Context

Having concluded the general survey of the understanding and its ideas in Part I, the author turns in Part II to an examination of two specific ideas: those of space and of time. The author gives no rationale for fixing on these ideas rather than others.

Background

The philosophical challenges posed by the concept of the infinite were known from the time of the Eleatic philosopher Zeno, who advanced his famous paradoxes of the infinite. One of the most difficult questions of modern philosophy was whether physical objects are divisible to infinity. On the one hand, there is nothing in the nature of space that gives any reason for the division to stop at a certain size. On the other hand, it seems that a division to infinity is impossible. The author’s interest here is ideas of space and time. If they are to represent space and time, we can ask whether they themselves are infinitely divisible. Berkeley had answered in the negative in his An Essay Towards a New Theory of Vision. “[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 65).

The Treatise

1. The author notes that what has “the air of paradox” is loved by philosophers and their disciples. One reason is that dealing with such matters shows the superiority of the philosophers, in that they pass beyond the conceptions of the common person. The other is that the contemplation of paradoxes is agreeable to us, causing “surprize and admiration,” which makes it appear that there is a foundation for the pleasure they give. Philosophers supply the paradoxes, and their disciples believe them readily, each providing pleasure to the other. The most evident instance of such a paradox the author can give is the “doctrine of indivisibility,” the examination of which begins his account of the ideas of space and time.

2. Everyone agrees (and even without agreement, it is evident from “the plainest observation and experience”) that the capacity of the mind is limited and thus that the mind cannot reach a “full and adequate conception of infinity.” We may take that to mean that the mind cannot represent an infinite number of objects. It is also obvious that whatever can be divided infinitely contains an infinite number of parts: “’tis impossible to set any bounds to the number of parts without setting bounds at the same time to the division.” It is supposed to follow that the idea of a finite quality is not infinitely divisible, but “by proper distinctions and separations we may run up this idea to inferior ones, which will be perfectly simple and indivisible.” If the mind could not end the division with simple and indivisible parts, it would have to represent infinitely many parts, which it cannot do.

3. The author claims that it is “therefore certain” that there is a minimum raised by the imagination that cannot be sub-divided and cannot be reduced in size without being totally annihilated. We have ideas of numbers and proportions like the 1/1,000th or 1/10,000th part of a grain of sand, but the images of them are the same as each other, and indeed they are the same as the image of the grain of sand itself. The idea of the grain of sand is not distinguishable into parts, and so is not separable into different ideas, given the principle that whatever is not distinguishable is not separable. [See Part I, Section 3 and Part I Section 7 statements of the converse of this principle, that what is distinguishable is separable.]

4. The same holds for impressions of sensation. The author proposes the following experiment. A spot of ink held at a distance is gradually moved away from one’s eyes. It will appear to be smaller and smaller, and then it will suddenly vanish. The author concludes that at the last moment when the spot is visible, an indivisible minimum is reached. The author rejects the alternative explanation that the object vanishes because no more light reflected from it reaches the eye. The light does reach a telescope which reveals the spot beyond the point at which the naked eye cannot detect it, and we can form an impression of the spot through the medium of the telescope. Given that the quantity of light entering the eye and the lenses of the telescope is the same, it must be explained why there is a small impression, or no impression, formed from the naked eye and a larger impression, or an impression at all, formed from the telescopically-aided eye. The author’s explanation of the difference is that a telescope (or a microscope) “spreads” rays of light that flow from objects. The spreading-out of the rays allows for the enlargement of the naked-eye image by giving parts to impressions that appear uncompounded to the naked eye. And it allows for the perception of an image that is inaccessible to the naked eye.

5. Common opinion is thus in error when it holds that the imagination cannot form an adequate idea of what is beyond a certain degree of smallness or largeness. We do form ideas of which nothing can be smaller, since they are minima. But the “only defect of our senses” is the way they represent things as uncompounded when they really consist of parts. The mistake we make (and of which we are unaware) is in taking the impressions to be more or less equal in size to the objects they represent, so that the object is thought to be small and uncompounded when it is actually composed of many parts and large relative to those parts. Reasoning tells us that there are vast numbers of imperceptible parts of bodies, so we think that the fault in our difficulty in representing them lies in the impressions, that, it is held, are unable to represent any object smaller than the minimum. But these impressions can in fact represent the parts of objects and we can build a representation of the object from those parts, by enlarging our conceptions of it. The author’s example is that of an insect a thousand times smaller than a mite, whose smallest parts (components of the animal spirits) are extraordinarily small. To form a just notion of a very small object, we need distinct ideas representing all its parts, which is extremely difficult to produce, due to their vast number. The theory of infinite divisibility has an insoluble problem, however, as we can never have a just notion if the parts are infinitely divisible, since a just notion requires representation of all the parts.

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