Godelling in the Valley

by rsbakker

Either mathematics is too big for the human mind or the human mind is more than a machine” – Kurt Godel

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Okay, so this is purely speculative, but it is interesting, and I think worthwhile farming out to brains far better trained than mine.

So BBT suggests that the ‘a priori’ is best construed as a kind of cognitive illusion, a consequence of the metacognitive opacity of those processes underwriting those ‘thoughts’ we are most inclined to call ‘analytic’ and ‘a priori.’ The necessity, abstraction, and internal relationality that seem to characterize these thoughts can all be understood in terms of information privation, the consequence of our metacognitive blindness to what our brain is actually doing when we engage in things like mathematical cognition. The idea is that our intuitive sense of what it is we think we’re doing when we do math—our ‘insights’ or ‘inferences,’ our ‘gists’ or ‘thoughts’—is fragmentary and deceptive, a drastically blinkered glimpse of astronomically complex, natural processes.

The ‘a priori,’ on this view, characterizes the inscrutability, rather than the nature, of mathematical cognition. Even without empirical evidence of unconscious processing, mathematical reasoning has always been deeply mysterious, apparently the most certain form of cognition when performed, and yet perennially resistant to decisive second order reflection. We can do it well enough—well enough to radically transform the world when applied in concert with empirical observation—and yet none of us can agree on just what it is that’s being done.

On BBT, our various second-order theoretical interpretations of mathematics are chronically underdetermined for the same reason any theoretical interpretation in science is underdetermined: the lack of information. What dupes philosophers into transforming this obvious epistemic vice into a beguiling cognitive virtue is simply the fact that we also lack any information pertaining to the lack of this information. Since they have no inkling that their murky inklings involve ‘murkiness’ at all, they simply assume the sufficiency of those inklings.

BBT therefore predicts that the informational dividends of the neurocognitive revolution will revolutionize our understanding of mathematics. At some point we’ll conceive our mathematical intuitions as ‘low-dimensional shadows’ of far more complex processes that escape conscious cognition. Mathematics will come to be understood in terms of actual physical structures doing actual physical things to actual physical structures. And the historical practice of mathematics will be reconceptualized as a kind of inter-cranial computer science, as experiments in self-programming.

Now as strange as it might sound, you have to admit this makes an eerie kind of sense. Problems, after all, are posed and answers arise. No matter how fine we parse the steps, this is the way it seems to work: we ‘ponder,’ or input, problems, and solutions, outputs, arise via ‘insight,’ and successes are subsequently committed to ‘habit’ (so that the systematicities discovered seem to somehow exist ‘all at once’). This would certainly explain Hintikka’s ‘scandal of deduction,’ the fact that purported ‘analytic’ operations regularly provide us with genuinely novel information. And it decisively answers the question of what Wigner famously called the ‘unreasonable effectiveness’ of mathematical cognition: mathematics can so effectively solve nature—enable science—simply because mathematics is nature, a kind of cognitive Swiss Army Knife extraordinaire.

On this picture, there is only implementation, implementations we ‘generalize’ over via further implementations, and so on and so on. The ideality, or ‘software,’ is simply an artifact of our metacognitive constraints, the ‘ghost’ of what remains when multiple dimensions of information are stripped away. Not only does BBT predict that the ‘foundations of mathematics’ will be shown to be computational, it also predicts that, as the complexities pile up, mathematics will become more and more the province of machines, until we reach a point where only our machines (if the possessive even applies at this point) ‘understand’ what is being explored, and the imperial mathematician dwindles to the status of technician, someone charged with translating various machine discoveries for human consumption.

But I ain’t no mathematician, so I thought I would open it up to the crowd: Does this look like the beginning?