Eric Temple Bell Quotes (1883-1960)

“Nearly always it is the recondite and complicated which is elaborated first; and it is only when some relatively unsophisticated mind attacks a problem that its deep simplicity is revealed.”

“Whatever its source, mathematics has come down to the present by the two main streams of number and form. The first carried along arithmetic and algebra, the second, geometry. In the seventeenth century these two united, forming the ever-broadening river of mathematical analysis.”

“Between the workable empiricism of the early land measurers... of ancient Egypt and the geometry of the Greeks in the sixth century before Christ, there is a great chasm. ...and the chasm is bridged by deductive reasoning applied consciously and deliberately to the practical inductions of daily life. Without the strictest deductive proof from admitted assumptions, explicitly stated as such, mathematics does not exist. This does not deny that intuition, experiment, induction, and plain guessing are important elements in mathematical invention. It merely states the criterion by which the final product of all guessing, by whatever name it be dignified, is judged to be or not to be mathematics. It is not known where or when the distinction between inductive inference—the summation of raw experience—and deductive proof from a set of postulates was first made, but it was sharply recognized by the Greek mathematicians as early as 550 B.C.”

“In the entire history of Greek mathematics, all but the incomparable Archimedes and a few of the more heterodox sophists appear to have hated or feared the mathematical infinite. Analysis was thwarted when it might have prospered.”

“Fourier's analytical theory of heat (final form, 1822), devised in the Galileo-Newton tradition of controlled observation plus mathematics, is the ultimate source of much modern work in the theory of functions of a real variable and in the critical examination of the foundation of mathematics.”

“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?”

“Descartes' method of finding tangents and normals... was not a happy inspiration. It was quickly superseded by that of Fermat as amplified by Newton. Fermat's method amounts to obtaining a tangent as the limiting position of a secant, precisely as is done in the calculus today. ...Fermat's method of tangents is the basis of the claim that he anticipated Newton in the invention of the differential calculus.”

“Had the early Greek mind been sympathetic to the algebra and arithmetic of the Babylonians, it would have found plenty to exercise its logical acumen, and might easily have produced a masterpiece of the deductive reasoning it worshipped logically sounder than Euclid's greatly overrated Elements. The hypotheses of elementary algebra are fewer and simpler than those of synthetic geometry. ...they could have developed it with any degree of logical rigor they desired. Had they done so, Apollonius would have been Descartes, and Archimedes, Newton. As it was, the very perfection... of Greek geometry retarded progress for centuries.”

“All computation was greatly simplified early in the seventeenth century by the invention of logarithms.”

“Science makes no pretension to eternal truth or absolute truth; some of its rivals do. That science is in some respects inhuman may be the secret of its success in alleviating human misery and mitigating human stupidity.”

“In the entire history of Greek mathematics, all but the incomparable Archimedes and a few of the more heterodox sophists appear to have hated or feared the mathematical infinite. Analysis was thwarted when it might have prospered.”

“René Descartes is more widely known as a philosopher than as a mathematician, although his philosophy has been controverted while his mathematics has not. ...In accordance with the ideals of his age, when experimental science was first seriously challenging arrogant speculation, Descartes set a greater store by his philosophy than his mathematics. But he fully appreciated the power of his new method in geometry.”

“The third and last exception to general sterility connects the arithmetic of forms with that other major outgrowth of ancient diophantine analysis, the Gaussian concept of congruence. Dickson in 1907 began the congruencial theory of forms, in which the coefficients of the forms are either natural integers reduced modulo p, p prime, or elements of a Galois field. The linear transformations in the theory, corresponding to those in the classical problem of equivalence, were similarly reduced, and hence modular invariants and covariants were defineable. By 1923 the theory was practically worked out, except for two central difficulties, by Dickson and his pupils. Simplified derivations for some of the results were given (1926) by E. Noether by an application of her methods in abstract algebra.”

“After his own fashion, Desargues discussed... Kepler's principle (1604) of continuity, in which a straight line is closed at infinity and parallels meet there...”

“This work of Diaphantus... was the first Greek mathematics, if indeed it was Greek, to show a genuine talent for algebra. ...He had begun to use symbols operationally. This long stride forward is all the more remarkable because his algebraic notation... was almost as awkward as Greek logistic. That he accomplished what he did with the available techniques places him beyond question among the great algebraists.”

“Modern mathematics was born with Archimedes and died with him for all of two thousand years. It came to life again with Descartes and Newton.”

“Fourier's analytical theory of heat (final form, 1822), devised in the Galileo-Newton tradition of controlled observation plus mathematics, is the ultimate source of much modern work in the theory of functions of a real variable and in the critical examination of the foundation of mathematics.”

“With the completion of Euclid's Elements... For the first time in history masses of isolated discoveries were unified and correlated by a single guided principle, that of rigid deduction from explicitly stated assumptions. ...If the Pythagorean dream of a mathematized science is to be realized, all of the sciences must eventually submit to the discipline that geometry accepted from Euclid.”

“Descartes' method of finding tangents and normals... was not a happy inspiration. It was quickly superseded by that of Fermat as amplified by Newton. Fermat's method amounts to obtaining a tangent as the limiting position of a secant, precisely as is done in the calculus today. ...Fermat's method of tangents is the basis of the claim that he anticipated Newton in the invention of the differential calculus.”

“Pearson's enthusiasm for W. K. Clifford's intuitive dynamics and physics, also for Clifford's violent hostility to traditional beliefs, influenced at least his earlier thinking. Both Clifford and Pearson were creative mathematicians; neither fitted the milk-and-water, namby-pamby 'great man' ideal of the 'great mathematician' which seems to be the accepted norm in historical accounts of mathematicians; and at least one of them would have hooted at the idea that he was, or was to become, an object of reverence for generations of students.”

“The sixth century [BC] was the time, and Greece the place, for human beings to reject once and for all the pernicious number mysticism of the East. Instead, Pythagoras and his followers eagerly accepted it as the celestial revelation of a higher mathematical harmony. Adding vast masses of sheer numerological nonsense of their own to an already enormous bulk, they transmitted this ancient superstition to the golden age of Greek thought, which passed it on to the first century A.D. to the decadent arithmologist Nicomachus. He, enriching his already opulent legacy with a wealth of original rubbish, left it to be sifted by the Roman Boethius, the dim mathematical light of the Middle Ages, thereby darkening the mind of Christian Europe with the venerated nonsense, and encouraging the gemaria of the Talmudists to flourish like a weed.”

“Objections... inspired Kronecker and others to attack Weierstrass' "sequential" definition of irrationals. Nevertheless, right or wrong, Weierstrass and his school made the theory work. The most useful results they obtained have not yet been questioned, at least on the ground of their great utility in mathematical analysis and its implications, by any competent judge in his right mind. This does not mean that objections cannot be well taken: it merely calls attention to the fact that in mathematics, as in everything else, this earth is not yet to be confused with the Kingdom of Heaven, that perfection is a chimaera, and that, in the words of Crelle, we can only hope for closer and closer approximations to mathematical truth—whatever that may be, if anything—precisely as in the Weierstrassian theory of convergent sequences of rationals defining irrationals.”

“He was admired for his science fiction and his Men of Mathematics. I was shocked when, just a few years later, Walter Pitts told me the latter was nothing but a string of Hollywood scenarios; my own subsequent study of the sources has shown me that Pitts was right, and I now find the contents of that still popular book to be little more than rehashes enlivened by nasty gossip and banal or indecent fancy.”

“By the time I was a student in high school I was reading the classic Men of Mathematics by E. T. Bell and I remember succeeding in proving the classic Fermat theorem about an integer multiplied by itself p times where p is a prime.”