⏳On History of Mathematics (2010-)

"The history of mathematical ideas is often very difficult to untangle, and because ideas evolve gradually over a long period of time it is impossible to draw an exact boundary between a given theory and its offspring. It is consequently a painful task to credit some developments to a small number of authors." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"The most amazing event in the history of Greek mathematics has to have been the discovery of irrational numbers. This was not merely a fact about real numbers, which didn’t exist yet. It was a blow to Pythagorean philosophy, one of the main tenets of which was that all was number and all relations were thus ratios. And it was a genuine foundational crisis: the discovery of irrational numbers invalidated mathematical proofs. More than that, it left open the question of what one even meant by proportion and similarity." (Craig Smoryński, "History of Mathematics: A Supplement", 2008)

"The characterization of mathematics as a deductive discipline is accurate but incomplete. It represents the finished and polished consequences of the work of mathematicians, but it does not adequately represent the doing of mathematics. It describes theorem proofs but not theorem proving. Moreover, the history of mathematics is not the emotionless chronology of inventions of evermore esoteric formalisms that some people imagine it to be. It has its full share of color, mystery, and intrigue." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)

"The history of mathematics can be studied chronologically, thematically, topically, and biographically. I have used in this course elements of each approach." (Israel Kleiner, Excursions in the History of Mathematics", 2012)

"The issue of rigorous foundations for calculus began with gropings in the early seventeenth century and concluded with a 'final' resolution in the 1870s. This rather slow evolution toward a logical grounding is not atypical in the history of mathematics. Rigor, formalism, and the logical development of a concept, result, or theory usually come at the end of a process of mathematical evolution. In the case of calculus, mathematicians achieved very impressive results during the seventeenth and eighteenth centuries by intuitive, heuristic reasoning, and therefore had no compelling reasons to put their subject on firm foundations. This does not mean that there was no concern during these two centuries for the logic behind the algorithms of calculus; and there were attempts, albeit unsuccessful, to supply it." (Israel Kleiner, Excursions in the History of Mathematics", 2012)

"The early history of group theory and Galois theory are closely related - after all, Galois was the person who introduced the term 'group' into mathematics. So it is not surprising that notions like Abelian equations from Galois theory influenced the terminology of group theory." 

"When a mathematical conjecture eventually turns out to be correct, its history often follows a standard pattern. Over a period of time, various people prove the conjecture to be true provided special restrictions apply. Each such result improves on the previous one by relaxing some restrictions, but eventually this process runs out of steam. Finally, a new and much cleverer idea completes the proof." (Ian Stewart, "Visions of Infinity", 2013)

"The history of mathematical ideas is often very difficult to untangle, and because ideas evolve gradually over a long period of time it is impossible to draw an exact boundary between a given theory and its offspring. It is consequently a painful task to credit some developments to a small number of authors." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"History of mathematics is done by mathematicians as well as historians. History models mathematics as a segment of the ongoing story of human culture. Mathematicians are likely to see the past through the eyes of the present, and ask, ‘Was it important? natural? deep? surprising? elegant?’ The historian sees mathematics as a thread in the ever-growing web of human life, intimately interwoven with finance and technology, with war and peace. Today's mathematics is the culmination of all that has happened before now, yet to future viewpoints it will seem like a brief, outmoded stage of the past." (Reuben Hersh, "Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)