People that are color blind don't have the concept of color, but can
get the concept from those that aren't color blind. So the question is
whether there is some concept that humans don't know about that can
never be discovered.

--
Ron

I think a popular usage of mathematics would accord with your description.
However, Wolf tied his usage into Godel's Incompleteness Theorem (GIT).
That means he is using the standard definition of formal Mathematics:
A set of formal languages.
Wolf asserted that natural language is incomplete due to GIT.
I asserted that GIT only applies to formal languages and the
usual description of GIT is a formalized result about number theory.
Peano Arithmetic is not a natural language.
Natural language is not axiomatized nor congruent to a
formal repesentation constructed around natural language such as Cyc.

"Stephen Harris" <stephen.p.harris@sbcglobal.net> wrote in message
news:<%5zTb.19094$_j3.15019@newssvr27.news.prodigy.com>...
"Wolf Kirchmeir" <wwolfkir@sympatico.can> wrote in message
news:jbysxveflzcngvpbpna.hsgsja1.pminews@news1.sympatico.ca...
On Fri, 30 Jan 2004 00:18:54 GMT, Stephen Harris wrote:

The issue under discussion is whether language is adequate. Language
is
composed of words representing concepts.

I wish it were that simple.


I think it is that simple. I will present again the statement to which I
objected:

Wolf wrote;
"You may object that mathematics isn't what you mean by an extension
of language. But I would argue that mathematics is an extension of
language,
and then some."

Do you guys realize that the communicative written and oral language
of mathematics is just another natural language while mathematics
itself is more than just communication?

Likewise, common sense processing is more than a computational means
to transfer concise common-sense statements (language). What's
surprising about that?

The logicist tradition has almost entirely failed on these accounts
because it turned out that mathematics is not the syntax of narrow
language. Neither do the truth values of semantic expressions have any
significance beyond being a tiny aspect of meaning in the analyzing
entity's mind. Therefore, I'm having great difficulty in following
your discussion. I'm going to have a deeper look at this last fringe
of the thread and make some comments, though.

Regards,

--
Eray

On Tue, 03 Feb 2004 18:11:54 GMT, Stephen Harris wrote:

But saying that is not the same as saying that language is inadequate - which
I take it was your point, and which seems to me to be true, regardless of
what we mean by in/adequacy of language.

What exactly do we mean by saying that language is "adequate"? In the context
of problem solving, I suggest "capable of properly posing any problem." But
that raises the question of what we mean by "properly posing a problem," and
AFAIK only in formal languages does that question have a clear answer. A
properly posed problem consists of well formed formulas, to use one
formalism. So if we say that ordinary language is capable of posing any
problem, aren't we implying that ordinary language is in fact a formal
language? Or are we saying that formal languages are contained in ordinary
language, and when we pose a problem, we explicitly or implicitly invoke a
formal language?

On Tue, 03 Feb 2004 18:11:54 GMT, Stephen Harris wrote:

I think a popular usage of mathematics would accord with your
description.
However, Wolf tied his usage into Godel's Incompleteness Theorem (GIT).
That means he is using the standard definition of formal Mathematics:
A set of formal languages.
Wolf asserted that natural language is incomplete due to GIT.
I asserted that GIT only applies to formal languages and the
usual description of GIT is a formalized result about number theory.
Peano Arithmetic is not a natural language.
Natural language is not axiomatized nor congruent to a
formal repesentation constructed around natural language such as Cyc.

I think I'm beginning to accept your point, but I'm muddled as to why,
exactly. And then again, maybe I can put my claim in terms that make it
both
clearer and closer to sustainable.

It seems to me that when you "pose a problem" you are formalising ordinary
language. Mathematics simply makes the formalism(s) explicit. Hence GIT
should apply, but after reading your comments, I'm not at all sure what
that
would mean. My first try at an answer would be something like this: We can
pose problems the validity of whose solutions we cannot decide.

On 27 Jan 2004 21:28:01 -0800, Ron Peterson wrote:
Human languages are extensible, so I think that any problem that can
be posed can be explained. Does someone have an example of a problem
that can't be explained?

Human language is adequate for posing the problem and for posing the
solution whether or not the solution is achievable.

Nice question. I'm not sure what you mean by "explaining" a problem.
It seems to me that posing a problem is equivalent to explaining it. So
I'll
shift the ground, perhaps unfairly, to solving problems rather than
explaining
them. Granting this shift, we see that posing a problem is not the same as
solving it.

You may object that mathematics
isn't what you mean by an extension of language. But I would argue that
mathematics is an extension of language, and then some.

You initially refuted my claim that human languages are inadequate to
express
all problems by stating that "any problem that can be posed can be
solved",
or words to that effect.

But saying that is not the same as saying that language is inadequate -
which
I take it was your point, and which seems to me to be true, regardless of
what we mean by in/adequacy of language.

What exactly do we mean by saying that language is "adequate"? In the
context
of problem solving, I suggest "capable of properly posing any problem."
But
that raises the question of what we mean by "properly posing a problem,"
and
AFAIK only in formal languages does that question have a clear answer. A
properly posed problem consists of well formed formulas, to use one
formalism. So if we say that ordinary language is capable of posing any
problem, aren't we implying that ordinary language is in fact a formal
language? Or are we saying that formal languages are contained in ordinary
language, and when we pose a problem, we explicitly or implicitly invoke a
formal language?

You initially refuted my claim that human languages are inadequate to
express
all problems by stating that "any problem that can be posed can be
solved",
or words to that effect. It seems to me that this statement is limited to

formal languages. It's clear enough that human languages aren't formal
languages, so I think your refutation doesn't really answer my claim, it
sidesteps it. Not that my claim is irrefutable; but it's an ill-posed
problem, so it needs redefinition. I haven't gotten very far with doing
that,
as my above remarks no doubt demonstrate, so I'll stop here.

BTW, I withdraw my claim that mathematics is (are?) an extension of
ordinary
language. IMO, the claim fails because of the vagueness of
"extension/extensible," and anyhow, it would be better to say that
mathematics are included in ordinary language. But that claim, also
suffers
from vagueness, so I don't think it's much of an improvement.

Best Wishes,

Wolf Kirchmeir, Blind River, Ontario

.............................................................................

You can observe a lot by watching. (Yogi Berra, Phil. Em.)
Remove 1st and last letters from address to reach me

"Wolf Kirchmeir" <wwolfkir@sympatico.can> wrote in message
news:jbysxveflzcngvpbpna.hsj1ao1.pminews@news1.sympatico.ca...
On Tue, 03 Feb 2004 18:11:54 GMT, Stephen Harris wrote:

I think a popular usage of mathematics would accord with your
description.
However, Wolf tied his usage into Godel's Incompleteness Theorem (GIT).
That means he is using the standard definition of formal Mathematics:
A set of formal languages.
Wolf asserted that natural language is incomplete due to GIT.
I asserted that GIT only applies to formal languages and the
usual description of GIT is a formalized result about number theory.
Peano Arithmetic is not a natural language.
Natural language is not axiomatized nor congruent to a
formal repesentation constructed around natural language such as Cyc.

I think I'm beginning to accept your point, but I'm muddled as to why,
exactly. And then again, maybe I can put my claim in terms that make it
both
clearer and closer to sustainable.

It seems to me that when you "pose a problem" you are formalising
ordinary
language. Mathematics simply makes the formalism(s) explicit. Hence GIT
should apply, but after reading your comments, I'm not at all sure what
that
would mean. My first try at an answer would be something like this: We
can
pose problems the validity of whose solutions we cannot decide.


Hello,

These long discussions can get off track, which is ok as long
as it doesn't transfer to my stated position. So one interpretation
of the statement(s) I took issue with was:

On 27 Jan 2004 21:28:01 -0800, Ron Peterson wrote:
Human languages are extensible, so I think that any problem that can
be posed can be explained. Does someone have an example of a problem
that can't be explained?

Lester first wrote:
Human language is adequate for posing the problem and for posing the
solution whether or not the solution is achievable.

Wolf responded:
Nice question. I'm not sure what you mean by "explaining" a problem.
It seems to me that posing a problem is equivalent to explaining it. So
I'll
shift the ground, perhaps unfairly, to solving problems rather than
explaining
them. Granting this shift, we see that posing a problem is not the same
as
solving it.

SH: I thought their rewording of what you wrote as too much off target.
And then you contributed to shifting the focus of _my_ responses.
I will post what I objected to again. I don't mind talking about problem
posing but I think that is a slightly different issue than your orignal
assertion:

Wolf wrote
You may object that mathematics
isn't what you mean by an extension of language. But I would argue that
mathematics is an extension of language, and then some.

SH: So below you withdraw this point. Now you say:

You initially refuted my claim that human languages are inadequate to
express
all problems by stating that "any problem that can be posed can be
solved",
or words to that effect.

SH: I am pretty sure that was Ron and Lester. I may have touched on it.
But my refutation involved what you wrote about mathematics/language
quoted above, which I repeated in some posts so that it wouldn't be lost.
So then moving on...

But saying that is not the same as saying that language is inadequate -
which
I take it was your point, and which seems to me to be true, regardless
of
what we mean by in/adequacy of language.


I think the culprit is a lack of information, inadequate knowledge.
Suppose
Fermat posed his last theorem rather than claimed the napkin was too
small.
Mathematics had to invent at least two new fields to answer Fermat's
claim.
Concepts were ready to describe the solution, once it was known. IMO,
lack of knowledge was responsible for words representing concepts, which
had not been discovered yet, not to be represented in our current
language.
To blame this shortcoming on language then imposes the responsibility
of describing the nature of reality without knowledge of reality informing
a description of reality which the words of language provide. To call
language inadequate, it needs to not be fulfilling some function, and to
me
the function not being performed leading to a lack of words, is obtaining
the
knowledge which leads to the production of new concepts --> new words.
Well, I've belabored that point enough. Maybe we will just disagree.


What exactly do we mean by saying that language is "adequate"? In the
context
of problem solving, I suggest "capable of properly posing any problem."
But
that raises the question of what we mean by "properly posing a problem,"
and
AFAIK only in formal languages does that question have a clear answer. A
properly posed problem consists of well formed formulas, to use one
formalism. So if we say that ordinary language is capable of posing any
problem, aren't we implying that ordinary language is in fact a formal
language? Or are we saying that formal languages are contained in
ordinary
language, and when we pose a problem, we explicitly or implicitly invoke
a
formal language?


By strange coincidence this reminds me of a question Hilbert posed
which I found on http://math.stanford.edu/~feferman/papers/deciding.pdf.

"Is the axiom of the solvability of every problem a peculiar
characteristic
of mathematical thought alone, or is it possibly a general law inherent in
the
nature of the mind, that all questions which it asks must be
answerable?...This
conviction of the solvability of every mathematical problem is a powerful
incentive to the worker. We hear within us the perpetual call: There is
the problem. >Seek its solution. You can find it by pure reason, for in

ignoramibus. ...


From this line of argument, it appears that there will be sensible
mathematical propositions which we can never prove or disprove, and that,
contrary to Hilbert's general view: Ignoramus et ignorabimus (We do not
know

Emil du Bois Reymond). Is this conclusion really justified by G¨odel's
incompleteness results? ....

In G¨odel's view, the "true reason" for his incompleteness theorems lies
in
the fact that

beyond any system S, we must accept new axioms concerning arbitrary
subsets
of the universe of objects with which S deals. If S is itself a system of
set
theory, these new axioms are called "axioms of higher infinity," since the
new sets >obtained will be
infinitely larger in a suitable sense than the sets which can be shown to
exist in S. It is indeed the case that by adding such new axioms one is
able to
establish the consistency of S. Of course, we then obtain a new system
S -which is >again incomplete-and then the process of adding new axioms must

must be iterated indefinitely. All this accords with G¨odel's underlying
belief in

the Platonic reality of set theory and that the kind of informal reasoning
which led us to accept the Zermelo-Fraenkel axioms, as true of this
reality, can be
continuedto expand these axioms indefinitely to settle hitherto undecided
propositions.

SH: Perhaps you will agree this is related to the original P & Q post
which
some of us challenged as recapitulating the "set of all sets, not a member
of itself
paradox.

IMO, this weighs on my side in our debate about whether the solution to a
problem

is implicit in the posing the question (which I did talk about). I think
that assumes some sort of causal connection between input question and
output >solution.

Gregory Chaitin, co-founder of Algorithmic Information Theory has made an
advance on Goedel Incompleteness and Turing's Halting Problem. He says
something very close to this: "This leads to my first result on the limits
of axiomatic reasoning, namely that most numbers are uninteresting or
random,
but we can never be sure, we can never prove it, in individual cases. And
these
ideas culminate in my discovery that some mathematical facts are true for
no reason, >they are true by accident, or at random.
http://www.umcs.maine.edu/~chaitin/summer.html Paradoxes of Randomness.

That sounds to me like, the solution is always contained in the question,
bites the dust.

You initially refuted my claim that human languages are inadequate to
express
all problems by stating that "any problem that can be posed can be
solved",
or words to that effect. It seems to me that this statement is limited
to

Nope not me. I said the solution was not intrinsic. I said that any
thought
which is conceived, can be named by a concept word, that it can always
be invented. That there were no such things as unconceivable concepts just
as there were no irresistable forces and immovable objects in the same
universe.

formal languages. It's clear enough that human languages aren't formal
languages, so I think your refutation doesn't really answer my claim, it
sidesteps it. Not that my claim is irrefutable; but it's an ill-posed
problem, so it needs redefinition. I haven't gotten very far with doing
that,
as my above remarks no doubt demonstrate, so I'll stop here.


My refutation is about the original language-->mathematics-->Godel
Inc.Thoerem remark. I only touched on one aspect of your modified claim
about
posing questions; which was the answer is not intrinsicly contained in the
question.

In case Eray is reading a dabble of information regarding Penrose and AI:

4.7 Penrose reports in section 3.1 on what Gödel took the significance of
his incompleteness theorems to be, via a quotation which had circulated
some
time back from Gödel's unpublished Gibbs lecture of 1951. That piece is
now
available in full as *1951 in Gödel (1995), with an illuminating
introductory note by George Boolos. More cautious than Penrose, Gödel
there
comes to the conclusion that "either...the human mind (even within the
realm
of pure mathematics) infinitely surpasses the powers of any finite
machine,
or else there exist absolutely unsolvable diophantine problems." (op.cit.,
p. 310). Boolos' discussion of this is tonic:

"There is a gap between the proposition that no finite machine meeting
certain weak conditions can print a certain formal sentence (which will
depend on the machine) and the statement that if the human mind is a
finite
machine, there exist truths that cannot be established by any proof the
human mind can conceive.... it is certainly not obvious what it means to
say
that the human mind, or even the mind of some one human being is a finite
machine, e.g. a Turing machine. And to say that the mind (at least in its
theorem-proving aspect), or a mind, may be represented by a Turing machine
is to leave entirely open just how it is so represented." (Boolos (1995)
p.
293).
or else there exist absolutely unsolvable diophantine problems

Feferman:
The solution of three of Hilbert's problems were to involve mathematical
logic and thefoundations of mathematics in an essential way, and it is
these
I want to tell you somethingabout in this lecture. They are the problems
numbered 1, 2 and 10 in his list, but for reasonsthat you'll see, I want
to
discuss them in reverse order.2Problem 10 called for an algorithm to
determine of any given Diophantine equationwhether or not it has any
integer
solutions. The solution of three of Hilbert's problems were to involve
mathematical logic and thefoundations of mathematics in an essential way,
and it is these I want to tell you somethingabout in this lecture. They
are
the problems numbered 1, 2 and 10 in his list, but for reasonsthat you'll
see, I want to discuss them in reverse order.2Problem 10 called for an
algorithm to determine of any given Diophantine equationwhether or not it
has any integer solutions. The solution of three of Hilbert's problems
were
to involve mathematical logic and thefoundations of mathematics in an
essential way, and it is these I want to tell you somethingabout in this
lecture. They are the problems numbered 1, 2 and 10 in his list, but for
reasonsthat you'll see, I want to discuss them in reverse order.2Problem
10
called for an algorithm to determine of any given Diophantine
equationwhether or not it has any integer solutions. The solution of three
of Hilbert's problems were to involve mathematical logic and the

foundations of mathematics in an essential way, and it is these I want to
tell you something about in this lecture. They are the problems numbered
1, 2 and 10 in his list, but for reasons that you'll see, I want to discuss
them in
reverse order.


Problem 10 called for an algorithm to determine of any given Diophantine
equation whether or not it has any integer solutions.

The most famous Diophantine equation is that addressed in the
so-called Fermat'sLast Theorem.Contrary to Hilbert's expectations, Problem
10 was eventually solved in the negative.This was accomplished in 1970 by
a
young Russian mathematician, Yuri Matiyasevich.
The most famous Diophantine equation is that addressed in the so-called
Fermat's Last Theorem.Contrary to Hilbert's expectations, Problem 10
was eventually solved in the negative. This was accomplished in 1970 by a
young >Russian mathematician, Yuri Matiyasevich. The general problem of the

or else there exist absolutely unsolvable diophantine problems,

Stephen

BTW, I withdraw my claim that mathematics is (are?) an extension of
ordinary
language. IMO, the claim fails because of the vagueness of
"extension/extensible," and anyhow, it would be better to say that
mathematics are included in ordinary language. But that claim, also
suffers
from vagueness, so I don't think it's much of an improvement.


Mathematicians invent formal systems which are unrelated to known reality.


Best Wishes,

Wolf Kirchmeir, Blind River, Ontario


.............................................................................
.................
You can observe a lot by watching. (Yogi Berra, Phil. Em.)
Remove 1st and last letters from address to reach me

Wolf Kirchmeir" <wwolfkir@sympatico.can> wrote in message
news:jbysxveflzcngvpbpna.hsj1ao1.pminews@news1.sympatico.ca...
On Tue, 03 Feb 2004 18:11:54 GMT, Stephen Harris wrote:

What exactly do we mean by saying that language is "adequate"? In the
contextof problem solving, I suggest "capable of properly posing any
problem."

Babies have minds and think and associate before they learn to speak.
So though we often think in words, perhaps more than non-verbal,
language is a tool of the process that we use to formulate problems.
Conceiving is not a function of language but a function of mind.
If you conceive with clarity, then language is always up to the task
of expressing the concept. I don't think you can find an example of
a concept that cannot be expressed in words. It isn't the job of language
to invent words to describe concepts which are as yet unknown or
non-existentent. Once the concept is cognized, language answers the call.

that raises the question of what we mean by "properly posing a problem,"
and
AFAIK only in formal languages does that question have a clear answer. A
properly posed problem consists of well formed formulas, to use one
formalism. So if we say that ordinary language is capable of posing any
problem, aren't we implying that ordinary language is in fact a formal
language? Or are we saying that formal languages are contained in
ordinary
language, and when we pose a problem, we explicitly or implicitly invoke
a
formal language?


I have other ideas about how this situation should be described so I will
provide some background information about mathematics and the real world.
Also I have a cold and don't feel like restating and retyping this stuff.

Travis Norsen:

Thus a dilemma is set up: How can science explain the orderliness of the
physical world, if it rejects the religious notion of a supernatural
orderer?

Immanuel Kant provided a way out of this dilemma which has had tremendous
influence on twentieth century physics. His solution was that we are the
orderer: perceptual awareness is not a direct grasp of the external world,
but a distortion -- it consists of sense data that have been processed by
our conceptual consciousness, twisted to fit certain innate conceptual
"categories" which order the chaotic data of the senses.

This unavoidable processing means that we do not perceive reality as it
really is, but rather only reality as it appears after processing. This
idea
was the basis of Kant's splitting between the world "as it really is"
(which
we can never know) and the world "as it appears to us," that is, as
filtered
through the distorting lens of our conceptual consciousness.

Physicists, then, are relegated to looking at experimental data,
formulating
mathematical principles to describe and quantify it, and making
calculations
and predictions on that basis -- without ever having a chance to discover
the true nature of the entities they are studying, i.e., without ever
understanding the underlying causes of the observed mathematical
relationships. In short, physics for Kant consists of categorizing and
cataloging mere appearances -- which physicists can never truly
understand,
since the underlying causal processes are inaccessible to human
consciousness. The best we can hope for is some mathematical equation
which
relates one appearance to the next.

Niels Bohr, one of the founding fathers of quantum mechanics and the man
who
is almost single-handedly responsible for the standard interpretation of
its
formalism, summarized this Kantian perspective eloquently: "There is no
quantum world. There is only an abstract quantum mechanical description.
It
is wrong to think that the task of physics is to find out how Nature is.
Physics concerns what we can say about Nature." [7]

Finally, Sir James Jeans summarizes the relationship between Kantian
philosophy and the Primacy of Mathematics: "[W]e can never understand what
events are, but must limit ourselves to describing the patterns of events
in
mathematical terms; no other aim is possible. Physicists who are trying to
understand nature may work in many different fields and by many different
methods; one may dig, one may sow, one may reap. But the final harvest
will
always be a sheaf of mathematical formulae. These will never describe
nature
itself... Thus our studies can never put us into contact with reality."
[9]

This kind of superficial description of appearances is what -- and all --
quantum mechanics and relativity offer. The equations themselves, of
course,
seem to be correct; they do indeed correctly describe the "appearances".
But
the fact that physicists have rejected the possibility of explaining
them --
that is purely Kant's influence. Kant convinced physicists that no causal
explanation of the equations is possible. [11]

SH: The author is probably overstating Kant's influence. And perhaps
"appearances" are not that distant from what reality "really is". But, I
get the impression when you use the word "explanation" that you think
there is some type of adequacy or completeness available in that
explaining; and that position is a minority philosophical position, and
I'm not sure you realized that when you wrote about this. Also Goedel
and Penrose are both Platonists, which is relevant to understanding
their views.

"Notice again the shared premise of the Platonist and Kantian versions of
the Primacy of Mathematics. Both regard the material world of entities as
not fully real, so that something external to physical matter must be
responsible for the orderly behavior of physical objects. The Platonists
find this orderer in a supernatural realm, while the Kantians find it in
the
inherent conceptual categories of our minds. Both find it in
consciousness."

SH: So compare this with the quote from the webpage below and limitations
of
"posing poroblems".

"Meanwhile, on the empirical front, mathematics continued to be a
spectacular success as a theory-building tool. The great successes of
20th-century physics (general relativity and quantum mechanics) wandered
so
far from the realm of physical intuition that they could be understood
only
by meditating deeply on their mathematical formalisms, and following
through
to their logical conclusions, even when those conclusions seem wildly
bizarre.

What irony. Even as mathematical 'perception' came to seem less and less
reliable in pure mathematics, it became more and more indispensable in
phenomenal science!"

SH: So there are 8 major interpretations of Quantum Theory(not Bohm).
The interpretations all use the same quantum mathematical formalism.
The interpretations are different explanations of physical reality,
all based on the same equations. Some of these interpretations (MWI)
are in principle not subject to verification by physical observation of
the
speculated reality. So the interpretations may indeed be "wildly bizarre"
and the logical processes which generated these interpretations, may
contains faults.

So the scope of explanatory power is dubious. We seem to share
some of the same starting premises but do not reach the same conclusion:
This situation is due to the inadequacy of language.

http://www.catb.org/~esr/writings/utility-of-math/ which I think is less
slanted, continued:

Against this background, Einstein's famous quote wondering at the
applicability of mathematics to phenomenal science poses an even thornier
problem than at first appears.

The relationship between mathematical models and phenomenal prediction is
complicated, not just in practice but in principle. Much more complicated
because, as we now know, there are mutually exclusive ways to axiomatize
mathematics! It can be diagrammed as follows (thanks to Jesse Perry for
supplying the original of this chart):

The key transactions for our purposes are C and D -- the translations
between a predictive model and a mathematical formalism. What mystified
Einstein is how often D leads to new insights.

We begin to get some handle on the problem if we phrase it more precisely;
that is, "Why does a good choice of C so often yield new knowledge via D?"

The simplest answer is to invert the question and treat it as a
definition.
A "good choice of C" is one which leads to new predictions. The choice of
C
is not one that can be made a-priori; one has to choose, empirically, a
mapping between real and mathematical objects, then evaluate that mapping
by
seeing if it predicts well.

For example, the positive integers are a good formalism for counting
marbles. We can confidently predict that if we put two marbles in a jar,
and
then put three marbles in a jar, and then empirically associate the set of
two marbles with the mathematical entity 2, and likewise associate the set
of three marbles with the mathematical entity 3, and then assume that
physical aggregation is modeled by +, then the number of marbles in the
jar
will correspond to the mathematical entity 5.

The above may seem to be a remarkable amount of pedantry to load on an
obvious association, one we normally make without having to think about
it.
But remember that small children have to learn to count...and consider how
the above would fail if we were putting into the jar, not marbles, but
lumps
of mud or volumes of gas!

One can argue that it only makes sense to marvel at the utility of
mathematics if one assumes that C for any phenomenal system is an a-priori
given. But we've seen that it is not. A physicist who marvels at the
applicability of mathematics has forgotten or ignored the complexity of C;
he is really being puzzled at the human ability to choose appropriate
mathematical models empirically."

SH: Penrose of Penrose tiles, builds on the italicized comment in his
attack
on AI.

Concluding quote:

"There are many things for which mathematical modeling leads at best to
fuzzy, contingent, statistical results and never successfully predicts
'new
entities' at all. In fact, such systems are the rule, not the exception.
So the proper answer to the question "Why is mathematics is so marvelously
applicable to my science?" is simply "Because that's the kind of science
you've chosen to study!"

SH: Lanaguage enables us to communicate our shared common perception of an
underlying reality.

The mind works by processing change; perceiving cause and effect
sequences unfurling through time.

Popular mathematics works by quantifying some of these observed
relationships.

There is no mathematical description of how we evolved and obtained a
consciousness which proceeded to display language. There is no
algorithmic description of natural language. If such a formal description
of consciousness were invented there is no way it can be verified against
how it actually happened.

Projects like Cyc may produce usable approximations of "understanding
language in context" but there is no way of determing how close that
approximation is to the actual reality of the experience, when the
trajectories will diverge, what question a Turing Test passing
machine will stumble over.

Regards,

Stephen

Mathematicians invent formal systems which are unrelated to known reality.