FOM: Connections between mathematics, physics and FOM

[Jeffrey John Ketland]

[I apologize in advance for the length of this message].
 
Recently there has been some discussion on fom about the 
relations between mathematics and physics. In particular, the 
attitude of some leading theoretical physicists, like Glashow and 
Weinberg. When I was trained as a physicist, we always thought 
that maths was a game, at best a useful instrument, to which the 
notion of truth simply doesn't apply. I think this still is the dominant 
view. Maths is nothing more than a useful instrument (neither true 
nor false) which can be "interpreted" to provide a language for the 
formulation of physical laws.

However, following Quine and Putnam, I now think that this 
formalistic position is completely wrong. Maths is interpreted all 
along, and is about (i.e., quantifies over) such things as numbers, 
functions, sets and structures. If maths were just an uninterpreted 
game (formalism), then it couldn't be applied to make *true* 
predictions about the world. In short, science contradicts 
formalism. Indeed, much maths has to be assumed to be true in 
order for some of its axioms to be included in the formulation of 
scientific theories, from which physical phenomena may be 
predicted and explained. (Hartry Field has argued against this 
claim in a beautiful book called "Science Without Numbers" (1980) 
and in a series of papers, anthologised as "Realism, Mathematics 
and Modality" (1989). However, I think that Field's program doesn't 
work. He requires that every mathematiczed physical theory T can 
be reformulated as a nominalistic theory N without using any 
quantification over mathematical entities. He also thinks that 
adding mathematical axioms - say the axioms of ZF with ur-
elements -- to such a nominalistic theory N always yields a 
conservative extension. I think that both claims are wrong.)
 
The questions I think need asking and answering are:
 
(i) What are the relations between maths and physics?
(ii) Why does maths apply to the physical universe?
(iii) What implications does the applicability of maths to science 
have for philosophical interpretations of maths?
(iv) In fact, what does a theory of application look like?
 
I wrote my PhD on exactly these topics. Here are my own favoured 
answers:
 
(i)+(ii) Mathematics is the science of patterns and structures. The 
physical universe is, as Martin Gardner and zillions of others have 
put it, mathematically structured. (Of course, it is a contingent and 
not a priori fact that the physical universe exemplifies one structure 
rather than another, but it is still a fact). So, the relation is that 
mathematicians study abstract structures which the physical 
universe might exemplify (for all we know). The mathematical 
physicist then picks these already studied structures and guesses 
that certain kinds of physical system exemplify this or that 
structure. That, in a nutshell, is why mathematics applies to the 
physical universe.
 
(iii) The following statement is logically true (as Quine more or less 
pointed out in 1948 and Putnam fiercely argued for on 1971).
IF you are realist about any particular scientific theory T, THEN you 
have to believe that the mathematical entities quantified over in that 
theory T really exist (even though they are abstract).

Of course, whether you're a *realist* about such a theory T is up to 
you. A realist is someone who thinks that (i) there is a fact of the 
matter as to whether T is true and (ii) also that T is a very good 
approximation to the truth. You can try to be some kind of 
instrumentalist or positivist about scientific theories, but that 
position is very implausible. The idea that GR and QFT are nothing 
more than convenient tools for organizing our voltmeter readings, 
and do not provide extremely accurate descriptions of the universe, 
is crazy to my mind. GR and QFT are approximately true theories, 
which get to grips with the largely unobservable structure of the 
physical universe.
It is also implausible (but an open research problem) that 
theoretical physics can get by either without mathematics at all 
(Field thinks it can) or with just constructive or predicative 
mathematics (Feferman thinks it can). If that is right, it follows that 
some scientific theories probably require mathematics which only 
has a platonistic interpretation (completed infinite sets, sets 
thereof, and so on). In a nutshell, (if Field and Feferman are wrong), 
we have: SCIENCE ENTAILS PLATONISM.
 
(iv) A theory of application needs to explain:
(a) How arithmetic is applied: long ago (1884, in fact)), Frege 
solved this problem: each concept F is associated with a number 
(its cardinality) #F. The (consistent and true) axiom for this is 
Hume's Principle:
	HP: #F = #G iff F and G are equinumerous
Equipped with the comprehension scheme for second-order logic, 
one can then introduce definitions and prove all the axioms of 
second-order arithmetic (Z_2) from HP. This idea was first 
conjectured by Crispin Wright (1983) and demonstrated in full detail 
by George Boolos (1987).
Some concepts are empirical concepts (like the concept "horse" or 
"space-time region"). So, (abstract) numbers are "glued" to these 
concepts. Then, some relations between concepts are connected 
to relations between numbers. (E.g., if the concept F is a subset of 
the concept G, then the number of Fs is less than the number of 
Gs). That's how arithmetic is applied to non-mathematical subject 
matters.
 
(b) How analysis (real numbers) is applied: this problem is largely 
solved by the theory of Fundamental Measurement (going back to 
Holder's Theorem). Put a load of physical objects into a structure, 
with empirical relations between them (e.g., "heavier than"). The 
resulting structure A is sometimes called an impure structure. 
Then one can sometimes find a representing homomorphism h 
from A to some pure mathematical structure B whose domain is 
the reals and whose basic relations are < and the usual operations 
on reals. This h is a measurement scale. E.g., the measurement 
scale mass-in-kg is a homomorphism from the set of physical 
objects to the reals such that if x is less massive than y, then h(x) 
< h(y). Fundamental Measurement is concerned with proving 
theorems to the effect that if an empirical impure structure A 
satisfies axioms T, then there exists an h to some pure structure B 
(and this h is unique up to re-scalings, etc.). That's how real 
numbers get applied to non-mathematical subject matter.
 
(c) How set theory is applied: this is crucial if we are to get 
structures built from physical domains. Set theory with ur-elements 
contains an axiom which says that there exists a collection of ur-
elements (e.g., chickens, space-time points, regions of space-
time, electrons, etc.). This collection is probably a set (not 
equinumerous with a proper class). We need such a set theory to 
apply the required representation theorems above, and we need 
such a set theory to talk about various physical functions (like 
measurement scales, tensor fields on space-time) which assign 
values (as masses, temperatures, lengths, etc.) to physical 
objects or to space-time points (as co-ordinates in a chart, or as 
values of the various curvature and quantum fields).
 
Once we got all of this sorted out-and I think that the above sketch 
is more or less right--we still have lots of internal questions left. 
Like: why particular theories (quantum field theory, relativity) uses 
certain mathematical structures (e.g., Lie groups, differentiable 
manifolds). In fact, I think Professor Steiner has written a book 
called "The Applicability of Mathematics as a Philosophical 
Problem" which discusses in a very valuable way such problems 
(my copy is in London, so I can't quickly check the main contents).

I think that Steiner discusses the (amazing) fact that what look like 
formal, purely mathematical, considerations can lead to the 
postulation of physical laws which then turn out to be very 
important (and possibly true: they are strongly confirmed by 
predicting and explaining new phenomena). The famous example is 
the Dirac equation (which is one of the 4 fundamental equations of 
physics: Maxwell's, Einstein's, Schrodinger's, and Dirac's). Dirac 
always said he got his equation by "playing about" with "pretty 
mathematics", with the requirement that (i) the equation be a linear 
and first-order, and (b) its "square" should be the Klein-Gordon 
equation. The trick was to introduce Dirac's anti-commuting 
gamma matrices, and spinor field functions. Oddly enough, this is 
nowadays all closely connected to anti-commutative Grassman 
algebras, which Frege once criticized for not meeting his standards 
of rigour.
 
I suspect that there is no predicting what pieces of pure 
mathematics might turn out to be useful in describing the physical 
universe. As Steiner mentions in an earlier posting, modular 
functions are needed in string theory, in calculating scattering 
amplitudes. Hermann Weyl's deep insights about the relevance of 
group theory (to physics) was long referred to as the "Group Pest". 
(Of course, in those days, Gottingen was big on this: Weyl's 
colleague, Emmy Noether's theorem is a deep theorem about the 
relationship between symmetries of a physical system and 
conservation laws). Weyl was right. Nobody ever writes down a 
lagrangian these days without discussing its symmetries.
 
Finally, what about FOM maths (set theory, model theory, 
recursion theory, proof theory)? Could that be of relevance to 
theoretical physics? I believe it could, and discussed this issue a 
little bit on fom last year (Joe Shipman took part).
First make two assumptions:

(i) The physical universe is infinite (i.e., contains infinitely many 
space-time points) and exemplifies a structure which contains at 
least one standard model of the natural numbers (one of our best 
physical theories, GR, does say this: it says spacetime is a 
differentiable manifold, and within this theory one can define space-
time regions which are omega-sequences of space-time points)

(ii) Our theories must be recursively axiomatizable in order for them 
to be usable by the presumably finite human mind.
 
It follows from Godel's theorem that some of our physical theories 
will be deductively incomplete with respect to the set of physical 
statements. For some of our physical theories, there will be godel 
sentences which cannot be proved within the theory, but can be 
proved by some set-theoretical extension. An example would be 
the consistency of a precise formalization of GR. The consitency 
statement Con(GR) could be coded, using a reference omega 
sequence of spacetime points, as a statement about spacetime. 
Assuming that GR is true (and thus consistent), this new 
statement would be another true sentence about spacetime, but it 
would not be provable in GR. However, presumably, the 
consistency of GR *is* provable in some suitable set theory. 
Stewart Shapiro discusses something like this (in very careful 
detail) in a paper in Journal of Philosophy 1983, called 
"Conservativenesss and Incompleteness", criticizing Field's idea 
that adding maths to a nominalistic theory (e.g., a synthetic theory 
of space-time) must be a conservative extension.
 
The central questions, I think, are:
 
A: Could the addition of presently undecided highly abstract *set-
theoretical* principles lead to the prediction of new *physical 
phenomena*? (I.e., could adding CH or a large cardinal axiom lead 
to the prediction of a new particle field, or fix the dimensionality of 
space-time, or something amazing like that).
 
B: Could the physical universe contain non-algorithmic processes 
which might be harnessed by humans to "compute" non-recursive 
functions (such as the characteristic function of the set of truths in 
arithmetic)?
 
Of course, these questions are highly speculative (some of my 
philosophical colleagues use the technical term "mad" to describe 
them). Geoffrey Hellman discusses the question (A) in his book 
Mathematics without Numbers (1989), citing some of the work of 
Harvey Friedman on the necessary uses of higher axioms of set 
theory. I don't know what answers to (A) and (B) would look like, 
but if these (mad?) ideas are right, then very important connections 
could be forged between theoretical physics and fom mathematics.

Jeff Ketland


Dr Jeffrey Ketland
Department of Philosophy C15 Trent Building
University of Nottingham NG7 2RD
Tel: 0115 951 5843
E-mail: Jeffrey.Ketland at nottingham.ac.uk

Dr Jeffrey Ketland
Department of Philosophy, C15 Trent Building
University of Nottingham, University Park,
Nottingham, NG7 2RD. UNITED KINGDOM.
Tel:    0115 951 5843
Fax:    0115 951 5840
E-mail: <Jeffrey.Ketland at nottingham.ac.uk>