Sunday, February 6, 2011

Five Reasons to be a Modal Carnapian

I currently believe mathematics is best understood in terms of combinatorial possibility plus Carnapian framework stipulations. One reason for thinking this is, of course, that understanding mathematical objects in this way lets you tell a nice story about access to abstract mathematical objects, like the one I tell in my dissertation! But here are 4 other reasons.

1. Thinking in terms of combinatoiral possibility lets you solve the bad company objection for neo-carnapianism about objects.

You can say: It is OK to stipulate that there are sets but not to make the stipulations for tonk, or for Boolos parities because...

One can safely extend one's language by introducing object stipulations S if and only if it would be combinatorially possible for the objects one's language currently acknowledges at each possible world to be supplimented by new objects in such a way as to satisfy S.

2. Thinking about things in terms of combinatorial possibility + carnapian framework stipulations explains why certain mathematical questions are substantive questions while others are not (substantive ones involve disagreement over combinatorial possibility as well as over which from among the many objects which it would be combinatorially possible to stipulate you actually do stipulate)

For example:

There is no substantive mathematical question about whether there are sets or categories or both, because everyone agrees that it would be combinatorially for there to be objects satisfy Carnapian framework stipulations phrased in terms of combinatorial possibility for the sets, and for the categories.

There is a substantive mathematical question about whether the Goldbach conjecture is true, because it is combinatorially impossible that the Fs and the Gs should satisfy the informal (non-first order logical!) framework stipulations for sets could yield different answers to Goldbach.

3. "Mathematical Hypothesis"-based alternatives like structuralism, fictionalism and plenetudinous platonism already need *some* powerful modal notion or a sense of possible and impossible which can take into account vocabulary that goes beyond the connectives of first order logic, if they want to capture intuitive claims about all questions in arithmetic having right answers.

Intuitively there are right answers to at least all questions about arithmetic. Each first order logical hypothesis about the numbers necessitates right answers to all questions about arithmetic. Therefore, any mathematical hypothesis that claims to cash out the truth conditions for our claim that phi in terms of the claim that if H then phi must crucially use non-first order vocabulary. Therefore if you want to make some hypothetical view of math work, you must either bite the bullet that some simple questions of arithmetic have no answer, or invoke a more mathematically powerful notion of possibilty/consequence

4. The notion of combinatorial possibility is very crisp and simple, indeed it maybe well the crispest and simplest notion that yields enough mathematical power to reconstruct intuitive verdics about truthvalues in arithmetic as per (3).

For example, if you accept Tarski's definition of logical truth in terms of facts about how it would be possible to reinterpret all properties and relations involved (where possible doesn't mean that a person could give a rule for how to do it, or somehow list the new extensions) then you pretty much already accept the powerful notion of how it would be in principle possible for an arbitrary n-place relation to apply to some objects which I have in mind when I talk about combinatorial possibility. So if you're not too much of a finitist/intuitionist to accept the very broad sense of in principle possible reassignment of extensions to predicates used in the Tarski definition, I think you should be OK with my notion of combinatorial possibility.

Secondly, combinatorial possibility faces none of the quandires that make people skeptical about the notion of metaphysical possibility e.g. it is possible for something to be both red all over and green all over, or for there to be zombies? We avoid these problem because facts about combinatorial possibility ignore all metaphysical facts about particular properties and relations involved. In some cases one might perhaps argue that some natural language sentences are vague with regard to what pattern of relationships between objects they assert. But once you specify that you are asking whether e.g. combiposs( Ex Redallover(x)&Greenallover(x)), it is perfectly clear that this is combinatorially possible, since it would be combinatorially possible to choose extensions for Redallover() and Greenallover() that make this true.

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