Creative Intelligence

The Essays which follow represent an attempt at intellectual cooperation. No effort has been made, however, to attain unanimity of belief nor to proffer a platform of "planks" on which there is agreement. The consensus represented lies primarily in outlook, in conviction of what is most likely to be fruitful in method of approach. As the title page suggests, the volume presents a unity in attitude rather than a uniformity in results. Consequently each writer is definitively responsible only for his own essay.

Larger Print, 13 point font


**************

Excerpt:

A characteristic of Greek social life is responsible both for the next phase of the development of mathematical thought and for the misapprehension of its nature by so many moderns. "When Archytas and Menaechmus employed mechanical instruments for solving certain geometrical problems, 'Plato,' says Plutarch, 'inveighed against them with great indignation and persistence as destroying and perverting all the good that there is in geometry; for the method absconds from incorporeal and intellectual or sensible things, and besides employs again such bodies as require much vulgar handicraft: in this way mechanics was dissimilated and expelled from geometry, and being for a long time looked down upon by philosophy, became one of the arts of war.'

In fact, manual labor was looked down upon by the Greeks, and a sharp distinction was drawn between the slaves who performed bodily work and really observed nature, and the leisured upper classes who speculated, and often only knew nature by hearsay. This explains much of the naïve dreamy and hazy character of ancient natural science. Only seldom did the impulse to make experiments for oneself break through; but when it did, a great progress resulted, as was the case of Archytas and Archimedes. Archimedes, like Plato, held that it was undesirable for a philosopher to seek to apply the results of science to any practical use; but, whatever might have been his view of what ought to be in the case, he did actually introduce a large number of new inventions" (Jourdain, The Nature of Mathematics, pp. 18-19). Following the Greek lead, certain empirically minded modern thinkers construe geometry wholly from an intellectual point of view.

History is read by them as establishing indubitably the proposition that mathematics is a matter of purely intellectual operations. But by so construing it, they have, in geometry, remembered solely the measuring and forgotten the land, and, in arithmetic, remembered the counting and forgotten the things counted.

Arithmetic experienced little immediate gain from its new association with geometry, which was destined to be of momentous import in its latter history, beyond the discovery of irrationals (which, however, were for centuries not accepted as numbers), and the establishment of the problem of root-taking by its association with the square, and interest in negative numbers.

The Greeks had only subtracted smaller numbers from larger, but the Arabs began to generalize the process and had some acquaintance with negative results, but it was difficult for them to see that these results might really have significance. N. Chuquet, in the fifteenth century, seems to have been the first to interpret the negative numbers, but he remained a long time without imitators. Michael Stifel, in the sixteenth century, still calls them "Numeri absurdi" as over against the "Numeri veri."

However, their geometrical interpretation was not difficult, and they soon won their way into good standing. But the case of the imaginary is more striking. The need for it was first felt when it was seen that negative numbers have no square roots. Chuquet had dealt with second-degree equations involving the roots of negative numbers in 1484, but says these numbers are "impossible," and Descartes (Geom., 1637) first uses the word "imaginary" to denote them. Their introduction is due to the Italian algebrists of the sixteenth century.

Geometry, however, among the Greeks passed into a stage of abstraction in which lines, planes, etc., in the sense in which they are understood in our elementary texts, took the place of actually measured surfaces, and also took on the deductive form of presentation that has served as a model for all mathematical presentation since Euclid. Mensuration smacked too much of the exchange, and before the time of Archimedes is practically wholly absent. Even such theorems as "that the area of a triangle equals half the product of its base and its altitude" is foreign to Euclid (cf. Cajori, p. 39). Lines were merely directions, and points limitations from which one worked. But there was still dependence upon the things that one measures. Euclid's elements, "when examined in the light of strict mathematical logic, ... has been pronounced by C. S. Peirce to be 'Riddled with fallacies'" (Cajori, p. 37). Not logic, but observation of the figures drawn, that is, concrete symbolization of the processes indicated, saves Euclid from error.

Roman practical geometry seems to have come from the Etruscans, but the Roman here is as little inventive as in his arithmetical ventures, although the latter were stimulated somewhat by problems of inheritance and interest reckoning. Indeed, before the entrance of Arabic learning into Europe and the translation of Euclid from the Arabic in 1120, there is little or no advance over the Egyptian geometry of 600 B. C. Even the universities neglected mathematics. At Paris "in 1336 a rule was introduced that no student should take a degree without attending lectures on mathematics, and from a commentary on the first six books of Euclid, dated 1536, it appears that candidates for the degree of A. M. had to give an oath that they had attended lectures on these books. Examinations, when held at all, probably did not extend beyond the first book, as is shown by the nickname 'magister matheseos' applied to the Theorem of Pythagoras, the last in the first book.... At Oxford, in the middle of the fifteenth century, the first two books of Euclid were read" (Cajori, loc. cit., p. 136). But later geometry dropped out and not till 1619 was a professorship of geometry instituted at Oxford. Roger Bacon speaks of Euclid's fifth proposition as "elefuga," and it also gets the name of "pons asinorum" from its point of transition to higher learning. As late as the fourteenth century an English manuscript begins "Nowe sues here a Tretis of Geometri whereby you may knowe the hegte, depnes, and the brede of most what erthely thynges."

The first significant turning-point lies in the geometry of Descartes. Viete (1540-1603) and others had already applied algebra to geometry, but Descartes, by means of coordinate representation, established the idea of motion in geometry in a fashion destined to react most fruitfully on algebra, and through this, on arithmetic, as well as enormously to increase the scope of geometry. These discoveries are not, however, of first moment for our problem, for the ideas of mathematical entities remain throughout them the generalized processes that had appeared in Greece. It is worth noting, however, that in England mechanics has always been taught as an experimental science, while on the Continent it has been expanded deductively, as a development of a priori principles.