By H. L. Haywood
Alas, this latter idea, so fruitful in the thought of Plato and Aristotle, did not take root, but slumbered for some two thousand years, waiting for a generation of men capable of conceiving and developing a conception so profound. Save for the dream of the Freemasons, mathematics continued to be considered a merely technical affair, useful to engineers and carpenters, but nobody guessed that it might possess any usefulness outside that narrow technical field. Mathematics a possible philosophy of human life, useful everywhere, to everybody, in all possible fields of human activity! It was impossible for men during those two thousand years even to formulate such an idea. It has remained for our own generation, building on the work of two or three generations immediately proceeding, to make the old dream come true. That it has at last come true everybody knows who knows anything at all about present day mathematics, more especially mathematical logic and mathematical philosophy; the story of how it has come true is a romantic story of great achievement—the greatest achievement, many believe, in all the long annals of mankind; and it is a story of grave moment to every Mason who in any degree cherishes that old ideal of the Craft that led our forbears to hang above the Master’s station the letter “G.” If only space permitted, and if this were the medium proper to it, it would be a pleasure to repeat in condensed form the history of those astonishing discoveries which have made of mathematics a new thing. Such a history would begin, perhaps, with the great Leibnitz, a contemporary of Sir Isaac Newton, who said, “Mathematics is my philosophy.” It would touch upon Spinoza, the mighty Dutch philosopher, to whom “belongs the credit of having been the first important thinker to see clearly that the method of Euclid’s ‘Elements’ was far more general than its matter,” and who tried to erect a system of ethics on the basis of Geometry, failing because the time was not ripe. It would tell of the discovery of the non-Euclidean Geometries by Bolyai, Lobachevski, Riemann, Klein, Lie, etc., and of the epoch-making definition of infinity made by Dedekind and Cantor. It would touch upon the work of scores of other thinkers to whom the world is infinitely more indebted than to most of its popular heroes. It would reach its climax with the crowning discovery of all, the discovery that at bottom mathematics and logic are one and the same thing—“one,” as Keyser has put it, “in the sense in which the roots the trunk and the branches of a tree are physically one.” And it would then only remain to show that on the basis of that system of thought which is at once pure mathematics and pure logic the best thinkers of our own day are even now painstakingly building a new philosophy of life, destined as surely to replace the older philosophies, as mathematics itself replaced the crude rule-of-thumb methods of primitive men. This, however, as I have already said, is not the place for such a history (though it is one of a kind of things that should be more discussed in Masonic journals); the studious reader must be referred to the works of Keyser, Russell and Whitehead, especially the “Mathematical Philosophy” of Keyser and “The Principles of Mathematics” of Russell. But how, you may well be asking, has it been possible to make a philosophy out of mathematics? The answer lies in the fact—at first of seeming small importance—that mathematics is a system of thought without content; the system holding true whatever kind of content you may be able to pour into it. If you pour into it the problems that bother you as a human being, problems of thought, or of conduct, or of the nature of the world, the result will be a philosophy; and just because the system you will be using is the system of mathematics you will find yourself in possession of a mathematical philosophy. One example will suffice. Euclid believed—like everybody else until less than a century ago, and like the vast majority of people even now—that Geometry has a subject matter peculiar to itself, consisting of space, points, lines, surfaces, angles, planes, circles, volumes, spheres, spirals, and similar entities. These entities were supposed to be the content of Geometry, and there was supposed to be no Geometry apart from such contents. This is known to be false. Geometry may have to do with such entities, but there are countless other kinds of entities that will serve equally well. Geometry is a self-consistent system of axioms or postulates, postulation functions, and doctrinal functions—in other words, a system of logical forms. Fit anything you can into those forms and the result will be Geometry of that thing. It is a mere accident that Euclid selected the set of entities he did; he might have selected any one of an infinite number of other sets, and the result would have been the same. What is true of pure Geometry is true of all pure mathematics. As written in the usual text-books it appears to deal with numbers, figures, letters, and a great variety of strange diagrams and bewildering formulate. But all that symbolic material is merely the technical machinery of the science; you may substitute for the numbers, figures, x’s, y’s, z’s, etc.—any suitable things you wish— and the result will be a mathematical treatment of those things. This means that many of the experiences, problems, ideas, and ideals of what we call our everyday life are all mathematizable. It was mathematics that enabled the physicists, chemists, biologists, astronomers and workers in other applied sciences to give us the steam-engine, the dynamo, the aeroplane, telephone, radio, modern surgery and all the other wonders; there is no reason why it should not lead to results equally marvellous if ever we learn to use it in dealing with all those momentously serious matters we call our problems of human life. As a matter of fact, mathematics is closer to everyday life than we are accustomed to think. Each of us is a mathematician unaware. A man may not be able to divide one fraction by another; he may be unable to count to ten; nevertheless he understands somewhat of mathematics if he understands anything at all, because mathematics is involved in the act of understanding itself. Without those ideas which mathematics makes the object of its special study, and to which its name is therefore given, no ideas at all would be possible; thinking itself would fall to pieces. Consider such ideas as these: counting, comparing, grouping, relating, serializing, adding, dividing, separating, order, dimensions, variation, dating, function, relations, infinity, etc., etc. Mathematics is nothing other than a rigorously precise definition and use of these and similar ideas and of the organization of them in various combinations. If we struck from our familiar English language every word used to express such ideas in all their multitudinous forms and ramifications our noble speech would be instantly reduced to an unintelligible chaos of meaningless noises. Mathematics is intelligence in its purest form. In it our most useful ideas are comprehended with complete thoroughness and defined with ultimate precision. Nay, one may justly say even more! One may say that it is in mathematics and in mathematics alone, that some of these ideas receive the only meaning they have, so that if a man is to use them at all he must know them as the mathematician knows them. When therefore our modern mathematicians released their science from its forbidding technical apparatus, setting mathematical ideas free, so that they, in all their precision, purity, self-consistency, and luminousness became available to all men, and for all purposes, they achieved for us all a service of unlimited usefulness, and they made possible a new future, a new development, a new hope for the world. In a newer manner, and in a way that only their rich modern equipment made possible, they have hung the letter “G” within reach of all who would become masters of the art of life, thereby fulfilling for us the old dream of our Masonic forefathers who dared to hope that their beloved art of Masonry might become as universal as the light. |