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Marvin Chester

Citation: Chester, M. (2002) Is symmetry identity?
International Studies in the Philosophy of Science 16, 111-124


Wigner found unreasonable the "effectiveness of mathematics in the natural sciences". But if the mathematics we use to describe nature is simply a carefully 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 constitutes 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

In his famous paper1 entitled, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", Eugene Wigner wrote:

"... the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious ... there is no rational explanation for it."

Mark Steiner thinks there might be a rational explanation. In his book2 "The Applicability of Mathematics as a Philosophical Problem", Steiner finds that the use of mathematics "cannot avoid being an anthropocentric strategy"3. It rests ultimately on the Human experience of nature. He is thus led to "explore the implication for our view of the universe"4 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"5 . He shows us that the status of 'physical law' is conferred by symmetry - invariance under the transformations of nature.

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 theoy 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 posseses an objective reality. On this assumption - 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 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.

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 Roche6. 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"7.

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 only a mathematical or physical but a universal epistomological interest .." he wrote in 19458. 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 Castellani9. 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 explicitely stated the idea is implied - as it is in the work of others10 - 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.

What is the case for this idea? Is it true? There is a good case for it but it may not be true!

If, indeed, group theory describes the world in such an elemental way then we must be able to provide the detailed correspondance 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 reasonable.

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 captures group theory as applied to the physical world is grounded in Section 8 of the paper.

But the same phrase - a perceived sameness under altered scrutiny - is just what captures 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'"11 noted M.L.G. Redhead. Here we posit that it is precisely invariance that identifies identity.

Group Theory has an innate taxonomic structure - a taxonomy for behavior. It assigns labels for behavior (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 review the elemental structure of the mathematics in 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 correspondance between the physical world and the formalism.

3. The Observer, the System, its States and its Rules

This and all the other thirteen sections have not been posted. A pdf of the full paper may be downloaded here. The paper appears in International Studies in the Philosophy of Science vol. 16, pages 111-124, 2002.

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February 2012
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