So far much o f the discussion has focused on Kant’s view as it bears on the
interrelated issues o f the representational scope o f the diagram and the generality of the justification offered by Euclid’s argument. These topics have received some
attention in the Kant l i t e r a t u r e / B u t there is an important further aspect, which bears on the general nature of the inferences relating to the diagram, and this has received much less attention from Kant’s commentators to date; perhaps from a faulty background assumption that these inferences are not, or cannot be, valid or
knowledge-yielding. ^
We can approach this topic by asking what Kant means by his insistence in the Doctrine o f Method that the geometer follows Euclid’s argument by reasoning “through a chain o f inferences that is always guided by intuition”. One way to understand the notion of “guidance” here might be in terms o f a justificatory appeal by “reading o f f ’ properties from the diagram. Michael Potter comes close to this reading, in suggesting that Kant believed that what underwrites the existence of a constructed point C in the famous equilateral triangle o f Prop. I.l is the fact that the lines o f the diagram or imagined figure actually cross. But if this is just Kant’s view, then it is in serious difficulty from the outset, since we have already noted that the justificatory appeal to the figure by itself is fallacious. On the other hand, if “guidance” is to be taken purely psychologically and as having no epistemic
significance, then it is not clear why we should think o f the diagram as contributing to justification, according to Kant.
Can we do better than this? Kant seems to offer as part o f his explanation of the role of intuition a contrast between the type o f construction to be found in algebra, and that to be found in geometry:
But mathematics does not merely construct magnitudes {quanta), as in geometry, but also mere magnitude (quantitatem), as in algebra, where it abstracts entirely from the constitution of the object that is to be thought on accordance with such a concept of magnitude. In this case it chooses a certain notation for all construction of magnitudes in general (numbers), as well as addition, subtraction, extraction of roots etc., and, after it has also designated the general concept of quantities in accordance with their different relations, it then exhibits all the procedures through which magnitude is generated and altered in accordance with certain rules in intuition; where one magnitude is to
Notably in Friedman 1992; but see also Howell 1973, Parsons 1983 Ch. 5 and Postscript, and Smit
2000.
An assumption typically made by exponents o f the “logical” interpretation o f Kant’s philosophy of geometry. I examine some aspects of this interpretation further in Postscript 1.
be divided by another, it places their symbols together in accordance with the form of notation for division, and thereby achieves by a symbolic construction equally well what geometry does by means of an ostensive or geometrical construction (of the objects themselves), which discursive cognition could never achieve by means of mere concepts.
Knowledge of algebra and geometry alike requires intuition for Kant, for both algebraic and geometrical reasoning rest on constructions in pure intuition. But the construction procedures are, he claims, of different kinds: in the case of geometry the procedure is “ostensive”, and the construction is “of the objects themselves”; while in the case of algebra the procedure is symbolic and “abstracts from the constitution o f ’ the object(s) represented.
I interpret Kant’s point here to relate to two different ways in which construction procedures can be used to represent mathematical objects or functions. Symbolic representation uses symbols such as numerals and function signs; and it is indirect, in that the reasoner knows what a certain symbol represents only in virtue of background knowledge as to what assumptions link it with its object or target of representation. These assumptions are conventional in nature: any symbol can in principle, given a suitable set of background assumptions, be taken to represent any object. Thus, for example, all of “x”, and “.” are standardly used to represent multiplication, but so for example could
“a”, “û”,
or“X ”,
if suitable conventions could be established. There is no in-principle constraint (though there may be constraints of ease of use etc.) on the actual form of representation or notation to be adopted. So symbolic representation permits a choice of notation between alternatives, all of which can bear the requisite representational relation to what they represent; and there are different notations available for numbers and arithmetical operations, for example, as Kant would have been aware.Ostensive representation, by contrast, is direct: the diagram must be such that it can itself reliably be recognised as a representation of an instance of the intended
category. This is an important in-principle constraint on ostensive representation, which has the effect of restricting the availability of alternative representational
Ibid.; emphasis added in the last sentence.
I have one serious reservation as to Kant’s account of symbolic construction, but this does not bear on the distinction as such, and need not concern us here. See Postscript 3 below.
forms; there will be few if any alternatives to a diagram or figure o f a triangle as a means to represent a geometrical triangle ostensively. Contextual assumptions notwithstanding, ostensive representation is not a merely conventional relation between diagram and object(s).
Euclid’s geometry, as given in what I have termed the Euclidean Presentation, does not use purely ostensive representational forms. First, it uses letters as labels, and these are symbolic, not ostensive. Secondly, words o f natural language (as in the sentences of the Propositions) are generally symbolic in the sense above. Euclid’s geometry is, then, strictly a mixed or heterogeneous system, in that it uses both ostensive and symbolic representations, the latter including the words o f the text. But the use o f diagrams gives it a heavily ostensive character.
We can now see what Kant appears to have in mind in referring to ostensive
constructions as constructing “the objects themselves”. An initial instruction bids the reasoner draw a diagram o f the relevant object(s): in Prop. 1.32, this is in the opening sentence of the Setting-Out (“Let ABC be a triangle”). Every construction procedure used to draw a given diagram can in principle be used to depict each o f the objects represented by the diagram. There is thus, given the relevant background
assumptions, a correspondence between the elements and properties o f the diagram and those o f its target or object(s): a correspondence that preserves in the diagram what we should now term the mathematical structure o f the object(s). The reason why Kant claims that ostensive presentation is of “the objects themselves” is that this structure preserves all the relevant geometrical properties o f the class o f geometrical objects in question. Thus in looking at the diagram it is in each case “as i f ’ a reasoner is looking at a geometrical object (or spatial configuration of geometrical objects). Such reasoning can be general in that it relates to a class o f objects all o f which conform to conditions o f construction, and thus possess the relevant structure. So any claim made by reasoning with the diagram will apply, mutatis mutandis, to any member of that class.
What, then, is the force o f Kant’s insistence on the “guiding role” o f intuition in geometrical reasoning? Should we understand this guidance psychologically, as
referring to the role o f the diagram in prompting inferences, or epistemically, as referring to the role o f the diagram in contributing to justification?
I suggest that Kant has both of these roles in mind. Psychologically, he reminds us that on drawing the auxiliary line the geometer “sees that here there arises” an angle adjacent to an existing angle— and it is by seeing the alternate and opposite angles on the diagram that the geometer is prompted to apply the relevant rules (e.g. Prop. 1.29). Epistemically, Kant stresses that it is intuition, via ostensive construction, that permits the reasoner to gain an understanding of the diagram on which the diagram can serve to justify claims about geometrical objects. For the diagram conveys structural information about the objects it represents, when understood in accordance with appropriate background conventions, and this feature is exploited by the reasoner in making inferences with the diagram.
On this reading, Kant takes the reasoning here—including the reasoning with the diagram—to be valid reasoning. Euclid’s argument is then “illuminating” because the diagram is used ostensively, to represent geometrical “objects themselves”; and it is valid “universally” because a reasoner’s grasp o f construction procedures underwrites the argument’s general conclusion. We can now better appreciate, for Euclid’s geometry at least, why Kant claims that it is construction— and in particular ostensive construction—that generates the “apodeictic” certainty o f mathematics. For it is in ostensive construction, and in the capacities that underlie it, that Kant locates the source o f the illumination, directness and generality o f this kind o f reasoning.
But we should again note that there are two critical gaps in Kant’s exposition: first, he needs to show, not merely that a diagram can preserve structural information about its objects as described, but that the inferences with the diagram in Euclid’s argument are valid inferences; second, he needs to show how a reasoner can be justified, indeed justified a priori, in believing the angle sum claim in full generality. I return to these
questions in Chapter 10.
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