David Deutsch's claim about "mathematicians' misconception"

[Lew Gordeew]

Zitat von "Timothy Y. Chow" <tchow at math.princeton.edu>:

> Jose Manuel Rodriguez Caballero wrote:
>> Reading [1], I found the following claim, due to David Deutsch, about the
>> relevance of the laws of physics in foundations of mathematics:
>>
>>> there was a widespread assumption -- which I shall call the  
>>> mathematicians' misconception -- 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.
> [...]
>> What could be the status of what David Deutsch calls the  
>> "mathematicians' misconception" in the framework of foundations of  
>> mathematics? Could be in the same category as Platonism, Formalism,  
>> and Intuitionism?
>
> Very interesting...thanks for mentioning this.
>
> The following link might work better:
>
> https://arxiv.org/pdf/1210.7439.pdf
>
> To quote a little bit more:
>
>    The theory of computation was originally intended only as a
>    mathematical technique of studying proof (Turing 1936), not a branch of
>    physics.  Then, as now, there was a widespread assumption---which I
>    shall call the mathematicians' misconception---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.  This is analogous
>    to Kant's (1781) misconception that he knew with certainty what the
>    geometry of space is.  In fact proof and computation are, like
>    geometry, attributes of the physical world.  Different laws of physics
>    would in general make different mathematical assertions provable.  (Of
>    course that would make no difference to which mathematical assertions
>    are *true*.)  They could also make different physical states and
>    transformations simple---which determines which computational tasks are
>    tractable, and hence which logical truths can serve as rules of
>    inference and which can only be understood as theorems.
>
> I don't feel like dissecting Deutsch's view in detail here---I have  
> voiced objections to similar ideas in the context of discussions of  
> hypercomputation [*]---but will just say that the misconceptions  
> seem to me to be on Deutsch's part and not on the mathematicians'  
> part.  If we want to attach an "ism" then I would attach it not to  
> the "mathematicians' misconception" itself, but rather to Deutsch's  
> own misconceptions; I'd propose the term "Deutschism" since I don't  
> think his views on this point are widely shared.
>
> [*] See for example https://cstheory.stackexchange.com/a/4838 and
>     https://cs.nyu.edu/pipermail/fom/2004-February/007932.html
>
> Tim

In support of Tim's opinion, recall that (even in propositional logic)  
there is no general consensus on "what the rules of logical inference  
are". After all, apart from classical model of inference (implication)  
there are other ideas coming from non-classical logics -- from minimal  
and intuitionistic to modal and fuzzy (and even paraconsistent) ones.  
Analogously there are different ideas about "what constitutes a  
proof". Proof theory is treated differently by math. logicians and its  
modern versions are virtually unknown to most experts on the laws of  
physics. Actually it's the other way around -- apparently the laws of  
physics are derived from earlier, abstract mathematical concepts of  
inference.

LG