Re-examining the history of mathematics requires also a re-examination of
the philosophy of mathematics, since the current philosophy of mathematics- as-proof excludes the possibility of any mathematics in non-Western cultures.
I
The Nature of Mathematical Proof
1 Euclid and Hilbert
3
History of geometry and the genesis of the current notion of mathematical proof
The currently dominant notion of mathematical proof is re-examined in
a historical perspective, to bring out the religious and political considera- tions that have led to the present-day belief in the certainty of mathematical knowledge and the Greek origins of mathematics. In the absence of any evidence for Euclid, Proclus' religious understanding of the Elements is con- trasted with Hilbert's synthetic interpretation, and with traditional Indian geometry--which permitted the measurement also of curved lines, facilitat- ing the development of the calculus in India.
2 Proof vs Pramāņa
59
Critique of the current notion of mathematical proof, and comparison with the tra-
ditional Indian notion of pramāņa
The currently dominant notion of mathematical proof is re-examined in a
philosophical perspective, in comparison with the traditional Indian notion of pramāņa. The claimed infallibility of deduction or mathematical proof is rejected as a cultural superstition. Logic varies with culture, so the logic underlying deduction can be fixed only by appealing to cultural authority or the empirical. In either case, deduction is more fallible than induction.
In preparation for the next chapter, a brief introduction is here provided
also to the understanding of numbers in the context of the philosophy of
śun- yavāda, which acknowledges the existence of non-representables--necessary also to be able to represent numbers on a computer. This is unlike Platonic idealism or formal mathematics, which introduces supertasks in the under- standing of numbers, whether integers or reals.