On Axioms (1800-1899)

"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 we have gone the same steps as the author." (John Keats, [Letter to John Hamilton Reynolds] 1818)

"Scientific Ideas can often be adequately exhibited for all the purposes of reasoning, by means of Definitions and Axioms; all attempts to reason by means of Definitions from common Notions, lead to empty forms or entire confusion." (William Whewell, "History of the Inductive Sciences from the Earliest to the Present Time", 1837)

"These sciences, Geometry, Theoretical Arithmetic and Algebra, have no principles besides definitions and axioms, and no process of proof but deduction; this process, however, assuming a most remarkable character; and exhibiting a combination of simplicity and complexity, of rigour and generality, quite unparalleled in other subjects." (William Whewell, "The Philosophy of the Inductive Sciences", 1840)

"The reasoning of mathematicians is founded on certain and infallible principles. Every word they use conveys a determinate idea, and by accurate definitions they excite the same ideas in the mind of the reader that were in the mind of the writer. When they have defined the terms they intend to make use of, they premise a few axioms, or self-evident principles, that every one must assent to as soon as proposed. They then take for granted certain postulates, that no one can deny them, such as, that a right line may be drawn from any given point to another, and from these plain, simple principles they have raised most astonishing speculations, and proved the extent of the human mind to be more spacious and capacious than any other science." (John Adams,"Diary", 1850)

"A physical theory, like an abstract science, consists of definitions and axioms as first principles, and of propositions, their consequences; but with these differences:—first, That in an abstract science, a definition assigns a name to a class of notions derived originally from observation, but not necessarily corresponding to any existing objects of real phenomena, and an axiom states a mutual relation amongst such notions, or the names denoting them; while in a physical science, a definition states properties common to a class of existing objects, or real phenomena, and a physical axiom states a general law as to the relations of phenomena; and, secondly,—That in an abstract science, the propositions first discovered are the most simple; whilst in a physical theory, the propositions first discovered are in general numerous and complex, being formal laws, the immediate results of observation and experiment, from which the definitions and axioms are subsequently arrived at by a process of reasoning differing from that whereby one proposition is deduced from another in an abstract science, partly in being more complex and difficult, and partly in being to a certain extent tentative, that is to say, involving the trial of conjectural principles, and their acceptance or rejection according as their consequences are found to agree or disagree with the formal laws deduced immediately from observation and experiment." (William J M Rankine, "Outlines of the Science of Energetics", Proceedings of the Philosophical Society of Glasgow, 1855)

"An axiom is proposition more general than the propositions or the science in which it employed as an axiom; or, an axiom is a proposition which is true of more subjects than the subject or the science in which it is quoted as an axiom. Hence. Geometry ought to admit as axioms all Algebraic truths. The simple truths of this kind, which are commonly called axioms, ore corollaries from the definitions of such terms as equal, whole, part, sum, etc." (The Pennsylvania School Journal, 1856)

"The maxim is, that whatever can be affirmed (or denied) of a class, may be affirmed (or denied) of everything included in the class. This axiom, supposed to be the basis of the syllogistic theory, is termed by logicians the dictum de omni et nullo [the maxim of all and none]." (John S Mill, "A System of Logic, Ratiocinative and Inductive", 1858)

"Induction and analogy are the special characteristics of modern mathematics, in which theorems have given place to theories and no truth is regarded otherwise than as a link in an infinite chain. 'Omne exit in infinitum' is their favorite motto and accepted axiom." (James J Sylvester, "A Plea for the Mathematician", Nature Vol. 1, 1870)

"When we consider that the whole of geometry rests ultimately on axioms which derive their validity from the nature of our intuitive faculty, we seem well justified in questioning the sense of imaginary forms, since we attribute to them properties which not infrequently contradict all our intuitions." (Gottlob Frege, "On a Geometrical Representation of Imaginary forms in the Plane", 1873)

"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)

"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)

" […] the naive intuition is not exact, while the refined intuition is not properly intuition at all, but arises through the logical development from axioms considered as perfectly exact." (Felix Klein, [lectures] 1893)

"Geometry, like arithmetic, requires for its logical development only a small number of simple, fundamental principles. These fundamental principles are called the axioms of geometry." (David Hilbert, "The Foundations of Geometry", 1899)