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[The following is summary, up to the end where I reproduce this section in full. My commentary is in brackets.]
David Hume
A Treatise of Human Nature
Book I: Of the Understanding
Part II: Of the Ideas of Space and Time
Section II: "Of the Infinite Divisibility of Space and Time"
§66
Objects have relations to each other.
An idea might adequately represent its object. If so, then all the relations, contradictions, and agreements that hold between the ideas hold also for their objects. This correspondence between ideas and objects is "the foundation of all human knowledge." (29)
We may have ideas of the smallest parts of extension. We noted previously that these are adequate ideas. We might arrive at them by dividing more complex ideas. However, that does not make them any less of an idea. Therefore, whatever proves to be contradictory or impossible between these simple ideas must also be impossible and contradictory in reality.
§67
Imagine that we are dividing something continuously. And then we eventually arrive at indivisible parts. This means our division must stop. Thus if something contains indivisible parts, then it is not infinitely divisible.
However, imagine instead we never arrive at indivisible parts. That means we also never stop making divisions. Thus if something does not contain indivisible parts, then it is infinitely divisible. Likewise, if something is infinitely divisible, it cannot contain indivisible parts.
This then holds for finite extensions:
1) if a finite extension is infinitely divisible, then it contains an infinite number of extensive parts.
If the consequent is false, then the antecedent is false too, so
2) if a finite extension does not contain an infinite number of parts, then it cannot be infinitely divisible.
So if Hume can show that 2 is absurd, then consequent of one is false too. And by denying the consequent, that will make its antecedent false too, thereby proving that finite extensions are not infinitely divisible.
Hume falsifies 2 by beginning with our original assumption: we may have an indivisible idea of the least part of extension.
a) Begin by conceiving the least idea we can have of an extensive part. Such an idea is a full idea. But it is the idea of the smallest conceivable part of extension. There can be no idea of a smaller part. This is because we begin with simple ideas, then build-up to more and more complex ones. So we are looking for the least complex idea of extension. If someone were to say, 'yes, but I can think of a smaller part of extension,' then it cannot be that we started with a simple idea. For, the idea of a simpler part of extension implies that the second simplest was compound. However, we are presuming that we start with a simple idea, hence there cannot be any simpler ones.
b) we take this simplest idea of extension, and we repeat it. This gives us a compound idea of a larger part of extension.
c) Then we repeat it a third time, fourth, and so on. But each time it is still the same idea that we keep adding.
d) We find that nothing stops us from going on to infinity. If we did go on to infinity, then we would have an infinite number of parts. But then we would also have the idea of an infinite extension.
e) Thus no finite extension can contain an infinity of parts. For, if an extension contained an infinity of parts, it would be infinite and not finite. Hence proposition 2 is false.
f) But if 2 is false, then so is 1. Hence no finite extension is infinitely divisible. (30a)
§68
Hume then adds an argument by Nicholas de Malezieu, which Hume finds "very strong and beautiful." (30c)
Consider 20 men. If only 19 men existed, then 20 men would not exist. All 20 exist only because each of the 20 exists. Suppose we claimed that
1) the quantity 20 exists, and
2) the quantity 19 exists, but
3) both have an infinity of parts.
Then what would make 20 one more unit greater than 19? So if something has a quantity of 20, it must be made up of 20 indivisible units. Hence we cannot suppose some number to exist while also denying the existence of indivisible units. If there is quantity, there must be indivisible units.
But supposedly extension is a number and it is also infinitely divisible. But if it is infinitely divisible, then it is not made up of units. But numbers must be made up of units. Hence infinitely divisible extension cannot exist.
Some might counter-argue that an extension has determinate parts that themselves have an infinity of subdivisions. But then we are regarding multiplicities as units. So then the 20 men would be one unit, and not 20.
The whole globe of the earth, nay the whole universe may be considered as an unite. That term of unity is merely a fictitious denomination, which the mind may apply to any quantity of objects it collects together. (30d)
If we group unities, then they cannot exist by themselves, for their value depends on their constituents. The only unity that can exist by itself is an indivisible unit that cannot be resolved into smaller unities. (30-31)
§69
The year 1737 cannot occur with the present year 1738. This is because no matter how contiguous moments of time are, they cannot be co-existent.
every moment must be distinct from, and posterior or antecedent to another. (31b)
Now also, consider if we could divide time infinitely. Then we would not have moments that are unitary. They would not be perfectly single. Hence they would not be different from the ones around them. Thus there would be an infinity of co-existent moments. This is absurd. So time also must be made up of indivisible units.
§70
Motion involves a proportioned relation between extension and time. It is inconceivable that one be infinitely divisible and the other not. So motion demonstrates that the indivisibility of space implies the indivisibility of time.
§71
Some might counter-argue that Hume is only showing difficulties with the concept of infinite divisibility. And these difficulties are of a sort that we cannot give a perfectly clear and satisfactory reply to them. Hume responds that if no counter-argument is admitted, then he has demonstrated a truth.
Demonstrations may be difficult to be comprehended, because of the abstractedness of the subject; but can never have any such difficulties as will weaken their authority, when once they are comprehended. (32a)
§72
Yet, many mathematicians argue that there really are equally strong counter-arguments, and that Hume's argument of indivisible points itself has problems. Later Hume will examine these arguments in detail. For now he will give a brief reason why they have no foundation.
§73
We cannot form the idea a mountain without a valley. Hence it is an impossibility. But we can form an idea of a golden mountain. Hence golden mountains could possibly exist. Metaphysicians express this in a maxim:
That whatever the mind clearly conceives includes the idea of possible existence, or in other words, that nothing we imagine is absolutely impossible. (32bc)
§74
We have been talking about extension. Thus we have an idea of it. But we do not have the capacity to conceive its infinite number of parts, were it to be infinitely divisible. So infinitely divisible extension might be an impossibility. However, we can conceive of the indivisible parts of extension. This idea implies no contradiction. So it is possible that extension is made up of indivisible parts. (32d) So if there are arguments that say extension cannot possibly be made of indivisible parts, then these argument are at best "mere scholastick quibbles." (32d)
§75
So, because we can prove the possibility of indivisible points, all arguments that say that indivisibles are impossible are false arguments. If one were to argue for infinite divisibility, then one is thereby asserting that indivisibles are impossible. Thus anyone who argues for the infinite divisibility of extension is expounding an "equally sophistical" argument.
[Directory of other entries in this series.]
From Hume's original text:
PART II. OF THE IDEAS OF SPACE AND TIME
SECT. II. OF THE INFINITE DIVISIBILITY OF SPACE AND TIME.
Wherever ideas are adequate representations of objects, the relations, contradictions and agreements of the ideas are all applicable to the objects; and this we may in general observe to be the foundation of all human knowledge. But our ideas are adequate representations of the most minute parts of extension; and through whatever divisions and subdivisions we may suppose these parts to be arrived at, they can never become inferior to some ideas, which we form. The plain consequence is, that whatever appears impossible and contradictory upon the comparison of these ideas, must be really impossible and contradictory, without any farther excuse or evasion.
Every thing capable of being infinitely divided contains an infinite number of parts; otherwise the division would be stopt short by the indivisible parts, which we should immediately arrive at. If therefore any finite extension be infinitely divisible, it can be no contradiction to suppose, that a finite extension contains an infinite number of parts: And vice versa, if it be a contradiction to suppose, that a finite extension contains an infinite number of parts, no finite extension can be infinitely divisible. But that this latter supposition is absurd, I easily convince myself by the consideration of my clear ideas. I first take the least idea I can form of a part of extension, and being certain that there is nothing more minute than this idea, I conclude, that whatever I discover by its means must be a real quality of extension. I then repeat this idea once, twice, thrice, &c., and find the compound idea of extension, arising from its repetition, always to augment, and become double, triple, quadruple, &c., till at last it swells up to a considerable bulk, greater or smaller, in proportion as I repeat more or less the same idea. When I stop in the addition of parts, the idea of extension ceases to augment; and were I to carry on the addition in infinitum, I clearly perceive, that the idea of extension must also become infinite. Upon the whole, I conclude, that the idea of all infinite number of parts is individually the same idea with that of an infinite extension; that no finite extension is capable of containing an infinite number of parts; and consequently that no finite extension is infinitely divisible3. 3 It has been objected to me, that infinite divisibility supposes only an infinite number of PROPORTIONAL not of ALIQIOT parts, and that an infinite number of proportional parts does not form an infinite extension. But this distinction is entirely frivolous. Whether these parts be calld ALIQUOT or PROPORTIONAL, they cannot be inferior to those minute parts we conceive; and therefore cannot form a less extension by their conjunction.
I may subjoin another argument proposed by a noted author [Mons. MALEZIEU], which seems to me very strong and beautiful. It is evident, that existence in itself belongs only to unity, and is never applicable to number, but on account of the unites, of which the number is composed. Twenty men may be said to exist; but it is only because one, two, three, four, &c. are existent, and if you deny the existence of the latter, that of the former falls of course. It is therefore utterly absurd to suppose any number to exist, and yet deny the existence of unites; and as extension is always a number, according to the common sentiment of metaphysicians, and never resolves itself into any unite or indivisible quantity, it follows, that extension can never at all exist. It is in vain to reply, that any determinate quantity of extension is an unite; but such-a-one as admits of an infinite number of fractions, and is inexhaustible in its sub-divisions. For by the same rule these twenty men may be considered as a unit. The whole globe of the earth, nay the whole universe, may be considered as a unit. That term of unity is merely a fictitious denomination, which the mind may apply to any quantity of objects it collects together; nor can such an unity any more exist alone than number can, as being 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 resolved into any lesser unity.
All this reasoning takes place with regard to time; along with an additional argument, which it may be proper to take notice of. It is 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. For the same reason, that the year 1737 cannot concur with the present year 1738 every moment must be distinct from, and posterior or antecedent to another. It is certain then, that time, as it exists, must be composed of indivisible moments. For if in time we could never arrive at an end of division, and if each moment, as it succeeds another, were not perfectly single and indivisible, there would be an infinite number of co-existent moments, or parts of time; which I believe will be allowed to be an arrant contradiction.
The infinite divisibility of space implies that of time, as is evident from the nature of motion. If the latter, therefore, be impossible, the former must be equally so.
I doubt not but, it will readily be allowed by the most obstinate defender of the doctrine of infinite divisibility, that these arguments are difficulties, and that it is impossible to give any answer to them which will be perfectly clear and satisfactory. But here we may observe, that nothing can be more absurd, than this custom of calling a difficulty what pretends to be a demonstration, and endeavouring by that means to elude its force and evidence. It is not in demonstrations as in probabilities, that difficulties can take place, and one argument counter-ballance another, and diminish its authority. A demonstration, if just, admits of no opposite difficulty; and if not just, it is a mere sophism, and consequently can never be a difficulty. It is either irresistible, or has no manner of force. To talk therefore of objections and replies, and ballancing of arguments in such a question as this, is to confess, either that human reason is nothing but a play of words, or that the person himself, who talks so, has not a Capacity equal to such subjects. Demonstrations may be difficult to be comprehended, because of abstractedness of the subject; but can never have such difficulties as will weaken their authority, when once they are comprehended.
It is true, mathematicians are wont to say, that there are here equally strong arguments on the other side of the question, and that the doctrine of indivisible points is also liable to unanswerable objections. Before I examine these arguments and objections in detail, I will here take them in a body, and endeavour by a short and decisive reason to prove at once, that it is utterly impossible they can have any just foundation.
It is an established maxim in metaphysics, That whatever the mind clearly conceives, includes the idea of possible existence, or in other words, that nothing we imagine is absolutely impossible. We can form the idea of a golden mountain, and from thence conclude that such a mountain may actually exist. We can form no idea of a mountain without a valley, and therefore regard it as impossible.
Now it is certain we have an idea of extension; for otherwise why do we talk and reason concerning it? It is likewise certain that this idea, as conceived by the imagination, though divisible into parts or inferior ideas, is not infinitely divisible, nor consists of an infinite number of parts: For that exceeds the comprehension of our limited capacities. Here then is an idea of extension, which consists of parts or inferior ideas, that are perfectly, indivisible: consequently this idea implies no contradiction: consequently it is possible for extension really to exist conformable to it: and consequently all the arguments employed against the possibility of mathematical points are mere scholastick quibbles, and unworthy of our attention.
These consequences we may carry one step farther, and conclude that all the pretended demonstrations for the infinite divisibility of extension are equally sophistical; since it is certain these demonstrations cannot be just without proving the impossibility of mathematical points; which it is an evident absurdity to pretend to.
From:
Hume, David. A Treatise of Human Nature. Ed. L.A Selby-Bigge. Oxford: Clarendon Press, 1979.
Text available online at:
http://ebooks.adelaide.edu.au/h/hume/david/h92t/
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