FOM: Theorizing About Theories II: Formality as Language Acquisition

[Dennis E. Hamilton]

This is the second of three notes on informal appreciation of Cantor's
development of a theory for transfinite sets and their equivalence.  I am
exploring informal theories about theories and what formal (deductive)
theories do and do not provide.  My purpose is to convey an appreciation for
the power of such methodology in giving us some purchase on the invisible
and ungraspable: namely the transfinite.  My goal is an informal depiction
of theorizing that is useful in an account of computers as formal mechanisms
the operation of which manifest abstract objects in interesting and
revealing ways.  I am still searching for some kind of diagrammatic
presentation.

The first of these notes launched off the posting by Bill Everdell on June
23.
http://www.math.psu.edu/simpson/fom/postings/0206/msg00082.html

For this one, I'm backing up to a posting by Bill on June 14.
http://www.math.psu.edu/simpson/fom/postings/0206/msg00058.html

This is even more informal than the first note (of July 18).  I don't talk
about Cantor or sets at all, except just here:

1.	CLEANING UP SUBSETS AND IDENTITY OF SETS - ONE MORE TIME

I did find where (A is-a-subset-of B) and (B is-a-subset-of A) *implies* (A
is-identical-to B).

It is in Abraham Fraenkel's discussion of Zermelo's system, as modified by
Fraenkel, in Bernays, pp.5-9.  I remembered more than was actually there.
The account by Fraenkel is rather pleasant to read, and I recommend it for
insight into what the concerns were and how an approach was selected for
axiomatization of set theory.

[Fraenkel1968]
Bernays, Paul.  Axiomatic Set Theory.  With a historical introduction by
Abraham A. Fraenkel. 2nd edition.  Studies in Logic and The Foundations of
Mathematics.  North-Holland (Amsterdam: 1958, 1968).  Unabridged and
unaltered republication by Dover Publications (New York: 1991).  ISBN
0-486-66637-9 pbk.


2.	FORMALITY AS LANGUAGE ACQUISITION (ANECDOTAL DISCOURSE)

I consider that learning and use of formal symbolisms is just another kind
of language acquisition.  It is neither more nor less conceptual than
ordinary language.  The power of formal symbolisms, to me, is that it
provides a way to grasp the essentials of something and not be distracted by
an inessential.  That is, it has been found to be an indispensable companion
to abstraction.  The simplicity of formalism as an apparent meaningless game
with symbols, is misleading.

2.1  I was annoyed, for some reason, with the emphasis on symbol
manipulation in Gödel-Escher-Bach.  I really put myself off about it.  What
I was reading into it, was that the symbol manipulation was an end in
itself.  I felt that was losing the whole point of the power of formality.
I am sure that I was misinterpreting Hofstadter in my reaction, yet it has
tainted my view of that work to this day.  Maybe that is what concerns
people who think we lose something in appealing to formalism.  I say we gain
worlds, and that is something to appreciate, even when it looks as if it
couldn't be so.

2.2  But I was very much impressed when I read Richard Feynman's Nobel
Lecture.  In it, he described the power of a particular symbolism and theory
for gaining some grasp on an important area in his investigations.  Without
the notation and model he devised, he doesn't see how he would have got to
the place he arrived in his thinking.  And he had since turned to other
formalizations because the one that he credited with so much wasn't
sufficient for the next stages of his theoretical investigations.  When I
read that I remembered that my first exposure to matrix algebra was through
a handy little notation that Einstein had introduced for making particular
relationships visible and easy to check.

2.3 Every month, I receive a bank statement that has a little form with
instructions for how to balance my bank account.  It is a very formal
procedure, offered without any explanation at all.  It works perfectly,
though before I finally trusted computer software to reconcile my bank
statement I used a version from a different bank that was more
understandable to me.  Actually, I can reconstruct why the procedure I am
being instructed to follow works.  But mostly it works, and it works even
for those who don't understand it, so long as there are reliable practices
for resolving discrepancies.  (My concern about the computer software I use
is precisely around discrepancies now being inscrutable and I might not be
able to resolve them.)  There's a theory behind the formality of reconciling
my bank statement, and there is more power in knowing what it is than in
not.  People can acquire the practice, just like facility at language,
without knowing how it works.  The power of having the appropriate theory is
that I can derive the procedure, or one for other circumstances, when I want
to.  I didn't surrender any other faculties in learning to do that.

2.4 After working with a formalism for a time, one begins to "think in it."
While doing that, it is difficult to "think out of it," at least for me, but
so long as I realize I am simply thinking in a particular language, I can
shift out of a particular way of thinking when I want to.  Sometimes there
is some effort to make the transition.

2.5 My wife and I are studying Italian.  I work at understanding the
principles of Italian (and I love the beautiful structure of the language,
double negatives and all), and my wife is far more proficient at using the
language.  I am told that I'll know I've reached an important plateau when I
discover I am dreaming in Italian.  I also appreciate that having an
understanding of Italian is not a particularly useful way to develop
facility using the language.  And that trying to define everything I learn
in Italian in terms of English translations is an ineffective approach.
People who teach languages tell me all of those things, as do the books I
have on learning Italian.  I still attempt that, yet if I simply used
Italian regularly, made mistakes and was corrected, I would be farther along
now.  So I'm told and so I believe based on the headway I made in a few
short but intensive classes in Italy, where what there was to do was speak
Italian.  (I also remember being in a tavern in Sienna, and realizing that I
couldn't understand what the group at a nearby table was saying.  They were
Americans, speaking English, and I was listening for Italian.  As soon as I
step off a plane in the U.S., I start listening for English and it is almost
as if I never knew any Italian.  That's what I mean by shifting.)

2.6 I have dreamt in "computer speak," though.  That is, around the age of
20 I would talk in my sleep about some symbolic computational process as if
I was in the computer being/doing that.  It was long ago, when I was first
immersed in computer programming.  I could "think in" the instruction set of
my first computer, and it was years before I lost all recognition of the
particular symbols and syntactic forms that were used.  (Oh, oh, it started
coming back to me ...)

2.7 When I engage in mathematics or logic for a sustained time, I find that
I start thinking in those terms.  I don't have to think about the next step
in a deduction, I can just write it down.  And I can leave more steps out,
knowing that the skipped steps are easy to fill in.  Except I don't even
think about it that much.  I can make up little formal grammars at the drop
of a hat, after having struggled with that years ago in developing grammars
for programming languages.  I may be clumsy at first, being out of practice,
but then it becomes second nature and it is as if I am operating in that
"language," the one that has that formalism and subject matter as part of
natural language.  And more and more I can operate in a reliable way at the
metatheory level, or some level above the formalism.  I don't get lost
between the two, but there was a time when I didn't recognize the
difference.

2.8 But fundamentally, I think our use of formal, symbolic language is just
additional to language.  A highly-refined addition.

3. RIGOR IN LANGUAGE

	'I find that every symbol we use must be shared,
	and this requires each symbol to be defined, and
	this in turn inevitably requires clear philosophical
	thinking and a translation into a current language
	in which, as Frege might say, people may communicate,
	agree and thus be "objective" about the objects of
	thought.'

3.1 It occurs to me that it is not the case that "this requires each symbol
to be defined."  We don't seem to require it for ordinary language.  There's
something more elusive in the building of an agreed formalism that is useful
for communication within a community.  There is the usual making of
mistakes, having misunderstandings, and working them out.  All of those
things about developing a rigorous language for a refined technical purpose
must occur. I don't think it is by requiring each symbol to be defined.  I
don't think it happens by definition of each symbol, even though it is
commonplace to say that.

3.2 I think this is one of the great insights around formalism.  I can state
rules to follow that don't have there be any need to define the symbols, as
it would seem to be if we were talking about "something."  I can have it be
that we aren't talking about anything, yet requiring there to be a
particular structure to honor in expressions in the formalism.  And in doing
that, I can create a shared structure for communication about many things
that are all possible interpretations of those symbols.  So we can have it
be that something is communicated beside the seemingly-meaningless formal
rules.  There is some conceptualization that we each build that is
sufficient for applying the formalism in "meaningful" discourse.  Now I can
turn to the formal system when I want to appraise something as a structure
of the afforded kind, whether as sets or the natural numbers or of the
attraction of gravity or any other invention of the human mind.

4.	TURTLES ALL THE WAY DOWN?

	'One might even argue, in Wittgensteinian fashion,
	that the meaning of any symbol is no more than the
	resultant of (or response to) all uses made of it,
	which means symbols are no more to be trusted than
	words or phrases--and indeed may turn out to be not
	even philosophically *distinguishable* from words
	and phrases in our joint pursuit of rigor.'

4.1   I would say that it is all language in use.  I don't think that is a
reason to deny symbolism nor to require that it be translatable into
ordinary language with the same economy, clarity, etc.

4.2   I do see that our formalisms, in any field, including private language
in the workplace, cuts us off from communicating with others.  I don't see
any inclination to put an end to that.  We are inventing language all of the
time.  At least the logician and mathematician may be doing it on purpose!
And it is valuable, especially for concepts that do not have much clarity in
ordinary language.

4.3   It would be great if there was a handy depiction of what is involved
in the incorporation and employment of formal symbolisms in our language,
though.  Some way so people could see the relationship and also appreciate
the context for communication about and reasoning about abstractions,
abstractions that have value when interpreted in the world (or elsewhere).
So we could see what it is that deductive theories provide.

4.3   Michael Polanyi proposed that there were two ways that we apprehend
language, as when reading or listening.  There is "focal attention" which is
on the concept that the text or utterance evokes.  Then there is
"subordinate attention" which is on the utterance or text itself.  Focal
attention is immediate and automatic, and it takes conscious effort to even
consider that all that is there is the utterance or text.  I think that is
an useful metaphor.  I see formal theories as providing subordinate
structures that guide our focal attention to abstracted concepts.  It is
like creating some kind of stencil or portal through which one reliably
accesses the conceptual where being particular would fail us.  And where we
can't even be certain what we are talking about, but we are confident that
it doesn't matter so long as we follow the rules.  I think that is what the
symbolic expression of abstract theories gains for us.

4.4   I say mathematical thinking is thinking and mathematical language is
language.  And we use it because it is valuable. A colleague of mine likes
to say that "It is turtles all the way down."   It's the punchline of a kind
of metaphysical joke.   There is no escape.  So, I guess the next question
might be, what is it all the way up?

-- Dennis


-----Original Message-----
From: Dennis E. Hamilton [mailto:dennis.hamilton at acm.org]
Sent: Thursday, July 18, 2002 22:48
To: fom at math.psu.edu
Cc: Everdell at aol.com
Subject: Cantor's subsets and the beauty of theories


[ ... ]

1.	CLEANING UP SUBSETS AND IDENTITY OF SETS

In a related note, I observed that

1.1	A is-a-subset-of B is equivalent to every member of A is a member of B.

1.2   And that there is a nice harmony in having

	(A is-identitical-to B) is-equivalent-to (A is-a-subset-of B) and (B
is-a-subset-of A)

1.3   I failed to notice that 1.2 can be an *axiom* for the identity of two
sets.  Now I can't find where I ran across that.  It basically has the
identity of sets be based on having the same members, which is always the
principle.  This principle has nothing to do with the cardinality of the
sets involved.  It applies for all sets.

[ ... ]

-----Original Message-----
From: owner-fom at math.psu.edu [mailto:owner-fom at math.psu.edu]On Behalf Of
Everdell at aol.com
Sent: Friday, June 14, 2002 20:26
To: fom at math.psu.edu
Subject: Re: FOM: precursors of Cantor (&Transfinite Logic)

[ ... ]

It has
always seemed to me that symbolism was suspiciously simple and somewhat
overrated as an aid to foundational thinking.  My reason for this heresy is
similar to Prof. Buckner's.  I find that every symbol we use must be shared,
and this requires each symbol to be defined, and this in turn inevitably
requires clear philosophical thinking and a translation into a current
language in which, as Frege might say, people may communicate, agree and
thus
be "objective" about the objects of thought.  One might even argue, in
Wittgensteinian fashion, that the meaning of any symbol is no more than the
resultant of (or response to) all uses made of it, which means symbols are
no
more to be trusted than words or phrases--and indeed may turn out to be not
even philosophically *distinguishable* from words and phrases in our joint
pursuit of rigor.  Unable, therefore, to profess the Leibnizian faith in
symbols, I am tempted to return to outer darkness to listen to the weeping
and gnashing of teeth.

[ ... ]