Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs.
David HilbertRelated topics
assumption completeness consistency faculty found induction infinity intuition mathematics need primal science understanding whitehead RussellRelated quotes
In asking forgiveness of women for our mythologizing of their bodies, for being unreal about them, we can only appeal to their own sexuality, which is different but not basically different, perhaps, from our own. For women, too, there seems to be that tangle of supplication and possessiveness, that descent toward infantile undifferentiation, that omnipotent helplessness, that merger with the cosmic mother-warmth, that flushed pulse-quickened leap into overestimation, projection, general mix-up.
John Updike
O God, enlarge within us the sense of fellowship with all living things, even our brothers, the animals, to whom Thou gavest the earth as their home in common with us. We must remember with shame that in the past we have exercised the high dominion of man with ruthless cruelty so that the voice of the earth, which should have gone up to thee in song, has been a groan of pain. May we realize that they live, not for us alone, but for themselves and for Thee and that they love the sweetness of life.
Basil of Caesarea
Fasts and vigils, the study of Scripture, renouncing possessions and everything worldly are not in themselves perfection, as we have said; theyare its tools. For perfection is not to be found in them; it is acquired through them. It is useless, therefore, to boast of our fasting, vigils, poverty, and reading of Scripture when we have not achieved the love of God and our fellow men. Whoever has achieved love has God within himself and his intellect is always with God.
John Cassian
Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning... The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings.
Alan Turing
I was early regarded as having unusual intellectual capacity. I was an omnivorous reader, and I added to that a desire to systematize my understanding. As a result, history, for example, was not merely a set of dates and colorful stories; I could understand it as a sequence in which one event flowed out of another. This sense of order crystallized during my high-school and college years into a predominant interest in mathematics and mathematical logic.
Kenneth Arrow
