From mathematics to philosophy Quotes

“It is highly likely that there are different levels of mathematical activity which can be measured by the ease of mechanization. For example, Euler told of how his theorems were often first discovered by empirical and formalistic experimentations. While these experimentations are probably easy to mechanize, the steps of deciding what experimentations to make and of finding afterwards the correct statement and proof of the theorems suggested are of a higher level and much harder to mechanize”
― Hao Wang, From Mathematics To Philosophy

“In the direction of formalization, there are two major successes in modern logic. First, the fairly well established conclusion that all of mathematics is reducible to axiomatic set theory and that, if one takes enough trouble, mathematical proofs can be reproduced in this system completely formally in the sense of mechanical checkability. Second, the results of Skolem and Herbrand according to which we can, by construing mathematical theorems as conditional theorems (viz. that the axioms imply the theorem) in the predicate calculus, search for each mathematical proof in a mechanical (in principle) way to determine whether a related Herbrand expansion contains a contradiction. Impressive as these results are, and encouraging as they are for the project of mechanizing mathematical arguments, they are only theoretical results which do not establish the strong conclusion that mathematical reasoning (or even a major part of it) is mechanical in nature.”
― Hao Wang, From Mathematics To Philosophy