We can now turn to the paradox of a mathematics based on a system of axioms which are not regarded as self-evident and indeed cannot be known to be mutually consistent. To apply the utmost ingenuity and the most rigorous care to prove the theorems of logic or mathematics, while the premisses of these inferences are cheerfully accepted, without any grounds being given for doing so, as ‘unproven asserted formulae’, might seem altogether absurd…if the acceptance of any proof requires the acceptance without proof of some presuppositions from which the proof is ultimately derived, it follows that the principle of rejecting any unproven statement in mathematics implies also the rejection of all proven statements and therefore of all mathematics.
The solution lies in rejecting the rule which denies acceptance to unproven statements, by admitting that our belief in logically anterior maxims of mathematical procedure is based on our previous acceptance of the procedure as valid…we should declare instead candidly that we dwell on mathematics and affirm its statements for the sake of its intellectually beauty, which betokens the reality of its conceptions and the truth of its assertions.
– Michael Polanyi Personal Knowledge p. 191-192
Ah Polanyi! For those of you who aren’t familiar with him, he was a scientist turned philosopher (of science) in the middle of the last century. His work in general and Personal Knowledge in particular is an attempt to remove the “cult of objectivity” from science. He wanted to replace the idea of science as impersonal with the recognition that the scientist is personally involved in the “art of knowing”.
So, this particular quote is interesting because Polanyi is extending his argument to mathematics, a branch of science that is in a sense easy to regard as the purest and most objective of all (XKCD believes it too). The theorems and proofs of mathematics are not things that can be turned out mechanically as if through a machine (well, not usually). They are statements of “intellectual beauty” that are affirmed by the personal commitment of the mathematician.
The popular understanding (and indeed my base understanding still) of mathematics is that of bits of freewheeling truth that we can derive if only given the right axioms and enough time. While it may still be in some sense true, the process (at least according to Polanyi) is much more subjective since it must always involve a subject (aka the mathematician).