Commentators who today talk of 'The Dark Ages' when faith instead of reason was said to ruthlessly rule, have for their animadversions only the excuse of perfect ignorance. Both Aquinas' intellectual gifts and his religious nature were of a kind that is no longer commonly seen in the Western world.

Commentators who today talk of 'The Dark Ages' when faith instead of reason was said to ruthlessly rule, have for their animadversions only the excuse of perfect ignorance. Both Aquinas' intellectual gifts and his religious nature were of a kind that is no longer commonly seen in the Western world.

An axiomatic system establishes a reverberating relationship between what a mathematician assumes (the axioms) and what he or she can derive (the theorems). In the best of circumstances, the relationship is clear enough so that the mathematician can submit his or her reasoning to an informal checklist, passing from step to step with the easy confidence the steps are small enough so that he cannot be embarrassed nor she tripped up.

Validity is the touchstone of inference, and truth of judgment: the fact that vichyssoise is cold ratifies the judgment that vichyssoise is, indeed, cold, and the judgment that vichyssoise is cold expresses the fact that vichyssoise is cold.

No man is obliged to be what he might have been.

The world of shapes, lines, curves, and solids is as varied as the world of numbers, and it is only our long-satisfied possession of Euclidean geometry that offers us the impression, or the illusion, that it has, that world, already been encompassed in a manageable intellectual structure. The lineaments of that structure are well known: as in the rest of life, something is given and something is gotten; but the logic behind those lineaments is apt to pass unnoticed, and it is the logic that controls the system.

If moral statements are about something, then the universe is not quite as science suggests it is, since physical theories, having said nothing about God, say nothing about right or wrong, good or bad. To admit this would force philosophers to confront the possibility that the physical sciences offer a grossly inadequate view of reality. And since philosophers very much wish to think of themselves as scientists, this would offer them an unattractive choice between changing their allegiances or accepting their irrelevance.