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Kitcher’s third objection is the Exactness Objection:

How can we resist the challenge that the presented entities do not have exactly the properties we take them to have?... Just as our powers o f ordinary

perception are limited and fallible, so too are our powers of mental perception. Because o f this we cannot assume that mental perception will give us exact knowledge even o f the particular figures we construct. We should concede that we might be unable to distinguish a straight line from one that is very slightly curved. The concession is dangerous. For imagine that we follow K ant’s procedure to arrive at a belief in a geometrical truth. The warranting power o f the procedure can be undermined by experiences involving deceptive

measurement which seem to show that the statement is only a close

approximation to the truth. Given such experiences, it would be rational fo r us to suppose that our mental visual acuity had failed us, and thus to inhibit formation o f the belief.

We can separate out three distinct questions for the neo-Kantian view from this passage. The first is this: why is the conclusion o f Euclid’s argument not true just of those (geometrically imperfect) triangles that exactly resemble the diagram actually drawn, and false o f geometrical triangles (which do not)? The neo-Kantian should accept that the reasoner’s mental powers are finite and fallible, and that a reasoner might in some circumstances be unable to distinguish a straight line from one that is

Kitcher 1984, p. 51 and p. 53f. Emphasis added.

very slightly curved. But even so, she can readily respond to this question by noting the ambiguity in the phrase “presented entities” in the first line: if “presented entities” refers to geometrical objects, then she can deny that geometrical objects as

represented by figures or diagrams may not have exactly the geometrical properties she takes them to have, for reasons already described. If “presented entities” refers to the diagrams or figures themselves, then the neo-Kantian can again deny that

diagrams or figures may not have exactly the physical or visualised properties she takes them to have; for she does not take them to have geometrical properties. As noted, a problem would only arise here if she made the erroneous assumption that she can reason validly by appealing directly to the figure or diagram and trying to “read o ff’ from it properties o f the geometrical object(s) represented.

The second question concerns what conclusion we are supposed to derive from what Kitcher terms the “dangerous concession”: that a reasoner might in some

circumstances be unable to distinguish a straight line from one that is very slightly curved. We can imagine situations in which such an inability to discriminate might indeed be dangerous: for example, given a school teacher’s request for pupils to measure the internal angles of a diagram of a triangle using a protractor, an

approximately correct answer would be highly sensitive to a number of factors: for example, the accuracy o f the drawing process, the flatness o f the surface, and the pupils’ measuring skills. Any deviation from a required norm on these three counts would unsettle the pupils’ reasoning, indeed that o f any human reasoner, to the desired conclusion (cf. the discussion in Chapter 4). But o f course this empirical and inexact process has no bearing on Euclid’s argument, to follow which a quite different type of reasoning is required. In the case of Prop. 1.32, the reasoning operates by leading the reasoner to grasp certain equalities between angles, without regard to the exact size o f those angles. And the reasoning appears to be much more robust than in the empirical case just considered, as a result. Why should this be? In the first place, there is no question o f any measuring here. Secondly, it seems that even a very inaccurate diagram or figure—one in which lines are bent, overlap or do not meet, for example—can nevertheless be taken by a reasoner to represent a geometrical triangle; and an inaccurate implementation of the construction procedure using such a diagram or figure— line CE drawn slightly but noticeably non-parallel to AB, for example— need not affect the reasoning. The diagram below illustrates this point:

c

So the final diagram constructed can be visibly inaccurate, without a reasoner being consciously aware o f or attending to and correcting the inaccuracy, and yet still be of use for purposes o f reasoning.

As yet, then, the concession is not dangerous to the neo-Kantian. What about the case in which the reasoner draws a diagram on what is, without her being consciously aware o f it, a curved surface? Isn’t there a danger here that she might mistakenly infer that the angles o f a triangle sum for example, to more than two right-angles? Note that this is not an unusual or limiting case: this is the situation in which a reasoner who follows Euclid’s arguments normally finds herself. But again, we need to bear in mind the crucial distinction between claims about the structure o f the physical universe, and claims about space as it is represented in Euclid’s geometry. Let us exclude the “reading-off’ view, since it is not a genuine way o f following Euclid’s argument. Then there seems no reason to think that the hidden curvature of the diagram has any bearing at all on the justification for the reasoner’s belief. Recall that the belief does not relate to the physical space occupied by the diagram, but— again—to what the reasoner takes the diagram to represent. And a reasoner who takes the diagram to represent a geometrically curved triangle— a triangle in hyperbolic or elliptical space—has made a mistake. So it does not seem as though the concession is dangerous here either.

What might make the above worry more plausible is that a diagram drawn relatively “large” on a sphere can have internal angles that are plainly more than two right angles, and perhaps closer or equal to three right angles. This does not seem to be Kitcher’s worry: this is not a case o f a reasoner’s inability to distinguish straight from curved, or o f “deceptive measurement”. But it does suggest that the reasoning here

might be rationally defeated by the appearance of the diagram. Hence the third question: is the neo-Kantian view committed to the claim that the justification here cannot be rationally defeated by the visual appearance o f the diagram?

The worry is this. Let us imagine hypothetically that justification can be defeated by the appearance o f the diagram. Then the way seems open for an opponent to claim, by parity o f reasoning, that the neo-Kantian has not made out the distinction between her view and “reading o f f ’ views that appeal to the diagram; if she accepts that belief can be defeated by the appearance o f the diagram, the claim might go, then she must accept that her positive belief is partly underwritten by the mere appearance of the diagram. But if the neo-Kantian accepts this last claim, then the reasoning must be fallacious. And even if she can avoid this outcome, it would seem hard for the neo- Kantian to retain her view that the reasoning is a priori in the face o f this challenge.

So the neo-Kantian must reject the possibility of epistemic defeat. But why think this is difficult? In the quotation above Kitcher talks of “undermining experiences ... which inhibit formation o f the belief’. However, we need to distinguish here between two possible inhibitors o f belief-formation. Psychological blockage or hindering

occurs when, in following an argument, a reasoner has a visual experience o f the diagram as a result o f which she is unable to form (or retain) a given belief, but without having any reason to doubt the (putative) justification o f that belief. Perhaps it is just not clear to her how to proceed. Epistemic blockage or hindering occurs when, in following an argument, a reasoner has a visual experience of the diagram as a result of which she is unable to form (or retain) a given belief, not because of any psychological difficulty, but because the (putative) justification o f that belief is weakened or destroyed by the experience.

Now take the case above, in which a reasoner draws a diagram on a sphere, and let us think o f locations on the sphere analogously with the Earth; as having a “North Pole” at the top as viewed by us, and an “Equator” running around the middle, etc. With this in mind, consider a reasoner who imagines drawing a diagram o f a triangle on such a sphere, with point A on the “North Pole” and the base lying on the “Equator” running from point B on “the Greenwich meridian” to point C at “90 degrees East”. Now imagine that she tries to follow Euclid’s argument in relation to such a diagram.

It will be straightforward for her to extend the base line BC further around the “Equator” to a point D (provided she does not go all the way round), as requested by the first step of the construction procedure. But how to draw a line through C parallel to AB? There is no way to do this: a line segment through C would have to be extendable to a great circle parallel to the great circle extending AB. But two great circles cannot be drawn parallel to each other; they must intersect at two points. So the reasoning to line 1 (the claim that ZBAC and ZACE are alternate) cannot

proceed; the reasoner has had a psychologically blocking experience. Such a reasoner can psychologically “unblock” herself by visualising a Euclidean plane figure o f a triangle, and reason on that basis.

Note that a reasoner’s knowledge (or rather, meta-knowledge) that she has been psychologically hindered or blocked in an earlier inference can rationally ground a worry about the epistemic reliability o f later inferences; the worry may be that she has failed to take in all the epistemically relevant information as a result. But this is not a case of epistemic defeat, as defined above; it is not a situation in which a visual experience of the diagram is defeating the justification o f a given belief. And what the example above brings out is the primacy of the text o f the argument over the diagram; it is this that prevents the non-existence of angle ACE in the case above from having epistemic force. If the representational properties of the diagram were not specified by the text, then inconsistencies between the diagram and the text could in principle count against justification. But since the function o f the diagram is to represent a situation described in the rubric o f the argument, only two possibilities exist: either it does so, in which case the argument as such may proceed; or it does not do so, in which case there is psychological blockage or hindering. In neither case is there epistemic blockage.

8.6 Summary

The goal o f this chapter has been to show that we can make room for a positive neo- Kantian view o f our target reasoning. The chapter first identified the neo-Kantian view, and distinguished it from Kant’s own views. It then reviewed in detail three

well-known lines o f potential criticism of the neo-Kantian view, beyond the Generality Objection discussed in Chapter 7.

1. The Irrelevance Objection suggested that the neo-Kantian could not treat the generalisation in Euclid’s argument as conceptual, on pain o f making the role o f intuition irrelevant, and the figure or diagram epistemically unnecessary. However, this criticism misses its intended target, which is not the acquisition of geometrical knowledge in general, but its acquisition using the kind(s) of reasoning required to follow Euclid’s argument. Once this further constraint is acknowledged, the claim of epistemic necessity for the diagram (or figure) is very plausible. By distinguishing between spatial and non-spatial concepts, I argued that neo-Kantian view can both treat the generalisation as conceptual and retain the idea that there is something intuitive about this reasoning, without conceding that the diagram is irrelevant to it. But this in turn requires a more detailed treatment o f the relevant concepts; I turn to this in Chapter 10 below.

2. The Practical Impossibility Objection suggested that, in the absence of infinite powers o f visualisation, the neo-Kantian lacked an account of the justification of general mathematical claims. In response, I argued that the neo-Kantian can argue for the general claim by using the accepted principle of

mathematical induction.

3. Lastly, the Exactness Objection raised a series o f worries relating to the justification afforded by reasoning using the figure or diagram. The neo- Kantian response was to insist on two distinctions: one between the properties o f the representing diagram or figure and the properties of the geometrical entity or entities represented; and one between psychological and epistemic blockage. Once these are understood, Euclid’s reasoning was found to be significantly more robust than a parallel but distinct empirical process of reasoning in geometry, consideration of which may erroneously be part o f the motivation for these worries.

The effect o f this discussion is to start to bring out the character of, and the specific commitments incurred by, what I have termed the neo-Kantian view. In particular, it highlights the extent to which the neo-Kantian view recognises a spatial aspect to certain geometrical concepts, and an underlying commitment to what might be termed a fine-grained approach to such concepts, within which the spatial/non-spatial

distinction can be articulated. It is here, in the specification o f concepts, that the deepest contrast is to be found with Kant’s own views.

In the next chapter, I turn to consider the logic o f Euclid’s argument.