Julius Caesar and the Number 2
Jill Dieterle
[1] What are numbers? What is, say, the number 2? Is it a property? Is it some sort of object? If so, what kind of object is the number 2? One of Frege's goals in Die Grundlagen der Arithmetikwas to answer such questions. He eventually came to the conclusion that numbers are extensions; i.e., kinds of objects. Ultimately, Frege's overall project failed, for Russell's paradox follows directly from the Fregean theory of extensions. But is there something salvageable in Frege's system? In the course of this paper, I will argue that there is.
[2] In Frege's Conception of Numbers as Objects, Crispin Wright argues that Frege's context principle -- "Never to ask for the bedeutung of a word in isolation, but only in the context of a sentence" -- should be interpreted as a principle concerning reference (as opposed to sense). Reading the principle this way, Frege's claim is that it is illegitimate to ask whether there really are objects of a given kind once it has been determined that expressions that stand for them function as singular terms in true sentences. Wright calls this the "syntactic priority thesis."1
[3] In the Grundlagen, Frege discusses three "symptoms" that signal that an expression is functioning as a singular term and thus refers to an object (if it refers at all): (i) It is a Fregean proper name (i.e., a proper name, a definite description, or a personal or demonstrative pronoun2 ); in Frege's terminology, it is a saturated expression.3 (ii) We are willing to willing to prefix the expression with the definite article. (iii) The expression can legitimately flank the identity sign in significant statements of identity. Frege argues that numbers are objects; his argument for this claim depends wholly on the fact that numerical expressions meet conditions (i) to (iii) above.
[4] In Part I of the paper, I examine sections 55 - 86 of the Grundlagen, discussing why Frege concluded that numbers were a certain kind of extension. In Part II, I discuss how parts of Frege's project can be salvaged.
1. Frege's Numbers
[5]What kinds of objects do numerical expressions denote? Frege attempts to answer this question in sections 55 - 86 of the Grundlagen. In section 55, Frege tentatively proposes the following definition:
the number 0 belongs to a concept, if the proposition that a does not fall under that concept is true universally, whatever a may be.This is an explanation of numbers as cardinality quantifiers. In symbolic notation:
the number 1 belongs to a concept F, if the proposition that a does not fall under F is not true universally, whatever a may be, and if from the proposition "a falls under F" and "b falls under F" it follows universally that a and b are the same.
the number (n + 1) belongs to a concept F, if there is an object a falling under F and such that the number n belongs to the concept "falling under F, but not a." (Frege (1884), section 55)
(E0x)Fx means (Ax)¬FxFrege eventually rejects this as a characterization of numbers, for we have not established that numbers are objects; this "definition" characterizes numbers as adjectives:
(E1x)Fx means ¬(Ax)¬Fx & (Ax)(Ay)((Fx & Fy) --> x = y)
(En+1)Fx means (Ex)(Fx & (Eny)(Fy & ¬(x = y)))
It is only an illusion that we have defined 0 and 1; in reality, we have only fixed the senses of the phrases "the number 0 belongs to" and "the number 1 belongs to"; but we have no authority to pick out 0 and 1 here as self-subsistent objects that can be recognized as the same again. (Frege, (1884) section 56.)4[6] The second attempt begins in section 62. After showing that numbers are objects and that they are not subjective, Frege asks, "How, then, are numbers given to us...?" (Frege (1884) section 62). This is an epistemological question in that it asks how we apprehend numbers, but Frege's strategy in answering it is to turn to the meanings of sentences in which numerical expressions occur, and so he invokes the context principle. If we can fix the senses of these sentences, then we will have explained what it is for such sentences to be true, and how we are able to understand such sentences. But then, given the context principle, we will be able to determine the reference on numerical expressions.5
[7] One might object at this point that the context principle couldn't allow us to determine the reference of an expression since it does not establish some kind of extra-linguistic association between the expression and its referent. The context principle may allow us to conclude that a particular term has reference, but it doesn't help us determine its referent. But this objection misses the point of the principle. As Dummett notes:
According to [the context principle], it is only as occurring in the context of a sentence that a name can so much as be said to stand for an object. It follows that to determine what 'the number 1' stands for, and hence what the number 1 is, involves fixing the senses of sentences in which that proper name occurs, and does not involve anything beyond this. (1981: 366)6So if we can understand sentences in which 'the number 1' occurs and we can determine the truth values of such sentences, then we will have grasped the reference of 'the number 1'.
[8] Since Frege has argued that numbers are objects (sections 57 - 61), identity statements in which numerical expressions occur must have a sense. Identity is a relation that applies to objects, and all objects must have identity conditions. Using the concept of identity, then, Frege's goal is to explain what it is that is identical. Thus he turns his attention to identity statements containing numerical expressions:
In our present case, we have to define the sense of the proposition "the number which belongs to the concept F is the same as that which belongs to the concept G"; that is, we must reproduce the content of this proposition in other terms, avoiding the use of the expression "the Number which belongs to the concept F". (Frege (1884) section 66.)Stripped of Fregean terminology, the goal is to find a way to explain the meaning of "the number of Fs is the same as the number of Gs" without using the phrase "the number of". Frege's idea is to use a principle of David Hume:
When two numbers are so combined as that the one has always a unit answering to every unit of the other, we pronounce them equal.7This is the notion of one-to-one correlation, and Frege intended to use it as his criterion of identity. So this gives us:
The number of Fs is the same as the number of Gsis equivalent to
There are just as many Fs as Gs.Notice that 'the number of' does not occur in the second sentence. It has been contextually defined in terms of the identity conditions for numbers. Henceforth, I will refer to this as "the contextual definition".
[9] The contextual definition initially seems to fulfill Frege's goals. It fixes the sense of statements of identity in which numerical expressions occur, and given the context principle, would seem to enable us to determine the reference of numerical expressions. Frege is not, however, content with the contextual definition, for it does not solve what has been come to be called "the Julius Caesar problem". In short, we are not able to determine the truth value of sentences of the form
the number of Fs = qwhere q is not of the form "the number of Gs". So we cannot determine whether the number of Fs is Julius Caesar.8
[10] This objection may seem odd, but for Frege, all identity statements must have determinate truth values. In fact, all concepts and relations (all functions) must admit every object as argument. For any first level concept F and any object a, the sentence "Fa" must be either true or false, for the range of the variables is not restricted at all -- the first-level quantifiers in Frege's system range over all objects. Given this, the proposed definition fails since it does not determine the truth value of sentences of the form "the number of Fs = q" where q is not an expression of the form "the number of Gs".
[11] Thus Frege comes to his final (and fatal) definition. He says:
The Number which belongs to the concept F is the extension of the concept "equal to the concept F." (Frege (1884) section 68)In other words, the number which belongs to the concept F is the class of all classes that are equinumerous with the concept F.
[12] After defining the notion of equinumerosity, the remaining sections of the Grundlagen are devoted to proofs using the definition of number. In section 73, Frege proves Hume's principle; in section 78 he proves what are in effect Peano's postulates.9 In sections 82 and 83, Frege outlines a proof that every finite number has a successor. Thus, there is no last number and the natural numbers are infinite.
[13] On Frege's view, extensions (or classes) are objects, so the final definition does not run afoul of the claim that numbers are objects. However, it is the notion of extension that leads to Russell's paradox. Axiom V of Frege's formal system says that for any concept F, the extension of F is the same as the extension of G if and only if for any object x, Fx iff Gx. Part of the problem with this axiom is that certain extensions are in the range of the variable x; the 'for any object x' includes extensions. Consider the concept R: 'extensions which are not members of themselves'. Let x be the extension of this concept. Is x in R? If so (i.e., it is a member of itself), then it is not a member of itself. But if it is not a member of itself, then it is a member of itself.10
[14] It is widely held that Russell's paradox was devastating to Frege's entire project. Of course, the paradox arose from the formal system laid out in the Grundgesetze, not from the Grundlagen itself. Frege's ontology in the latter work, however, offers him little protection against said paradox. But can Frege be saved from contradiction? Can any of Frege's program be saved? I now turn to this question.
2. Frege and Structuralism
[15] From a structuralist standpoint, Frege went too far, for the contextual definition of number is enough to characterize the structure of the natural numbers.11 In Chapter 4 of (1983), Crispin Wright derives Peano's postulates in second-order logic with Hume's principle as an added axiom.12 Hume's principle is, of course, Frege's contextual definition of number. Peano's postulates explicitly define the structure of the natural numbers, and on a structuralist view, at this point Frege had done enough. Suppose that he had stopped with the contextual definition. What would have happened to the rest of his program?
[16] The important question is whether Hume's principle is satisfiable.13 If it is, then at least some of Frege's project can be saved. George Boolos, John Burgess, and Harold Hodes have shown that the principle is, in fact, satisfiable. 14 Hodes' project is to show that a version of logicism is true: mathematics really is (third-order) logic. He rejects Frege's claim that numbers are objects, but much of the rest of Frege's program survives. Boolos' project is a bit different, and he actually shows more than just that Hume's principle is satisfiable. Boolos proves that all of the principles Frege actually utilizes in the Grundlagen are consistent.
[17] The only extensions Frege makes use of in the Grundlagen are extensions of higher-level concepts of the form 'equinumerous with the concept F'. Boolos shows that it is consistent to assume the existence of these extensions and thus the mathematical program can be completed without contradiction. Boolos shows that
Numbers: (AF)(E!x)(AH)(Hnx iff H eq F)is consistent, where 'n' is a two-place predicate letter that connects concept variables and object variables. 'Hnx' can be read 'H is in the extension x'. 'H eq F' should be read 'H is equinumerous with F'. Boolos then goes on to show that Hume's principle follows and the rest of Frege's program in the Grundlagen can be completed; i.e., Peano's postulates can be derived using standard axiomatic second-order logic.
[18] Since Numbers is satisfiable, Frege's program can be completed without contradiction. But the only extensions that can be countenanced are those that Numbers gives us license to include, since to accept a general principle involving the existence of extensions leads to Russell's paradox. This seems to be rather arbitrary; it would be preferable to find an alternate characterization of number.15
[19] As noted earlier, Frege's contextual definition characterizes a structure and thus, on a structuralist view, he had done enough. It has been shown that Hume's principle is satisfiable and that Peano's postulates can be derived in second-order logic with Hume's principle as an added axiom. But Fregean questions remain: Is Julius Caesar a number? If so, which one?
[20] Recall that on the strict Fregean view, all identity statements must have determinate truth values; if a term has objectual reference, it can legitimately flank the identity sign in any statement of identity. Furthermore, for Frege, "all objects" refers to a domain which includes all objects. Objects and the expressions that stand for them are saturated, whereas functions and the expressions that stand for them are unsaturated. With the goal of salvaging much of Frege's system, I propose that we alter this position somewhat. The Julius Caesar problem can be solved by taking the notion of saturation as relative -- relative to a coherent theory or area of discourse.16 (I will call this the "Neo-Fregean" position.) Since saturation carries with it the related notions of singular termhood and objecthood, these latter notions become relative as well. On the Neo-Fregean view, what is object from one perspective is function from another.
[21] The Neo-Fregean suggestion is to take the Fregean principle:
(F) If an expression functions as a singular term in true sentences, then there is an object denoted by that expressionand alter it in the following way:
(NF) If an expression functions as a singular term in true sentences relative to a theory T, then there is an object denoted by that expression.[22] The idea is to leave most of Frege's project intact; the only alteration is to relativize the notion of saturation, and thereby the corresponding notions of singular termhood and objecthood. Instead of claiming that there is one domain which includes all objects, we allow various domains. Whether or not an expression refers to an object is relative to which domain we are talking about.
[23] The Neo-Fregean suggestion respects Frege's syntactic priority thesis: expressions that function as singular terms in true sentences of the theory refer to objects. The structure of true sentences of the language plays the dominant role in settling ontological questions. But the Neo-Fregean diverges from Frege insofar as he or she denies that there is one universal domain of objects. The Neo-Fregean claims that ontology differs from theory to theory; objecthood is theory relative. The program attempts to codify the intuitively plausible idea that the world can be divided into objects in different ways. For example, consider the question, "How many objects are on the table?" asked when "all there is" on the table is a deck of cards. Is the correct answer "One", "Fifty-two", or some number x, where x is the number of molecules in the deck? The Neo-Fregean's claim is "the correct answer" is context dependent and theory relative. What counts as an object depends on the expressive resources of the language one employs and how that language individuates.
[24] Of course, the principle (NF) raises a number of questions. Most prominently, how does an expression function as a singular term relative to a particular theory or area of discourse? Suppose that Frege's "three symptoms" (i.e., that the expression be a Fregean proper name, that we can prefix the expression with the definite article, and that the expression can legitimately flank the identity sign) were adequate to non-circularly demarcate the class of expressions that refer to objects.17 Suppose further that we limit ourselves to the expressive resources of a particular theory; i.e., its concepts and corresponding predicates, constants, and general individuative apparatus. These resources will determine which expressions function as singular terms by expressly limiting (or expanding) what we are able to say relative to that theory, what things we are able to talk about, and what kinds of inferences are licensed.
[25] Suppose, for instance, that a theory T lacks the resources to discriminate between what, from a wider perspective, are different tokens of the same type. The vocabulary of T contains no predicate (monadic or relational) truly applicable to one token but not another; T is incapable of discriminating more finely than types. Relative to T, terms referring to the type will function as singular terms to which we apply the definite article and that we allow to flank the identity sign. Consider, for example, a language that is incapable of discriminating between various individuals who have held the office of President of the United States. Call this language Lp. Lp contains the usual logical vocabulary plus monadic predicates such as '...has veto power' and '... is elected every four years'. But Lp lacks the linguistic resources for distinguishing between what, from a wider perspective, are distinct individuals of whom we can truly predicate '... is/was a President of the United States'. Distinctions between Jimmy Carter, Ronald Reagan, and Bill Clinton are inexpressible in Lp; relative to Lp, 'President' will function as a singular term, and inferences treating it as such will be licensed. From 'The President is elected every four years' we can validly infer that there is something such that it is elected every four years. But the referent of 'President' is the office of President. Relative to Lp -- the language in which the inference is embedded -- we cannot even "see" the various individuals who have held the office of President; i.e., Lp does not have the expressive resources to make the distinctions which allow us to individuate between those individuals.
[26] The Neo-Fregean suggestion is that the objects of a theory or area of discourse are those things to which its singular terms refer. Relative to Lp, the office of President is an object. But relative to a theory with finer grained discriminative apparatus, '...is/was a President' functions predicatively, true of a number of individuals. From this perspective, 'President' does not refer to an object, but rather a Fregean function which takes other objects as arguments.
[27] We are moving toward the structuralist conception of objecthood. On the structuralist view numbers are objects, but what it is to be an object is structurally defined: to be a mathematical object is to be a place in a structure. Anything at all can have a mathematical structure; any denumerably infinite system of objects whatsoever can display the structure of the natural numbers. But structures themselves are types, and mathematical objects are places in structures. What it is to be the natural number two is to be the appropriate place in the structure of the natural numbers.
[28] To fully arrive at the structuralist conception of objecthood, we need to add one thing to the Neo-Fregean strategy: a coherent theory characterizes a structure. Before turning to mathematics, it will be useful to look at a non-mathematical example. Consider, again, the office of the President of the United States. This office is part of a structure that is characterized by the Constitution. The political structure of the United States has the positions President, Vice President, Senators, Representatives, and Supreme Court Justices. The Constitution gives us a formula for determining how many places the political structure contains. There are two Senators for every state and one Representative for every voting district; currently, the political structure of the United States has 100 Senators, 435 Representatives, and 9 Supreme Court Justices. Every two years when elections are held, the system changes.18 New individuals are elected and take on the role of their predecessors. But the overall structure -- the way the various individuals are related to one another vis a vis their role in the political process -- remains the same. The President has veto power regardless of who occupies the office of President, the Congress has the power to override the President's veto no matter who sits in the House and the Senate, and the Supreme Court has the power to declare acts of Congress unconstitutional no matter who wears the robes. The structure of the United States government is just what all possible systems that could instantiate this structure have in common.19
[29] In political science textbooks, the language pertaining to the political structure of the United States is often employed. When we teach students how the government works, we use sentences whose variables range over the positions as such, without regard to any system that might instantiate the political structure. The instructor will say things like, "The President has the power to veto a bill". Were a student to ask, "Do you mean George Bush or Bill Clinton?" the instructor would take this as evidence that the student did not understand. The President, as such, has the power to veto a bill; it is a property of the position in the political structure, and not merely of an individual who occupies that position.
[30] Let us return to mathematics; specifically, to arithmetic and the natural number structure. The language of second-order arithmetic characterizes the structure of the natural numbers and implicitly defines its objects. The language contains names ('1', '2', '3', ...) which refer to the objects of arithmetic, and predicates ('successor', 'less than', 'plus', etc.) which describe the arrangement of the objects. Peano's postulates (stated in the metalanguage) give an explicit definition of the structure of the natural numbers by describing a distinguished element (0) and a successor function for generating the rest of the structure:
(1) 0 is a number.(1), (2), (3), and (4) characterize an infinite progression in which each element is a number. (1) and (4) give us the unique element with which we begin. (5) is the induction axiom. The natural number structure is thus characterized, and we define the relations of addition, multiplication, etc. on this structure. Notice that the language described does not have the expressive resources to refer to, say, the square root of 2. This is because the square root of 2 is not a part of the natural number structure, and the only objects that the language of arithmetic is capable of referring to are those that are places in the structure of the natural numbers. Relative to the natural number structure, natural numbers are objects. Relative to a more encompassing structure and a richer language, however, numerical expressions may function predicatively.
(2) If n is a number, then the successor of n (s(n)) is a number.
(3) If s(n) = s(m), then n = m.
(4) 0 is not the successor of any number.
(5) Any property which belongs to 0, and also to the successor of any number which has that property, belongs to all numbers.
[31] The informal language of set theory is one language in which numerical expressions function predicatively. Relative to set theoretic discourse, we can say that particular systems of objects (such as the finite von Neumann ordinals) have the structure of the natural numbers; i.e., they are tokens of the natural number structure type. In such systems, particular sets play the role of the numbers and numerical expressions can be predicated of them. They are not, however, identical to the natural numbers, just as individuals who have held the office of President of the United States are not identical to the office itself.
[32] The distinction I am pointing to is that between offices and office holders. The offices can be thought of as classes of the individuals who have held/currently hold the offices. If we restrict our discourse in such a way that we cannot discriminate between the members of the class, then, relative to that discourse, the classes themselves are objects. But if a discourse can discriminate between the individuals, then expressions that refer to the offices/classes function as complex predications, true of a number of individuals. Consider again the President of the United States. Relative to a full background language, we can discriminate between individuals who have played the role of President. In the sentence "Bill Clinton is the President", '... is the President' is functioning as a complex predicate; the Clinton government is a token of the type of political structure of the United States, and Clinton himself occupies the appropriate place in the government.
[33] Analogously, the finite von Neumann ordinals are a token of the natural number structure type. {Ø, {Ø}} occupies the appropriate place in this token and '... is two' is predicable of it. Again, {Ø, {Ø}} is not identical to 2; {Ø, {Ø}} has the property of filling the two-office. But if we limit our discourse to the language of arithmetic, we cannot discriminate between different tokens of the natural number structure type -- we lack the conceptual resources to distinguish between objects based on any properties other than their arithmetical structural properties -- and these properties are held by all tokens of that type.
[34] Earlier, I noted that the relativity of ontology is a central thesis of structuralism. Whether we classify something as an object depends on the point of view from which we are speaking. It follows directly from this that there is no absolute domain of objects. Relative to the structure of the natural numbers, 2 is an object in its own right; relative to the political theory of the United States, President is an object. But neither are objects from the point of view of a more encompassing theory, whose corresponding language has the capacity to refer to sets in the former case and people in the latter who play those roles in particular systems. In the more encompassing theories, they are functions picked out by predicates, which take objects as arguments.
[35] We are now ready to return to the original problem. The goal was to salvage as much of Frege's project as possible. I suggested that the Neo-Fregean take Frege's notion of saturation as relative. Using the examples of arithmetic and the political theory of the United States, we can see how this relative notion of saturation will differ from Frege's original notion.
[36] From the Neo-Fregean position, relative to political theory, 'President' is a saturated expression. It displays all of the Fregean symptoms of an expression that stands for an object: we readily apply the definite article to it ("the President"), it is a Fregean proper name, and identity conditions can be given for its referent independently of any individual who occupies that office. The same can be said for the expression '2'. It displays all of the Fregean symptoms of an expression that stands for an object, and identity conditions can be given for its referent without reference to any particular system that instantiates the natural number structure.
[37] Relative to another theory or area of discourse, however, both 'President' and '2' are unsaturated expressions. In a more encompassing theory, '... was a President of the United States' and '... is 2' require arguments to complete them. There are certain systems which display the structure of the natural numbers (e.g., the finite von Neumann ordinals), and there are systems which instantiate the structure of our political system (particular governments). With regard to the systems themselves, the expressions are functions; '... was a President of the United States' can be completed by the saturated expressions 'Jimmy Carter', 'Bill Clinton', and so on. '... is 2' is a function which takes {Ø, {Ø}} as one of its arguments.
[38] Frege's Julius Caesar problem stems from the orientation that the structuralist program rejects; namely, that the phrase 'all objects' refers to a fixed domain which includes all objects. The Julius Caesar problem arose for Frege because he thought that all terms with objectual reference must be admissible in argument place in every statement of identity. The contextual definition of number did not determine the truth value of identity statements of the form
the number of Fs = qwhere Julius Caesar, for example, is the value of q. The structuralist rejects the claim that all objects can be taken as arguments for identity statements (if the claim is take in the sense that Frege intended) because the phrase 'all objects' makes no sense if the 'all' really means all. Whether or not something is an object is theory relative. Of course, a structuralist can say that identity statements must take all objects as arguments, but in so saying, the 'all' is restricted; it refers to all objects in a particular structure.
[39] We can meaningfully ask questions concerning the identity of places in one structure, but any question of identity between the objects that comprise systems that have that structure and the places in the structure itself are illegitimate. For example, "2 + 2 = 4" and "42 = 16" are legitimate because all of the constants involved refer to objects in the natural number structure. Identity statements like "{Ø, {Ø}} = 2", however, are illegitimate. Why is this? Well, suppose that "{Ø, {Ø}} = 2" was a legitimate identity statement. Then, no matter what one's view of ontology, it would have to be the case that {Ø, {Ø}} and 2 were both objects. But if one's language has the discriminative capacity to recognize properties of the structure of which {Ø, {Ø}} is a part other than its arithmetical structural properties, then 2 is not an object but rather a function ("... is two") which takes objects as arguments. Relative to the natural number structure and its diminished language, one cannot even formulate the question of whether {Ø, {Ø}} is two. Relative to a more encompassing structure, the question can be formulated, but it is not a question of identity. It is, instead, a question of office occupancy; it is asking whether, in a particular system, {Ø, {Ø}} plays the role of two. The question must be asked from outside the language of arithmetic. From the Neo-Fregean position, '... is two' is an unsaturated expression in this context, and needs an argument to complete it. '... is two' is a function which can take objects in the more encompassing structure as its arguments. And we do not countenance identity claims between objects and functions.
[40] Returning to Frege's Julius Caesar problem, one could imagine a system that instantiates the natural number structure in which Julius Caesar plays the role of, say, two. The question, "Is Julius Caesar two?" would make sense in this context, but it would not be a question of identity because in this context, '... is two' is an unsaturated expression. Relative to the area of discourse in which one could even formulate the question, 'two' is not a saturated expression; it is a function ('... is two') that needs completion. In Fregean terminology, if Julius Caesar does play the role of two in a particular system, then Julius Caesar falls under the concept two, for the sentence "Julius Caesar is two" is true in that context.
3. Conclusion
[41] From a structuralist standpoint, Frege went too far in his definition of number, for the contextual definition is enough to characterize the natural number structure. Frege took the fatal step because of the Julius Caesar problem, which resulted from his view of objecthood. The Neo-Fregean position avoids this problem without running into contradiction by claiming that saturation is theory relative. On the Neo-Fregean view, the locution "all objects" makes no sense if the 'all' really means all, because objecthood is relative. We can say that identity statements must take all objects as arguments, but the 'all' here is restricted to a particular structure.
The Julius Caesar problem is avoided by the Neo-Fregean because in the context in which one could even formulate the question "Is Julius Caesar two?" '... is two' is a function which takes objects as arguments. To say that Julius Caesar is two from this perspective is not to make a claim of identity.
[42] It has been shown that Hume's principle is satisfiable, and that Frege's project can be completed in a second-order system with Hume's principle as an added axiom. Frege's fatal step was treating extensions as objects that can fall under concepts, and this has been avoided by the Neo-Fregean position. But what has survived? How much of Frege's program has been salvaged?20
[43] On the Neo-Fregean view that I have advocated, numbers are objects. This is a central claim of Frege's Grundlagen, and it has survived. Furthermore, we have kept the syntactic priority thesis intact. The only fundamental change was the relativization of saturation, and the corresponding notions of singular termhood and objecthood. This is a somewhat radical change, but it saves the Fregean program from contradiction while providing an answer to the Julius Caesar problem.
[44] Finally, it is worth pointing out that Frege himself would not have liked the Neo-Fregean answer. I have suggested that Peano's postulates explicitly characterize the natural number structure, and this implicitly defines the objects of arithmetic. In a letter to Hilbert dated December 27, 1899, Frege writes:
... axioms and theorems can never attempt to fix the bedeutung of a sign or word which appears in them; rather, this must already stand fixed.21On Frege's view, we have not adequately characterized mathematical objects as objects unless we are able to specify of what their essence consists. This must be done independently of the axioms that apply to them -- we have to be able to say what numerical constants refer to in some direct, absolute way. But although this was Frege's view, is it essential to his program in the Grundlagen? Michael Hallett argues that it is not. He writes:
First, it seems clear from Dedekind's work and, say, from Boolos' recent elementary reconstruction of the theory using just second-order logic, that this extra, ontological component [specifying the essence of mathematical objects] is not necessary, even if partially 'theory driven'. Second, even though it produces some more theory to answer the further ontological question, the Frege account does not succeed in answering the question in any suasive way. For, even after being told that "numbers are logical objects of such-and-such kind", one might still ask, "How do you know that you really have got hold of 'purely logical objects'?" (Hallet 1990: 198)22And, furthermore, how do we know that Julius Caesar is not an extension?
Jill Dieterle
Eastern Michigan University
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Notes
1 See Wright (1983), (1988), and (1990). See also Hale (1984) and (1987).
2 Of course, Frege eventually came to regard complete declarative sentences as Fregean proper names as well, but I omit discussion of them in this paper.
3 Concepts, functions, and the expressions that stand for them are unsaturated.
4 After having said this, Frege devotes section 57 through 61 to showing that numbers are objects.
5 As Dummett notes, this is the first example of the "linguistic turn" in the history of philosophy. See (Dummett 1991: 111-112).
6 See also (Dummett 1991: 156)
7 A Treatise of Human Nature, Book I, part iii, section 1. Quoted in Grundlagen, section 63.
8 The Julius Caesar example actually occurs in section 56 of the Grundlagen; the example Frege uses in section 66 has to do with directions. He is discussing how to contextually define directions using lines. His objection is that the definition will not tell us whether England is identical to the direction of the Earth's axis. Frege says, "Naturally, no one is going to confuse England with the direction of the Earth's axis, but that is no thanks to our definition of direction".
9 After defining '0', '1', and 'successor'.
10 This is, of course, Russell's Paradox.
11 See e.g., Shapiro (1989) and Parsons (1990) for a discussion of the structuralist philosophy of mathematics. See also Kraut (1980) for a similar view of objecthood.
12 George Boolos and Harold Hodes both show something similar. See Boolos (1986/7) and (1987) and Hodes (1984).
13 This model theoretic question is, of course, not one Frege would have asked, for he had no vantage point from which to ask it. From our perspective, however, it is the important question. We want to know if Frege's project can be completed without contradiction, and if Hume's principle is satisfiable, then it can be.
14 Boolos (1986/7) and (1987), Hodes (1984), and Burgess (1983). Relevant portions quoted in Boolos (1987).
15 I do not intend for this to be a criticism of Boolos. He does not advocate that we actually take this route; Boolos' goal is to show that Frege's principles in the Grundlagen are, by themselves, not inconsistent.
16 The use of 'theory' here is unfortunately yet unavoidably imprecise. I mean something along the lines of an area of discourse with a high degree of conceptual homogeneity.
17 They actually are not adequate; the specification is circular. However, the fact that Frege's three symptoms are inadequate will not affect the discussion here; as long as we can give conditions that fulfill the goal, the following position is sustainable. Dummett (1983), Hale (1979) and (1984), and Wright (1983) have improved on Frege's conditions, and the resulting criteria do seem to fulfill the goal.
18 That is, the office holders. It also changes whenever someone resigns or dies or a new Supreme Court Justice is appointed, etc. The point is that the system changes whenever a new individual (or individuals) occupies an office.
19 The 'possible' is used to avoid including irrelevant features that all actual systems have had in common. For example, all actual Presidents have been male, but that is not essential to the political structure of the United States.
20 Of course, I have said nothing about Frege's logicism, which was a large part of his overall program.
21 This letter is reproduced in Frege (1980).
22 Emphasis in original. The reference to Boolos is (1987); the reference to Dedekind is (1888).
©1997 Jill Dieterle
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