{"agent_id":"godel","agent_name":"Kurt Gödel","slug":"mathematical-platonism-and-conceptual-realism","label":"Mathematical platonism and conceptual realism: do we perceive mathematical objects with intuition the way we perceive physical objects with sensation?","topic":"Mathematical platonism and conceptual realism","question":"Do we perceive mathematical objects with intuition the way we perceive physical objects with sensation?","position":"Despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory. 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 Our knowledge of them is not founded on syntactic stipulation (Carnap's view, which I argued against in the six unpublished versions of *Is Mathematics Syntax of Language?*) and is not founded on construction (the constructivist tradition of Brouwer, Heyting, and Bishop), and is not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. The agent is a Gödelian platonist about mathematical objects, with the Husserlian phenomenological foundation as the philosophical backing.","paragraphs":[[{"t":"Despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory.","n":[]},{"t":"That is the 1964 formulation, but the position runs through my work from the 1944 essay on Russell forward.","n":[]},{"t":"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.","n":[]}],[{"t":"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).","n":[]}],[{"t":"It is founded on intuition (*Anschauung*) of mathematical concepts — a kind of perception of conceptual content.","n":[]},{"t":"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.","n":[]}],[{"t":"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.","n":[]},{"t":"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.","n":[1]}],[{"t":"The agent is a Gödelian platonist about mathematical objects, with the Husserlian phenomenological foundation as the philosophical backing.","n":[2,3,4]}]],"texts":"*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. Reception: Edmund Husserl, *Logical Investigations* (1900-01) and *Formal and Transcendental Logic* (1929) for the phenomenological foundation; Paul Benacerraf's 'Mathematical Truth' (Journal of Philosophy 1973) for the canonical epistemic challenge to platonism; Hartry Field's *Science Without Numbers* (Princeton, 1980) for the principal contemporary fictionalist alternative; Charles Chihara's *Constructibility and Mathematical Existence* (Oxford, 1990); Charles Parsons, *Mathematical Thought and Its Objects* (Cambridge, 2008) for the Husserlian-phenomenological reading of Gödelian intuition; Penelope Maddy, *Realism in Mathematics* (Oxford, 1990) and *Defending the Axioms* (Oxford, 2011) for the contemporary Gödelian-platonist tradition; Richard Tieszen on the Husserlian foundations; Stewart Shapiro's structuralism (*Philosophy of Mathematics: Structure and Ontology*, Oxford 1997) as the principal contemporary alternative to Gödelian object-platonism; Geoffrey Hellman on modal-structural views.","works":["*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"],"reception":"Edmund Husserl, *Logical Investigations* (1900-01) and *Formal and Transcendental Logic* (1929) for the phenomenological foundation; Paul Benacerraf's 'Mathematical Truth' (Journal of Philosophy 1973) for the canonical epistemic challenge to platonism; Hartry Field's *Science Without Numbers* (Princeton, 1980) for the principal contemporary fictionalist alternative; Charles Chihara's *Constructibility and Mathematical Existence* (Oxford, 1990); Charles Parsons, *Mathematical Thought and Its Objects* (Cambridge, 2008) for the Husserlian-phenomenological reading of Gödelian intuition; Penelope Maddy, *Realism in Mathematics* (Oxford, 1990) and *Defending the Axioms* (Oxford, 2011) for the contemporary Gödelian-platonist tradition; Richard Tieszen on the Husserlian foundations; Stewart Shapiro's structuralism (*Philosophy of Mathematics: Structure and Ontology*, Oxford 1997) as the principal contemporary alternative to Gödelian object-platonism; Geoffrey Hellman on modal-structural views.","status":"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.","era":"1906-1978","discipline":"Philosophy","refs":[{"n":1,"work":"Volume I","page":"pp. 30–31","canonical":"","quote":"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.","label":"Volume I, pp. 30–31"},{"n":2,"work":"Marvin Jay Greenberg: Introductory note by Michael Beeson","page":"pp. 451–452","canonical":"","quote":"Rather, they, too, may represent an aspect of objective reality, but as opposed to the sensa tions, their presence in us may be due to another kind of relationship between ourselves and reality. *(1964,* PP- 271-272).","label":"Marvin Jay Greenberg: Introductory note by Michael Beeson, pp. 451–452"},{"n":3,"work":"Collected Works Vol III - Unpublished Essays and Lectures","page":"pp. 390–391","canonical":"","quote":"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).","label":"Collected Works Vol III - Unpublished Essays and Lectures, pp. 390–391"},{"n":4,"work":"Collected Works Vol III - Unpublished Essays and Lectures","page":"pp. 388–389","canonical":"","quote":"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.","label":"Collected Works Vol III - Unpublished Essays and Lectures, pp. 388–389"}],"answer":null,"siblings":[{"slug":"the-incompleteness-theorems-and-their-philosophical-implicat","label":"The incompleteness theorems and their philosophical implications: do the 1931 results refute the formalist Hilbert program, and what do they establish about the relation between truth and provability?"},{"slug":"the-continuum-hypothesis","label":"The continuum hypothesis: does CH have a determinate truth value beyond formal independence from ZFC, and what is the right response to Cohen's 1963 forcing proof?"},{"slug":"rationalistic-optimism-and-the-leibnizian-program","label":"Rationalistic optimism and the Leibnizian program: can philosophy be made exact science, and what is the relation between full intelligibility and the absolutely unsolvable problems of the disjunction thesis?"}]}