By Stanislaw Saks
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Extra resources for Analytic Functions
He is showing mathematics alive, in development. He does not try to explain what math is or how it works. Instead, he shows what mathematicians really do, with an eye to methodology, motivation, goals and criteria. I was fortunate in establishing a partnership with Philip J. Davis. He wanted to write math for the intelligent non-mathematician, while I wanted to write philos¬ ophy of math, so we agreed to work in parallel, for mutual support. Unfortunately, I was overwhelmed by deep personal problems.
I claim that mathematical entities are notions, or ideas, or concepts— not private incommunicable ones, of course, but public, shared ideas. I had to cope with the unwillingness of philosophers to accept this information. This is an Is¬ sue broader than the philosophy of mathematics, it is an issue of general ontology. What kinds of entities are there? In philosophy of mathematics I encountered three sorts of entities. Material, or physical, of course. Also mental, in the sense of the private, internal, subjective world of the individual philosopher.
We could go on to the octahedron, which has three internal diagonals, and then the other Platonic solids, and then regular polytopes in four dimensions, and on and on. My mental models of these other mathematical entities would also enable me to answer questions. Mathematical research is based on reasoning about our mental models of math¬ ematical entities, starting from established mathematics, and seeking to add to it or improve it (not by syntactic transformations of formal “axioms”). It is carried out in human brains and bodies, historically developed in societies on the surface of this planet, and assisted by scraps of paper, blackboards, printed matter, and digital computers.