The Foundations of Arithmetic (Die Grundlagen der Arithmetik)

„If a concept fundamental to a mighty science gives rise to difficulties, then it is surely an imperative task to investigate it more closely until those difficulties are overcome.”

„It is sad and discouraging to observe how discoveries once made are always threatening to be lost again in this way, and how much work promises to have been done in vain, because we fancy ourselves so well off that we need not bother to assimilate its results.”

„There are different kinds of laws of thought to suit the different kinds of objects thought about.”

„A proposition may be thought, and again it may be true; let us never confuse these two things. We must remind ourselves, it seems, that a proposition no more ceases to be true when I cease to think of it than the sun ceases to exist when I shut my eyes.”

„If everything were in continual flux, and nothing maintained itself fixed for all time, there would no longer be any possibility of getting to know anything about the world and everything would be plunged in confusion.”

„What is known as the history of concepts is really a history either of our knowledge of concepts or of the meanings of words.”

„In the enquiry that follows, I have kept to three fundamental principles: always to separate sharply the psychological from the logical, the subjective from the objective; never to ask for the meaning of a word in isolation, but only in the context of a proposition; never to lose sight of the distinction between concept and object.”

„It is in the nature of mathematics always to prefer proof, where proof is possible, to any confirmation by induction.”

„The aim of proof is, in fact, not merely to place the truth of a proposition beyond all doubt, but also to afford us insight into the dependence of truths upon one another.”

„The procedure of induction, we may surmise, can itself be justified only by means of general propositions of arithmetic — unless we understand by induction a mere process of habituation, in which case it has of course absolutely no power whatever of leading to the discovery of truth.”

„Empirical propositions hold good of what is physically or psychologically actual, the truths of geometry govern all that is spatially intuitable, whether actual or product of our fancy. The wildest visions of delirium, the boldest inventions of legend and poetry, where animals speak and stars stand still, where men are turned to stone and trees turn into men, where the drowning haul themselves up out of swamps by their own topknots — all these remain, so long as they remain intuitable, still subject to the axioms of geometry.”

„It marks, therefore, an important difference between colour and Number, that a colour such as blue belongs to a surface independently of any choice of ours. […] The Number 1, on the other hand, or 100 or any other Number, cannot be said to belong to the pile of playing cards in its own right, but at most to belong to it in view of the way in which we have chosen to regard it.”

„It ought to be considered that number… is nothing fixed and settled, really existing in things themselves. It is entirely the creature of the mind, considering, either an idea by itself, or any combination of ideas to which it gives one name, and so makes it pass for a unit.”

„Any conclusions drawn from a false assumption are liable to be false.”

„Number is but another name for diversity. Exact identity is unity, and with difference arises plurality.”

„It has often been said that units are units in respect of being perfectly similar to each other; but though they may be perfectly similar in some respects, they must be different in at least one point, otherwise they would be incapable of plurality. If three coins were so similar that they occupied the same place at the same time, they would not be three coins, but one.”

„A thing is called one or single simply with respect to its existence, and not with respect to its essence; for we only think of things in terms of number after they have first been reduced to a common genus.”

„A proposition to be true is just not the same thing as for it to be thought.”

„Mathematics is not concerned with the nature of our mind, and the answer to any question whatsoever in psychology must be for mathematics a matter of complete indifference.”

„The mathematician cannot create things at will, any more than the geographer can; he too can only discover what is there and give it a name.”

„The reason's proper study is itself. In arithmetic we are not concerned with objects which we come to know as something alien from without through the medium of the senses, but with objects given directly to our reason and, as its nearest kin, utterly transparent to it.”