DISEÑO Y ELABORACIÓN DE LA TARJETA DE CONTROL DE SECUENCIA.

The second o f Kitcher’s objections is the Practical Impossibility Objection. This runs as follows:

How do we determine that sequences o f presentations which we cannot in practice achieve are in principle possible for us? ... Kant claims that pure

intuition can yield the knowledge that line segments are infinitely divisible. Now it is evident that we cannot attain this knowledge by observing a line

segment infinitely divided. So what Kant must intend is that we give ourselves a sequence o f presentations, showing a continued process of subdivision. Since there are practical limits on our ability to do this, we shall face an awkward question: are these limits reflections o f a structural property of experience? To resolve this issue we need, again, that same insight into the structure o f experience which pure intuition was supposed to provide.

This objection refers to Proposition 1.10 in Euclid, which provides a construction procedure for bisecting a given line. Now on Kitcher’s reading of Kant, geometrical truths are “about some particular feature of the world—that feature in virtue of which they are true”.^^^ The neo-Kantian, by contrast, takes Euclid’s geometry to be a piece of pure mathematics, as I have mentioned. In relation to the neo-Kantian view, the Practical Impossibility Objection asks whether following Euclid’s argument can >ield

a priori knowledge o f a general property of finite geometrical lines—here, the property o f being infinitely divisible—given that we cannot perform the apparently requisite infinity o f acts o f construction. We cannot actually draw, and it seems we cannot visualise, the infinite divisibility of a finite geometrical line. So how can drawing diagrams or visualising figures here contribute to the justification of a geometrical belief?

Questions o f infinite divisibility are complex, and some way removed from the argument of Prop. 1.32; so it could be readily argued that little in the main line of our discussion hangs on this worry. But the general worry is sufficiently relevant to our wider concerns to deserve consideration here.

In fact, there is a fairly straightforward response to be made: the reasoner can use mathematical induction. Recall that the “weak” principle o f induction on the positive integers states that for any property P, in order to prove that all numbers have P, it is sufficient to prove two things: first, that the number 1 has P; and secondly, that for every positive integer n, if n has the property P, then its successor n + 1 also has the property P .‘^’ Thus, where “n” ranges over the positive integers, the inference is of the form:

Kitcher 1984, p. 51.

Kitcher 1975, p. 29. I discuss Kitcher’s argument for this view, and argue that it again misreads Kant, in the Postscript to this chapter.

PI

Vn(Pn -> (Pn + 1))

Vn(Pn)

Let Pn be defined as the property for any line L of having 2" parts, where parts have positive length. Then the argument can proceed by induction on the positive integers. The basis step is given by Euclid’s construction: Let a visualised line represent L; then the construction given in Prop. 1.10 shows that L has 2 (= 2^) parts. The induction hypothesis is: Let L have 2" parts. Then the argument runs:

Let I be any part of L. Letting the visualised line represent /, Euclid’s construction shows that I has 2 parts.

So each o f L ’s 2" parts has 2 parts. But any part o f a part o f L is a part o f L. So L has 2 x 2 " parts, i.e. 2”^* parts So, by induction, for any n, L has 2" parts.

Hence, for any finite line segment L, for any n, L has 2" parts.

This establishes the desired conclusion, and it does so by a process that involves reasoning with a visualised figure. Note that a reasoner can follow a very similar process in relation to a line drawn on paper. But the conclusion as to infinite

divisibility will not relate to the physical composition of the line that has been drawn; it would be irrelevant to object that, at some suitably sub-atomic level, the line might “run out o f parts”, so to speak. Rather, the reasoner is using a physical line to

represent a geometrical line.

In relation to Prop. 1.10, the neo-Kantian claim is that here too— as in the earlier discussion o f Prop. 1,32—what justifies the general conclusion is first, that this is valid reasoning, which includes reasoning with the diagram, and secondly, that the reasoner’s grasp o f the relevant construction procedure is sufficient to justify her in

believing that the conclusion holds for all such lines. So, provided that these claims as to validity and generality can be made good, the fact that she may not actually be able to bisect the physical line precisely by using the procedure, nor know whether she has done so, is irrelevant; she can be justified in believing that such a procedure, applied to the geometrical line she takes the diagram to represent, would bisect it. I shall argue for the validity and generality claims in relation to Prop. 1.32 in Chapter 10.

I suggest, then, that the neo-Kantian view can readily meet the Practical Impossibility Objection.