Re: Robot Evolution

Phil Roberts, Jr. wrote:
Tim Tyler wrote:

John Lucas's 'Godel' argument has been
much-criticized - and Penrose's views in this
area are essentially a variation on it.

I concede that there is a clear majority who
disagree with the Lucas/Penrose position. On the
other side of the equation, however, we have:

a. Hofstadter, Dennett, Penrose, Clarke and
Chaitin, in various ways acknowledging that Godel
at least SUGGESTS a disconnect between formalism
and mathematical reasoning.

Well, no. Goedel proved a very limited thing about
limitations on the power of proof productions
generated from systems of axioms "at least as
powerful as the Peano postulates". That proof in and
of itself had nothing to convey about how _humans_
accomplish mathematical reasoning.

That humans would _like_ to appear somehow superior
to such productions lacks a first demonstration to
satisfy that liking.

So far, no human has _ever_ proved any mathematical
proposition that can be proved _unprovable_ using
Goedel's mechanism.

Quite on the contrary, again and again humans employ
the work done by Goedel to identify _other_
unprovable propositions.

In each case, as with the Hilbert program or, after
the fact, Frege's work, such a "proof of
unprovability" serves as an excellent "stop wasting
time here" signpost [not that a universe of kooks
won't keep trying to square the circle or trisect
the angle, not all are literate enough to read "give
up, what you seek cannot be done" and take it as
solid advice].

b. Little unanimity as to what exactly is wrong
with the Godel argument, with dozens and dozens of
different sorts of objections, many based on
impenetrable confabulations.

There's nothing "wrong with" the Goedel arguement,
unless you and I are talking at cross purposes and
you mean something other than Goedel's proof.

That proof is simple enough to follow that anyone
with a minimally competent education in mathematics
can work through it in an evening, and once worked
through, it becomes "self-evident"; too simple to
challenge.

If on the contrary you are using "the Goedel
argument" as the false hypothesis that humans are
doing something "super-computational" in their
mathematical reasoning, I'll just refer you back to
my prior posting: no, we aren't.

c. Papers still being published criticizing the
Godel argument against mechanism almost 80 years
after Godel first published his theorem.

So? Twenty years ago or more, "special relativity
scientists" supposedly universally agreed to stop
using "mass" to describe the combination of rest
mass and energy of motion represented as mass
[because the term used that way is dependent on the
frame of reference of the observer], yet just last
year there was a paper published arguing that
"scientists" should do just what, for the most part,
they have done, change their terminology.

"Publish or perish" is a terrible blight, since it
results in so much re-publication of what is already
known.

d. The universal abandonment of Hilbert's program
of formalizing mathematical reasoning by
mathematicians all over the world subsequent to
Godel's proof.

Right -- Goedel proved that the goal was
unattainable, everyone who could read his proof with
understanding immediately realized that he was
correct, and diverted their attention elsewhere.

This in NO WAY argues for the superiority of human
mathemaatical reasoning to computational
mathematical reasoning.

To the contrary, each mathematician who changed
courses was thereby agreeing that indeed his/her
work was _not_ going to somehow "beat Goedel", and
that trying to do so is a fool's game.

Each such diversion was a vote by an intelligent and
well informed participant that human reasoning _was_
limited to the same limits as "computational
reasoning".

e. Intersubjectively reproducible empirical
evidence (feelings of worthlessness) suggesting
that not even Mother Nature herself seems to be
able to constrain rationality within a formalism
(the program for "trying to stay alive").

That's not, and never has been, how nature
"programs" species. Evolution doesn't work that way.
In fact, it cannot, since it works at the level of
gene allele frequencies of occurrance in an entire
population.

Protection and propagation of genes shared in common
with ones own genome can lead to some wonderfully
counterintuitive behaviors. Study any good writeup
on the genetic basis of altruism for more details.

f. Evidence (e.g., Parfit, 'Reasons and Persons',
p. 12) that any theory that attempts to constrain
rationality within a formal structure (e.g., a
fixed objective) can be shown to sanction rational
irrationality (i.e., can be shown to be
self-defeating).

But that is still agreement with Goedel's proof,
recast as "there are some good behaviors that cannot
be proven (computationally) to be good behaviors".

Again, no argument that humans are "better than
computational", more an argument for some findings
of game theory, that sometimes the only way to win
is to randomize your choices in some careful way.

If you always jink left in fleeing from the
tiger, the tiger will learn the shortcut
that makes you lunch. If you randomize your
choices between left and right options, the
tiger cannot improve over following your
trail as you followed it, and your chance of
making the tiger into a robe improves.

Brief version of what's wrong:

"A mathematician often makes judgments about
what mathematical statements are true. If he
or she is not more powerful than a computer,
then in principle one could write a (very
complex) computer program that exactly
duplicated his or her behavior.

Assumes what is being questioned.

No, it doesn't. That's the subjective portion of a
longer argument.

But any program that infers

Programs don't infer,

Before correcting the language of someone speaking
in his own field of expertise, you'd be well advised
to check your knowledge of the meanings a word can
take. Computer programs doing mathematical
reasoning, which for the most part use formal
deductive logic to draw conclusions, very much _do_
"infer":

<quote>
infer
v 1: reason by deduction; establish by deduction [syn: deduce,
deduct, derive]
2: draw from specific cases for more general cases [syn:
generalize,
generalise, extrapolate]
3: conclude by reasoning; in logic [syn: deduce]
4: guess correctly; solve by guessing; "He guessed the right
number of beans in the jar and won the prize" [syn: guess]
5: believe to be the case; "I understand you have no previous
experience?" [syn: understand, gather]
</quote>
http://dict.die.net/infer/

they model logical relations that have been found
to underly human inferences on most occasions.

Nonsense.

As to whether these relations are actually being
followed or simply EMBEDDED IN our inferences
remains to be seen.

Bafflegab.

mathematical statements can infer no more
than can be proved within an equivalent
formal system of mathematical axioms and
rules of inference,

True,

Then why did you intrude the above line noise?

but Lucas/Penrose assumes we can go beyond this,
that the intuiting of mathematical truth is not
simply a matter of logical proof:

So? They're wrong, or so all those who voted with
their feet when they abandoned the Hilbert program
concluded.

Continuing to cite their arguments, without
balancing them with the known rebuttals, doesn't
seem particularly useful or integrous when they are
fairly universally considered to be incorrect.

The immediate consequence is that truth cannot
be defined in terms of provability.

Nonsense.

The issue isn't "we must take as truth what we
really only have on faith". The issue is (and it is
good for humankind to be so humbled) that the list
of things we can _know_ to be "true" [identical for
intellectually honest persons to the list of things
we can _prove_ to be "true"] is severely
circumscribed, and there's no wriggle room to go
around that circumscription.

We have, for example, no way to prove that the
lights in the sky will not spell out "Drink Coca
Cola" starting tomorrow, but we can, if we are sane,
live our lives in perfect peace without such a proof.

The supposition that pretty much everything should
be subject to being known is an especially
pernicious form of hubris in a universe where
Heisenberg's uncertainty principle holds sway.

In any serious intellectual endeavor we shall
be able to formulate simple mathematical
arguments, and thus shall be subject to
Godel's incompleteness theorem. However far
we go in formalizing our canons of proof, we
shall be able to devise propositions which are
not, according to those canons, provable, but
are none the less, true. So it is one thing
to be provable, and a different thing to be
true. Truth outruns provability. (J.R.
Lucas).

Lucas makes the mistake of assuming that truth
exists separate from provability. This is of course
the error of theism. What Goedel proved is that
there are propositions whose truth _or_ falsehood
cannot be determined.

The assumption that some of those propositions must
therefore be "true ones" misunderstands what "true"
should mean.

Lucas wants it to have a separate meaning from
"provable", but I cannot see how allowing such a
separate meaning can be anything but incredibly
dangerous.

Allowing that, allows frauds and mountebanks to
assign the token "true" to any proposition which
they can cast as a Goedel-style unprovable
proposition, merely on their self-interested say-so.

This argument won't fly if the set of axioms
to which the human mathematician is formally
equivalent is too complex for the human to
understand.

What is the basis for the assumption that the
intuiting of mathematical truth is based on a set
of axioms, let alone that they must be too complex
to understand?

That was the assumption to which you objected above
as:

> Assumes what is being questioned.

on which this onging argument is being _still_ being
argued by the prior poster.

"If human reasoning about mathematics is
computational, then the argument that the human can
somehow understand and then by computational means
exceed his own axiom set is incorrect if that axiom
set is too complex for the human to understand" --
to summarize badly.

[It's worth noticing that any computational
device, including the human mind, is
probably storage-limited from entirely
understanding its own operation, in any
case. It would probably need to know, for
example, the bit by bit storage reliability
(probability of failure) of its entire
storage mechanism to accomplish such a
task, more than a bit of data per bit of
available storage.]

These are amazing claims, which Penrose hardly
bothers to defend. Reviewers knowledgeable
about Godel's work, however, have simply
pointed out that an axiom system can infer
that if its axioms are self-consistent,

An axiom system can infer?

That's shorthand. The longhand is something like
"any correct production of sentential calculus using
the rules of logic and axioms of a logical system
can infer". Live with the usual form, please, rather
than arguing vocabulary to distract from the
weakness of your arguments.

then its Godel sentence is true.

That's a very strong claim of equivalence.

An axiom system just can't determine its own
self- consistency.

That's just another way of stating what Goedel
proved (as Tim already knows, I'm just saying that
for the rest of the audience).

But then neither can human mathematicians know
whether the axioms they explicitly favor (much
less the axioms they are formally equivalent
to) are self-consistent.

That's a bit overstated; _if_ you can derive a
contradiction from them, they definitely _are_
inconsistent. Once proved inconsistent in that way,
one _knows_ them to be inconsistent.

The problem is that no _guaranteed_ plan for
deriving such a contradiction exists; to claim that
one does violates Goedel's proof.

To the best of my understanding, the situation with
which one is confronted is that mathematics becomes
just another branch of science whose theorems are
subject to issues of falsifiability.

_If_ what you proved leads to a contradiction, _and_
if your proof is formally correct when cast in terms
of your base axioms, _then_ your system of axioms is
inconsistent and what is derived from it is _all_
cast into suspicion.

The nationfulls of mansions of mathematics
that will fall _if_ the Riemann hypothesis
proves incorrect would be hard to count by
today, I'd guess, but mathematicians have,
slowly, gotten used to dealing with such
uncertainty.

Cantor and Frege's proposed axioms of set
theory turned out to be inconsistent, and this
sort of thing will undoubtedly happen again.''

Agreed.

Why then do you go on to say:

But we can nonetheless "know" them to be true in
the sense that we all agree we have good reason to
believe.

You are undergoing a massive failure to understand
what it meant for Frege's system of axioms to be
"inconsistent".

That's not just something we can, or he could, or
his colleagues would, let slide by with a "but we
know it's true anyway, nudge, nudge, wink, wink".

Starting from an inconsistent set of axioms,
anything at all can be proved.

That his axioms proved to be inconsistent, meant
that his life's work crumbled like a house of
cards, and he died self-perceived to be a failure
[despite that today is he greatly respected for
having established much of the logical and
philosophical foundation of the concept of "number",
up until then a concept with no particular rigor
attached to it].

That partial success in achieving rigor didn't leave
the set of his axioms that proved inconsistent the
least bit acceptable to mathematicians, then or now.

As to what this has to do with evolution - if
humans can do things no machine can do - or will
ever be able to do - that may impact the
hypothesis that machine-based organisms may
replace humans as the dominant life form on earth
over the next century or so.

And on the contrary, if humans are "computational"
in a very rich sense, then a steady program of
replicationg those computational capabilities into
integrated mechanisms is a reasonable prospect to
accomplish, if not our replacement, then our
augmentation by peer much faster intellects,
especially as such a program can feed on its own
successes as it goes along, by using its outputs
as intellect augmenters to create the next
generation of outputs.

More importantly, it would mean that there is
reason to suspect that E. O. Wilson may have
gotten it wrong in asserting genetic determinism:

Can the cultural evolution of higher ethical
values gain a direction and momentum its own
and completely replace genetic evolution? I
think not. The genes hold culture on a leash.
The leash is very long, but inevitably values
will be constrained in accordance with their
effects on the human gene pool (E. O.
Wilson).

That is a wholly separate issue, a discussion whose
demesne is meme-space, not cyberspace.

Cultural evolution is not about robots [except
perhaps by coincidence of mechanisms likely to be
employed by cultural evolution], but about evolution
at the level of ideas despite slowing (by medical
advances and pacification of the environment) of
genetic evolution to something running short on
forces of natural selection.

I find that Wilson is perhaps correct that we cannot
become more moral than our genome will allow, but
long term confinement of the most massively immoral
of our population is an extremely effective
contraceptive.

So, that particular kind of genetic evolution may
perhaps be among the most strongly "unnaturally"
selected ones which humankind is presently
experiencing [though I don't know quite how well
"chop off the distribution tail" works in natural
selection as opposed to "bias the whole
population distribution" mechanisms].

and that Dawkins may have actually gotten it
right in asserting the converse:

We, alone on earth, can rebel against the
tyranny of the selfish replicators" (Dawkins,
1976, p. 215).

The most immediate question would be: as a
beneficiary of the dice tosses of the selfish
replicators, how wise or secure should we feel in
engaging in such a rebellion?

I know that there are many, many misfeatures
in selfish replication. Forced sex will
easily stand for the lot as it is fairly
easy to comprehend.

But we need to be very careful in what we
are willing to give away in forbidding the
best rapists their reproductive spoils:
bigger, stronger, smarter, healthier,
sneakier, more dexterous -- how much of that
can humanity afford to forego promoting in
its genome?

That's flame bait, and I have a daughter and
a granddaughter that make that an
uncomfortable issue to raise, but still, it
exists whether we discuss it or not.

The other side of that issue of course is
that the target who successfully resists
forced sex gets (usually "her") choice of
mates, and may select one on other, better
criteria for "best genes to propagate in a
cooperative society", since we humans are
the biggest part of our own environment
these days.

We, as a society, may (and usually do)
choose to remove forcers of sex from the
gene pool, but we should be doing that
advisedly, with evolutionary as well as
social consequences fully exposed to
discussion.

It would be ironic if humankind in rebelling against
its "selfish replicators" hastened its own
extinction by losing competition to species choosing
and undergoing no such rebellion, species like fire
ants or killer bees, for example.

However, this particular argument for the
qualitative superiority of humans is simply wrong
- and (IMO) rather obviously so for anyone who
knows anything about Godel's work.

Why then is one of the papers you referenced
written in 2004? Shouldn't this have all been over and
done with decades ago for a flaw that is so "obvious"?

See above in my response to your item "c". This
situation is rife throughout science. Second raters
rewrite what first raters first publish in more
arcane terms, and sometimes decades later.
Sometimes, also, papers merely summarize the "state
of an art" and are a more convenient citation than
are the original sources.

[quote from Penrose]

The many arguments that computationalists and
other people have presented for wriggling around
Godel's original argument have become known to me
only comparatively recently; perhaps we act and
perceive according to an unknowable algorithm,
perhaps our mathematical understanding is
intrinsically unsound, perhaps we could know the
algorithms according to which we understand
mathematics, but are incapable of knowing the
actual roles that these algorithms play. All
right, these are logical possibilities. But are
they really plausible explanations?

For those who are wedded to computationalism,
explanations of this nature may indeed seem
plausible. But why should we be wedded to
computationalism?

Primarily because the alternative seems to be
mysticism, and descents into mysticism are the
historical stopping point of scientific progress.

I do not know why so many people seem to be.

Yet, some apparently hold to
such a view with almost religious fervour.

Typical theistic BS, trying to call "atheism" a
religion, trying to call maintaining that "humans
compute like machines do" a religious claim.

Each is of course the exact opposite, but tarring
them with the historical record of abject idiocy of
theism reduces them in the minds of naive observers
of the discussion to "mere alternatives" rather than
the "deliberate diametric opposites" they are.

(Indeed, they may often resort to unreasonable
rudeness when they feel this position to be
threatened!)

Like Penrose accusing anyone who opposes _his_ view
of exercising "religous fervor"? The irony is
intense here.

Perhaps computationalism can indeed explain the
facts of human mentality -- but perhaps it
cannot. It is a matter for dispassionate
discussion, and certainly not for abuse!

Yet when only mysticism is proposed as an
alternative, why would Occam's Razor not come into
immediate employment?

I find it curious, also, that even those who
argue passionately may take for granted that
computationalism in some form -- at least for
the objective physical universe -- HAS to be
correct. Accordingly, any argument which seems
to show otherwise MUST have a "flaw" in it.

Certainly any argument whose only basis is mysticism
_does_ have a flaw in it, by that very fact.

Why would Penrose contend otherwise?

Even Chalmers, in his carefully reasoned
commentary, seeks out "the deepest flaw in the
Godelian arguments". There seems to be the
presumption that whatever form of the argument
is presented, it just HAS to be flawed. Very
few people seem to take seriously the slightest
possibility that the argument might perhaps, at
root, be correct! This I certainly find
puzzling.

Okay, for sure here he is using "Goedelian
arguments" to be shorthand standing for "because
Goedel is correct, therefore human (mathematical)
reasoning must be other (and better) than
computational".

That's incredibly sloppy usage, seemingly
attributing to Goedel something he had no part in
formulating, but merely inspired by work elsewhere.

As to his finding opposition to descent into
mysticism among scientists (mathematicians, here)
"puzzling", he need merely re-read a history of the
end of the life of Galileo to cure his puzzlement.

Mysticism is the avowed enemy of science, and not
the least bit shy when allowed to acquire power
about imposing the death penalty for use of science
in preference to mysticism.

Nevertheless, I know of many who (like myself)
do find the simple "bare" form of the Godelian
argument to be very persuasive. To such
people, the long and sometimes tortuous
arguments that I have provided in 'Shadows of
the Mind' may not add much to the case -- in
fact, some have told me that they think that
they effectively weaken it! It might seem that
if I need to go to lengths such as that, the
case must surely be a flimsy one. (Even
Feferman, from his own particular
non-computational standpoint, argues that my
diligent efforts may be "largely wasted!) Yet,
I would claim that some progress has been made.

Why? Mysticism has been promoted, benefiting exactly
whom?

I am struck by the fact that none of the
present commentators has chosen to dispute my
conclusion G (in 'Shadows', p. 76) that "Human
mathematicians are not using a knowably sound
algorithm in order to ascertain mathematical
truth". (Roger Penrose, 'Psyche' Vol 2)

Nothing humans do is "knowably sound" [among other
reasons, because there is no absolute metric for
"right behavior"]. Why should mathematics be an
exception?

The success of evolutionary algorithms in optima
search shows beyond refutation that "looking for an
answer by wandering around lost" is a perfectly
functional mechanism if one applies appropriate
biases to the process.

Peer review, collegeal cooperation, and
"backtracking search on perceived failure" are
several splendid such biasing mechanisms that turn a
random search methodology into a frequently
converging one.

That's how meme evolution happens, among many other
ways.

FWIW

xanthian.


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