Numbers and Extension

122. In Arithmetic therefore we regard not the Things but the Signs.

These signs are not regarded for their own sake, but because they direct us how to act with relation to Things, and dispose rightly of them.

Abstract Ideas are thought to be signified by Numeral Names or Characters, while they do not suggest Ideas of particular Things to our Minds.

I shall not at present enter into a more particular Dissertation on this Subject; but only observe that it is evident from what hath been said, those Things which pass for abstract Truths and Theoremes concerning Numbers, are, in reality, conversant about no Object distinct from particular numerable Things, except only Names and Characters; which originally came to be considered, on no other account but their being Signs, or capable to represent aptly, whatever particular Things Men had need to compute. Whence it follows, that to study them for their own sake would be just as wise, and to as good purpose, as if a Man, neglecting the true Use or original Intention and Subserviency of Language, should spend his time in impertinent Criticisms upon Words, or Reasonings and Controversies purely Verbal.

123. From Numbers we proceed to speak of Extension, which considered as relative, is the Object of Geometry. The Infinite Divisibility of Finite Extension, though it is not expresly laid down, either as an Axiome or Theoreme in the Elements of that Science, yet is throughout the same every where supposed, and thought to have so inseparable and essential a Connexion with the Principles and Demonstrations in Geometry, that Mathematicians never admit it into Doubt, or make the least Question of it.

And as this Notion is the Source from whence do spring all those amusing Geometrical Paradoxes, which have such a direct Repugnancy to the plain common Sense of Mankind, and are admitted with so much Reluctance into a Mind not yet debauched by Learning: So it is the principal occasion of all that nice and extreme Subtilty, which renders the Study of Mathematics so difficult and tedious. Hence if we can make it appear, that no Finite Extension contains innumerable Parts, or is infinitely Divisible, it follows that we shall at once clear the Science of Geometry from a great Number of Difficulties and Contradictions, which have ever been esteemed a Reproach to Humane Reason, and withal make the Attainment thereof a Business of much less Time and Pains, than it hitherto hath been.

124. Every particular Finite Extension, which may possibly be the Object of our Thought, is an Idea existing only in the Mind, and consequently each Part thereof must be perceived. If therefore I cannot perceive innumerable Parts in any Finite Extension that I consider, it is certain they are not contained in it: But it is evident, that I cannot distinguish innumerable Parts in any particular Line, Surface, or Solid, which I either perceive by Sense, or Figure to my self in my Mind: Wherefore I conclude they are not contained in it. Nothing can be plainer to me, than that the Extensions I have in View are no other than my own Ideas, and it is no less plain, that I cannot resolve any one of my Ideas into an infinite Number of other Ideas, that is, that they are not infinitely Divisible.

If by Finite Extension be meant something distinct from a Finite Idea, I declare I do not know what that is, and so cannot affirm or deny any thing of it. But if the terms Extension, Parts, and the like, are taken in any Sense conceivable, that is, for Ideas; then to say a Finite Quantity or Extension consists of Parts infinite in Number, is so manifest a Contradiction, that every one at first sight acknowledges it to be so. And it is impossible it should ever gain the Assent of any reasonable Creature, who is not brought to it by gentle and slow Degrees, as a converted Gentile to the belief of Transubstantiation.

Ancient and rooted Prejudices do often pass into Principles: And those Propositions which once obtain the force and credit of a Principle, are not only themselves, but likewise whatever is deducible from them, thought privileged from all Examination. And there is no Absurdity so gross, which by this means the Mind of Man may not be prepared to swallow.

125. He whose Understanding is prepossest with the Doctrine of abstract general Ideas, may be persuaded, that (whatever be thought of the Ideas of Sense,) Extension in abstract is infinitely divisible.

And one who thinks the Objects of Sense exist without the Mind, will perhaps in virtue thereof be brought to admit, that a Line but an Inch long may contain innumerable Parts really existing, though too small to be discerned.

These Errors are grafted as well in the Minds of Geometricians, as of other Men, and have a like influence on their Reasonings; and it were no difficult thing, to shew how the Arguments from Geometry made use of to support the infinite Divisibility of Extension, are bottomed on them. At present we shall only observe in general, whence it is that the Mathematicians are all so fond and tenacious of this Doctrine.

126. It hath been observed in another place, that the Theoremes and Demonstrations in Geometry are conversant about Universal Ideas. Sect. 15. Introd. Where it is explained in what Sense this ought to be understood, to wit, that the particular Lines and Figures included in the Diagram, are supposed to stand for innumerable others of different Sizes: or in other words, the Geometer considers them abstracting from their Magnitude: which doth not imply that he forms an abstract Idea, but only that he cares not what the particular Magnitude is, whether great or small, but looks on that as a thing indifferent to the Demonstration: Hence it follows, that a Line in the Scheme, but an Inch long, must be spoken of, as though it contained ten thousand Parts, since it is regarded not in it self, but as it is universal; and it is universal only in its Signification, whereby it represents innumerable Lines greater than it self, in which may be distinguished ten thousand Parts or more, though there may not be above an Inch in it. After this manner the Properties of the Lines signified are (by a very usual Figure) transferred to the Sign, and thence through Mistake thought to appertain to it considered in its own Nature.

127. Because there is no Number of Parts so great, but it is possible there may be a Line containing more, the Inch-line is said to contain Parts more than any assignable Number; which is true, not of the Inch taken absolutely, but only for the Things signified by it.

But Men not retaining that Distinction in their Thoughts, slide into a belief that the small particular Line described on Paper contains in it self Parts innumerable. There is no such thing as the ten-thousandth Part of an Inch; but there is of a Mile or Diameter of the Earth, which may be signified by that Inch.

When therefore I delineate a Triangle on Paper, and take one side not above an Inch, for Example, in length to be the Radius: This I consider as divided into ten thousand or an hundred thousand Parts, or more. For though the ten-thousandth Part of that Line considered in it self, is nothing at all, and consequently may be neglected without any Error or Inconveniency; yet these described Lines being only Marks standing for greater Quantities, whereof it may be the ten-thousandth Part is very considerable, it follows, that to prevent notable Errors in Practice, the Radius must be taken of ten thousand Parts, or more.

128. From what hath been said the reason is plain why, to the end any Theoreme may become universal in its Use, it is necessary we speak of the Lines described on Paper, as though they contained Parts which really they do not. In doing of which, if we examine the matter throughly, we shall perhaps discover that we cannot conceive an Inch it self as consisting of, or being divisible into a thousand Parts, but only some other Line which is far greater than an Inch, and represented by it. And that when we say a Line is infinitely divisible, we must mean a Line which is infinitely great. What we have here observed seems to be the chief Cause, why to suppose the infinite Divisibility of finite Extension hath been thought necessary in Geometry.

129. The several Absurdities and Contradictions which flowed from this false Principle might, one would think, have been esteemed so many Demonstrations against it.

But by I know not what Logic, it is held that Proofs `a posteriori are not to be admitted against Propositions relating to Infinity. As though it were not impossible even for an infinite Mind to reconcile Contradictions.

Or as if any thing absurd and repugnant could have a necessary Connexion with Truth, or flow from it. But whoever considers the Weakness of this Pretence, will think it was contrived on purpose to humour the Laziness of the Mind, which had rather acquiesce in an indolent Scepticism, than be at the Pains to go through with a severe Examination of those Principles it hath ever embraced for true.