triangles made of steel neither would prove that the sum of the angles of the triangle the mathematician is talking about is 180° nor would it disprove it. The reason is again that the theorem is independent of the object. If we prove a theorem as to a triangle or as to the number 2, the proof is nowhere dependent upon the material of an object, or upon its chemical constitution, or upon the day of the month, or upon the weather. This fact, which is obvious, is a sufficient reason for asserting that these theorems are therefore valid, irrespective of the material, or the time, or the weather. A more subtle question is raised if we view these universal theorems as Enriques does, that is, as observed invariants of experience. If this is all they are, then, as experience proceeds through the ages, they may turn out ultimately to be only relative invariants, and might even be only approximately invariant. It is true that in a series of changes of form, in which we find an element that nevertheless does not change with the form, there is in this invariant element an independence from those features that accompany the changes. As an example, the harmonic ratio of four points is not disturbed by a projective transformation. But the ratio in question may be studied without considering it to be an invariant of such a transformation. We may study geometry, to be sure, from the projective point of view and reach the usual metrical theorems, but the more natural way to arrive at them is to study them directly. The fact that they may be looked at as invariants is a fact of which they are indeed independent. There is, so to speak, a higher degree of independence than that of invariance, namely, independence of an absolute type. This is the kind of independence that we will find in most mathematical developments, if they are carefully analyzed. Consequently we can be certain that in studying those things which are independent in the absolute sense of time, place, and person, and not merely invariant as to time, place, and person, we really arrive at a permanent structure in the highest sense possible. We find in the following a clear summary of the answer to the question:1 The mathematical laws presuppose a very complex elaboration. They are not known exclusively either a priori or a posteriori, but are a creation of the mind; and this creation is not an arbitrary one, but, owing to the mind's resources, takes place with reference to experience and in view of it. Sometimes the mind starts with intuitions which it freely creates; sometimes, by a process of elimination, it gathers up the axioms it regards as most suitable for producing a harmonious development, one that is both simple and fertile. Thus mathematics is a voluntary and intelligent adaptation of thought to things, it represents the forms that will allow of qualitative diversity being surmounted, the moulds into which reality must enter in order to become as intelligible as possible. It was C. S. Peirce2 who defined Mathematics to be the "study of ideal constructions." He adds the remark: "The observations being upon objects of imagination merely, the discoveries of mathematics are susceptible of being rendered quite certain." The importance of viewing mathematics as a tremendous structure is brought out by this definition, from the humble magic square to vast systems such as projective geometry, functions of complex variables, theory of numbers, analysis in general. If I E. Boutroux, Natural Law in Science and Philosophy, trans. by Rothwell, p. 40. 2 Century Dictionary; article “Mathematics.” the mathematician were engaged only in ideal building, the definition might be sufficient. But we have already seen that he is interested in ranges and in multiple ranges, which may be considered to be the materials of building, as well as in the synthesis of these materials. And we shall see also that he is furthermore equally interested in the study of types of synthesis aside from the structures themselves. Like a master architect, he must study his stones and metals, he must design beautiful and useful structures, but he must do more. He must investigate the possible orders under various limitations. And, most of all, he is obliged to consider the actual processes of construction, which leads him into dynamic mathematics. REFERENCES MacMahon, Combinatory Analysis, 1915–16. Bragdon, Projective Orament, 1912. CHAPTER V LOGISTIC AND THE REDUCTION OF MATHEMATICS TO LOGIC In the year 1901 we find in an article by Bertrand Russell: "The nineteenth century, which prides itself upon the invention of steam and evolution, might have derived a more legitimate title to fame from the discovery of pure mathematics. One of the chiefest triumphs of modern mathematics consists in having discovered what mathematics really is. . . . . Pure mathematics was discovered by Boole in a work which he called The Laws of Thought. . . . . His work was concerned with formal logic, and this is the same thing as mathematics." Again, Russell says, "The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of Mathematics consists in the analysis of Symbolic Logic itself." Also in Keyser's address3 we find: " . . . the two great components of the critical movement, though distinct in origin and following separate paths, are found to converge at last in the thesis: "Symbolic Logic is Mathematics, Mathematics is Symbolic Logic, the Twain are One." 2 On the other hand, we find Poincaré4 saying after his various successful attacks on logistic: "Logistic has to be made over, and one is none too sure of what can be saved. It is unnecessary to add that only Cantorism 1 International Monthly, 4 (1901), pp. 83-101. 2 Principles of Mathematics, p. 5. 3 Columbia University Lectures. 4 Science et méthode, p. 206. and Logistic are meant, true mathematics, those which serve some useful purpose, may continue to develop according to their own principles without paying any attention to the tempests raging without them, and they will pursue step by step their accustomed conquests which are definitive and which they will never need to abandon." What, then, is this logistic which made such extravagant claims in 1901 and in 1909 was dead? In order to understand it we must go back to the third century B.C., when Aristotle was developing the study usually called logic. The logic of Aristotle is well enough defined when it is called the logic of classes. A class may be defined in the following terms. Let us suppose that we start with a proposition about some individual, as for example, "8 is an even number," or as another case, "Washington crossed the Delaware." If, now, we remove the subject and substitute the variable x, we shall have the statements: "x is an even number, x crossed the Delaware," which are called propositional functions, from analogy to mathematical functions. In this case the functions have but one variable or undetermined term, x. If we let x run through any given range of objects, the resulting statements will be some true, some false, some senseless. Those that are true or false constitute a list of propositions. For example, we may say: "6 is an even number, 9 is an even number, this green apple is an even number," the first a true proposition, the second a false proposition, the third an absurdity. So I might say: "Washington crossed the Delaware, the Hessians crossed the Delaware, the North Pole crossed the Delaware," which are, respectively, true, false, and absurd, the first two cases being propositions. The propositional function with one vari |