Algebra, Theory of Numbers and Mathematical Logic

Adherents of formal logic may be compared to a maker of porcelain dishes who would contend that he was simply paying attention to the form of his dishes, pots, and vases, but that he did not have anything to do with the raw material.

All competent thinkers agree with Bacon that there can be no real knowledge except that which rests upon observed facts. This fundamental maxim is evidently indisputable if it is applied, as it ought to be, to the mature state of our intelligence. But, if we consider the origin of our knowledge, it is no less certain that the primitive human mind could not, and indeed ought not to, have thought in that way. For if, on the one hand, every Positive theory must necessarily be founded upon observations, it is, on the other hand, no less true that, in order to observe, our mind has need of some theory or other. If in contemplating phenomena we did not immediately connect them with principles, not only would it be impossible for us to combine these isolated observations, and therefore to derive profit from them, but we should even be entirely incapable of remembering facts, which would for the most remain unnoted by us.
Thus there were two difficulties be overcome: the human mind had to observe in order to form real theories, and yet had to form theories of some sort before it could apply itself to a connected series of observations. The primitive human mind, therefore, found itself involved in a vicious circle, from which it would never have had any means of escaping, if a natural way of the difficulty had not fortunately found by the spontaneous development of Theological conceptions. ...chimerical hopes ..exaggerated ideas of man's importance in the universe to which the Theological Philosophy ...at the commencement, ...afforded an indispensable stimulus without the aid which we cannot, indeed, conceive how the primitive human mind would have been induced to undertake any arduous labours.

It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better.