Context

In the previous Section, the author argued that no idea that is formed of a finite quality can be divided infinitely, due to the finite capacity of the mind. In the present Section, he builds on this conclusion in arguing that the essential finitude of our ideas of finite quantities of space and time implies the possibility that space and time themselves are only finitely divisible, and hence that the infinite divisibility of space and time cannot be demonstrated, which would require that its opposite (mere finite divisibility) is impossible.

Background

Zeno and Aristotle

The question of whether space and time are infinitely divisible arose in ancient philosophy in the context of Zeno’s paradoxes. The paradox known as the dichotomy introduces the problem. Aristotle describes it briefly: The first asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal (Physics, Book VI, Chapter 9) Suppose there is a finite distance from A to B. This distance is divisible into halves. So in order for someone to get from A to B, he would have to travel 1/2 the distance. But this new distance is divisible into halves, so he would have to travel half of that distance, which is 1/4 the total distance from A to B. If the distances are divisible to infinity, then there are infinitely many distances to traverse. To pass over these distances would require an infinite amount of time, which is not available to us. Aristotle responded to this argument by claiming that in fact we do have an infinite amount of available time to traverse a finite distance. The reason is that time is infinite in the sense of being infinitely divisible, as is the space to be traversed. But the space is finite the sense of having a beginning and end. So, a space that is finite in extent but infinite in its divisibility can be passed over in a time that is finite in extent but infinite in divisibility (Chapter 2). Aristotle in fact argued that no continuous quantity, such as length and time, could be made up of indivisible points (Chapter 1). Aristotle used his denial of the composition of time by moments to counter Zeno’s arrow paradox. The paradox is as follows. If an object, say an arrow, is in motion, then at any given moment it occupies a given space, say one that stretches from A to B. But something that occupies a space at a moment is at rest with respect to that space. So the arrow is not in fact moving (Chapter 9). Aristotle points out that this reasoning depends on the assumption that there are moments in time, but time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles (Chapter 9).

Descartes

In the Principles of Philosophy (1644), Descartes described the nature of God to be infinite, though the notion of the infinite is not comprehensible by the finite human mind (Part I, Article XIX). He reserved that notion for God and proposed using the notion of the indefinite in its place when describing extended things.

We will thus never embarrass ourselves by disputes about the infinite, seeing it would be absurd for us who are finite to undertake to determine anything regarding it, and thus as it were to limit it by endeavouring to comprehend it. We will accordingly give ourselves no concern to reply to those who demand whether the half of an infinite line is also infinite, and whether an infinite number is even or odd, and the like, because it is only such as imagine their minds to be infinite who seem bound to entertain questions of this sort. And, for our part, looking to all those things in which in certain senses, we discover no limits, we will not, therefore, affirm that they are infinite, but will regard them simply as indefinite. Thus, . . . because a body cannot be divided into parts so small that each of these may not be conceived as again divided into others still smaller, let us regard quantity as divisible into parts whose number is indefinite . . . .

Locke

Locke’s account of the infinite is found in Book II, Chapter 17 of his Essay concerning Human Understanding (1690), Of Infinity. He takes the idea of infinity to apply to composites only.

Finite and infinite seem to me to be looked upon by the mind as the modes of quantity, and to be attributed primarily in their first designation only to those things which have parts, and are capable of increase or diminution by the addition or subtraction of any the least part: and such are the ideas of space, duration, and number . . . . (Section 1)

Bayle

Bayle devoted a large entry in his Historical and Critical Dictionary (1697) to Zeno and his paradoxes. In Note F to that article, he writes that the indivisibility of time leads directly to the arrow paradox. You cannot find an instant when the Arrow leaves its place; for if you find one, it will be at the same time in that place, yet not there. He argues for the indivisibility of time in the following way. His first claim is that no two parts of time can exist together.

I say then that what suits Monday and Tuesday with respect to succession, suits every portion of Time whatsoever. Since then it is impossible for Monday and Tuesday to exist together, and that of necessity Monday must cease to be before Tuesday begins to be, there is no part of time whatsoever, which can co-exist with another; each must exist alone; each must begin to be, when the precedent ceaseth to be; and each must cease to be before the following can begin to exist.

Berkeley

In his Essay toward a New Theory of Vision of 1709, Berkeley appealed to experience to support his claim that sensible (that is, visible and tangible) extension cannot be infinitely divided. [W]hatever may be said of Extension in Abstract, it is certain sensible Extension is not infinitely Divisible. There is a Minimum Tangibile, and a Minimum Visibile, beyond which Sense cannot perceive. This every ones Experience will inform him (Section 54). In the 1710 Principles of Human Knowledge, Berkeley attempted to turn the doctrine of the infinite divisiblity of matter against those materialists who assert that matter exists independently of the mind. His general claim is that matter is an incomprehensible somewhat, which hath none of those particular qualities whereby the bodies falling under our senses are distinguished from one another (Section 47). To bolster this claim, he notes that the most approved and considerable philosophers allow universally that matter is infinitely divisible, a view which is thought to be demonstrated beyond all exception on the basis of the received principles. The consequence of this view is that each particle of matter has infinitely many parts, which are not perceived because our senses are not acute enough to sense them, though it would if they were to become sharper. The observed properties of the object would change as it were more closely scrutinized, until when the sense becomes infinitely acute the body shall seem infinite. There is no difference in the body in this case, but only one in perceptions. The body itself, on this view, must be infinitely extended, and consequently void of all shape and figure. Berkeley concluded that on the materialist view, all the perceivable qualities of objects are framed by the mind and depend on the mind for their existence. Regarding the infinite divisibility of time, Berkeley notes that they lead him to some odd thoughts about his own existence, that doctrine lays one under an absolute necessity of thinking, either that he passes away innumerable ages without a thought, or else that he is annihilated every moment of his life, both which seem equally absurd (Section 98).

The Treatise

1. The author begins by describing what he claims to be in general the foundation of all human knowledge. This is the principle that if ideas represent objects adequately, then the relations, contradictions and agreements of the ideas are all applicable to the objects. [This principle seems to be based on the meaning of ‘adequate,’ as used by Locke (see above). If a collection of ideas adequately represents a collection of objects, then all the relations, etc. among the former must apply to the latter. If the principle applies to individual ideas and objects, then these ideas must represent all the relations of the objects as well, perhaps by representing all their relational properties.] The principle is now applied to the case of the ideas we have of the most minute parts of extension. The author claims that these ideas [presumably of very small parts of extension] are adequate. There is a limit to the smallness that can be represented by an idea, no matter how much we suppose the parts of extension to be divided and sub-divided. Thus, as a plain consequence, when we compare these ideas, whatever appears impossible and contradictory regarding them must be really impossible and contradictory, without any farther excuse or evasion.

2. Regarding the process of dividing and sub-dividing, if there is a division of a thing that admits of no further sub-division, then the division is not infinite. From this the author infers that if there is an infinite division, the divided object must have an infinite number of parts. It follows then from the supposition that a finite extension is infinitely divisible that there is no contradiction in the notion that it has an infinite number of parts. Conversely, if it is a contradiction in the notion that a finite extension has an infinite number of parts, then no finite extension can be infinitely divisible. The author maintains that there is in fact a contradiction or absurdity in the latter supposition, as he can see by the consideration of my clear ideas. Take the smallest idea of an extension that you can have. There can be no idea smaller than it. [Given that it adequately represents a part of extension,] it follows [by the principle in the previous paragraph] that whatever I discover by its means must be a real quality of extension. The discovery is as follows. If the idea is repeated, it produces a compound idea of a greater extension, in proportion to the number of repetitions. The extension stops expanding when the process of repetition is stopped, but if it is repeated infinitely, I clearly perceive, that the idea of extension must also become infinite. [Because the original idea is repeated infinitely, the compound idea of the extension must have infinitely many parts.] Upon the whole, I conclude, that [1] the idea of an infinite number of parts is individually the same idea with that of an infinite extension. It also follows that [2] no finite extension is capable of containing an infinite number of parts. [A finite extension would have to be produced by a finite repetition of the original idea, in which case it would have only finitely many parts, each one either the original idea or a repeat of it.] Finally, it follows that [3] no finite extension is infinitely divisible. [Suppose a finite extension were infinitely divisible. Then it must have an infinite number of parts. But by [2], it cannot have infinitely many parts. Therefore, a finite extension cannot be infinitely divisible.]

[Footnote. The author notes an objection that had been made to his reasoning. The objection would distinguish between two kinds of parts of extension: proportional parts and aliquot parts. [An aliquot part is an exact divisor of a thing. Thus, consider a length of four inches. Its (proper) aliquot parts would be lengths of two inches and of one inch, but not a length of three inches. A proportional part would (presumably) take the extension as a unit and be expressed as a proportion of that unit. Thus, a unit may be divided into halves, quarters, eights, etc.] To be divisible infinitely, the objection goes, requires only that the object have infinitely many proportional parts. [There need not be infinitely many exact divisors of a quantity for it to be divisible infinitely.] However, the objection continues, the addition of an infinite number of proportional parts does not form an infinite extension. [Perhaps the reason is that anything composed of any number of fractional parts could always be bigger if the parts were bigger. The objection would be that the author’s comparison of infinite division and infinite aggregation, on which his argument depends, is inapplicable.] The author regards this distinction as being entirely frivolous. Whatever the parts are called, they cannot be smaller than those minute parts we conceive. Therefore, it is not possible to put them together to form a quantity of extension smaller than that of the smallest conceivable part of extension.

2. A second argument, which seems to the author to be very strong and beautiful, is added in support of his thesis.

[Footnote. This argument is attributed to Nicolas de Malézieu (in Elements of Geometry, 1705).]

The starting-point for the argument is the evident proposition that existence in itself belongs only to unity and never applies to number except because of the unities [unites] of which the number is compos’d. For example, the reason we say that twenty men exist is because of the existence of one man, two men, etc. Denying the existence of the unitary components of the aggregate of twenty men requires the denial of the aggregate itself. Thus, it is utterly absurd to suppose any number to exist, and yet deny the existence of unites. Metaphysicians commonly suppose that extension is a number and not a unity or indivisible quantity. The consequence of this supposition is that extension does not exist [because it is not composed of unities, in that each component is itself extended, and hence a number]. One might object that [rather than extension per se,] a determinate quantity of extension [such as an inch] is a unity [and hence exists], while at the same time allowing that this quantity can be sub-divided inexhaustibly into an infinite number of fractional quantities. If this were to be allowed, then we would also have to allow that an aggregate of twenty men, the whole earth, or the whole universe is a unity. It is true that we can think of these aggregates as unities, but this thought is only a fictitious denomination, which the mind may apply to any quantity of objects it collects together. Such a unity cannot exist alone any more than a number can exist alone [without the component numbers that make it up]. Any such fictitious unity is in reality a true number. But the unity, which can exist alone, and whose existence is necessary to that of all number, is of another kind, and must be perfectly indivisible, and incapable of being resolv’d into any lesser unity.

4. The two arguments just given apply to a parallel thesis concerning time. [No finite quantity of time is infinitely divisible into times.] There is in addition an argument that applies to only to time. The author takes it to be a property inseparable from time, and which in a manner constitutes its essence, that each of its parts succeeds another, and that none of them, however contiguous, can ever be co-existent. Just as no period of time can coincide with another (as the year 1737 cannot concur with the present year 1738), no moment of time can co-exist with any other moment, but must be distinct from it and occur either before or after it. From this principle the author concludes that existing time must consists of indivisible moments. The argument is by reductio ad absurdum. Suppose that the division of time had no end, and that each succeeding moment were not perfectly single and indivisible. It follows [or so the author claims, without explanation] that there woul’d be an infinite number of co-existent moments, or parts of time; which I believe will be allow’d to be an arrant contradiction. [The argument seems to be this. Suppose m is a divisible moment of time. Divide m into two moments m₁ and m₂. Now there are two moments co-existing with m. Since the division can be made to infinity, there are infinitely many moments co-existing with m. In that case, m is not really a moment, and the only feature that excludes its being a moment is its divisibility. So no moment is divisible.]

5. [The addition of this third argument makes a stronger case for the indivisibility of moments of time than is the case for the indivisibility of the parts of space. However, the indivisibility of moments of time can be used to support the thesis of the indivisibility of the parts of space.] The nature of motion makes it evident that [t]he indivisibility of space implies that of time. [No argument is given for this evident claim, but the reasoning seems to go as follows. Suppose an object moves from point A to point B. The distance from A to B is infinitely divisible. Therefore, in the motion from A to B, the object would have to traverse an infinite number of parts of space. But it could do so only if there were a moment of time for each part of space traversed, in which case there would have to be infinitely many moments of time. If there are infinitely many moments of time, then time is infinitely divisible.] Therefore, if the moments of time cannot be indivisible, neither can the parts of space.

6. The defenders of the doctrine of the infinite divisibility of space and time might admit that these arguments present difficulties, to which no perfectly clear and satisfactory answer can be given. But the author attributes this response to the custom of referring to what is intended to be a demonstration of the falsehood of a thesis as being a difficulty for it. He contends that nothing could be more absurd than trying to evade in this way the force and evidence of the demonstration. Difficulties can arise only in probable reasoning, in which one reason in favor of a proposition may be counter-balanced by a reason opposed to it, thus diminishing its force. If an argument is proposed as a demonstration, it is all or nothing: either the argument is fallacious or it is just and can admit of no difficulties. ’Tis either irresistible, or has no manner of force. Anyone, then, who treats an attempted demonstration as something open to objections and replies, and ballancing of arguments is either reducing human reasoning to nothing but a play of words or else is not competent to understand the subject-matter properly. While the abstract nature of a subject-matter may make a demonstration difficulty to comprehend, once it is comprehended, its authority cannot be undermined by so-called difficulties.

7. Still, mathematicians will say that the positive doctrine of the indivisibility of the parts of space and time is itself subject to fatal objections. [Later, Immanuel Kant would claim that the arguments against both the thesis of the indivisibility of parts of matter and the antithesis of their infinite divisibility are equally strong and destroy each other. See the Critique of Pure Reason, First Antinomy.] The author will examine these arguments in detail [in Section 4], but in the meantime, he will try to show here, with a short and decisive reasoning, that they cannot have any just foundation.

8. The author appeals to what he calls an establish’d maxim in metaphysics. This maxim is stated in two ways, the second being a re-wording of the first. (1) that whatever the mind clearly conceives includes the idea of possible existence, and (2), that nothing we imagine is absolutely impossible. [The connection between conception of an object and that of its existence is examined in Section 6.] An example is the formation of the idea of a gold mountain, from which we infer the possible existence of a gold mountain. Conversely, we cannot form the idea of a mountain without a valley, so such a mountain is not possible.

9. We talk and reason about extension, so it is certain that we have an idea of it. Equally certain is that the idea we have, while divisible into parts (inferior ideas), is not divisible infinitely and does not have infinitely many parts. The reason for this inability to divide infinitely is that the capacity of the human mind is finite and cannot therefore carry out an infinite division. [This argument was given in Section 1, paragraph 2.] So we have an idea of extension which consists of indivisible parts or inferior ideas. Since nothing we imagine is absolutely impossible, extension composed of indivisible parts is not impossible (i.e., implies no contradiction). Since it is possible for indivisible parts of extension to exist, it is not possible to demonstrate that they do not exist. And consequently all the arguments employ’d against the possibility of mathematical points are mere scholastick quibbles, and unworthy of our attention.

10. A further consequence of this argument is that every attempt to demonstrate the impossibility of indivisible parts of space and time is equally sophistical or fallacious. Any such attempted demonstrations, if successful, would be applicable to mathematical points [rather than merely to parts of space and time]. But the demonstration of the impossibility of mathematical points is evidently an absurd enterprise.

The Enquiry

In Part II of the concluding Section XII, Hume discusses attempts by skeptics to destroy our confidence in abstract reasoning. He cites as the chief objection a conflict between our common-sensical ideas of space and time and those of the geometers, who argue that space and time are infinitely divisible—a view that shocks the clearest and most natural principles of human reason. The notion that extension and time are composed of infinitely many parts, each of which is divisible to infinity, appears absurd on the face of it, yet there are compelling demonstrations of its truth. However, rather than repose in skepticism as the result of this conflict, reason is troubled by it and is skeptical of its own skepticism. How any clear, distinct idea can contain circumstances, contradictory to itself, or to any other clear, distinct idea, is absolutely incomprehensible; and is, perhaps, as absurd as any proposition, which can be formed.

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