Kitcher on Abstract Objects

1. Contra Plato.

On the presentation of his empiricist account for the origins of mathematical knowledge, Kitcher uses Platonism as his illustrative antagonist. Platonism is the rival theory against which Kitcher tries to prove the virtue of his position. Particularly, his theory aims at overcoming the problems of Platonism derived from the postulation of abstract objects to account for mathematical truth. At the very end of the first section of the sixth chapter, he synthesizes this aim in the following sentences:

I shall attempt to work out an interpretation which will . . . overcome the difficulties I have uncovered in Platonism. . . . There is no suggestion of a gap between these ordinary objects and other, more ethereal, entities which lurk behind them. . . . Finally, the avoidance of abstract objects will free us from those troublesome questions . . . which platonists seem forced to answer. [Kitcher, Philip: The Nature of Mathematical Knowledge, Oxford University Press, New York, 1984. p. 107. All bibliographical references belong to this text, unless otherwise indicated.]

The main question I want to raise in this brief paper is whether Kitcher's empiricist account actually succeeds in avoiding the problems of Platonism generated by the postulation of abstract objects as ultimate sources of mathematical truth. I would like to know whether Kitcher's deviation from the platonist path to follow a trail of empiricism does not end up leading him back to the same pitfalls he was trying to avoid. The main question is not so much whether his empiricism is not some sort of Platonism. After all, the answer depends on what we understand for "Platonism." Kitcher calls 'Platonism' any theory which holds that "true mathematical statements are true in virtue of the properties of abstract objects." [p. 102 ] This definition is lax enough to content Kitcher's brand of empiricism, but it is not the only possible one. There are obvious similarities as well as differences between Kitcher's position and Platonism. The relevant enquiry, therefore, is to explore these differences to see how far from the pitfals of Platonism do they take Kitcher.

Kitcher's empiricist account of mathematical reality relies on the articulation of two different answers to the question of "in virtue of what are mathematical truths true?" The first answer is that "Mathematics is true in virtue of human operations," and the second is that "mathematics is true in virtue of the structural features of reality." The purpose of my brief paper will be to trace the first of these answers to see if it provides an actual alternative to the postulation of abstract objects as ultimate sources of mathematical truth. The reason I avoid talking about the second thesis is that it belongs to Kitcher's view of mathematical change in relation to scientific change, which is the theme of the next session of our conference, and I do not want to invade their area.

It is important to bear in mind that Kitcher does not want to do away with the tenet that mathematical propositions are descriptive and that, to account for their truth, we need to specify the objects whose properties it describes. What he rejects is the platonist answer in terms of a mathematical heaven of sets and numbers. Even if we leave aside the details of his criticism of Platonism and accept the claim that mathematical truths are not true in virtue of the properties of some abstract objects, the question that remains is: "If not abstract objects, what then?"

One central idea of my proposal is to replace the notions of abstract mathematical objects ...with the notion of mathematical activity... [p. 110]

Kitcher wants to replace the traditional platonist picture of mathematics as describing abstract mathematical entities [p. 106] with the picture of mathematics describing some sort of mathematical activity. Specifically, he invites us to interpret mathematics as describing the operations of collecting and correlating. Asserted this way, there seems to be a vast gap between the platonist position and Kitcher's. Collecting and correlating seem to be as concrete a set of activities as there are. They do not belong in any ethereal heaven far over our heads, but in our everyday life. If Kitcher succeeds in showing that mathematics owes its truth to these concrete activities, he would have actually shown us a way out of the platonist bottle. However it would be too early to claim victory at this point in so far as, almost as soon as Kitcher states this and other equally bold statements, he starts backing down on them. Since the beginning of the chapter, Kitcher qualifies his stronger remarks by saying things like: "I shall later qualify this thesis and explain more carefully what it amounts to" [p. 108] or "let us note explicitly that construing the structure of reality to be manifested in the operations we actually perform is obviously inadequate." [p. 109] It is important, therefore, not to stop our analysis at the level of slogans, and take a closer look at the complete theory in order to detect the actual mechanisms doing the work in Kitcher's explanation.

The main trope upon which Kitcher builds his explanation of mathematical truth is the notion of an ideal subject, "whose status as an ideal subject resides in her freedom from certain accidental limitations imposed on us." There are two questions I want to raise about this idea that the ideal subject and her ideal operations arise by abstraction from certain of our accidental limitations. First, in what sense can we say that, after we have removed these so-called accidental limitations, we are still talking about us or that which "the world lets us do to it"? And, second, what are we actually abstracting? How do we decide which limitations are accidental and which are not, what to abstract and what not? The following section will deal with the first question, while the third one will try to present Kitcher answer to the second. After that, the fourth section will draw conclusions from those two section to paint a critical picture of Kitcher's empiricism. Finally, I will give a brief, but detailed criticism of one of his examples of how his theory can be applied to a branch of mathematics: Mill Arithmetic.

2. The ideal mathematician and us

Along with his presentation of the notion of 'ideal subject', Kitcher never tells us exactly how much of 'us' is still there in the ideal subject. He never tells us how much of us is accidental with respect to mathematical activity. At most, we can see through some of his examples to some features that distinguish us from our 'ideal' counterpart: In the first instance, there is mortality. The span of human lifetime does not affect the ideal subject, who can perform mathematical operations continuously until the end of time. She is also free from all our biological limitations. Actually, she is not even physical at all, so she has none of our physical limitations either. There might be other differences, but these few are enough to make us wonder how feasible is it, not just to theorize, but even to talk about this ideal subject. Can you really imagine a subject with no physical, biological or temporal limitations? If so, could you tell me anything else about her? Could you, for example, tell me how does she collect or correlate things? I raise this question because this is what Kitcher's explanation finally boils down to. To appraise his theory, we must be able to look at the ideal subject, see how it behaves and recognize mathematics in what she does. We must be able to turn to her and say things like 'Oh, yes, that is a Grothendieck construction! Look at it! '

Regarding intuitionism, Kitcher maintains that his position is similar to it in as far as both hold that mathematical truths "owe their truth (at one level) to the operations we perform." It is important to ask at which 'level' exactly are the mathematical activities described by mathematics 'ours'. Though he maintains that mathematical truths are true in virtue of human operations, Kitcher is not such a constructivist as to think that the ultimate references of mathematical terms are the actual human operations we perform, but instead, the operations that we could perform. What mathematics aims at describing is not actual human mathematical behavior, but what Kitcher calls the "permanent possibilities of manipulation." [p. 108] In the end, what we know about mathematical activities, i.e., the operations of the ideal subject, is that they are not actual, but possible. Not concrete, but abstract [Pp. 117, 134]. Not real, but ideal. Not existing, but postulated [p. 134]. Additionally, since they are performed by the ideal subject, they neither happen in time nor are bounded by any physical or biological constraints. For a theory that flatters itself as emerging "from consideration of the ways in which humans actually infer and from the knowledge claims which we actually make," [p. 97] it is hard to see mathematical activity, as described by Kitcher, as something very close to our human ways. Furthermore, if Kitcher is serious about his claim to describing 'what the world let us do to it', he cannot avoid taking in consideration temporal, biological and physical constraints. What else could 'what the world let us do to it' mean, if not whatever physical, biological and temporal constraints it imposes on us?

3. Abstraction to the Test

To the second query ­how do we decide which limitations are accidental and which are not, what to abstract and what not?­, Kitcher would reply that it is not a suitable question. In a certain sense, there is no answer. If what we are asking is whether or not there is a way of recognizing prima facie which features of our actual collecting and gathering are accidental or not, the answer is not. There is no way of recognizing which features of our concrete operations of gathering and correlating should be abstracted. However, to posit the question in this sense is to miss the point of Kitcher evolutionism. What he is trying to convey is precisely that we cannot tell in advance which features are to be abstracted. We have to look at our actual mathematical knowledge and see what has already been abstracted. If we understand the question in this sense, then the answer lies in the history of mathematics. Which features should be abstracted? Those which would leave us with the mathematical activity which corresponds to the way mathematics is actually performed.

To clarify this point, let us look at a real example close to us: non-well founded sets. In order to understand non-well founded sets as describing some sort of 'gathering' operation, we would have to abstract the feature of well-foundness from our concrete notion of gathering. In this world, whenever we assemble a group of things, the very group we are producing cannot be one of the things gathered. We can put three apples in a pile, and we can group this pile with other things to make another pile. However, we cannot make a group that contains these three apples and itself. It can contain some other group of apples, but not this very same group of three apples. Even our everyday language fails us when it comes to describing this sort of operations. What should we say? Make a group with these three apples and the group of these three apples and the group of these three apples and the group.... All of our actual concrete gatherings are well founded. However, we could imagine the ideal subject as being able to fold back in time and perform this sort of non-well founded gatherings. Before there were non-well founded sets, this feature had not been abstracted. It had not been considered an accidental feature, but a necessary one. However, now we have to consider it so. Nevertheless, we know this because of the way mathematics has evolved, not because there is something about the well foundness of gathering that makes it accidental now and not before.

That is why the performance of the ideal subject is not arbitrary. It could not be, in so far as it is inferred [p. 118] from our actual (current and historical) manipulations of reality. We cannot abstract arbitrarily any feature or character of us or our activity to construct those of the ideal subject or her operations. We have to do a lot of history (and anthropology) to come out with the answers. Furthermore, the results also have to be empirically tested with respect to our human operations. This is where Kitcher marries mathematics and the natural sciences: both operate through abstract models which are induced from facts in the world and whose efficiency as models is also empirically tested against them.

4. Platonism and Social Constructivism

My criticism in this paper is not as naive as to say that the mere postulation of abstract objects or the use of the concept of 'ideal' is sufficient to dismiss Kitcher's position as Platonist. As I already said at the very beginning of this paper, there are obvious similarities between the platonist position and Kitcher's. For one thing, both hold that mathematical truths are true in virtue of the properties of abstract objects. However, the differences are also pretty obvious. Mostly, we have to recognize that Kitcher abstract entities do not lie behind the world making it the way it does, as platonist ideas do. On the contrary, they represent regularities and generalities of our world in an approximate way. The relation of representation runs the opposite direction than in Platonism. Our operations are not imperfect copies of the operations of the ideal subject. On the contrary, the ideal operations areapproximative generalizations inferred from our operations. In a certain sense, we could say that Kitcher's abstract objects play the epistemological role of platonist ideas, but they do not have their ontological status. This is what Kitcher calls, following Chihara, "mythological platonism". A kind of paltonism, if you like, but not the traditional aprioristic one.

Kitcher thinks that the major advantage of his unorthodox paltonism over aprioristic one is that it explains how its abstractions stem from our everyday practice and are tested against them, just as any abstract model in the natural sciences. However, there is still some danger lurking behind Kitcher's naturalist answer. There is still the risk of Kitcher naturalism becoming mere social constructivism. If the abstract model of an ideal subject performing ideal operations is induced out of our concrete instances of mathematical truths, and these truths are the only empirical data against which we test the abstract model, in what sense can we say that we have explained what makes a mathematical truth true? Surely, there must be something else in Kitcher's explanation. There must be something 'corrective' about the operations of the ideal subject which distinguish it from our operations. If we want to scape this circularity, the ideal subject must be able to operate in ways which we are yet to be described by our current mathematics. Otherwise, what would it mean for the ideal subject to be ideal? For a subject to be ideal, it does not suffice to be abstract, it has to be at least less fallible in her operations that us. Or else, it could play no explanatory role.

This blade cuts both ways. On the one hand, if Kitcher does not accept that the operations of the ideal subject cover more field than that described by the mathematics of any historical period (like today), he might well fall into social constructivism where truth is nothing besides that which the mathematical community of that period takes to be true. On the other hand, if he does, his theory faces strong problems explaining historical error. Take, for example, the case of naive set theory. At that moment, whatever made the truths of naive set theory true must have been the operations of the ideal subject. However, as we well know now, such theory led into paradoxes. What does this fact mean for the ideal subject? Should we conclude that she paradoxical in its behavior? Was she, for example, able to 'operate' Russell's paradox? To explain what happened at that moment, should we 'abstract' also the notion of consistency from her operations? I do not think Kitcher would be happy with an inconsistent ideal subject. To take the other horn in the dilemma would be to say that the operations of the ideal subject had lead her to discover such paradoxes, and to somehow avoid them. In this sense, we could now say that all her operations were already in the Zermelo Fraenkel framework and that naive set theory had missed at describing them. But the question remains. In what sense is this an explanation of mathematical truth? Is it actually telling us what it is for mathematical truths to be true? For example, has it tell us anything about why are the claims of Zermelo Fraenkel set theory are true, and those of naive set theory are not? There is no explanation in Kitcher's theory, besides the historical fact that now we know of the advantages of Zermelo Fraenkel set theory. But how do we know now that Zermelo Fraenkel's set theory is not false itself. Should we just wait for mathematical history to prove so? It seems like Kitcher's theory is just changing the shift of the problem from our mathematical practice here and now to talk about abstract operations by the abstract subject. Maybe this is all Kitcher wants, an illustrative image that helps us better understand the problems of mathematical truth. In that sense, there might be many virtues to his proposal. Nevertheless, they do not ammount to an explanation of what makes mathematical truths true.

5. On Kitcher's reconstruction of Mill's Arithmetic

We do not need to go as far as contemporary developments of set theory and talk about competing theories to find difficulties in Kitcher's explanation. Even his first example, Mill Arithmetic, is a bundle of flaws and imperfections.

In order to accurately understand Kitcher reconstruction of Mill Arithmetic, we must take seriously his claim that it is operations which make mathematical truths true. In the particular case of arithmetic, the relevant operation is that of segregation. Notice, therefore, that the primitive notions of his language are not segregations themselves, but predicates defined on segregative operations. Consequently, the variables of the language range over segregations. When, for example, in Axiom (13), Kitcher writes («x)(Ux), this must be interpreted as "there is a segregation which is a one-operation". Kitcher's ontology, therefore, is inhabited by operations. In this sense, it distinguishes itself from the platonist ontology of arithmetic, which is populated by natural numbers.

There is an enormous temptation to read Kitcher's axioms of Mill's arithmetic as talking about objects or group of objects instead of operations. We are very tempted, for example, to read axiom (6) (x)(y)(z)(w)((Sxy & Szw & Myw) Æ Mxz) as saying that if x is the group obtained by adding one object to i, and z is the group obtained by adding one object to w, where y and w have the same number of objects, then x and z also have the same number of objects. But, in strict sense, this reading is wrong. A more accurate reading would be "if, when we perform x we segregate all of the objects segregated in y, together with a single object extra, and when we perform z we segregate all of the objects segregated in w, together with a single object extra, where the objects segregated in y and w can be made to correspond to one another, then the objects segregated in x and z can also be made to correspond to one another".

This temptation is so big, that even Kitcher finds it hard to resist. Addition, for example, is defined by Kitcher the following way:

When we combine the objects collected in two segregative operations on distinct objects we perform an addition on those operations. [ p. 107]

In this definition, Kitcher confuses talk of objects and collections of objects with talk of operations. Here, Kitcher defines an operation (addition) on operations (two segregative operations on distinct objects) as an operation (combination) on objects (those collected in the aforementioned two segregative operations). If addition is performed when we combine objects, then addition is a kind of combination, which is an operation on objects, not on operations, as the definition presumes. Someone might argue that this objection relies only on the surface form of the definition. They might argue that addition so defined is still an operation on operations in so far as it is a function which takes two segregative operations as arguments and assigns them another operation, the addition properly speaking, as value. But even this is not enough, because the resulting operation is no longer a segregation, but a combination and the predicates of the language are explicitly defined only on segregations.

However, there are deeper problems with Kitcher definition of the primitive notions of Mill Arithmetic in terms of objects. All of the primitive notions of Mill Arithmetic are defined in terms of objects. A one-operation is defined as the segregation of a single object. The same happens in the definitions for successor (the addition of a single extra object), addition (the combination of the objects segregated by two operations) and matchability (the matching of the objects of two distinct segregations) [p. 112]. The first problem, is that it fails to avoid ontological commitment of objects. On the contrary, to make sense of Mill Arithmetic, Kitcher still needs objects, an infinite number of them. This difficulty stems directly from the definitions of one-operation and successor. A one-operation, for example, is defined as the segregation of 'a single object" [p. 112]. If there is a one-operation, there has to be at least one object. A successor operation, on the other hand, is defined as the addition of a single 'extra object' to a segregation. Hence, if there is a successor for the one-operation, there has to be at least an extra object; and so forth until we are committed to an infinite number of objects, besides an infinite number of segregations.

The second and deeper difficulty is circularity. In Mill Arithmetic, predicates on operations are defined through similar arithmetic predicates on objects. What is a one-operation? The segregation of one object. What is to match two operations? To match their objects. These definitions do not explain what it is to add, match or any of the other primitive arithmetic operations. On the contrary, they require us to understand them before making any sense of the definitions. How can we recognize, for example, two segregations as matchable if we do not know already what it is to match two groups of objects? Furthermore, how can we recognize a segregation as a one-operation if we cannot tell when the object it segregates is one? It is impossible.

The definitions of the primitive notions of Kitcher Mill Arithmetic are filled with notions as that of 'a single object', 'all of the objects', 'a single extra object' and 'distinct objects' which are arithmetical themselves. Frege had already warned us about the problems involved with using the notions of 'single object' and 'distinct object' as primitive. In order to recognize an object as one, we need to answer the previous questions of 'one what?'. In order to recognize two objects as distinct, we need to answer the question 'distinct whats?'. Under different concepts, the same object can be seen as one or many. What under the concept of 'beach' can be seen as a single object, can be seen as many under the concept of 'grain of sand'. The same can be said about the notions of 'distinct' and 'the same'. Sonny and Cher were different persons, but the same performing act, at least for some years. This is in no way a mere imprecision from Kitcher, but a serious flaw in his theory. Kitcher needs to give us a criterion of identity under which we can recognize the objects segregated as one or many, different or the same, in order to make sense of his definitions. Whatever criterion he provides, it will seriously affect his theory. First of all, it will affect his ontology, in so far as it will tell what sorts of objects his theory is committed to. However, it might also affect his claim to empiricism. In order for his arithmetic to be thoroughly empiricist, Kitcher has to offer a criterion of identity which cannot be reduced to a non-empiricist account.

The need of providing a criterion of identity is not exclusive of Kitcher talk of objects. It is a problem also for his explicit ontology of segregations. In order to be able to quantify over segregations, Kitcher has to provide us with a criterion of identity for them. He has to tell us how to recognize a segregation as one and the same, and how to distinguish it from other segregations. If I have a pile of apples on the table and I push one off the pile and over the edge of the table, how many segregations have I performed? This operation can be seen both as the segregation of one apple from the pile on the table, or as two sequential segregations: the segregation of the apple from the pile and the segregation of the apple from the group of apples on the table. Furthermore, if I put the apple back on the pile and then push it off again, have I performed a new segregation or repeated the previous one? Kitcher has to answer these questions if he wants his Mill Arithmetic to make any sense at all.



Axel Arturo Barceló Aspeitia