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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 I: "Of the Infinite Divisibility of our Ideas of Space and Time"
§62
There is a reason why philosophers "greedily embrace" paradoxical concepts and counter-intuitive notions.
Because these ideas surprise and stimulate students' minds, philosophers easily obtain followers who are eager to receive their far-fetched conceptions.
From these dispositions in philosophers and their disciples arises that mutual complaisance betwixt them; while the former furnish such plenty of strange and unaccountable opinions, and the latter so readily believe them.
The doctrine of infinite divisibility is one such "strange and unaccountable" notion.
§63
It is universally admitted that the human mind's limitations disallow it to conceive infinity.
Although we have complex ideas, we may continue dividing them until arriving at simple ideas. But if there is a limit to our divisions, then there cannot be infinite parts. Hence we cannot have any idea that is itself infinitely divisible.
§64
It is therefore certain, that the imagination reaches a minimum, and may raise up to itself an idea, of which it cannot conceive any sub-division, and which cannot be diminished without a total annihilation.
We imagine a grain of sand. Now we envision a thousandth part of the grain of sand and a ten-thousandth part of the grain of sand. Is our image of that thousandth part a thousand times smaller of an image? and the ten thousandth part: is its image ten thousand times smaller? Certainly not.
When you tell me of the thousandth and ten thousandth part of a grain of sand, I have a distinct idea of these numbers and of their different proportions; but the images, which I form in my mind to represent the things themselves, are nothing different from each other, nor inferior to that image, by which I represent the grain of sand itself, which is supposed so vastly to exceed them.
If something consists of parts, then we may distinguish these parts with it. But we cannot analyze the idea of a grain of sand into ten-thousand simpler ideas. We may conceive the grain of sand, and we may conceive of its ten-thousandth parts. But the idea for any one of these parts is in no way somehow ten-thousand times less of an idea.
§65
The image of the grain of sand is an idea of the imagination. We found that we may divide-up the grain of sand infinitely many times, but the idea we form for each smaller part is no more diminished. We find something similar with sense impressions.
Hume has us conduct a demonstration. We put an ink spot on a piece of paper. We hold our gaze on the spot while moving backward from it. At some point, we are so far away that we no longer see it. Right before that point it was as small as it could be before disappearing.
it is plain, that the moment before it vanished the image or impression was perfectly indivisible.
The number of light rays hitting the dot will remain the same no matter how far away we stand from it. So the reason we fail to distinguish the dot's parts is not because there is only one ray of light reflecting to our eyes. The reason we cannot distinguish its parts is because we have moved to a distance where our impressions of the image are reduced to their minimum, "and were incapable of any farther diminution."
Now imagine that we are looking up close at a mite that is just large enough for us to see, but still so small that we cannot yet discern any of its parts. Placing it under a microscope does not increase the number of light rays striking it. Instead, the microscope spreads the rays reflecting off the mite. This allows us to see more of the mite's parts, and it creates numerous additional perceptual minima. Before, we could not perceive them. But under the microscope, each of these smallest perceptible parts of the magnified mite is as large as the mite previously was to our naked eye.
§66
We think that there are things that are incredibly large and small, and that our imagination has limits to how much of something's greatness or smallest it can imagine. For, it seems that no matter how small we imagine something to be, it should have parts that are even smaller.
We form this misconception by means of an analogy. The microscope allows us to see that there are perceptual minima within other perceptual minima. But our microscopes have limits to their powers of magnification. So at their highest magnification, we see perceptual minima. But we rightly presume that those minima have smaller parts which our microscopes cannot enarge enough for us to discern.
The first analogy is:
just as the second smallest minimum has its minima that we can perceive, so too must the smallest perceptual minima have their minima that we cannot perceive.
The second analogy takes this to our imaginations:
just as there are small things we know to exist but are incapable of seeing, so too are there things smaller than the smallest things we can imagine.
This mistake we are not sensible of: but taking the impressions of those minute objects, which appear to the senses, to be equal or nearly equal to the objects, and finding by reason, that there are other objects vastly more minute, we too hastily conclude, that these are inferior to any idea of our imagination or impression of our senses.
And yet, Hume argues that still we can form ideas of something no matter how small:
This however is certain, that we can form ideas, which shall be no greater than the smallest atom of the animal spirits of an insect a thousand times less than a mite.
For,
Nothing can be more minute, than some ideas, which we form in the fancy; and images, which appear to the senses; since there are ideas and images perfectly simple and indivisible.
So conceiving any one of these smaller and smaller parts of the mite is not the problem. The difficulty is conceiving the mite as whole in all its completion. For that requires we know all its parts. We may conceive any one of these parts by themselves, but there are infinitely many.
Hence for Hume, the problem of infinite divisibility is not the problem of conceiving the smallest parts, or conceiving of indivisible parts. For these we may conceive. The problem of infinite divisibility is that if things are infinitely divisible, then everything is made up of an infinity of parts. And if we must know all these parts to know the thing, then we must know an infinity of parts to know the whole thing, which seems absurd. So the problem of infinite divisibility is not conceiving the infinitely smallest parts, but of conceiving the infinitely composed whole.
[Directory of other entries in this series.]
From Hume's original text:
PART II. OF THE IDEAS OF SPACE AND TIME
SECT. I. OF THE INFINITE DIVISIBILITY OF OUR IDEAS OF SPACE AND TIME.
Whatever has the air of a paradox, and is contrary to the first and most unprejudiced notions of mankind, is often greedily embraced by philosophers, as shewing the superiority of their science, which coued discover opinions so remote from vulgar conception. On the other hand, anything proposed to us, which causes surprize and admiration, gives such a satisfaction to the mind, that it indulges itself in those agreeable emotions, and will never be persuaded that its pleasure is entirely without foundation. From these dispositions in philosophers and their disciples arises that mutual complaisance betwixt them; while the former furnish such plenty of strange and unaccountable opinions, and the latter so readily believe them. Of this mutual complaisance I cannot give a more evident instance than in the doctrine of infinite divisibility, with the examination of which I shall begin this subject of the ideas of space and time.
It is universally allowed, that the capacity of the mind is limited, and can never attain a full and adequate conception of infinity: And though it were not allowed, it would be sufficiently evident from the plainest observation and experience. It is also obvious, that whatever is capable of being divided in infinitum, must consist of an infinite number of parts, and that it is impossible to set any bounds to the number of parts, without setting bounds at the same time to the division. It requires scarce any, induction to conclude from hence, that the idea, which we form of any finite quality, is not infinitely divisible, but that by proper distinctions and separations we may run up this idea to inferior ones, which will be perfectly simple and indivisible. In rejecting the infinite capacity of the mind, we suppose it may arrive at an end in the division of its ideas; nor are there any possible means of evading the evidence of this conclusion.
It is therefore certain, that the imagination reaches a minimum, and may raise up to itself an idea, of which it cannot conceive any sub-division, and which cannot be diminished without a total annihilation. When you tell me of the thousandth and ten thousandth part of a grain of sand, I have a, distinct idea of these numbers and of their different proportions; but the images, which I form in my mind to represent the things themselves, are nothing different from each other, nor inferior to that image, by which I represent the grain of sand itself, which is supposed so vastly to exceed them. What consists of parts is distinguishable into them, and what is distinguishable is separable. But whatever we may imagine of the thing, the idea of a grain of sand is not distinguishable, nor separable into twenty, much less into a thousand, ten thousand, or an infinite number of different ideas.
It is the same case with the impressions of the senses as with the ideas of the imagination. Put a spot of ink upon paper, fix your eye upon that spot, and retire to such a distance, that, at last you lose sight of it; it is plain, that the moment before it vanished the image or impression was perfectly indivisible. It is not for want of rays of light striking on our eyes, that the minute parts of distant bodies convey not any sensible impression; but because they are removed beyond that distance, at which their impressions were reduced to a minimum, and were incapable of any farther diminution. A microscope or telescope, which renders them visible, produces not any new rays of light, but only spreads those, which always flowed from them; and by that means both gives parts to impressions, which to the naked eye appear simple and uncompounded, and advances to a minimum, what was formerly imperceptible.
We may hence discover the error of the common opinion, that the capacity of the mind is limited on both sides, and that it is impossible for the imagination to form an adequate idea, of what goes beyond a certain degree of minuteness as well as of greatness. Nothing can be more minute, than some ideas, which we form in the fancy; and images, which appear to the senses; since there are ideas and images perfectly simple and indivisible. The only defect of our senses is, that they give us disproportioned images of things, and represent as minute and uncompounded what is really great and composed of a vast number of parts. This mistake we are not sensible of: but taking the impressions of those minute objects, which appear to the senses, to be equal or nearly equal to the objects, and finding by reason, that there are other objects vastly more minute, we too hastily conclude, that these are inferior to any idea of our imagination or impression of our senses. This however is certain, that we can form ideas, which shall be no greater than the smallest atom of the animal spirits of an insect a thousand times less than a mite: And we ought rather to conclude, that the difficulty lies in enlarging our conceptions so much as to form a just notion of a mite, or even of an insect a thousand times less than a mite. For in order to form a just notion of these animals, we must have a distinct idea representing every part of them, which, according to the system of infinite divisibility, is utterly impossible, and, recording to that of indivisible parts or atoms, is extremely difficult, by reason of the vast number and multiplicity of these parts.
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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