ONT Re: Inquiry into Inquiry
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Howard Pattee wrote (HP):
John Collier wrote (JC):
Jon Awbrey wrote (JA):
JC, on a parallel thread:
| I've argued that supervenience is basically an empty concept that avoids
| the difficult issues (empty formalism of the worst kind in Jon's sense).
As far as I can tell from my records, it was JC who introduced the phrase "empty formalism"
into this discussion. Perhaps it was intended to be a para-phrase. At any rate, it seems
to have served its purpose as a straw-person and should now be considered utterly sans wit.
After a few days of idle reflection in widely-scattered airports I eventually arrived
at the potential insight that John and I were simply talking on totally skew lines this
whole time, as I was talking about the life of inquiry and mathematics in particular as
cultural systems and not as formal systems.
JC: Peirce would argue that formalization in this sense leads
to loss of meaningfulness, and is intrinsically impossible.
Wittgenstein, in his later work, held out for a similar
position. Barwise and Perry, in Situations and Attitudes,
take a similar position to Peirce and Wittgenstein on
formalizability, but they conceal this for practical
reasons (most analytic philosophers reject the position
a fortiori -- it is incompatible with their methodology).
Hertz is a nice example, since he is not as positivistic
as some of his cohorts, but as Stan pointed out, the strict
distinction between syntax and semantics and interpretation
of signs is not supported by Peirce. The reason is that to
be a sign in the fullest sense requires all three, and one
without the other can only be separated in abstraction
(meaning simply that we consider partially by focusing
on one aspect or another).
I do not think that Peirce used the words "formal" and "formalizable"
in this way, at least not much, and so I am guessing that this is yet
another interpolation that will only add to the confusion over form.
HP: It really makes no difference to physicists if philosophers like
Peirce, Wittgenstein, Barwise and Perry, or anyone else, supports
or rejects their normative strategy of separating syntax and semantics,
that is, separating the formal mathematical expression of laws and the
interpretation of laws (including measurement of initial conditions).
Nobody believes that perfect objectivity and totally meaningless and
precise formalization are attainable, anyway, so that is not an issue.
But that does not make these ideals a bad normative strategy. Just
because absolute beauty, truth, and justice don't exist does not make
striving for them a bad normative strategy.
Where I left off here was asking Howard what he meant by "strict separation"
of syntax and semantics, and I have yet to get an answer to this. I do not
think it makes much sense to say that Peirce did or did not separate them
until you specify what sense you mean. For example, would you say that
the X-axis and the Y-axis in a cartesian coordinate system are strictly
separate or not? What about the lines of latitude and longitude?
JC: Much as I agree with the gist of what you wrote in what followed, I think it does matter
for two reasons. The first is simply that I think it is important to understand what
scientists are actually doing, not just what they think they are doing, or what they
are trying to do. The other reason is that my studies of the incommensurability
issues leads me to think that over-striving is a big source of the sort of
communication problems that arise across paradigms. I think this also
applies in more mundane practices like culture and politics. In fact
one of the reasons that I studied the problem for ten years or so was
that I believe science is one of the most rational if not the most
rational of our cultural pursuits, and that if it has elements of
irrationality and even dogged stubbornness, as Hanson, Kuhn, and
Feyerabend suggested, then the hope in politics is limited.
Growing up in Quebec, I was all to aware of how different
cultures could avoid engaging each other except in the
meanest terms, so I hoped to understand why scientists
are sometimes mean in order to get a broader view.
JC: I also think that a more modest view of science is appropriate for finding
common ground in the science wars. I'm currently re-evaluating Sellars
views of the Manifest and Scientific Images from this perspective for a
book to be published in Australia on his view of our schizophrenic
intellectual heritage.
JC: I am also not at all sure that it is off the present topic,
concerning the true nature of inquiry.
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Subj: OCA: Re: Mathematics As Formalism -- Glass Beads & Jewels Of Denial
Date: Mon, 03 Sep 2001 01:46:08 -0400
From: Jon Awbrey <jawbrey@oakland.edu>
To: Organization Complexity Autonomy <oca@cc.newcastle.edu.au>
CC: Generic Ontology Group <ontology@ieee.org>, Arisbe <arisbe@stderr.org>
John Collier wrote (JC):
Jon Awbrey wrote (JA):
JA, casting Runes:
| Formalism (Mathemetical) is a name which has been given to any one
| of various accounts of the foundations of mathematics which emphasize
| the formal aspects of mathematics as against content or meaning, or which,
| in whole or in part, deny content to mathematical formulas. The name is often
| applied, in particular, to the doctrines of Hilbert, although Hilbert himself
| calls his method axiomatic, and gives to his syntactical or metamathematical
| investigations the name Beweistheorie ('proof theory', q.v.).
|
| Alonzo Church, in Runes, DOP, page 111.
| Dagobert D. Runes, 'Dictionary of Philosophy',
| Littlefield, Adams, & Co., Totowa, NJ, 1972.
JC: My point is that these are always abstractions from more concrete cases, and
always occur in context. "Just follow the rules" is not possible in general
unless the rules are embedded within some system whose context specifies what
the rules are. A "meaningless formal calculus" is possible, but that is not
propositional logic or any other formal system with which I have worked.
I only work with formal systems that have a semantics. It seems to me
rather pointless, to say the least, to engage in meaningless formal
calculus. I sincerely doubt that anyone has proposed either as
useful in anyway, but I am willing to look at examples.
JA: Vide Supra.
JC: Hibert's formalism fails as a foundation for mathematics
just because it can't be given a semantics completely.
This is not true of propositional calculus, and would
not be true of mathematics if Hilbert's program had
been successful, so I maintain that you have not
given me an example. The very fact that Hilbert's
program had clear criteria for failure shows that
it was not empty formalism. Chaitin remarks on
some interesting work that pushes the limits of
formalism into new and surprising areas, but
again there is a clear semantics, and these
cases do not provide counterexamples either.
JA: I think that you must have been somewhere over the Pacific when this discussion
started. You are repeating, more prosaically, all of the things that I have
already said, if a bit more rhapsodically, about what a Big Lie this brand
of self-styled "Formalism" is as a philosophy of practicing mathematics.
Then you seemed to deny that anybody had ever proposed this philosophy,
and so I gave you an example to remind you that some people really had.
Any work that pushes this Formal Ism to the brink of having semantics
is work that has converted this Formal Ism into its Formal Apostasy.
JC: I must admit, that I often felt that some of my professors at UCLA
were engaged in one or the other or both practices, but that is
really not very fair to them, and was more my own frustration
with their interests, and their lack of concern with what
cannot be formalized than anything else.
JC: One would get bored at having to deal with nonsense for very long.
My understanding of this distinction comes from reading Russell on
proper names and on incomplete symbols, as well as Kaplan and Perry
on demonstratives and identification.
JA: Beauty is Form, and Form Beauty,
And that's all you need to know.
There's more to follow for sure,
Quasi modo gratuitous corollary.
In life as in mathematics, only
Beauty can render it worthwhile.
JA: Still, by posing things in all-or-none terms,
saying "X cannot be formalized" when we need
to say "X cannot be exhaustively formalized",
we characteristically miss the whole point
of the exercise, partially to formalize X,
and deny ourselves the practical benefits
of doing just that. I see a certain type
of psychodynamic here that needs to be
interpreted in a therapeutic remedium.
JC: Granted, but there is a tendency to resort to technique
when it is available to the ignorance of everything else.
JA: In math as in art, the name of that is poor technique.
JC: I don't think so. It is poor practice, but can yield excellent technical work.
JA: De gustibus, and so on, ...
JC: Yeh, that's what I meant above about my revision
of my evaluation of what some of my professors
were doing.
JC: This has led many contemporary philosophers of logic,
some of them friends of mine, to argue that truth and
validity are restricted to what we can constructively
define through one or another specific technique. I
believe that, much as formalization is useful, it must
systematically push out issues of truth and validity
for any logic as strong as 1st order.
JA: The name of that is poor formalization.
JC: No, unless all formalization is poor formalization.
See my paper on the dynamics of information and the
origins of semiosis for my reasons.
JA: I think that we just attach different meanings to the word "formalization".
JC: I mean by formalization the abstraction from concrete cases to their
relational structure (where structure is understood in the mathematical
sense that fully determines a set of models, realizations, or instantiations).
What do you mean?
JA: Reflection on conduct that produces an explicit description or a partial expression
of an aspect of its form that had until that time only implicitly been informing it.
JC: That sounds to psychological for me. Was that implication intended?
If not, could you rephrase to remove any potential psychological implication?
Sepcifically, reflection is used by both Locke and Hume as a psychological activity.
JA: Do you really intend to say that using one's mind makes one sound psychological?
Do you really intend to say that using ones' mind implicates one in some action
the evidence of which one ought to remove, or could remove, by a mere rephrasal?
JA: If using the mind is outlawed,
Only outlaws will have minds.
JA: Formalization is a process. What sort of process is it?
JC: Truth functional logic and modal logic, on the other hand, are complete.
As I have said before on this list, that is one of the reasons I favour
an information based logic. It gives us both sides of the coin within
the logic itself, as Greg Chaitin argues admirably, in my opinion,
but not in the opinion of many logicians who do not see intuitively
how distinctions are relevant to logic. I do think that they need
therapy, and Wittgenstein is as good as Peirce for this, I think.
JA: Logical systems can be "complete" in their own terms,
but sign systems are never complete when you wake up
and recall their due function in describing a world,
JC: I don't believe the first phrase makes sense in contrast
to the second (see my remarks on "uninterpreted" above).
JA: The word "complete" is ambiguous here, if not a complete cipher.
I cannot say yet whether the ambiguity can be made systematic.
JC: Sorry, I don't get this.
JC: Funny, I would take it that once we have done the latter,
it follows that the complete formal systems are complete.
I wouldn't say that they are complete in their own terms,
because I couldn't even venture to say what that meant.
JA: I informally use phrases like "complete relative to itself" to mean that
all expressions with universal model sets in some universe are provable.
Propositional calculus is like that, and this means that one can check
its theorems quasi modo model theory over a suitably chosen universe.
JC: More than that, Prop Calc applies to all possible universes,
so one does not have to check it in particular universes.
JA: I have no practical and usable conception of "all possible universes" --
I have always had to take my universes one or two or three at a time.
I am talking about the measures of computational work that it takes
in real terms to check whether a given proposition is a tautology.
From my practical experience in writing theorem checking, proving,
and applied logic programs, I have learned that model-theoretic
methods are very often more efficient, useful, and all-round
much more informative for the types of problems that arise
in real applications. I am engaged in applied logic here,
not just decorating the study with tautologies, but using
propositions to describe situations of interest and then
computing derived facts about these situations. Now, it
does makes a lot of practical sense to factor out a core
of pure logic at the heart of this applied enterprise,
but unless it helps to keep the engine running it is
not really that interesting for the sake of a' that.
JC: Well, the antecedent of my statement above is a consequence of definition,
or at least it is usually taken that X is possible iff it has a consistent
model, and Prop Calc determines consistency, if anything does (a non-trivial
condition, given the non-formalizability of the truth concept, except relatively
to a language in some meta-language). Validity is also only relatively formalizable.
In both cases we know that there is no definable limit of embedded meta-languages that
converges on the desired absolute concept. I realize that this is contrary to Peirce's
convergent realism, but with all due respect to Peirce, he did not understand these issues.
JA: Peirce understood his aim well enough to know that inquiry is not an arrow
that can be nocked by a purely deductive enbowing, which is better than the
Flatlander Tourists In Babel have ever understood about the Logic Of Science.
JC: I said, "with all due respect", but this still does not address my concern.
JA: Whether any of us knows how much respect is due
is a thing that remains to be seen, but let me
attempt a paraphrase.
JA: You are trying to carve out a "deductive idol of science" (DIOS),
in syllogistics, to figment a "barbaric image of science" (BIOS).
This is the same picture that Carnap et alii et aliae tried to paint,
but worse than that, it is the same serpentine idol that Kant attempted
to invest with the full power of his 'False Subtelty', of thinking that
thinking all basely comes down to yet another application of transitivity,
which worm of ourobore Peirce had already demolished while still yet a babe
in his philosophic cradle. For as almost any child can see, this dim image,
this all 2-dim idol, lacks the pragmatically essential dimension that it
would need to come into its own a viable model of inquiry.
JA: As for the business about convergence,
you are reading Peirce backwards here.
I will have to explain this bit later.
JC: This is why I maintained before that Goedel's results are
of much more significance than you were willing to grant them.
Forgive me, but I really don't see how the more specific cases
in applied logic can alter this more general result in any way.
JA: I am talking about how to conserve the environment while spraying for mosquitoes.
You are suggesting how the right sort of syntactical nuclear device oughta do it.
JC: I won't accept your metaphor as apt, but the nuclear device applies to the spray as well.
It is just one of the nuclear devices. I also don't think that Godel's results even make
sense if they are treated as entirely syntactic. You have to have the concept of validity
and consistency (semantic notions) for the arguments to have any consequent for mathematics
or logic.
JA: You are telling me why the sky is blue.
I already know why the sky is blue.
What I need to know is whether to
plant corn or beans or alfalfa
in the Spring and what sort
of fertilizer is best to
use for each crop, and
knowing this will take
a very large number of
manurial comparisons
to find out, that is,
if a former really
wants to know his
business.
JC: Now, Goedel saw the way out as a sort of transcendental idealism, as did Kant.
In neither case do I see that it leads to the resolution of issues in science
or the philosophy of science, though Goedel thought that his closed temporal
loops showed the ideality of time. Not many have agreed with him. Now Peirce's
style of completeness at the ideal limit seems to me to make sense only within
this sort of framework, but, as I said, personally I can't see any way to give
it an interpretation that I can make meaningful (by which I mean, which I can
use in any way that makes a difference to my understanding, let alone to my
experience).
JA: This is all beside the point, my point anyway, in several directions at once:
JC: Your point doesn't exist, as far as I can see, or at best you have set up
a straw formalist. My point is: don't trust Peirce so much; he was off
his metaphysical rocker. I think his more mature work is more stable,
but it also suffers from a yearning for completeness that just can't
be satisfied. This presents problems for him in his characterizations
of truth and sound method.
JA: 1. The working philosophy of all mathematicians is platonic realism,
not any kind of trance-idealism. Numbers exist. Things follow.
It is all so taken for granted that they do not even bother to
discuss it, and this led some dis-contented civilians to think
that they could speak for them,
JC: Wait a second. Hilbert (your example) was a discontented civilian?
JA: All tinkers toy with ideas. All tinkers try out the passing fancies.
As a rule, none of this tinkering has much impact on the regulative
principles that are embodied in their workaday practices. The rule
of platonic realism is so preëminent in mathematics that I find all
apparent exceptions are generally accountable to anomie, bad faith,
concessions to fashion, or self deception. What these anomalies
have in common is a lack of adequate reflection on actual praxis.
So I will not call Hilbert a civilian, obviously, but a person
who denies content I take to be dis-contented, axiomatically.
JA: but what got said was so silly that it stirred Gödel to speak up
and shoot it down. You can call it a therapeutic advance to turn
a virus against itself, but nobody had that particular bug before
those Principians infected the population with it.
JC: Well, I don't think that Russell knew about the completeness of first order logic.
Certainly not explicitly. He knew there were problems with unrestricted second
order logic. He found the problems in Frege's work. Now, since anything that
can be said explicitly can be said in first order logic, and first order logic
is complete, I am not at all sure what the silliness you refer to is supposed
to be.
JA: The folly is in thinking that inquiry into reality can be carried out in FOL.
JA: There is a type of person I know who is always saying one of these bits:
JA: | Everything <pick one:
| worth saying,
| that can be said
| >
| can be said in <pick one:
| FOL,
| prose,
| set theory,
| forty words or less
| >
JA: What this type person always seems to forget is how utterly retro-spective
is the POV from which they condescend to say this, forgetting all the work
and the woe that it took some poor sucker to discover what they wrap up so.
It is the science of discovery versus the science of review all over again.
JA: It's a little like certain software manufactors we know who want to
bill you for fixes to bugs you never had until you unwarily bought
their wares. The misconceit of mathematics that got itself embodied
in 'Principia Mathematica' is not the sort of mistake that an earlier,
realist generation would ever have made in the first place, having never
even thought to arrogate to themselves such an overweaning over-extension
of a lame idea as to to think that the entire foundation of mathematics
could ever be finitely axiomatized.
JC: I repeat: Goedel's proof is a refutation of Hilbert's formalism,
which had clear criteria for falsification. Goedel found these.
There was a bug in the works. There is no known bug in the works
for propositional calculus, and there is good reason to believe
that one cannot be found. Hilbert's formalism was not a lame idea.
It works perfectly well for limited (but ever more expanded) parts
of mathematics, Peirce notwithstanding.
JA: I happen to find ZOL interesting. Most folks do not think there's much to it.
JA: Thank you for continuing to explain to me that Formal Ism is dead. Tell others.
My interest leads me to ask this: Aside from whatever it reputedly did for us,
will it tell us where our next interesting and useful axiom set will come from?
JA: 2. You seem to be blithely oblivious to the way that pragmatic factors
affect what we can actually do with logical languages in practical
computing terms. The question is not "how the more specific cases
in applied logic can alter this more general result in any way".
The question is how this more general result might bear on the
practical details of what we need to do in specific cases of
applied logic, while the circumstance is that the high-flown
principle does not reach the ground on which help is needed.
Theorem-cranking is a purely incidental part of the task --
contingent propositions come into far greater prominence
when the expressions of the language are actually being
used as descriptions of pragmatic objects or situations.
And here you want to be able to generate models to spec.
This really takes a whole different order of technique
than the sort of gamey methods found in be-wise-theory.
JC: No doubt this is last sentence is true. Nonetheless,
one can eliminate large numbers of blind alleys with
some simpler general arguments.
JA: This is exactly the sort of stuff that I used to say before
I started to try it out on the computer. Now I know better.
JC: However, you have over-generalized the specific result beyond
its area of applicability, which has been my complaint all along.
JA: Did I try every blind alley that could be tried? No I did not. Nor do I plan to.
JC: I think that the reason why you do this is a mistaken characterization
of formalism, and I suspect a confusion or mixing of aspects of natural
and formal language semantics.
JA: I have never used any characterization of Formal Ism
but the ones that its exponents themselves expounded.
All the straw I strew in my construal is their straw.
JC: Oh dear, I seem to have entered
those ominous troublesome regions
must faster than I had intended to do.
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