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[The following is summary, up to the end where I reproduce this section in full. My commentary is in brackets. Paragraph headings are my own.]

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David Hume

A Treatise of Human Nature

Book I: Of the Understanding

Part III: Of Knowledge and Probability

Section I: Of Knowledge

§161 Relations between ideas, and ideas in relation

Previously we discussed the seven sorts of philosophical relations between ideas: resemblance, identity, relations of time and place, proportions in quantity or number, degrees in any quality, contrariety, and causation.

Hume illustrates 2 classes for these relations.

1) Ideas in Relation:

Certain experiences will lead us to conceive triangles. We learn from observing a triangle that its three angles equal two right angles. We also see that no matter which triangle we imagine, this relation invariably holds. If we were to consider instead a square and say that three angles of a square equals two right angles, clearly this is wrong, for each angle of a square is already a right angle. Hence the first category of relations are ones that depend entirely on the ideas being related.

2) Relations between ideas:

But consider instead the distance between two bookends. If we add more books, that will not change the ideas we have of the bookends. But it will change their relationship of distance from each other. This is the same for two books standing contiguously. If we remove one to read it, that does not change either book, but it does change the relation of contiguity between them.

Now consider for example two of the same book, both freshly pressed, and alike in every way. We may say that they are identical. However, we might also consider them as numerically different. Hence the objects stayed the same, but the relation between them changed.

This time consider the fact that you have had indigestion occasionally throughout your life. Now imagine some foreign dish you have frequently seen other people eat, but never until now did you want to try it. After eating it, you have indigestion. So now you conceive these ideas to have a causal relation. For, the dish causes you indigestion. The dish itself never changed, nor did the sensation of indigestion. However, they obtained new relations between them.

Distance, contiguity, identity, and causation are relations of the second sort: relations that may change even without also there being a change in the ideas related.

The relations between objects are never inherent to them. Rather, they form themselves in our mind. And in the case of cause and effect, their relations are obtained by means of our memories.

There is no single phaenomenon, even the most simple, which can be accounted for from the qualities of the objects, as they appear to us; or which we cou'd foresee without the help of our memory and experience. (69-70)

§162 Intuiting most qualitative values

We considered the equality between the triangle's three angles and two right angles to depend on the ideas being compared. We may confirm this by counting the degrees and seeing that they are quantitatively the same. Equality is more generally a relation of proportion in quantity and number. There are only three other of the seven philosophical relations that likewise can be objects of knowledge. All four together are:

1) resemblance,

2) contrariety,

3) degrees in quality, and

4) proportions in quantity or number.

If two books are the same, we notice immediately. If they are different books, that we notice right away as well. And if one is a red book and the other thing a red apple of about the same shade and hue, we notice that relation instantly too. We need go no further to confirm whether or not the objects relate to each other in these ways. For, we immediately intuit their relations. So of these four relations, three are "discoverable at first sight, and fall more properly under the province of intuition than demonstration:" resemblance, contrariety, and degrees of a quality. (70b) We can easily tell that such ideas existence and non-existence are contrary to each other. And it is only difficult to discern differences in heat, color, or taste when the objects are very similar. Otherwise we can tell the difference right away. And these judgments never require our thoughtful reflection.

this decision we always pronounce at first sight, without any enquiry or reasoning. (70b)

§163 Intuiting some quantitative values

We noted that we are usually able to intuit differences in degrees of a quality. Also, when something is clearly greater or lesser than another thing, we might easily discern the difference. So when there is a noticeable difference, we might also intuit differences in proportions of quantity or number. However, judging things to be equal usually requires additional efforts or means.

§164 Geometry is imprecise

In geometry we establish the proportional relations between the lines and angles of shapes. Hence geometry is the art of fixing a figure's proportions. We might also use our senses and imagination to try to determine how many right angles equal all the triangle's angles. Yet we find that our senses and imagination are not very precise in these matters. However, geometry as a science proves much more precise in making such determinations. Nonetheless, some considerations call the precision of this art into question:

1) Are the origins of our geometrical principles precise? Consider that we are not born with the idea of a triangle. First triangles are shown to us. Then we study the proportional relations of the triangles given to us to see what their defining properties are. Yet, we still base these precise mathematical determinations on imprecise sense impressions. Hence geometry's

first principles are still drawn from the general appearance of the objects; and that appearance can never afford us any security, when we examine, the prodigious minuteness of which nature is susceptible. (71a)

For example, consider that our ideas of straight lines lead us to conclude that they can never share a common line segment. But in fact, we base this principle on a difference of angle between the lines which we may sensibly detect. However, we could also imagine the angle between the lines being less than our senses can perceive. [see §120 for Hume's description of this problem.] But we do not have a good way to conceive the geometrical consequences of such a minute difference. So really we cannot assure the truth of this principle. In fact, most mathematical principles are limited in this way.

§165 Algebra and arithmetic are more precise

In algebra and arithmetic, we preserve exactness and certainty throughout our intricate chains of reasoning. And, we may use a precise standard for assessing equalities and proportions between numbers. By these exact means, we may determine relations between values "without any possibility of error" (71bc). And we may determine equalities in these sciences, even though we have little means to do so in geometry.

When two numbers are so combined, as that the one has always an unite answering to every unite of the other, we pronounce them equal; and it is for want of such a standard of equality in extension, that geometry can scarce be esteemed a perfect and infallible science. (71c)

§166 But geometry is fine for simple cases

So algebra and arithmetic are more precise than geometry. Yet, before we noted that geometry is more precise than our sense impressions and imagination. The defect in geometry we noted was that its principles are always based on imprecise sense impressions. And yet, do not these principles also set a high standard of exactness that we can never discern with our senses? Consider if we saw a plane figure with a thousand angles and sides, called a chiliagon. Our eyes alone will not be able to judge what geometry tells us: that its angles equal 1996 right angles. However, our eyes almost always are able to determine that we cannot draw more than one straight line between two points. And really geometry is meant to be applied in these simple cases where we almost cannot be wrong. And if we are wrong, there will not be significant consequences to our error.

§167 Geometry is not abstract

Hume will now address the argument that mathematical notions are not to be grasped by the imagination, but rather must only be conceived by our reason. According to this view, we may use abstract definitions to conceive triangles without specifying the lengths and proportions of their sides, for example. Such mathematicians consider these notions as "spiritual and refined perceptions." In this way, they replace clear and precise ideas for ones that are obscure and uncertain, in attempt to abstract from particulars.

To counter-argue, Hume reminds us that all ideas are copied from our impressions [see §17 for more on this principle.] We also know that all impressions are clear and precise. Hence the ideas copied from them cannot "contain anything so dark and intricate." (72-73)

An idea is by its very nature weaker and fainter than an impression; but being in every other respect the same, cannot imply any very great mystery. (73a)

[Directory of other entries in this series.]

From the original text:

## Sect. i. Of Knowledge.

There are seven [Part I. Sect. 5.] different kinds of philosophical relation, viz. RESEMBLANCE, IDENTITY, RELATIONS OF TIME AND PLACE, PROPORTION IN QUANTITY OR NUMBER, DEGREES IN ANY QUALITY, CONTRARIETY and CAUSATION. These relations may be divided into two classes; into such as depend entirely on the ideas, which we compare together, and such as may be changed without any change in the ideas. It is from the idea of a triangle, that we discover the relation of equality, which its three angles bear to two right ones; and this relation is invariable, as long as our idea remains the same. On the contrary, the relations of contiguity and distance betwixt two objects may be changed merely by an alteration of their place, without any change on the objects themselves or on their ideas; and the place depends on a hundred different accidents, which cannot be foreseen by the mind. It is the same case with identity and causation. Two objects, though perfectly resembling each other, and even appearing in the same place at different times, may be numerically different: And as the power, by which one object produces another, is never discoverable merely from their idea, it is evident cause and effect are relations, of which we receive information from experience, and not from any abstract reasoning or reflection. There is no single phaenomenon, even the most simple, which can be accounted for from the qualities of the objects, as they appear to us; or which we coued foresee without the help of our memory and experience.

It appears, therefore, that of these seven philosophical relations, there remain only four, which depending solely upon ideas, can be the objects of knowledge said certainty. These four are RESEMBLANCE, CONTRARIETY, DEGREES IN QUALITY, and PROPORTIONS IN QUANTITY OR NUMBER. Three of these relations are discoverable at first sight, and fall more properly under the province of intuition than demonstration. When any objects resemble each other, the resemblance will at first strike the eve, or rather the mind; and seldom requires a second examination. The case is the same with contrariety, and with the degrees of any quality. No one can once doubt but existence and non-existence destroy each other, and are perfectly incompatible and contrary. And though it be impossible to judge exactly of the degrees of any quality, such as colour, taste, heat, cold, when the difference betwixt them is very small: yet it is easy to decide, that any of them is superior or inferior to another, when their difference is considerable. And this decision we always pronounce at first sight, without any enquiry or reasoning.

We might proceed, after the same manner, in fixing the proportions of quantity or number, and might at one view observe a superiority or inferiority betwixt any numbers, or figures; especially where the difference is very great and remarkable. As to equality or any exact proportion, we can only guess at it from a single consideration; except in very short numbers, or very limited portions of extension; which are comprehended in an instant, and where we perceive an impossibility of falling into any considerable error. In all other cases we must settle the proportions with some liberty, or proceed in a more artificial manner.

I have already I observed, that geometry, or the art, by which we fix the proportions of figures; though it much excels both in universality and exactness, the loose judgments of the senses and imagination; yet never attains a perfect precision and exactness. It’s first principles are still drawn from the general appearance of the objects; and that appearance can never afford us any security, when we examine, the prodigious minuteness of which nature is susceptible. Our ideas seem to give a perfect assurance, that no two right lines can have a common segment; but if we consider these ideas, we shall find, that they always suppose a sensible inclination of the two lines, and that where the angle they form is extremely small, we have no standard of a right line so precise as to assure us of the truth of this proposition. It is the same case with most of the primary decisions of the mathematics.

There remain, therefore, algebra and arithmetic as the only sciences, in which we can carry on a chain of reasoning to any degree of intricacy, and yet preserve a perfect exactness and certainty. We are possest of a precise standard, by which we can judge of the equality and proportion of numbers; and according as they correspond or not to that standard, we determine their relations, without any possibility of error. When two numbers are so combined, as that the one has always an unite answering to every unite of the other, we pronounce them equal; and it is for want of such a standard of equality in extension, that geometry can scarce be esteemed a perfect and infallible science.

But here it may not be amiss to obviate a difficulty, which may arise from my asserting, that though geometry falls short of that perfect precision and certainty, which are peculiar to arithmetic and algebra, yet it excels the imperfect judgments of our senses and imagination. The reason why I impute any defect to geometry, is, because its original and fundamental principles are derived merely from appearances; and it may perhaps be imagined, that this defect must always attend it, and keep it from ever reaching a greater exactness in the comparison of objects or ideas, than what our eye or imagination alone is able to attain. I own that this defect so far attends it, as to keep it from ever aspiring to a full certainty: But since these fundamental principles depend on the easiest and least deceitful appearances, they bestow on their consequences a degree of exactness, of which these consequences are singly incapable. It is impossible for the eye to determine the angles of a chiliagon to be equal to 1996 right angles, or make any conjecture, that approaches this proportion; but when it determines, that right lines cannot concur; that we cannot draw more than one right line between two given points; it’s mistakes can never be of any consequence. And this is the nature and use of geometry, to run us up to such appearances, as, by reason of their simplicity, cannot lead us into any considerable error.

I shall here take occasion to propose a second observation concerning our demonstrative reasonings, which is suggested by the same subject of the mathematics. It is usual with mathematicians, to pretend, that those ideas, which are their objects, are of so refined and spiritual a nature, that they fall not under the conception of the fancy, but must be comprehended by a pure and intellectual view, of which the superior faculties of the soul are alone capable. The same notion runs through most parts of philosophy, and is principally made use of to explain oar abstract ideas, and to shew how we can form an idea of a triangle, for instance, which shall neither be an isoceles nor scalenum, nor be confined to any particular length and proportion of sides. It is easy to see, why philosophers are so fond of this notion of some spiritual and refined perceptions; since by that means they cover many of their absurdities, and may refuse to submit to the decisions of clear ideas, by appealing to such as are obscure and uncertain. But to destroy this artifice, we need but reflect on that principle so oft insisted on, that all our ideas are copyed from our impressions. For from thence we may immediately conclude, that since all impressions are clear and precise, the ideas, which are copyed from them, must be of the same nature, and can never, but from our fault, contain any thing so dark and intricate. An idea is by its very nature weaker and fainter than an impression; but being in every other respect the same, cannot imply any very great mystery. If its weakness render it obscure, it is our business to remedy that defect, as much as possible, by keeping the idea steady and precise; and till we have done so, it is in vain to pretend to reasoning and philosophy.

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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