Formal Constructing of a Physical Theory
Taking Spacetime as an Example
Thomas Benda
ABSTRACT. The concrete physical world, which is obvious to everyone, in fact allows
mental access only by aspects and in models. It is consequently argued that a physical
theory ideally talks about a formal model as a faithful representative of the physical
world and has a purely formal axiomatic base. A set-theoretical ontology is outlined.
A formal theory is then presented which has a subject rich enough to represent an
important aspect of the concrete world, namely, spacetime.
Motive of a formal construction
At first glance, there is nothing more evident than the ontology of the concrete,
physical world, which apparently consists of objects occupying space and extending
over periods of time, assigned properties and related to each other. On a closer look,
however, objects of the concrete world are hard to individuate, resistant to classification and seemingly impossible to reduce to primitives. Furthermore, as soon as we
undertake a few steps to classify and reduce physical objects, entities are generated
that are just theoretical, their existence being derived from the acceptance of certain
physical theories. Examples of theoretical objects are particles and fields, like the
electromagnetic field, atoms of certain types and elementary particles, and - possibly
and in a more subtle way - space and time.
The existence of physical objects, be they phenomenal or theoretical, presupposes a
realistic attitude, rightly called “robust realism”1, which we are willing to adopt
throughout this talk.
Physics did not achieve its famed advances before a quantitative approach was
systematically undertaken that maps physical objects on mathematical objects and
employs a mathematical language to express physical properties, which then may
suitably be called “physical magnitudes”. Physical laws mainly express relations
between physical magnitudes. Any physical magnitude is characterized by certain
mathematical properties, e.g., by being a real number, and having physical content. In
practice, physical theories are formulated in a mathematical language before a
background of intended physical meanings of objects and properties, then theorems
1
W.V.O. Quine, Ontological Relativity and Other Essays, Columbia University Press, New York
(1969).
are found and applied to certain parts and aspects of physical reality. Mathematical
representations, while usually restricted to certain contexts, have been hugely
successful for explaining the physical world and in turn guide human intuition for
understanding the concrete world, witness recent progress made in particle physics. If
we take mathematical representations of physical objects and properties serious, the
concrete and abstract worlds are more intertwined than it seems at first.
Yet success is not truth. A philosopher should be more interested in truth than in
success and may therefore resist the urge of a mathematical formulation of the
concrete world. However, no convincing non-mathematical path of our understanding
thereof has revealed itself so far.
With what has been said above, for mentally accessing the world we cannot but devise
a set of suitable models of the world. A physical theory codifies what we know about
an aspect of the physical world. It is - in a somewhat traditional definition - a set of
hopefully true statements partly infered from each other. If we say that theories are
there to give the world a law-like behavior and if we suppose the latter to be a
condition for experiencing and understanding the world, then to experience and
understand the world we need characterizing physical theories. This is less trivial than
it sounds. The world might also have a unique history, being no model of any laws,
with all recurring patterns just being of a statistical nature. This possibility has been
discussed in the context of quantum theory. However, we do not follow it up.
Preferably, the statements of a theory allow to be based on a set of primary statements,
i.e., axioms. Since the history of science advises us better not to aspire to find a theory
of the entire physical world, we may as well direct our attention only to physical
theories that are axiomatizable, that is, the true statements of which are infered from
an identifiable set of axioms. All axiomatizations of physical theories undertaken so
far have physical axioms and mathematical axioms and inference rules, which are
normally not stated, an approach that seems sensible since we do not know what
physical inference rules would be.
Axiomatizations of physical theories, even of classical physical theories, were undertaken only in the last few decades, when formal tools became available and interest in
the philosophy of science and foundational studies grew. No axiomatization of a
physical theory undertaken as yet has done without mathematical representations of
physical objects and properties. Interestingly, mathematical representations blur, even
call into question the distinction between physical objects and properties suggested by
natural language. We reasonably expect any axiomatization of a physical theory to
specify the mathematical representations used. Thus a set of axioms2 would contain a,
usually larger, part enumerating physical entities and stating their mathematical
properties. The remaining axioms would provide laws applying to those physical
entities. Of the former, a typical axiom of classical mechanics would state “m(p) is a
real number”, wherein m(p) is thought of, but not said to mean the mass of the object
denoted by p. This axiom apparently stipulates that the set of masses of all bodies we
can think of satisfies the known axioms of arithmetic of real numbers, implying an
arithmetic of masses. But we see that the question what physical objects and properties really are is left open, only the mathematical properties of physical objects are
talked about. This is less of a loss than it seems because, as noted above, we anyway
can think of physical reality only in terms of models.
Axiomatizations of the kind just described usually employ a semi-formal set-theoretical language similar to the customary language of mathematicians. A set-theoretical
language talks about a set of individuals, allowing to express all physical properties as
extensions and thus avoiding introducing of new primary notions. Regarding
semi-formality, it has indeed been argued3 that strictly formal languages employed in
physical theories are unnecessary, even distracting from a supposed less precise nature
of physics. In our view, however, keeping in mind that axiomatiations as described
above do not use concepts beyond set-theory and mathematics, there is nothing being
lost in using a strictly formal theory to express physical reality. As noted, we anyway
have no way of talking about the concrete world except in models which involve
universes of strictly individuated objects and their properties. There is no better way
to talk about these than in a formal language joined by a formal axiomatic system. If
we further agree that models of physical theories should be mathematical, it is but a
small step to insist on a set-theoretical model for the physical world, because the
set-theoretical universe is the unifying model of vast parts of mathematics.
Like the axiomatizations of the kind described above, we do not discuss the physical
meaning of any of the objects mentioned therein. In contrast to those axiomatizations,
we intend clearly and expressively a model which is a set-theoretical universe, in fact
the standard hierarchy of sets based on urelemente explained below. Physical objects
and magnitudes are defined, their mathematical properties are not given by axioms,
but are proved. Throughout this effort, a strictly formal language is used.
2
An influential axiomatization of classical mechanics is found in J.C.C. Mc Kinsey, P. Suppes,
"Axiomatic Foundation of Classical Particle Mechanics", Journal of Rational Mechanics and Analysis
2, p. 253 (1953).
3
See J.C.C. Mc Kinsey, P. Suppes, "On the Notion of Invariance in Classical Mechanics", British
Journal for the Philosophy of Science 5, pp. 290-302 (1954).
A set-theoretical universe, bare of physical meaning, may not seem the most appropriate way to stand for the physical world, and one may feel reluctant to employ this
Platonic entity as the chosen representative for the physical world. Like previous axiomatizations, however, we are content to see a set-theoretical universe as an adequate
representation of at least parts of the physical world and the theorems we will be able
to derive as a faithful reconstruction of physical laws. Certainly a thorough
set-theoretical formulation of physical properties helps to make these more precise.
A model which is a set-theoretical universe is simple and precise. It represents at least
aspects of a genuine physical world. For this it does not matter if we are Platonists. A
mental construct is no less a faithful representative than something real. Models like
these have been traded under the label of “formal reconstructions” of physics. We
prefer to be bolder and to talk of “formal constructions”.
A modest beginning of this undertaking is to represent the accommodating vessel of
all things concrete, spacetime itself, and formally to express its properties. This involves a strictly formal theory talking about a universe that represents spacetime and
in practice boils down to a formal reconstruction, including a formal axiomatization,
of the currently established theory of spacetime, the General Theory of Relativity,
which has not been undertaken so far.
In the following, we give a review of the standard hierarchy of sets, establish a logical
framework based on Zermelo's theory Z, and finally provide a theory called T which
allows to be interpreted physically as delivering spacetime.
The standard hierarchy of sets4
Formal theories are characterized by dealing with objects that have clear identities,
each being identical with itself and nonidentical with other objects. Any attempt to
define identity in a noncircular way does not succeed, yet we understand identity.
Apparently, only ideal, abstract entities have clear identities. Prime examples are
natural numbers and sets.
Like natural numbers, sets, that is, collections of objects to aggregates, are intuitively
grasped. Once we have accepted the notion of identity, but another small step is
needed to bring up two judgements that are both unprovable and unrefutable. Let us
call objects with clear identities “individuated” and let us call sets that are individuated, contain individuated elements only and are not the set of everything “extensionally determined”. The two judgements are:
(S1) Every set is uniquely determined by its elements and is individuated provided
all of its elements are individuated.
4
See U. Blau, Die Logik der Unbestimmtheiten und Paradoxien (in German), unpublished manuscript,
Munich (2000ff.).
(S2)
The collection of subsets of any extensionally determined set is extensionally
determined.
(S2) allows to form for every set S the set of subsets, the power set of S. Having declared two sets S1 and S2 uniquely by their elements according to (S1), a set containing elements of S1 or S2, i.e., the union set of S1 and S2, is readily formed. These
operations of forming power sets and union sets allow to construct a hierarchy of sets
by the following self-referential imperative:
(V0) Consider a set, in a special case, the empty set.
(V1) Consider the power set of the set considered in the last step and repeat (V1).
(V2) Consider the union of all sets thus considered and repeat (V1) and (V2).
Starting out from the empty set in step (V0) leads us to consider pure sets, i.e., the
empty set and sets generated therefrom by forming power sets and union sets. These
simple operations carry the set hierarchy further and further, approximating an
intended, but inexpressible set of everything, which is absolutely unlimited. The
standard model of the Zermelo-Fraenkel theory ZF is this hierarchy generated from
the empty set. Beginning from a broader base, a nonempty set, the elements of which
are called “urelemente”, leads to a hierarchy of sets, a small initial section of which
looks suitable to be the standard model of a physical theory.
The Platonist acknowledges the hierarchy of sets as real, and even the non-Platonist
will concede that the hierarchy of sets is a natural and simple mental construct, offering itself as a useful intended model of any theory with a set-theoretical language.
The spacetime theory T
We now discuss a theory called T, which delivers a construction of spacetime. The
theory T is the result of several steps of extending the classical bivalent logic, the
indispensable base of any stronger, applied logical system.
L  Z  ZU  T
 indicates strengthening and specifying,  indicates conservative extending.
L is the classical-bivalent logic with identity introduced in the usual manner. We take
 (negation symbol) (disjunction symbol) and V (existential quantifier) to be the
connectives in primitive notation. The axioms of L are (L1) to (L5), the inference
rules, (R1) to (R5).
Z is Zermelo's theory, which is obtained from L by specifying a single predicate
constant  and by introducing the additional set-theoretical axioms (M1) to (M7). We
leave out the replacement scheme of ZF as well as the axiom of choice. Z allows to
construct mathematics, as far as it is physically relevant, particularly calculus and
linear algebra. The hierarchy of sets outlined above is a standard model of the theory
Z. Natural and real numbers are sets within this hierarchy.
ZU is an extension of Z by adding the unary predicate constant W (“is urelement”)
and the axiom (M8). A nonempty set of urelemente requires that identity not be
defined.
The language of T, as compared to the language of ZU, has an additional ternary
predicate constant ∤ (“intersects at”). The axioms (P1) to (P9) are added to the axioms
of ZU. The theory T is formally described by a metatheory MT, which is ontologically
stronger than T by postulating the additional axiom (M9).
The next few paragraphs give a more technical description of what the theory T implies. With the axiomatic base (P1) to (P9), spacetime points are defined as sets of
worldlines, a topology on the set of spacetime points is defined, coordinate systems
are introduced, and it is proved that the set of spacetime points is a smooth Hausdorff
manifold. This is a somewhat lengthy, but straightforward process, which leads to the
usual starting point of spacetime theories, a manifold of spacetime points. Thus the
theory T, apart from being strictly formal, starts at a more basic level than previous
theories of spacetime. Having obtained a manifold, tangent vectors, which are crucial
for its geometrical characterization, are readily defined. Within the theory T, we are
able to define tangent vectors along worldlines, without resorting to a metric. It turns
out that tangent vectors along worldlines are precisely those that have positive coordinates in all coordinate systems, so that being a tangent vector along a worldline becomes an intrinsic property. Furthermore, this allows naturally to define a Lorentzian
metric tensor in a way that tangent vectors along worldlines and only these have
positive lengths in any coordinate system. In the intended interpretation, worldlines
(urelemente) are the formal primitive constituents of spacetime (a manifold of sets of
worldlines). The formal universe of T is a representation of spacetime as we know it
from the General Theory of Relativity. Indeed, in the intended interpretation of T, an
ab-initio axiomatization of the General Theory of Relativity has been found.
In an axiomatization along the lines given above, it would seem natural to make
spacetime points themselves primitive objects and have worldlines be sets of certain,
causally connected spacetime points. This is the point of view usually taken. An axiomatization like this would employ a mathematical language, which would be
semi-formal, but may be with some effort made formal; would state spacetime to be a
semi-Riemannian manifold; state worldlines to be chains of spacetime points with
positive distances between neighbors; characterize causally connected spacetime
points as being elements of the same worldline; and would finally state a list of laws.5
5
Spacetime points are primitive entities in J. Ehlers; R.A.E Pirani; A. Schild, "The Geometry of Free
Fall and Light Propagation". In: L.O. Raifeartaigh (ed.), General Relativity, Papers in Honour of
J.L. Synge, pp. 63-84. Oxford: Clarendon Press (1972). This paper, however, is not a strictly formal
axiomatization of the General Theory of Relativity.
Thereby, physical spacetime is mainly described in terms of its intended model,
making heavy use of a sophisticated mathematical language.
Translated to our attempt to use the hierarchy of sets as a model, we would make
spacetime points be (the interpretations of) urelemente and have worldlines be sets of
causally connected spacetime points.
The present work takes a different approach, regarding worldlines as being primary to
spacetime points, the latter being set-theoretically constructed out of the former. The
present approach acknowledges the importance of causal connection of spacetime
points and carries this further by giving worldlines their primary status. This allows
for a relatively small number of axioms and gives a law-like description of spacetime
in contrast to the model-like description sketched above. Furthermore, the - rather
technical but essential - tangent vectors are introduced without even having defined a
metric, making the theory T presuppose less and thus be somewhat more fundamental
than its predecessors.
It should be pointed out that, while a law-like description of physical reality is given
preference here, a model-like description has its virtues from a pragmatic point of
view. Given the short lifetimes of physical laws but the comparative stability of
intended models, a model-like description requiring less drastic changes looks
attractive to the scientific community.6
While physics is unthinkable without spacetime, the physical world with its material
objects obviously consists of more than just spacetime. In the General Theory of
Relativity, Einstein's field equations account for material qualities of the physical
world, like mass, energy and momentum, establishing a relation of these with the
geometry of spacetime. Indeed, Einstein's field equations may be taken as definitory
equations, reducing material qualities to geometry. In the context of the present work,
we see that those parts of physics that are successfully geometrized would allow for a
formal foundation. As is well known, with the arrival of quantum theory and its
overwhelming predicting successes, the program of the geometrization of physics has
stalled, so that the project of an axiomatization of physics in its entirety seems as
hopeless as ever.
Nevertheless, a faithful representation of spacetime - but no more than that - has been
obtained. On the one hand, this shows the expressive power of formal languages; on
the other hand it guides our intuition to get an ontological picture of an important part
of physics.
6
This point of view is adopted by Structuralists, see, e.g., J.D. Sneed, The Logical Structure of
Mathematical Physics, Dordrecht (1979); W. Stegmüller, The Structuralist View of Theories, Springer,
Berlin (1979).