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"Blessed are they that have not seen and yet believe."

Mathematical Logic used to beexpressed in Geometric or quasi-Geometric diagrams. We now write it usually in a terminology borrowed from Algebra.

Modern Mathematical Logic may be said to be a tree, whose root is represented by the Newtonian Fluxion-method; and its trunk by the Logique of Gratry[1]; while its main branches are such works as those of Babbage, De Morgan, Boole, and Hinton; and the Sciences of Quaternions and of the so-called Fourth-Dimensional Mathematics. Carrying on the same simile, we might add that the numerous little twigs of special methods thrown off from the main boughs would have all the more chance of being fruitful if they were not so ready to sever their connection with the stock from which they sprang. As a matter of chronology, the treatises of Babbage, De Morgan, and Boole were published before the first appearance of the Logique. But then it must be remembered that Science is not created by the printer; books merely represent, in visible form, a thought-growth which has its actual existence in the Mind of Humanity ; and the chronological order in which the several parts of a new Science are projected on to the surface of Literature is not always identical with the order in which they were evolved. Mathematical Logic will be best understood by those who master the ideas of Gratry before entering upon the study of the more detailed but less philosophically comprehensive treatises of the other writers referred to above. As a matter of fact, neither Babbage nor Boole could have done what he did had he not perceived the truth of some principle analogous to that of Gratry.

And there is no greater hindrance to human progress than lack of piety towards Humanity's best teachers. The statement may be considered a truism; but unfortunately we are all too prone to neglect truisms. Principles so self-evident that no one disputes them, occupy less attention than do theories about which it is possible to get up a controversy; and the settlement of a fierce dispute is often best effected by insisting on the practical recognition of some principle which everybody concerned is disregarding, precisely because nobody has attempted to deny its truth. I repeat:—

There is no greater hindrance to progress than lack of piety towards great teachers; this is well shown by the whole history of Logic since the publication of Gratry's Logique in 1855.

Any Seer of a great comprehensive Truth, such as Gratry, is necessarily ahead of his age. He may be, in a sense, appreciated by his contemporaries; that is to say, he may be loved for his goodness, and admired because of certain secondary results of the intellectual Vision which has been vouchsafed to him; but the very essence of such a man's position is that he is seeing truths for which the world around him is not yet ready. When Gratry was writing his Logic, he was looking through the intellectual débacle which was then only beginning, and beyond it, to the new-created thought-world which might arise from amid the chaos, if the next generation would wisely consider its ways. No reader can properly understand Gratry who does not realize the nature of the cataclysm of which he was watching the beginning; and few persons in his time had any adequate conception of what was coming on the intellectual world. The condition of confusion which he foresaw in prophetic vision, we know as an indisputable and terrible fait accompli. He wrote, therefore, rather for us than for his contemporaries; to them he brought, chiefly, a warning, unintelligible to the majority, of stormy weather ahead; to us he offers practical pilotage through an actual hurricane.

But, alas! how shall we persuade young men trained in "the newest methods," that an Oratorian monk of fifty years ago was a better logician than they? "Gratry? oh! no doubt he was very clever for his time. But—he had not read any of our modern authorities; what can he have known of our new methods?" The young teacher, who is preparing pupils for examinations by the help of the last new text-book, and who perhaps hopes to make a name for himself by compiling a still newer one, resents being told of one greater than the author most in repute. So it happens that the writer, who was hardly understood in his own day because he was so far ahead of the age, is shunted into oblivion, when the time comes when he could be understood, because it suits the purposes of interested persons to believe that he is behind their age. The main cause of the chaotic condition into which our thought-life has fallen, is the feverish impatience of teachers, each of whom wants to make his own voice heard. But any teacher who is weary of the conflict between Science and Religion, between Knowledge and Faith, between the valuable Lessons of Materialism and the consoling belief in the spiritual world; any teacher who sincerely desires, not to make a cheap and noisy fame for himself, but to find Truth and Peace for the young souls committed to his care, might do well to devote a quiet vacation to the study of Gratry's Logique.

Our author begins by clearly recognizing that mathematics is not so much a department of human thought as the ground-plan of all sane thought; he treats mathematical Science, not as a special set of truths, but as a map of the country in which Truth is to be found. He of course entirely repudiates the profane notion that mathematical Logic can afford any proof or disproof of religious truths; but he shows that mathematical reasoning throws light on the nature of valid proof of every kind. The faculties by which the existence of God is revealed to us are hyper-intellectual; the processes by which we learn spiritual truth are extra-syllogistic; and it pleases a certain school of reasoners to deny their validity. Gratry shows that, but for the exercise of similar faculties and the use of similar processes, we could have discovered nothing of the Higher Mathematics. It pleases certain Agnostics to assert that "it is impossible to reason from the finite to the Infinite." The same objection was formerly made against the Integral Calculus, but the Integral Calculus is now an accepted fact; and any one who cares to learn can know exactly how to "integrate a finite expression to Infinity," and can make sure that his results are absolutely correct. No true mathematician depends on testimony for his knowledge of mathematics; each has within himself as absolute a personal conviction of the validity of the processes by which he works out his results, as if those processes were purely syllogistic. None the less is that conviction arrived at by a mode of mental action essentially of the same nature as that far more vigorous and sustained mental exercise by which we arrive at a sound and rational faith in God.

"Reasoning is not one process only, it contains two processes, equally valid ; but the second is by far the most powerful and most fertile. The syllogistic process deduces, passing from like to like. It never leaves its original stand-point; it never rises above that stand-point. It develops what it originally possessed. The other process, the dialectic, rises above its own starting-point; it does not merely develop the material which it originally possessed, it acquires fresh material. . . . This process passes from the finite to the Infinite. . . . This is the process of prayer. . . .Whence it follows that the human mind is capable of a process which, when it is taught as a part of Logic, will give wings to Logic which before had only feet. . . . The Logic of the highest philosophers, both practical and theoretical, has always had wings. But the class of average thinkers exerts itself chiefly in trying to cut those wings, and seems to have succeeded. They must learn to recognize their error, or rather their crime." The author then quotes the following passage from Cournot:—"The process by which the mind seizes new truths is often quite distinct from that by which it connects known truths together; and most important truths have been first seen by the help of that philosophic sense which precedes rigorous proof." Gratry objects to Cournot speaking as if one process was less rigorous than the other. And, in fact, we are quite as sure what is the result of an "integration to Infinity," as of what is the answer to a Rule of Three sum.

Suppose that a marble is running round in an elliptical groove under our eyes. Geometry enables us to investigate the relations between the various portions of the course which we see it pursue. But we know its course by direct inspection. Geometry can but arrange into a convenient form, information which we have gathered by observation; or, at most, it shortens the processes by which we acquire information. But when an Astronomer has observed a small portion of the orbit of a heavenly body, it disappears from his ken. He has to construct in imagination its future path, guided by knowledge of the hidden law of its motion. In order to do so, he must resort to the method of the Infinitesimal Calculus; the method thus described by Gratry:—"We have analyzed the finite in order to know the infinitesimal. From the knowledge given by the study of the finite, we have eliminated the quality of finiteness; what remains is true for the infinitesimal; that is to say, for the analysis and knowledge of the Indivisible and the Infinite. We have analyzed the discontinuous, the divisible, the finite; and have found therein the law of the Continuous, the Indivisible, the Infinitesimal."

But when we know the theoretical law of the planet's course, how do we know that, when we have lost sight of it, it still continues to obey the Laws of Mathematics? No Syllogistic Logic can prove that it does not wander, lawlessly, into space. Yet the mathematician ventures to construct, in imagination, its whole course. He is enabled to do so by an act of Faith—faith in the Great Unity Who governs Nature; faith which is the evidence of things not seen. This act of Faith is so elementary and spontaneous, that only a few deep thinkers have recognized it for what it is. But slight as it is, all the delights and powers conferred by the Higher Mathematics are the reward of that simple trust! See how The Unity keeps His ancient covenant with man! Eye hath not seen, nor ear heard, nor hath it yet entered into the heart of man to conceive, what He hath in store for those that love Him!

And slight as is the act of faith needed to understand transcendental mathematics, many are unable to perform it. There are so-called mathematicians who confess themselves unable to see the validity of the reasoning on which the Higher Mathematics rests; they only see that it must be somehow right because predicted eclipses and comets appear in due season; and they therefore assume the legitimacy of the stand-point whence predictions are made; but their own mental action goes no deeper than working out the syllogistic consequences of a knowledge which they do not properly possess. They believe because they have seen prediction fulfilled. Blessed are they who believe in The Unity before they see. No possession of any fact which we know only through the intellect or the senses can equal the intimate and rapturous sense of Union with the Great Unity which is given to him who is enabled by faith to trace in imagination the exact course of a planet out of sight.

Now if persons incapable of understanding transcendental mathematics should presume to claim from true mathematicians a profession of ignorance about all of which they themselves are ignorant, such arrogance would faintly picture that of much (so-called scientific) Agnosticism.

Gratry seems to have foreseen that the Science of Induction would never be developed except by the unselfish co-operation of several men, each of whom should be willing to subordinate his own personal work to the carrying out of a common aim. Mr. Boole's work was, as he gratefully acknowledged, made possible by the generous and self-forgetting aid freely given to him by many contemporary mathematicians; and in particular by Professor De Morgan, who seemed to take a pleasure in effacing himself to bring forward the man whom he might have been expected to feel a rival. And one of my pleasantest recollections of Mr. Boole himself, is his studying with loving and almost rapturous delight the pages of the Oratorian Father, who had stated with masterly and exhaustive completeness the fundamental principle which he himself had been vainly endeavouring to express.

  1. Logique, Gratry, 2 vols. Douniol, Paris.