Beyond the History of Mathematics III

“The history of mathematics is important […] as a valuable contribution to the history of civilisation. Human progress is closely identified with scientific thought. Mathematical and physical researches are a reliable record of intellectual progress. The history of mathematics is one of the large windows through which the philosophic eye looks into past ages and traces the line of intellectual development.” (Florian Cajori, “A History of Mathematics”, 1893)

 “This history constitutes a mirror of past and present conditions in mathematics which can be made to bear on the notational problems now confronting mathematics. The successes and failures of the past will contribute to a more speedy solution of notational problems of the present time.” (Florian Cajori, “A History of Mathematical Notations”, 1928)

“There are no absolutes [...] in mathematics or in its history.” (Eric T Bell, The Development of Mathematics, 1940)

“Unfortunately, the mechanical way in which calculus sometimes is taught fails to present the subject as the outcome of a dramatic intellectual struggle which has lasted for twenty-five hundred years or more, which is deeply rooted in many phases of human endeavors and which will continue as long as man strives to understand himself as well as nature. Teachers, students, and scholars who really want to comprehend the forces and appearances of science must have some understanding of the present aspect of knowledge as a result of historical evolution.” (Richard Curand [forward to Carl B Boyer’s “The History of the Calculus and Its Conceptual Development”, 1949])

"All followers of the axiomatic method and most mathematicians think that there is some such thing as an absolute ‘mathematical rigor’ which has to be satisfied by any deduction if it is to be valid. The history of mathematics shows that this is not the case, that, on the contrary, every generation is surpassed in rigor again and again by its successors.” (Richard von Mises, “Positivism: A Study in Human Understanding”, 1951)

“It is paradoxical that while mathematics has the reputation of being the one subject that brooks no contradictions, in reality it has a long history of successful living with contradictions. This is best seen in the extensions of the notion of number that have been made over a period of 2500 years. From limited sets of integers, to infinite sets of integers, to fractions, negative numbers, irrational numbers, complex numbers, transfinite numbers, each extension, in its way, overcame a contradictory set of demands.” (Philip J Davis, “The Mathematics of Matrices”, 1965)

“Students enjoy […] and gain in their understanding of today's mathematics through analyzing older and alternative approaches.” (Lucas N H Bunt, Phillip S Jones & Jack D Bedient, “The Historical Roots of Elementary Mathematics”, 1976)

“[…] how completely inadequate it is to limit the history of mathematics to the history of what has been formalized and made rigorous. The unrigorous and the contradictory play important parts in this history.” (Philip J Davis & Rueben Hersh, “The Mathematical Experience”, 1985)

"We who are heirs to three recent centuries of scientific development can hardly imagine a state of mind in which many mathematical objects were regarded as symbols of spiritual truths or episodes in sacred history. Yet, unless we make this effort of imagination, a fraction of the history of mathematics is incomprehensible.” (Philip J Davis & Rueben Hersh, “The Mathematical Experience”, 1985)

“Like anything else, mathematics is created within the context of history […]” (William Dunham, “Journey Through Genius”, 1990)

Previous Post <<||>> Next Post