In his essay, “On ‘Insolubilia’ and their Solution in Symbolic Logic,” Russell has a brief discussion about the role of “intuition” as evidence for truths in math and logic. I’m tempted to quote the whole two pages or so of Russell on this topic, but I’ve decided to sum up most of it, and quote sparingly. Russell begins with a shocker: “The method of logistic is fundamentally the same as that of every other science.” Intuition is not an infallible source of knowledge according to Russell. “The object is not to banish ‘intuition’,” Russell explains, “but to test and systematise its employment, to eliminate errors to which its ungoverned use gives rise, and to discover general laws from which, by deduction, we can obtain true results never contradicted, and in crucial instances confirmed, by intuition.”
How does one confirm basic truths in math and logic? According to Russell, “The ‘primitive propositions’ with which the deductions of logistic begin should, if possible, be evidence to intuition; but that is not indispensable, nor is it, in any case, the whole reason for their acceptance.” The basis for accepting primitive propositions is justified inductively. For example, all of the known consequences of such propositions seems to be true, while none of the known consequences are not.
“Among several systems fulfilling all these conditions, that one is to be preferred, aesthetically, in which the primitive propositions are fewest and most general; exactly as the law of gravitation is to be preferred to Kepler’s three laws as the starting-point of mathematical deductions.” Ah, simplicity and beauty are important as well. Interesting…
Of course, Russell thinks we need some form of correction and testing of our logical/mathematical intuitions because paradoxes reveal that these inuitions are fallible. Rather than junking our intuitions, Russell proposes a method of systematization and testing them—making logic & math an inductive science like astronomy!