IntroductionWe are surely living in a strange intellectual age. It is far from being one of stagnation; remissness and languor are not among its characteristics. It is rather one of stirring activity, of hard-working and indefatigable pursuit of science. To the onlooker it presents the appearance of a busy ant-hill, with all the nimble running to and fro that makes its denizens the very symbol of industry. But withal it is a very revolutionary ant-hill. It is one where the workers are as busy as ever, or even busier, but where they have refused to form ant-hills on the traditional lines, and are frantically exerting themselves to destroy all vestige of existing construction in the intense desire not only to upbuild in a new manner, but with entirely new materials. It is no new thing in human thought to find revolutionary attempts to overthrow the more generally accepted knowledge gathered by the human race throughout the ages. The world has long been familiar with religious scepticism; it is nothing modern. Ages ago philosophy after philosophy arose in the effort to destroy the accepted metaphysical basis of knowledge. This is a fact as old as philosophy itself. Physical science has had its revolutions that threw overboard what previously was held. But his age has gone further in this respect than any other; it has extended its attacks to the utmost bounds of science. The mutineers against the old order have seized the ship of knowledge and nailed the flag of dissent to the mast; they have driven the defenders of all manner of orthodoxy below decks and battened down the hatches over them, and have left in their administration not a single department of science. It has thus remained for the present age to go beyond all others in this respect. It has left nothing not upturned and uprooted in its search after truth. It has done what ever even was thought of before; it has attacked the basis of the supposedly most secure of all sciences, mathematics, and has dissolved its very elements in the strong acid of modern sceptical rationalism. Euclid has not fared better than the Bible or the philosophy of the schools, and the trimly classic but solid edifice of the geometry of the Greeks that has come down unchanged and unshaken for over two thousand years, has been pulled down and its fragments thrown into the discard together with other intellectual instruments shaped by the human mind in ages past. Investigation of the basis of our knowledge and an occasional stock-taking of our intellectual acquirements is doubtless a most useful and even necessary process, and one with which we certainly would not wish to find fault. But it is a question whether or not the rage of our century for novelty has not carried it too far; whether much of value has not been uprooted and cast aside along with that which ought to be rejected; whether the thinning process destined to promote growth has not become wasteful destruction; whether the intense intellectual activity of our day may not be in great part but futile effort and useless bustle. When normally sound criticism turns into destructive bolshevism, it is time to inquire whether the criticism is as sound as that which it criticises. Our present aim is to investigate this matter as far as the long established geometry of Euclid is concerned. A question that has puzzled mathematicians for over two thousand years, and that up to date has never received a satisfactory solution, is Euclid’s famous Postulate on converging lines, called the fifth postulate in the editions of Heath and of Heiberg; termed axiom 13 in the edition of Clavius; afterwards axiom 12 by Simpson; and axiom 11 by John Bolyai. Euclid deduced his whole geometry from five axioms and five postulates, all of them short and simple and generally admitted, except the one mentioned above. This reads: “If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” This is usually referred to as Euclid’s parallel postulate. For over two thousand years geometers tried to demonstrate this postulate and failed, and we still have many of these attempts, from the first on record, that of the astronomer, Ptolemy, down to those of Legendre, who tried all his life to prove it. The reason of these attempts was not precisely because there ever was any doubt about the truth of the postulate, which until lately was never challenged. It was rather a question of method. This postulate of Euclid is so different from his other postulates in simplicity of expression and immediateness of evidence, and so much more in the order of one of his propositions to be proved, that assuming it as a postulate seemed a blot on his method. Without doubt, it was his inability to prove this proposition by itself with the aid of the other postulates and axioms, and thus connect it in the general framework of his system, that led Euclid, when once he found he needed it for further deduction (which first occurs in the twenty-ninth proposition of the First Books), to assume it once for all as a postulate. His successors, while seeing the difficulty as well as he, through successive generations unsuccessfully sought to remove it, but they wee never able to improve his position, and make the proposition a theorem instead of an assumption. As a result of this failure, certain mathematicians of the last century came to the conclusion that the postulate was indemonstrable, certainly a very easy way to cut the Gordian knot of the difficulty; and then, with the utmost inconsequence, and with more mental agility than either poise or balance, jumped to the other and much more radical and subversive conclusion, that the proposition itself was not valid. Having made the jump, they immediately began to busy themselves with constructing Geometries, not only without its aid, but with another postulate contradictory to it. Such is the solid and deeply. dug base on which rests much of modern mathematical science,0r what is more and more passing for such. It was indeed quite a victory for scepticism to be able to found by such slight intellectual effort on a purely negative foundation the most positive of all sciences. Prestidigitator and magician have nothing at a11 on the founders and supporters of the so called non-Euclidean Geometry. The method followed too is largely the same. The gentlemen who wish to deceive our senses by their manual dexterity rely to a great extent on the psychological element; they try to distract our attention, and turn it to something else, while performing the trick that is to mystify us. The neo-mathematicians do the same. They depend on the advertising value and peculiar psychological effect of broad assertions uttered with all assurance, of ridicule heaped on attempts to conserve what the world has held for thousands of years with great benefit to itself, and what furthermore is the instinctive conclusion of the commonest of common sense, in order to distract the inattentive mind from the real point at issue, and impose credulity if not conviction, as to the superior enlightenment and greater depth of the new mathematicians. Just let us recall a few samples of the method. Here is what a mathematician of this kind, to whom great authority has been allowed in this generation, Poincare, (he is called by Halstead one of the two greatest mathematicians of the day; the other is Hilbert, whose work we shall also have occasion to consider) has to say of these efforts to prove the Postulate. “What a vast effort has been wasted in this chimeric hope is truly unimaginable.”1 Elsewhere he adds: “The postulate of Euclid then cannot be demonstrated; and this impossibility is as absolutely certain as any mathematical truth whatsoever.”2 Lobatschewsky, along with Bolyai, the founder of the new Geometry, says: “The futility of the efforts which have been made since Euclid’s time during the lapse of two thousand years, awoke in me the suspicion that the ideas employed might not contain the truth sought to be demonstrated.”3 John Bolyai thus speaks of his own short monograph: “The Science of Absolute Space independent of the truth or falsity of Euclid’s Axiom XI (which can never be decided a priori).”4 The English translator of this work of John Bolyai calls it, “The most extraordinary dozen pages in the whole history of thought.”5 Hoüel, a translator and vulgarizer of the new Geometry and one of those who are supposed to have proved the impossibility of establishing the Postulate, says: “We must place among the chimeras the hope that so many geometers still entertain of being able to demonstrate otherwise than by experience the axiom of Euclid. Henceforward, such attempts should be put on the same footing as the squaring of the circle and perpetual movement.”6 This is all very strong language. What in the beginning was a mere desire to perfect a method, first became a suspicion of its truth, then suddenly developed into the most absolute certitude of the opposite, expressed in the most exaggerated terms, as in Poincare’s words quoted above. Poincare not only speaks of the futile endeavors, but alludes ironically to the one or two attempts still presented yearly to the Academy in proof of Euclid’s postulate. Perhaps he wishes the authors to be considered as still living in the dark ages of mathematics, almost as backward as if they had not yet heard that the world is turning on its axis. No doubt such strong language is meant as a sort of scarecrow symbol to keep off trespassers from this domain of thought which has been seized upon as their own by the newer school of mathematicians. No doubt we ought to be duly impressed when those who at least speak so authoritatively, have their authority so nearly universally granted, and that we ought to fear to trespass where there is so much sound and waving of arms to keep us off. But perhaps some may suspect that so much noise is not at all a sign of complete sureness or even assurance, but rather a cloak for a certain amount of misgiving, and an effort to keep up a valiant front, as a boy is supposed to whistle noisily to keep up his courage in passing through a cemetery by night. But at nay rate there are those who may wish to hold their right of going where they please in the public domain of thought, which is a common that cannot be fenced in. So long as Euclid remained the basis of all mathematical teaching, one might allow the neo-Geometers, in perfect freedom and without interruption or aggression, to amuse themselves with fabricating any kind of fantastic Geometry they might wish. At most it was a harmless exercise, and perhaps a gentle and amusing recreation. But that is no longer the case. Persistent advertising, so well understood in our days, is having its effect. Non-Euclidean Geometry is no longer kept by itself on the back seat, but has come with vociferous voice to the front, and has seized upon as its own by right not only some of the best mathematical minds, but the front page of the newspapers as well. It is no longer a harmless mental exercise or a mere intellectual curiosity, but it is made the foundation of a new theory of the universe, that of Relativity, which is heralded wherever the printed work reaches. Non-Euclidean Geometry is no longer represented by two obscure men, Lobatschewsky and John Bolyai, who died disappointed and perhaps embittered because their theories met with no recognition from the erudite. Its standard bearer now is Einstein, who rises to a class by himself, and is sitting on the top of the learned world, and who himself doubts whether there are twelve mathematicians who can follow him to the heights of the new Parnassus. We are not putting any weight on the authenticity or signification of this statement; it is enough for us that it has been so vulgarized, as it shows at least a tendency. With all the heralding of the new mathematics, perhaps it would be well for us before we attempt the laborious climbing to such great heights, to see on what base the mountain rests, and find out whether it is real, or only a mirage of cloud which it would be a useless beating of arms and stamping of feet to attempt ascending. If it is only a cloud mountain, we may look for it to dissolve in the first breath of wind that will strike it. There have been sophists in every age of the history of thought, and none better known than those who gave us the name among the clear-minded Greeks, who used the acumen of their intellects on every subject of human thought, which they. tested with the acid of scepticism and rationalism. They attacked the most elementary and obvious truths of common sense, as much from the love of argument and the love of intellectual battle as from anything else. But we can always imagine them with their tongues in their cheeks, knowing that they were but playing a clever game, and themselves not believing in its results. Moreover, sophism never stopped the healthy trend of Greek thought, but, if anything, only brought it out more sharply and clearly. But the modern sophists are of a different kind. They take themselves so seriously that the world has been willing to take them at their word. It cannot believe that such portentous solemnity is anything but the badge of the highest genius and truth. The statement by Halstead concerning John Bolyai, that twenty-four pages written by a young man of twenty-one are the most extraordinary two dozen pages in the history of human thought, could hardly be excelled for naive seriousness. Not that we hold the matter of age against him, but when we consider that this young man still lived a matter of thirty-seven years and contributed nothing further in this line or in any other to the history of thought, there might be some prudence in hesitating to acclaim his utterance as the most extraordinary of all time. Extraordinary utterances are usually made by men who are able to show themselves extraordinary, and that more than once in a lifetime. What we think of the extraordinariness of this utterance and others we shall have occasion to make clear later on, but it seems that the history of human thought is scarcely to be weighed and balanced in such a hasty and cavalier manner. These new geometers climb and climb, at least they go through the motions, and they never worry whether or not they have anything to climb; they weave and weave their web, and never bother whether it is a solid cable attached to reality, or as a spun gossamer floating on the autumn breeze. Their whole aim is in the climbing and the weaving. That their theories combat common sense, that they contradict the elementary facts of intuition, causes them no concern; so much the worse for common sense; overboard with intuition. Their system stands by itself on nothing else. They need no previous foundation, and no basis at all but hypothesis. If they can weave from an hypothesis, the hypothesis is established. The only proof they ever exacted or proclaimed as necessary for their Geometry is that it do not contradict itself. But if it contradicted anything else, the anything else had to go. We may seem to be exaggerating; but this is not a rhetorical presentation, but cold fact, as we shall see. It goes without saying that the real scientist will always be ready to abandon any theory that facts prove to be false; that he will, moreover, be ever on the lookout for whatever is unsatisfactory or incomplete in his scientific reasonings. He knows that science is a growth, and like all growth subject to change, and to rejection of the useless and outworn. But on the other hand he will not lightly reject what has stood the test of ages, and has received the full sanction of the talent and genius of all time. He knows that growth is likewise continuous, and the new can be acquired only on the basis of the old. He is aware that a thousand difficulties do not create a doubt, nor a thousand doubts the certainty of the opposite proposition. He is assured that this is a correct methodological, metaphysical and scientific principle. But when we are asked to cut loose from all that has been previously held, to scrap all our acquired ideas and notions, to start out completely anew, as if nothing had been hitherto attained; to construct new notions of fundamental geometric concepts, of distance, and direction, of space and time, all because of a simple difficulty in finding a satisfactory proof for what was after all simple enough and clear enough to be accepted as a postulate, and, moreover, when we find that such a method has been adopted by serious-minded scientists, it is time to call a halt. The strange part of it all is that while the most subtle and exaggerated hypercriticism is applied to the old-time geometry, and to the principles on which it is founded, the bases of the new Geometry are allowed to stand without the least attempt to examine them critically, and it is welcomed with open arms, and no questions asked as to its provenance. It is unfortunate that in so much modern work more effort must be given to the labor of debunking than to that of actual development. The field must always be kept clear of inordinate growth of weeds, if we are to produce the useful. This is a process that is always required, but in no age more than the present, and in all branches of science. It seems that the law of labor made for man when he was driven from the terrestrial Paradise applies to his intellectual as well as to his physical life. Thorns and thistles are as large a part of his work as the useful crops. Even here he is bound to earn his living by the sweat of his brow. For the labor of debunking is what makes him sweat. Growth itself is painless and pleasant. There are even enemies that sow their cockle in the field of science, but here at least it may not be left till the time of the harvest before being bound into bundles and burned. But this is not our main purpose. There would be as much sport in chasing the will-o’ -the-wisp as in following all the vagaries of the new mathematicians. It would be just about as possible, and just about as profitable. Our aim is primarily to undertake the duty of showing that the so-called fifth postulate of Euclid is as capable of valid geometric proof as any other proposition of traditional geometry, and that if the efforts of over two thousand years mean anything, they show at least that the consensus of thinkers on this matter is that the proposition is demonstrable, if not entirely evident. As we hope to show, the demonstration is easy, simple and clear. Many no doubt failed because they took the wrong road, the long and tortuous path of a reductio ad absurdum - “proof leading to the impossible.” What to the ancients appeared as a near axiom should scarcely require a whole series of propositions tending to a contradiction, many of which have been taken over bodily by the new Geometers and incorporated into their system. But we shall leave this for the treatise itself. Naturally we shall have to pay some attention to the theories built on assumptions contradictory to Euclid’s maxim, for we shall at least investigate where and how they depart from the still generally accepted doctrine, and on what the hypotheses at the bottom of each are founded. We are not opposed to hypotheses. They are surely a legitimate method of seeking to account for the facts of the physical universe, at least as a temporary arrangement. But hypotheses are legitimate only when they start from knowledge that has already been acquired by the human race, and when they are distinctly kept as hypotheses. They must at least be constructed out of ideas that are conceivable and the result of our previous knowledge. Systems that are so revolutionary that they jettison even the simplest facts of intuition can scarcely claim for themselves any overweening authority, and need more than one or two probabilities in their favor to establish them. The other requirement for a legitimate use of a hypothesis is that it be professed as a hypothesis, and not as an absolute dogma, whose very existence must make the holding of anything in the least contrary a sign of reaction and unintelligent adherence to outworn conservatism. There can only be true progress when we hold what we have and further increase it. Real science wants to know the starting point and also the point of arrival. Intelligent working requires some knowledge of where we are going to land when we leap in the dark. It is not a leap from the solid ground of reality into the chasm of the unknown in the hope that we may land on something new. The old geometers at least kept their feet on the solid ground, a thing the new Geometers do not do. These have made a clear jump over the precipice, and only expect justification from their survival. Moreover, in considering Relativity as a system, we cannot be satisfied with the fragmentary and unconnected notes of the mathematician or even the physicist, because we can scarcely find there a really unified conception. For this we must have recourse to the exposition of the philosopher-for he at least must attempt to rationalize and unify his concepts. Even though his final results may not be more satisfactory so far as building a consistent system is concerned, his unification offers a better object to dissect, and contradictions are more easily detected and pointed out, since he, more than the mathematician and the scientist, drags them from their dark lurking places and puts them face to face. But even here analysis must be limited, if by nothing else than by the weight that should be allowed to the essentially absurd. This last is the other point we undertake to show in analyzing Metageometry and Relativity.
|