Two issues have interested me for a long time. One is Kant's perception of metaphysics as an illusion-prone area, while the other involves the intriguing way mathematicians expand their concepts. Although mathematicians may talk about the sine of a complex number, they do not try to define the sine function to apply to the moon. The connection between the two areas becomes clearer when one recalls that Kant has argued that the antinomies of reason derive from illegitimate expansions of concepts beyond their range of application (e.g., applying the categories of causality to the whole world of phenomena). It is important to note that this connection has reappeared and even intensified in contemporary thought. One cannot imagine modern mathematics and physics without the procedure of expansions of concepts, and the analyses of Russell's paradoxes by Russell, Gödel, and others are echoes of Kant's view that one cannot view certain totalities as genuine objects. A parallel approach, with certain changes, may even be attributed to Wittgenstein, whose aim in philosophy was to “bring words back home,” as well as Brouwer's diagnosis that classical logic was derived from a careless expansion of logical laws that are valid for finite collections to unbounded ranges.

These developments raise an interesting question about Kant's analysis of the source of illusions, as they make it clear that modern scientists and mathematicians do not respect the boundaries within which the concepts they use were originally defined.

Once we accept the idea that forced expansions are an important manifestation of human rationality, we need a theory of concepts, reference, and thought to accommodate them. In this chapter I present a picture of this sort for the notions of concepts and reference, trying to make it as close as possible to Frege's realism and extensionalism for concepts. In chapter 8 I shall examine its implication for the notion of thought. Let me state at the outset that I do not pretend that the picture I shall propose in the following chapter is the best that could be formulated, but I do claim that it is a better idealization than Frege's. Later I will discuss the implications of forced expansions for definitions, focusing on Wittgenstein's thesis that definitions are not always feasible because of the ever-present possibility of expansion. I shall argue that the view he offers in connection with his notion of family resemblance cannot be derived from the expansion of concepts. Afterwards I will briefly examine two pictures of concepts that give an important place to non-arbitrary expansions but which are not faithful to Frege's realism.

STAGES OF CONCEPTS

If the development of concepts is not to be described merely as the replacement of one concept by another, must we accept the strange idea that the same concept can have different extensions?

The concept of square root was expanded to include the negative numbers; the concept of power, originally defined only for the natural numbers, was expanded to include zero, fractions, and real and complex numbers; the logarithm function, which was originally defined only for positive numbers, was expanded to the negative numbers; in general, nearly every mathematical function has been expanded in a non-arbitrary way. But this is not only true of mathematics; in physics as well there are expansions of concepts that were originally defined only for a restricted range. The expansion of the concept of temperature to black holes, the notion of instantaneous velocity, the idea of imaginary time, and perhaps even the idea of determining the age of the universe are a few examples of this process. Metaphors and analogies can also be considered expansions of concepts beyond the sphere in which they were first used. Moreover, philosophy has always been suspected of expanding concepts beyond their legitimate range of applicability. It seems that every area that contains concepts also contains expansions of concepts.

Various incidental remarks about expansions of concepts that have taken place throughout the development of modern mathematics were made by Leibniz, Pascal, Bernoulli, and Gauss. The first attempts to deal with this phenomenon systematically, however, were George Peacock's (1791–1858) “principle of permanence of equivalent forms” and Peano's requirement that logical notation must leave room for functions to develop. With Frege a crucial turn took place.

The history of mathematics and the sciences is replete with examples of the expansion of concepts. Nowadays we are witness to a growing interest in the history of mathematics which has given rise to a range of essays on the history of specific concepts and theories. In this chapter, I should like to concentrate on several turning points and dilemmas in the development of the idea of expanding concepts and domains. This will require tracing the emergence of expansions as a general process from specific examples, and distinguishing these developments from the history of other general and basic notions such as algebraic structures and deduction. At the end of this chapter I briefly survey the state of the art in the study of expansions in mathematical logic and philosophy.

EARLY DEBATES

Expansions of concepts began to occur in seventh-century India, with negative numbers, the irrational numbers, and the zero. In sixteenth-century Europe a great number of expansions occurred one after another, giving Western mathematics a unique status. The first signs of this phenomenon were apparently the introduction of the zero and the beginnings of algebra, which were brought to the West by the Arabs.

When Western mathematicians developed these ideas, they did not follow pure logic; in fact, they had to make some compromises on rigor. If they had not done so, their expansions would have been blocked by the ancient Greek conception of mathematics, just as this conception had first blocked the acceptance of the rational numbers and then of the irrational numbers.

As we saw in the Introduction, Frege summarily dismissed any notion of conceptual expansions. Since his theory is the simplest and the strongest, it will be useful to begin with it. There is another reason for beginning with Frege's position. Even though Frege started out by questioning the nature of numbers and attempting to understand deduction, he was the first to do so on the basis of a broader view of language, thought, and reference. In this respect no logician prior to Frege was better able to provide a philosophical dimension to the issue of expansions, which can be useful for the philosophy of language. Thus, examining Frege's opposition to expansions should be a natural way of getting into the philosophical discussions of expansions, some of which were mentioned in the previous chapter.

Frege presents three arguments to demonstrate that the idea of the expansion of concepts is incoherent. These arguments can be separated from Frege's inner motivations for abolishing the idea, and therefore deserve careful study. I present them in order of importance, beginning with the least important.

THE ARGUMENT FROM REALISM

Frege's realism about concepts and their place in the world of reference leads naturally to the notion that concepts cannot change, and thus cannot be expanded. Once concepts are detached from the thinking subject, they do not undergo the developments that subjects do.

As we have seen, Frege absolutely excluded expansions from the realm of logic, because he claimed that logic involves the laws of truth, and truths are not developed, but can only be derived from other truths. But then we remain with the question about what to do with the process that other mathematicians and philosophers describe as “expanding a concept.” Two alternatives were suggested by Frege. The first, which considers expansions to be changes within the thinking subject, was criticized above. The second alternative is that the original concept has simply been replaced by a new one.

My intention in this chapter is to present an analysis of non-arbitrary expansions. I will begin with a criticism of Frege's second alternative, which can then be generalized to a description of the development of concepts in terms of embedding a given model into a richer one. This emendation leads to the main section of this chapter, which presents a framework for introducing several kinds of non-arbitrary internal expansions, and closely examines the issue of the completeness of certain logics that arise naturally out of the consideration of expansions.

BETWEEN REPLACEMENT AND GROWTH OF CONCEPTS

The third argument discussed in the previous chapter makes it clear why Frege insists that concepts do not develop, but rather that one concept is replaced by another. But even though this description provides somewhat of a solution to the problem, it raises even more difficult problems.

In this chapter I will try to advance the ideas presented in the previous chapter. First I will investigate the connection between forced expansions and logical deductions, discuss the general applicability of forced expansions, and analyze their relation to the rule-governed completion of sequences or matrices. This will support what I asserted in the previous chapter – that not only can forced expansions be described formally, but they are a special kind of rational, logical procedure. In other words, I propose not only that we can write a formalism of non-arbitrary expansions but that such expansions are immanently linked to logic. I complete the chapter with a section on the source of the productivity of forced expansions, which will help us judge which of several possible expansions is the most promising one.

THREE CHARACTERISTICS

The claim that forced expansions are rational, logical procedures seems to require a discussion of the nature of rationality and logic. For that purpose, however, it would be necessary to take a stand on controversial issues that would take us far away from the limited topic of the present book. Instead, I have chosen to list three characteristics of forced expansions that are uncontroversially accepted as characteristics of rational procedures. If anyone prefers to keep the term “logic” for something narrower, I am willing to concede that forced expansions do not constitute a logical procedure par excellence, but only something resembling such a procedure.

The concept of forced expansion has been sufficiently articulated by now so as to facilitate finding such expansions in other settings. In the present chapter I will examine one of Gödel's arguments, which has not yet been given the attention it deserves, but which is easy to disclose once we are aware of non-arbitrary developments of concepts (I shall call it “Gödel's second argument”).If we reformulate this argument in the terminology that has been developed here, we can understand Gödel as claiming that the very existence of forced expansions proves (a) the existence of concepts and at the same time (b) naturally leads to the possibility of perceiving them. In the first part of the present chapter I set forth Gödel's argument in broad outline.

In the second part I examine the notion of the perception of concepts as it arises from Gödel's descriptions. Here I try to show how this argument is relevant to the lively discussion about this notion of Gödel's, which has developed around Gödel's more famous argument deriving our intuition of objects from the fact that the axioms of set theory are forced on us (call this “Gödel's first argument”).Familiarity with the phenomenon of expansions allows us to refine Gödel's picture and make his views seem more plausible.

In the third part of the chapter I discuss Gödel's argument in favor of realism for concepts.

If the truth can be stretched, what does this tell us about the meanings of sentences? This question must arise in any approach that connects the meanings of sentences with the concept of truth. But, as we have seen again and again, Frege's position gives this question a unique significance. Frege distinguished between science and fiction, calling the latter a realm where sentences have meaning without having truth-value, or, more precisely, where it does not matter what the truth-value of a sentence might be. Stretching the truth thus entails refiguring the distinction between science and fiction. The possibility of such a refiguring does, however, have a favorable implication. Just as true sentences can be used in literature, because they may be interesting for reasons other than their truth-value, there can also be sentences which lack a reference in Frege's view, yet are worthy of scientific discussion because they may potentially have a truth-value.

I therefore suggest that it is worth distinguishing a set of inchoate thoughts within the world of thoughts. These are partial senses of sentences that may either develop into complete thoughts or be discovered to be meaningless as the result of a non-arbitrary expansion. If we want to take seriously the idea that truth can be stretched, then we must amend several basic laws of Frege's involving thoughts and judgments. Subsequently I hope to sharpen my view by comparing it with that of Wittgenstein, the only thinker I know of who expresses a similar opposition to Frege.

The weird problems discussed by philosophers and the paradoxes encountered in philosophical discourse have often been connected with the expansion of concepts. As I mentioned in chapter I, this idea was forcefully expressed by Kant, who considered the problems of dogmatic metaphysics to be the result of the incautious expansion of concepts. Questions about the cause of the world or the age of the universe come from expansions of the concepts of cause and age which are valid for what can be perceived by intuition – beyond what can be considered as phenomena, leading to illusions. We saw a variation of this view in the twentieth century in the wake of the crisis in the foundations of mathematics. Here too leading philosophers and logicians made remarks reminiscent of Kant's position. The most prominent was Russell, who suggested, in his theory of types, that Russell's paradox can be dealt with by avoiding the claim that all properties are meaningful for all objects. This analysis, like Kant's, asserts that we become involved in antinomies when we try to expand our concepts to where they are not applicable (see, for example, the first page of the first edition of the first Critique). This move has had a number of variations, some of which I present below. In fact, remnants of this view guide various projects in the foundations of mathematics, including that of ZFC (Zermelo–Fraenkel and the axiom of choice).

So far I have discussed expansions of concepts that do not involve the addition of new objects. Now I will examine whether it is possible to generalize this discussion to external expansions, where new objects are involved. The point I suggest here is basically a formalist one, claiming that words such as “-3” and “√(-1)” can play a crucial role in external expansions, but this approach is given a new sense here: the important external expansions such as negative and complex numbers are viewed as the result of stretching the identity relation. I shall then move to the debate between Frege and the formalists, trying to find a way to retain the intuitions of both sides. Corresponding to the transition from words to objects that occurs in external expansions, there is a subtler transition in which constraints on potential entities are transformed into axioms on a well-defined realm of objects. This is the subject of the third section of this chapter.

This leaves us with the question of where to start. Do we assume, like Kronecker, that the natural numbers are at the basis of all expansions of numbers? It seems to me that this assumption is not necessary. One way that we can begin, which is probably not the only possible way, is to construct the ordinal numbers and the relations between them, hoping that further expansions will take us to richer structures.

In this book I have analyzed a certain type of expansion of concepts – non-arbitrary changes of extension – and investigated some implications for the philosophy of language and of mathematics. I hope that I have managed to prove my central claim about the importance of the phenomenon of expansions to fundamental questions in mathematics and logic. But I am far from believing the interest in this phenomenon is exhausted in this book. In fact, since my interest here was to present forced expansions, and their relevance to important questions that have been discussed in philosophy since Frege, I have not exhausted the subjects discussed in the previous nine chapters. Instead of summarizing what I have presented so far, I wish to illustrate the directions in which the phenomenon of conceptual change should be researched.

I present the simplest issues first and work my way up to the more difficult ones.

A. Modern logic, founded by Frege, gives us no tools for understanding concept development, for it forces us to claim of the developed concept that either it is identical to the old one or it is completely different from it. When we get to philosophy, we feel that this sharp division is insufficient to get to the bottom of the problem, but dealing with a specific problem of this sort does not let us return to logic, or to simpler situations and to see what happens there. The first cluster of questions I propose is to take examples of expansions from mathematics and science, preferably as simple as possible, and to analyze them. Thus, removed from the heat of the philosophical debate we are discussing, we can see what is happening in a clearer way.

How should we define physicalism or minimal physicalism? In my view, this question calls for stipulation because these are theoretical terms without a uniform use. Different views of psychophysical relations are physicalistic in different ways and to different degrees, and there is an obvious interest in clarifying and distinguishing these views and determining which are true. My aim in this chapter will be to do some of the clarifying and distinguishing. Stipulation of a unique thesis as physicalism or minimal physicalism must come with a rationale, and as I have none to offer I shall not pursue this.

Some regard physicalism as the thesis that all first-order properties instantiated in the spatiotemporal world are physical properties. I shall refer to this as type physicalism or property physicalism. It can be presented in the form of a supervenience thesis – another popular way of defining physicalism – on the assumption that a property is physical if and only if it logically supervenes on microphysical properties. This is one way, or a first approximation of a way, of characterizing a physical property in terms of microphysical properties. But as I think that the following discussion will apply on any reasonable view of physical property, I will mostly continue to talk of physical properties without getting more specific. I shall assume that it will be clear enough to think of a microphysical property as an assignment of fundamental microphysical parameters in some type of spatial or spatiotemporal region, where the fundamental microphysical parameters are those featuring in an ultimate microphysical theory.

According to Jeffrey Poland, the “basic conviction” of the physicalist is that:

I agree with Poland that this sort of explanatory claim is central to the physicalist idea. In this chapter I explore how we may fill out and clarify that explanatory claim in terms of its modal implications. In particular, I focus on two sufficiency claims to which physicalism is committed. Ultimately, there is no way to replace the explanatory claim with sufficiency claims; nonetheless, there are good reasons to focus on them. They help cash out the demands of physicalism, giving us a better grasp on its content. In so doing, they also put us in a better position to evaluate apparent counterexamples to physicalism, think further about the implications of physicalism for science, and, perhaps most urgently, get clear on what sense there is to be made of so-called naturalization projects. Further, they provide a dialectically useful route to justifying physicalism itself. As I shall argue at the end of this chapter, if they are true, it is overwhelmingly likely that the overarching explanatory claim is true.

Physicalistic Explanation and Sufficiency

In what sense is the physical realm supposed to explain everything else?

Every era has its weltanschauung and in much contemporary philosophy the doctrine of ‘physicalism’ plays this role. Such foundational assumptions exert considerable influence by subtly configuring philosophical debates, problems, and projects, though the precise nature of these assumptions is often hard to clearly discern. Happily, however, philosophers have recently begun to subject physicalism to sustained scrutiny, both positive and negative. Exactly how the doctrine of physicalism should be formulated, what its significance is, and whether it has a justification, in addition to a range of other issues, have all recently been discussed.

We believe such attention is intellectually a very healthy development. There are always parts of the reigning weltanschauung that it would be productive to discard and those that ought to be retained. But this process of critically evaluating various assumptions only proceeds in lockstep with the illumination of their natures. The present anthology was conceived with these thoughts in mind and the chapters that follow all seek to further the project of critically illuminating various aspects of physicalism and its implications. Though we shall let the authors speak for themselves, and the variety of their concerns precludes easy summary, it may help the reader to have a general guide to the volume's contents. The largest section, “Part 1, Physicalism,” comprises chapters generally sympathetic to the doctrine of physicalism. Among the work they undertake, is that of articulating the historical genesis and present justification of physicalism; its most adequate formulation; modal status; and its implications for mental causation and the special sciences. A range of more or less critical chapters comprise the next section, “Part 2, Physicalist Discontents,” and take as their focus the metaphysical presuppositions of physicalism, …