Saturday, October 28, 2017

Trivializing Benacerraf and Monstrous Moonshine

[note: Sorry that this post is a little wordy. I'm trying to get back into blogging, and forcing myself to post stuff I'm not quite happy with is part of that. Also see this paper for a way more detailed and less aggro take on this issue.]

Remember that the access problem (aka the Benacerraf problem) for realists goes something like this: If moral/mathematical/etc realism is true, how can human accuracy and reliability about moral/mathematical facts be anything but a miracle or a mystery?

People like Justin Clarke-Doane and perhaps David Enoch, (call them The Trivializers) have been suggesting that we can answer all legitimate access worries besetting for realists about necessary domains like mathematics morals etc. just by “stapling together” two things to explain why we couldn't have easily been wrong (and providing a similarly trivial explanation for the sensitivity of our I won't discuss here):
  • an  explanation (e.g.historical or evolutionary) for why we reliably believe certain moral/mathematical/etc claim,
  • the fact that these claims are necessary truths
So, for example, a classic platonist might try answer access worries by explaining human reliability about mathematics like so:

TRIV: Mathematicians reliably believe truths because they reliably believe only those mathematical claims which can be proved in a certain formal system (e.g. ZFC) and this formal system is (necessarily) truth preserving w.r.t. the platonic mathematical objects.

Such an explanation is not likely to satisfy anyone who feels an intuitive access worry. For TRIV just explains one intuitively mysterious match between human psychology and objective which intuitively “cries out for explanation” (our acceptance of theorems that match whats going on in Plato's heaven) by positing another such mysterious match (our acceptance of axioms that match what's going on in Plato's heaven).

But it does (in some sense) suffice to explain the safety of human beliefs by deriving our reliability about realist facts concerning the relevant domain (i.e., that in all close possible worlds, our beliefs about these domains match up with the truth) from more general premises which the realist (if not their deflationary opponents) accepts.

And I take Trivializers to be suggesting that our intuitions that certain regularities involving necessary truths "cry out for" explanation in a deeper/more unified sense, which mere deductions of reliability/safety/sensitivity like TRIV need not provide, is an illusion.

There are two reasons why I don't buy this.

First, conceptually analyzing anything (from tablehood to justice) is infamously hard, but there are plenty of good paradigms for how to think about ”crying out for explanation” which seem to apply equally well to necessary and contingent regularities (e.g., norms that say we should preferring theories that have fewer degrees of freedom and or Kitcher’s idea of scientific explanation as unification).

Much more importantly though, embracing the Trivializers' position seems to have deeply implausible revisionary consequences for mathematical practice, since mathematicians sure seem to think that some regularities involving necessary truth truths can cry out for (non-trivial) explanation.

For example, consider this quote from a popularizing article about the history of John Conway's `Monstrous Moonshine' conjecture.

``Strangely enough, [the j- function]’s first important coefficient is 196,884, which McKay instantly recognized as the sum of the monster’s first two special dimensions.
Most mathematicians dismissed the finding as a fluke, since there was no reason to expect the monster and the j-function to be even remotely related. However, the connection caught the attention of John Thompson, a Fields medalist now at the University of Florida in Gainesville, who made an additional discovery. The j-function’s second coefficient, 21,493,760, is the sum of the first three special dimensions of the monster: 1 + 196,883 + 21,296,876. It seemed as if the j-function was somehow controlling the structure of the elusive monster group.'’’

Note that mathematicians already had separate proofs of facts about the j-function and monster group (and hence an `explanation' of for the match between these facts of the kind which TRIV provides, i.e., a deduction, from more general premises, of the fact that this relationship holds in all close possible worlds). But (once this match between apparently unrelated domains proved striking enough) they expected to find some further/deeper unifying explanation and this expectation guided choices for further research.

So I'm prima facie pretty suspicious of the idea that felt intuitive demands for unifying/satisfying/non-trivial explanations of regularities involving necessary truths are generally misguided. But maybe I'm not being charitable to Clarke Doane and/or he can find some way of separating the "this regularity cries out for further explanation" intuitions he wants to dismiss as unreliable from those which obviously do good work in mathematics.

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