Motion

116. It follows that the Philosophic Consideration of Motion does not imply the being of an absolute Space, distinct from that which is perceived by Sense, and related to Bodies: Which that it cannot exist without the Mind, is clear upon the same Principles, that demonstrate the like of all other Objects of Sense.

If we inquire narrowly, we shall find we cannot even frame an Idea of pure Space, exclusive of all Body. This I must confess seems impossible, as being a most abstract Idea.

When I excite a Motion in some part of my Body, if it be free or without Resistance, I say there is Space: But if I find a Resistance, then I say there is Body: and in proportion as the Resistance to Motion is lesser or greater, I say the Space is more or less pure. So that when I speak of pure or empty Space, it is not to be supposed, that the Word Space stands for an Idea distinct from, or conceivable without Body and Motion. Though indeed we are apt to think every Noun Substantive stands for a distinct Idea, that may be separated from all others: Which hath occasioned infinite Mistakes.

When therefore supposing all the World to be annihilated besides my own Body, I say there still remains pure Space: Thereby nothing else is meant, but only that I conceive it possible, for the Limbs of my Body to be moved on all sides without the least Resistance: But if that too were annihilated, then there could be no Motion, and consequently no Space. Some perhaps may think the Sense of Seeing doth furnish them with the Idea of pure Space; but it is plain from what we have elsewhere shewn, that the Ideas of Space and Distance are not obtained by that Sense. See the Essay concerning Vision.

117. This ends all those Disputes and Difficulties, which have sprung up amongst the Learned concerning the nature of pure Space.

But the chief Advantage arising from it, is, that we are freed from that dangerous Dilemma, to which several who have employed their Thoughts on this Subject, imagine themselves reduced, to wit, of thinking either that Real Space is God, or else that there is something beside God which is Eternal, Uncreated, Infinite, Indivisible, Immutable. Both which may justly be thought pernicious and absurd Notions.

Many Divines and notable Philosophers have, from the Difficulty they found in conceiving either Limits or Annihilation of Space, concluded it must be Divine.

And some of late have set themselves particularly to shew, that the incommunicable Attributes of God agree to it. Which Doctrine, how unworthy soever it may seem of the Divine Nature, yet I do not see how we can get clear of it, so long as we adhere to the received Opinions.

118. Hitherto of Natural Philosophy: We come now to make some Inquiry concerning that other great Branch of speculative Knowledge, to wit, Mathematics.

These, how celebrated soever they may be, for their Clearness and Certainty of Demonstration, which is hardly any where else to be found, cannot nevertheless be supposed altogether free from Mistakes; if in their Principles there lurks some secret Error, which is common to the Professors of those Sciences with the rest of Mankind.

Mathematicians, though they deduce their Theoremes from a great height of Evidence, yet their first Principles are limited by the consideration of Quantity: And they do not ascend into any Inquiry concerning those transcendental Maxims, which influence all the particular Sciences, each Part whereof, Mathematics not excepted, doth consequently participate of the Errors involved in them. That the Principles laid down by Mathematicians are true, and their way of Deduction from those Principles clear and incontestible, we do not deny.

But we hold, there may be certain erroneous Maxims of greater Extent than the Object of Mathematics, and for that reason not expresly mentioned, though tacitly supposed throughout the whole progress of that Science; and that the ill Effects of those secret unexamined Errors are diffused through all the Branches thereof. To be plain, we suspect the Mathematicians are, as well as other Men, concerned in the Errors arising from the Doctrine of abstract general Ideas, and the Existence of Objects without the Mind.

119. Arithmetic hath been thought to have for its Object abstract Ideas of Number.

Of which to understand the Properties and mutual Habitudes is supposed no mean part of speculative Knowledge. The Opinion of the pure and intellectual Nature of Numbers in Abstract, hath made them in esteem with those Philosophers, who seem to have affected an uncommon Fineness and Elevation of Thought.

It hath set a Price on the most trifling numerical Speculations which in Practice are of no use, but serve only for Amusement: And hath therefore so far infected the Minds of some, that they have dreamed of mighty Mysteries involved in Numbers, and attempted the Explication of natural Things by them. But if we inquire into our own Thoughts, and consider what hath been premised, we may perhaps entertain a low Opinion of those high Flights and Abstractions, and look on all Inquiries about Numbers, only as so many difficiles nugæ, so far as they are not subservient to practice, and promote the benefit of Life.

120. Unity in Abstract we have before considered in Sect. 13, from which and what hath been said in the Introduction, it plainly follows there is not any such Idea. But Number being defined a Collection of Unites, we may conclude that, if there be no such thing as Unity or Unite in Abstract, there are no Ideas of Number in Abstract denoted by the numeral Names and Figures. The Theories therefore in Arithmetic, if they are abstracted from the Names and Figures, as likewise from all Use and Practice, as well as from the particular things numbered, can be supposed to have nothing at all for their Object. Hence we may see, how intirely the Science of Numbers is subordinate to Practice, and how jejune and trifling it becomes, when considered as a matter of mere Speculation.

121. However since there may be some, who, deluded by the specious Shew of discovering abstracted Verities, waste their time in Arithmetical Theoremes and Problemes, which have not any Use: It will not be amiss, if we more fully consider, and expose the Vanity of that Pretence; And this will plainly appear, by taking a view of Arithmetic in its Infancy, and observing what it was that originally put Men on the Study of that Science, and to what Scope they directed it. It is natural to think that at first, Men, for ease of Memory and help of Computation, made use of Counters, or in writing of single Strokes, Points or the like, each whereof was made to signify an Unite, that is, some one thing of whatever Kind they had occasion to reckon.

Afterwards they found out the more compendious ways, of making one Character stand in place of several Strokes, or Points. And lastly, the Notation of the Arabians or Indians came into use, wherein by the repetition of a few Characters or Figures, and varying the Signification of each Figure according to the place it obtains, all Numbers may be most aptly expressed: Which seems to have been done in Imitation of Language, so that an exact Analogy is observed betwixt the Notation by Figures and Names, the nine simple Figures answering the nine first numeral Names and Places in the former, corresponding to Denominations in the latter. And agreeably to those Conditions of the simple and local Value of Figures, were contrived Methods of finding from the given Figures or Marks of the Parts, what Figures and how placed, are proper to denote the whole or vice versa.

And having found the sought Figures, the same Rule or Analogy being observed throughout, it is easy to read them into Words; and so the Number becomes perfectly known. For then the Number of any particular Things is said to be known, when we know the Name or Figures (with their due arrangement) that according to the standing Analogy belong to them. For these Signs being known, we can by the Operations of Arithmetic, know the Signs of any Part of the particular Sums signified by them; and thus computing in Signs, (because of the Connexion established betwixt them and the distinct multitudes of Things, whereof one is taken for an Unite,) we may be able rightly to sum up, divide, and proportion the things themselves that we intend to number.