INTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUM
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INTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUM
INTUITIONS OF THREE KINDSIN GODEL’S VIEWS ON THE CONTINUUMABSTRACT: Godel judges certain consequences of the continuum hypothesis to be implausible, a INTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUMand suggests that mathematical intuition may be able to lead us to axioms from which that hypothesis could be refuted. It is argued that Gode I must take the faculty that leads him to his judgments of implausibility to be a different one from the faculty of mathematical intuition that is supposed to INTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUM lead US to new axioms. It is then argued that the two faculties are very hard to tell apart, and that as a result the very existence of mathematicalINTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUM
intuition in Godel’s sense becomes doubtful.John p. BurgessDepartment of PhilosophyPrinceton UniversityPrinceton. NJ 08544-1006 USAINTUITIONS OF THREEINTUITIONS OF THREE KINDSIN GODEL’S VIEWS ON THE CONTINUUMABSTRACT: Godel judges certain consequences of the continuum hypothesis to be implausible, a INTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUMinuum problem.1 have been much discussed, and yer some questions have perhaps nor received all the attention they deserve. 1 will address two here.First, an exegetical question. Late in the paper Gõdel mentions several consequences of the continuum hypothesis (CH), most of them asserting the existen INTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUMce of a subset of the straight line with the power of the continuum having some properly implying the "extreme rareness" of the set.2 He judges all thINTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUM
ese consequences of (’11 to be implausible. The question 1 wish to consider is this: What is the epistemological status of Godet's judgments of implauINTUITIONS OF THREE KINDSIN GODEL’S VIEWS ON THE CONTINUUMABSTRACT: Godel judges certain consequences of the continuum hypothesis to be implausible, a INTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUMquestion. Godel makes much of the experience of the axioms of set theory "forcing themselves upon one as true," and at least in the continuum problem paper makes this experience the main reason3for positing such a faculty as "mathematical intuition." After several senses of "intuition" have been dis INTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUMtinguished and examined, however. I wish to address the question: In order to explain the Godelian experience, do we really need to posit "mathematicaINTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUM
l intuition," or will some more familiar and less problematic type of intuition suffice for the explanation? I will tentatively suggest that Godel docINTUITIONS OF THREE KINDSIN GODEL’S VIEWS ON THE CONTINUUMABSTRACT: Godel judges certain consequences of the continuum hypothesis to be implausible, a INTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUMonIn the broadest usage of "intuition" in contemporary philosophy, the term may be applied to any source (or in a transferred sense, to any item) of purported knowledge not obtained by conscious inference from anything more immediate. Sense-perception fits this characterization, but so does much els INTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUMe, so we must distinguish sensory from nonsensory intuition. Narrower usages may exclude one or the other. Ordinary English tends to exclude sense-perINTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUM
ception, whereas Kant scholarship, which traditionally uses "intuition" to render Kant's "Anschauung." makes sense-perception the paradigm case.34If wINTUITIONS OF THREE KINDSIN GODEL’S VIEWS ON THE CONTINUUMABSTRACT: Godel judges certain consequences of the continuum hypothesis to be implausible, a INTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUMough all objects of outer sense have spatial features and all objects of outer and inner sense alike have temporal features, space and time are features only of things as they appear to US. not of things as they are in themselves. They are forms of sensibility which we impose on the matter of sensat INTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUMion, and it is because they come from US rather than from the things that we can have knowledge of them in advance of interacting with the things. OnlINTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUM
y empirical, a posteriori intuition can provide specific knowledge of specific things in space and time, but pure intuition, spatial and temporal, canINTUITIONS OF THREE KINDSIN GODEL’S VIEWS ON THE CONTINUUMABSTRACT: Godel judges certain consequences of the continuum hypothesis to be implausible, a INTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUMand of arithmetic amounts to.Or so goes Kant’s story, simplified to the point of caricature. Kant claimed that his story alone was able to explain how we are able to have the a priori knowledge of three-dimensional Euclidean geometry and of arithmetic that we have. But as is well known, not long aft INTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUMer Kant's death doubts arose whether we really do have any such a priori knowledge in the case of three-dimensional Euclidean geometry, and later doubINTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUM
ts also arose as5to whether Kant’s story is really needed to explain how we are able to have the a priori knowledge of arithmetic that we do have. GõdINTUITIONS OF THREE KINDSIN GODEL’S VIEWS ON THE CONTINUUMABSTRACT: Godel judges certain consequences of the continuum hypothesis to be implausible, a INTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUMtinguishes mathematical geometry and physical geometry; and while the one may provide a priori knowledge and the other knowledge of the world around US. neither provides a priori knowledge of the world around US. Mathematical geometry provides know ledge only of mathematical spaces, w hich are usual INTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUMly taken to be just certain set-theoretic structures. Physical geometry provides only empirical knowledge, and is inextricably intertwined with empiriINTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUM
cal theories of physical forces such as electromagnetism and gravitation.And for neither mathematical nor physical geometry docs three-dimensional EucINTUITIONS OF THREE KINDSIN GODEL’S VIEWS ON THE CONTINUUMABSTRACT: Godel judges certain consequences of the continuum hypothesis to be implausible, a INTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUMo longer thought to be a good model of the w orld in which we live and move and have our being. Already with special relativity physical space and time are merged into a four-dimensional physical spacetime, so that it is only relative to a frame of reference that we may6 INTUITIONS OF THREE KINDS IN GÖDEL''S VIEWS ON THE CONTINUUMINTUITIONS OF THREE KINDSIN GODEL’S VIEWS ON THE CONTINUUMABSTRACT: Godel judges certain consequences of the continuum hypothesis to be implausible, aGọi ngay
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