[FOM] Reply to Franzen, Heck, Davis

[Harvey Friedman]

Reply to Buckner 4/20/PM 8:25PM.

I don't think you have answered Heck and Davis adequately, and you
need to modify your remarks concerning lawyers.

I trust you will get
to responding to my previous posting (in response to you) as well,
where I question whether your postings deal with any issues in the
foundations of mathematics. See, e.g.,
http://www.cs.nyu.edu/pipermail/fom/2003-April/006392.html and
http://www.cs.nyu.edu/pipermail/fom/2003-April/006409.html

Heck in his posting http://www.cs.nyu.edu/pipermail/fom/2003-April/006410.html
has very specifically contradicted your assertion that "existential
conservatism" is representative of current philosophy of language, or
philosophers of language. I hope you will respond to that.

Also, you have not begun to answer Martin Davis' entirely correct
assertion that, at least prima facie, virtually the entire modern
literature in mathematics and science goes well beyond the depth and
scope of ordinary language concepts ordinary language thinking. See
http://www.cs.nyu.edu/pipermail/fom/2003-April/006404.html

Unless you can somehow indicate how any significant portion of the
entire modern literature in mathematics and science can be naturally
recast in terms of your apparently strict standards from ordinary
language, you should promptly concede the point to Davis.

Bringing in lawyers and demonstrably erroneous information about
their compensation doesn't seem to me to have any bearing on the
issue.

See below for more detailed comments.
>
>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.

When I teach I want to know how many students are in the class, which
is an integer and determined, theoretically and practically, by
counting. When I weigh myself I want to know the number of pounds,
including fractional parts. This is viewed as a real number, as least
ideally. Less ideally, a rational number. Still less ideally, some
conditions are placed on the size of the denominator.

This situation is not normally viewed as aspects of language, but
important conceptual differences that are fully and properly
reflected in sharply different mathematical notions that go far
beyond ordinary language and ordinary reasoning. I.e., the
mathematical number systems of the integers, the rationals, and the
reals.

You can see here, very vividly, the power and clarity of mathematics
over ordinary language and ordinary reasoning for basic elementary
scientific contexts.

If you want to claim that this is a mirage, this is an interesting
claim, but in order to have such a claim be taken seriously, you need
to give some interesting interpretation.

Maybe I can help steer this into some possibly productive line. For
scientific purposes, and sometimes commercial purposes, one wants
very accurate weight of items. One needs what is called an "analog to
digital converter" to get such weights. This is abbreviated A-D. One
uses some very serious physics and engineering, involving theories of
electricity and/or light in order to build appropriately accurate A-D
and/or support them intellectually.

In an A-D, one transforms the weight input into a physical process
involving electricity and/or light which has a discrete character
that can be counted - i.e., electrical pulses go into a counter
circuit.

You could take the very idea of the need for an A-D, and ignore all
of the interesting modern science and technology behind it, and  try
to say that weighing things amounts to nothing more than counting
something!!

However, this would be an interesting but questionable move that
needs some serious defense - given the substantial
mathematical/scientific infrastructure to support the workings and
implementations of modern state of the art A-D's.

You have a great deal of work to do to convince anybody that (what I
understand of) your approach is not hopelessly out of touch with
mathematics and science; and in particular, with the foundations of
mathematics and science. You could try to do this, and if you get
anywhere it would be quite interesting.

The points I am making can also be made with respect to accurate
measurements of speed. Accurate measurements are even needed to
maintain detailed sports records, but also - and more intensively -
to conduct the wars against Iraq. Regardless of political views, one
is rather awestruck at what the USA did, when compared with the
history of warfare. Do you think it could be done with strict
adherence to ordinary language concepts and ordinary language
thinking?
>
>(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 scientist=
s
>deluded in their evident belief that the use of technical mathematical
>language is needed to properly deal with their concerns? Will he show us th=
e
>power of ordinary language by writing the equations of general relativity
>defining the
>gravitational field of the universe or Schr=F6dinger's equation for the
>evolution of the wave forms of quantum mechanics in those terms".
>
>I know nothing about general relativity or Schrodinger.

What parts of the above apply equally well to high school mathematics?

>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,

Difficult is relative. To experts and serious students, Black-Scholes
is now trivial, as it  has been around for some time and is standard
material in courses. If this was an e-mail list devoted to financial
economics - rather than to f.o.m. - then the vast majority of the
subscribers would regard it as utterly trivial to them.

>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 wa=
y
>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.

Good interview technique. However, there is enough in the use and
implementation of such equations - including their implementation on
a computer - that involves a great deal of not very ordinary language
and not very ordinary thinking. Can you build a state of the
art computer by purely ordinary language and thinking skills?
>
>Btw I've noticed that many trained mathematicians can find this very hard.
>They are excellent at rearranging equations, and seem to have a supernatura=
l
>ability to do this correctly and rapidly.  But they are often hopeless at
>explaining what these formula really are.

Are you interviewing mathematicians who are seeking jobs at
btopenworld? Why is this group of mathematicians representative of
mathematicians generally? Or is it?

>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?).

One can go on and on with citing special groups of highly paid people 
in order to justify some claim or another. Here is one.

"By contrast, CEOs who really know the mathematics and science behind
their financial investment companies' research are compensated far
more than your average lawyer whose job it is to frame financial
contracts."

Two examples: Edward Thorp ran a well known investment firm which
traded in a lot of complex investment contracts, and has a Ph.D. in
Mathematics. Also Jim Simons who currently runs a very big financial
investment firm which trades in all kinds of investment contracts,
and who also has a Ph.D. in Mathematics.

I should add that Bill Gates reportedly has a big advantage over
competitors in that he has an unusually good handle on the
math/science done at Microsoft. He might not answer a phone for what a
lawyer of the kind you are referring to earns in a year. He started off doing
some computer science, and years ago wrote an interesting mathematical joint
paper with a well known computer scientist now at UC Berkeley, on the
performance of a computer algorithm.

By the way, what does "orders of magnitude" mean in what you wrote?

The definition of order of magnitude is "ten times". Two orders of
magnitude means "one hundred times". Three orders of magnitude means
"one thousand times".

What you wrote is (on average) false under even "ten times". Are you 
thinking of
some ordinary language concept of "order of magnitude"?

And what "moral were you to draw?" with this (erroneous) comparison?