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We generally hold that a mathematical proposition such as "2 + 2 = 4" is necessarily true; it is difficult to imagine a possible world in which it is false. However, is it possible that "2 + 2 = 4" is not a statement that expresses a mathematical necessity (or an operation involving numeric values that must provide a certain result), but rather presents an inductive inference based on how people currently "define" the number "2", and the operator "+"? We could, for example, someday come to discover that "2" does not represent "2 things or ideas"; what we call 2 things may turn out to be 3 things, or 1 thing, etc. If this is possible then it would seem that "2 + 2 = 4" is an empirical, not a rational truth. Is this intelligible?

I realize that this last statement, that we could discover 2 to refer to 3 things, etc., entails a theory of what a number is, i.e. a number "represents a quantity or amount of something". It seems, though, that in order to conclude that "2 + 2 = 4" is a necessary truth we must hold that a number is a "fixed" value; for example, a number is a (theoretical) quantity which is not grounded in any emprical relation (e.g. "2" represents a theoretical value that is "fixed"). Surely, though, the way we use numbers seems to indicate (as does applied mathematics) that we do not (always) mean that "2" is theoretical or rational; if this were true then we might (ironically) lose all meaning that mathematics provides in everday use.

I realize there is more to this issue than what I present here; what, for example, does it mean for a number "to represent" to something or another. I think, however, there is enough here to make the question clear.

October 14, 2008

Response from Richard Heck on October 17, 2008

Perhaps the first thing to say here is that we need to distinguish the question whether it is necessary that 2+2=4 from the question whether the sentence "2+2=4" is necessarily true. It seems to me that no sentence is necessarily true. Any sentence might have been false, simply because that sentence might have meant something other than what it in fact means. For example, "2+2=4" might have meant that 3+3=4, and then it would have been false. And that is what it would have meant had "2" meant 3 rather than what it does mean, namely, 2. So I agree absolutely the whether "2+2=4" is true depends upon how we what "2" and "+" and "4" and "=" all mean, not to mention the grammatical rules that govern the significance of combining them in a certain way. And if you want to put that by saying that the truth of this sentence depends upon how we "define" the numeral "2", I won't object. Not too strongly, anyway.

But it is an entirely different question whether it is necessary that 2+2=4. That is not at all a question about what our words mean. It is a question about addition, numbers, and the like. The general view is indeed that 2+2 could not have been other than 4, and the reason is pretty simple: It's very difficult to conceive how 2+2 could have been other than 4.

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