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. (shrink)
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. (shrink)
The paper examines the roots of Husserlian phenomenology in Weierstrass's approach to analysis. After elaborating on Weierstrass's programme of arithmetization of analysis, the paper examines Husserl's 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. Husserl's 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. (shrink)
This paper discusses Jean van Heijenoort’s (1967) and Jaakko and Merrill B. Hintikka’s (1986, 1997) distinction between logic as auniversal language and logic as a calculus, and its applicability to Edmund Husserl’s phenomenology. Although it is argued that Husserl’s phenomenology shares characteristics with both sides, his view of logic is closer to the model-theoretical, logic-as-calculus view. However, Husserl’s philosophy as transcendental philosophy is closer to the universalist view. This paper suggests that Husserl’s position shows that holding a model-theoretical view of (...) logic does not necessarily imply a calculus view about the relations between language and the world. The situation calls for reflection about the distinction: It will be suggested that the applicability of the van Heijenoort and the Hintikkas distinction either has to be restricted to a particular philosopher’s views about logic, in which case no implications about his or her more general philosophical views should be inferred from it; or the distinction turns into a question of whether our human predicament is inescapable or whether it is possible, presumably by means of model theory, to obtain neutral answers to philosophical questions. Thus the distinction ultimately turns into a question about the correct method for doing philosophy. (shrink)
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. (shrink)
The paper traces the development and the role of syntactic reduction in Edmund Husserl’s early writings on mathematics and logic, especially on arithmetic. The notion has its origin in Hermann Hankel’s 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 formal mathematics. (shrink)
In his 1896 lecture course on logic–reportedly a blueprint for the Prolegomena to Pure Logic –Husserl develops an explicit account of logic as an independent and purely theoretical discipline. According to Husserl, such a theory is needed for the foundations of logic (in a more general sense) to avoid psychologism in logic. The present paper shows that Husserl’s conception of logic (in a strict sense) belongs to the algebra of logic tradition. Husserl’s conception is modeled after arithmetic, and respectively logical (...) inferences are viewed as analogical to arithmetical calculation. The paper ends with an examination of Husserl’s involvement with the key characters of the algebra of logic tradition. It is concluded that Ernst Schröder, but presumably also Hermann and Robert Grassmann influenced Husserl most in his turn away from psychologism. (shrink)
This paper discusses Jean van Heijenoort’s and Jaakko and Merrill B. Hintikka’s distinction between logic as auniversal language and logic as a calculus, and its applicability to Edmund Husserl’s phenomenology. Although it is argued that Husserl’s phenomenology shares characteristics with both sides, his view of logic is closer to the model-theoretical, logic-as-calculus view. However, Husserl’s philosophy as transcendental philosophy is closer to the universalist view. This paper suggests that Husserl’s position shows that holding a model-theoretical view of logic does not (...) necessarily imply a calculus view about the relations between language and the world. The situation calls for reflection about the distinction: It will be suggested that the applicability of the van Heijenoort and the Hintikkas distinction either has to be restricted to a particular philosopher’s views about logic, in which case no implications about his or her more general philosophical views should be inferred from it; or the distinction turns into a question of whether our human predicament is inescapable or whether it is possible, presumably by means of model theory, to obtain neutral answers to philosophical questions. Thus the distinction ultimately turns into a question about the correct method for doing philosophy. (shrink)
It is beginning to be rather well known that Edmund Husserl, the founder of phenomenological philosophy, was originally a mathematician; he studied with Weierstrass and Kronecker in Berlin, wrote his doctoral dissertation on the calculus of variations, and was then a colleague of Cantor in Halle until he moved to the Göttingen of Hilbert and Klein in 1901. Much of Husserl’s writing prior to 1901 was about mathematics, and arguably the origin of phenomenology was in Husserl’s attempts to give philosophical (...) foundations first for analysis and later for the formal sciences in general. However, what exactly Husserl’s thoughts about mathematics were is relatively little known. Stefania Centrone’s book Logic and Philosophy of Mathematics in the Early Husserl fills this lacuna. Centrone deciphers Husserl’s early texts about mathematics and logic somewhat selectively, but also extremely accurately and carefully into lucid English and in terms we know, without however reading contemporary views into Husserl’s views. Her analysis reveals for example that Husserl’s view of logic is close to what is today taught in the standard textbooks and that he viewed abstract mathematics as a theory of structures. Her work also uncovers Husserl’s relationship to the algebraists of logic as well as to Bolzano, Frege, and Hilbert.The book is composed of three rather independent chapters. In contrast to most of the other commentaries on early Husserl, the organization of the book is thematic rather than an attempt to document the various stages in the development of Husserl’s views. The first chapter discusses Husserl’s major work on arithmetic, the second the idea of pure logic, and the last the imaginary in mathematics. (shrink)
The aim of this volume is to offer an updated account of the transcendental character of phenomenology. The main question concerns the sense and relevance of transcendental philosophy today: What can such philosophy contribute to contemporary inquiries and debates after the many reasoned attacks against its idealistic, aprioristic, absolutist and universalistic tendencies—voiced most vigorously by late 20th century postmodern thinkers—as well as attacks against its apparently circular arguments and suspicious metaphysics launched by many analytic philosophers? Contributors also aim to clarify (...) the relations of transcendental phenomenology to other post-Kantian philosophies, most importantly to pragmatism and Wittgenstein’s philosophical investigations. Finally, the volume offers a set of reflections on the meaning of post-transcendental phenomenology. (shrink)
