Re: Penrose vs the Robot


"abo" <dkfjdklj@xxxxxxxxx> wrote in message
news:1134031410.589521.105800@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>
> Stephen Harris wrote:
>
>> I think Daryl's argument is that all brain processes, whether analog or
not
>> can be simulated by a computable process that there is a corresponding
>> digital function to produce the same output, even if the inner process
>> differs. Recall that a continuous process can be simulated by discrete
>> pulses of very high frequency.
>

> Strictly speaking you are correct. We can grid the four-dimensional
> Universe (or at least the Universe to some large finite limit) in a
> very fine way, so finely that for all pratical purposes, our
> perceptions are unable to distinguish the difference between the grid
> and continuous reality. Because points on the lattice would be finite,
> then the behaviour of what goes on is necessarily computable (assuming
> the usual axioms).
>

I meant the simulation of a continuous process by a discrete process to
be an analogy -- suppose part of a human's behavior or even thinking is
non-computable, this thinking might be simulated or approximated by a
program so that we couldn't distinguish/perceive a difference that might
be part of a human variation or we could attribute the behavior to that.

> Because points on the lattice would be finite,
> then the behaviour of what goes on is necessarily computable (assuming
> the usual axioms).

This is news to me! An algorithm is defined as a terminating process so
it is finite. But in the other direction, where is the rule or axiom which
says
that any finite process must be computable or computably representable?

I think another begged question is that finiteness is intrinsic to
computability.
So I will quote Knuth, a bit long, just to introduce the term "computational
method" which is exactly like a finite computable algorithm except the
requirement for finitness. I don't mean that for practical purposes,
finiteness
isn't a hand tool, but that Daryl started his plausibility argument with

"To say that something is computable doesn't imply that it is something
that is computed. Human behavior results from electrochemical reactions
inside the human brain, not from computation. The claim I am making
is that the *result* of these electrochemical processes is not something
that allows us to recognize mathematical truth any better than a
sufficiently powerful computer program. ...

abo says...

>You now seem to be limiting human behavior to one single type -
>recognizing mathematical truth. Is this intended?

Daryl says:
That's the context of Penrose' argument. He was specifically
addressing the question of whether humans have some mathematical
ability that is beyond that of any computer program. ...
That's right. Recognizing mathematical truth is just a special case."

Earleir Daryl said:
Computability is not about deduction from axioms. That's
one particular computable process, but not all computable
processes are of that form.

My argument that human behavior is computable is this:

Consider a human being as an input/output system. I claim
that

1. The inputs---what we hear, what we see, what we feel
when we touch objects, the strain against muscles,
etc---have finite precision, finite range and finite
processing rate. Thus there are only finitely many
*distinguishable* inputs that we can possibly receive
during any given time interval.

2. The outputs---how we move our muscles to respond to
those inputs---are similarly finite in precision and
range.

3. Our memories have finite capacity for details. Past
a certain point, we cannot distinguish between input
sequences. If I look at a mathematical formula with
10,000 characters, and then look at a second formula
that differs from the first by only one character,
I may not be able to detect the difference.

These three claims, which I think are both empirically
and theoretically justified, imply that we can't meaningfully
compute anything that isn't computable by a *finite* automaton.
The set of mathematical formulas that we can come to know
are valid is not only r.e., it is probably a regular expression,
or even a finite set.

I believe that this is actually true for any physical system
of bounded size, energy, number of particles, etc."

[SH: I don't quibble much with 1 thru 3. I would appreciate an
expansion on why this implies "that we can't meaningfully compute
anything that isn't computable by a *finite* automaton." More to
the point how does this implication if valid, mean that there is a
computable method for acquiring or translating this information
("capturing all true formulas and nothing else")?

Another thing, shifting the territory mathematical language to natural
language. Humans have finite life spans. They can only produce a
finite number of sentences during a lifetime. Yet, most linguists do
not use that finiteness to answer the question "Is Natural Language
Infinite?" No. They answer yes, it it countably infinite. That means they
do not consider the finite physical category of a human life sufficient
to determine the vastness (cardinality) of a logical abstract category.

Daryl: The set of mathematical formulas that we can come to know
are valid is not only r.e., it is probably a regular expression,
or even a finite set.

Changing this to sentences: A natural languge can have a formalized
subset and this has rules. Both a human and a robot could construct
sentences in this formal language. But humans speak and have
created natural language which does not have a set of rules. So
there is going to be no rule for a robot to use to utter (through a
microphone) any sentence which does not have a rule to generate it.
You might argue that a human uses such a rule even if he doesn't
know what it is. Or that it doesn't exceed a finite combinatorial
expansion. And that even if natural language is uncountably infinite,
uncomputable, that a random generation of sentences could potentially
duplicate any sentence of finite length that a human could also utter.

Daryl: These three claims, which I think are both empirically
and theoretically justified, imply that we can't meaningfully
compute anything that isn't computable by a *finite* automaton.

SH: I found the gap in this reasoning. It certainly seems to
presuppose that humans only compute and they do this by a
rule (humans are finite) so that a finite automaton can match that.

But how do we know that humans only compute? Penrose says
that humans don't only compute, they do more. His argument
to support his view fails. That does not mean his primary assumption
that humans do not just compute is wrong.

Computability is an algorithmic process and Turing covered that,
so there is no dispute that robots can compute what humans can
compute and probably a lot more.

So how do we know that humans only compute? Why can't
unlearned non-computable instinctual responses impact synapse
firing potentials, which are very time dependent and also impacted
by emotional input. It would seem that you would have to make a
further claim that emotions and instincts are computable.

Right now your claim seems to be that if it is finite then it is
computable and if it is computable then it is finite. Your view
also suffers from abstracting from physical processes to
more abstract notions. Like all humans are composed of atoms
does not mean that all atom compositions are humans. As you
know there are papers on incomputability in the universe. Geroch

Also you don't provide a means for the robot to algorithmically
find a method to construct sentences in a particular context so
that they are fraught with meaning and intentionality. Proabably,
you adopt the view that a very finely grained level (or is it very
course grained fundamental) that the universe is completely
deterministic so that intentionality has no inherent meaning. ]
------------------------------------------------------------------------------

> Strictly speaking you are correct. We can grid the four-dimensional
> Universe (or at least the Universe to some large finite limit) in a
> very fine way, so finely that for all pratical purposes, our
> perceptions are unable to distinguish the difference between the grid
> and continuous reality. Because points on the lattice would be finite,
> then the behaviour of what goes on is necessarily computable (assuming
> the usual axioms).

SH: So I am interested in how finiteness and axioms actually apply.
I want to know what axioms you mean below. I quote Knuth to show
that a process can be computable without being finite, one just changes
the term to computation method. I think arguments which impute that a
(logical) formal description of the universe constrains or predicts
(ignoring
emergence) the behavior of the universe are putting the cart before the
horse.
Penrose had no business saying what can exist in the physical universe, such
as physical robots running physical programs cannot simulate the behavior
of physical humans. Godelian Inc. (a result about formal abstract systems)
does not have the scope to impinge upon determining what is possible in the
physical universe. Since we are agreed that other human behavior such as
the primitive brain processes which produce thinking which is essential to
motivating speech and language, is involved besides just creative
mathematical discovery, I'm thinking of including some ideas that come from:
Is Natural Language Infinite? Most linguists say natural language is
countably infinite,
despite the fact that there are a finite number of humans living a finite
number
of years in a finite? universe. Why should a change in theory that the
universe
is not finite in the sense of the big bang reverses direction and the
universe
ends to the universe may expand forever be used as a basis for deciding
what is computable. Post and Langendoen (experts) for instance believe that
natural language is uncountably infinite which I think means uncomputable.
Would that mean in theory that humans could produce a different
quality/range
of sentences than robots using a computable program, while the robots could
produce a greater quantity/vastness of sentences
practically(physically)speaking.
Any way Post has something interesting to say about generative grammars
which is contested by model-theoretic frameworks. I think that the more
closely
the implicit assumptions of Daryl's position are examined, then the more
philosophically nebulous (or debatable) the position becomes and his view
loses its veneer of plausibility.

Two key properties of natural languages
"The invented languages of logic, mathematics, and computer science | hence-
forth, formal languages | are stipulated sets of strings (or other
structures)
defined over a finite vocabulary of symbols. Post production systems and the
generative grammars that are based on them are ideally suited to stating the
explicit grammars of formal languages, and were invented for exactly that
purpose.
But natural languages have a number of properties that clearly differentiate
them from formal languages. In this section we review two illustrative
phenomena:
first, the fact that being grammatically ill-formed is a matter of degree,
and second,
the fact that there is no fixed lexicon for a natural language. We point out
that
model-theoretic frameworks immediately suggest appropriate ways to describe
the relevant phenomena. Generative frameworks do not."

> Because points on the lattice would be finite,
> then the behaviour of what goes on is necessarily computable (assuming
> the usual axioms).

Knuth, Vol. 1, Sec. 1.1:

"The modern meaning for algorithm is quite similar to that of
_recipe_,_process_, _method_, _technique_, _procedure_, _routine_,
_rigmarole_, except that the word "algorithm" connotes something
just a little different. Besides merely being a finite set of
rules that gives a sequence of operations for solving a specific
type of problems, an algorithm has five important features:

1) Finitness. An algorithm must always terminate after a finite
number of steps. [...] (A procedure that has all of the
characteristics of an algorithm except that it possibly lacks
_finitness_ may be called a _computational method_. [...])


(A procedure which has all the characteristics of an algorithm
except that it possibly lacks finitness may be called a
"computational method". Besides his algorithm for the greatest
common divisor of two integers, Euclid also gave a geometrical
construction that is essentially equivalent to algorithm E,
except that it is procedure for obtaining the ``greatest common
measure'' of the lengths of two line segments; this is a
computational method that does not terminate if the given
lengths are ``incommensurate''). [SH: for intererested constructivists]


2) Definitess. Each step of an algorithm must be precisely
defined; the actions to be carried out must be rigorously and
unambiguously specified for each case. [...]


3) Input. An algorithm has zero or more _inputs_: quantities
that are give to it initially before the algorithm begins, or
dynamically as the algorithm runs. [...]


4) Output. An algorithm has zero or more _outputs_: quantities
that have a specified relation to the inputs. [...]


5) Effectiveness: An algorithm is also generally expected to be
_effective_, in the sense that its operations must all be
sufficiently basic that they can in principle be done exactly
and in a finite length of time by someone using pencil and
paper. [...] "

(> But this begs the question, in the sense that ultimately the question
> is not whether a theoretical computer can imitate human behaviour, but
> whether an actual computer, with finite resources, can. Certainly it
> is of little interest if you are endowing the computer with more
> resources than posited in the Universe. I think in such cases it also
> becomes reasonable to throw out the Successor Axiom, in which case the
> assertion that there is a computable function mimicking a lattice does
> not follow automatially.
>

Gee, I think the situation is the opposite. Not getting into physical things
like digestion but intellectual functions. In areas of computability,
computer are already superior to humans. They play better chess by brute
force. They can locate misspelled words faster (proofread) and correct them.
The question is if there is a human intellectual function that can't be
simulated. So that gets into organization, context and semantics. Producing
a transaltion of "The Illiad" which is considered "gifted" by human readers
who don't know the source of the translation/translator. That means the
robot may need to simulate emotions and instinctual motivations.
Psychologists cannot enumerate the basic emotions much less describe how
they interact and establish (mixed) priorities.

There is already a computer in California which has 50% of human potential
in terms of pathways corresponding to snyapses (a few trillion). Physical
resources is not the issue with AI (maybe you mean exponential explosion
which happens with some approaches) but learning to learn, and organizing
priorities, which is why they just can't conglomerate a bunch of human
expert systems can come up with a near human functioning system. Anyway,
your point about a theoretical computer being needed could benefit from some
expansion, I've never heard of it. I think you may mean something like
learning neural networks. It is also true to simulate the output of a Turing
machine computing Pi exceeds the resources of memory for the physical
universe. But that isn't needed to practically simulate a human by a
computer since computers can already remember billions of more digits of Pi
than any human.

> I think in such cases it also
> becomes reasonable to throw out the Successor Axiom, in which case the
> assertion that there is a computable function mimicking a lattice does
> not follow automatially.

I don't understand this. How this could determine the capability of a
physical device? The fundamental motivation for a successor is the human
perception of time as a series of instances which causally follow each
other. But before humans existed, the universe existed, which proceeded to
manifest humans and their intellectual abstract notions such as a Successor
Axiom and the universe did this completely independently of such a notion,
unless one wants to conjure up a Platonic reality of eternal verities.

Theory predicted that there would be no forbidden symmetry but nonetheless
it was observed (related to Penrose tiles). I don't see any reason to
believe that the universe _is_ a formal system which has its behavior
determined by some abstract set of axioms/rules. When a prediction generated
by a formal theory fails to match an observation, it is the theory which is
wrong, not reality. But this issue of how to describe the universe; Wolfram
speculates that the universe is an evolved cellular automata. Yet another
disturbance in the Force of truth.


> Now maybe a computer will be able to imitate a muscle's behaviour. I
> don't know of any valid argument against, but I don't see any valid
> argument for, other than actually constructing the thing. Basically I
> think arguments *for* in this domain (either Penrose's or Daryl's) are
> almost always invalid. Peoples' fancies catch hold and they almost
> always try to establish too much.
>

I agree with your conclusion although it doesn't seem like we arrived at it
the same way, perhaps like a human and a robot producing the same output :-)


>
> We do not know, we will probably not know for some time, and we may
> never know.
>

"All r. e. sets have transformational grammars; but r. e. sets do not
necessarily have r. e. complements." 'There may be some answer to a general
question, but no general effective procedure for finding it.'

I don't think there is a computational method for building a robot to
simulate Penrose. I think the robot can be built to approximate Penrose ala
Myhill,

"Not every feature of the world is either listable or computable. For
example, the property of being a true statement is neither listable nor
computable. One can approximate the truth to greater and greater
accuracy by introducing more and more rules of reasoning and adding
further axiomatic assumptions, but it can never be captured by any finite
set of rules. These attributes that have neither the property of listability
nor that of computability--the "prospective" features of the world--are
those that we cannot recognize or generate by a series or sequence of
logical steps. They witness to the need for ingenuity and novelty; for
they cannot be encompassed by any finite collection of rules or laws.
Beauty, simplicity, truth; these are all properties that are prospective."

Prospecting for information by trial and error,

Stephen

http://mathworld.wolfram.com/CreativeSet.html

"A recursively enumerable set A is creative if its complement is productive.
Creative sets are not recursive. The property of creativeness coincides with
completeness. Namely, set A is creative iff if it is many-one complete.

Elementary arithmetic formulas are built up from 0, 1, 2, ..., , , ,
variables, connectives, and quantifiers. The set of all true arithmetic
formulas is productive. Informally speaking, this means that no
axiomatization of arithmetic can capture all true formulas and nothing else.
For example, consider Peano arithmetic. Under the assumption that no false
arithmetic formulas are provable in this theory, provable Peano arithmetic
formulas form a creative set."




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