[FOM] Reply to Franzen, Heck, Davis

[Dean Buckner]

I've just returned from (very sunny) trip to West Scotland, and find over 50
FOM-related mails in my inbox.  Apologies for bunching these replies
together.  As a point of interest, the famous Eilean Donan castle, which
appears on almost any postcard picture of the Highlands, and which
represents for many people (including me) the Platonic ideal of a romantic
Highland castle, is in fact a complete fake.  It was built in the early
years of the 20C, thus is no older than set theory itself.

(1) Torkel (2) Richard Heck (3) Martin Davis

(1)  On Torkel's point that we can choose whether a variable stands in for
the name of a sentence, or the sentence itself, I'm a little surprised.  OK
then, let's choose to substitute a sentence for the first variable, and the
name of a sentence for the second variable in  "G iff G is true" giving:

    grass is green iff "grass is green" is true

Which makes sense, except it fails to support the redundancy story of truth
that I thought Torkel wanted, and which is implied by "G iff G is true".  On
the contrary: the quotation marks convert a sentence into a name of a
sentence, and thus remove the element of assertion.  The words "is true" put
this back again, and so are not redundant at all.  The redundancy story
requires that we substitute the same thing for the same variable, and
therefore cannot "choose".

(2)  On Richard Heck's point that the mass/count distinction does not
originate Jespersen.  I claimed this because I read it somewhere, but have
lost the reference.  The idea it could be found in the Port Royal is
plausible, as Arnauld is an expert grammarian (indeed, wrote a separate book
on grammar).  But I could not find it. Nor could I find any reference in
Mill's System of Logic, which follows Port Royal quite closely.  The closest
thing is the distinction between concrete and abstract nouns, but that is
not the same thing.

As for Aristotle, all the early discussions illustrate how we tend to
confuse semantical phenomena with something in the real world.  The message
of natural langauge philosophy is that we shouldn't.  For example (from a
neo-scholastic textbook) "Abstract terms have no plural.  The plurality that
a quality may have in the real order, it acquires in virtue of the concrete
individuals in which it inheres.  Hence when we conceive it in isolation, we
have no means of conceiving it multiplied.  Where plural forms are used, as
when we speak of "enthusiasms" or "ineptitudes", we merely mean the various
instances in which the quality was realised.  The quality as abstract, is
incapable of multiplication."

Which is of course all rubbish: and admixture of grammar, psychology and
Aristotelian "science".  An Aristotelian would say that the "substantial
form" of gold gives it its ductility, weight, colour.  Every kind of stuff
in Nature has this substantial form, which, joined to materia prima, makes
it the kind of thing it is.  You can read the entire Metaphysics as a
treatise on language, but written with the strange belief that  the
phenomena of language, were somehow embedded in the world.

It's not enought to make the distinction between mass and count terms.  The
trick is to see that this is a distinction of language alone.  This is a 20C
innovation.

"Des Cartes must be allowed the honour of being the first who drew a
distinct line between the material and the intellectual world, which, in all
the old systems, were so blended together that it was impossble to say where
the one ends and the other begins" (Reid p.270)

"The distinction, therefore, between Differentia, Proprium, and Accidens, is
not grounded in the nature of things, but in the connotation of names; and
we must seek it there if we wish to find what it is"  (Mill)

Mill also writes "A fundamental error [i.e. Aristotelian realism] is seldom
expelled from philosophy by a single victory.  It retreats slowly, defends
every inch of ground, and often, after it has been driven from the open
country, retains a footing in some remote fastness."

(3)  Martin writes  "Let Mr. Buckner pick up a scholarly journal devoted to
any of the sciences or even economics. Does he believe that these scientists
deluded in their evident belief that the use of technical mathematical
language is needed to properly deal with their concerns? Will he show us the
power of ordinary language by writing the equations of general relativity
defining the
gravitational field of the universe or Schrödinger's equation for the
evolution of the wave forms of quantum mechanics in those terms".

I know nothing about general relativity or Schrodinger.  But I know a little
about financial economics, a discipline which has revolutionised our
understanding of money.  There is a very famous formula (for which its
discoverers recently won the Nobel Prize) called the "Black-Scholes"
equation.  It is difficult to derive, and is related to certian equations in
Physics.  As an interview question it's always useful to test a candidate's
knowledge by asking them to explain the meanings (in English, obviously) of
the different terms of which the equation is composed.  This is the only way
to understand what the formula really means, and it's very important that
people do understand it, because it's used in financial risk management,
where black boxes are dangerous things.


Btw I've noticed that many trained mathematicians can find this very hard.
They are excellent at rearranging equations, and seem to have a supernatural
ability to do this correctly and rapidly.  But they are often hopeless at
explaining what these formula really are.  By contrast, lawyers, whose job
it is to frame financial contracts in a way that is both precise and
intelligible, do have a knack for translating mathematical ideas into plain
language.  (But then they are paid several orders of magnitude more than
your average graduate mathematician - is there a moral to be drawn?).

Dean