“The Elements has been considered as offering models of scrupulously rigorous demonstrations. It is certainly true that in point of rigour it compares favourably with its modern rivals; but when examined in the light of strict mathematical logic, it has been pronounced by C. S. Peirce to be "riddled with fallacies." The results are correct only because the writer's experience keeps him on his guard.”

— Florian Cajori, book A History of Mathematics

Source: A History of Mathematics (1893), p. 37. Reported in Moritz (1914)

Adopted from Wikiquote. Last update June 3, 2021. History

Help us to complete the source, original and additional information

Do you have more details about the quote "The Elements has been considered as offering models of scrupulously rigorous demonstrations. It is certainly true that …" by Florian Cajori?

Florian Cajori13

American mathematician 1859–1930

Related quotes

“The life of the law has not been logic; it has been experience… The law embodies the story of a nation's development through many centuries, and it cannot be dealt with as if it contained only the axioms and corollaries of a book of mathematics.”

Oliver Wendell Holmes Jr. book The Common Law

The Common Law (1881), p. 1.
1880s

“Mathematics has also been developed as a philosophy, in the sense in which this term is defined by A. N. Whitehead as 'the endeavor to frame a coherent, logical and necessary system of general ideas in terms of which every element of our experience can be interpreted'. Substitute 'mathematics' for 'experience' and we have an admirable description of its speculative and philosophic development…. Philosophy of mathematics… has its paradoxes and antimonies, and also diverse schools of thought…”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

“So much of modern mathematical work is obviously on the border-line of logic, so much of modern logic is symbolic and formal, that the very close relationship of logic and mathematics has become obvious to every instructed student. The proof of their identity is, of course, a matter of detail: starting with premisses which would be universally admitted to belong to logic, and arriving by deduction at results which as obviously belong to mathematics, we find that there is no point at which a sharp line can be drawn, with logic to the left and mathematics to the right. If there are still those who do not admit the identity of logic and mathematics, we may challenge them to indicate at what point, in the successive definitions and deductions of Principia Mathematica, they consider that logic ends and mathematics begins. It will then be obvious that any answer must be quite arbitrary.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

Source: 1910s, Introduction to Mathematical Philosophy (1919), Ch. 18: Mathematics and Logic

“We shall see that the mathematical treatment of the subject [of electricity] has been greatly developed by writers who express themselves in terms of the 'Two Fluids' theory. Their results, however, have been deduced entirely from data which can be proved by experiment, and which must therefore be true, whether we adopt the theory of two fluids or not. The experimental verification of the mathematical results therefore is no evidence for or against the peculiar doctrines of this theory.”

James Clerk Maxwell book A Treatise on Electricity and Magnetism

A Treatise on Electricity and Magnetism (1873), §36.

“The word experiment derives from 'experimentum' - that which has been experienced and what, for the writer, has been experienced is the work of his great predecessors”

Vernon Scannell (1922–2007) British boxer and poet

A Proper Gentleman, 1977

“At this time it is difficult to put one's finger on any single contribution in the decade 1950 - 1960 which is comparable to those above, and yet progress has probably been even greater. From the point of view of an educator, one cannot overlook the wide distribution which has been given to these ideas. There has been remarkable progress from analysis to synthesis, always a sign of maturity in a field of analytic endeavour. There has been consolidation, for example in the establishment of a more rigorous basis for information theory; there has been unification, for example in the demonstration of the formal similarity between game theory and linear programming; there has been application to mathematically more difficult situations, for example nonlinear servo systems and information channels with memory; there has been implementation, as in commercially available computers which by any reasonable measure are hun- dreds of times more powerful than the primitive devices of 1950; there has been de-limitation of the boundaries of many of these fields.”

Robert E. Machol (1917–1998) American systems engineer

Source: Information and Decision Processes (1960), p. viii

“With respect to the mathematical part of optics, the curvature of the images, formed by lenses and mirrors, has been correctly investigated, and the inaccuracy of some former estimations has been demonstrated.”

Thomas Young (scientist) (1773–1829) English polymath

Preface
A Course of Lectures on Natural Philosophy and the Mechanical Arts (1807)

“Mathematics, in the development of its ideas, has only to take account of the immanent reality of its concepts and has absolutely no obligation to examine their transient reality.”

Georg Cantor (1845–1918) mathematician, inventor of set theory

From Kant to Hilbert (1996)

“The first demonstration (Disq. Arith., Arts. 125-145) which is presented by Gauss in a form very repulsive to any but the most laborious students, has been resumed by Lejeune Dirichlet in a memoir in Crelle's Journal… and has been developed by him with that luminous perspicuity by which his mathematical writings are distinguished.”

Henry John Stephen Smith (1826–1883) mathematician

Report on the Theory of Numbers (1859) Part I, p. 59.
The Collected Mathematical Papers of Henry John Stephen Smith (1894) Vol. 1

“Because of their very nature, science and logical thinking can never decide what is possible or impossible. Their only function is to explain what has been ascertained by experience and observation.”

Rudolf Steiner (1861–1925) Austrian esotericist

Cosmic Memory, Prehistory of Earth and Man.

Help us to complete the source, original and additional information

Florian Cajori13

Related quotes

Related topics