Eric Temple Bell on History of mathematics

“In their lack of common mathematical curiosity, the algebraists of Islam and the European Renaissance were contemporaries of the ancient Egyptians. They wondered and were perplexed, of course; but there they stopped, because they lacked the Greek instinct for logical completeness and generality.”

“If the early Greeks were cognizant of Babylonian algebra, they made no attempt to develop or even to use it, and thereby they stand convicted of the supreme stupidity in the history of mathematics. ...The ancient Babylonians had a rare capacity for numerical calculation; the majority of Greeks were either mystical or obtuse in their first approach to number. What the Greeks lacked in number, the Babylonians lacked in logic and geometry, and where the Babylonians fell short, the Greeks excelled. Only in the modern mind of the seventeenth and succeeding centuries were number and form first clearly perceived as different aspects of one mathematics.”

“Nothing is easier... than to fit a deceptively smooth curve to the discontinuities of mathematical invention. Everything then appears as an orderly progression... with Cavalieri, for instance, indistinguishable from Newton in the neighborhood of the calculus, or Lagrange from Fourier in that of trigonometric series, or Bhaskara from Lagrange in the region of Fermat's equation. Professional historians may sometimes be inclined to overemphasize the smoothness of the curve; professional mathematicians, mindful of the dominant part played in geometry by the singularities of curves, attend to the discontinuities. ...That such differences should exist is no disaster. Dissent is good for the souls of all concerned.”

“From Pythagoras and Zeno to Hilbert and Brouwer, mathematicians have reveled in the flexibility of their reasoning, and some few have sought to understand the sources of its power. For centuries after the first great age of mathematics in ancient Greece, it was accepted without question that deductive reasoning, if properly applied, would never lead to inconsistencies. ...With the intrusion of irrational numbers to disrupt the integral harmonies of the Pythagorean cosmos, a controversy that has raged off and on for well over two thousand years began: is the mathematical infinite a safe concept in mathematical reasoning, safe in the sense that contradictions will not result from the use of this infinite subject to certain prescribed conditions? (The infinities of religion and philosophy are irrelevant for mathematics).”

“The recent period, that of modern mathematics, extends from 1801 to the present. Some might prefer 1821... Perhaps the most significant feature of this century was the beginning of the abstract, completely general attack. ...Each of five men—Lobachewsky, Bolyai, Plücker, Riemann, Lie—invented as part of his lifework as much (or more) new geometry as was created by all the Greek mathematicians in the two or three centuries of their greatest activity. There are good grounds for the frequent assertion that the nineteenth century alone contributed about five times as much to mathematics as had all preceding history. This applies not only to quantity but, what is of incomparably more importance, to power. ...the advances of the recent period have swept up and included nearly all the valid mathematics that preceded 1800 as very special instances of general theories and methods.”