Abstract Reasoning Quotes

How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?

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.

Nobody really thinks who does not abstract from that which is given, who does not relate the facts to the factors which have made them, who does not - in his mind - undo the facts. Abstractness is the very life of thought, the token of its authenticity.

Mathematics is a language plus reasoning. It's like a language plus logic. Mathematics is a tool for reasoning.

In the first place a philosophical proposition must be general. It must not deal specially with things on the surface of the earth, or within the solar system, or with any other portion of space and time. . . . This brings us to a second characteristic of philosophical propositions, namely that they must be a priori. A philosophical proposition must be such as can neither be proved nor disproved by empirical evidence. . . . Philosophy, if what has been said is correct, becomes indistinguishable from logic as that word has now come to be used.

Reason is man's faculty for grasping the world by thought, in contradiction to intelligence, which is man's ability to manipulate the world with the help of thought. Reason is man's instrument for arriving at the truth, intelligence is man's instrument for manipulating the world more successfully; the former is essentially human, the latter belongs to the animal part of man.

Algebra is the intellectual instrument which has been created for rendering clear the quantitative aspects of the world.

The intellectual life of man consists almost wholly in his substitution of conceptual order for the perceptual order in which his experience originally comes.

Mathematics is concerned only with the enumeration and comparison of relations.

The universe is governed by science. But science tells us that we can't solve the equations, directly in the abstract.

A learned parson, rusting in his cell at Oxford or Cambridge, will reason admirably well upon the nature of man; will profoundly analyze the head, the heart, the reason, the will, the passions, the senses, the sentiments, and all those subdivisions of we know not what ; and yet, unfortunately, he knows nothing of man... He views man as he does colours in Sir Isaac Newton's prism, where only the capital ones are seen; but an experienced dyer knows all their various shades and gradations, together with the result of their several mixtures.

Mathematics is entirely free in its development, and its concepts are only linked by the necessity of being consistent, and are co-ordinated with concepts introduced previously by means of precise definitions.

... mathematics is the science of skillful operations with concepts and rules invented just for this purpose.

Do not imagine that mathematics is hard and crabbed, and repulsive to common sense. It is merely the etherealization of common sense.

Philosophy can only be approached with the most concrete comprehension.

A philosophy is characterized more by the formulation of its problem than by its solution of them.

When you think about it, 12:15 P.M. is actually 11:75 A.M.

Mathematics is the source of a wicked intellect that, while making man the lord of the earth, also makes him the slave of the machine.

You can go through your whole life telling yourself that life is logical, life is prosaic, life is sane. Above all sane. And I think it is. I've had a lot of time to think about that. And what I keep coming back to is [her] dying declaration: 'So you understand that when we increase the number of variables, the axioms themselves never change.'

Before he has seen the whole, how unusually perceptive and imaginative the person must be to evolve the entire sequence by meditating on its single, pair or triplet of essential images.

But men love abstract reasoning and neat systematization so much that they think nothing of distorting the truth, closing their eyes and ears to contrary evidence to preserve their logical constructions.

What makes philosophy so tedious is not the profundity of philosophers, but their lack of art; they are like physicians who soughtto cure a slight hyperacidity by prescribing a carload of burned oyster-shells.

Mathematics has the completely false reputation of yielding infallible conclusions. Its infallibility is nothing but identity. Two times two is not four, but it is just two times two, and that is what we call four for short. But four is nothing new at all. And thus it goes on and on in its conclusions, except that in the higher formulas the identity fades out of sight.

Brutes abstract not. -- If it may be doubted, whether beasts compound and enlarge their ideas, that way, to any degree; this, I think, I may be positive in, that the power of abstracting is not at all in them; and that the having of general ideas is that which puts a perfect distinction betwixt man and brutes, and is an excellency which the faculties of brutes do by no means attain to.

When I dealt with set theory, I could never make it be the music that I wanted.