Berkeley and the “Generality Objection”

We can introduce Kant’s view by considering a central objection to any account on which reasoning with a diagram is taken to justify. The Generality Objection is

simply this: How can the reasoner who follows Euclid’s argument be justified in believing that the angle sum claim holds for all triangles?

Berkeley advances this point in the form of an imagined counter to his claim that we cannot have abstract general ideas, in the Introduction to the Principles o f Human Knowledge".

But here it will be demanded, how we can know any proposition to be true of all particular triangles, except we have first seen it demonstrated o f the

abstract idea of a triangle which equally agrees to all? For, because a property may be demonstrated to agree to some one particular triangle, it will not thence follow that it equally belongs to any other triangle, which in all respects is not the same with it. For example, having demonstrated that the three angles o f an isosceles rectangular triangle are equal to two right ones, I cannot therefore conclude this affection agrees to all other triangles which have neither a right angle nor two equal sides. It seems therefore that, to be certain this proposition is universally true, we must either make a particular

demonstration for every particular triangle, which is impossible, or once for all demonstrate it o f the abstract idea o f a triangle, in which all the particulars do indifferently partake and by which they are all equally represented.^®

It will be recalled that in Section 13 Berkeley famously (and perhaps unfairly)^' attacked Locke’s claim that we can form general ideas in geometry, such as a general idea o f a triangle that is (in Locke’s words) “neither oblique nor rectangle, equilateral, equicrural nor scalenon, but all and none of these at once”. As Berkeley notes, this rejection creates an apparent difficulty for his own view o f geometry, for he concurs in the view that geometrical theorems are quite general (“universally true”), and if geometrical arguments do not employ abstract ideas, then it is unclear how they may be general at all. In particular, there seems to be no means in principle to prevent the reasoner from erroneously over-generalising properties from the diagram that relevant triangles may not possess.®^

Berkeley’s response is as follows:

Berkeley 1988, Introduction, Section 16. On Locke here, see Ayers 1991, Chs. 5 and 27.

Note that a reasoner might also over-restrict from the diagram: that is, from following the argument in relation to a diagram o f a rectangular isosceles triangle, she might conclude that all rectangular isosceles triangles had the angle sum property. This would, strictly speaking, be an inaccuracy, but not

To which I answer, that, though the idea I have in view whilst I make the demonstration be, for instance, that o f an isosceles rectangular triangle whose sides are o f a determinate length, I may nevertheless be certain it extends to all other rectilinear triangles, of what sort or bigness soever. And that because neither the right angle, nor the equality, nor determinate length of the sides are at all concerned in the demonstration. It is true the diagram I have in view includes all these particulars, but then there is not the least mention made of them in the proof o f the proposition. It is not said the three angles are equal to two right ones, because one of them is a right angle, or because the sides comprehending it are o f the same length. Which sufficiently shows that the right angle might have been oblique, and the sides unequal, and for all that the demonstration have held good. And for this reason it is that I conclude that to be true o f any obliquangular or scalenon which I had demonstrated of a particular right-angled equicrural triangle, and not because I demonstrated the proposition o f the abstract idea of a triangle.^^

Here Berkeley’s point seems to be this. Imagine someone who follows Euclid’s argument in relation to a diagram of a right-angled isosceles triangle, perhaps as below:

For such a reasoner, there is no valid generalisation available merely from a visual experience o f this diagram to a claim about all triangles, i.e. including all those that are neither right-angled nor isosceles, for reasons noted in Chapter 4. However, though the diagram above is o f a right-angled isosceles triangle, these particular properties “are not mentioned in the proof of the proposition”. That is, it is only qua triangle that it plays any role in Euclid’s argument.

Berkeley says that the argument “does not mention” the specific properties of being right-angled and isosceles, but this is insufficient, since an argument may fail to mention a property or claim and yet still implicitly rely on it. But bearing this in mind, his point is surely that Euclid’s argument may be understood as justifying the angle sum property generally, because it does not rely on any claim about the triangle represented that is not a general property of all triangles. By contrast, the reasoner

who, at the conclusion of the reasoning, takes the concluding claim only to apply to those triangles that the diagram visually resembles, has failed to follow Euclid’s argument correctly.

Now this line o f thought is hardly original to Berkeley: it can be found in Proclus.^"^ It offers an intriguing and potentially plausible account o f how the Generality Objection can be met in relation to Euclid’s argument.^^ However, Berkeley’s account itself might be thought to face a serious difficulty here.^^ For his account is vague at the crucial point, as to what if anything determines the representational scope of the diagram. There seem to be three candidate answers in principle here. The first is that it is the visual features o f the diagram alone that determine its scope; but Berkeley rejects this fallacious appeal to the diagram, as we have noted. The second is that the diagram’s scope is determined by the intentions o f the reasoner, and these might in principle extend to triangles that the diagram did not visually resemble. But this is patently insufficient, since such intentions may vary from reasoner to reasoner; what is required is some independent yardstick that can be reliably used to determine scope by any suitably informed reasoner. The third is that the scope of the diagram is determined by the construction procedure specified in the text o f the argument. But then it starts to seem as though Berkeley is tacitly appealing to an abstract general idea o f a triangle. For it is quite unclear in virtue o f what, if not abstract ideas, a reasoner may grasp ex ante the generality o f a given construction procedure.