Notes on Hume’s Treatise

by G. J. Mattey

Book 1
Of the UNDERSTANDING
PART 3
Of knowledge and probability.

Sect. 1. Of knowledge.

1. The author recounts the seven kinds of philosophical relations that were discussed in Part 1, Section 5. These are:

The kinds of relations on this list are then divided into two classes: An example of a relation in the first class is that of the equality of the three angles of a triangle and two right angles: a relation of proportion in quantity. So long as we have in mind a triangle and two right angles, the relation of equality cannot change. An example of a relation in the second class is that of the pair contiguity and distance between two objects. The two may be touching and then removed to a distance away from each other, all without there being any change in the objects themselves. The author notes that the place of the two “depends on a hundred different accidents, which cannot be foreseen by the mind.” A second kind of relation of the second class is that of identity. In Part 1, Section 5, this relation was said to hold in the strictest sense of an object when it endures in a constant and unchanging way. In Part 4, Section 6, where the author discusses personal identity, identity is attributed to “any mass of matter, of which the parts are contiguous and connected, . . . provided all the parts continue uninterruptedly and invariably the same, whatever motion or change of place we may observe either in the whole or any of its parts.” Thus, if we have the idea of a composite object, the ideas of its components might remain the same, while their relation of contiguity and connectedness might change. For example, it might be broken in half, in which case we have two objects and not an enduring identical object. Quite unexpectedly, the author does not describe identity, and the reason it belongs to the second class of relations, in this way. Here is what he says: “Two objects, tho’ perfectly resembling each other, and even appearing in the same place at different times, may be numerically different.” He seems to mean that there is nothing in the ideas of the two objects which is sufficient to constitute their identity, since even perfect resemblance is not enough. They may, for example, occupy different places at the same time. What is not clear is why it is that a change in the relation of identity could take place without change in the ideas of the two objects. Perhaps this means that the two perfectly resembling objects could interpenetrate and occupy the same place, thus passing from being different to being identical without any change in themselves. But this would conflict with the author’s suggestion about the nature of interpenetration in Part 2, Section 4, where it is described as the annihilation of one body and the preservation of the other. Annihilation would certainly be a “change in the ideas.” Moreover, the two objects would not thereby become identical. So it seems that the example in the present section is not the best one the author could have offered. The third and final kind of relation in the second class is that of causation. No inspection of the ideas of the cause and the effect can reveal “the power, by which one object produces another.” Something else is required, and it can only be found in experience, “and not from any abstract reasoning or reflection.” In no cases whatsoever, even the most simple one, can this power “be accounted for from the qualities of the objects, as they appear to us; or which we coul’d foresee without the help of our memory and experience.”

2. Only the philosophical relations which depend solely upon the ideas of what is related qualify as “objects of knowledge and certainty.” The four kinds of relations in this first class are: resemblance, contrareity, degrees in quality, and proportions in quantity and number. The first three of these are objects of “intuition,” rather than “demonstration,” since they are “discoverable at first sight.” Resemblance will strike the mind in such a way that detecting it “seldom requires a second examination.” The same holds for contrareity, as in the case of existence and non-existence (described in Part 2, Section 6), which “destroy each other, and are perfectly incompatible and contrary.” This is something no one can doubt. The case of degrees of quality (color, taste, heat, cold) is not so clear cut. Take heat as an example. One would have a difficult time distinguishing the impression of heat of 100° Fahrenheit with an impression whose degree is 101°F. On the other hand, there seems to be no problem in determining that the degree of heat in an impression at 101° is greater than that of one of 70°. “And this decision we always pronounce at first sight, without any enquiry or reasoning.”

3. In the case of proportions of quantity or number, the situation is similar to degrees of quantity. There will be some cases where the proportion (one quantity is greater than another) can be detected at a glance, as when one is considerably greater than the other. In cases of equality or exact proportions (say, of two lines), for the most part “we can only guess at it from a single consideration.” There will be times when “we perceive an impossibility of falling into any considerable error,” where “very short umbers, or very limited proportions” are detectable at first sight. For the rest, either we have to be very loose with our judgments (one line looks to be about a foot longer than the other) or “proceed in a more artificial manner” (that is, resort to artifice, such as measuring or applying mathematical techniques).

4. The author now recounts the conclusions drawn in Part 2, Section 4 that are relevant to the present discussion. Geometry is the art (or artificial means) by which the proportions of figures are determined. It never attains perfect precision and exactness, though “it much excels both in universality and exactness, the loose judgments of the senses and imagination.” The “first principles” of geometry are “drawn from the general appearance of objects,” which is not adequate to allow us to deal with very small objects. The author recounts his example of the intersection of two lines which meet at a very inclined angle. This appearance is such that it does not allow confirmation of the general geometrical principle that no two straight lines can have a segment in common. “’Tis the same case with most of the primary decisions of mathematics.”

5. Algebra and arithmetic are the only two sciences which allow intricate chains of argument while preserving “perfect exactness” in a way that we can be certain of the results. The reason is that they deal with discrete quantities, which allow precise comparison of equality and proportion (two quantities are equal if and only if they are made up of the same number of units, which has come to be known as “Hume’s principle.”) Because this standard cannot be applied in geometry, due to the infinite divisibility of extension, “geometry can scarce be esteem’d a perfect and infallible science.”

6. At this point, the author considers an objection to his assertion that geometrical judgments are more universal and exact than those based merely on the senses or imagination. The problem with geometry is that its basic principles are drawn from appearances, which in turn are present to the senses. Thus, it is argued, the inexactitude of the appearances must be transferred through the chain of geometrical reasoning to its conclusions. The author allows that the defect in geometry keeps us from full certainty, but he notes a way in which geometry maintains its superiority over the bare use of the senses. It allows us to draw exact conclusions far beyond any that we would be able to draw from sensible appearances without its aid. For example, the senses could never judge or even conjecture that the angles of a chiliagon (one-thousand-sided regular figure) is equal to 1,996 right angles, though this is deducible from geometrical principles. The only problem with this conclusion is that it is too exact, because the basic principles are too exact for the appearances on which they are based. On the other hand, these appearances are “the easiest and least deceitful,” as well as being relatively simple. As such, they cannot lead us into any great error.

7. The author concludes the section with a second observation about demonstrative reasoning, as suggested by the case of mathematics. According to the author, all mathematical principles are based on appearances. In contrast, most mathematicians have claimed that the ideas that are the objects of mathematics “are so refin’d and spiritual a nature” that they are understood only by “a pure and intellectual view, of which the superior faculties of the soul are alone capable.” The intellect is claimed to be capable of the perfect precision that the senses and imagination cannot attain. Most parts of philosophy are infected with this same view. It is the basis of the standard doctrine of abstract ideas, according to which we can have an idea of a triangle, for example, without its being endowed with any particular properties, such as congruity of angles, particular length, or proportion of sides--properties which must be present in any image of a triangle. The reason the philosophers embrace the alleged pure and intellectual view is that it covers up the absurdity of their reasoning. Their conclusions cannot be criticized except through the use of clear ideas, but the alleged abstract ideas to which they appeal are “obscure and uncertain.” The “artifact” of abstract ideas is destroyed by the copy principle: “that all our ideas are copy’d from our impressions.” Because all impressions are “clear and precise,” the ideas which are copies of them are clear and precise as well, although they are weaker and fainter. For this reason, our ideas are “dark and intricate” only through our own fault, as in the case of the invention of abstract ideas. Now it is the case that insofar as our ideas are weaker than our impressions, they may be hard to make out with exactness. This problem can be remedied, at least to some extent, by paying steady attention to them. “And till we have done so, ’tis in vain to pretend to reasoning and philosophy.”

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