Philosophy Of Math Quotes

As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.

Do mathematics have a relation to reality or are they only a mathematical symbol?

It is the merest truism, evident at once to unsophisticated observation, that mathematics is a human invention.

We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori.

God made the integers, man made the rest.

In the pure mathematics we contemplate absolute truths which existed in the divine mind before the morning stars sang together, and which will continue to exist there when the last of their radiant host shall have fallen from heaven.

Logic, too, also rests on assumptions that do not correspond to anything in the real world, e.g., on the assumption that there areequal things, that the same thing is identical at different points in time: but this science arose as a result of the opposite belief (that such things actually exist in the real world). And it is the same with mathematics, which would certainly never have arisen if it had been understood from the beginning that there is no such thing in nature as a perfectly straight line, a true circle, and absolute measure.

Logic and mathematics are nothing but specialised linguistic structures.

It can be shown that a mathematical web of some kind can be woven about any universe containing several objects. The fact that our universe lends itself to mathematical treatment is not a fact of any great philosophical significance.

...mathematics is distinguished from all other sciences except only ethics, in standing in no need of ethics. Every other science, even logic, especially in its early stages, is in danger of evaporating into airy nothingness, degenerating, as the Germans say, into an arachnoid film, spun from the stuff that dreams are made of. There is no such danger for pure mathematics; for that is precisely what mathematics ought to be.

... the laws of physics and of logic ... the number system ... the principle of algebraic substitution. These are ghosts. We just believe in them so thoroughly they seem real.

As far as I know, only a small minority of mathematicians, even of those with Platonist views, accept the idea that there may be mathematical facts which are true but unknowable.

The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilised of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought.

Mathematics, in the development of its ideas, has only to take account of the immanent reality of its concepts and has absolutely no obligation to examine their transient reality.

There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, to which we have access only with our mind, just as a world of physical reality exists, the one like the other independent of ourselves, both of divine creation.

But it is certainly not possible to insist on one hand that the formalism is complete and to insist on the other hand that its application to 'the actual' actually demands a step which cannot be derived from it.