## Abstract

Traditionally, Henri Poincaré (1845-1912) has been recognized as a pioneer in the discovery of predicative set theory. Less attention, however, has been attributed to the importance of intuition to his interpretation of what it means to exist in the domain of pure mathematics. This imbalance is hard to justify in terms of Poincaré's own philosophical writings, as his contentions with his contemporaries - most notably, the logicists - are typified by appeals to the indispensability of intuition as a foundation for mathematics. The value of predicativism as a means of circumventing the set-theoretic paradoxes and indeed, of rendering mathematical definitions and proofs more transparent is well-recognized. Nevertheless, the associated systems are also noted for their restrictive nature in terms of proof-theoretic strength. It is important, therefore, to take a fresh look at Poincaré's philosophical writings to establish how the divergence between his foundational arguments and his reputation as a predicativist may have emerged. In this chapter, I shall offer new insight into the relevance of intuition to Poincaré's criteria for set existence and in particular, his requirement that infinite sets should be surveyable. On this basis, I shall argue that Poincaréan set theory ought to be more liberal than has traditionally been assumed. In exploring the above criteria, I shall also revisit Poincaré's unexplained affinity towards classical logic. By appealing to the influence of Aristotle's writings on Poincaré's interpretation of logic, I shall show that his ambiguous reference to the intuition of pure number as that of pure logical forms invites further exploration. Thus, I shall argue that Poincaré's treatment of the principles of Non-contradiction and Excluded Middle as necessarily true has its roots in logical intuition - a variety of intuition which he neglects at the cost of his own anti-logicist stance. These findings serve as an extension of my recent work on the nature of mathematical intuition. As such, they provide some essential groundwork for developing an alternative set theory with the potential to create a more successful synergy between Poincaré's philosophy and practice in pure mathematics.

Original language | English |
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Title of host publication | Philosophy of Science |

Editors | Samuel Pintuck, Colin Reynolds |

Publisher | Nova Science Publishers Inc |

Pages | 33-66 |

Number of pages | 34 |

ISBN (Print) | 9781621002765, 1621002764 |

Publication status | Published - Jan 2012 |