A Quote by Alan Turing
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.
Related Quotes
Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity.
As an exercise of the reasoning faculties, pure mathematics is an admirable exercise, because it consists of reasoning alone and does not encumber the student with any exercise of judgment.
Intuition is the discriminative faculty that enables you to decide which of two lines of reasoning is right. Perfect intuition makes you a master of all.
Intuition is a combination of insight and imagination that was once attributed to spiritual communication. Mathematicians call it 'fuzzy logic,' drawing conclusions from vague or subjective input. The mind becomes aware without the direct intervention of reasoning. Once you can imagine something you can begin the process of creating it. Executives use intuition to make many product, investment, and hiring decisions, even if they deny it. Success in business may depend on an accurate gut.
I would call the attention of the reader to the difference between "reason" and "reasoning." Reason is a light, reasoning a process. Reason is a faculty, reasoning an exercise of that faculty. Reasoning proceeds from one truth to another by means of argumentation. This generally involves the whole mind in labor and complexity. But reason does not exist merely in order to engage in reasoning. The process is a means to an end. The true fulfillment of reason as a faculty is found when it can embrace the truth simply and without labor in the light of single intuition.
Intuition comes from the whole person, from a place that includes the conscious and the unconscious. The total result of all feelings and perceptions manifests spontaneously through intuition. Intuition gives expression to the feelings; that expression is unique and perfectly fitted to the needs of the moment.
The apprehension of... values is intuitive; but it is not a built-in intuition, not something with which one is born. Intuition in art is actually the result of... prolonged tuition.
There are two modes of acquiring knowledge, namely by reasoning and experience. Reasoning draws a conclusion and makes us grant the conclusion, but does not make the conclusion certain, nor does it remove doubt so that the mind may rest on the intuition of truth, unless the mind discovers it by the path of experience.
Mathematics is a logical method. . . . Mathematical propositions express no thoughts. In life it is never a mathematical proposition which we need, but we use mathematical propositions only in order to infer from propositions which do not belong to mathematics to others which equally do not belong to mathematics.
When you're an actress you are in a very specific position, cerebrally, I would think you know. You are a thinking person but you are in a certain perception of the world between really being conceptual and completely stupid. So you are in between, with a lot of intuition and instinct. And that's not completely being intellectual because I think intuition is more blurred. It comes from intuition. Instinct. More than really reasoning.
For most problems found in mathematics textbooks, mathematical reasoning is quite useful. But how often do people find textbook problems in real life? At work or in daily life, factors other than strict reasoning are often more important. Sometimes intuition and instinct provide better guides; sometimes computer simulations are more convenient or more reliable; sometimes rules of thumb or back-of-the-envelope estimates are all that is needed.
The Reader may here observe the Force of Numbers, which can be successfully applied, even to those things, which one would imagine are subject to no Rules. There are very few things which we know, which are not capable of being reduc'd to a Mathematical Reasoning, and when they cannot, it's a sign our Knowledge of them is very small and confus'd; and where a mathematical reasoning can be had, it's as great folly to make use of any other, as to grope for a thing in the dark when you have a Candle standing by you.
Common sense is merely unaided intuition, and unaided intuition is reasoning performed in the absense of instruments and the tested knowledge of science. Common sense tells us that massive satellites cannot hang suspended 36,000 kilometers above the one point on the earth's surface, but they do.
An absolute can only be given in an intuition, while all the rest has to do with analysis. We call intuition here the sympathy by which one is transported into the interior of an object in order to coincide with what there is unique andconsequently inexpressible in it. Analysis, on the contrary, is the operation which reduces the object to elements already known.
I'm saying that we should trust our intuition. I believe that the principles of universal evolution are revealed to us through intuition. And I think that if we combine our intuition and our reason, we can respond in an evolutionary sound way to our problems.
Mathematical reasoning may be regarded.
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