Postscript 1 : Kant and the “Logical Interpretation”

The interpretation advanced here treats the Doctrine o f Method seriously: as, in effect, advancing a coherent and in many respects rather persuasive claim for the epistemic value o f a particular kind o f reasoning. By contrast, the consensus amongst Kant’s commentators has long been that Kant’s claims in the Doctrine of Method are

obviously mistaken. This view was famously expressed by Bertrand Russell, as noted in Chapter 1 :

Kant, having observed that the geometers o f his day could not prove their theorems by unaided arguments, but required an appeal to the figure, invented a theory o f mathematical reasoning according to which the inference is never strictly logical, but always requires the support o f what is called intuition. Though this view has been extremely influential among philosophers o f many different stripes, it has been developed in recent years into what has been termed the “logical interpretation” o f Kant’s philosophy of geometry. On the logical

interpretation, the function o f intuition for Kant here is, in effect, to compensate for deficiencies in the then-available logic. In this Postscript, I want to contrast this interpretation with the view developed above.

The logical interpretation has been developed most explicitly in the work o f Michael Friedman. Friedman claims, of Kant’s discussion o f the Euclidean Presentation:

In contending that construction in pure intuition is essential to this proof, Kant is making two claims that strike us as quite outlandish today. First, he is claiming that (an idealised version) o f the figure we have drawn is necessary to the proof. The lines AB, BC, CE, and so on are indispensable constituents; without them the proof simply could not proceed. So geometrical proofs are themselves spatial objects. Second, it is equally important to Kant that the lines in question are actually drawn or continuously generated, as it were. Proofs are not only spatial objects, they are spatio-temporal objects as w ell... Kant’s conception of geometrical proof is of course anathema to us. Spatial figures, however produced, are not essential constituents o f proofs, but, at best, aids (and very possibly misleading ones) to the intuitive comprehension o f proofs. Whatever the intended interpretation o f the axioms or premises of a geometrical proof may be, the proof itself is purely “formal” or “conceptual” object; ideally, a string of expressions in a given formal language.

Although Friedman regards the claims in the Doctrine o f Method as mistaken, his overall approach is sympathetic to Kant. It recalls that, on a standard view, the reason for many if not all o f the logical gaps in Euclid’s geometry is that it lacks existence

Russell 1919, p. 145.

See especially Friedman 1992, Chapters 1 and 2; and Friedman 2000.

Friedman 1992, pp. 57-58. I am assuming here that Friedman is speaking “in his own voice”; that is, that he holds the views described here.

axioms, a theory o f order governing points in the line, and modem concepts of continuity, denseness etc. On Friedman’s view, Kant is implicitly aware that he has no means in the (canonically syllogistic, subject-predicate) logic then available to express certain desired concepts of continuity, infinity and infinite divisibility. The function o f intuition is, in this regard, to permit such concepts to be represented. A concept o f continuity is, the suggestion goes, given representation for Euclid by the motion o f a mathematical point (an idealised stylus) drawing a line; a concept of infinity is given representation by the reasoner’s (idealised) ability to iterate, for example, the production o f a line segment an indefinite number of times by

application o f Postulate 3; and a concept of infinite divisibility is given representation by the (idealised) iterable bisection of a line segment according to Prop. 1.10. These actions all are, or involve, implementations o f construction procedures. Moreover, construction procedures are required, according to Friedman, if diagrams in Euclid are to give rigorous representation of the relevant geometrical concepts. Thus Friedman says o f Euclid’s constmction o f a circle in Postulate 3:

The underlying idea [behind Kant’s claim that this constmction “first generates the concept of a figure” at A234/B287] ... is that the existential proposition corresponding to this constmction—that for any point and any line there is a circle with the given point as centre and the given line as radius— cannot be conceptually expressed for Kant. In mere syllogistic logic this existential proposition cannot, strictly speaking, even be stated (as we would now put it, it involves the form of quantificational dependence VV3). The only way even to think or represent this proposition—so as, in particular, to engage in rigorous geometrical reasoning thereby—is by means o f the constmction itself.^ ^

Thus on this view Kant believes that it is the possibility that the Euclidean

Presentation provides to constmct diagrams on paper or figures in imagination that allows representation o f these foundational geometrical concepts, and other concepts from complex diagrams constmctible fi*om the basic ones; and it is by means of intuition that a reasoner can come to possess these concepts.

On this interpretation, then, the role of intuition for Kant is primarily as a substitute for what would in modem terms be logical forms of representation using

q u a n tif ie r s .T h e stage is now set for Friedman to claim that this view has a crucial drawback, however; one that we can appreciate but Kant could not. This comes in Kant’s mistaken belief that construction procedures are justificationally required to secure the relevant order properties of the geometrical line; mistaken, since these properties can only be rigorously formalised, as discussed, via a modem theory of order in the style o f Hilbert. We can know, thus, as Kant could not, that the diagrams in Euclid are “inessential constituents of proofs, but at best aids... to the intuitive comprehension o f proofs”, which are purely formal objects.

Friedman’s interpretation o f Kant is subtle and illuminating; and I have only been able to sketch a small part of the overall line of thought. Nevertheless, I think this much is enough to suggest that it—and the logical interpretation generally—faces a number of difficulties as an account of Kant’s thinking here, on both internal and external grounds.

The first difficulty relates to a question of exegesis. Friedman introduces his

discussion via a consideration o f the Doctrine o f Method; he uses this to raise modem worries about the order properties o f the line in Euclid, and then seeks to show that these in tum generate questions as to the representation of continuity and infinity for Kant, to which the latter is alive (although perhaps only implicitly) in the Aesthetic and Analytic. But these logical issues, though important, are plainly not ones that Kant has principally in mind in the Doctrine of Method, and do not help, at least in any direct way, to explain his thought there.

By contrast, the Doctrine o f Method passage is much more clearly focused on the

epistemology of the different processes o f reasoning themselves. Kant is asking: given that there are these two apparently different processes o f reasoning—

mathematical and philosophical—how do they differ, and how does each justify and guide us in coming to certain conclusions? What seems to distinguish the geometer’s reasoning is, at least in the first place, that it relates to a particular diagram, has a visual component, and yields “evident and universally valid” conclusions. It is these phenomena that require explanation; and the putative role of intuition in guaranteeing

representation o f certain concepts of continuity etc., while it may be relevant overall, is not the target here. Indeed, as its opening words make clear, the wider point of the Doctrine o f Method as a whole is, precisely, to provide the plan and methods by which the various constituents (the “elements” of pure reason) can be combined into an entire system o f pure speculative knowledge. How they are to do so centrally concerns the epistemology of the methods of reasoning involved.

The second difficulty may seem rather fussy; but it has a wider point. Recall that Friedman interprets the Doctrine of Method passage as making two apparently “outlandish” claims. The first of these is that “(an idealised version) of the figure we have drawn is necessary to the proof. The lines AB, BC, CE, and so on are

indispensable constituents; without them the proof simply could not proceed. So geometrical proofs are themselves spatial objects.” But this is surely not Kant’s claim. Rather, his claim relates to the proof (strictly speaking: argument) as presented by Euclid', he is claiming that the figure is necessary to the Euclidean

Presentation of the argument, that the presentation could not be understood without it. Far from being outlandish, this claim seems to be true, for reasons discussed in Chapter 6. Does it mean that the presentation must be spatial? If the presentation takes a written form and contains a diagram, then it will occupy physical space. If the presentation is simply thought through in the mind, then it will not occupy physical space. But even in this case it is nevertheless plausible that the figure as visualised presents spatial, specifically two-dimensional, information. So Kant’s point here is— in line with the focus of the Doctrine o f Method on the epistemology of reasoning—to focus on the nature o f our interaction with a specific presentation o f the argument. That is, Kant is seeking an explanation that conforms to what he takes to be the facts, and specifically the phenomenology, of the way in which the reasoner can correctly follow Euclid’s argument.

Finally, and most crucially, it remains quite unclear why intuition is supposedly required to compensate for deficiencies in logic. Why, for example, should the absence o f a formal language of quantifier logic within which to represent the

existential proposition corresponding to Postulate 3 have any bearing on the ability of Kant—or a geometrical reasoner—to give that proposition conceptual expression?

the sentence “For any point and any line there is a circle with the given point as centre and the given line as radius”? It is not clear that Kant is moved at all by the

expressive limitations of logic as he then knew it. Quite the contrary: if the reading of Kant’s distinction between ostensive and symbolic construction given here is correct, a sentence such as that above—or a sentence in a formal language of quantifier logic—would be deemed symbolic, not ostensive, precisely because it did not present, and preserve through a given process of reasoning, what we might term the spatial content o f the intuition of the diagram. Moreover, if we regard intuition as a mere substitute for logical representation, the main thrust o f Kant’s insistence on the guiding role o f intuition in geometrical reasoning, which allows the reasoner to construct and reason as to “the objects themselves”, is lost.

Notice that I am not here disputing the positive claim that intuition may in principle be invoked to play a quasi-logical role for Kant in the way Friedman describes. However, on the interpretation I propose Kant’s concerns are not purely logical, and the role o f the diagram is not merely to compensate for deficiencies in his logic. Rather, we need to take what he says in the Doctrine o f Method at face value. If we do so, we can see that Kant is here primarily seeking to explain, not the logical

presuppositions of a Euclidean argument, but the distinctive patterns o f reasoning that occur in relation to the diagrams within the Euclidean Presentation. If this is so, then the role o f intuition in reasoning is distinguishable from any role it may have in, for example, guaranteeing the order properties o f the line; one could, as it were, add Hilbert’s theory o f order (in a suitable form) explicitly as background assumptions to the Euclidean Presentation and still reason wholly or partly in the way Kant has in mind in the Doctrine of Method.