FOM: Arbitrary Objects

[Insall]

Professor Kanovei wrote:


At this moment, either Y submits x as required with which
X maintains the argument F(x)--> C
or Y refuses and quits, then X wins by default.

Thus, the trick is to charge the opponent with all burden
of thinking about what is an arbitrary object.


My response:
     I agree completely with this ``advocate-adversary'' presentation, which
may rightly be considered a part of game theory, I would think.  A
case-by-case treatment may even be required, since the use of the term
``arbitrary'' in mathematics is somewhat murky.  Thus, one may read ``let x
be an arbitrary wff of the FOPC'' and be completely certain what the author
intended, but find it impossible to mechanically implement, because the most
syntactically astute observer, in their most obtuse moments, might ask the
following, quite rightly, but also quite infuriatingly, indicating his/her
idiosyncratically pedantic approach to all of this:  ``But which FOPC?'' -
for in fact, there are many, are there not?  Yet, if we restrict to cases in
which the language one has in mind is denumerably generated, and if the most
pedantic of us agrees to this restriction, then between any two FOPCs one
may have in mind, there is a natural isomorphism, speaking both loosely and
categorically, so the teacher may look the syntactic pedanticist in the eye
and say ``You knew what I meant.'' and mean it.  When the student subsides,
or when the instructor carefully delineates which symbols are fundamental in
his or her FOPC,and redesigns it from the empty set to the highest level, we
may continue to discuss properties commonly held by all wffs of the
particular FOPC in question, and probably of all wffs of any FOPC any member
of the class amy come up with, according to a pre-defined notion of an FOPC.
But since this process almost inevitably leads one to a more than twice as
long as the class period regress, the instructor is most likely to refuse to
give the detail asked by the wonderfully pedantic syntacticist, point out
the beauty of this bright one's question, and move on to discuss the
properties all wffs have in common, leaving open pandora's wormy box of
arbitrary denials of the meaningfulness of the concept of ``an arbitrary wff
of the FOPC''.



Professor Friedman wrote that ``arbitrary'' is not an adjective.  I guess a
linguist might disagree.  To settle this question, one may reasonably, I
think, consult a dictionary of the English language.  On page 67 of the
dictionary I have available at the moment, namely the New College Edition of
the American Heritage Dictionary of the English Language, the word
``arbitrary'' is referred to as an adjective.  Of course, dictionary writers
revise their work frequently to account for usage, but never for
mathematical usage above most other usages.  Thus one may look to usage, and
the definition of the term ``adective''.   However, this becomes the
grammatical version of the unreasonable (though, perhaps not infinite)
regress I mentioned above.  Professor Friedman would rightly point out that
I should know what he means.  For in fact, I think I do know what professor
Friedman means, in the same sense that Kant pointed out long ago that the
(German?) noun ``existence'' is not properly considered in logic to be a
predicate.  In this case, the adjective ``arbitrary'' should not be a
predicate.  It seems to me that it is merely a part of speech used to
emphaisize the application of a universal quantifier.  I submit that the
same holds of the term ``random''.  In the natural language we call English,
the term ``random'' is an adjective.  However, in the most careful
treatments of mathematical statistics, it is, I think, not even a term, but
always part of a noun.  For example, a ``random variable'' is not a variable
in the sense you and I, as logicians, think of a variable, a ``random
number'' is not, in the sense understood by a number theorist, a number, but
it is a random variable whose range is a set of numbers.  As I understand
it, a ``random variable'' is a measurable function from one measure space to
another.  Thus one may speak reasonably of an ``arbitrary random variable''
as opposed to a ``specific random variable'', but, as I can tell, it makes
no sense to speak of a ``random arbitrary variable''.  At least, I have not
yet seen this term defined.  (However, I expect that many of us already are
figuring out how we would define it, so as to make it as useful as possible,
in our respective opinions.  Personally, I would probably consider it as a
random variable whose range is a subset of the set of variables of some
specific language of the FOPC - but that is just me.)


Professor Silver has, I think, brought up this question of the term
``arbitrary'' out of a reasonable curiosity about the mathematical use of
terms that originate in a natural language such as English.  I recall
concerning myself with the usage of the term ``arbitrary'' when I was a
Mathematics graduate student.  It seemed to me to be redundant in every case
in which it occurs, because in each case I saw, the apparent intent of the
author was to emphasize the universal quantification of some hypothesis or
conclusion.  This seemed so obvious I felt no need to explain it to anyone,
except the occasional student who asks.  Now that the question has been
brought up here, am I to assume that it is reasonable for me to begin to
tell you how I interpret such statements?  I guess it may be reasonable, as
long as this forum tolerates my long-winded replies to what appear to be
simple questions.  The trouble is that since I have not read Suppes and Fine
on this subject, I am likely to cover some old ground, or, as some say,
``re-invent the wheel''.  Is it worth our time to do so?  If professor
Silver is interested in it, I guess it is worth our time to do this.  Is it
just an ``infinite'' regress?  If all we do is disagree, it is likely to not
appear to me to be much of a success.


Others have commented on this term arbitrary from other mathematical
subjects, such as category theory (Bauer, for example), and they all have
some merit.  But is this not the case just because our use of the term
``arbitrary'' in mathematics is fairly clear and somewhat redundant?  For in
fact, it seems to me that the models I have seen here for the term
``arbitrary'' are themselves somehow redundant as well.  This is not
necessarily meant as a negative comment, about either the usage of the term
``arbitrary'' or about the models of its usage, or about this current
discussion.  However, it brings me to the question of how much about such a
topic is anyone else willing to read from me, or from anyone else, if you do
not agree with the things being said, or if you agree but think the things
being said need not be said?


Matt Insall