Quotable Mathematics

"An axiom is proposition more general than the propositions or the science in which it employed as an axiom; or, an axiom is a proposit...

"[...] hence upon the principle that even for an infinite totality of sets there are always mappings that associate with every set one ...

"Mathematics has been trivialized, derived from indubitable, trivial axioms in which only absolutely clear trivial terms figure, and fr...

"Mathematics is an aspect of culture as well as a collection of algorithms." (Carl B Boyer, "The History of the Calculus and ...

"A proof in mathematics is a compelling argument that a proposition holds without exception; a disproof requires only the demonstration...

"All followers of the axiomatic method and most mathematicians think that there is some such thing as an absolute ‘mathematical rigor’ ...

"If we wish to express our ideas in terms of the concepts synthetic and analytic, we would have to point out that these concepts ar...

"Pure mathematics is a collection of hypothetical, deductive theories, each consisting of a definite system of primitive, undefined, co...

"Mathematics is not placid, static and eternal. […] Most mathematicians are happy to make use of those axioms in their proofs, although...

"One very important genus of complex ideas that we encounter everywhere are those in which the idea of collection (Inbegriff ) appears....

"Axioms in philosophy are not axioms until they are proved upon our pulses: we read fine things but never feel them to the full until w...

"It cannot be that axioms established by argumentation should avail for the discovery of new works, since the subtlety of nature is gre...

"If geometry is to serve as a model for the treatment of physical axioms, we shall try first by a small number of axioms to include as ...

"I see with much pleasure that you are working on a large work on the integral Calculus [...] The reconciliation of the methods which y...

"He who seeks for methods without having a definite problem in mind seeks for the most part in vain." (David Hilbert, 1902) “Stati...

"So is not mathematical analysis then not just a vain game of the mind? To the physicist it can only give a convenient language; but is...

"The use of a mathematical definition is, to deduce from it the properties of the thing defined […]" (Robert Woodhouse," On t...

"The world of ideas which it [mathematics] discloses or illuminates, the contemplation of divine beauty and order which it induces, the...

"Metaphor consists in giving the thing a name that belongs to something else; the transference being either from genus to species, or f...

"Reflexion is careful and laborious thought, and watchful attention directed to the agreeable effect of one's plan. Invention, on t...