MATHEMATICS.

Zeno wrote a book against the Mathematics. We are informed of this by Proclus, who adds that Possidonius refuted it. Huetius having told us that Epicurus rejected geometry, and the other parts of the Mathematics, because he believed, that they, being founded on false principles, could not be true, adds, that Zeno attacked them another way. This was by alleging that in order to render them certain, some things should have been added to their principles which were not joined to them. Mathematics are the most evident and certain of all human sciences, and yet they have met with opposers. If

Zeno had been a great metaphysician, and followed different principles from those of Epicurus, he might have composed a book not very easy to be refuted, and cut out more work for the geometricians than they imagine. All sciences have their weak side; nor are the mathematics free from that defect. Indeed very few people are able to oppose them well, because to succeed in this engagement it is requisite not only to be a good philosopher, but also a very profound mathematician. But those endued with the latter quality are so ravished with the certainty and evidence of their inquiries, that they never think of examining whether there be any illusion in them, or whether the first foundation be well established. They rarely think of suspecting any deficiency in them, although it is very certain, that several disputes prevail amongst the most famous mathematicians. They refute one another, and answers and replies multiply among them as well as among other learned men. We observe this among the moderns, and it is certain that the ancients were not more unanimous. It is a proof that there are in this road several dark paths, and that a man may wander and lose the track of truth. This must of necessity be the lot of one side or the other, since one affirms what the other denies. It may be urged that this is the fault of the artificer, but not of the art, and that all these disputes proceed from some mathematicians mistaking that for a demonstration which is not so; but that very thing shews that there are some obscurities in this science. Besides, the same thing may be urged, with respect to the disputes of other learned men. It may be said, that if they closely followed the rules of logic, they would avoid the wrong consequences, and false assertions which mislead them. Nevertheless we must confess, that there are many philosophical subjects, concerning which the best logicians are incapable of coming to a certainty, by reason of the want of evidence in the object; but the object of the Mathematics is free from this inconvenience. Be it so; yet there is in this object a very great and irreparable defect; for it is a mere chimera which cannot possibly exist. Mathematical points, and consequently the lines and surfaces, globes and axes, of the geometricians are fictions which never can have a being: they are therefore inferior to those of the poets; for the latter commonly contain nothing that is impossible, but have at least probability and possibility.

Gassendus made an ingenious observation. Gasseud. Phys. s. i. lib. iii. cap. v. pag. 264, Oper. Tom. i. ⁸⁴ He says that the mathematicians, and especially the geometricians, have established their empire in the land of abstractions and ideas, where they walk at their ease; but that when they get into the country of realities, they soon meet with an invincible resistance. “The mathematicians, and especially the geometricians, by abstracting quantity from matter, have erected for themselves a sort of empire, where they enjoy the greatest liberty, as they never meet with any obstacle from the grossness and stubbornness of matter. Accordingly they have, in the first place, supposed that in quantity thus abstracted, there are such dimensions as a point, which hath no parts, and a line, or length without breadth, generated by the flowing of a point, &c. Such are the suppositions, by virtue of which the mathematicians retire within the limits of pure and abstract geometry, and as it were into a separate kingdom, where they compose their famous demonstrations. In a word, it is the mathematicians, who in this their empire of abstractions, suppose those things to be indivisible, which are without parts, without length, or breadth, and that where there is a division of parts, it may be carried on in infinitum. It is not thus with the

natural philosophers, who having to do with the material world, and living, as it were, in the empire of matter, cannot take such liberty.” He gives an instance of the vanity of their pretended demonstrations, in two subtle mathematicians who attempted to prove that a finite and an infinite quantity are equal. “Nuper Viri præclari Cavalerius, et Torricellius ostenderunt de acuto quodam solido infinitè longo et cuipiam tamen parallelepipedo, cylindrove finite æquali.” Others prove that there are infinite quantities bounded on each side. If they find evidence in demonstrations of that kind, ought it not to be suspected, since after all, it doth not overbalance the evidence with which common sense informs us, that what is finite can never be equal with what is infinite, and that infinity as infinity cannot have any bounds? I add, that it is not true, that evidence can accompany these gentlemen in all their speculations. For proof of this I shall quote an author who is very well acquainted with their subtilties. “It were to be wished,” saith he, Journal de Trevoux, for May and June, 1701, Art. xxxiii. ⁸⁵ “that the analysis of infinitely small quantities, which is pretended to be so admirably fertile, should carry in its demonstrations that evidence which is, and of right ought to be expected in geometry. But when they argue on the infinite, on the infinite of infinite, on the infinite of infinite of infinite, and so on, without ever finding terms to stop them, and when they apply to infinite magnitudes these infinites of infinites, those whom they would either instruct or convince have not always the penetration requisite to see clearly into such profound abysses. Those who are accustomed to the old ways of reasoning in geometry, do not easily quit them to follow such abstracted methods; they choose rather not to go so far than to engage in the new paths of the infinite of infinite of infinite, where we do not always see very clearly about us, and where it is easy to go astray without perceiving it. For it is not sufficient, in geometry to conclude, we must evidently see that the conclusion is just.”

It is a pretty good prejudice against the Mathematics, that Pascal despised them even before he gave himself up to devotion. He had been passionately fond of the Mathematics, and had made extraordinary progress in them. He was, besides, endued with a very solid judgment, and very few people were capable of knowing the value of things better than he. It was not by his conversion to the only thing necessary, that he became disgusted at these sciences which had so charmed him. It was the examination of the thing itself; and the reflection he made on the discourse of a layman, which cured him of his prepossession. It would be foolish to imagine that the Chevalier de Mere attacked him with pious reflexions. He undoubtedly made use of no other than philosophical considerations.

Observe, that it is highly proper that those, who endeavour to shew the weak side of the Mathematics, should convince the public that they understand them, that they have studied them, that they acknowledge the usefulness of them, and that they have no design to rob them of their just value. The learned bishop of Avranches, above cited, hath acted according to this rule, after having said several excellent things concerning the uncertainties and illusions of that science.

I believe that the Chevalier de Mere intended to recommend the philosophy of ideas, the most subtile metaphysics, which only tend to the contemplation of spirits, and the intelligible world which is in the mind of God; but the characteristics, which distinguish that science from the Mathematics, have not been considered by him; nor did he remember that the Mathematics have this principal property of

considering extension as separated from matter and every sensible quality. Extension or intelligible matter, is their object as sensible matter is that of physics. Their excellency, according to the ancients, consists in disengaging us from perishable corporeal things, and raising us to those which are spiritual, immutable, and eternal. Thence it proceeded that Plato disliked the conduct of some mathematicians who attempted to verify on matter their speculative propositions. I shall here transcribe a most excellent passage of Plutarch, which turns on a saying of Plato, that God is continually employed in geometrical exercises. Plutarch. in Marcello. ⁸⁶ “This saying, intimates what he himself has often mentioned, when he praised geometry as a science that takes men off from sensible objects, and fixes their thoughts on such as are intellectual and eternal, the contemplation of which is the end of all philosophy, as the viewing of mysteries is the end of mystical religion: for of all the mischiefs which arise to us from that sympathy in pain and pleasure which fastens the soul to the body, this is the greatest, that it renders sensible things more evident to us than intellectual, and forces the understanding to judge, rather according to passion than reason. Man being accustomed by his feeling of pain or pleasure, to regard the mutable and uncertain nature of bodies as a thing actually subsisting, grows blind, and loses the knowledge of that which really subsists, and destroys that instrument and light of the soul, which is worth a thousand bodily eyes, and by which alone the Deity can be discerned. Now in all the mathematical sciences, as in plain and smooth mirrors, the images and marks of the truth of intellectual objects appear, but geometry chiefly, as being the parent of all the rest, withdraws, and, as it were, purifies and sets loose the understanding from the thoughts of sensible objects; and therefore Plato himself reprehended Eudoxus, Archytas, and Menechmus, for endeavouring to reduce the doubling of the cube to mechanical operations, as if it were impossible to find out two lines which could be demonstrated to be mean proportionals. He objected to them that all that was good in geometry would be lost and corrupted, if it were made to fall back again to sensible things, instead of rising higher and contemplating those immaterial and immortal images, to which God was always attentive, and of which he was also the cause.”

Several passages of Aristotle also inform us, that quantity, as disengaged from whatever falls under the senses, is the object of the Mathematics; and the greatest part of the mathematicians own, that this object exists no where but in the mind. Dr Barrow disapproves the granting this. Isaac Barrow, Lect. V, pag. 85. ⁸⁷ His censure falls expressly on the Jesuit Blancanus and on Vossius; but it is certain that Blancanus was in the right, and ought not to have been censured for any thing besides asserting the possibility of the existence of the globe and triangle, &c. of the geometricians. There is no need of a long discourse to shew it impossible that this globe, or that triangle, &c. should really exist; we need only remember that such a globe placed on a plane, would only touch it in one indivisible point, and that rolling on the plane, it would touch it always in a single point. Whence it will result, that it must be wholly composed of unextended parts; but that is impossible, and manifestly includes this contradiction, that an extension would exist and not be extended. It would exist according to the supposition, and it would not be extended, since it would not be at all distinct from a being not extended. All philosophers agree that

the material cause is not distinct from its effects; wherefore what would be composed of unextended parts, would not be distinct from them; but whatever is the same thing with an unextended being, is necessarily an unextended thing. Our divines, when they teach that the world was produced out of nothing, do not mean that it is composed of nothing: the word nothing doth not signify the material cause of the world, “materiam ex qua but the state antecedent to the creation of the world, which they call “terminum à quo,” and they acknowledge that taking the word nothing in the first sense, it is absolutely impossible that the world should have been made of it. Now it is not more extravagant to assert, that the world was made of nothing as of its material cause, than to affirm that a foot of extension is composed of unextended parts. It is not therefore possible that either an angel, or God himself, should ever produce the triangle, the plane, the circle, the globe, &c. of the geometricians; and therefore Blancanus on this head deserved to be censured.

Art. Zeno