On Numbers (1880-1889)

"The old and oft-repeated proposition ‘Totum est majus sua parte’ [the whole is larger than the part] may be applied without proof only in the case of entities that are based upon whole and part; then and only then is it an undeniable consequence of the concepts ‘totum’ and ‘pars’. Unfortunately, however, this ‘axiom’ is used innumerably often without any basis and in neglect of the necessary distinction between ‘reality’ and ‘quantity’, on the one hand, and ‘number’ and ‘set’, on the other, precisely in the sense in which it is generally false." (Georg Cantor, "Über unendliche, lineare Punktmannigfaltigkeiten", Mathematische Annalen 20, 1882)

"If we now notice that all of the numbers previously obtained and their next successors fulfill a certain condition, [that the set of their predecessors is denumerable,] then this condition offers itself, if it is imposed as a requirement on all numbers to be formed next, as a new third principle [...] which I shall call principle of restriction or limitation and which, as I shall show, yields the result that the second number-class" (II) defined with its assistance not only has a higher power than [the first number-class]" (I), but precisely the next higher, that is, the second power." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883)

"The foregoing account of my researches in the theory of manifolds has reached a point where further progress depends on extending the concept of true integral number beyond the previous boundaries; this extension lies in a direction which, to my knowledge, no one has yet attempted to explore. My dependence on this extension of number concept is so great, that without it I should be unable to take freely the smallest step further in the theory of sets." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883)"

"To the thought of considering the infinitely great not merely in the form of what grows without limits - and in the closely related form of the convergent infinite series first introduced in the seventeenth century-, but also fixing it mathematically by numbers in the determinate form of the completed-infinite, I have been logically compelled in the course of scientific exertions and attempts which have lasted many years, almost against my will, for it contradicts traditions which had become precious to me; and therefore I believe that no arguments can be made good against it which I would not know how to meet." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883)

"When you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely in your thoughts advanced to the state of science." (Lord Kelvin, "Electrical Units of Measurement", 1883)

"How are complex numbers to be given to us then […]? If we turn for assistance to intuition, we import something foreign into arithmetic; but if we only define the concept of such a number by giving its characteristics, if we simply require the number to have certain properties, then there is no guarantee that anything falls under the concept and answers to our requirements, and yet it is precisely on this that proofs must be based." (Gottlob Frege, "Grundlagen der Arithmetik" ["Foundations of Arithmetic"], 1884)

"The basis of arithmetic lies deeper, it seems, than that of any of the empirical sciences, and even than that of geometry. The truths of arithmetic governs all that is numerable. This is the widest domain of all; for to it belongs not only the existent, not only the intuitable, but everything thinkable. Should not the laws of number, then, be connected very intimately with the laws of thought? " (Gottlob Frege, "The Foundations of Arithmetic", 1884)

"To apply arithmetic in the physical sciences is to bring logic to bear on observed facts, calculation becomes deduction. The laws of number, therefore, are not really applicable to external things, they are not laws of nature. They are, however, applicable to judgements holding good of things in the external world they are laws of the laws of nature. They assert not connections between phenomena, but connections between judgements, and among judgements are included the laws of nature." (Gottlob Frege,"The Foundations of Arithmetic", 1884)

"Number is but another name for diversity." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1887)

"In calling arithmetic" (algebra, analysis) just a part of logic, I declare already that I take the number-concept to be completely independent of the ideas or intuitions of space and time, that I see it as an immediate product of the pure laws of thought." (Richard Dedekind, "Was sind und was sollen die Zahlen?", 1888)

"In science nothing capable of proof ought to be accepted without proof. Though this demand seems so reasonable yet I cannot regard it as having been met even in […] that part of logic which deals with the theory of numbers. In speaking of arithmetic" (algebra, analysis) as a part of logic I mean to imply that I consider the number concept entirely independent of the notions of intuition of space and time, that I consider it an immediate result from the laws of thought." (Richard Dedekind, "Was sind und was sollen die Zahlen?", 1888)

"These primitive propositions […] suffice to deduce all the properties of the numbers that we shall meet in the sequel. There is, however, an infinity of systems which satisfy the five primitive propositions. [...] All systems which satisfy the five primitive propositions are in one-to-one correspondence with the natural numbers. The natural numbers are what one obtains by abstraction from all these systems; in other words, the natural numbers are the system which has all the properties and only those properties listed in the five primitive propositions." (Giuseppe Peano, "I fondamenti dell’aritmetica nel Formulario del 1898" ["The Principles of Arithmetic, presented by a new method"], 1889)