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

[Robbie Lindauer]

I wrote:

>> Since, for someone who rejects the Lucas theory, the human can not
>> potentially decide the sentence for the given machine in question.
>> The problem is that the onus for explaining WHY the human can not
>> decide this question -not even potentially- lies clearly with the
>> mechanist.  Without ASSUMING that a particular brand of materialism is
>> true, call it Turing Machine Materialism, (as if there were no other
>> choices!) there's no particular reason to assume the
>> conclusion either.
>

Arnon replied:

> You totally confuse things here, and forgot who is claiming
> what. So let me remind you: the debate is *not* whether the
> "Turing Machine Materialism" is true or not. The debate is
> whether it is refuted by Lucas Theory or not.

It appears, in particular, to be about whether or not the following 
statement is true:

The Questionable Assumption (QA)

	"If a sentence is decidable by anyone, then it is POTENTIALLY 
decidable by a human."

While a formal proof of this is yet to have been presented, I'm willing 
to defend the following:

Take some theory subject to Godel's incompleteness theorem, call it T.  
Let T be consistent.
Let G be a statement whose truth is not decided by T and which is 
syntactically correct in T.
Let Platonism be the rule that T is either True or False, and whichever 
one, necessarily so, and let it hold for the sake of our discussion.
In particular, either G or !G.

For any formal system of the previous kind, there is another formal 
system T' for which the decision about G is a theorem (namely T + G or 
T + !G).  Call T' the G-Successor of T.  This is trivially true - if T 
is a formal system defined by a set of axioms and G is independent of 
T, then adding G to T results in another formal system.  Suppose it 
didn't.  Then the resulting formal system T' is identical with T, but 
since T' decides G and T does not, we have a contradiction, QED

We know T' is consistent since if T doesn't decide G, then there is no 
proof of either G or not-G in T and there are no other axioms added to 
T but G to make T'.

T' also has a G-Successor, call it T''.  (see above)

There is, therefore, a (potentially non-unique) progression of 
G-Successors:

(T, T', T'', T''', ...) for any given T.

Let G be the distinguished undecidable sentence of T which is an axiom 
of T', and G' be the same for T'', etc. by which T' is distinguished 
from T, and T'' distinguished from T''', etc.

There are therefore the distinguished undecidable sentences of the 
G-Succession-Progression of T which are axioms (and therefore theorems) 
of the G-succeeding theory.

Let P be a person.  Let P' be another person, P'', yet another, etc.

Let T(P) be the assertion "T formalizes P" or "Person P is represented 
by the formalism T", T''(P) the sentence "T'' formalizes P", T''(P') 
the sentence "T'' formalizes P'", etc.

A semi-formalization of the Questionable Assumption:

	"For any t in (T, T', T'', T''', ...) and any p in (P, P', P'', P''', 
...), Pos(t(p)"

	That is, it's possible that there is a formalism which is any given 
person, and it could be one of the many of them.

	That is, the Questionable assumption is roughly, that Turing 
Materialism could be true.

The Turing Materialist will want to be more particular about how he 
phrases the proposition.  He'll want to retreat to "It's possible that 
there is a single formal system which corresponds to every single 
person"

So :

		"Ap in (P, P', P'', ...), Pos Et in (T, T', T'', T''', ...): t(p)"

(roughly)
"For all people, it's possible that there exists a formalism which is 
their formalism."

Now here obviously a lot turns on the meaning of "possible".  We're not 
talking, obviously, about physical possibility since it's not even a 
reasonable assumption that I could, physically speaking, even decide 
all the theorems of PRA, let alone ZF or what-have-you.  Alas, nobody 
has the time to do such things.  Similarly with any given computer and 
any given infinite formalism supposedly represented by it.  There will 
never be a time T at which it will have proved all of the the theorems 
of PRA.

However, we're talking LOGICAL possibility.

For the time being, let's use the possible-worlds model (even though 
it's particularly distasteful).  A world is LOGICALLY possible if no 
contradiction is proposed to be true in that world.

Let's imagine that in world W (our world), T(P).
Let W' be the world in which T'(P).

For the sake of particularity, let W' be the world in which the twin 
prime conjecture is true and I can prove it (though I don't actually), 
but W be the world in which T' is true and I can not prove it.  (Or, 
"it could be proved using the formalism which is me.")  These worlds 
will be Lewis "adjacent" in Lewis' sense.  It may not even be necessary 
that they be physically different (since we are talking about the 
potential for proving it).

Have we imagined here a contradiction (other than that I could exist in 
a different world)?  Well, sort-of, a contradiction of the assertion 
that T is a formalization of P, since potentially P can decide G 
(because P could be formalized by T' in world W') WHEREAS it's simply 
provably false that T could be identical with T' since G is undecidable 
in T and an axiom of T'.

A Turing Materialist must respond "It's impossible that P be formalized 
by T' because if P is formalized by T, then it is NECESSARY that P is 
formalized by T."

But this is absurd since not only can we imagine cases where some 
person P's formalism changes, but we have actual real cases where some 
person's apparent behavior, physical structure, experience, knowledge, 
and etc.,  has been expanded or diminished, eg. AGING, learning, 
Alzheimer's, etc.

A Turing Materialist may respond:

"It's necessary that a the formalism with which a person is born is the 
same as the one with which the person dies and apparent behavior, 
experience and physical capacity are irrelevant to it."

(an obvious empirical aside:  The only reason I could think of for 
believing something like that, though, would be the assumption of 
Turing Materialism.  Given the fact that we do change over time (and 
here I assume that we can agree -we do change over time-), the 
existence of the phantom formalism guiding it all throughout our 
history that is unchanged AS A MATTER OF LOGICAL NECESSITY, is 
ridiculous, as materialistically distasteful as the Spirit (as distinct 
from the Soul).  This is to say, if one is going to believe in Ghosts 
why not believe in ghosts that make sense at least?)

However, aside from the observational and anecdotal evidence, we also 
have a nifty proof:

1) Let it be a contingent fact that I exist at all.   World War II 
could have ended in the death of my grandparents, for instance.
2) for some T,  Nec(T(me))


1 -> ! Nec(T(me))
.............
1->!2


The Turing Materialist may prefer a more finely grained modal logic 
allowing the fabled necessary a posteriori:

In every possible world where I exist, necessarily, have formalism T.

Now the Ghost objection is more real.  Apparently my formalism is 
beyond causal interaction since, according to the Turing Materialist:

Nec("If I exist, then I have formalism T")
Then for all physical interactions of mine, PI, if not necessarily PI, 
then PI has no effect on my formalism.

Hence, either strong PERSONAL determinism is true OR my formalism is 
not effected by physical factors.  (say for instance, whether or not my 
brain is removed).

If my formalism is not POSSIBLY effected by physical factors (e.g. 
which formalism I have), then, I conclude, my formalism is not 
physical, and we're back to the ghost theory.

Alternatively, if Strong Determinism is true, then it's irrelevant 
whether or not I have a "turing formalism" since I have the trivial 
"Everthing that will happen to robbie will happen to robbie" 
"formalism" which is strongly finitistic and not subject to godelian 
objections anyway.


>  To refute
> your claim for having  a *proof* of your beliefs one does not have to
> disproof any of your assumptions, or even to give any reason
> why s/he thinks a certain assumption is false. It suffices to isolate
> just one assumption you use which you cannot prove, and your
> claim for proving your thesis is destroyed.

I agree wholeheartedly.  So for instance, since there is no proof of 
the Axiom of Choice, we should also reject any "theorems" that are 
supposedly "proved" on that basis.  Similarly with replacement, the 
axiom of infinity, etc.

Naturally, we can generalize, any axiomatic proof which relies on what 
someone thinks is a reasonable assumption, e.g. the axiom, should be 
considered unproved until the questionable assumption on which it is 
based is proved, and consequently provisionally disregarded until the 
axiom is proved.

However, "real life reasoning" is slightly different.  We have a kind 
of common sense approach to basic assumptions.  Like successor - it's 
natural to say that every number has a successor because we know what 
it's like for any given number to give its successor "that plus 1".  
Can we PROVE that every number has a successor?  Well, if it's an 
axiom, we can deduce it directly from the axioms.  If its contradiction 
were contradicted by a theorem, we could say it was proved, but then 
we'd have to reasonably ask what the grounds of the theorem are, and 
they in turn, will be the axioms.   Where will we get the proofs for 
our axioms?

If someone takes a ridiculous assumption as an axiom, we can laugh it 
off - oh, that's ridiculous...  But if their assumption is prima facie 
plausible, of course it remains to the interlocutors to decide among 
themselves which one is the most reasonable assumption if nothing can 
be proved EITHER WAY.

In this particular case, though, what I've done is identify the 
ridiculous assumption upon which Turing Materialism must rest, e.g. the 
assumption that it's IMPOSSIBLE that I may have a different algorithm 
than I supposedly do in fact have who's only possible motivation would 
be turing materialism anyway.  The double-whammy of fallacies - the 
ridiculous assumption AND the circular argument.

Consequently, the argument is not structured as you suspected:

> We have proved A->B (let us assume that you did)
> There is no particular reason to doubt A
> Hence B.

The form of the proof is this:

Let "Turing Materialism" be TM and the Questionable Assumption be QA:

TM -> ! Pos(QA)
Pos(QA)
_____
!TM

I believe that is not, at least, a well known logical fallacy.

Robbie Lindauer


ps - taking some "philosophical" comments out of order:

On Oct 11, 2006, at 4:07 PM, Arnon Avron wrote:

>>
>> As for the "no expert" clause, what one person considers an expert,
>> another considers a lunatic.
>
> There are some objective criterions, you know. Goedel theorems
> are *theorems* of mathematical logic, and so the experts on them
> are the mathematical logicans, especially those who made contributions
> to the subject (like generalizing and/or improving the results
> and their proofs). I guess that according to you, Goedel himself
> would not be considered as an expert on his theorem, but as
> a lunatic ...


I would say that every proof which lies on questionable assumptions is 
a "proof-in-a-system" and anyone who asserts something stronger than 
this, for instance you're argumentum ad verecundiam, is either unaware 
that it is a fallacy or is pushing some political program hoping that 
no one will notice OR thinks they are in possession of a transcendental 
argument (in which case, I love to hear it).  If they've been notified 
of their fallacy and continue on in it, yes, they end up in the lunatic 
basket.

Hypothetically, Godel was aware that his theorems applied only to 
infinitary systems of a particular kind, so no, Godel wouldn't end up 
in the loony-basket, I think.   I know that Godel had some musings 
about God and Platonism and such, but I don't know whether or not he 
related these directly to his ability to prove them.  Perhaps someone 
on the list knows more than I do about it.

>> Hence my universal disdain for
>> axiomatic  mathematics as fairy tales.
>>
>> Welcome to Philosophy.
>
> Only to a certain part of the present world of philosophy:
> the people  who think that one need not know much about some area, or
> have a deep understanding  of it, in order to have strong
> opinions about it. Luckily, not all philosophers are like this.

I don't have a strong opinion about this matter, it's irrelevant to me 
whether or not Lucas' argument is effective -I don't care-.  My 
interest is social - I'm interested in what makes some people defend it 
vehemently and others attack it vehemently.  As a result, it's 
interesting to me how intensely you state your attitudes and to what 
extent your vehemence is supported by an underlying rationality.