FOM: Connections between mathematics, physics and FOM

[Mark Steiner]

        It was worth joining this list, just to find out that the
subject to
which I have devoted the last 15 years of my life, the application of
mathematics to the natural sciences, is the subject of the dissertation
of a colleague.  I hope this dissertation appears as a book, making it
the second book in the twentieth century (which, as has been pointed
out, ends on December 31 this year) on the subject, at least to my
knowledge.

        It took me many years to get up the courage to write on the
subject:
I soon realized that with this material I would never get a Ph.D., so I
put it aside and waited till I got a job, and could write whatever I
wanted.  I soon found out, that a job without tenure means that you
CAN'T write "whatever you want."  So I tried to get tenure.  This took
many years and a 5,000 mile journey, namely to the Hebrew University,
which promised to give me tenure on arrival (one of the many broken
promises of University life).  By the time I got tenure, I was too tired
to write anything, much less a book on the Applicability of
Mathematics.  I needed a sabbatical.... The invention of the Apple
Computer, along with (finally!) a sabbatical, was what saved this
project from going into the grave with me.

        Though it is true that I have written a book with the title that
Jeffrey mentions, "The Applicability of Mathematics as a Philosophical
Problem," (sorry about the title--Harvard University Press wanted me to
call it "The User-friendly Universe" but my daughter said that the book
would end up on the computer science shelves) I'll be happy to let him
field the questions on this subject.  (I'm still thinking of a
Wittgenstein/Goedel posting.)  I'd really like the book to stand by
itself.  If anybody wants to ask me privately about anything I write
below, I'll try to answer.  (And, if even two people on this list buy
the book, it will soar from its present place of 144,501 on the
best-seller list of amazon.com to around 40,000.)

        But since the book has been mentioned, let me make some points
from
it,which are relevant to Jeffrey's ideas.  I'd be flattered if he would
take them into account in working his dissertation into a book (which,
as I say, I'm looking forward to--meanwhile, could you send me a copy of
it?).

        First, one of the main points in my book is that there is no
such
thing
as "the" problem of the applicability of mathematics, because there is
no one thing called the "applicability" of mathematics. So some of
Jeffrey's comments apply to one problem, and some to another (I'm sure
Jeffrey is aware of all this, but it doesn't come through clearly enough
in the posting).

        Thus, Frege, Field, Wigner, and I are NOT dealing with the same
question about the applicability of mathematics, and in fact they are
not talking about the same concept of "applicability."

        Frege is raising a "semantic" question about arithmetic: Take
the
argument, "There are three planets within 93,000,000 miles from the sun;
there are six planets not within this distance from the sun; 6+3=9;
hence, there are nine planets."  The problem with this argument is that
"three" in the first premise, and 3 in the third seem to have different
semantic roles; "three" as a physical predicate and 3 as (apparently)
the name of an abstract object.  Frege solved this problem, as all
readers of this list know, by showing that "three" can also be regarded
as the name of an abstract object, as the first premise can be
formalized Nx(x is a planet within...)=3.  Cardinality, like the
derivative, and Frege's quantifiers, is an operator.  Note that Frege
was NOT particularly interested in the applicability of mathematics to
the physical world, but rather to anything, including itself (i.e. we
count the roots of equations as well as planets).
Two points about Frege: (a) of course his argument assumes that 3 is an
object, and nominalists (like Field) don't believe in such objects.  But
this is not about applicability, but about Platonism.  (b) Frege's idea
that numerical attributions can be written NxFx=n, (n finite or infinite
cardinals) also DISsolves another standard question, how can abstract
entities be related to the world?  Answer: they aren't, n is related to
F, a concept, which may or may not be a physical concept.  Thus two old
questions about the applicability of mathematics were answered by Frege
over 100 years ago.  Here I very much agree with Dr. Ketland and applaud
his point.

        Field is asking the following two questions:
(a) Since, as he sees it, there are no numbers, so that all statements
of the form NxFx=n are false, how then can we make true predictions
using such statements?
His answer to this is similar to Hilbert's answer concerning
"infinitary" statements--mathematics is a "conservative extension" of
nominalistic physics, so that using arithmetic instrumentally to make
deductions from true nominalistic theories (those that don't refer to
mathematical "objects") will preserve nominalistic truth.  This leads to
the following research project:

(b) How can we characterize specific physical concepts without referring
to mathematical entities?

Really, mathematicians are best suited for this project, but since
"core" mathematicians are not likely to solve philosophical questions on
request, Field has to do it himself.  (To his credit, my colleague
Saharon Shelah is willing to answer any question of any colleague, if it
can be phrased as a mathematical question--he usually does this with
amazing speed.)  For example, Field shows how you can conceive of the
classical gravitational field as a theory of properties of spacetime
(and points in spacetime he regards as "kosher" for the nominalist--I
don't, by the way, and besides, there may not be any such thing as space
time, if some of the incomprehensible lectures on 56 dimensional lattice
structures I heard recently are saying what I think they said).

Now I see question (b) as about a question which Field doesn't even
raise explicitly, because it is not about "nominalism" but about the
DESCRIPTIVE ADEQUACY of (certain) mathematical concepts: namely the
schema:

(b'(C)) What is the NONMATHEMATICAL explanation for the fact that
mathematical concept C can be used in describing physical phenomena?
For the Poisson equation, Field lays down conditions in geometrical
language for this equation to describe something.  This project has
value IRRESPECTIVE OF NOMINALISM.  I don't think anybody has the answer
for C=group representations on a Hilbert space, i.e. as far as we know
NOW, there is no other language in which, even qualitatively, we can
even express this concept.  Scheme b'(C) is a very important question
(or set of questions) which has to be answered piece by piece.  Note,
that the question "What is mathematics?" does not enter into the special
cases of this scheme.

Wigner is asking yet another question (in his famous essay, The
Unreasonable Applicability of Mathematics in the Natural Sciences,
reprinted in Symmetries and Reflections, etc.) about mathematics as a
whole: why is it that MATHEMATICS AS SUCH is uniquely suited to DESCRIBE
physical events?  This question cannot be answered, without an answer to
the question What is Mathematics, which has been discussed on this
list.  Wigner gives a partial characterization of mathematics
(f.o.m.-ers don't get excited, he's not talking about mathematical
reasoning, but what makes a concept mathematical in the first place) in
terms of aesthetic properties (of what IN FACT counts as a mathematical
concept--Harvey has already expressed his view that mathematical
concepts should NOT in the future be so judged), which leads him to the
view that the applicability of mathematics in physics is a "miracle,"
since there is no reason the universe (even at the subatomic level)
should "care" about what we find beautiful.

Note, by the way, that if we regard the applicability of ("core")
mathematics AS SUCH as a real phenomenon, then the view of Gardiner that
the reason is that the universe is mathematically structured, or
exemplifies mathematical structures, is an EMPTY PSEUDOEXPLANATION.  I
have been trying to persuade my colleagues of this point for years.  The
term structure is just another word for "set theoretical predicate" (if
we take the common dogma that mathematics can all be formalized in ZFC),
BUT NOT EVERY SET THEORETICAL PREDICATE COUNTS AS "CORE" MATHEMATICAL!
I have never yet spoken to a core mathematician or read one who would
agree to the proposition that chess, for example, when defined as a set
theoretical predicate, is a mathematical structure (I'm speaking of the
first-order theory of chess, not Game Theory).  I recently asked Kazhdan
this question (he's a chess player of almost grandmaster rank) and he
said, emphatically, NO.  (His interesting explanation was that beginners
in chess speak the same language as the grandmasters, which is not true
in mathematics.)  It's the core mathematicians who, for better or for
worse, decide what is mathematics.

So is the remark that "Mathematics deals with symmetries and so does
physics" a pseudoexplanation, which trades on an equivocation between
the modern definition of a symmetry as an invariance under a group
action and the older definition as "balance" of some visual sort.  Under
the modern definition, this statement just restates the problem to be
solved.

However, most philosophers have dismissed Wigner's article for another
reason, not Gardiner's.  They simply deny the phenomenon altogether.
Skeptics say that there  is no such phenomenon as Wigner's descriptive
applicability.  Rather, sometimes mathematical concepts describe
physical phenomena and sometimes they don't.  In fact, most attempts to
apply mathematical concept C to physics fail.

Aside from this, Wigner does not make clear the connection between the
"mathematicality" of concepts C and their applicability.  He just cites
a number of examples (modestly, leaving out his own, most remarkable
contributions) such as the inverse square law, which happen to be
mathematical.  But he doesn't show how the fact that they were
mathematical contributed to the applicability.


My own contribution is a second-order version of Wigner's
problem--rather than asking why mathematical concepts C turned out to
describe physical phenomena, I discuss (I don't really ask anything) the
strategy of using the very concept "mathematics" in DISCOVERING these
very descriptions.  I can't go into this without reproducing my book in
its entirety, but the strategy I discuss is that of assuming that
phenomena yet to be described correctly utilize MATHEMATICALLY SIMILAR
(including identical)CONCEPTS to the INCORRECT descriptions hitherto
used.  Now SU(3) "resembles" SU(2)--for the mathematician, but, prior to
the discovery of quarks, this resemblance had no real physical content;
rather, just the opposite, the success of this analogy led to the
discovery of quarks.  Hence the theory of quarks cannot be used to
explain why scientists were rational to guess at SU(3) from the success
of SU(2).  (And, if you will permit me to repeat myself, Gardiner's
thought that the "universe is mathematical" is just a restatement of the
problem, assuming it is a problem, not a solution.)  Or take the
strategy of assuming that every classically described system with a
symmetry group G (which acts on "phase space") corresponds to a "quantum
analogue" with exactly the same symmetry group, acting on a vector
space.  (This has nothing to do with going to the "classical limit"
because we are talking about the behavior of the system in regions where
classical physics doesn't work.)  This assumption is pure
Pythagoreanism, and depends on the general concept of "mathematics"--the
projection that once a mathematical concept has been discovered to be
useful in physics, "it" will appear again, or SOMETHING SIMILAR.  Pauli
used this idea to derive the energy levels of hydrogen and many other
facts by assuming that the hydrogen atom, though not obeying Kepler's
laws, obey the SYMMETRY of Kepler's laws, namely O(4).  As Dr. Ketland
pointed out to us, Emmy Noether pointed out the relation between
continuous symmetries and conserved quantities; Lenz (Pauli's department
chairman) applied this to the Kepler problem and discovered a "hidden"
extra ABSTRACT O(3) symmetry, leading to "conservation of the Lenz
vector" (vector that lies in the plane of the orbit, in the line between
the two foci), which together with the O(3) symmetry responsible for
angular momentum conservation, leads to a O(3)+O(3)=O(4) (a small
mathematical miracle in its own right) symmetry.  Pauli simply
"quantized" the Lenz vector.  (Of course, as I pointed out already, he
had no idea what he was "doing", since he didn't know what a Lie group
is.)  For a "reconstruction" of this argument, see Sternberg, Variations
on a Theme By Kepler, or his remarkable book, Group Theory and Physics
(Cambridge).  Even under the reconstruction, the argument is pure
Pythagoreanism (Sternberg doesn't mind this: he's a Pythagorean)--there
is no rational nonPythagorean reason why, just because in classical
physics group G operates on phase space, it must in quantum physics
operate on a complex vector space (this is surely NONPROJECTIBLE in the
sense of Goodman--unless you're a Pythagorean).

The Yang-Mills implementation of this strategy is quite remarkable,
because to implement it they had to INVENT a fictitious "classical"
theory of the nuclear force, by "analogy" to that of electromagnetism
(but by the "natural" (?) analogy between U(1) and the abovementioned
SU(2)).  Since nobody knew about fiber bundles (physicists being
intolerant, as before, of pure mathematics, though Yang later gave the
excuse that he was "incapable" of mastering the theory of fiber
bundles), the analogy between classical theory of electromagnetism and
the (projected by fictitious) classical theory ofnuclear force
field was mediated by quantum mechanics itself!  Using quantum
mechanical formalism and a purely mathematical analogy, Yang and Mills
invented a phony differential equation for the nuclear force field.
They then promptly "quantized" it--quantization here is a syntactical
transformation which turns a false classical equation into a (hopefully)
true quantum one.  I discuss this case in my book also.  I note that
Mills himself, writing years later, bemoaned the fact that the gauge
field theory program they initiated still involved a detour into
"classical" equations.

In the case of Pauli, really, his arguments are not even Pythagorean,
but purely formal.  He simply substituted self-adjoint matrices for
variables in various physical magnitudes; then he tried to "solve" the
resulting equations, dividing by matrices he hadn't first proved were
invertible (for example).  There is also the little problem that the
product of two self-adjoint matrices is not necessarily self-adjoint,
and the Lenz vector happens to be a cubic polynomial in classical
physics.  You can read my book for the rest of the details.

The conclusion of my book is that modern physics was discovered largely
[logicians take note: I mean here a NECESSARY not a SUFFICIENT
condition] by the assumption that the way to make guesses is by analogy
to what works but what is known to be false, and that the analogies were
not defined by anything but the structure of the family of mathematical
concepts, and (what is truly bizarre) EVEN BY THE STRUCTURE OF
MATHEMATICAL NOTATION.  (Jeffrey gave an example extensively discussed
in my book: the discovery of Dirac's equation.)  The idea that you can
discover things by the structure of mathematical notation, even when (in
Harvey's view) it's only "pre-mathematical" notation (as in the case of
the Feynman path integral) is anthropocentric (again, see my book for
the full argument).

My view is that views such as Gardiner's therefore collapse into arrant
anthropocentrism.  This would be fine (some of my best friends...),
except that Gardiner is well known for his vocal and even scatalogical
denunciations of the so-called "anthropic principle" in physics.  (For
the record, I do not hold this principle, mainly because I don't know
what it is.)  What I have written is an attempt to persuade Jeffrey to
take this criticism of Gardiner's views into account as he pursues his
study of the fascinating story of the application of mathematics.  I
hope he also discusses what I have not, namely, why the application of
mathematics to physics is so different to that of economics (even
mathematical economics).

        Actually, however, I didn't join this list to discuss my views
on
the application of mathematics--I'm not even sure that they are relevant
to
f.o.m.  Rather I would like to get back to studying questions of f.o.m.
and their relevance to philosophy after many years of working on the
above topic.  Some of the material I've seen here encourage me to think
that there is still much for philosophy to learn from the experts on
foundations.  I do hope that at least some readers of the above, even if
they are not persuaded by what I have written, at least come to the
conclusion that there is also something to learn from philosophical
analysis by people who have been trained in it--if not by me, than by
others.