A Quote by Alfred North Whitehead
The whole of mathematics consists in the organization of a series of aids to the imagination in the process of reasoning.
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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.
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.
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.
Mathematics is a language plus reasoning. It's like a language plus logic. Mathematics is a tool for reasoning.
With the exception of the geometrical series, there does not exist in all of mathematics a single infinite series the sum of which has been rigorously determined. In other words, the things which are the most important in mathematics are also those which have the least foundation.
One of the chief triumphs of modern mathematics consists in having discovered what mathematics really is.
First, the probable cause of AIDS has been found: a variant of a known human cancer virus. Second, not only has the agent been identified, but a new process has been developed to mass produce this virus. Thirdly, with the discovery of both the virus and this new process, we now have a blood test for AIDS. With a blood test, we can identify AIDS victims with essentially 100% certainty.
A woman's chastity consists, like an onion, of a series of coats. You may strip off the outer ones without doing much mischief, perhaps none at all ; but you keep taking off one after another, in expectation of coming to the inner nucleus, including the whole value of the matter. It proves, however, that there is no such nucleus, and that chastity is diffused through the whole series of coats, is lessened with the removal of each, and vanishes with the final one which you supposed would introduce you to the hidden pearl.
Reasoning is the pastime of my whole household, and all this reasoning has driven out Reason.
Mathematics is not just a language. Mathematics is a language plus reasoning.
Thus metaphysics and mathematics are, among all the sciences that belong to reason, those in which imagination has the greatest role. I beg pardon of those delicate spirits who are detractors of mathematics for saying this . . . . The imagination in a mathematician who creates makes no less difference than in a poet who invents. . . . Of all the great men of antiquity, Archimedes may be the one who most deserves to be placed beside Homer.
Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems-general and specific statements-can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences.
As we speak of poetical beauty, so ought we to speak of mathematical beauty and medical beauty. But we do not do so; and that reason is that we know well what is the object of mathematics, and that it consists in proofs, and what is the object of medicine, and that it consists in healing. But we do not know in what grace consists, which is the object of poetry.
As of 2013, according to the World Health Organization, 35 million people were estimated to be living with HIV or AIDS globally, and 39 million have died from the disease. The epidemic of denial won, and now everyone knows there is money in the making of drugs for AIDS.
What would mathematics have amounted to without the imagination of its devotees-its giants and their followers? There never was a discovery made without the urge of imagination-of imagination which broke the roadway through the forest in order that cold logic might follow.
But the development of human society does not go straight forward; and the epic process will therefore be a recurring process, the series a recurring series - though not in exact repetition.
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