Is symmetry identity?
ABSTRACT Wigner found unreasonable the "effectiveness of mathematics in the natural
sciences". But if the mathematics we use to describe nature is simply a coded
expression of our experience then its effectiveness is quite reasonable. Its
effectiveness is built into its design. We consider group theory, the logic of
symmetry. We examine the premise that symmetry is identity; that group theory
encodes our experience of identification. To decide whether group theory
describes the world in such an elemental way we catalogue the detailed
correspondence between elements of the physical world and elements of the
formalism. Providing an unequivocal match between concept and mathematical
statement completes the case. It makes effectiveness appear reasonable. The
case that symmetry is identity is a strong one but it is not complete. The
further validation required suggests that unexpected entities might be
describable by the irreducible representations of group theory.
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p. 1 Is Symmetry Identity
Is symmetry identity?
Physics emeritus, UCLA, Los Angeles, California, USA
Abstract Wigner found unreasonable the "effectiveness of mathematics in the natural
sciences". But if the mathematics we use to describe nature is simply a coded expression of
our experience then its effectiveness is quite reasonable. Its effectiveness is built into its
design. We consider group theory, the logic of symmetry. We examine the premise that
symmetry is identity; that group theory encodes our experience of identification. To decide
whether group theory describes the world in such an elemental way we catalogue the detailed
correspondence between elements of the physical world and elements of the formalism.
Providing an unequivocal match between concept and mathematical statement completes the
case. It makes effectiveness appear reasonable. The case that symmetry is identity is a
strong one but it is not complete. The further validation required suggests that unexpected
entities might be describable by the irreducible representations of group theory.
1. Effectiveness Speaks
p. 2 Is Symmetry Identity
In his famous paper entitled, "The Unreasonable Effectiveness of Mathematics in the
Natural Sciences", Eugene Wigner (1979, p. 223) wrote:
"... the enormous usefulness of mathematics in the natural
sciences is something bordering on the mysterious ... there
is no rational explanation for it."
Some think there might be a rational explanation. In his book The Applicability of
Mathematics as a Philosophical Problem, Steiner (1998, p. 5) finds that the use of
mathematics "cannot avoid being an anthropocentric strategy". It rests ultimately on the
human experience of nature. He is thus led to "explore the implication for our view of the
universe" (p. 2) of the evident applicability of mathematics to the physical world. Steiner
turns Wigner's plaint inside out asking what the evident effectiveness of mathematics tells us.
In what follows I will support and expand on this seminal notion.
Wigner was an acknowledged master of group theory, the mathematical theory of
symmetry. Laws of nature - physical laws - are governed by group theory. Bas van Fraassen
demonstrates this in his book, Laws and Symmetry (van Fraassen, 1989). He shows us that
the status of 'physical law' is conferred by symmetry - invariance under the transformations
But what is symmetry that it should underlay the very foundation of natural law?
Alternatively put: Why is group theory so effective in describing the physical world? The
answer is that it codifies the basic axioms of the scientific enterprise. The logic of group
theory is the logic of scientific inquiry so that the mathematics we use to describe nature is a
carefully coded expression of our experience.
Group theory is the mathematical formulation of internal consistency in the description
of things. We assume that the system being observed has an intrinsic character independent
of the observer's perspective. It's there. It possesses an objective reality. On this assumption
p. 3 Is Symmetry Identity
- that it's there - how the system is perceived under altered scrutiny must be a matter of
logic. Its appearance follows the logic of intrinsic sameness (Section 7). The codification of
that logic is a matter of group theory. And its success in portraying the physical world is what
validates the assumption.
Rather than ponder its efficacy we take instruction from the mathematics. We know -
by experiment - that the physicist's mathematical description of the physical world is
approximately correct. From the very efficacy of mathematics in describing it we may derive
a message about the nature of the physical world. It is this inversion of Wigner's quest that
we pursue here.
2. How symmetry is identity
I propose that, as used to describe the physical world, symmetry is so elemental that
it coincides with the concept of identity itself. The theory of symmetry is the mathematical
expression of the notion of identification and that is why it is so effective as the basis of
science. By identity is meant the end result of identification, not the other sense of the word
pertaining to identical.
That symmetry has played a substantive role in thinking about nature has a venerable
history dating back to antiquity. This history is nicely outlined by Roche (1987). He discusses
many examples. All of them demonstrate how symmetry considerations have helped solve
physical problems. For example Descartes deduces that "two equal elastic bodies which
collide with equal velocities rebound with these velocities exactly reversed because of
symmetry". (Quote from Roche, 1987, p. 18)
It was Ernst Cassirer who first articulated the idea that the mathematical theory of
symmetry - group theory - may transcend its problem-solving utility. Group theory "has not
p. 4 Is Symmetry Identity
only a mathematical or physical but a universal epistemological interest .." he wrote
(Cassirer, 1945, p. 273). The suggestion was that it has something to do with how we know;
how we evaluate perceptions.
We know objects by their properties. Constitution is what "confers to the carrier of a
set of properties the dignity of an object", says Elena Castellani (1998, p. 182). She
'constitutes' an elementary object of physics - electron, nucleon - from group theory by
showing that invariance under the spatio-temporal transformation group yields as
characterization of the object its energy, its linear momentum, its angular momentum and its
mass. To constitute something, then, is to assign to it labels of significance. The significance
arises from invariance properties.
This exercise exemplifies a broader view of symmetry. Although not explicitly stated
the idea is implied - as it is in the work of others (Van Fraassen, 1989 and Toraldo di Francia,
1998)- that symmetry may be literally equated with identity. To constitute is to identify. We
explore the notion that symmetry is identity; that group theory is the theory of identity.
If, indeed, group theory describes the world in such an elemental way then we must
be able to provide the detailed correspondence between elements of the physical world and
elements of the formalism. This is the way we accept Wigner's challenge to make reasonable
the effectiveness of the mathematics. To display associations between mathematical notions
and physical ones we need concise verbal expressions to match the concise mathematical
ones. Expressions which generate images are the ones that make the mathematics
Using non-technical language, here is a synopsis of the case. The essential quality that
characterizes symmetry is this: the appearance of sameness under altered scrutiny. That this
definition corresponds to group theory as applied to the physical world is grounded in Section
8. But the same phrase - a perceived sameness under altered scrutiny - is just what captures
p. 5 Is Symmetry Identity
the notion of identity. When something is recognizably the same under many perspectives we
grant it identity. An identification is made by labelling. The label tags what it is that we
perceive as invariant. "If there were no invariants we could not define 'identity'" noted M.L.G.
Redhead (1975, p. 78). Here we posit that it is precisely invariance that identifies identity.
Group Theory has an innate taxonomic structure - a taxonomy for behaviour. It
assigns labels for behaviour (appearance) under altered scrutiny. The irreducible
representation labels of Group Theory are the identification markers - the labels of identity.
This is explored in Sections 9, 11 and 12. A central concept in this exposition is altered
scrutiny. It is examined in Section 4. In applications to the physical world it is precisely
altered scrutinies that are the group elements.
To ground the case synopsized in the preceding paragraphs we use a notation that
iconizes the philosophical content: Dirac notation. It is the natural tool for the group theory of
physical processes. What follows is a review of material that every group theory scholar
knows, but recast so as to expose how the mathematics encodes our axioms about how
nature works. To do this we focus on the correspondence between the physical world and the
3. The observer, the system, its states and its rules
In examining anything one is an observer. So in asking questions about nature we
focus on what an observer might find. The observer divides the world into distinguishable
parts. Each one, perceived as a discernibly distinct entity, may be called a 'system'. The
system is what is studied by the observer; something accessible to measurement. The
difficulty about the concept lies in the matter of isolation; that the system not be coupled to
the rest of the universe. Evidently such systems are not to be found in nature. It is an