“Pure Mathematics is the class of all propositions of the form “p implies q,” where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as may be involved in the general notion of propositions of the above form. In addition to these, mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth.”

—  Bertrand Russell

Principles of Mathematics (1903), Ch. I: Definition of Pure Mathematics, p. 3
1900s

Adopted from Wikiquote. Last update Sept. 29, 2023. History

Help us to complete the source, original and additional information

Do you have more details about the quote "Pure Mathematics is the class of all propositions of the form “p implies q,” where p and q are propositions containing …" by Bertrand Russell?
Bertrand Russell photo
Bertrand Russell 562
logician, one of the first analytic philosophers and politi… 1872–1970

Related quotes

“There is the definition [of mathematics], boldly proposed by Pierce that 'Mathematics is the science which draws necessary conclusions', and more explicitly formulated by Russel that 'Pure Mathematics is the class of all propositions of the form "p implies q"… it was… the purpose of Russell's treatise to provide a complete, exact and convincing justification of this definition… instead, he and Whitehead collaborated to give a magisterial account of the Principia Mathematica.”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

“After defining semantical concepts like logical truth and related ones, I proposed to interpret the modalities as those properties of propositions which correspond to certain semantical properties of the sentences expressing the propositions. For example, a proposition is logically necessary if and only if a sentence expressing it is logically true.”

Rudolf Carnap (1891–1970) German philosopher

Source: Carnap’s intellectual biography (1963), p. 62

“Because of the "all-or-none" character of nervous activity, events and the relations among them can be treated by means of propositional logic. It is found that the behaviour of every net can be described in these terms, with the addition of more complicated logical means for nets containing cycles; and that for any logical expression satisfying certain conditions, one can find in net behaving ever fashion it describes.”

Warren S. McCulloch (1898–1969) American neuroscientist

Source: A logical calculus of the ideas immanent in nervous activity (1943), p. 115

Auguste Comte photo

“Social positivism only accepts duties, for all and towards all. Its constant social viewpoint cannot include any notion of rights, for such notion always rests on individuality.”

Auguste Comte (1798–1857) French philosopher

Le Catéchisme positiviste (1852)
Context: Social positivism only accepts duties, for all and towards all. Its constant social viewpoint cannot include any notion of rights, for such notion always rests on individuality. We are born under a load of obligations of every kind, to our predecessors, to our successors, to our contemporaries. These obligations then increase or accumulate, for it is some time before we can return any service. … Any human right is therefore as absurd as immoral. Since there are no divine rights anymore, this concept must therefore disappear completely as related only to the preliminary regime and totally inconsistent with the final state where there are only duties based on functions.

“Discontinuity of its linguistic and logical terms is for the conscious analytical intellect psychologically and logically prior to notions of continuity.”

Lancelot Law Whyte (1896–1972) Scottish industrial engineer

Essay on Atomism: From Democritus to 1960 (1961)
Context: Discontinuity of its linguistic and logical terms is for the conscious analytical intellect psychologically and logically prior to notions of continuity.... This functional priority... may not have been reflected in the history of the development of reason in all human communities.... But it is relevant for the West that the Pythagoreans, with their discrete integers and point patterns, came before Euclid, with his continuous metrical geometry, and that physical atomism as a speculative philosophy preceded by some two thousand years the conception of a continuous physical medium with properties of its own.<!--pp.13-14

“I would not want you to suppose that my rejection of Allen Forte's theory of pitch-class sets implies a rejection of the notion that there can be such a thing as a pitch-class set. It is only when one defines everything in terms of pitch-class sets that the concept becomes meaningless.”

George Perle (1915–2009) American composer

Page 67
See: Allen Forte
The Listening Composer

William John Macquorn Rankine photo

“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 John Macquorn Rankine (1820–1872) civil engineer

Source: "Outlines of the Science of Energetics," (1855), p. 121; Second paragraph

“In Foundations of the Theory of Signs (p. 6), the three terms in question were defined follows: pragmatics as the study of "the relation of signs to interpreters", semantics as the study of "the relations of signs to the objects to which the signs are applicable", syntactics as the study of "the formal relations of signs to one another."”

Charles W. Morris (1903–1979) American philosopher

Source: Writings on the General Theory of Signs, 1971, p. 301

“The Arithmetica… is deficient, sometimes pardonably, sometimes without excuse, in generalization. The book of Porismata, to which Diophantus sometimes refers, seems on the other hand to have been entirely devoted to the discussion of general properties of numbers. It is three times expressly quoted in the Arithmetica… Of all these propositions he says… 'we find it in the Porisms'; but he cites also a great many similar propositions without expressly referring to the Porisms. These latter citations fall into two classes, the first of which contains mere identities, such as the algebraical equivalents of the theorems in Euclid II…. The other class contains general propositions concerning the resolution of numbers into the sum of two, three or four squares…. It will be seen that all these propositions are of the general form which ought to have been but is not adopted in the Arithmetica. We are therefore led to the conclusion that the Porismata, like the pamphlet on Polygonal Numbers, was a synthetic and not an analytic treatise. It is open, however, to anyone to maintain the contrary, since no proof of any porism is now extant.”

James Gow (scholar) (1854–1923) scholar

p, 125
A Short History of Greek Mathematics (1884)

Willem de Sitter photo

“The field equations, in their most general form, contain a term multiplied by a constant, which is denoted by the Greek letter λ… sometimes called the "cosmical constant." This is a name without any meaning… We have, in fact, not the slightest inkling of what its real significance is. It is put in the equations in order to give them the greatest possible degree of mathematical generality, but, so far as its mathematical function is concerned, it is entirely undetermined: it may be positive or negative, it might also be zero.”

Willem de Sitter (1872–1934) Dutch cosmologist

Kosmos (1932), Above is Beginning Quote of the Last Chapter: Relativity and Modern Theories of the Universe -->

Related topics