Mathematical Foundations Quotes

Mathematics, in its widest significance, is the development of all types of formal, necessary, deductive reasoning.

I accept no principles of physics which are not also accepted in mathematics.

For mathematics, even to the logical forms in which it moves, is entirely dependent on the concept of natural number.

No matter what engineering field you're in, you learn the same basic science and mathematics. And then maybe you learn a little bit about how to apply it.

The only thing that might have annoyed some mathematicians was the presumption of assuming that maybe the axiom of choice could fail, and that we should look into contrary assumptions.

And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities...?

Mathematics is the gate and key to science.

Regarding the fundamental investigations of mathematics, there is no final ending ... no first beginning.

I have in later years taken to Euclid, Whitehead, Bertrand Russell, in an elemental way.

All results of the profoundest mathematical investigation must ultimately be expressible in the simple form of properties of the integers.

Before a kid learns how to use a computer that can solve mathematical problems, he or she should know how to do arithmetic without a computer.

Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.

Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what the infinitely little might be.

I am not a mathematician, but I was aware that for centuries, mathematics was considered the queen of the sciences because it claimed certainty. It was grounded on some fundamental certainties - axioms - that led to others.

I do not want to presuppose anything as known. I see in my explanation in section 1 the definition of the concepts point, straight line and plane, if one adds to these all the axioms of groups i-v as characteristics. If one is looking for other definitions of point, perhaps by means of paraphrase in terms of extensionless, etc., then, of course, I would most decidedly have to oppose such an enterprise. One is then looking for something that can never be found, for there is nothing there, and everything gets lost, becomes confused and vague, and degenerates into a game of hide and seek.

Every one who understands the subject will agree that even the basis on which the scientific explanation of nature rests is intelligible only to those who have learned at least the elements of the differential and integral calculus, as well as analytical geometry.

[I]f in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics, in so far as disposed through it we are able to reach certainty in other sciences and truth by the exclusion of error.

The social sciences, I thought, needed the same kind of rigor and the same mathematical underpinnings that had made the 'hard' sciences so brilliantly successful.

Mathematics as a science, commenced when first someone, probably a Greek, proved propositions about "any" things or about "some" things, without specifications of definite particular things.

Maths is like a ladder - if you miss the first few rungs, you can't jump up, so start with the basics.

The primes are the raw material out of which we have to build arithmetic, and Euclid's theorem assures us that we have plenty of material for the task.

But as the work proceeded I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was not more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.

One of the chief triumphs of modern mathematics consists in having discovered what mathematics really is.

All the mathematical sciences are founded on relations between physical laws and laws of numbers.