Tuesday, June 2, 2020

Arbitrariness and contingency

I’ve come to be impressed by the idea that where there is apparent arbitrariness, there is probably contingency in the vicinity.

The earth and the moon on average are 384400 km apart. This looks arbitrary. And here the fact itself is contingent.

Humans have two arms and two legs. This looks arbitrary. But it is actually a necessary truth. However there is contingency in the vicinity: it is a contingent fact that humans, rather than eight-armed intelligent animals, exist on earth.

Ethical obligations have apparent arbitrariness, too. For instance, we should prefer mercy to retribution. Here, there are two possibilities. First, perhaps it is contingent that we should prefer mercy to just retribution. The best story I know which makes that work out is Divine Command Theory: God commands us to prefer mercy to just retribution but could have commanded the opposite. Second, perhaps it is necessary that we should prefer mercy to retribution, because our nature requires it, but it is contingent that we rather than beings whose nature carries the opposite obligation exist.

Now here is where I start to get uncomfortable: mathematics. When I think about the vast number of possible combinations of axioms of set theory, far beyond where any intuitions apply, axioms that cannot be proved from the standard ZFC axioms (unless these are inconsistent), it’s all starting to look very arbitrary. This pushes me to one of three uncomfortable positions:

  • anti-realism about set theory

  • Hamkins’ set-theoretic multiverse

  • contingent mathematical truth.


Martin Cooke said...
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Benjamin Stowell said...

I hope arbitariness implies contingency, because then I think it follows that God exists.

If imperfection implies arbitrariness implies contigency, then an imperfect foundation of reality is a contingent foundation. If brute necessity is preferred over brute contingency (it is), then the foundation of reality must be perfect.

Martin Cooke said...

I have rewritten my observation that anti-realism about set theory ought not to be uncomfortable. Basically, standard set theory gives us good mathematical models of mathematics; so, although arbitrariness enters into how there are lots of possible set theories, there is a lack of arbitrariness about standard set theory, which comes from the necessities of mathematics.

Axiomatic sets are defined by the axioms of set theories, much as fictional characters are defined by their canonical stories. Lots of stories are possible. The big ones seem to involve archetypes, but that is human psychology, not word combinatorics. Similarly, there are lots of set theories. But the big one is standard set theory.

The von Neumann ordinals, within standard set theory, are a good mathematical model of whole numbers. Whole numbers are as they are in the real world. Each and every thing is one thing, while numbers greater than one are properties of collections of things. There is a kind of logical necessity about the properties of whole numbers. So, there are only a few good models of them, in standard set theory. And there are only a few set theories that can give us good models of everything in old-school mathematics.

It is a bit like the history of a famous person. That story is probably quite accurate, but it is hardly the same kind of thing as that person's life. That life is the way it was. There are lots of stories, but relatively few good histories of that life are possible. And the arbitrariness of set theories in general might be more like the arbitrariness of a monkey hitting keys on a typewriter.