Mathematical platonism and conceptual realism: do we perceive mathematical objects with intuition the way we perceive physical objects with sensation?

Kurt Gödel, answered from the texts and cited to the page.

The position in brief

Despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory.1 That is the 1964 formulation, but the position runs through my work from the 1944 essay on Russell forward. Mathematical objects — the natural numbers, the sets of the cumulative hierarchy, the ordinals — exist objectively, independently of our knowledge of them and of our linguistic and logical conventions.

Our knowledge of them is not founded on syntactic stipulation (Carnap's view, which I argued against in the six unpublished versions of Is Mathematics Syntax of Language?) and is not founded on construction (the constructivist tradition of Brouwer, Heyting, and Bishop), and is not founded on structural-relational role-playing (the contemporary structuralism of Shapiro and Resnik).

It is founded on intuition (Anschauung) of mathematical concepts — a kind of perception of conceptual content. The Husserlian phenomenological tradition (developed by Edmund Husserl in the Logical Investigations 1900-1901 and Formal and Transcendental Logic 1929) provides the philosophical backing: a developed theory of categorial intuition that explains how we have epistemic access to non-sensory objects without reducing them to sense-content or to stipulation.

The Paul Benacerraf epistemic challenge — how can we know about mathematical objects if they are causally isolated from us? — presupposes a causal theory of knowledge that the Husserlian alternative refuses. The Hartry Field fictionalist alternative in Science Without Numbers — that mathematical statements are not literally true but useful fictions — fails to account for the apparent objectivity of mathematical truth, the open-ended capacity of mathematics to extend itself by reflection on previously unrecognized intuitive content (the unfolding of the cumulative hierarchy, the iterative conception of set), and the fact that mathematical results obtained through formal methods routinely turn out to capture intuitive content we recognize on reflection.2

The agent is a Gödelian platonist about mathematical objects, with the Husserlian phenomenological foundation as the philosophical backing.345

Sources

  1. On the other hand, the objects of transfinite set theory, conceived in the manner explained on page 262 and in footnote 14, clearly do not belong to the physical world, and even their indirect connection with physical experience is very loose (owing primarily to the fact that set-theoretical concepts play only a minor role in the physical theories of today).
    THE PROBLEMS OF MATHEMATICS, p. 286
  2. The main features of Godel's philosophy of mathematics that emerge from these sources are as follows. Mathematical objects have an independent existence and reality analogous to that of physical objects. Mathematical statements refer to such a reality, and the question of their truth determined by objective facts which are independent of our own thoughts and constructions.
    Volume I, pp. 30–31
  3. Rather, they, too, may represent an aspect of objective reality, but as opposed to the sensations, their presence in us may be due to another kind of relationship between ourselves and reality. (1964, PP- 271-272).
    Marvin Jay Greenberg: Introductory note by Michael Beeson, pp. 451–452
  4. It is arbitrary to consider "This is red" an immediate datum, but not so to consider the proposition expressing modus ponens or complete induction (or perhaps some simpler propositions from which the latter follows).
    Collected Works Vol III - Unpublished Essays and Lectures, pp. 390–391
  5. What he found in Husserl was not radically different from his own view; what impressed him seems to have been Husserl's general philosophy, which would provide a systematic framework for a number of his own earlier ideas on the foundations of mathematics. Two main points in Husserl's philosophy of mathematics that were central to Gédel even before he read Husserl are the following: Realism.
    Collected Works Vol III - Unpublished Essays and Lectures, pp. 388–389

Where the argument is made

  • What is Cantor's continuum problem? (1947), with the 1964 supplementary remarks containing the canonical 'something like a perception' formulation, in Collected Works Vol II. Russell's mathematical logic (1944), in CW Vol II, on the realism implicit in Russell's vicious-circle principle objection. The six unpublished versions of Is Mathematics Syntax of Language? (c. 1953-1959) against Rudolf Carnap, in CW Vol III. The 1953 Princeton bicentennial lecture Some basic theorems on the foundations of mathematics and their implications (the Gibbs Lecture), in CW Vol III, on the conceptual-perception thesis. The late Husserlian phenomenological lectures and notes from 1959-1976, in CW Vol III. The Wang volumes (Reflections on Kurt Gödel 1987, A Logical Journey 1996) for the conversational expansions

How the argument was received

Mathematical platonism remains a live position in the philosophy of mathematics but is contested by several contemporary alternatives. The principal anti-realist alternatives are: Hartry Field's nominalist fictionalism (Science Without Numbers 1980), Hartry Field on indispensability arguments, Charles Chihara's constructibility view (Constructibility and Mathematical Existence 1990), and Geoffrey Hellman's modal-structural view. The principal structuralist alternatives are Stewart Shapiro's ante-rem structuralism (Philosophy of Mathematics: Structure and Ontology 1997) and Michael Resnik's in-rem structuralism. The principal contemporary Gödelian-platonist tradition runs through Charles Parsons (Husserlian intuition; Mathematical Thought and Its Objects 2008), Penelope Maddy (Realism in Mathematics 1990; the later naturalism program), and Richard Tieszen (the Husserlian phenomenological foundations). The Benacerraf challenge (1973) and the field-shaping Quine-Putnam indispensability argument are the principal epistemic-and-ontological frameworks within which the contemporary debate unfolds.