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.
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"Systems of Logic Based on Ordinals," section 11: The purpose of ordinal logics (1938), published in Proceedings of the London Mathematical Society, series 2, vol. 45 (1939)
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In a footnote to the first sentence, Turing added: "We are leaving out of account that most important faculty which distinguishes topics of interest from others; in fact, we are regarding the function of the mathematician as simply to determine the truth or falsity of propositions."Alan Turing
We know truth, not only by the reason, but also by the heart, and it is in this last way that we know first principles; and reason, which has no part in it, tries in vain to impugn them. The sceptics, who have only this for their object, labour to no purpose. We know that we do not dream, and however impossible it is for us to prove it by reason, this inability demonstrates only the weakness of our reason, but not, as they affirm, the uncertainty of all our knowledge. For the knowledge of first principles, as space, time, motion, number, is as sure as any of those which we get from reasoning. And reason must trust these intuitions of the heart, and must base them on every argument. (We have intuitive knowledge of the tri-dimensional nature of space, and of the infinity of number, and reason then shows that there are no two square numbers one of which is double of the other. Principles are intuited, propositions are inferred, all with certainty, though in different ways.) And it is as useless and absurd for reason to demand from the heart proofs of her first principles, before admitting them, as it would be for the heart to demand from reason an intuition of all demonstrated propositions before accepting them.
This inability ought, then, to serve only to humble reason, which would judge all, but not to impugn our certainty, as if only reason were capable of instructing us. Would to God, on the contrary, that we had never need of it, and that we knew everything by instinct and intuition! But nature has refused us this boon. On the contrary, she has given us but very little knowledge of this kind; and all the rest can be acquired only by reasoning.
Therefore, those to whom God has imparted religion by intuition are very fortunate, and justly convinced. But to those who do not have it, we can give it only by reasoning, waiting for God to give them spiritual insight, without which faith is only human, and useless for salvation.Blaise Pascal
...it is rare that mathematicians are intuitive, and that men of intuition are mathematicians, because mathematicians wish to treat matters of intuition mathematically, and make themselves ridiculous, wishing to begin with definitions and then with axioms, which is not the way to proceed in this kind of reasoning. Not that the mind does not do so, but it does it tacitly, naturally, and without technical rules; for the expression of it is beyond all men, and only a few can feel it. 1
Blaise Pascal
Many people think that the progress of the human race is based on experiences of an empirical, critical nature, but I say that true knowledge is to be had only through a philosophy of deduction. For it is intuition that improves the world, not just following a trodden path of thought. Intuition makes us look at unrelated facts and then think about them until they can all be brought under one law. To look for related facts means holding onto what one has instead of searching for new facts. Intuition is the father of new knowledge, while empiricism is nothing but an accumulation of old knowledge. Intuition, not intellect, is the 'open sesame' of yourself.
Albert Einstein
All great men are gifted with intuition. They know without reasoning or analysis, what they need to know.
Alexis Carrel
Indeed, it is not intellect, but intuition which advances humanity. Intuition tells man his purpose in this life.
Albert Einstein
Turing, Alan
Turk, Roy
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