Mathematical Quotes

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Heisenberg now approached the problem from a new philosophical angle. He discarded all models, pictures and parables, and made a clear distinction between sure knowledge we gain from observation of nature and the conjectural knowledge we introduce when we use models, pictures and parables. Sure knowledge... can only be numerical, so that Heisenberg's results were inevitably mathematical in form, and could not disclose anything about the true nature of physical properties or entities.

All mathematical calculations about the course of nature must start from some assumed law of nature... Accordingly, however accurately we have calculated that some event must occur, the doubt always remains — Is it true?

Alfred North Whitehead

Indeed it is paradoxical that abstractions so remote from reality should achieve so much. Artificial the mathematical account may be, a fairy tale perhaps, but one with a moral.

How small is the number of individuals who are qualified to think justly on metaphysical, moral, or political subjects, in comparison of those, who may be trained by practice to follow the longest processes of mathematical reasoning.

To examine the varied relations of necessary truth, and to trace through its successive developments, the simple principle to its ultimate result, is the peculiar province of Mathematical Analysis.

Charles Babbage

Mathematicks may help the naturalist, both to frame hypotheses, and to judge of those that are proposed to them, especially such as relate to mathematical subjects in conjunction with others.

Our inner weighing of evidence is not a careful mathematical calculation resulting in a probabilistic estimate of truth, but more like a whirlpool blending of the objective and the personal. The result is a set of beliefs - both conscious and unconscious - that guide us in interpreting all the events of our lives.

Leonard Mlodinow

It must not be supposed... that it is to a gift of Nature that I owe such mathematical power as I possess. No, it has come by hard work, hard work. Not mere industry, but brain-splitting thinking — hard work; hard work that has often endangered my health.

Carl Gustav Jacob Jacobi

Modern algebra has exposed for the first time the full variety and richness of possible mathematical systems.

Saunders Mac Lane

It must be admitted that among a certain number of advanced mathematical physicists whose work has lain mainly in the twentieth century, the ether is regarded with suspicion, or even with contempt. And some of the opponents go so far as to as to say that the nineteenth century idea of the ether has failed to establish itself, and that in consequence the whole idea of the ether is under a cloud, and that it is only upheld by a few antiquated supporters, who, though are willing to admit many modifications in the original nineteenth century notions of an ether, feel the need of a medium capable of performing the functions attributed to it.

It is from this absolute indifference and tranquillity of the mind, that mathematical speculations derive some of the most considerable advantages; because there is nothing to interest the imagination; because the judgment sits free and unbiased to examine the point. All proportions, every arrangement of quantity, is alike to the understanding, because the same truths result to it from all; from greater from lesser, from equality and inequality.

Mathematical Knowledge adds a manly Vigour to the Mind, frees it from Prejudice, Credulity, and Superstition.

Astronomers... if they would be more than mere artisans, must be more or less mathematicians, and must separately study the mathematical grounds of their science...

William Rowan Hamilton

Mathematics is found to be the special, necessary lens through which nature must be observed since nature is defined as being exactly a structure of mathematical entities and relations.

A book on the new physics, if not purely descriptive of experimental work, must essentially be mathematical.

The constitution of the universe, so far as it can be clearly apprehended by our intellect, thus assumes a shape involving an assemblage of mathematical propositions: certain algebraical formulae, and the knowledge when and how to apply them, constitute the last step of the physical science to which we can attain.

William Whewell

The physicist analyzes matter into particles, and finds that their forces and motions can be described in mathematical terms. The materialist pushes this scientific result into philosophy, and says that there is no other reality.

William Cecil Dampier

If we approach the Divine through symbols, then it is most suitable that we use mathematical symbols, these have an indestructible certainty.

Nicholas of Cusa

The peculiarity of the evidence of mathematical truths is that all the argument is on one side.

John Stuart Mill

The potential usefulness of a mathematical concept or technique in helping to advance scientific understanding has very little to do with what one can foresee before that concept or technique has appeared.

Saunders Mac Lane

If you attack a mathematical problem directly, very often you come to a dead end, nothing you do seems to work and you feel that if only you could peer round the corner there might be an easy solution. There is nothing like having somebody else beside you, because he can usually peer round the corner.

Diophantos lived in a period when the Greek mathematicians of great original power had been succeeded by a number of learned commentators, who confined their investigations within the limits already reached, without attempting to further the development of the science. To this general rule there are two most striking exceptions, in different branches of mathematics, Diophantos and Pappos. These two mathematicians, who would have been an ornament to any age, were destined by fate to live and labour at a time when their work could not check the decay of mathematical learning.

Thomas Little Heath

I have often noticed that when people come to understand a mathematical proposition in some other way than that of the ordinary demonstration, they promptly say, "Oh, I see. That's how it must be." This is a sign that they explain it to themselves from within their own system.

Georg Christoph Lichtenberg

It is possible, of course, to operate with figures mechanically, just as it is possible to speak like a parrot: but that hardly deserves the names of thought. It only becomes possible at all after the mathematical notation has, as a result of genuine thought, been so developed that it does the thinking for us, so to speak.

The belief that the underlying order of the world can be expressed in mathematical form lies at the very heart of science. So deep does this belief run that a branch of science is considered not to be properly understood until it can be cast in mathematics.