### Is Julius Ceasar Identical to the number 2?

Suppose we got a bunch of arithmatical truths like 2+7=9 and 5+7=12 and so on. Those truths have referring expressions such as '1' and '2' and so on. Some people wonder what those expressions refer to. I guess I really don't know much about this, but I think there are a few things that people tend to accept about arithmatical truths. First, we have an incomplete axiomatization of arithmatic. Those axioms have certain structural properties that arithmatic must satisfy. But, other things satisfy those axioms as well. For example, under one interpretation of the axioms, the number 2 is identical to the set containing the set containing the null set. On another interpretation, the number 2 is identical to Julius Ceasar. Just like with other logical axioms, there is nothing in the structure that guarantees that the referring expressions refer to the some things rather than others. This is supposed to be problematic for arithmatic because it is hard to see what kind of causal relations would help to fix the referent of the referring expressions in arithmatic. So, how do we solve this problem?

Some people solve it by claiming that arithmatical expressions are quantified expressions that quantify over the things that satisfy the axioms of arithmatic. So, for example, 2+2=4 turns out to say something like for any interpretation of the referring expressions and function in '2+2=4' that satisfies the axioms of arithmatic, the function expressed by '+' on that interpretation applied to the referents of '2' and '2' on that interpretation yeilds the referent of '4' on that interpretation (or something like that). Views like this are structuralist views. I think structuralist view is mistaken. Simply put, '2+2=4' just doesn't have a quantificational structure. So, structuralist views are mistaken.

However, I wonder if structuralist views are on the right track. Suppose that we have some arithmatical axioms. We might simply supervaluate over those axioms. Thus, 2+2=4 turns out to be supertrue in virtue of the fact that on any admissiable interpretation of the terms, '2+2=4' expresses a truth. However, it does not express a quantificational truth. One happy consequence of this view is that the Julius Ceasar problem goes away. Although 2=Julius Ceasar on some interpretation, it does not on others. So, it is not supertrue that 2=Julius Ceasar. In fact for any thing whatsoever, it is not supertrue that 2= that thing. However, it is supertrue that 2=something. Supervaluationists allow for supertrue existential statements without true instances. Now, if we simply identify truth in arithmatic as supertruth, we have a solution to the Julius Ceasar problem and we can have all the supposed benefits of structuralism (epistemic access to arithmatical truths, no causal contact with any abstract entities, etc).

Like I said, I don't know much about philosophy of math. So, maybe this view has already been expressed. If it has, I would like to know. If it hasn't I'd like to know what others think of it.

## 10 Comments:

It's super(duper)true that ~(Julius Caesar = 2).

It is not super true that ~(JC=2) since there are some interps on which it is true. I am not sure about super duper true.

Let me first make an off-topic comment about structuralism. Then I'll say something about the view Joshua proposes.

Joshua gives the following characterization of structuralism:

"[A]rithmatical expressions are quantified expressions that quantify over the things that satisfy the axioms of arithmatic. So, for example, 2+2=4 turns out to say something like for any interpretation of the referring expressions and function in '2+2=4' that satisfies the axioms of arithmatic, the function expressed by '+' on that interpretation applied to the referents of '2' and '2' on that interpretation yeilds the referent of '4' on that interpretation (or something like that). Views like this are structuralist views."

I don't know whether this is an accurate characterization, since I don't know much about structuralism. However, if it is an accurate characterization, it seems to me that Church's translation test shows that structuralism is false. '2+2=4' does not say the same thing as 'For any interpretation of the referring expressions and function in "2+2=4" that satisfies the axioms of arithmetic, the function expressed by "+" on that interpretation applied to the referents of "2" and "2" on that interpretation yields the referent of "4" on that interpretation'. The former does not say the same thing as the latter because a content-preserving translation of the latter into a language that does not employ Arabic numerals will clearly refer to the expression '2+2=4' (as well as to the expressions '2' and '4'), while a content-preserving translation of the former into such a language will not.

OK, now something about the view that Joshua proposes. Joshua expresses this view as follows: 'Suppose that we have some arithmatical axioms. We might simply supervaluate over those axioms. Thus, 2+2=4 turns out to be supertrue in virtue of the fact that on any admissiable interpretation of the terms, '2+2=4' expresses a truth.'

I think this is certainly a view that someone might hold. The idea, I take it, is this: First, an interpretation of an arithmetical statement is admissible if and only if it satisfies the axioms of arithmetic. Second, an arithmetical statement is supertrue if and only if every admissible interpretation of that statement is true. Finally, an arithmetical statement is true if and only if it is supertrue.

I think, however, that the fact that our axiomatization of arithmetic is incomplete is going to result in problems for the view. Our axiomatization of arithmetic is incomplete because there are some true arithmetical statements that are not entailed by the arithmetical axioms. Let S be one of these statements. Then either (i) S is true under all admissible intepretations (i.e., under all interpretations under which the arithmetical axioms are true), (ii) S is true under some admissible interpretations and false under others, or (iii) S is false under all admissible interpretations.

But if the view under discussion is true, neither (ii) nor (iii) can be the case. After all, it was stipulated that S is a true arithmetical statement and according to that view, if either (ii) or (iii) is true, then S isn't a true arithmetical statement. Thus, if the view under discussion is true, then (i) must be true: S is true under all interpretations under which the arithmetical axioms are true.

Now either S just happens to be true under all interpretations under which the arithmetical axioms are true--in other words, S is true under each of those interpretations but needn't have been--or it's necessary that S is true under each of those of interpretations. If the former and the view under discussion is true, then some arithmetical statements aren't necessarily true. On the other hand, if the latter, then given the standard supervaluationist characterization of validity (see Williamson's Vagueness, pp. 147-8), the argument from the arithmetical axioms to S is valid, in which case the arithmetical axioms entail S. But it was stipulated that the arithmetical axioms don't entail S. Thus, given the standard supervaluationist characterization of validity, if the view under discussion is true, then some arithmetical statements aren't necessarily true.

Here, then, are the options for the proponent of the view that Joshua proposes:

(a) Deny that arithmetic is incomplete

(b) Accept that some arithmetical statements aren't necessarily true

(c) Reject a supervaluationist treatment of the validity of arithmetical arguments despite accepting a supervaluationist treatment of the truth of arithmetical statements

None of these options seems very palatable to me.

Hi Greg,

Let me respond to your first comment first and then I'll respond to your second comment in a little while. You are absolutely right that the view I expressed fails Church's translation test. I think this indicates that I have failed to correctly characterize Structuralism. I think what we need to do is quantify over the interpretations that satisfy the axioms of arithmetic. so, let's start out with the various interpretations of our logical language that satisfy the arithmetical axioms. There will be a number of such interps. call them, interp 1, interp 2, interp 3 and so on. Let the interpretation of '2' under interp 1 be called '2 interp 1'. Let the interpretation of '2' under interp 2 be called '2 interp 2' and so on. Do the same thing with the interps of '4' and '+' and so on. These expressions and intended to be complex expressions. Now, we can say that 2+2=4 means that for any x if interp x is a satisfier of arithmetic, then the function + interp x applied to {2 interp x, 2 interp x) yields 4 interp x.

Now, I think this will not fail the church test and I think this will be roughly the view that I thought was Structuralism in arithmetic.

Greg,

You present an argument against the view that I propose. You suggest that the proponent of my view is committed to accepting one of (a)-(c). But, you also think that none of (a)-(c) is acceptable. Hence, the view I suggest is mistaken.

I believe that there is another option for the proponent of my view. The proponent of my view should distinguish between an arithmetical statement being a semantic consequence of the axioms of arithmetic and an arithmetical statement being provable from the axioms of arithmetic.

An arithmetical statement is a semantic consequence of the axioms of arithmetic just in case under any interpretation under which the axioms are true, the statement in question must be true as well. An arithmetical statement is provable from the axioms of arithmetic just in case there is a finite series of steps from the axioms using only acceptable rules of inference, to the statement in question.

Now, if I understand Godel's incompleteness theorem correctly (and admittedly it has been a while since I did mathematical logic and I might be misremembering how theorem works), but if I understand it correctly, then the theorem says that there are certain statements in arithmetic for which there is no proof or disproof. But, I believe it is consistent with this that either the statements or their negations are semantic consequences of the axioms of arithmetic.

Hi Joshua,

Let me address your responses in turn.

You respond to my objection to your original formulation of structuralism about arithmetic by conceding that the formulation in question is defective and proposing an alternative. I agree with you that the alternative isn't subject to the objection to the original, which is good.

However, I'm worried about the alternative because I take it that one of the motivations for structuralism is to explain how arithmetical truths can true even though there are no numbers.

Why do I think this causes a problem for the alternative formulation? Because it says "let's start out with the various interpretations of our logical language that satisfy the arithmetical axioms. There will be a number of such interps. call them, interp 1, interp 2, interp 3 and so on. Let the interpretation of '2' under interp 1 be called '2 interp 1'. Let the interpretation of '2' under interp 2 be called '2 interp 2' and so on. Do the same thing with the interps of '4' and '+' and so on. These expressions and intended to be complex expressions. Now, we can say that 2+2=4 means that for any x if interp x is a satisfier of arithmetic, then the function + interp x applied to {2 interp x, 2 interp x) yields 4 interp x". Now presumably '2+2=4' is not supposed to turn out vacuously true, so it must be the case that there is an x such that interp x is a satisfier of arithmetic. But it seems like we're existentially quantifying here over numbers and thus that this formulation of structuralism reintroduces numbers through the back door, as it were.

Does this seem right?

Let me now turn to your response to my objection to the view you propose.

First of all, let me say that I suspected that this might be the correct response to the objection, which is why I said in the e-mail I sent you that I wasn't entirely sure of the objection.

I don't have a reply to your response at this time, but I think that there is an unresolved issue. So let me raise it here.

I think that we should really distinguish between three types of consequence: metaphysical consequence, semantic consequence, and syntactic consequence. Non-supervaluationists, at least, can roughly state what these three types of consequence are supposed to amount to as follows:

-Statement P is a metaphysical consequence of set S of statements iff necessarily, if all the members of S are true, then P is true.

-P is a semantic consequence of S iff necessarily, there is no interpretation I of the non-logical constants in the members of S and in P under which all the members of S are true but P is false. (NOTE: "Interpretation" is being used here in a different sense than it is used by supervaluationists.)

-P is a syntactic consequence of S iff there is a finite proof from the members of S to P that uses only acceptable inference rules.

Now I take it that the distinction you make between "semantic consequence" and "provability" is roughly equivalent to my distinction between metaphysical consequence and syntactic consequence. (Though, of course, you gave a supervaluationist gloss on metaphysical consequence.)

Here's the relevance of all this. I'm no expert on Godel's incompleteness theorem, but I would have thought that if it is true, then there are some arithmetical statements that are not semantic consequences (in my sense) of the arithmetical axioms. Assuming this is so (which it may not be) here's what I'm wondering: Is there a supervaluationist-friendly account of semantic consequence? (Does the account above work, for instance? I'm really not sure, but I haven't thought about it too hard.) Is the conjunction of this account with the view you propose about arithmetic consistent with the claim that there are some arithmetical statements that are not semantic consequences of the arithmetical axioms?

Just to make myself entirely clear, I don't have any idea what the answers to these questions are, since I haven't yet made any attempt to answer them myself. I just thought I'd get the issues out there.

Hi Joshua,

Old post - and topic - but I'm curious what you think.

'2+2=4' expresses a necessary truth. On this view, though, I'm worried that it won't be.

Given this sort of semantics, it looks like it will be indeterminate which proposition the sentence '2+2=4' expresses. Given one way of making precise the relevant expressions, it will express a proposition with Julius Caesar as a constituent. Given another, it won't.

What do we say about whether a sentence, s, expresses a necessary truth when it is indeterminate which of a number of propositions it expresses?

I can see two options:

(1) s expresses a necessary truth just in case, for any legitimate way of making s precise, s expresse s a necessarily true proposition.

(2) s expresses a necessary truth just in case, necessarily, for any legitimate way of making s precise, s expresses a truth.

If (1) is correct, then '2+2=4' won't express a necessary truth. On one legitimate precisification of that sentence, it expresses a proposition with JC as a constituent. But, JC is a contingent thing. So, in worlds at which JC doesn't exist, the relevant proposition is false (or, not-true (we might deny this, I guess)).

If (2) is correct, then I'm less sure what to say. Is it necessarily true that, for any legitimate way of making '2+2=4' precise, it would express a truth? I'm not sure why we should think so. If things had been sufficiently different (with, for instance, our linguistic practices), ways of making that sentence precise which are actually illegitimate would have been legitimate. Some of those ways, it seems, may be ways in which s turns out false.

If that's right, then, if (2) is right, '2+2=4' does not express a necessary truth. (Nor, I think, does any sentence which is indeterminate in this way.)

Andrew,

I guess I prefer the second of your two options. but, I'm not sure why we should have to say that

"If things had been sufficiently different (with, for instance, our linguistic practices), ways of making that sentence precise which are actually illegitimate would have been legitimate."

I wonder if things had been sufficiently different if the words would have been the same words that we are actually using. If they would be different words, then we shouldn't care about making those words precise when determining the truth value of our mathematical sentences. But, I really don't know about this response.

There might be other options available though. Consider this option:

(3) S expresses a necessary truth just in case for some way of making the terms of S precise, S expresses a necessary truth.

That one doesn't seem to have the problems of (1). Alternatively, we might say that necessity is a gerrymandered concept. Perhaps, mathematical sentences are necessary in different ways than other statements. Perhaps what makes mathematical sentences necessary is that they express a truth on every legitimate precisification.

Finally, we might just be conventionalists about necessity. We just have the appropriate conventions for treating mathematical statements as necessarily truths.

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