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 TuringRelated topics
aiding call facilities intuition making mathematics perhaps resultRelated 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
Philosophy is written in this grand book, which stands continually open before our eyes (I say the 'Universe'), but can not be understood without first learning to comprehend the language and know the characters as it is written. It is written in mathematical language, and its characters are triangles, circles and other geometric figures, without which it is impossible to humanly understand a word; without these one is wandering in a dark labyrinth.
Galileo Galilei
You are angry with your neighbor, you despise him, do not like to speak peaceably and lovingly to him, because there is something harsh, abrupt, careless, unpleasant to you in his character, in his speech, in his manners-because he is more conscious of his dignity than perhaps is necessary; or because he may be somewhat proud and disrespectful; but you yourself, your neighbor's physician and teacher, are more guilty than him.
John of Kronstadt
...Once you entomb mathematics in an artificial language à la Hilbert, once you set up a completely formal axiomatic system, then you can forget that it has any meaning and just look at it as a game that you play with marks on paper that enable you to deduce theorems from axioms. You can forget about the meaning of the game, the game of mathematical reasoning, it's just combinatorial play with symbols! There are certain rules, and you can study these rules and forget that they have any meaning!
Gregory Chaitin
When we went to school we were told that we were governed by laws, not men. As a result of that, many people think there is no need to pay any attention to judicial candidates because judges merely apply the law by some mathematical formula and a good judge and a bad judge all apply the same kind of law. The fact is that the most important part of a judge's work is the exercise of judgment and that the law in a court is never better than the common sense judgment of the judge that is presiding.
Robert H. Jackson