Gottlob Frege

"Often it is only after immense intellectual effort, which may have continued over centuries, that humanity at last succeeds in achieving knowledge of a concept in its pure form, by stripping off the irrelevant accretions which veil it from the eye of the mind."

"The ideal of strictly scientific method in mathematics which I have tried to realise here, and which perhaps might be named after Euclid I should like to describe in the following way... The novelty of this book does not lie in the content of the theorems but in the development of the proofs and the foundations on which they are based... With this book I accomplish an object which I had in view in my Begriffsschrift of 1879 and which I announced in my Grundlagen der Arithmetik. I am here trying to prove the opinion on the concept of number that I expressed in the book last mentioned."

"It really is worth the trouble to invent a new symbol if we can thus remove not a few logical difficulties and ensure the rigour of the proofs. But many mathematicians seem to have so little feeling for logical purity and accuracy that they will use a word to mean three or four different things, sooner than make the frightful decision to invent a new word."

"Our representative absolutist is Gottlob Frege, whose writings did as much as anything to revive the mathematizing approaches of the Platonist tradition around 1900, and did so—quite explicitly—as a means of protecting philosophy from subordination to the facts of history and psychology. ...The Platonist strand in Descartes philosophy was revived... by... Frege, who promulgated the original programme of conceptual analysis in his Foundations of Arithmetic. ...Frege ...was rebelling ...against the tendency to telescope formal and prescriptive laws of thought, which were the proper concern of logic, with the empirical and descriptive laws of thinking, which were the business of cognitive psychologists... [W]e should ignore all merely empirical discoveries, whether about the development of understanding in the individual mind or about the historical evolution of our communal understanding. ...Philosophers must concern themselves with concepts only as timeless, intellectual ideals, towards which the human mind struggles, at best, painfully and little by little. ...[A]ctual conceptions current in any existing community are philosophically significant only as an approximation to the eternal system of ideal concepts. ...[A]ny actual, historical set of conceptions has a legitimate intellectual claim on us, only to the extent that it approximates that ideal."

"Being true is different from being taken as true, whether by one or by many or everybody, and in no case is it to be reduced to it. There is no contradiction in somethings being true which everybody takes to be false. I understand by laws of logic not psychological laws of takings-to-be-true, but laws of truth. ...If being true is thus independent of being acknowledged by somebody or other, then the laws of truth are not psychological laws: they are boundary stones set in an eternal foundation, which our thought can overflow, but never displace. It is because of this that they have authority for our thought if it would attain truth. They do not bear the relation to thought that the laws of grammar bear to language; they do not make explicit the nature of our human thinking and change as it changes."

"Gottlob Frege created modern logic including "for all," "there exists," and rules of proof. Leibniz and Boole had dealt only with what we now call "propositional logic" (that is, no "for all" or "there exists"). They also did not concern themselves with rules of proof, since their aim was to reach truth by pure calculation with symbols for the propositions. Frege took the opposite track: instead of trying to reduce logic to calculation, he tried to reduce mathematics to logic, including the concept of number."