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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"

Section IV: "Objections Answer'd"

§91

Hume's system of space and time consists of two parts. The first follows this chain of reasoning:

1) The mind's capacity is not infinite.

2) Thus, neither our ideas of extension nor of duration contain an infinite number of parts or inferior ideas.

3) Rather, they could only consist of a finite number of simple and indivisible ideas.

4) It is possible that our ideas of space and time are made-up of indivisible simple ideas.

5) But, it is not possible that our ideas of space and time are infinitely divisible.

6) Hence, our ideas of space and time are certainly made-up of a finite number of indivisible simple ideas.

§92

The second part of Hume's systems follows from this first part.

1) The smallest parts of our ideas for space and time are indivisible.

2) If these indivisible parts had no content, then we could not conceive them.

3) But we can conceive them.

4) Thus these indivisible parts are filled with something real and existent.

5) Thus every time we conceive of space or time, we must be conceiving of something that is in space or time.

6) "The ideas of space and time are therefore no separate or distinct ideas, but merely those of the manner or order, in which objects exist: Or, in other words, 'tis impossible to conceive either a vacuum and extension without matter, or a time, when there was no succession or change in any real existence." (39-40)

Objections to Hume indivisibility of space and time have been leveled against both parts of his system. And both parts are connected. Hume will begin by examining the objections against the finite divisibility of extension.

§93

I.

The first objection that Hume addresses actually does more to prove the connection and dependence of the two parts of his system than it does succeed at counter-arguing either of them.

Many argue that indivisible mathematical points are absurd. Because these points do not exist, we cannot conceive of extension being made-up of them [see this entry for the distinction between mathematical points and physical points].

Now, we know that we have impressions of objects' color and solidity. These impressions are made-up of small simple indivisible impressions of color and solidity.

Physical points are divisible. So the ideas of physical points would be compound. They would be made-up of simpler ideas. But our indivisible impressions are not made-up of lesser-ideas. So we do not have impressions of the object's physical points. Rather, we have sense impressions of the object's indivisible inextensive parts. These impressions must then be of the object's mathematical points. [This position bears a strong resemblance to Deleuze's interpretation of Spinoza's intensities, which are non-extensive qualitative variations. And their degree of qualitative variation is a calculus differential. Hence it is purely quantitative. But also as a differential, it is not made-up of (extensive) terms. It could also be that Deleuze would regard Hume's simple ideas in the same light, that is, as intensities. For more on intensities as inextensive differentials of qualitative variation, see in particular Deleuze's "Cours Vincennes: 10/03/1981" and "Cours Vincennes: 20/01/1981".]

So Hume thinks that the infinite divisibility of matter is an absurd extreme position. He thinks the same for the argument that mathematical points are non-entities. There is a "medium" position between the two. When we regard the mathematical points as corresponding to simple impressions, then we neither regard matter as infinitely divisible, nor mathematical points as non-entities. (40b)

Now, extensions always have parts. Thus physical points, which are extensions, must also have parts. These parts must be different from one another. And anything that is different, the mind can conceive separably [see §46 for more on this principle.] But, say we are conceiving of one physical point. It is different from another. But within this idea must be more ideas of simpler parts. But if every part has another part to infinity, then every time we conceive any physical part whatsoever, we are thereby conceiving its smaller parts together in the part they make-up.

Thus the mind could not possible conceive all the subparts separate from each other. In other words, we would always be part that we are conceiving not separately from each other. Hence extension must instead have fundamental indivisible parts that we may conceive apart from each other.

§94

II.

Hume then addresses a second objection to his theory of simple extensive parts as being mathematical points:

If there is no extensive space between things, then their boundaries must overlap. And, if there are spaces between indivisible things, then there are extensive parts between them. So if there are only indivisible atoms, there cannot be extensive spaces between them. Thus if extension is made-up of atoms, then their boundaries must overlap. That means, the atoms must penetrate each other's space.

For one thing to have both its own place, and to penetrate another's space, requires that part of it be in something else, and part of it be in itself. But then the thing would have parts. Thus such things would not be perfectly simple. Hence because indivisible atoms would have to penetrate each other, they would not be indivisible. So extension, argue some, cannot be made-up of indivisible mathematical points. (40-41) [This is much like Aristotle's argument for the continua of space and time.]

§95

Hume will counter-argue by giving a better definition for penetration. Imagine two solid bodies approaching each other. They collide. Thereby they unite into one body that extends no further than either did alone. This is really what penetration means. But for this to happen, one body mustbe annihilated, and the other preserved, although we are not very able to determine which one survived and which one did not. So before they collide we have the idea of two bodies. After they collide we have the idea of only one body. Penetrating bodies are no longer distinguishable:

'Tis impossible for the mind to preserve any notion of difference betwixt two bodies of the same nature existing in the same place at the same time. (41b)

§96

Thus penetration is now defined as "the annihilation of one body upon its approach to another."

Hume has us imagine a blue point and a red point in our field of vision. Now we imagine them moving toward each other, and colliding. We are able to envision ourselves seeing the red point and the blue point lying next to each other, without one being annihilated, and without there being any other colored points between them.

Hume's claim is that a colored or tangible point need not be annihilated if it encounters another such point. Rather, what we find happening instead is that the two points unite to form a compound and divisible body. Each part is perfectly contiguous. No parts lie between them. But despite their union, both parts remain distinct, separate, and distinguishable.

§97

Now Hume has us conduct a demonstration much like the previous one involving the ink dot [see §65]. Place a dot of ink on a piece of paper. Then move backward until we no longer are able to see the spot. Now return towards the spot. We find that it becomes increasingly visible by short intervals. At some point, only its color seems to grow in strength, and not so much its size.

Now, at one stage of our moving away from the dot, it was no more than an indivisible point. So we know that it is possible to conceive such points. However, consider when we see the dot up close. And imagine now that we are asked to differentiate in it all its indivisible points. We will not be able to, because we believe that it is inconceivable to distinguish the indivisible parts of the dot. But, our demonstration showed us that we are able to make out indivisible points.

Because we have difficulty in making-out indivisible points in extended things, we also have difficulty seeing how indivisible points lie contiguous without annihilating each other.

§98

III.

Other objections to extension's indivisible parts come from mathematics. Hume will show that his theory conforms to their definitions, even if it does not agree with their demonstrations.

§99

A surface is length and breadth without depth.

A line is length without breadth or depth.

A point has neither length, breadth, nor depth.

Now, we consider the world around us to extend. If it were infinitely divisible, we would continually make smaller and smaller extensive pieces. And because we presume it is infinitely divisible, then each time we make a smaller piece, it can be divided again. That means that everything in an infinitely divisible extensive world would have at least some length, breadth, and depth so that it can always be divided again. But, we just defined a point as something without breadth, depth, or length.

Consider instead Hume's theory. There are smallest indivisible points, and these by themselves do not extend. Now, everything extensive has parts. And, as soon as you add two of these "mathematical points" together, then you have something with parts, and hence something extensive. So Hume says that the only way these geometrical definitions could hold for the world around us is if it were divisible to non-extensive atoms [which are something akin to Deleuzean intensities.] Hume then asks if there are any defenses for extensity's infinite divisibility which may account for these definitions for point, line, and plane.

§100

Hume says there are two such defenses. In the first defense, some argue that points should not be thought to exist in the first place. This is because they are objects of the imagination, and not of reality. [compare to Évellin's theory of the difference between imaginary and real points.]

§101

We know that whatever can be clearly and distinctly conceived may possibly exist in actuality [see §73 for more on this principle.] So, to say that something is impossible is really to claim that we have no clear idea of it.

'Tis in vain to search for a contradiction in any thing that is distinctly conceiv'd by the mind. Did it imply any contradiction, 'tis impossible it cou'd ever be conceiv'd. (43b)

§102

So either

1) we may conceive divisible points, and hence they are possible, or

2) we cannot conceive divisible points, and hence they are impossible.

There is no in-between. So we cannot say that divisible points are impossible but thinkable. But we saw above that some who argue for extension's infinite divisibility say that points, lines, and planes are only geometrical abstractions. In actuality there can be no such things. Rather, we can only think them.

Consider for example that we want to imagine a line. Those taking this position would like us to imagine length without breadth, "in the same manner as we may think of the length of the way betwixt two towns, and overlook its breadth." (43bc) Now, if a line has no breadth, then we can no longer divide it lengthwise. So then extension in the world around us would cease being infinitely divisible.

What we need to do instead is grasp these terms in our reason, rather than imagination. [this is much like how Spinoza avoids Zeno's paradoxes. Spinoza explains that substance is infinite. It is infinitely indivisible. But, when we consider substance under certain modes of thinking, we might regard it in terms of measure, time, and motion. These are infinitely divisible in our imaginations. But when we use our reason, we find that substance is not infinitely divisible. In fact, it is not divisible at all. Although Spinoza comes to a different conclusion, he also bases his argument on the difference between the modes of understanding quantity. By the imagination we get Zeno's paradoxes. By the intellect we do not.] So, we will never find length without breadth in the world around us. Nor can we picture it in our minds. However, consider that when we see the white marble globe, our minds tend to associate it in two directions: towards round things and towards white things. So even though we never find colors without forms, we make what Hume calls a 'distinction of reason.' We may reason that there are two qualities of the white globe, but we never find them apart in the world or in our minds [for more, see §61]. So according to this argument, when we see something with length and breadth, then our minds tend in two directions of association: 1) with things with length, and 2) with things with breadth.

§103

Previously Hume gave the following argument against this kind of reasoning:

1) Our minds do not have infinite capacities.

2) A line that has length but not breadth would have a minimum of breadth.

3) Arriving at a minimum requires infinite divisions, which requires an infinite capacity of the mind.

Hume will not resort again to the above argument. Rather, he will show new absurdities in their defense.

§104

A surface terminates a solid.

A line terminates a surface.

A point terminates a line.

Now we will try to conceive of these terminations, presuming that extension is infinitely divisible. And if extension is infinitely divisible, then anything in it is infinitely divisible. Which means the ideas of anything in extension would be infinitely divisible.

So we imagine for example a square. Now we want to imagine its termination, namely, a line making up one of its sides. When we think of that line, we find that it is made-up of smaller parts. Thus it is divisible. So we divide it into more precise ideas of the termination. But again, if extension is infinitely divisible, we will find this idea also to be composite. So long as we presuppose infinite divisibility, we will never arrive at the termination.

For let these ideas be suppos'd infinitely divisible; and then let the fancy endeavour to fix itself on the idea of the last surface, line or point; it immediately finds this idea to break into parts; and upon its seizing the last of these parts, it loses its hold by a new division, and so on in infinitum, without any possibility of its arriving at a concluding idea. The number of fractions bring it no nearer the last division, than the first idea it form'd. Every particle eludes the grasp by a new fraction; like quicksilver, when we endeavour to seize it. (44a)

However, we do presume that something terminates the surface. This means there is something determinate that terminates the surface. If this termination consists of parts or inferior ideas, then the real termination is found within another internal idea. Thus the idea of the termination cannot consist of parts.

this is a clear proof, that the ideas of surfaces, lines and points admit not of any division; those of surfaces in depth; of lines in breadth and depth; and of points in any dimension. (44b)

§105

Because this argument is so convincing, defenders of infinite divisibility have conceded that nature is at least partly endowed with indivisible mathematical points. Others "elude the force of this reasoning by a heap of unintelligible cavils and distinctions." But this is just hiding from superior arguments rather than counter-arguing them directly.

Both these adversaries equally yield the victory. A man who hides himself, confesses as evidently the superiority of his enemy, as another, who fairly delivers his arms. (44c)

§106

Thus if we presuppose infinite divisibility, we cannot conceive the termination of a figure. But this is required for performing geometrical demonstrations. However, if we presuppose extension to be made-up of indivisible points, then we may conceive of a figure's termination. Thereby we may conduct geometrical demonstrations.

§107

Those who would want to give a demonstration of infinite divisibility will need to do so based on inexact ideas, such as infinitely divisible terminations, and also maxims that are not precisely true, such as there may be length without breadth. Those who presuppose infinite divisibility seek an absolute precision. But geometry needs to regard proportions roughly "and with some liberty."

Its errors are never considerable; nor would it err at all, did it not aspire to such an absolute perfection. (45b)

§108

Mathematicians may regard extension as infinitely divisible, or made-up of indivisible parts. Either way, they will have difficulty explaining what they mean when they say that one line or surface is equal to, greater than, or less than another.

§109

Those mathematicians who presuppose indivisible points have the "readiest and justest answer." They may reply by saying that lines or surfaces are equal when they have the same number of indivisible points. This answer is just. But when we do see these smallest points, they are so tiny that we confuse them with each other. Thus it is impossible for our minds to compute their number. Hence never in our lives do we measure things by counting their indivisible points. So this is a useless answer.

§110

If extension were infinitely divisible, then both a small and a large line will both have infinitely many points. But we cannot say that there is a greater, lesser, or equal quantity of points between two infinities. So those who presuppose infinite divisibility cannot say that we compare sizes by counting points. But some say that a yard is made of 3 feet, and that is how we compare sizes, for example. But the question is, how do we know how to make our feet equal so that we may add 3 together? They say, there are 12 inches in each foot. Again we want to know what makes the inches equal. So long as extension is infinitely divisible, there will be no absolute standard for comparisons of size.

§111

Some argue that we may determine the equality of lengths by placing them in contact, and seeing if all their parts correspond and touch each other. In other words, we base these comparisons on relations of congruity. Now, to be sure that all their parts correspond, we would need to look at their very smallest parts at the edges. That means we will need to conceive of their smallest parts. But the smallest parts we can conceive are mathematical points. Then we will be counting smallest indivisible parts. Yet, we saw already this is a useless standard. So we must find some other solution.

§112

Consider that we may easily tell just by looking that a yard is longer than a foot. So we are often able to use our vision alone to correctly determine when objects are equal to, greater than, or less than one another.

§113

Yet, our judgments are sometimes incorrect. So we rectify them by further review and by use of absolute standards of measure. But even these corrections are susceptible to further correction. (47d)

§114

So we often compare sizes first by making a judgment, then correcting it. When we judge it, we use a looser method of comparison. When we correct it, we use a stricter method.

Yet we have reason to believe that there are bodies vastly smaller than our ability to perceive them. So this method of comparison will not be able to determine their equalities of size. And, were we to remove such a minute body from something we see, we would not be able to detect that subtraction. But we must imagine these smaller bodies. So this is an imaginary standard.

For as the very idea of equality is that of such a particular appearance corrected by juxtaposition or a common measure, the notion of any correction beyond what we have instruments and art to make, is a mere fiction of the mind, and useless as well as incomprehensible. (48b)

§115

The same applies to curves and straight lines. Our senses are able to differentiate the two, but we have difficulty defining them.

Suppose that we wanted to be sure that what looks like a straight line is really not in fact curved. If we presupposed indivisible points, we could perhaps say that the one end of the line is so many indivisible points off-center from the other end of the line. But we cannot use our senses to count such points. So this is a just a "distant notion of some unknown standard." But if we presuppose infinite divisibility, then we would continue dissecting the difference between the alignment of the one end from the other end. If the line is perfectly straight, this method would never arrive at that conclusion, because it would need to continually make its comparisons more-and-more precise. So we are worse off if we presuppose infinite divisibility.

Thus we can offer no definition to distinguish lines from curves. Yet, we still are able to make corrections in our visual comparisons. Now, we often imagine these corrections continuing at a level beyond our senses. Thereby we mistakenly conclude that there could be a perfect standard that results from an infinite process of correction. But because it is a fictional standard, we are unable to explain or comprehend it. (49d)

§116

Nonetheless, mathematicians pretend to define a straight line as 'the shortest way between two points' (49-50).

Now imagine that we see a line that is the shortest distance between two things. And suppose this is our only impression of both a line and of the shortest distance between points. Then, our definition, 'a line is the shortest distance between two points,' is a tautology. It is like also saying 'the shortest distance between two points is always the shortest.' In order for this definition to make sense, we also need to have an impression of a distance that is not the shortest between two points. Then we may compare it with the shortest. Only after that comparison would it make sense that a line is the shortest, and curves are less short. So in other words, before we may define lines in this geometrical way, we must already presuppose that we know the difference between lines and curves. And also we must obtain this knowledge by means of visual comparisons and not geometrical abstractions. (50a.b)

§117

Hume reiterates his point that we do not have precise ideas for equality & inequality, shorter & longer, and straight & curve. Hence we cannot use one term in the pair to find a standard for the other. Exact ideas cannot be founded on loose indeterminate ones.

§118

Mathematicians try to define 'plane surface' as the flowing of a straight line. [This is much like Spinoza's notion of a genetic definition. A circle is produced by a straight line pivoting on a point. The closed curve that the swinging endpoint traces is a circle.]

Hume notes some objections to this definition for a plane:

1) On the one hand, we have ideas of surfaces. And on the other, we have this particular idea of how to form a surface. But these are two entirely different ideas. So one term should not define the other.

2) Our idea of a straight line is no more precise than our idea of a plane. So we will not arrive at a precise idea by defining one in terms of the other.

3) A straight line may flow irregularly, and thereby create some figure different from a plane. So we must define the flowing of the right line as following the path between two parallel straight lines. But such a condition already creates a planar path for the flowing line to traverse, hence this would be a circular definition.

§119

So there is a common method for conceiving some of the most essential terms in geometry, namely, equality and inequality, straight line and plane surface. However, by this method, the terms prove to be inexact and indeterminate. We can neither determine when real objects fit these definitions, nor can we conceive them in our minds. We must always resort to our judgment, even though it is fallible and weak. To claim there to be more precise determinations than we are able to make is to give a useless or imaginary standard. We also obtain no clearer of a conception of these terms by presupposing there is an omnipotent deity who alone may form or conceive them.

As the ultimate standard of these figures is derived from nothing but the senses and imagination, it is absurd to talk of any perfection beyond what these faculties can judge of; since the true perfection of any thing consists in its conformity to its standard. (51b)

§120

So some mathematicians presuppose a precise standard arrivable somehow by means of infinite divisibility. But we see that such a presupposition leads to loose and uncertain definitions for the most essential and fundamental terms in geometry. So such a mathematician would have difficulty for example showing that there can be no more than one straight line between two points. So we see that he has lost his means to create convincing demonstrations.

Consider another geometrical fact that the mathematician will have difficulty demonstrating: two different straight lines cannot share a common segment [they either

a) lie along the same path, and are thus one line, because only one line may be found between points,

b) they intersect at a point, which is not a segment, or

c) do not interesect, hence could not possibly have a common segment.]

Hume says that given the imprecission of the mathematician's definitions, we may say that it is conceivable that two lines that are apart by great distances and nearing each other only very slowly, will eventually come together to form a common segment. In order to counter-argue, the mathematician would need to have some different idea of a straight line. If he says that on account of infinite divisibility, there will always be some angle of intersection, then he is using a standard of comparison that we never actually use in real life. For we never succeed in conducting infinitely many divisions. Rather, we have a method of visually comparing lines. And according this means, we might judge two lines as straight but find that they may also meet and share a segment. It is only when we imagine some impossible standard do we conclude that our senses must always be wrong in this regard.

§121

On account of these problems, we see that there can be no convincing geometrical demonstration for extension's infinite divisibility. For these demonstrations are based on the definitions of points, lines, and planes, which can only be understood if extension is made-up of indivisible parts.

§122

The second sort of defenses for infinite divisibility are based on the idea of point of contact.

Mathematicians tend to consider their drawings imperfect renderings of perfect figures.

Hume then asks the mathematician to imagine a perfect circle with a tangent line. Hume asks, does the mathematician conceive them touching in an indivisible mathematical point, or do both the circumference and the line share some extent of space?

Either

a) he imagines them meeting at an indivisible mathematical point. We know that whatever can be imagined can also possibly be real in actuality. But supposedly this mathematician claims that indivisible points are not possible in actuality. Or,

b) if they share the same space, then at their point of contact, either the circle is straight, or the line is curved. Hence even this conception confuses two ideas that should be kept distinct. (53c)

[Directory of other entries in this series.]

From Hume's original text:

## Sect. iv. Objections Answered.

Our system concerning space and time consists of two parts, which are intimately connected together. The first depends on this chain of reasoning. The capacity of the mind is not infinite; consequently no idea of extension or duration consists of an infinite number of parts or inferior ideas, but of a finite number, and these simple and indivisible: It is therefore possible for space and time to exist conformable to this idea: And if it be possible, it is certain they actually do exist conformable to it; since their infinite divisibility is utterly impossible and contradictory.

The other part of our system is a consequence of this. The parts, into which the ideas of space and time resolve themselves, become at last indivisible; and these indivisible parts, being nothing in themselves, are inconceivable when not filled with something real and existent. The ideas of space and time are therefore no separate or distinct ideas, but merely those of the manner or order, in which objects exist: Or in other words, it is impossible to conceive either a vacuum and extension without matter, or a time, when there was no succession or change in any real existence. The intimate connexion betwixt these parts of our system is the reason why we shall examine together the objections, which have been urged against both of them, beginning with those against the finite divisibility of extension.

I. The first of these objections, which I shall take notice of, is more proper to prove this connexion and dependence of the one part upon the other, than to destroy either of them. It has often been maintained in the schools, that extension must be divisible, in infinitum, because the system of mathematical points is absurd; and that system is absurd, because a mathematical point is a non-entity, and consequently can never by its conjunction with others form a real existence. This would be perfectly decisive, were there no medium betwixt the infinite divisibility of matter, and the non-entity of mathematical points. But there is evidently a medium, viz. the bestowing a colour or solidity on these points; and the absurdity of both the extremes is a demonstration of the truth and reality of this medium. The system of physical points, which is another medium, is too absurd to need a refutation. A real extension, such as a physical point is supposed to be, can never exist without parts, different from each other; and wherever objects are different, they are distinguishable and separable by the imagination.

II. The second objection is derived from the necessity there would be of PENETRATION, if extension consisted of mathematical points. A simple and indivisible atom, that touches another, must necessarily penetrate it; for it is impossible it can touch it by its external parts, from the very supposition of its perfect simplicity, which excludes all parts. It must therefore touch it intimately, and in its whole essence, SECUNDUM SE, TOTA, ET TOTALITER; which is the very definition of penetration. But penetration is impossible: Mathematical points are of consequence equally impossible.

I answer this objection by substituting a juster idea of penetration. Suppose two bodies containing no void within their circumference, to approach each other, and to unite in such a manner that the body, which results from their union, is no more extended than either of them; it is this we must mean when we talk of penetration. But it is evident this penetration is nothing but the annihilation of one of these bodies, and the preservation of the other, without our being able to distinguish particularly which is preserved and which annihilated. Before the approach we have the idea of two bodies. After it we have the idea only of one. It is impossible for the mind to preserve any notion of difference betwixt two bodies of the same nature existing in the same place at the same time.

Taking then penetration in this sense, for the annihilation of one body upon its approach to another, I ask any one, if he sees a necessity, that a coloured or tangible point should be annihilated upon the approach of another coloured or tangible point? On the contrary, does he not evidently perceive, that from the union of these points there results an object, which is compounded and divisible, and may be distinguished into two parts, of which each preserves its existence distinct and separate, notwithstanding its contiguity to the other? Let him aid his fancy by conceiving these points to be of different colours, the better to prevent their coalition and confusion. A blue and a red point may surely lie contiguous without any penetration or annihilation. For if they cannot, what possibly can become of them? Whether shall the red or the blue be annihilated? Or if these colours unite into one, what new colour will they produce by their union?

What chiefly gives rise to these objections, and at the same time renders it so difficult to give a satisfactory answer to them, is the natural infirmity and unsteadiness both of our imagination and senses, when employed on such minute objects. Put a spot of ink upon paper, and retire to such a distance, that the spot becomes altogether invisible; you will find, that upon your return and nearer approach the spot first becomes visible by short intervals; and afterwards becomes always visible; and afterwards acquires only a new force in its colouring without augmenting its bulk; and afterwards, when it has encreased to such a degree as to be really extended, it is still difficult for the imagination to break it into its component parts, because of the uneasiness it finds in the conception of such a minute object as a single point. This infirmity affects most of our reasonings on the present subject, and makes it almost impossible to answer in an intelligible manner, and in proper expressions, many questions which may arise concerning it.

III. There have been many objections drawn from the mathematics against the indivisibility of the parts of extension: though at first sight that science seems rather favourable to the present doctrine; and if it be contrary in its DEMONSTRATIONS, it is perfectly conformable in its definitions. My present business then must be to defend the definitions, and refute the demonstrations.

A surface is DEFINed to be length and breadth without depth: A line to be length without breadth or depth: A point to be what has neither length, breadth nor depth. It is evident that all this is perfectly unintelligible upon any other supposition than that of the composition of extension by indivisible points or atoms. How else coued any thing exist without length, without breadth, or without depth?

Two different answers, I find, have been made to this argument; neither of which is in my opinion satisfactory. The first is, that the objects of geometry, those surfaces, lines and points, whose proportions and positions it examines, are mere ideas in the mind; I and not only never did, but never can exist in nature. They never did exist; for no one will pretend to draw a line or make a surface entirely conformable to the definition: They never can exist; for we may produce demonstrations from these very ideas to prove, that they are impossible.

But can anything be imagined more absurd and contradictory than this reasoning? Whatever can be conceived by a clear and distinct idea necessarily implies the possibility of existence; and he who pretends to prove the impossibility of its existence by any argument derived from the clear idea, in reality asserts, that we have no clear idea of it, because we have a clear idea. It is in vain to search for a contradiction in any thing that is distinctly conceived by the mind. Did it imply any contradiction, it is impossible it coued ever be conceived.

There is therefore no medium betwixt allowing at least the possibility of indivisible points, and denying their idea; and it is on this latter principle, that the second answer to the foregoing argument is founded. It has been pretended [L’Art de penser.], that though it be impossible to conceive a length without any breadth, yet by an abstraction without a separation, we can consider the one without regarding the other; in the same manner as we may think of the length of the way betwixt two towns, and overlook its breadth. The length is inseparable from the breadth both in nature and in our minds; but this excludes not a partial consideration, and a distinction of reason, after the manner above explained.

In refuting this answer I shall not insist on the argument, which I have already sufficiently explained, that if it be impossible for the mind to arrive at a minimum in its ideas, its capacity must be infinite, in order to comprehend the infinite number of parts, of which its idea of any extension would be composed. I shall here endeavour to find some new absurdities in this reasoning.

A surface terminates a solid; a line terminates a surface; a point terminates a line; but I assert, that if the ideas of a point, line or surface were not indivisible, it is impossible we should ever conceive these terminations: For let these ideas be supposed infinitely divisible; and then let the fancy endeavour to fix itself on the idea of the last surface, line or point; it immediately finds this idea to break into parts; and upon its seizing the last of these parts, it loses its hold by a new division, and so on in infinitum, without any possibility of its arriving at a concluding idea. The number of fractions bring it no nearer the last division, than the first idea it formed. Every particle eludes the grasp by a new fraction; like quicksilver, when we endeavour to seize it. But as in fact there must be something, which terminates the idea of every finite quantity; and as this terminating idea cannot itself consist of parts or inferior ideas; otherwise it would be the last of its parts, which finished the idea, and so on; this is a clear proof, that the ideas of surfaces, lines and points admit not of any division; those of surfaces in depth; of lines in breadth and depth; and of points in any dimension.

The school were so sensible of the force of this argument, that some of them maintained, that nature has mixed among those particles of matter, which are divisible in infinitum, a number of mathematical points, in order to give a termination to bodies; and others eluded the force of this reasoning by a heap of unintelligible cavils and distinctions. Both these adversaries equally yield the victory. A man who hides himself, confesses as evidently the superiority of his enemy, as another, who fairly delivers his arms.

Thus it appears, that the definitions of mathematics destroy the pretended demonstrations; and that if we have the idea of indivisible points, lines and surfaces conformable to the definition, their existence is certainly possible: but if we have no such idea, it is impossible we can ever conceive the termination of any figure; without which conception there can be no geometrical demonstration.

But I go farther, and maintain, that none of these demonstrations can have sufficient weight to establish such a principle, as this of infinite divisibility; and that because with regard to such minute objects, they are not properly demonstrations, being built on ideas, which are not exact, and maxims, which are not precisely true. When geometry decides anything concerning the proportions of quantity, we ought not to look for the utmost precision and exactness. None of its proofs extend so far. It takes the dimensions and proportions of figures justly; but roughly, and with some liberty. Its errors are never considerable; nor would it err at all, did it not aspire to such an absolute perfection.

I first ask mathematicians, what they mean when they say one line or surface is EQUAL to, or GREATER or LESS than another? Let any of them give an answer, to whatever sect he belongs, and whether he maintains the composition of extension by indivisible points, or by quantities divisible in infinitum. This question will embarrass both of them.

There are few or no mathematicians, who defend the hypothesis of indivisible points; and yet these have the readiest and justest answer to the present question. They need only reply, that lines or surfaces are equal, when the numbers of points in each are equal; and that as the proportion of the numbers varies, the proportion of the lines and surfaces is also varyed. But though this answer be just, as well as obvious; yet I may affirm, that this standard of equality is entirely useless, and that it never is from such a comparison we determine objects to be equal or unequal with respect to each other. For as the points, which enter into the composition of any line or surface, whether perceived by the sight or touch, are so minute and so confounded with each other, that it is utterly impossible for the mind to compute their number, such a computation will Never afford us a standard by which we may judge of proportions. No one will ever be able to determine by an exact numeration, that an inch has fewer points than a foot, or a foot fewer than an ell or any greater measure: for which reason we seldom or never consider this as the standard of equality or inequality.

As to those, who imagine, that extension is divisible in infinitum, it is impossible they can make use of this answer, or fix the equality of any line or surface by a numeration of its component parts. For since, according to their hypothesis, the least as well as greatest figures contain an infinite number of parts; and since infinite numbers, properly speaking, can neither be equal nor unequal with respect to each other; the equality or inequality of any portions of space can never depend on any proportion in the number of their parts. It is true, it may be said, that the inequality of an ell and a yard consists in the different numbers of the feet, of which they are composed; and that of a foot and a yard in the number of the inches. Bat as that quantity we call an inch in the one is supposed equal to what we call an inch in the other, and as it is impossible for the mind to find this equality by proceeding in infinitum with these references to inferior quantities: it is evident, that at last we must fix some standard of equality different from an enumeration of the parts.

There are some [See Dr. Barrow’s mathematical lectures.], who pretend, that equality is best defined by congruity, and that any two figures are equal, when upon the placing of one upon the other, all their parts correspond to and touch each other. In order to judge of this definition let us consider, that since equality is a relation, it is not, strictly speaking, a property in the figures themselves, but arises merely from the comparison, which the mind makes betwixt them. If it consists, therefore, in this imaginary application and mutual contact of parts, we must at least have a distinct notion of these parts, and must conceive their contact. Now it is plain, that in this conception we would run up these parts to the greatest minuteness, which can possibly be conceived; since the contact of large parts would never render the figures equal. But the minutest parts we can conceive are mathematical points; and consequently this standard of equality is the same with that derived from the equality of the number of points; which we have already determined to be a just but an useless standard. We must therefore look to some other quarter for a solution of the present difficulty.

There are many philosophers, who refuse to assign any standard of equality, but assert, that it is sufficient to present two objects, that are equal, in order to give us a just notion of this proportion. All definitions, say they, are fruitless, without the perception of such objects; and where we perceive such objects, we no longer stand in need of any definition. To this reasoning, I entirely agree; and assert, that the only useful notion of equality, or inequality, is derived from the whole united appearance and the comparison of particular objects.

It is evident, that the eye, or rather the mind is often able at one view to determine the proportions of bodies, and pronounce them equal to, or greater or less than each other, without examining or comparing the number of their minute parts. Such judgments are not only common, but in many cases certain and infallible. When the measure of a yard and that of a foot are presented, the mind can no more question, that the first is longer than the second, than it can doubt of those principles, which are the most clear and self-evident.

There are therefore three proportions, which the mind distinguishes in the general appearance of its objects, and calls by the names of greater, less and equal. But though its decisions concerning these proportions be sometimes infallible, they are not always so; nor are our judgments of this kind more exempt from doubt and error than those on any other subject. We frequently correct our first opinion by a review and reflection; and pronounce those objects to be equal, which at first we esteemed unequal; and regard an object as less, though before it appeared greater than another. Nor is this the only correction, which these judgments of our senses undergo; but we often discover our error by a juxtaposition of the objects; or where that is impracticable, by the use of some common and invariable measure, which being successively applied to each, informs us of their different proportions. And even this correction is susceptible of a new correction, and of different degrees of exactness, according to the nature of the instrument, by which we measure the bodies, and the care which we employ in the comparison.

When therefore the mind is accustomed to these judgments and their corrections, and finds that the same proportion which makes two figures have in the eye that appearance, which we call equality, makes them also correspond to each other, and to any common measure, with which they are compared, we form a mixed notion of equality derived both from the looser and stricter methods of comparison. But we are not content with this. For as sound reason convinces us that there are bodies vastly more minute than those, which appear to the senses; and as a false reason would perswade us, that there are bodies infinitely more minute; we clearly perceive, that we are not possessed of any instrument or art of measuring, which can secure us from ill error and uncertainty. We are sensible, that the addition or removal of one of these minute parts, is not discernible either in the appearance or measuring; and as we imagine, that two figures, which were equal before, cannot be equal after this removal or addition, we therefore suppose some imaginary standard of equality, by which the appearances and measuring are exactly corrected, and the figures reduced entirely to that proportion. This standard is plainly imaginary. For as the very idea of equality is that of such a particular appearance corrected by juxtaposition or a common measure, the notion of any correction beyond what we have instruments and art to make, is a mere fiction of the mind, and useless as well as incomprehensible. But though this standard be only imaginary, the fiction however is very natural; nor is anything more usual, than for the mind to proceed after this manner with any action, even after the reason has ceased, which first determined it to begin. This appears very conspicuously with regard to time; where though it is evident we have no exact method of determining the proportions of parts, not even so exact as in extension, yet the various corrections of our measures, and their different degrees of exactness, have given as an obscure and implicit notion of a perfect and entire equality. The case is the same in many other subjects. A musician finding his ear becoming every day more delicate, and correcting himself by reflection and attention, proceeds with the same act of the mind, even when the subject fails him, and entertains a notion of a compleat TIERCE or OCTAVE, without being able to tell whence he derives his standard. A painter forms the same fiction with regard to colours. A mechanic with regard to motion. To the one light and shade; to the other swift and slow are imagined to be capable of an exact comparison and equality beyond the judgments of the senses.

We may apply the same reasoning to CURVE and RIGHT lines. Nothing is more apparent to the senses, than the distinction betwixt a curve and a right line; nor are there any ideas we more easily form than the ideas of these objects. But however easily we may form these ideas, it is impossible to produce any definition of them, which will fix the precise boundaries betwixt them. When we draw lines upon paper, or any continued surface, there is a certain order, by which the lines run along from one point to another, that they may produce the entire impression of a curve or right line; but this order is perfectly unknown, and nothing is observed but the united appearance. Thus even upon the system of indivisible points, we can only form a distant notion of some unknown standard to these objects. Upon that of infinite divisibility we cannot go even this length; but are reduced meerly to the general appearance, as the rule by which we determine lines to be either curve or right ones. But though we can give no perfect definition of these lines, nor produce any very exact method of distinguishing the one from the other; yet this hinders us not from correcting the first appearance by a more accurate consideration, and by a comparison with some rule, of whose rectitude from repeated trials we have a greater assurance. And it is from these corrections, and by carrying on the same action of the mind, even when its reason fails us, that we form the loose idea of a perfect standard to these figures, without being able to explain or comprehend it.

It is true, mathematicians pretend they give an exact definition of a right line, when they say, it is the shortest way betwixt two points. But in the first place I observe, that this is more properly the discovery of one of the properties of a right line, than a just deflation of it. For I ask any one, if upon mention of a right line he thinks not immediately on such a particular appearance, and if it is not by accident only that he considers this property? A right line can be comprehended alone; but this definition is unintelligible without a comparison with other lines, which we conceive to be more extended. In common life it is established as a maxim, that the straightest way is always the shortest; which would be as absurd as to say, the shortest way is always the shortest, if our idea of a right line was not different from that of the shortest way betwixt two points.

Secondly, I repeat what I have already established, that we have no precise idea of equality and inequality, shorter and longer, more than of a right line or a curve; and consequently that the one can never afford us a perfect standard for the other. An exact idea can never be built on such as are loose and undetermined.

The idea of a plain surface is as little susceptible of a precise standard as that of a right line; nor have we any other means of distinguishing such a surface, than its general appearance. It is in vain, that mathematicians represent a plain surface as produced by the flowing of a right line. It will immediately be objected, that our idea of a surface is as independent of this method of forming a surface, as our idea of an ellipse is of that of a cone; that the idea of a right line is no more precise than that of a plain surface; that a right line may flow irregularly, and by that means form a figure quite different from a plane; and that therefore we must suppose it to flow along two right lines, parallel to each other, and on the same plane; which is a description, that explains a thing by itself, and returns in a circle.

It appears, then, that the ideas which are most essential to geometry, viz. those of equality and inequality, of a right line and a plain surface, are far from being exact and determinate, according to our common method of conceiving them. Not only we are incapable of telling, if the case be in any degree doubtful, when such particular figures are equal; when such a line is a right one, and such a surface a plain one; but we can form no idea of that proportion, or of these figures, which is firm and invariable. Our appeal is still to the weak and fallible judgment, which we make from the appearance of the objects, and correct by a compass or common measure; and if we join the supposition of any farther correction, it is of such-a-one as is either useless or imaginary. In vain should we have recourse to the common topic, and employ the supposition of a deity, whose omnipotence may enable him to form a perfect geometrical figure, and describe a right line without any curve or inflexion. As the ultimate standard of these figures is derived from nothing but the senses and imagination, it is absurd to talk of any perfection beyond what these faculties can judge of; since the true perfection of any thing consists in its conformity to its standard.

Now since these ideas are so loose and uncertain, I would fain ask any mathematician what infallible assurance he has, not only of the more intricate, and obscure propositions of his science, but of the most vulgar and obvious principles? How can he prove to me, for instance, that two right lines cannot have one common segment? Or that it is impossible to draw more than one right line betwixt any two points? should be tell me, that these opinions are obviously absurd, and repugnant to our clear ideas; I would answer, that I do not deny, where two right lines incline upon each other with a sensible angle, but it is absurd to imagine them to have a common segment. But supposing these two lines to approach at the rate of an inch in twenty leagues, I perceive no absurdity in asserting, that upon their contact they become one. For, I beseech you, by what rule or standard do you judge, when you assert, that the line, in which I have supposed them to concur, cannot make the same right line with those two, that form so small an angle betwixt them? You must surely have some idea of a right line, to which this line does not agree. Do you therefore mean that it takes not the points in the same order and by the same rule, as is peculiar and essential to a right line? If so, I must inform you, that besides that in judging after this manner you allow, that extension is composed of indivisible points (which, perhaps, is more than you intend) besides this, I say, I must inform you, that neither is this the standard from which we form the idea of a right line; nor, if it were, is there any such firmness in our senses or imagination, as to determine when such an order is violated or preserved. The original standard of a right line is in reality nothing but a certain general appearance; and it is evident right lines may be made to concur with each other, and yet correspond to this standard, though corrected by all the means either practicable or imaginable.

To whatever side mathematicians turn, this dilemma still meets them. If they judge of equality, or any other proportion, by the accurate and exact standard, viz. the enumeration of the minute indivisible parts, they both employ a standard, which is useless in practice, and actually establish the indivisibility of extension, which they endeavour to explode. Or if they employ, as is usual, the inaccurate standard, derived from a comparison of objects, upon their general appearance, corrected by measuring and juxtaposition; their first principles, though certain and infallible, are too coarse to afford any such subtile inferences as they commonly draw from them. The first principles are founded on the imagination and senses: The conclusion, therefore, can never go beyond, much less contradict these faculties.

This may open our eyes a little, and let us see, that no geometrical demonstration for the infinite divisibility of extension can have so much force as what we naturally attribute to every argument, which is supported by such magnificent pretensions. At the same time we may learn the reason, why geometry falls of evidence in this single point, while all its other reasonings command our fullest assent and approbation. And indeed it seems more requisite to give the reason of this exception, than to shew, that we really must make such an exception, and regard all the mathematical arguments for infinite divisibility as utterly sophistical. For it is evident, that as no idea of quantity is infinitely divisible, there cannot be imagined a more glaring absurdity, than to endeavour to prove, that quantity itself admits of such a division; and to prove this by means of ideas, which are directly opposite in that particular. And as this absurdity is very glaring in itself, so there is no argument founded on it which is not attended with a new absurdity, and involves not an evident contradiction.

I might give as instances those arguments for infinite divisibility, which are derived from the point of contact. I know there is no mathematician, who will not refuse to be judged by the diagrams he describes upon paper, these being loose draughts, as he will tell us, and serving only to convey with greater facility certain ideas, which are the true foundation of all our reasoning. This I am satisfyed with, and am willing to rest the controversy merely upon these ideas. I desire therefore our mathematician to form, as accurately as possible, the ideas of a circle and a right line; and I then ask, if upon the conception of their contact he can conceive them as touching in a mathematical point, or if he must necessarily imagine them to concur for some space. Whichever side he chuses, he runs himself into equal difficulties. If he affirms, that in tracing these figures in his imagination, he can imagine them to touch only in a point, he allows the possibility of that idea, and consequently of the thing. If he says, that in his conception of the contact of those lines he must make them concur, he thereby acknowledges the fallacy of geometrical demonstrations, when carryed beyond a certain degree of minuteness; since it is certain he has such demonstrations against the concurrence of a circle and a right line; that is, in other words, be can prove an idea, viz. that of concurrence, to be INCOMPATIBLE with two other ideas, those of a circle and right line; though at the same time he acknowledges these ideas to be inseparable.

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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