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Monday, May 13, 2013

Possibility, Actuality and Necessity

In my last post, in response to Timothy Williamson's hesitations regarding naturalism, I rambled a bit about possible difficulties in sorting out a naturalistic understanding of mathematical truth.  I was reflecting on the problem of universality:  Logical and mathematical truths are not truths about the actual world, but extend to all possible worlds.  That would seem to be very hard to explain, if logical and mathematical truths are limited by local factors--factors which ground them in facts about our world.  Today I have worked out a possible solution.

The first step is to distinguish between two types of possibility:  logical possibility and physical possibility.  Another way of putting it is that possibility can be relative to a logical framework or a physical one.  When we say that something is physically possible, we mean it cannot be ruled out by the laws of physics (or, if you don't want to favor physics above other sciences, we can just say "laws of nature").  When we say that something is logically possible, on the other hand, we mean it is consistent with the rules of logic.

The next step is to consider that there is more than one possible set of physical laws.  (This can mean that there is more than one logically possible set of physical laws, or even that there is more than one possible set of laws which could adequately describe the known universe.)  I think of physical laws as predictive models or frameworks.  So, to say that something is physically possible is to say that it cannot be ruled out by our current predictive model for understanding the actual world.  Physical possibility, in so far as it is conceivable, is relative to a predictive model.

Similarly, there is more than one possible logical system.  To say that something is logically possible is to say that it cannot be ruled out by whatever logical system (or systems) we are using.  To put it more generally, we might say that logical possibility, in so far as it is conceivable, is relative to a logical system.

How then can we speak of all possible worlds, analytic truth--what is called "logical necessity?"  If something is true in all possible worlds ("logically necessary" or "analytically true"), it means that it is true according to the rules of a given language.  Analytic truth is truth by definition.  It is truth by language.  We can conceive of "all possible worlds" only so far as we have language.  Our linguistic capacities limit our understanding of all possible worlds.

Thus, we can distinguish between actuality and possibility, and speak of necessity, without postulating knowledge which is beyond the constraints of the actual world.  The actual world makes language possible.  The actual world places constraints on our language and thought.

A question arises:  Is it possible for there to be a world which is not constrained by any properties identical to any local (actual world) properties constraining our thinking?  If so, could we conceive of it?

Let X = such a world.  If X is conceivable, then we can conceive of something which is not constrained by anything local which constrains our thinking.  Therefore, either nothing local constrains our thinking or we arrive at a contradiction.  The contradiction may most easily be avoided by claiming that X is inconceivable, in which case we cannot say it is possible.  So either there is no possible world beyond the constraints of the actual world, or our thinking is not constrained by the actual world.

It is hard for me to comprehend the idea that our thinking could be unconstrained by the actual world.  So it seems more intuitive (to me) to say that there is no possible world beyond the constraints of the actual world.  However, this does not mean the actual world is the only possible world.  It rather means that we cannot conceive of a world that is completely unlike the world we inhabit.  Our powers of conception cannot completely transcend what we experience.

In my last post, I regarded science as testing claims about the actual world against claims about particular possible worlds (empirical hypotheses) and universals (all possible worlds:  the rules of language and logic).  I think it is more accurate to say that science tests empirical hypotheses (models of possible worlds) against the actual world (observation) via previously adopted models of possible worlds, the rules of logic and language.  What science gives us are possibilities, not actualities.  Philosophy, on the other hand, explores, deepens, illuminates and at times challenges the rules of logic and language.  What then of mathematics?  Does it provide models of possible worlds?  Does it explore the rules of logic and language?  Maybe it does something else entirely.  Maybe it gives us, not possibilities, but actualities.  Mathematical theorems are truths about the actual world.  However, they are truths which (at least partially) structure the rules of logic and language.  So they are also truths about all possible worlds.

The initial question was, How is it that we can have necessary truths in mathematics if we are constrained by the actual world?  The answer is:  Because mathematical truths are both truths about the actual world and truths which structure our thinking about logical and linguistic possibility itself.  And since necessary truths are truths by virtue of logic and language alone, then mathematical truths are by definition necessary truths.  They are no less empirically grounded for all that.