ONT Mathematics As Formalism -- Glass Beads = Jewels Of Denial
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
| 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.
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
John Collier wrote (JC):
Jon Awbrey wrote (JA):
JA: And this should not have been such a big surprise.
It is not humanely possible for healthy human beings to
devote their lives to an activity that they themselves
consider meaningless. I work with formal systems that
I conform to custom in calling "uninterpreted", but the
reason why these systems of forms are privileged, indeed,
so highly prized, is that they have so many splendored
and sundry interpretations, not because they have none.
I do not know how to convey this -- it is a fact of life
discovered, not a thing to be proved from any axioms --
I could write a story or a novel maybe, but Hesse already
wrote it, with far more skill than I could muster, and still
so few get it. I could try to relate my personal philosophy
of mythematics, but I have done that before and have seen
people think I am joking, and even before Howard explained
to me the nature of a joke I could have told you that it
does no good to explain why the joke is no joke. So I
must leave it at that ...
JC: If this is your notion of uninterpreted, I can see why you might have had trouble.
I see uninterpreted as abstracting from specific interpretation, but not lacking
an interpretation in the abstract. I see the alternative you propose as nonsense.
JA: Of course it's nonsense -- that's what I have been been saying the whole time.
But I am not the one who has been proposing it. If you have never heard a hint
before of the phrases "meaningless formal calculus" and "just follow the rules",
then I am sorry that I ever troubled you.
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.
Vide Supra.
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?
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: 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.
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: 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.
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: 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).
This is all beside the point, my point anyway, in several directions at once:
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, 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. 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.
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.
Jon Awbrey
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤