Real numbers and infinity ‘in the small’

We saw above something of Wittgenstein‘s quarrel with Cantor about the concept

‗real number‘, and connected this to questioning the development of Cantor‘s notion of infinity – ₀ and so on. We can link this with the development of incommensurability and the irrationality of 2.

Here is a diagram:

I have copied this from Wittgenstein‘s RFM. (See p. 291) He comments on it thus:

It is by combining calculation and construction that one gets the idea that there must be a point left out on the straight line, namely P, if one does not admit 2 as a measure of distance from O. ‗For, if I were to construct really accurately, then the circle would have to cut the straight line between its points.‘

And he goes on to say,

This is a frightfully confusing picture …

What is the application of the concept of a straight line in which a point is missing?!

The application must be ‗common or garden‘. The expression ―straight line with a point missing‖ is a fearfully misleading picture. The yawning gulf between illustration and application.

The ‗straight line with a point missing‘ is the (Wallis‘s) number line. The point that has gone missing is that representing the square root of 2, and the diagram shows how we can apply Pythagoras‘ Theorem to suggest the place where the missing number should go. Draw on the axes, as shown, a right-angled isosceles triangle of unit side, one unit side being placed along the horizontal axis. The hypotenuse of this triangle can then be transferred onto the horizontal … and its length is 2. The

‗number‘ is missing from the line precisely because 2 is irrational or incommensurable with the unit length – it is this that Wittgenstein points to in talking of ‗2 as a measure of distance from O‘. There is no (rational) number of units that measures the distance from O to the point at which the arc (of radius 2 by Pythagoras‘ Theorem) cuts the line. If all the numbers are rational then indeed, it seems, we have a ‗straight line with a point missing‘.

Wittgenstein describes this picture as ‗fearfully misleading‘. It will help us see what he means if we relate it to our earlier discussion. Recall the choice of criteria involved in deciding whether we should accept 2 as a number in good standing in spite of its irrationality, or alternatively deny that the diagonal of a unit square has a length that is expressible numerically. The picture seems to make the choice for us.

Looking at the picture in the knowledge that there is no rational number of units from O to P, it can seem that nevertheless there is a point there – in a gap just where the arc cuts the line. The idea that we can find this gap there, as it were waiting to be filled, connects plainly with the notions that the real number 2 can be discovered and that there is a matter of fact about whether there really is such a number.

One particular standard way of developing the notion of the real numbers involves taking, not the ‗cut‘ itself as point P in the diagram above, but rather thinking of such a cut as determining the number in terms of the two sets of numbers above and below. I am going to look at some passages where Wittgenstein takes this on. Here he can best be read as tackling the idea as it is expressed by his contemporary G.H.

Hardy in what was for a long time the standard textbook for mathematical analysis in English, Hardy‘s A Course of Pure Mathematics.¹¹¹ Thus, for example, here is Hardy considering the example of 2,

The square of any rational number is either less than or greater than 2. We can therefore divide the positive rational numbers … into two classes, one containing the numbers whose squares are less than two, and the other those whose squares are greater than two. (Hardy 1952 p. 8)

Hardy goes on to describe such a mode of division of the set of rational numbers into a lower and an upper set as an example of a ‗section‘. (op. cit. p. 11) The two properties, x² < 2 and x² > 2, are such that every rational number possesses one of them, and no rational number satisfies both. They define a section in Hardy‘s terminology, as will any pair of mutually exclusive properties defined on the set of rational numbers such that every rational number has one or other of the properties.

Now some such pairs of properties define lower or upper sets with, respectively, a greatest or least member.¹¹² (Consider x2 and x >2, for instance.) Such pairs of

111 I have no contemporary evidence, but it seems at least plausible reading the relevant sections of RFM alongside Hardy that Wittgenstein had Hardy‘s book to hand as he wrote. I offer some brief references to this below.

112 See op. cit. p. 12 for a proof that these are exclusive options, if that is not obvious.

properties can be associated with, or correspond to, that greatest or least rational number. This means, according to Hardy, that

… we are almost forced to a generalisation of our number system. For there are sections (such as that [derived from x² < 2 and x² > 2]) which do not correspond to any rational number. The aggregate of sections is a larger aggregate than that of the positive rational numbers; it includes sections corresponding to all these numbers, and more besides. It is this fact which we make the basis of our generalisation of the idea of number. (Hardy 1952 pp. 13-14)

That is, according to Hardy, and in terms I have used to characterise the conceptual development of our number system, consideration of the notion of a ‗cut‘ or

‗section‘ (‗almost‘) forces the choice of criterion on us. It is clear, however, that this

‗forcing‘ is non-mathematical – it derives, as Hardy himself admits, from the

‗geometrical representation‘ – from the ‗picture‘ Wittgenstein draws, a picture that Hardy also refers to in getting the development up and running:

The result of our geometrical representation of the rational numbers is therefore to suggest the desirability of enlarging our conception of ‗number‘ by the introduction of numbers of a new kind. (op. cit. p. 7)

Hardy – and the standard development – goes on to prove ‗Dedekind‘s Theorem‘, that the real numbers so defined as sections of the rationals (as ‗Dedekind cuts‘) form a continuum. Dedekind‘s Theorem, indeed, is a part of mathematics, as is the completeness of the set of real numbers as Hardy characterises it. It would be tempting to think that Dedekind‘s Theorem guarantees that there are no ‗further gaps‘ in the (Wallis‘s) number line once the ‗gaps‘ like that at irrational points like

2 have been filled. This would be a mistake, as Hardy himself carefully admits:

It is convenient to suppose that the straight line … is composed of points corresponding to all the numbers of the arithmetical continuum and no others.* …

* This supposition is merely a hypothesis … we use geometrical language only for purposes of illustration … (op. cit. p. 24)

Now, coming back to Wittgenstein, we have seen him to characterise illustrations in this arena as ‗misleading‘. And, he says,

The picture of the number line is an absolutely natural one up to a certain point; that is to say so long as it is not used for a general theory of real numbers. (RFM p. 286)

It may be tempting to read Wittgenstein on Cantor and Dedekind/Hardy as somehow challenging the mathematics of real numbers. However, that is not his intent.

Remember Philosophical Investigations §124,

Philosophy … also leaves mathematics as it is …

We are philosophically misled by the way illustrations work on our understanding here, Wittgenstein suggests. For example.

The misleading thing about Dedekind‘s conception is the idea that the real numbers are there spread out in the number line. They may be known or not; that does not

matter. And in this way all that one has to do is to cut or divide into classes, and one has dealt with them all.(RFM p. 290)

The idea that the numbers – real irrational as well as rational – are in some sense already there, waiting our discovery of them, is what Wittgenstein challenges. And this idea, of numbers somehow given in extension prior to the development of the mathematics that deals with them, is one that he points out as fostered by the manner of development Hardy espouses.

I mentioned at the end of the last section that Wittgenstein, as we saw Wallis and some of his peers did, emphasised the importance of applications of mathematics for determining the direction of determination – what I characterised above in terms of criterial choice – of the concepts of mathematics. Here he is contrasting ‗misleading‘

illustrations of analysis with ‗essential‘ illustrations that are at the same time applications:

The geometrical illustration of Analysis is indeed inessential¹¹³; not, however, the geometrical application. Originally the geometrical illustrations were applications of Analysis. Where they cease to be this they can be wholly misleading.

What we have then is the imaginary application. The fanciful application.

The idea of a ‗cut‘ is one such dangerous illustration.

Only in so far as the illustrations are also applications do they avoid producing that special feeling of dizziness which the illustration produces in the moment at which it ceases to be a possible application; when, that is, it becomes stupid. (RFM p. 285)

What might ‗illustrations [that] are also applications‘ be? Here is one such. Think of a standard way of illustrating mathematical analysis‘ way with instantaneous rates of change – via the tangent to a curve at a particular point. An illustration of how we find the gradient of this tangent might well go via a consideration of various secants to the curve drawn at the point of tangency, successive secants approaching closer and closer to the tangent as their points of intersection with the curve approach each other closer and closer. We can see such successively-drawn secants as heuristic in visualising the limiting process that gives us the gradient of the tangent on the curve.

Below are some snapshots of an animation illustrating this process as graphed on Cartesian axes for the function f (x) = x²; the point of tangency is (0.8, 0.64):

113 This is one of those instances where we might usefully think of Wittgenstein as responding directly to Hardy in A Course of Pure Mathematics. See, for instance, Hardy 1952: ‗It is convenient, in many branches of mathematical analysis, to make a good deal of use of geometrical illustrations. The use of geometric illustrations in this way does not, of course, imply that analysis has any sort of dependence upon geometry: they are illustrations and nothing more …‘ This is a point Hardy reiterates several times (e.g. P. 24, P. 30, …).

Wittgenstein looks for a diagnosis of where we might be misled, along with Hardy, by such illustrations; not mathematically, to be sure, but philosophically.

-2 -1 1 2 3 4 2

4 6 8 10

gradient 3.8

-2 -1 1 2 3 4

2 4 6 8 10

gradient 3.47

-2 -1 1 2 3 4

2 4 6 8 10

gradient 3.25

=

=

=

-2 -1 1 2 3 4 2

4 6 8 10

gradient 2.81

-2 -1 1 2 3 4

2 4 6 8 10

gradient 2.48

-2 -1 1 2 3 4

2 4 6 8 10

gradient 2.15

=

=

=

To reiterate, this geometric illustration is inessential to the actual analysis, while the geometric application – of finding the gradient of the tangent – is essential to what is going on. The mathematical illustration – the geometry – at the same time as suggesting the way the mathematics of a rate of change might best be formalised and worked out, does satisfyingly solve the problem of finding a tangent at a point on the curve.

More than that, though, what this application of the limit process to the problem of finding a tangent makes clear is the way in which we extend our concept of a rate of change as a ratio of intervals (an average rate of change, e.g. the average speed over a given time-interval) to an instantaneous rate of change (a rate of change at a point, e.g. the speed at a given instant of time). It may be possible to see this as a discovery, perhaps, of what instantaneous rates of change actually amount to … but seeing the development in action like this certainly predisposes us to see in this case at least the possibility that what counts as an instantaneous rate of change may well depend on the determination – extension in a different sense – of the concept of a

-2 -1 1 2 3 4

2 4 6 8 10

gradient 1.82

-2 -1 1 2 3 4

2 4 6 8 10

gradient = 1.6

=

rate of change rather than a discovery of the fact of what an instantaneous rate of change really is.

That is a good example of how an illustration can be at the same time an application.

The ‗yawning gulf between illustration and application‘ is bridged in this case. To spell it out again: finding the gradient of a tangent at a point is at the same time an application of the analytic process of differentiation as well as an illustration of that very process and why we take the process to be what it is. But the idea of a cut as determining the concept of a real number, we have seen Wittgenstein to claim, is

‗dangerous‘ in that it has no aspect of possible application built in to it. More, and worse, such free-standing illustrations predispose us to see mathematical concepts – of real numbers in this case – as given somehow in extension prior to any application of concepts involving them – a mistaken way of seeing that Wittgenstein convicts Cantor, Dedekind, Hardy – all of us – as being tempted into and as leading to further philosophical error.

With this examination of ‗infinity in the small‘, we have come almost full circle – back to questioning Cantor‘s ‘hocus pocus‘. We go wrong, I am suggesting with Wittgenstein, if we consider the development of mathematical (and other) concepts, including those of number and infinity, as driven by discovery of facts rather than as determinations of concepts. Further, an important aspect of how such concepts are determined is to be found in their application. Lacking application, we may find ourselves at a loss to see how – even implicitly – a particular criterial choice in conceptual development can usefully be made. The choice between factual discovery and conceptual determination as descriptions of mathematical development may itself not be wholly cut-and-dried. (We might consider asking whether ‗concept determination‘ as opposed to ‗factual discovery‘ gets the facts right or is an example of concept determination, should we need convincing of this.) Taken all-in-all, though, the examples I have offered make at least a plausible case for seeing Wittgenstein‘s distinction as important and illuminating. I have one final example now, to make the point and tie together some of the themes of this chapter.