Infinitesimal

“ The conceptions of the infinite and absolute as ordinarily understood are tiresome because they are unmeaning, but there is no peculiar sanctity or mystery in them. We might as well make an infinitesimal series of fractions or a perpetually recurring decimal the object of our worship. They are the widest and also the thinnest of human ideas, or, in the language of logicians, they have the greatest extension and the least comprehension. ”

“ These preliminary ideas being established, the spirit of the infinitesimal analysis consists in constantly neglecting the infinitely small quantities in comparison with finite quantities, and generally the infinitely small quantities of any order whatever in comparison with all those of an inferior order. ”

“ These, I say, assert there are infinitesimals of infinitesimals of infinitesimals, &c., without ever coming to an end; so that according to them an inch does not barely contain an infinite number of parts, but an infinity of an infinity of an infinity ad infinitum of parts. Others there be who hold all orders of infinitesimals below the first to be nothing at all; thinking it with good reason absurd to imagine there is any positive quantity or part of extension which, though multiplied infinitely, can never equal the smallest given extension. ”

“ But as we are not being, but existence, we are finite and limited in our very nature; remove the limitations, and we are not God, but nothing. Eliminate the finite, says Père Gratry, and you have God, in the same way and by the same process that the mathematician has his infinitesimals. But this process of elimination of the finite gives the mathematician only the infinitely less than the finite number or quantity, and it would give the theologian not the infinitely greater but the infinitely less than the finite existence. ”

“ It is an idea which implies contradiction; and all these ‘infinites’ and ‘infinitesimals’ are of use only in geometry, as imaginary roots are in algebra.” That which is ordinarily called the infinite, that is, the quantitative infinite, is in reality only the indefinite. “We involve ourselves in difficulty when we talk about a series of numbers extending to infinity; we imagine a last term, an infinite number, or one infinitely little. ”

“ The task is not without difficulty; it is not enough to open a book at random and to analyse any proof we may come across. First of all, geometry must be excluded, or the question becomes complicated by difficult problems relating to the rôle of the postulates, the nature and the origin of the idea of space. For analogous reasons we cannot avail ourselves of the infinitesimal calculus. We must seek mathematical thought where it has remained pure—i.e., in Arithmetic. ”

“ The same sort of thing happens in space. If any piece of matter be cut in two, and then each part be halved, and so on, the bits will become smaller and smaller, and can theoretically be made as small as we please. However small they may be, they can still be cut up and made smaller still. But they will always have some finite size, however small they may be. We never reach the infinitesimal in this way, and no finite number of divisions will bring us to points. ”

“ Increments or Decrements infinitely small, used by foreign Mathematicians instead of Fluxions or Velocities of nascent and evanescent Increments. VI. Differences of various Orders, i. e. Quantities infinitely less than Quntities infinitely little; and infinitesimal Parts, of infinitesimals of infinitesimals, &. without end or limit.
VII. Mysteries in faith unjustly objected against by those who admit them in Science.
VIII. Modern Analysts supposed by themselves to extend their views even beyond infinity: Deluded by their own Species or Symbols. ”

“ If φ′ (x) and ψ′ (x) have determinate finite limits when x is increased indefinitely, while φ (x) and ψ (x) are determinately (positively or negatively) infinite, we have the result expressed by the equation
For the meaning of the statement that φ (x) and ψ (x) are determinately infinite reference may be made to the article Function. The evaluation of forms of the type ∞/∞ leads to a scale of increasing “infinities,” each being infinite in comparison with the preceding. ”

“ Therefore it is no absurdity to say, that an Infinite may have both Finite and Infinite Parts, Finite in Quantity, Infinite in Number. But those that say, if there were an Infinite Body, that each of its Parts must of necessity be Infinite too, are much mistaken; for it is a contradiction in the same Terms to say One Infinite Part, for the very Name of a Part includes a Finiteness, but take all parts of an Infinite Body together, then you may rightly say they are infinite. ”

“ Moreover, when we say that the infinite cannot proceed from the finite, we do not deny that an infinite property may proceed from a substance finite in its essence.
When we admit the infinite property, we admit at the same time all that is necessary in the substance in order that this property may have its root in it, so long as we do not deny the character of finite which essentially belongs to every creature. When we deny that creatures are or can be infinite, we speak of essential infinity, of that infinity which implies necessity of being and absolute independence under every aspect ”

“ We know the conditions which must be fulfilled, but at the same time, we see the impossibility of fulfilling them. When any one tries to persuade us of the contrary, we reflect on the idea of the infinite, and say: "No; it is a contradiction of infinity; it is not infinite, but finite." We distinguish perfectly well between the absence of the perception of the limit and its non-existence. ”

“ Infinite is not-finite, and that means ‘end-less.’ What does endless mean? Not the mere negation of end, because a mere negation is nothing at all, and infinite would thus = 0. The endless is something positive; it means a positive quantity which has no end. Any given number of units is finite; but a series of units, which is produced indefinitely, is infinite. This is the sense of infinite which is in most common use, and which, we shall see, is what Hedonism believes in. It is however clear that this infinite is a perpetual self-contradiction, and, so far as it is real, is only finite. ”

“ An infinite quantity we have seen to be a self-contradiction, and the advance towards it fallacious; so that ‘more’ does not come any nearer to ‘most.’ In comparison of infinity, all finite sums are equal. When you ask for the difference between each and the infinite, in order to compare these differences one with the other, you get in every case the same answer, Between the infinite and each finite alike there is a quantity, about which in no case can we say more than that it is not any finite sum. ”

“ Although these arguments in favor of the existence of the idea of the infinite, if we examine our own mind, we cannot deny that we find there a certain vagueness and confusion which inspire strong doubts as to the reality of this idea. What is presented to our mind, when we think of the infinite? The imagination abandoned to itself, extends space, expands dimensions, multiplies numbers indefinitely, but it offers nothing to the intellect which has the marks of infinity. ”

“ Was it possible that three absolute infinites, which is the same as three Gods, could be contained in one and the self-same essence, why not as well any other number of infinites? But as certain as infinity cannot admit of addition, so certain a plurality of infinites cannot exist in the same essence; for real infinity is strict and absolute infinity, and only that, and cannot be compounded of infinities or of parts, but forecloses all addition. A personal or circumscribed God, implies as great and manifest a contradiction as the mind of man can conceive of ”

“ If the Finite and the Infinite can in turn be deduced from one another, neither conception can be truly called an axiom. The real axiom is existence, which includes both, and which is defined by mathematicians as that which is not self-contradictory.
Now the problem of Infinity includes also that of continuity; in other words, the problem of number includes that of cardinal and ordinal number. It is time to get to the mathematical definition of number, which we have found as a conception can be attached both to the Finite and to the Infinite. What is number in mathematics? ”

“ For, I think it is evident, that the addition of finite things together (as are all lengths whereof we have the positive ideas) can never otherwise produce the idea of infinite than as number does ; which, consisting of additions of finite units one to another, suggests the idea of infinite, only by a power we find we have of still increasing the sum, and adding more of the same kind ; without coming one jot nearer the end of such progression.
14. How we cannot have a positive idea of infinity in quantity. ”

“ Hence the ancients never assert either the infinitely possible or the infinite real. There is in all Gentile science, or Gentile philosophy, no conception of the infinite; there is only the conception of the indefinite.
This same reasoning disposes of the infinite divisibility of matter still taught in our text-books. The infinite divisibility of matter is an infinite absurdity; for it implies an infinity of parts or numbers, which is really a contradiction in terms. We know nothing that better illustrates the unsoundness of the method of the scientists. ”

“ We may infer from this that the possibility of assigning a finite difference to an infinite extension, does not prove any thing against its true infinity. The infinite, and because it is infinite, contains all that belongs to the order in which it is infinite. We may take any sure value, and considering it as a difference, and we shall obtain a finite difference. But far from proving the absence of infinity, this confirms its existence; for it shows that all the finite is contained in the infinite. ”

“ This may suffice to show how the idea of infinity may be applied under different conditions and produce different results, without any contradiction. What is infinite under one aspect is not so under another aspect; hence we have the orders of infinities which figure so largely in mathematics; but I say again that these contradictions are not susceptible of any explanation if we attribute an absolute value to the idea of the infinite, instead of considering it as the abstract representation of the negation of limit. ”

“ Infinite; we cannot under our present conditions know the whole Truth; if we could do that we should be able, as it were, to look all round the subject, and Infinity would then be seen to be a pseudo-conception of our finite thoughts. We can only think of one finite subject at a time, and, at that moment, all other subjects are cancelled ”

“ It may be admitted that human thought is so limited that we cannot picture infinity to our minds. The question is, simply, can we not so define the idea of infinity, that it may be validly employed in any process of reasoning? We have seen that the attempts to prove contradictions in the very conception of infinity, are based on the fallacy of using two distinct and contradictory definitions of the term. The correct definition recognises infinites as incapable of comparison; the erroneous and tacitly-assumed definition involves the idea that infinites must be equal. ”

“ The general conception of the negation of limit gives us an idea of infinity in the abstract, but not any infinite thing. Without the intuitive cognition of an infinite object, and with only a very imperfect idea of it, we may speak of infinity without falling into contradiction, and determine the cases in which it may be applied to a being or to an order of beings, whether real or possible. Man has many ideas of this vague kind, which nevertheless answer his necessities. ”

“ So that in fact we go through an infinite, [space] in an infinite [time] and not in a finite [time] , and we touch infinite things with infinite things, not with finite things.” Philoponus, a sixth-century commentator (R.P. 136A, Exc. Paris Philop. in Arist. Phys., 803, 2. Vit.) , gives the following illustration: “For if a thing were moved the space of a cubit in one hour, since in every space there are an infinite number of points, the thing moved must needs touch all the points of the space: it will then go through an infinite collection in a finite time, which is impossible.” ”

“ The answer is that infinity is perfection without lack, and if you say that it has power without limit and no power with limit, you are detracting from its perfection. And if you say that the world, which is lacking in perfection, was the initial limit, you are lacking its power. Therefore, we must say that it is a limit without limit, and the initial limit is the spheres, which are the complete power and the lacking power. When they receive influence from its perfection, they are complete power, and when they are lacking influence, they are lacking power. ”

“ That all things may be resolved into their minima's, and what is beyond them, is nothing, and that there is one maximum, or biggest, which is the world, and what is beyond that, is Infinite. Truly, Madam, I must ingeniously confess, I am not so high learned, as to penetrate into the true sense of these words; for he says, they are both divisible, and indivisible, and yet no atomes, which surpasses my Understanding; for there is no such thing, as biggest and smallest in Nature, or in the Infinite matter; for who can know how far this World goes, or what is beyond it? ”

“ Must we discard all belief in an infinite mind because we cannot understand it, and therefore say we are not a part of it because there is no Infinite? Should we discard the belief in the infinitude of number, because we cannot understand it, and therefore say that finite number is not a part of the infinite? ”

“ It is, of course, possible that there is an endless regress: that whatever appears as a particular is really, on closer scrutiny, a class or some kind of complex. If this be the case, the axiom of infinity must of course be true. But if it be not the case, it must be theoretically possible for analysis to reach ultimate subjects, and it is these that give the meaning of "particulars" or "individuals." It is to the number of these that the axiom of infinity is assumed to apply. ”