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 coördinate 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.
III
Contemporary Thought in Arithmetic and Geometry
To develop the complete history of arithmetic and geometry would be a task quite beyond the limits of this paper, and of the writer's knowledge. In arithmetic we were able to observe a stage in which spontaneous behavior led to the invention of number names and methods of counting. Then, by certain speculative and "play" impulses, there arose elementary arithmetical problems which began to be of interest in themselves. Geometry here also comes into consideration, and, in connection with positional number symbols, begin those interactions between arithmetic and geometry that result in the forms of our contemporary mathematics. The complex quantities represented by number symbols are no longer merely the necessary results of analyzing commercial relations or practical measurements, and geometry is no longer directly based upon the intuitively given line, point, and plane. If number relations are to be expressed in terms of empirical spatial positions, it is necessary to construct many imaginary surfaces, as is done by Riemann in his theory of functions, a construction representing the type of imagination which Poincaré has called the intuitional in contradistinction to the logical (Value of Science, Ch. I). And geometry has not only been led to the construction of many non-Euclidian spaces, but has even, with Peano and his school, been freed from the bonds of any necessary spatial interpretation whatsoever.
To trace in concrete detail the attainment of modern refinements of number theory would likewise exhibit nothing new in the building up of mathematical intelligence. We should find, here, a process carried out without thought of the consequences, there, an analogy suggesting an operation that might lead us beyond a difficulty that had blocked progress; here, a play interest leading to a combination of symbols out of which a new idea has sprung; there, a painstaking and methodical effort to overcome a difficulty recognized from the start. It is rather for us now to ask what it is that has been attained by these means, to inquire finally what are those things called "number" and "line" in the broad sense in which the terms are now used.
In so far as the cardinal number at least is concerned, the answer generally accepted by Dedekind, Peano, Russell, and such writers is this: the number is a "class of similar classes" (Whitehead and Russell, Prin. Math., Vol. II, p. 4). To the interpretation of this answer, Mr. Russell, the most self-consciously philosophical of these mathematicians, has devoted his full dialectic skill. The definition has at least the merit of being free from certain arbitrary psychologizing that has vitiated many earlier attempts at the problem. Mr. Russell claims for it "(1) that the formal properties which we expect cardinal numbers to have result from it; (2) that unless we adopt this definition or some more complicated and practically equivalent definition, it is necessary to regard the cardinal number of a class as indefinable" (loc. cit., p. 4). That the definition's terms, however, are not without obscurity appears in Mr. Russell's struggles with the zigzag theory, the no-class theory, etc., and finally in his taking refuge in the theory of "logical types" (loc. cit., Vol. III, Part V. E.), whereby the contradiction that subverted Frege and drove Mr. Russell from the standpoint of the Principles of Mathematics is finally overcome.
The second of Mr. Russell's claims for his definition adds nothing to the first, for it merely asserts that unless we adopt some definition of the cardinal number from which its formal properties result, number is undefined. Any such definition would be, ipso facto, a practical equivalent of the first. We need only consider whether or not the formal properties of numbers clearly follow from this definition.
Mr. Russell's own experience makes us hesitate. When he first adopted this definition from Frege, he was led to make the inference that the class of all possible classes might furnish a type for a greatest cardinal number. But this led to nothing but paradox and contradiction. The obvious conclusion was that something was wrong with the concept of class, and the obvious way out was to deny the possibility of any such all-inclusive class. Just why there should be such limitation, except that it enables one to escape the contradiction, is not clear from Mr. Russell's analysis (cf. Brown, "The Logic of Mr. Russell," Journ. of Phil., Psych., and Sci. Meth., Vol. VIII, No. 4, pp. 85-89). Furthermore, to pass to the theory of types on this ground is to give up the value of the first claim for the definition (quoted above), since the formal properties of numbers now merely follow from the definition because the terms of the definition are reinterpreted from the properties of number, so that these properties will follow from it. The definition has become circular.
The real difficulty lies in the concept of the class. Dogmatic realism is prone to find here an entity for which, as it is obviously not a physical thing, a home must be provided in some region of "being." Hence arises the realm of subsistence, as for Plato the world of facts duplicated itself in a world of ideas. But the subsistent realm of the mathematician is even more astounding than the ideal realm of Plato, for the latter world is a prototype of the world of things, while the world of the mathematician is peopled by all sorts of entities that never were on land or sea. The transfinite numbers of Cantor have, without doubt, a definite mathematical meaning, but they have no known representatives in the world of things, nor in the imagination of man, and in spite of the efforts of philosophers it may even be doubted whether an entity correlative to the mathematical infinite has ever been or can ever be specified.