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.
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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.
Mathematical reasoning may be regarded.
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.
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.
It may well be doubted whether human ingenuity can construct an enigma... which human ingenuity may not, by proper application, resolve.
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.
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.
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.
Regarded in isolation, an idea may be quite insignificant, and venturesome in the extreme, but it may acquire importance from an idea which follows it; perhaps, in a certain collocation with other ideas, which may seem equally absurd, it may be capable of furnishing a very serviceable link.
Tad Homer-Dixon is a rare kind of public intellectual, who combines real expertise with a commitment to communicate to the widest possible readership. In The Ingenuity Gap he wants us all to wake-up to the fearful possibility that our blithe trust in science and technology may be misplaced. Human ingenuity may not be capable of coping with two emerging crises of this century and the next: population growth and environmental despoliation. Read Homer Dixon's wake-up call and you will see the future very differently.
The way to a beautiful, strong, healthy body is to start by trusting yourself right now. Let go of the struggle and surrender to your body's needs. Your intuition will tell you what to eat. It may direct you to exercise vigorously, in which case you'll enjoy it, or it may tell you to slow down and rest. It may tell you to stay in bed all day, or it may tell you to get up early. There are no rigid rules. Your body knows perfectly well what's good for it.
I will make love my greatest weapon and none on whom I call can defend against its force.
My reasoning they may counter; my speech they may distrust; my apparel they may disapprove; my face they may reject; and even my bargains may cause them suspicion; yet my love will melt all hearts liken to the sun whose rays soften the coldest clay.
I will greet this day with love in my heart.
If the system exhibits a structure which can be represented by a mathematical equivalent, called a mathematical model, and if the objective can be also so quantified, then some computational method may be evolved for choosing the best schedule of actions among alternatives. Such use of mathematical models is termed mathematical programming.
I may not be funny. I may not be a singer. I may not be a damn seamstress. I may have diabetes. I may have really bad vision. I may have one leg. I may not know how to read. I may not know who the vice president is. I may technically be an alien of the state. I may have a Zune. I may not know Excel. I may be two 9-year-olds in a trench coat. I may not have full control of my bowels. I may drive a '94 Honda Civic. I may not “get” cameras. I may dye my hair with Hydrogen Peroxide. I may be afraid of trees. I may be on fire right now. But I'm a fierce queen.
I am fully assured, that no general method for the solution of questions in the theory of probabilities can be established which does not explicitly recognize, not only the special numerical bases of the science, but also those universal laws of thought which are the basis of all reasoning, and which, whatever they may be as to their essence, are at least mathematical as to their form.
As a man may be born with a mathematical faculty, and by training that faculty year after year may immensely increase his mathematical capacity, so may a man be born with certain faculties within him, faculties belonging to the soul, which he can develop by training and by discipline.
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