FOM: Connections between mathematics, physics and FOM

[Jeffrey John Ketland]

Connections between maths, physics and fom: continued

Mark Steiner and Matt Install have raised some questions about 
my earlier posting. Joe Shipman has sent me an excellent paper of 
his about relations between physics and computability theory.  So I 
have composed this point-by-point discussion of some of the topics 
raised. In particular, I want to say that there is fom significance in 
many of the topics below. There are also open research problems 
(most of which I lack the detailed requisite skill to work on).

(A) Plato’s dictum: Mathematics is a science of abstract patterns 
and structures
(B) Gardner’s dictum: The physical universe is mathematically 
structured
(C) Galileo’s dictum: the book of Nature is written in the language 
of mathematics
(D) Field’s program: Eliminating all (!) mathematics from science
(E) Feferman’s program: Eliminating non-constructive mathematics 
from science
(F) The Kazhdan problem: the difference between interesting and 
uninteresting 
mathematical structures.
(G) The Wigner problem: The “unreasonable applicability” problem
(H) The Kronecker problem: The finiteness of the human mind.
(I) The Laplace-Poincare problem: predictability and computational 
tractability.
(J) Quantum computation
(K) The Zeno Problem: supertasks in spacetime and infinity 
machines
(L) Godelian phenomena in physics: Reverse physics?
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(A) Plato’s dictum: Mathematics is a science of abstract patterns 
and structures

This is the slogan of the contemporary structuralists in the 
philosophy of mathematics, Michael Resnik and Stewart Shapiro. I 
am in broad agreement with this (realist) position. See:

[1] Michael Resnik 1997: Mathematics as a Science of Patterns
[2] Stewart Shapiro 1997: Philosophy of Mathematics – Structure 
and Ontology

At Matt Insall correctly observed, the English word “pattern” has a 
connotation of predictability, computational tractability or whatever. 
That is not what is usually meant here. A pattern can be any 
pattern at all – a random binary sequence, for example. And a 
structure can be any abstract structure at all, including non-
axiomatizable structures.
The crucial point is that these abstract structures exist outside us, 
are not “made” or “constructed” by us. They have an objective 
existence independent of the human mind, and even independently 
of their exemplification in the physical universe. So Plato’s dictum 
is meant to be realist or platonist slogan, to be chanted against 
constructivists and formalists.
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(B) Gardner’s dictum: The physical universe is mathematically 
structured
 
This is meant to provide a rough explanation for the very possibility 
of applying mathematics (never mind applying which bit of maths to 
which bit of the world). Again, the idea goes way back to Plato and 
before (Pythagoras, of course). 
Mathematical structures are Plato’s “Forms” and the physical 
systems are the “imperfect copies” of those Forms.
Not all Forms need to have instances in the world. Church’s 
famous example, the concept or form “purple cow” has no 
instances in the world. Similarly, not all mathematical structures 
need have instances in the physical world. For a start, some 
mathematical structures are just too big (like the natural 
models of ZFC)!
I think I disagree with Mark Steiner that Gardner’s dictum is a 
pseudo-explanation. Really, it’s just a very weak statement saying 
that some of the abstract structures that pure mathematicians 
study really do turn up in physics. It certainly doesn’t explain, 
for example, why spacetime is a four dimensional manifold or why 
the gluon wavefunctions are irreducible representations of SU(3)!
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(C) Galileo’s dictum: the book of Nature is written in the language 
of mathematics

Although (A) and (B) don’t solve all of the problems to do with 
applicability, they do explain Galileo’s dictum. Physical particles 
move along continuous curves in continuous spacetime. Their 
velocities are derivatives of their position functions and their 
accelerations are proportional to the local forces acting. So, the 
book of Nature will contain lots of analysis and reference to real 
numbers, and functions thereon.
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(D) Field’s program: Eliminating all (!) mathematics from science

The Quine-Putnam argument is a major hot topic in contemporary 
philosophy of mathematics. It says that our scientific theories 
make endless reference to mathematical entities, so we can’t be 
realists about those theories without also being realist about those 
mathematical entities. Quine says such things in tens of his 
articles and books, starting with

[3] W.V. Quine 1948: “On What There is”, in Quine 1980, From a 
Logical Point of View.
and finishing with
[4] W.V. Quine 1995: From Stimulus to Science

For Putnam’s version of the QP argument, see
[5] Hilary Putnam 1971: “Philosophy of Logic”, in Putnam 1979, 
Mathematics, Matter and Method: Collected Papers.

In 1980 Hartry Field transformed philosophy of mathematics by 
attacking the QP argument at its heart and claiming that all (!!!) of 
mathematics can be eliminated from science. See:

[6] Hartry Field 1980: Science Without Numbers

Field’s 2 main claims are that
	(i) we can always eliminate mathematics from any physical 	
	theory to obtain a “nominalistic” theory (i.e., no quantifiers 	
	ranging over numbers, or sets).
	(ii) if we add mathematics to a nominalistic theory, we always 
	get a conservative extension.

He substantiated these two claims by, indeed, giving a nominalistic 
version of Newtonian gravitational physics (including a nominalistic 
or synthetic treatment of spacetime) and by arguing that any model 
of a nominalistic theory N can be expanded to a model of the result 
of adding ZFC to N (if this were true, then N+ZFC would have to be 
a conservative extension of N).
Field’s point about conservative extensions is closely connected to 
Hilbert’s ideas about “real” and “ideal” mathematics. PRA is 
supposed to codify the “real” and adding “ideal” set theory should 
give a conservative extension. Then “ideal” set theory would just be 
a “useful, but dispensable, instrument” for finding out real things 
about the numbers. Similarly, Field’s idea is that adding 
mathematics (i.e., set theory) would just be “useful, but 
dispensable, instrument” for finding things out about the concrete 
world.

HOWEVER. Damn!!! It doesn’t work! Adding set theory to PRA is 
non-conservative and adding set theory to certain synthetic 
descriptions of spacetime is non-conservative. This is closely 
connected to Godel’s theorems. In the  mathematical case, all this 
is well-known to readers of this list. But the case for the non-
conservativeness of adding set theory to nominalistic spacetime 
theories is carefully developed in:

[7] Stewart Shapiro 1983 “Conservativeness and Incompleteness”, 
Journal of Philosophy 80.
And in
[8] John Burgess and Gideon Rosen 1997: A Subject With No 
Object.

In fact, the Introduction to Burgess and Rosen’s book above [8] is 
the best introductory discussion of modern nominalism that I know 
of.
Matt Insall makes a point which Stewart Shapiro also made (in the 
book [2] above) about Field’s program. Even if one could eliminate 
direct reference to numbers and sets from physical theory, we 
would still be studying a *structure exemplified in the world*.

Aside from the conservativeness issue, there are many technical 
problems in this program. How do you do the nominalizing for 
Quantum Theory? For General Relativity? My view is that Field’s 
program is beautiful and philosophically attractive. 
But I think it doesn’t work.
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(E) Feferman’s program: Eliminating non-constructive mathematics 
from science

Solomon Feferman wrote a paper in 1988

[9] Solomon Feferman 1988: “Weyl Vindicated: Das Continuum 70 
years later” (Temi e propetive della logica e della filosopfia della 
scienza contemporanee, Bologna).

developing a predicative set theory W (conservative over PA), 
finishing with an argument that all the mathematics needed for 
science could be developed within W.
Not many philosophers know much about this idea: that although 
we’ll use mathematics in science, we’re restricted to some kind of 
constructive mathematics. In fact, I’m not sure that many people 
have worked on it at all.
However, there are some interesting papers by Geoffrey Hellman 
arguing that certain more restrictive versions of constructivism are 
demonstrably inadequate to the needs of present science. In 
particular,

[10] Geoffrey Hellman 1993: “Gleason’s theorem is not 
constructively provable”, J.Philosophical Logic.
[11] Geoffrey Hellman 1998: “Mathematical constructivism in 
spacetime”, British Journal for the Philosophy of Science.

There was a storm of protest by a few constructivists about the 
claim about Gleason’s theorem’s unprovability. I don’t know the 
details. The points he makes in [11] are more convincing, however. 
According to GR, the (spacetime) continuum is right “out 
there”, exemplified in physical spacetime. So, we are permitted to 
use non-constructive analysis to study this genuinely real physical 
system. An example Hellman discusses is the proof of Hawking-
Penrose singularity theorems, which are non-constructive 
existence theorems.

As well as Feferman’s program (based on predicative foundations), 
there would also be “Bishop’s program”, and perhaps many others, 
corresponding to the various foundational positions. My own view is 
that just as constructive mathematics is inadequate for the needs 
of core mathematics, so it will turn out that constructive 
mathematics is inadequate for the needs of core theoretical 
physics.

Volker Halbach has asked me privately if ACA_0 would be 
sufficient for theoretical physics. I doubt it. By known results, it 
can’t prove such things as Ramsey’s Theorem (Simpson 1998, pp. 
210-215), and it is perfectly conceivable that combinatorial stuff 
might well be required in theoretical physics. It can’t prove the 
Sigma^1_0 axiom of choice (you need ATR_0: Simpson 1998, pp., 
205-206) and that might be needed in physics.
Moreover, I think that something like “reverse physics” could be 
developed. I’ll discuss this a bit more below (Section (L)). 
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(F) The Kazhdan problem: the difference between interesting and 
uninteresting mathematical structures.

Mark Steiner mentions that core mathematicians wouldn’t treat 
“the structure” of chess as a mathematical structure, ans cites 
Kazhdan. But that’s really a separate problem, about what it is that 
makes some structures more *interesting* than others. I don’t think 
they would say “Chess has no structure at all”. I think they would 
say “OK. It’s a structure, but a very boring one”.
I don’t know why some mathematical structures are more 
interesting to study in their own right than others. But it’s a bit like 
asking why physicists don’t study, say, pebbles on the beech. 
Pebbles are physical systems. Why not study them? Well, the 
fact that physicists don’t take much interest in studying pebbles 
doesn’t imply that they don’t, in general, study the behaviour of 
physical systems.
(By the way, I remember Feynman once describing in a TV 
interview how, one drunk night, he found himself making spaghetti 
and he noticed an interesting pattern in the way it snapped when 
bent. He spent a couple of hours trying to figure out physically why 
this happens, and then gave up). 
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(G) The Wigner problem: The “unreasonable applicability” problem

Mark Steiner discusses this problem in excellent detail in his book 
and in his posting. An example I would add to his list is Dirac’s 
analogy between Poisson brackets in classical mechanics, which 
are then taken over as commutation relations to quantum 
mechanics. In classical physics, {q, p} = 0. In quanatum physics, 
[Q, P] = ih.
[By the way, the quantum commutation relations [Q, P] = ih have 
no countable representations. The spectrum of the self-adjoint 
operators Q and P are forced (by the commutation relations) to be 
the whole real line. That demolishes the often-canvassed idea that 
quantum mechanics somehow suggest that space should be 
"discrete"].
Maybe the universe is mathematical in a way that is closely 
connected to the way that the human mind itself is mathematical, 
and so purely abstract structures that strike us as interesting to 
study (like SU(2)) are precisely those that appear in nature. But I 
really have no idea. It’s a big problem.
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(H) The Kronecker problem: The finiteness of the human mind.

The mathematician Leopold Kronecker was a famous early finitist, 
saying something like “God created the integers and the rest is 
Man’s work”. The allegedly finite nature of the mind is a massive 
topic, but Matt Insall asked about why we usually assume that the 
mind is finite (for the human brain takes up, say, 100 cubic cm of 
physical space, and that contains uncountably many spacetime 
points).
First, most agree these days that the mind is somehow “produced” 
by the brain. (But, of course, no one knows how). Second, the 
alleged finiteness of the mind is an empirical matter (and 
presumably contingent matter: we can imagine a possible world 
containing beings whose minds weren’t finite, and who could 
perform mental supertasks). But this is an assumption built into 
almost all discussions of the relation between the mind and 
computational models of the mind.
Turing himself (I think) said that, in developing the concept of a 
Turing machine, he had wanted to model the way an *idealized 
mind* would think when computing the solution to a problem, using 
pencil and paper. Flashes of “insight” or “intuition” are not allowed.

Chomsky (1957: Syntactic Structures) asked the question: how 
can the finite mind grasp an infinite number of meaningful 
sentences? He often mentions that this question had already 
occurred to von Humboldt, a 100 years before. One of his great 
insights was to use the notions of recursion and computability 
developed 20 years before by Godel, Post, Church and Turing. The 
set of meaningful sentences in humanly learnable language must 
be a decidable set. It is the standard view in theoretical lingusitics 
that a language containing a non-recursive set of meaningful 
sentences recursive would simply be unlearnable.
But it’s an empirical matter all the same.

Roger Penrose does believe that there are flashes of “insight” or 
“mathematical intuition” which cannot be modelled computationally 
or algorithmically. He cites the apparent human ability to recognize 
that the Godel sentence for Peano Arithmetic is true, even though 
G isn’t a theorem of PA. Of course, many, many people have 
written about this (including Martin Davis and even me), disputing 
the validity of this argument.
I don’t think that the fact that the human cranium encloses 
uncountably many spacetime points helps with this problem. There 
is a basic neuronal level of brain activity – neurotransmitters moving 
along synapses -- and I am very sceptical that the infinity in 
question could be “harnessed” by brain cells. Penrose thinks some 
funny quantum process, combing quantum gravity with a realistic 
interpretation of wavefunction collapse, is at work in generating 
human consciousness. Mmmm.
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(I) The Laplace-Poincare problem: predictability and computational 
tractability.

Laplace proposed that a creature which knew all the present 
conditions of the universe and its laws, could predict all future 
events and states. Poincare studied the predictability of n-body 
problems in classical phase space. There are many topics here, 
many of clear fom relevance. Joe Shipman has sent me his paper,

[12] Aspects of Computability in Physics

This is one area where there has been some important research. In 
particular, there is the work of Pour-El and Richards

[13] Pour-El and Richards 1987: Computability and Analysis in 
Physics

Results in this area are concerned with how (classical) dynamical 
systems evolve over time. In particular, Pour-El and Richards 
showed that for the conventional wave equation, a computable real-
valued function describing the initial data could evolve into a non-
computable function. I don’t know much more than that really.
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(J) Quantum computation

Matt Insall mentioned quantum computation in his posting. I don’t 
know too much about this topic either. David Deutsch – one of the 
pioneers of the idea -- has written a book (which I don’t possess 
unfortunately), including a discussion of this stuff, called

[14] David Deutsch 1998: The Fabric of Reality

Deutsch thinks that the possibility of quantum computation points 
towards to so-called many universes interpretation of quantum 
mechanics.
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(K) The Zeno Problem: supertasks in spacetime and infinity 
machines

I think there’s an even more interesting development than quantum 
computation. That’s the possibility of supertasks in physical 
spacetime, an idea which goes right back to Zeno’s paradoxes. A 
supertask is an infinite sequence of distinct operations, 
completed in a finite time. A Turing machine which doesn’t halt 
performs a supertask, but unfortunately takes infinitely long to 
“finish” its business (of course, there is no 
final state). “Counting” through all the numbers to see if and 
equation of the form f(n) = 0 has a solution is a supertask.

Recently a young philosopher at Cambridge, Mark Hogarth, has 
considered unusual space-times (now called Malament-Hogarth 
spacetimes) in which an infinitely long geodesic (i.e., which has an 
infinite proper time) can be viewed in its entirety by an observer 
over some finite proper time interval. Put the Turing machine M on 
this geodesic and wait until its “finished” (strictly speaking, the 
information you receive is either (i) M halted at some finite stage n 
or (ii) M actually didn’t halt at all. It is this latter information which 
is not available in the usual case). You have what Earman 
and Norton call an “infinity machine”. See:

[15] Earman and Norton 1996, “Infinite Pains: The Trouble with 
Supertasks”, in Morton (ed.) 1996. Benacerraf and his Critics.

This paper is an excellent introduction to supertasks, debunking 
some of the alleged paradoxes associated with supertasks, and 
discussing the possible implications of building “infinity machines”. 
In particular, using simple infinity machines, the decision problem 
for Pi^0_1 and Sigma^0_1 sentences is soluble, by getting the 
machine to run through all the numbers, looking for verifiers or 
falsifiers. For more complex formulas, you need an infinity of infinity 
machines, all chained together. Apparently this is discussed in,

[16] Mark Hogarth 1994: “Relativistic non-Turing machines and the 
failure of Church’s Thesis” (don't know where this appeared)

I’m not sure, but can’t the consistency of formal system be 
encoded as a Pi^0_1 sentence? If so, then presumably such a 
simple infinity machine could prove the consistency of any formal 
system, such as ZFC. The infinity machine simply determines the 
truth value of the Pi^0_1 sentence coding ZFC’s consistency. If the 
infinity machine doesn’t in fact halt (after its infinity of operations), 
then ZFC is, in fact, consistent.
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(L) Godelian phenomena in physics: Reverse physics?

Physical theories are really no different from mathematical 
theories, except that mathematicians have made much greater 
efforts to formalize and study mathematical theories proof-
theoretically and model-theoretically. In particular, physical 
theories can suffer from Godel-type incompleteness (if the universe 
is rich enough to contain a spacetime model of natural numbers, 
which is probably true). This means that GR, for example, will be 
deductively incomplete w.r.t. the sentences about spacetime. 
Adding say CH or a large cardinal axiom to GR may then generate 
some new consequences.

One might conceive of a kind of “reverse physics” with some basic 
geometrical spacetime theory Geom as its base theory. Then one 
might try to discover “reverse physics” results of the form:
	(i)   Geom  |- GH <-> A
Where A is some (preferably experimentally measureable) 
statement about spacetime. If we could determine, by experiment, 
that A is true, then that would be an excellent reason for accepting 
CH, even though a reason derived from physical experiment. 
This would be completely analogous to a standard reverse maths 
result, say:
	(ii)   RCA_0  |- ACA_0 <-> BW
Where BW is the Bolzano-Weierstrass Theorem.

Jeff Ketland
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>