David Deutsch's claim about "mathematicians' misconception"

[Joe Shipman]

Most of this discussion seems to miss what I find interesting about this.

Deutsch doesn’t distinguish between logic and mathematics here:
that what the rules of logical inference are, and hence what constitutes a proof, are a priori logical issues, independent of the laws of physics.
On one hand, one feels that what a human will accept as a valid argument is partly an issue of language and convention and partly an issue of a priori abstract thought, in neither case depending on the laws of physics beyond the minimal structure necessary to support the existence of minds and of physically instantiated symbolic expressions.

On the other hand, given that humans will accept as valid machine-generated proofs far too large to be humanly surveyable, depending on their understanding of the laws of physics to gain confidence that the machine performed as designed, the theoretical possibility of checking such a proof by hand seems no longer to be essential. One can imagine experimental verification of mathematical statements following from the axioms of a physical theory, without there being an algorithm to correctly generate such statements (for example, if some measurable dimensionless quantity had a value that was a definable but not a recursive real).

My view is that these are both valid ways of looking at it, the difference turning on the definitions of the terms “inference” and “proof”. The second viewpoint seems to be expanding the definition of “prove” to mean “obtain knowledge of with the highest achievable degree of scientific or practical certainty”, but I would rather adjust terminology and talk about “demonstration” or “verification” rather than “proof”, and “inductive” or “scientific” rather than “logical” inference.

With these distinctions established, the problem goes away as far as logic is concerned. But an issue remains for mathematics, because some mathematics is manifestly not related to scientific theories of any types ever usefully developed (CH and AC being obvious examples, Borel Determinacy being perhaps a better example). One may sidestep this by taking a formalist attitude and only talking about proofs from axioms in a formal language with the usual nice properties, but one would like to talk about whether a mathematical statement is “provable” informally without specifying an axiomatic system, using my second, broader sense which can also be rendered as “demonstrable” or “knowable” or “verifiable”. It may be a genuine and contingent property of our-universe -as-we-are-capable-of-experiencing-it that CH is simply *not knowable*, while universes with different constitutions might have such phenomenological plenitude that there would be a way in which its beings could come to *know* whether CH is true in a way that we can’t.

In this last way of looking at it, the laws of physics have something important to say about what is mathematically knowable, and therefore maybe about the logical concepts of inference and proof too, despite my earlier distinctions which tried to get rid of the issue.

— JS

-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20201125/4839d7d5/attachment-0001.html>