Chapter

Positing Mathematical Objects

Michael D. Resnik

in Mathematics as a Science of Patterns

Published in print December 1999 | ISBN: 9780198250142
Published online November 2003 | e-ISBN: 9780191598296 | DOI: http://dx.doi.org/10.1093/0198250142.003.0009
 Positing Mathematical Objects

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If, as I grant, mathematical objects are abstract entities existing outside of space and time, and if the idea of supernaturally grasping abstract entities is scientifically unacceptable, then we need to explain how we can attain mathematical knowledge using our ordinary faculties. I try to meet this challenge through a postulational account of the genesis of our mathematical knowledge, according to which our ancestors introduced mathematical objects by first positing geometric ideals and then postulating abstract mathematical entities. Since positing involves simply introducing a discourse about objects and affirming their existence, positing mathematical objects involves nothing more serious than writing fiction. For this reason, postulational approaches seem better suited for conventionalists; so in the second part of this chapter, I explain how positing in mathematics is different from positing in fiction, and how we can gain knowledge from the former. Finally, I try to make sense of the idea that mathematical postulates are about an independent mathematical reality and that we can refer to that reality through them, by giving an immanent and disquotational account of reference and contrasting it with a transcendent/causal account.

Keywords: conventionalists; disquotation; fiction; genesis; immanent; mathematical knowledge; mathematical object; mathematical postulate; mathematical realism; positing; postulational; reference; transcendent

Chapter.  8596 words. 

Subjects: Philosophy of Mathematics and Logic

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