🚀 go-pugleaf

In <1992Nov2.020038.19948@guinness.idbsu.edu> holmes@garnet.idbsu.edu (Randall Holmes) writes:

>In article <1992Nov1.102609.13247@black.ox.ac.uk> microsoc@black.ox.ac.uk (Microsoc) writes:
>>On Free will vs determinism...
>>
>>What do you netters think about Lucas' argument (which has been pinched by
>>Penrose now and will appear in his forthcoming book)? Put baldly it is:
>>
>>1.	We can do higher mathematics, therefore
>>2.	We have free will.
>>
>>Penrose argues to (2.	Artificial intelligence is impossible given current
>>conceptions of "computability")
>>
>>The bare bones of the argument (I can't do it justice in the time I have
>>acailable here  <-- excuse typo, my editor is broken) are that we can apply
>>a Go"del type arguemtn. It can be shown (fairly conclusively, at least
>>prima facie) that IF we do math by means of an algorithm THEN that algorithm
>>cannot be knowable,

>to us, certainly.

> otherwise we can construct a Stopping (sorry, Halting)
>>paradox a` la Turing. The point is that in doing math we can disprove
>>ceratin (whoops) putative theories (theorems, even - sheesh, my typing isbad
>>today) WITHOUT recourse to a computable (decidable, causal)
>algorithm.

>There is no reason to believe we do not use an algorithm; we just
>don't know what it is.

>>
>>Sorry this is so short - if there is sufficient demand I'll put the argument
>>in a little more detail when my )(*!@#$%^*&( editor is working properly again.
>>
>>Marc Read

>These arguments are unsound.

In such short form, yes, but could it be that Penrose has a much more elaborate
theory that might survive even your scrutiny?  I have not read his
arguments; have you?

>P.S.  I am a professional mathematical logician; I know exactly what
>Godel's Theorem says, and it does not prove that we have free will or
>can prove things non-algorithmically, nor does it disprove the
>possibility of AI.

I have some expertise in mathematical logic, being a set-theoretic
topologist.  We work all the time with axioms that are independent
of the usual "self-evident" ones like the axiom of choice.

[This is the axiom that for each collection of disjoint non-empty
sets, there is a set which intersects each member of the collection
in a single-element set.]

  But
occasionally we come across axioms that seem *almost* self-evident,
and we also have a strong intuition for the *consistency* of various
axioms ["large cardinal axioms"].   

Now, where do these intuitions come from, if we are *only* the
concatenation of atoms blindly following physical laws?  I admit
this is not exactly the same argument as the abbreviated Penrose
argument you see above, but it comes from the same deep wellsprings,
and fuels the speculation that there is more to our world than just
the scientifically explainable phenomena.

Peter Nyikos