[Arisbe] Re: Inquiry Into Inquiry

[Jon Awbrey]

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Howard Pattee wrote (HP):
Jon Awbrey wrote (JA):

JA: Howard, I continue to be at a loss for how to interpret many
    of the technical terms and phrases that you use in this text.
    I do not know what else to do but keep pressing for clarity.
    Many of my most familiar terms and phrases are being used in
    ways the meaning of which to you I can neither grasp, guess,
    nor for which invent any consistent sense.  Most of these
    problematic themes appear to arise in the neighborhood of
    the rootword "forma" -- which to me is an old Latin word
    for "beauty", and so not a conceit that I would gladly
    let go.

HP: I am using the concept of 'formal" only in the sense used in the areas you mention,
    formal math, formal logic, formal language.  I'm using the standard meaning found
    in any math dictionary:  "An uninterpreted symbolic system including non-logical
    axioms on which there is a relation of deducibility in purely syntactic terms."
    Hermann Weyl, more metaphorically, calls a formal system: "... a game of symbols
    played according fixed rules [where] the symbols are not meant to be symbols for
    something."

I believe that many different issues are being confounded here under the same name.
This is very typical of what happens when a good idea, like "let's pay attention to
the forms of what's happening here", gets turned into an "-ism", like "nothing but
the forms of what's happening here matters very much at all".  I understand about
uninterpreted formal systems, and I do a large part of my basic logical work with
abstract calculi that are even more uninterpreted than most you will typically see.
But not all formal systems are uninterpreted, nor meant to be valued most highly in
that very abstract form.  I tend to use the word "abstract" when I want to emphasize
the circumstance that the form of the proceedings is being emphasized almost to the
exclusion of interest in anything else, as in the phrase "pro forma", but neither
the word "abstract" nor the word "formal" implies that the concrete material has
somehow gone away.  The way I see it, the only sense of "formal" worth having
is one that highlights the facets of form without deluding us into thinking
that no other aspect matters, that is, up until we reach that pythagorean
perspective where we can truly see exactly how form and matter are one.

Your line from Weyl is typical of the formalist line.
What I did not find out for many years is that nobody,
especially no real mathematician, as Weyl most surely
was, ever really lives by that line.  Some of this is
very personal to me, as I bought this line for a very
long time, and only gradually came to the realization,
and with not just a little sense of personal betrayal
and deep intellectual disillusionment, that this line
is just a red herring that your average mathematician
will tend to hand a pesky student in order to get him
out of his office, to avoid a dreaded "philosophical"
discussion, or else that a famous virtuoso will hand
a tedious reporter in order to give her a nice story.
It is a convenient line in a certain type of social
situation, and maybe it is their idea of a harmless
joke or a pacifying fiction, but if I ever run into
a mathematician who says that he or she really and
truly buys it, then my initial hypothesis will have
be that the person is a victim of a self-deception.
 
HP: As I warned, I'm not talking stereotypic rationalizations.

HP: I'm simply defining my usage [of "formal"] by stating a simple,
    empirically testable condition for a formal process.
    Just follow the rules.

JA: I guess that the trouble starts right at the very beginning, but
    I did not realize before that you meant this as formal definition.
    Are you then saying that you're saying:
    |
    | Definition.  A "formal process" is "just follow the rules".
    |

HP: The class of formal system cannot be formally defined.
    "Just follow the rules" was as short a description
    as I could imagine that expresses the key idea
    that interpretation of the symbols (other than
    just following the rules) is not relevant in
    formal manipulations.

I am trying to point out the fact that this is not any kind of definition,
whether casual or formal, because the predicates "following a rule" (FAR)
and "following a set (sequence, series, system) of rules" (FASOR) are way
too ambiguous as casually used and way too far from being empirical ideas.

HP: Adding a column of numbers can be accomplished whether or
    not the numbers refer to apples, bytes, or nothing but number.
    It's called calculating.

JA: I see an act of abstraction, and then I see an activity of calculation.

HP: You see too much.  To add, a calculator does not have to abstract anything.
    It just follows the rules.

Normally speaking, we do the abstraction before we go to the calculator.
Then the calculator does the calculation?  You could say that, I guess,
but then you would be saying too much.  More carefully speaking, you'd
be nearer the truth to say that the calculator goes through the motion
that we interpret as being a calculation.  And then we do the reverse
of the initial abstraction step, to wit, we interpret the patterns of
colored lights, in order to make use of the whole operation.  I think
that it is necessary to see all of this in order to understand why in
the heck we bother to do it, to wit, to know the pragma of the conduct.

JA: Oh, you mean an algorithm?
    Id est, an effective procedure?

HP: Yes and Yes.  These are special cases of formal procedures.

JA: To wit, an effective description
    of a particular pattern of behavior?
    You regard this as empirically testable?

HP: Yes.  The test for formal operations (quoting Kleene):
    "It must me possible to perform the deductions treating
    the symbols as words without meaning.  For to say they
    have meaning necessary to the deductions ... amounts
    to saying that not all of the properties necessary
    for deduction have been expressed in the axioms."
    In other words, if you can't find a necessary rule,
    or you do have to act without a rule, then you don't
    have a formal system.

You must be aware of the circumstance that there are different
dispositions, inclinations, persuasions, and personal styles of
thinking even in the formal sciences like logic and mathematics.
For instance, there are model-theoretically inclined folks and
there are proof-theoretically bent folks, and all of the lines
that you are citing in this connection are skewed to the sorts
of things that the "proof and syntax is all" (PASIA) school is
very well known for preaching.  But not everybody attends that
particular Church.

JA: I worry about that.  Some things I know of
    say yes, okay.  Other things say no, no way.
    I will have to think about this for a while.

HP: Within the limits of error a formal system
    is probably more empirically testable than
    any other empirically testable system. But
    you are well-justified in worrying about that,
    since all empirical tests are noisy and have
    some error.  Therefore, on this general ground
    alone we never can verify with certainty that
    a "formal system" is more than just very good
    statistics.

JA: Again, I see that you are using this phrase "formal system" in a way
    that I cannot unify with the long-accustomed uses that I know for it --
    id est, I can see that you are enclausing it in express predications
    in which it would never occur to me to enclose it, and so this tends
    to make me suspect that you are using it to describe a subject other
    than those that I have formerly described, expressed, or known by it.

HP: Perhaps that is because I am pointing out the physical limitations
    of formal systems that nobody likes to mention.

In Classical AI, it used to be discussed much and mentioned almost to tears
in the hushed tones of the "Physical Symbol System Hypothesis" (PSSH), still
another of the annoying truths that Peirce used to mention, and more, to use.

But now I am guessing that you are speaking of the implementation
or the realization of an abstractly specified formal system, which
is certainly fine by me, but the properties of control, information,
or the lacks thereof, of which you are speaking, I am still guessing,
are only properly predicable of the total system, from which the form
is solely in our imaginations abstracted.

JA: Just by way of seeking a point of common origin and reference, the first things
    that pop into my mind when I see "formal system" are things that by other names
    many call "axiom systems", "formal grammars", "formal languages", and their ilk.
    Is that anything like what you intend to indicate here?

HP: That is exactly the way I have been thinking of "formal" all along.

JA: But I still perceive a residue of the attitude called "formalist"
    in this seemingly, all too seemingly innocent definition of "formal".
    It shows up in the tendency to use qualifiers like "just", "merely",
    "nothing but" in the invitation to follow.

HP: You are too suspicious.  I said I did not want to get involved here
    with the metaphysics of the "formalists" (versus "intuitionists",
    "logicists", Platonists") on the foundations of mathematics.
    To work with a formal system (math, logic, language, whatever)
    does not need a commitment to any philosophy.  You don't even
    have to like it.  Just follow the rules.

But you are the one who keeps quoting the formalist bible as if they
were the only ones who can speak for the one true teaching about form.
I already know -- that if you ask them -- they'll tell that this is so.

HP: Your perception of my attitude is incorrect.  You are mistaking my attempt
    to clearly communicate (by defining my words) with what I actually believe
    about formal systems.

Again, you are mistaking the teachings of formalists
with the truth, the whole truth, and nothing but the
truth about form.  You are using a particular school's
characterization of what constitutes the formal aspect,
and their doctrine, or fiction, or joke, about what it
all means to them, as if there were no other view of it.

There other ways of looking at form,
not to mention its relation to matters
of dynamics and styles of interpretation.

HP: I believe formal systems are an ideal that cannot be met in principle,
    but that can can be made to work well enough to safely bet on their
    results.  While you are thinking about this, I will summarize what
    I think about the physics of symbols.

HP: http://www.c3.lanl.gov/~rocha/pattee/

HP: The conceptual problem with formal systems is worse than simply the 
    fact that we cannot, with certainty, empirically verify their existence.

JA: Now here is an example of what I mean, a "formal system"-related sentence
    that gives me such a jolt when I read it that I do a double take and then
    conclude for the moment that we must be using radically diverse languages.
    Can you elaborate on what this sentence means?

HP: The only formal tests for a formal system are for its completeness and its consistency.
    But any formal test assumes that all the symbols are read and written correctly and that
    all the rules have been followed correctly.  This requires the informal observation and
    recognition of symbols which are only assumed to be error free, but which cannot be proved
    to be error free (since proof is only a formal concept).  Since physically all observations
    are noisy, there is always a finite probability of error.  But as I said, the recognition of
    symbols and rules is less noisy than most measurements (that is why we use bits and binary
    switches in computers).

This is the same problem that people get into when they confound logic and psychology.
I believe that it is a bad idea to confound the level of the implementation with the
level of the abstractly specified formal system, as you do when you say stuff like:

| HP: The conceptual problem with formal systems
|     is worse than simply the fact that we cannot,
|     with certainty, empirically verify their existence.

The first time that I read this I naturally parsed the substantive "existence" at the end
with an inflection that semantically agreed with the adjectives "conceptual" and "formal"
up front, but now I am getting the drift that you meant "existence" in a physical sense.
And that just mixes the levels.  The reason we abstract the form is to attend away from
the implementation.  It is fine and dandy to engineer the reverse of this abstraction --
I recommend it on a periodic basis -- but when one goes through the trouble to ginger
the inverse step, stepping down in the direction of the ground, then the predicates
that one hauls in are no longer the predicaments of the form.

HP: The deeper problem from the physicist's view is that every single step
    of a formal [process] is subject to empirical error.  This is assuming
    measurement and control are irreversible events, hence dissipative and
    noisy.  I'm ignoring Landauer, Bennett, et al. who claim "in principle"
    reversibility because nobody has done it.)  In this view, a mathematical
    proof is nothing more than a series of noisy empirical measurement
    (recognizing symbol vehicles) controlled by noisy rules (rewriting,
    storing, etc.).  This holds for all reading and writing of symbols
    (symbol vehicles being in part defined as arbitrary coded physical
    structures).

I would just prefer to say:

| The deeper problem from the physicist's view is that every single step
| of an actualized, implemented, or realized formal process is subject
| to empirical error.

Does that make sense to you?

JA: Okay, this more like stuff that I have at least thought about a little.

HP: The fact that the accumulated statistical errors may be very small is an
    important practical fact, but that would not satisfy Platonic formalism.

JA: But now you are jolting me again.  "Platonic formalism" clangs like
    a contradiction in terms to my mind's ear -- of course, if you were
    to capitalize the pun in the form "Platonic Formalism" then perhaps
    that would have telegraphed the punchline in a way that you thought
    was just too easy.  If this be not a joke, then can you tell me
    what it means?

HP: I meant only to contrast Platonic forms which are imagined to be "perfect forms"
    with physically realizable forms that are imperfect and statistical.  There is
    an "unbridgeable gulf" (Planck) between a probability, however small, and
    perfect determinism.

Okay, but I do not see a connection between platonic ideas and perfect determinism.

HP: There is also the related Lakatos-type problem that we cannot completely define sets,
    rules, and domains of applicability leaving no ambiguity.  The "formal" logic problems
    of infinite sets, consistency, and completeness are also real, but ignored by most
    physicists as "just formal."  They have  more pragmatic problems.

JA: And thereby hangs a tale.  Let me try pull it through the form of a riddle:
    Can you tell me what was the first widely-used "Virtual Reality" system in
    the field of computer science?

HP: Turing's original machine was a virtual reality model
    formalizing the brain trying to do arithmetic, but
    maybe Pythagorus was using numbers to make virtual
    models of God.

Good guess, but I was thinking "FORTRAN".

It was implemented by people who had to attend to the realities
of the information dimension so that physicists could forestall
by a few more years the need to do so and to keep on pretending
that they operated in a world of infinite precision arithmetic.

JA: When I say I do not understand your usage, I really mean it.
    Oh, it's not like I never heard this way of talking before,
    or even that I did not speak this way for years and years.
    No, I mean that I do not understand this form of talking
    in the way that I no longer understand expressions that
    I am beginning to suspect are irreducibly ambiguous
    or even irredeemably inconsistent.  It's still just
    a suspicion at this point, but it grows stronger.

HP: So we appear to agree about the inadequacy of formal systems, ...

JA: We might, just perhaps, but we would first have to arrive at
    an accommodation as to what the phrase "formal system" names.

HP: So we appear to agree about the inadequacy of formal systems,
    if perhaps for different reasons.  And yet, we all continue
    to use formal systems.  Why is this?  Well, it's because we
    have nothing better.  Noise is built in to the universe.
    It is everywhere, and the ideal of formal systems is not
    only to reduce noise to the minimum, it also frees us from
    the bounds of everyday experience.  How else but by formal
    systems could the imagination unambiguously manipulate (and
    communicate) imaginary numbers in infinite dimensional spaces?
    (That was the reason for my Hilbert space question.) Galileo and
    all physicists thereafter have followed Nicholas Cusanus's advice
    with unreasonable success:  "If the transcendental is accessible to
    us only through the medium of images and symbols, let the symbols at
    least be as distinct and unambiguous as mathematics will permit."

My new guess is that when you say "formal system" you mean the type of object system
that I would consider as an augmented system or a compound object under the formulas
of an "implemented formal system" or maybe a "formal system + concrete realization".

JA: Well, now I am getting the sense that you are posing all symbol systems
    to fall out under the banner of "formal systems".  Is that a good guess?

HP: Certainly not!  Almost all symbol systems, all semiotic systems,
    are usually thought of as interpreted systems.  You know that.
    Formal systems are rare just because all extrinsic meaning, all
    semantic reference, has been eliminated (as far as possible).

No, I am pretty sure that I would never say it that way.
You simply must realize that the particular description
of "What Is And What Should Be The Formal" that you are
using in all of these statements is only the picture of
a single historically circumscribed and situated school
of thought.  I personally think that the entire picture
is a well-intentioned joke that just got out of control,
like certain computer viruses that we have came to know
and to hate.

I really see no need to "eliminate" the meaning of a form of conduct
in order to arrive at an appreciation of its form, and I have worked
on formal semantics as well as on formal pragmatics.  I believe that
the adjective "formal" can have a focusing effect on the mind, while
remaining neutral with respect to many details of matter and meaning.

To use an analogy that I have put into practice in a very real way,
formal structure is derived from concrete conduct much in the same
way that tangents and normals are derived and produced from curves.

HP: It is also wise to remember that evolution by natural selection requires noise
    in the genetic symbol system.  There is good evidence that creative thought, and
    therefore inquiry, also require some noise.  As Spinoza and Martha Stewart always
    say, sub specie aeternitatis, a little noise is a good thing.  The greatest danger
    of formal language, as Stan would agree, is its premature or inappropriate application
    to imaginative, vague, or creative thought.

JA: Okay, I can warm to this a little, as it was always less the noise factor,
    the big hullaballo over ambiguity and all that -- than the sampling ratio,
    all the stuff that gets left out of our formal acounts, that brings me to
    the point, almost, of decrying how "inadequate" formal systems always are,
    though, on reflection, I'm apt to back off pressing this point when I wit
    of how "adequacy" is adequately judged only relative to an intendable aim.

HP: I agree.  As I said, you try to use formal systems
    all the time because they are much less noisy and
    ambiguous than natural language.

Perhaps, but does not formal abstraction involve a certain amount
of what is often so euphemystically called "systematic ambiguity"?

I guess if I tried to sum up what I have been trying to say in this cycle,
it would be that I view the point of view that we describe as the "formal"
as a way of looking at an object system and not as its exhaustive essence.

Jon Awbrey

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