Mirja Hartimo

Towards completeness: Husserl on theories of manifolds 1890–1901

Husserl’s notion of definiteness, i.e., completeness is crucial to understanding Husserl’s view of logic, and consequently several related philosophical views, such as his argument against psychologism, his notion of ideality, and his view of formal ontology. Initially Husserl developed the notion of definiteness to clarify Hermann Hankel’s ‘principle of permanence’. One of the first attempts at formulating definiteness can be found in the Philosophy of Arithmetic, where definiteness serves the purpose of the modern notion of ‘soundness’ and leads Husserl to a ‘computational’ view of logic. Inspired by Gauss and Grassmann Husserl then undertakes a further investigation of theories of manifolds. When Husserl subsequently renounces psychologism and changes his view of logic, his idea of definiteness also develops. The notion of definiteness is discussed most extensively in the pair of lectures Husserl gave in front of the mathematical society in Göttingen (1901). A detailed analysis of the lectures, together with an elaboration of Husserl’s lectures on logic beginning in 1895, shows that Husserl meant by definiteness what is today called ‘categoricity’. In so doing Husserl was not doing anything particularly original; since Dedekind’s ‘Was sind und sollen die Zahlen’ (1888) the notion was ‘in the air’. It also characterizes Hilbert’s (1900) notion of completeness. In the end, Husserl’s view of definiteness is discussed in light of Gödel’s (1931) incompleteness results.

Phenomenology and Mathematics

R. Tieszen, Mathematical Realism and Transcendental Phenomenological Idealism G.E. Rosado Haddock, Platonism, Phenomenology, and Interderivability C. Hill, Husserl on Axiomatization and Arithmetic D. Lohmar, Intuition in Mathematics. On the function of eidetic variation in mathematical proofs J. Hintikka, How Can a Phenomenologist Have a Philosophy of Mathematics? M. Hartimo, Development of Mathematics and the Birth of Phenomenology J.J da Silva, Beyond Leibniz: Husserls Vindication of Symbolic Knowledge R. Hanna, Mathematical Truth Regained O.K. Wiegand, On Referring to Gestalts.

Mathematical roots of phenomenology: Husserl and the concept of number

The paper examines the roots of Husserlian phenomenology in Weierstrasss approach to analysis. After elaborating on Weierstrasss programme of arithmetization of analysis, the paper examines Husserls Philosophy of Arithmetic as an attempt to provide foundations to analysis. The Philosophy of Arithmetic consists of two parts; the first discusses authentic arithmetic and the second symbolic arithmetic. Husserls novelty is to use Brentanian descriptive analysis to clarify the fundamental concepts of arithmetic in the first part. In the second part, he founds the symbolic extension of the authentically given arithmetic with stepwise symbolic operations. In the process of doing so, Husserl comes close to defining the modern concept of computability. The paper concludes with a brief comparison between Husserl and Frege. While Frege chose to subject arithmetic to logical analysis, Husserl wants to clarify arithmetic as it is given to us. Both engage in a kind of analysis, but while Frege analyses within Begriffsschrift, Husserl analyses our experiences. The difference in their methods of analysis is what ultimately grows into two separate schools in philosophy in the 20th century.

The Development of Mathematics and the Birth of Phenomenology

The article examines Husserl’s view of mathematics as a continuation of Weierstrass’s project. While Husserl adopts the more modern axiomatic approach to mathematics as opposed to Weierstrass’s genetic approach, he continues to be Weierstrassian in his preoccupation for foundational questions. The latter part of the article examines in what way the outcome is Platonistic in Husserl’s own usage of the term. By Platonism Husserl means both a belief in immutable and unchanging ideal structures, as well as, a search for critical justification in reflection. In the latter sense of the term Husserl’s “Platonism” can be seen as an outcome of Husserl’s Weierstrassian ethos.

From geometry to phenomenology

Richard Tieszen [Tieszen, R. (2005). Philosophy and Phenomenological Research, LXX(1), 153–173.] has argued that the group-theoretical approach to modern geometry can be seen as a realization of Edmund Husserl’s view of eidetic intuition. In support of Tieszen’s claim, the present article discusses Husserl’s approach to geometry in 1886–1902. Husserl’s first detailed discussion of the concept of group and invariants under transformations takes place in his notes on Hilbert’s Memoir Ueber die Grundlagen der Geometrie that Hilbert wrote during the winter 1901–1902. Husserl’s interest in the Memoir is a continuation of his long-standing concern about analytic geometry and in particular Riemann and Helmholtz’s approach to geometry. Husserl favored a non-metrical approach to geometry; thus the topological nature of Hilbert’s Memoir must have been intriguing to him. The task of phenomenology is to describe the givenness of this logos, hence Husserl needed to develop the notion of eidetic intuition.

Syntactic reduction in Husserl’s early phenomenology of arithmetic

The paper traces the development and the role of syntactic reduction in Edmund Husserl’s (1856–1938) early writings on mathematics and logic, especially on arithmetic. The notion has its origin in Hermann Hankel’s (1839–1873) principle of permanence that Husserl set out to clarify. In Husserl’s early texts the emphasis of the reductions was meant to guarantee the consistency of the extended algorithm. Around the turn of the century Husserl uses the same idea in his conception of definiteness of what he calls “mathematical manifolds.” The paper argues that the notion anticipates the notion of reduction in term rewrite theory in computer science. The role of the reduction for Husserl is, however, primarily epistemological: its purpose is to impart clarity to (at least parts of) formal mathematics.

Husserl on completeness, definitely

The paper discusses Husserl’s notion of definiteness as presented in his Göttingen Mathematical Society Double Lecture of 1901 as a defense of two, in many cases incompatible, ideals, namely full characterizability of the domain, i.e., categoricity, and its syntactic completeness. These two ideals are manifest already in Husserl’s discussion of pure logic in the Prolegomena: The full characterizability is related to Husserl’s attempt to capture the interconnection of things, whereas syntactic completeness relates to the interconnection of truths. In the Prolegomena Husserl argues that an ideally complete theory gives an independent norm for objectivity for logic and experiential sciences, hence the notion is central to his argument against psychologism. In the Double Lecture the former is captured by non-extendibility, that is, categoricity of the domain, from which, so Husserl assumes, syntactic completeness is thought to follow. In the so-called ‘mathematical manifolds’ the expressions of the theory are equations that are reducible to equations between elements of the theory. With such an equational reduction structure individual elements of the domain are given criteria of identity and hence they are fully determined.

Husserl and Hilbert

The paper examines Husserl’s (1859–1938) phenomenology and Hilbert’s (1862–1943) view of the foundations of mathematics against the backdrop of their lifelong friendship. After a brief account of the complementary nature of their early approaches, the paper focuses on Husserl’s Formale und transzendentale Logik (1929) viewed as a response to Hilbert’s “new foundations” developed in the 1920s. While both Husserl and Hilbert share a “mathematics first,” nonrevisionist approach toward mathematics, they disagree about the way in which the access to it should be construed: Hilbert wanted to reach it and show it consistent by his formalism on the basis of sensuous signs, Husserl held that there should be a reduction to elementary judgements about individuals. Husserl’s reduction does not establish the consistency of mathematics but he claims it is important for the considerations of truth.

HUSSERL AND GÖDEL’S INCOMPLETENESS THEOREMS

The paper examines Husserl’s interactions with logicians in the 1930s in order to assess Husserl’s awareness of Godel’s incompleteness theorems. While there is no mention about the results in Husserl’s known exchanges with Hilbert, Weyl, or Zermelo, the most likely source about them for Husserl is Felix Kaufmann (1895–1949). Husserl’s interactions with Kaufmann show that Husserl may have learned about the results from him, but not necessarily so. Ultimately Husserl’s reading marks on Friedrich Waismann’s Einfuhrung in das mathematische Denken: die Begriffsbildung der modernen Mathematik, 1936 , show that he knew about them before his death in 1938.

Phänomenologie und Mathematik

Nach dem Husserl in den Jahren 1884 bis 1886 die Vorlesungen von Brentano gehort hatte (s. Kap. II.1), entschied er, sein Leben der Philosophie zu widmen. Trotzdem hat er bis etwa 1902 eine Reihe detaillierter Werke im Gebiet der Mathematik geschrieben. Den grosten Teil dieser Zeit hat er in Halle verbracht, wo Cantor sein Kollege war. Im Jahr 1901 ist er nach Gottingen, dank Felix Klein und David Hilbert damals ein weltweit fuhrendes Forschungszentrum der Mathematik, umgezogen.