[FOM] The Lucas-Penrose Thesis vs The Turing Thesis
Arnon Avron
aa at tau.ac.il
Sat Oct 7 17:38:39 EDT 2006
I did not think that I'll return to this discussion, but Shipman's
message reveals the great confusion concerning the
Lucas-Penrose thesis. So this message is not about
the validity of the argument, but about *what* is the thesis.
> Define "Human mathematics" as the collection of formalized
> sentence in the language of set theory which are logical
> consequences of statements
> that will eventually come to be accepted by a consensus of human
> mathematicians as "true".
> Proposition A: There exists a recursively enumerable and consistent
> set of sentences which contains "Human mathematics".
>
> Proposition B: There exists a recursively enumerable
> and consistent set
> of sentences which equals "Human mathematics".
None of these two (very vague!) propositions has much to do with
the mechanist thesis that we are machines, or with the
Lucas-Penrose thesis (at least in the way I have so far understood it,
and the way I found it interesting, though false). The
Lucas-Penrose thesis is about every particular human being,
not about the totality of all human beings that ever live
or will (potentially) live in the future. It essence
was expreesed by someone here on FOM with the slogan "*I* am
not a machine". It is mainly in this form that the thesis
has relevance to the mind-body problem and other important
philosphical issues!
Personally, I believe that each of us is a machine. At the same
time it seems to me quite possible that every true arithmetical
sentence in the language of PA belongs to your "Human mathematics"
(I am not sure about it, but I see litle evidence for the contrary, and
most mathematicians somehow believe in this, or so it seems to me).
What can be achieved by each human mathematician alone is almost
certainly only a subset of what can be achieved by humanity
as a whole: I see no reason to think that all of us have the
same potential (and the same applies to you, I think: otherwise why
did you find it necessary to talk about consensus?
You could have simply talked about the potential
of a single mathematican, had he lived for ever).
Another way to see the point is to follow you word by word and
define "machine mathematics" as the collection of formalized
sentence in the language of set theory which are logical
consequences of statements
that will eventually come to be accepted by a consensus of machine
mathematicians as "true". Is there any reason to assume that
"machine mathematics" is r.e.?? (In fact there might be reasons to
believe that it includes all true arithmetical sentences: Call
any machine which produces arithmetical sentences "a machine
mathematician" iff all the arithmetical sentences it produces are
true. Let an arithmetical sentence be "accepted by a consensus of
machine mathematicians" once 1000 machine mathematicians
have produced it. Then obviously all true arithmetical sentences
belong to "machine mathematics" according to these definitions.
Now you might object of course that these are not good definitions
or characterizations of "mathematicians" and "consensus". Right.
Do you have better ones?).
Arnon Avron
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