Much of The Philosophy of Set Theory is an admirably clear and close examination of well-known questions both in set theory (the independence of GCH and AC from ZF, the status of constructible set s and impredicative definitions, etc.) and in the philosophy of set theory (finitism vs. transfinite set realism, etc.). I am going to use this paper to track another of Tiles' questions, however -- one about which she had interesting but quite unclear things to say. This question is a question about the status of the real number system as a representational system. I will not explicate Tiles on this question, but will try to amplify her suggestions in my own terms. Mary Tiles begins The Philosophy of Set Theory with a quotation from Wittgenstein's Philosophical Grammar:
Like the enigma of time for Augustine, the enigma of the continuum arises because language misleads us into applying to it a picture that doesn't fit. Set theory preserves the inappropriate picture of something discontinuous, but makes statements about it that contradict the picture under the impression that it is breaking with prejudices; whereas what should really have been done is to point out that the picture just doesn't fit, that it certainly can't be stretched without being torn, and that instead of it one can use a new picture in certain respects similar to the old one. (p 471)
Historically, the enigma of continuity has been a puzzle about the simultaneous coexistence of continuity and partition -- how can time be both continuous and made up of instants? How can space be both unmarked and measurable? The "picture that doesn't fit," Tiles says, is a conception of continua as being made of parts: as being wholes given after their parts. A continuum is a whole given before its parts: it is indefinitely divisible precisely because it is not an aggregate of atoms; there are no joints at which it must be carved.
Set theory attempts to axiomatize and study a specifically mathematical version of the part/whole relation, that of elements to sets. Tiles argues that the sort of part/whole relation implicit in Cantor's set-theoretical founding of the real numbers, and in the sets and subsets definable in ZF, is a relation where wholes are given neither before nor after their parts. This is a "'new picture in certain respects similar to the old one."
l. Framing the question
There are two main impulses in Tiles' examination of set/real number theory. One is a careful historicism which assumes that we're more likely to be able to sort out ambiguities in current concepts if we look at the concepts they have replaced. She defends this position at length in her (1984); in The Philosophy of Set Theory she acts on it by giving detailed attention to the sorts of part/whole relation present in the natural number series, in syllogistic logic, in ancient and coordinate geometry, in Bernoulli's theory of probability, in analytic geometry and analysis generally, and finally in the forms of logic Frege and Russell proposed as foundations for set theory itself.
The second impulse is a largely implicit broadly-conceived non-finitist constructivism. which sees set/real number theory as a representational system and not, as Cantor saw it, as a metaphysical system. She is not a formalist, because she believes set axiomatization is constrained by preaxiomatic intuitions -- for instance there are aspects of geometric representation and aspects of arithmetic representation which must be present together in a set theory purporting to unify representational domains. And she is also not strictly an anti-realist, because a representational system can be seen as more-or-less realist about structures -- i.e. there's nothing conventional about the results of a systematic calculus whose ordered elements and ordered operations will ensure an ordered structure that must be taken as given.
Tiles' constructivism is most plainly evident in her concluding paragraphs where she argues that set theory should not be thought of as a logical foundation for general mathematical activity, hut that it should be seen as a superstructure, a "representation of space", a "space in which to create structures" (222). This is an interesting suggestion, but it is given with an uncharacteristic lack of explicit support. What is the difference between a foundation and a superstructure? Is a superstructure something like the ground in a figure/ground relation? If it is a set-theoretic superstructure will it have to be granular in nature? If not, how is it different from the plain old homogeneous continuity of old-fashioned geometrical space?
I think the answers to these questions are implicit in what Tiles has to say about the way the real number system and algebraic functions work together to bridge continuity and discreteness by superimposing analog and digital systems of representation. The terms analog and digital are of course never used by Tiles; I have borrowed them from the universe of communications engineering. Analog representation systems use continuous magnitudes to represent othercontinuous magnitudes: the height of a column of mercury in a thermometer would be an example. Digital representation is of antecedently atomic units by similarly atomic symbols: Hockey players represented by numbers, for instance.
2 . Coordinate geometry as analog representation
A graph in coordinate geometry employs physical continuity (that of a sheet of paper, say) to represent continuous magnitudes. X and y axes are supplied with units that are indexed to some sort of discrete number, an integer or rational fraction. The space between unit markers is then assumed to be indexed to all of the numerical values 'between' these discrete markers. The graph thus appears to be able to represent unmarked and even incommensurable magnitudes along with those that are discrete and can be marked. Even when the value indexing some magnitude cannot be differentiated it is thought to be represented by the graph: because it is there, just there, at some exact location on the physical instantiation of the graph.
Coordinate representation thus seems to be a system that is analog and digital at the same time. This is partly an illusion, which results from the way a digital system based on discrete units can easily be translated into a grid which is an analog form using line lengths to represent commensurate intervals. So an analog version of a (well-ordered) digital system can be superimposed over any analog system using continuous line lengths to represent continuous magnitudes: a scale can be drawn against the mercury in a thermometer. In engineering terms the grid is an analog-to-digital transducer corresponding to sampling or coarse-graining methods by which continuous values are digitized.
But does the graph also give us a digital version of analog continuity? (Does it make sense to ask how many points there are in the real line? Are real numbers objects determinate enough so we can ask how many of them there are? Is Cantor's continuum hypothesis true or false independently of what the ZF axioms can tell us?) Can the points of potential division on an x or a y axis coherently be seen as numbers?
3. Functions and totalities
Algebraic notation and the resulting development of a theory of functions makes a crucial difference to the ways we can think of the relation of representational continuity and discreteness. A function expression is inherently ambiguous; it can be read as naming or characterizing a curve whose continuity is antecedently given, and it can be read as a rule which successively creates a curve out of points as numbers are substituted for variables.
There are geometrical curves which are easy to draw but very hard to describe in algebraic notation; if we think of the curve itself as being the function, we'd want to say the algebraic expression is an attempt to approximate the function. Here the curve is seen as a whole given before its parts. But algebraic notation also allows us to write a functional expression at random and then attempt to draw its graph. In this else the curve is a whole which can only be given after its parts have been plotted by substituting numbers for variables. Here we would be tempted to say it is the functional expression itself which is the function, since it seems to 'contain' or name a whole structure of arguments and values, whereas the graph is only a partial instantiation of that structure.
So there are three different ways we can see functions as representational entities. (1) We can see the geometric curve as an analog representation of the structure of continuously changing relations amongst continuous magnitudes. (2) We can see the graph of an arbitrary function as basically a digital representation with an analog look (since, as with line graphs of company profits, only some of the points on the line should be read). And (3) we can see the algebraic expression of a function as representing a whole structure of argument-value relations, quite independently of geometric interpretation.
This last version seems to enfold both the continuity of the geometric curve (because all possible values are included) and the inherent granularity of arithmetic systems (because the function is expressed in a form that demands completion by numbers). In this sense of it, a fraction is a whole which is not given after its parts, since the algebraic expression is not determined by points or numbers given before it. And neither is it a whole given before its parts, since it requires completion by actual numbers. Because it designates a structure of relations, it seems actively to pick out all of its values and to pick them out simultaneously. And yet, as a structure, it cannot exist before its parts and has no identity without them.
When we think algebraically and think in terms of sets of numbers, rather than sets of points, then it is clear that the set of numerical values of a function is unique. The numbers here serve as a means of specifying a structure which, in some cases, we have no other means of specifying. (p.83)
Functional notation also makes it possible for a function to enter into other mathematical expressions and operations. It can in this way behave like a number, and this reinforces the sense that it is a determinate, simultaneously given whole.
4. Pathological functions and point realism
In the course of investigating the use of infinite trigonometric series to represent functions, Fourier discovered 'pathological' functions algebraically expressed as infinite sums. A graph of a pathological function would have to show an infinite number of oscillations, or an infinite number of discontinuities within any arbitrarily small interval. Such a graph cannot be pictured, but:
... if the law can be written and by this means rationally investigated, the graph of the function must be presumed in some senses to exist and to be a totality of points over which our only hold is now algebraic. These points are the members of the set of values of a function f(x) for any x considered as a numerical argument. Thus it becomes necessary to think of the original, smoothly continuous line as itself a set of points, each indexed by a number, and which has an unimaginably, because infinitely, complex order structure. (p.82)
It seems misleading to talk about a graph existing here, but I think what Tiles is getting at is that if we think of a pathological function as a function in the third of the senses outlined above -- if we see it as a single structure containing all the relations it expresses -- then we seem to have to think of it as a structure explicitly articulated at an actual infinity of points. Even without the use of geometric intuition which would lead us to see these points of articulation as spatial markers, this seems to be a structure which is a unity (the simultaneous copresence of relations all of which are relations both to each other and to the whole) whose identity depends on an infinite number of differentiated structural elements.
It does seem. to be both continuous and discrete. There is something, it seems, inherently analog about the notion of structure, which is simultaneous, multidimensional, multiordinal relation. But whether the points of articulation in such a structure 'really exist' as discrete points, whether this sort of structure is also a digital system, will depend on whether the real number system provides really discrete representational elements.
5. The system of reals
When they are put into the form of potentially infinite binary or decimal fraction expansions it is evident that the real numbers have an indeterminacy like the indeterminacy of potential points of division on a continuous line. Binary trees show how this works:
[TREE DIAGRAM from Tiles, p.92]
Consider the set of real numbers between O and 1 as given by their binary decimal notation. All such sequences can be represented by the full binary tree which can be thought of both as the making of successive choices about whether to put O or 1 in the nth decimal place and as the making of successive divisions of the unit line. All the points corresponding to an infinite sequence with an initial segment .0001 ... lie in the interval [1/16, 1/8]. Each addition of a level. to the tree chops the tine up into smaller bits. (p.92)
Since this process is nonterminating there is no reason here to think any real number will ever be finally differentiated from other reals sharing initial segments of a branch on the binary tree. And there is no terminal grain in a continuous line to provide a rational stopping place for the process of base2 exponentiation of point totalities .
This indeterminacy led cantor to try to provide another sort of picture of the reals. If the real number system incorporates a model of the rational numbers, and if the rationals are thought of as indexing entirely precise positions in a continuous interval, then the limit of an converging series of rational numbers will also have to have an entirely precise position (because it is 'between' precise positions that 'push in' toward one and only one terminal point). Geometric intuition thinks of this point as 'being there' whether or not it can, in practice, be given any sort of unique name. So Cantor defined real numbers as (equivalence classes of) fundamental sequences of rational numbers. He went on to show how they could be used as numbers: how to define an order on them, how to define standard mathematical operations over them.
With the development of the classical theory of real numbers on the foundations suggested by Cantor and Dedekind it has become possible, without obvious incoherency, to think of a continuum as 'made up' of points (strictly as a set of points). Moreover, this view seems to be required by the way in which functions defined over real numbers are customarily associated with their 'graphs.' (p.92)
But here it's wise to make a distinction between kinds of continuum. The continuum 'made up' of limit points of real numbers is not the geometric continuum:
'Construction' of the real numbers, which is the initial vehicle for thinking about a point continuum, is not a geometrical construction. It is not a matter of building a continuum by distributing points in space, but of defining the real numbers and showing that these can be ordered so as to be order-isomorphic to the points on a line. (p.93)
Here Tiles is pointing out that the real number system is a representational system, a notational system. Sometimes it can he used to model another representational system, the (analog) geometrical system ( which itself sometimes is used to model physical continuity).
6. Foundations and superstructures
The structural complexity of pathological functions and the geometrically intuited independent existence of limits of converging sequences both seemed to suggest that "simple continuity could not be thought of as the background against which analytic operations were to be interpreted" (p.84). "Background" here shouldn't be taken as the ground of a figure/ground relation (and this is an answer to one of my earlier questions). It should not be taken in a spatial sense at all. It might be better to call it a language. I think what Tiles has in mind when she talks about a superstructure is something like the whole of a representing practice.
It might be that a particular representing practice could have a logical foundation which guaranteed consistency. A constructible inner model of ZF might be a logically founded subsystem of this sort. But Tiles is saying that she doesn't think set theory as a whole is a unitary logically founded system, because, for example, power set can be impredicatively understood. So set theory cannot be thought of as a foundation for the real numbers either.
Set theory and the real number system, then, are superstructures in the sense that they are languages allowing us to define more than one kind of structure. Then it would not make sense to look for the structure of set theory, to study it as an object. What world make sense is to find out what structures can be constructed given the resources of ZF, including its impredicative resources.
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