[FOM] The Lucas-Penrose Thesis vs The Turing Thesis

Arnon Avron aa at tau.ac.il
Sun Oct 8 04:51:58 EDT 2006

Quoting Robbie Lindauer <robblin at thetip.org>:

> If there is a machine that is consistent
>       and
> If that machine can perform mathematics of at least the complexity of PA
> Then there exists a (possibly true) sentence which that machine can not
> decide.
> This is not dependent on any aspects of the mechanics or physics of our
> world.  In NO POSSIBLE WORLD can the machine decide that sentence.  If
> machines were purely spiritual, if God were a machine, God could not
> decide the question.
> This is partially constitutive of what it is to be THAT machine.  If
> you can solve the problem in question, even potentially, then you are
> not that machine.

So far I agree, but I dont see the relevance to the claim
that a given human mathematician is or isn't a machine. Even less
do I see the relevance  to my original reply to Shipman,
which refers to the totalities of arithmetical sentences that can be
proved by the totalities of (potential) human mathematicians and (potential)
machine mathematicians. If these totalities exist in any sense
(which I doubt) then they are different from the set of potential
arithmetical theorems of any particular machine (including humane

I dont agree with the rest of your message, but as I declared not long
ago, I see no point to argue with you. Obviously, the fact that no expert
(whether that expert believes that we are machines or not)
accepts your arguments, does not cause you to try to understand what is
wrong with them --- and it never will.

Arnon Avron

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