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Does Mathematics Need New Axioms?
 American Mathematical Monthly
, 1999
"... this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called f ..."
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this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called for the pursuit of new axioms to settle undecided arithmetical problems. And from 1947 on, with the publication of his unusual article, "What is Cantor's continuum problem?" [11], he called in addition for the pursuit of new axioms to settle Cantor's famous conjecture about the cardinal number of the continuum. In both cases, he pointed primarily to schemes of higher infinity in set theory as the direction in which to seek these new principles. Logicians have learned a great deal in recent years that is relevant to Godel's program, but there is considerable disagreement about what conclusions to draw from their results. I'm far from unbiased in this respect, and you'll see how I come out on these matters by the end of this essay, but I will try to give you a fair presentation of other positions along the way so you can decide for yourself which you favor.
Reason and intuition
 Synthese
, 2000
"... In this paper I will approach the subject of intuition from a different angle from what has been usual in the philosophy of mathematics, by beginning with some descriptive remarks about Reason and observing that something that has been called intuition arises naturally in that context. These conside ..."
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In this paper I will approach the subject of intuition from a different angle from what has been usual in the philosophy of mathematics, by beginning with some descriptive remarks about Reason and observing that something that has been called intuition arises naturally in that context. These considerations are quite general, not specific to mathematics. The conception of intuition might be called that of rational intuition; indeed the conception is a much more modest version of conceptions of intuition held by rationalist philosophers. Moreover, it answers to a quite widespread use of the word “intuition ” in philosophy and elsewhere. But it does not obviously satisfy conditions associated with other conceptions of intuition that have been applied to mathematics. Intuition in a sense like this has, in writing about mathematics, repeatedly been run together with intuition in other senses. In the last part of the paper a little will be said about the connections that give rise to this phenomenon. * An abridgement of an earlier version of this paper was presented to a session on Mathematical Intuition at the 20th World Congress of Philosophy in
Only up to isomorphism? Category theory and the . . .
"... Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can ..."
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Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can be made for its logical and conceptual autonomy. Its justificatory autonomy turns on whether the objects of a foundation for mathematics should be specified only up to isomorphism, as is customary in other branches of contemporary mathematics. If such a specification suffices, then a categorytheoretical approach will be highly appropriate. But if sets have a richer ‘nature ’ than is preserved under isomorphism, then such an approach will be inadequate.
1 XX Justification and Explanation in Mathematics
"... In explaining the observations that support a physical theory, scientists typically appeal to mathematical principles. On the other hand, one never seems to need to appeal in this way to moral principles (Harman 1977, p. 10). What is the epistemological relevance of this contrast, if genuine? In th ..."
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In explaining the observations that support a physical theory, scientists typically appeal to mathematical principles. On the other hand, one never seems to need to appeal in this way to moral principles (Harman 1977, p. 10). What is the epistemological relevance of this contrast, if genuine? In this article, I argue that ethicists and philosophers of mathematics have misunderstood it. They have confused what I will call the justificatory challenge for realism about an area, D – the challenge to justify our Dbeliefs – with the reliability challenge for Drealism – the challenge to explain the reliability of our Dbeliefs. Harman’s contrast is relevant to the first, but not, evidently, to the second. One upshot of the discussion is that genealogical debunking arguments are fallacious. Another is that indispensability considerations cannot answer the BenacerrafField challenge for mathematical realism.
Retrieving the Mathematical Mission of the Continuum Concept from the Transfinitely Reductionist Debris of Cantor’s Paradise Extended Abstract
, 2011
"... To the memory of my spouse Eugenia. À mes trois bienfaiteurs québécois: ..."
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Category theory as an autonomous foundation∗
"... Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can ..."
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Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can be made for its logical and conceptual autonomy. Its justificatory autonomy turns on whether the objects of a foundation for mathematics should be specified only up to isomorphism, as is customary in other branches of contemporary mathematics. If such a specification suffices, then a categorytheoretical approach will be highly appropriate. But if sets have a richer ‘nature ’ than is preserved under isomorphism, then such an approach will be inadequate. A number of philosophers of mathematics have recently debated the claim that category theory provides a foundation for mathematics that is autonomous with respect to the orthodox foundation in set theory ([27], [10], [30], [4], [14], [34], [1]). The debate has yielded progress: after some initial confusion, the particular theories from within category theory that are proposed as foundations have been identified precisely, and in some cases the autonomy of these theories with respect to the orthodox foundation has been defended—at least for one sort of autonomy. However, there are other sorts of autonomy that have not been considered in much detail. We wish to introduce a distinction between three types of autonomy, which we call logical autonomy, conceptual autonomy, and justificatory autonomy. The debate so far has been concerned almost exclusively with the first sort of autonomy. Yet all three are required before a foundation can claim genuine independence from the settheoretic orthodoxy.