On Proofs (1875-1899)

"’Divide et impera’ is as true in algebra as in statecraft; but no less true and even more fertile is the maxim ‘auge et impera’. The more to do or to prove, the easier the doing or the proof." (James J Sylvester, "Proof of the Fundamental Theorem of Invariants", Philosophic Magazine, 1878)

"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)

"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)

"[…] it is not immaterial to the cogency of our proof whether 'a + bi' has a sense or is nothing more than printer's ink. It will not get us anywhere simply to require that it have a sense, or to say that it is to have the sense of the sum of a and bi, when we have not previously defined what 'sum' means in this case and when we have given no justification for the use of the definite article." (Gottlob Frege, "Grundlagen der Arithmetik" ["Foundations of Arithmetic"], 1884)

"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 one truth upon another. After we have convinced ourselves that a boulder is immovable, by trying unsuccessfully to move it, there remains the further question, what is it that supports it so securely." (Gottlob Frege, "The Foundations of Arithmetic", 1884

"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)

"That which is provable, ought not to be believed in science without proof" (Richard Dedekind, "Was sind und was sollen die Zahlen?", 1888)

"Pure mathematics proves itself a royal science both through its content and form, which contains within itself the cause of its being and its methods of proof. For in complete independence mathematics creates for itself the object of which it treats, its magnitudes and laws, its formulas and symbols." (Christian H Dillmann, "Die Mathematik die Fackelträgerin einer neuen Zeit" , 1889)

"The Descriptive-Geometry Theorem that any map whatsoever can have its divisions

properly distinguished by the use of but four colors, from its generality and intangibility, seems to have aroused a good deal of interest a few years ago when the rigorous proof of it appeared to be difficult if not impossible, though no case of failure could be found. The present article does not profess to give a proof of this original Theorem; in fact its aims are so far rather destructive than constructive, for it will be shown that there is a defect in the now apparently recognized proof [of Kempe’s]" (Percy J Heawood, 1890)

"Strictly speaking, the theory of numbers has nothing to do with negative, or fractional, or irrational quantities, as such. No theorem which cannot be expressed without reference to these notions is purely arithmetical: and no proof of an arithmetical theorem, can be considered finally satisfactory if it intrinsically depends upon extraneous analytical theories." (George B Mathews, "Theory of Numbers", 1892)

"With our notion of the essence of intuition, an intuitive treatment of figurative representations will tend to yield a certain general guide on which mathematical laws apply and how their general proof may be structured. However, true proof will only be obtained if the given figures are replaced with figures generated by laws based on the axioms and these are then taken to carry through the general train of thought in an explicit case. Dealing with sensate objects gives the mathematician an impetus and an idea of the problems to be tackled, but it does not pre-empt the mathematical process itself." (Felix Klein, "Nicht-Euklidische Geometrie I: Vorlesung gehalten während des Wintersemesters 1889–90", 1892)

"If men of science owe anything to us, we may learn much from them that is essential. For they can show how to test proof, how to secure fulness and soundness in induction, how to restrain and to employ with safety hypothesis and analogy." (Lord John Acton, [Lecture]" The Study of History", 1895)

"The folly of mistaking a paradox for a discovery, a metaphor for a proof, a torrent of verbiage for a spring of capital truths, and oneself for an oracle, is inborn in us." (Paul Valéry, 1895)

"In order to comprehend and fully control arithmetical concepts and methods of proof, a high degree of abstraction is necessary, and this condition has at times been charged against arithmetic as a fault. I am of the opinion that all other fields of knowledge require at least an equally high degree of abstraction as mathematics, - provided, that in these fields the foundations are also everywhere examined with the rigour and completeness which is actually necessary." (David Hilbert, "Die Theorie der algebraischen Zahlkorper", 1897)