implicit by Dedekind! Yet he insists on proving the existence of relevant systems of objects and functions (of a simple infinity, a complete ordered field). And he appeals to the “self”, “thoughts”, etc. for N; then further constructions, via power sets, equivalence classes, etc., for Z, Q, R, C. What, then, is the sense of “logic” underwriting such existence proofs and concepts? Not traditional logic; nor current “Tarskian” logic! 7 P HILOSOPHICAL I MPLICATIONS

8. Relevant works by Cassirer: “Kant und die moderne Mathematik” (1907) Substanzbegriff und Funktionsbegriff (1910) Philosophie der Symbolischen Formen, Vol. 3 (1929) The Problem of Knowledge, Vol. IV (1930s, published in 1950) Explicit adoption of: Dedekind’s structuralism (much earlier than Benacerraf, Resnik, etc.) Dedekind’s logicism (rather than Frege’s, Russell’s, or Couturat’s) Broader framework now: Incorporated into Marburg-style “critical” or “logical idealism”. Shift with respect to “concept formation” in mathematics (cf. Heis). Namely, shift from “substance concepts” to “function concepts”. Then embedded into an even broader theory of “symbolic forms”. 8 C ASSIRER ’ S R ECEPTION OF D EDEKIND

9. Kantian themes throughout: Against any kind of “naïve realism”, for mathematics and in general; emphasis on the crucial role of “synthesis” in all human cognition. Kant’s idea of the “construction of concepts” – but traditional geometric constructions (straight edge & compass) are to be replaced by “logical” ones. Away from Kant and back again (!?): Initially Cassirer rejects any appeal to “intuition”, relying on Dedekind here (and Russellian logic); especially “intuition” in any “psychological” sense. “Functional” mathematics, as represented by Dedekind, is a “fact of science” for which philosophy should account in terms of its conditions, its “origin”. In Cassirer’s later works, there are indications that “intuition” in a broader sense is not discarded – cf. Dedekind’s appeal to the “self”, “thoughts”, etc.? No appeal to “geometric evidence” in the traditional sense at that level, only to more basic, more general “intuitive” considerations (not fully clarified). A matter of “orientation” in thought (cf. Kant’s “Was heißt sich im Denken orientieren?”); especially: subject vis-à-vis objects; representations of both. Perhaps: “topo-logical” (as Pierre Keller has suggested to me)? 9 C ASSIRER ON D EDEKIND AND K ANT

10. Rethinking of mathematical concepts in the 19 th century, especially in the Göttingen tradition (later also: Hilbert, Minkowski, Noether, etc.). New foundational background: A general theory of sets/classes and of functions (later morphing into axiomatic set theory and category theory). From “geometricism” through “arithmetization” to “logicism”, the latter understood in a broad sense (beyond Frege, Russell, and Couturat). Rejection of earlier, strong appeals to “intuition” (traditional Euclidean geometry); a “conceptual” approach instead – see Dedekind/Cassirer. A remaining, or resurfacing, question about “intuition” in a different, perhaps “topological” sense (but not Brouwerian intuitionism/revisionism). An issue especially for the principles underlying the existence/construction results needed; also with respect to symbolic representation (later Cassirer). Perhaps rehabilitation of Kant, after all? At least if he is interpreted (re- interpreted?) as appealing to “intuition” along such lines (“orientation”)? In any case, there remains a question about the role of “intuition” (e.g., for the “cumulative hierarchy of sets”); or conversely, about the nature of “logic”. 10 C ONCLUSION