The Speculations of Infinities

130. Recently, the Speculations about Infinities have run so high.

They have grown to such strange Notions, as have occasioned no small Scruples and Disputes among the Geometers of the present Age. Some there are of great Note, who not content with holding that finite Lines may be divided into an infinite Number of Parts, do yet farther maintain, that each of those Infinitesimals is it self subdivisible into an Infinity of other Parts, or Infinitesimals of a second Order, and so on ad infinitum.

These, I say, assert there are Infinitesimals of Infinitesimals of Infinitesimals, without ever coming to an end. So that according to them an Inch doth 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 ever equal the smallest given Extension. And yet on the other hand it seems no less absurd, to think the Square, Cube, or other Power of a positive real Root, should it self be nothing at all; which they who hold Infinitesimals of the first Order, denying all of the subsequent Orders, are obliged to maintain.

131. Have we not therefore reason to conclude, they are both in the wrong, and that there is in effect no such thing as Parts infinitely small, or an infinite number of Parts contained in any finite Quantity? But you will say, that if this Doctrine obtains, it will follow the very Foundations of Geometry are destroyed: And those great Men who have raised that Science to so astonishing an height, have been all the while building a Castle in the Air.

To this it may be replied, that whatever is useful in Geometry and promotes the benefit of humane Life, doth still remain firm and unshaken on our Principles. That Science considered as practical, will rather receive Advantage than any Prejudice from what hath been said. But to set this in a due Light, may be the Subject of a distinct Inquiry.

For the rest, though it should follow that some of the more intricate and subtile Parts of Speculative Mathematics may be pared off without any prejudice to Truth; yet I do not see what Damage will be thence derived to Mankind. On the contrary, it were highly to be wished, that Men of great Abilities and obstinate Application would draw off their Thoughts from those Amusements, and employ them in the Study of such Things as lie nearer the Concerns of Life, or have a more direct Influence on the Manners.

132. If it be said that several Theoremes undoubtedly true, are discovered by Methods in which Infinitesimals are made use of, which could never have been, if their Existence included a Contradiction in it, I answer, that upon a thorough Examination it will not be found, that in any Instance it is necessary to make use of or conceive infinitesimal Parts of finite Lines, or even Quantities less than the Minimum Sensible: Nay, it will be evident this is never done, it being impossible.

133. By what we have premised, it is plain that very numerous and important Errors have taken their rise from those false Principles, which were impugned in the foregoing Parts of this Treatise. And the Opposites of those erroneous Tenets at the same time appear to be most fruitful Principles, from whence do flow innumerable Consequences highly advantageous to true Philosophy as well as to Religion. Particularly, Matter or the absolute Existence of Corporeal Objects, hath been shewn to be that wherein the most avowed and pernicious Enemies of all Knowledge, whether humane or divine, have ever placed their chief Strength and Confidence.

And surely, if by distinguishing the real Existence of unthinking Things from their being perceived, and allowing them a Subsistence of their own out of the Minds of Spirits, no one thing is explained in Nature; but on the contrary a great many inexplicable Difficulties arise:

If the Supposition of Matter is barely precarious, as not being grounded on so much as one single Reason: If its Consequences cannot endure the Light of Examination and free Inquiry, but skreen themselves under the dark and general pretence of Infinites being incomprehensible:

If withal the Removal of this Matter be not attended with the least evil Consequence, if it be not even missed in the World, but every thing as well, nay much easier conceived without it:

If lastly, both Sceptics and Atheists are for ever silenced upon supposing only Spirits and Ideas, and this Scheme of Things is perfectly agreeable both to Reason and Religion: Methinks we may expect it should be admitted and firmly embraced, though it were proposed only as an Hypothesis, and the Existence of Matter had been allowed possible, which yet I think we have evidently demonstrated that it is not.

134. True it is, that in consequence of the foregoing Principles, several Disputes and Speculations, which are esteemed no mean Parts of Learning, are rejected as useless. But how great a Prejudice soever against our Notions, this may give to those who have already been deeply engaged, and made large Advances in Studies of that Nature: Yet by others, we hope it will not be thought any just ground of Dislike to the Principles and Tenets herein laid down, that they abridge the labour of Study, and make Humane Sciences more clear, compendious, and attainable, than they were before.

135. Having despatched what we intended to say concerning the knowledge of Ideas, the Method we proposed leads us, in the next place, to treat of Spirits: With regard to which, perhaps Humane Knowledge is not so deficient as is vulgarly imagined. The great Reason that is assigned for our being thought ignorant of the nature of Spirits, is, our not having an Idea of it.

But surely it ought not to be looked on as a defect in a Humane Understanding, that it does not perceive the Idea of Spirit, if it is manifestly impossible there should be any such Idea. And this, if I mistake not, has been demonstrated in Sect. 27: To which I shall here add that a Spirit has been shewn to be the only Substance or Support, wherein the unthinking Beings or Ideas can exist: But that this Substance which supports or perceives Ideas should it self be an Idea or like an Idea, is evidently absurd.

136. It will perhaps be said, that we want a Sense (as some have ima gined) proper to know Substances withal, which if we had, we might know our own Soul, as we do a Triangle.

To this I answer, that in case we had a new Sense bestowed upon us, we could only receive thereby some new Sensations or Ideas of Sense.

But I believe no Body will say, that what he means by the terms Soul and Substance, is only some particular sort of Idea or Sensation. We may therefore infer, that all things duly considered, it is not more reasonable to think our Faculties defective, in that they do not furnish us with an Idea of Spirit or active thinking Substance, than it would be if we should blame them for not being able to comprehend a round Square.