Mathematicians are machines that are unable to recognize the fact that they are machines

[Evgenii Rudnyi]

I am reading a Russian book about the “no computer thesis” based on the
Gödel theorem. In the book there was a nice quote - see below - that
somewhat close to what Bruno says.

"And if such is the case, then we (qua mathematicians) are machines that
are unable to recognize the fact that they are machines. As the saying
goes: if our brains could figure out how they work they would have been
much smarter than they are. Gödel’s incompleteness result provides in
this case solid grounds for our inability, for it shows it to be a
mathematical necessity. The upshot is hauntingly reminiscent of
Spinoza's conception, on which humans are predetermined creatures, who
derive their sense of freedom from their incapacity to grasp their own
nature. A human, viz. Spinoza himself, may recognize this general truth;
but a human cannot know how this predetermination works, that is, the
full theory. Just so, we can entertain the possibility that all our
mathematical reasoning is subsumed under some computer program; but we
can never know how this program works. For if we knew we could
diagonalize and get a contradiction."

Haim Gaifman,
What Gödel’s Incompleteness Result Does and Does Not Show
http://www.columbia.edu/~hg17/godel-incomp4.pdf

Best wishes,

Evgenii

[Bruno Marchal]

It is good. Not as far as us, but this avoids the common mistake when
using Incompleteness outside logic. Of course the author seem ignore
(there is no references) that by going (quite) a bit farer, we get an
explanation of consciousness (true, knowable, undoubtable,
undefinable, and unjustifiable) *and* of the origin and shape of the
physical laws, making Digital Mechanism testable.

What still many miss, but that Post intuited, and Webb and Kleene
understood largely, is the unavoidable conflict between all the ways
of approaching truth enforced by incompleteness to the machine
reasoning about itself or selves (1p, 3p, ...). Indeed, incompleteness
forces the nuances between the 8 (!) nuances of "beweisbar":

p ((sigma) truth)
[]p (rationally justifiable, provable ("beweisbar" itself, when
translated in the arithmetical language)
[]p & p (first person, knowable)
[]p & ~[]f ("bet-table", predictable, observable, distinguishable)
[]p & ~[]f & p (sensible yet partially measurable, qualia)


The illumination, which is also the blasphemy if presented as being
true, is that

G1* proves p <-> []p <-> ([]p & p) <-> ([]p & ~[]f) <-> ([]p & ~[]f &
p),

but the Löbian, terrestrial, effective condition is that G does not
prove any of those equivalence, and the Löbian entity knows it
(machine and non-machine).
G1 proves p -> []p, indeed, it is taken as axiom, as it characterizes
in particular the machine aware of their own Turing universality. "p -
> []p" expresses "Turing universality" in arithmetic in some weak
technical sense: they can prove for any sigma_1 sentence that if they
are true then they can prove it: that is for each arithmetical
sentence p, they can prove p -> Beweisbar('p'), modally "p -> []p".
But the machine itself, as defined correctly (hopefully) by its
susbtitution level and personal relative description, is literally
enforced to see the truth (p) "masked" by the body ([]p). The Löbian
entity can't prove []p -> p, given that she can't already prove, even
taken as an axiom, []f -> f, that is its own consistency (~[]f). And
this makes all the other nuances obeying to different logics.

It match well, and that is plausibly not a coincidence, the discourse
of those open to the mystical experience and rationalism like the
neopythagoreans and neoplatonists.

Kind Regards,

Bruno



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[spudb...@aol.com]

So therefore, we are all machines, as we cannot recognize this "fact."  :-)  So are dogs, and bacteria, correct?

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[Bruno Marchal]

On 16 Oct 2017, at 22:14, spudboy100 via Everything List wrote:

So therefore, we are all machines, as we cannot recognize this "fact."  :-) 

Well, this does not follow. If we are machine we cannot recognize it. But if we cannot recognize it does not entail that we are machine (we might be god-like, and unable to recognize that we are machine because we would not be a machine.

So are dogs, and bacteria, correct?

Assuming we are machine, it is very reasonable that bacteria and dogs are too. But they will be unable to recognize the fact, not because they are machine, but because they lack the introspection ability.

The situation is similar with Robinson versus Peano-Arithmetic. Robinson arithmetic is unable to prove its consistency, because it has very poor provability ability. It cannot even prove that 0 + x = x. Peano-arithmetic, on the contrary has very powerful provability arithmetic, and so cannot prove its consistency due to incompleteness related to that power. Both are Turing universal though.

Bruno

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