- John D Barrow, "Theories of Everything" (1990 - since updated and re-released as "New Theories of Everything")
I've been trying for awhile to understand this connection between groups and symmetry.
Mathematical groups were (and still are) (sadly) blandly (i.e. axiomatically) defined in textbooks, as consisting of a set of objects, and a binary operation that combines pairs of these objects to make product objects also in the set - for example the objects could be numbers and the operation could be addition : 2 + 3 = 5. One of the elements of the set is a distinguished one, called the identity - for example, "0" ; and for each element there is a unique partner called its inverse, and when these two are combined they yield the identity. For example 3 + -3 = 0. And the associative law holds when we write statements that combine in turn 3 or more objects. For example 2 + 3 + 4 = (2 + 3) + 4 = 2 + (3 + 4).
While seeming plausible if somewhat pointless, as a conceptual cartoon highlighting some basic properties shared by addition, multiplication and similar algebraic processes, there was no hint that these "groups" had anything especially to do with symmetry, or any clue about why they might be so important not just in mathematics but beyond, in physics and maybe even further afield.
I'm a barking mad collector of introductory maths books, so lets have a look at how a few of these present groups, and how they explain the relationship with symmetry :
"Introduction to the Theory of Finite Groups" by Walter Lederman , 1961, - a lucid down-to-earth favourite, but the word "symmetry" does not even appear in the text at all as far as I and the index can see.
On the other hand another '60s text "A Brief Survey of Modern Algebra" by Garrett Birkhoff and Saunders MacLane, from 1965, begins its chapter on Groups with...
"1. Symmetries of the Square"
"...the idea of "symmetry" is familiar to every educated person. But fewer people realize that there is a consequential algebra of symmetry....".
Moving on to the 1970's - Herstein's "Topics in Algebra" (1975) describes groups as "one of the fundamental building blocks for the subject today called abstract algebra" - but does not mention symmetry.
And then a more modern introductory textbook - "Contemporary Abstract Algebra" (2002)by Joseph A. Gallian. This has very many examples of geometric symmetry in the chapter "Introduction to Groups", starting with the symmetries of the square as did Birkhoff and MacLane - i.e. reflections and rotations that map a square onto itself.
Finally, the Princeton Companion to Mathematics (2008), introduces groups in a section headed "Four Important Algebraic Structures" :
"...wherever symmetries appear, structures known as groups follow close behind".
So of these 5 books, 3 explicitly make the link from symmetry to group theory - but none actually makes the opposite link : is it also the case that wherever structures known as groups appear, symmetries follow close behind ? Are "symmetry" and "group theory" coextensive ?
One answer seems to be provided by "Representation Theory", and the answer is "yes" - wherever groups appear, symmetries do follow close behind. This is because for any group, "Group Representation Theory" shows how the elements of the group can be interpreted ("represented") as matrix operators in a linear algebra on a vector space, in such a way that the multiplication of these matrices precisely mirrors the composition of the elements of the group. And these linear matrix operators are just reflections, rotations etc. In a sense the group provides the syntax of symmetry - the internal structure of it - and the representation of the group as a set of matrices provides the semantics by which we see the group structure at large, as rotations , reflections etc, of actual spatial figures.
(It seems to me that even in introductory text books, it would be an idea to demystify this relationship between groups and symmetry by making at least a passing reference to the enterprise of representation theory)
My bit of mathematical phenomenology
However it seems to me that considerations of symmetry in algebra arise at a more fundamental level, than is suggested by the examples such as geometrical symmetries of the the square, as provided by the technology of group representation theory.
A bit of phenomenological navel gazing (see below) at what is going on as soon as we write down any sentence in algebra, reveals implicit assumptions of underlying symmetry, and that the group concept is part of the fabric of algebra, rather then being a garment built from that fabric, as suggested by the textbook axiomatisation.
And I conclude that groups are essentially about composition of "operators" - and the basic concept of an operator is a very interesting and powerful one - a thoroughgoing "operator phenomenology" is an intellectually stimulating way of thinking not just in mathematics and physics but beyond.
A simple example from arithmetic - the operator-phenomenological recasting of
2 + 3 = 5
(+2) o (+3) = (+5)
(where I am using the o symbol to mean composition of operators)
- i.e. , "the plus two operator composed with the plus three operator gives the plus 5 operator". The "group operation" is not really addition , or multiplication, or rotation - it is always composition. It is the members of the group - operators - that increment and decrement, multiply, rotate.
The (on reflection) fascinating, distinctive , unexpected thing about "operators" is that they can be studied and manipulated independently of any individual subject or situation being operated on - they completely depart their substrate (numbers to be incremented and decremented, squares to be rotated...) and we study their internal interactions without any reference to a substrate. This "operator phenomenology" is obscured in the usual examples of groups such as integers under addition - because numbers are not obviously operators. It is much more obvious in group examples such as symmetries of a square, where the elements of the group are quite clearly operators - rotation and reflection- acting on a symmetrical substrate - squares, hexagons, circles etc.
Now for the humble phenomenological navel gazing that leads to these conclusions.
Algebraic sentences such as
c = a + b
e = c + d
c = a ^ b
e = c ^ d
(for whatever ^ we choose to define)
e = a * b * d
by their very nature describe a process that has a subtle inherent symmetry - that is , the process involves a transformation, that leaves some fundamental aspect of the transformed substrate unchanged (- a reasonable definition of symmetry).
The symmetry inherent in these statements, consists in the fact that the products of the process are identically qualified to be inputs to further processes - i.e. simply, we can take the product of the process "a" + "b", and feed it into a new process of adding "d". While the process of adding "b" to "a" does transform "a" into something else, that "something else" is precisely concordant with "a" in the way that it may be further transformed exactly in the way that "a" was.
(....of course this is exactly what algebra is. As Herstein puts it, to operate algebraically is to "combine two elements of a set....to obtain a third element of the set". This seems to be a fairly trivial and obvious thing - but is actually a much more exotic and unlikely activity than it appears to be at first sight - it is extremely rare if not completely unknown, to see anything behaving algebraically in the natural world. Two atoms do not (normally) combine to make another atom (except in a fusion reaction) - they combine to make something else - a molecule. Language does not in general behave algebraically - two words do not combine to form another word, they combine to form a phrase, which is a different type of thing)
So - there is an inherent symmetry as soon as we do any algebra at all, and indeed we will find that the sorts of real world processes that may be usefully modelled by these types of algebraic statements, do posses clear symmetries not possessed by substrates that may not be so modelled.
Thus if "a" is a 10 degree clockwise rotation of a hoop about its center, and "b" is a 20 degree clockwise rotation, we can let c be (say) a 30 degree clockwise rotation and this working model makes perfect sense as a realisation of these algebraic statements, precisely because of the rotational symmetry of a hoop - a hoop rotated about its center is just as good a place from which to embark on further rotations, as it was at the start.
By contrast if, say, we let "a" = a bolt and "b" = a nut and try to express composing these algebraically : "c=a+b" == "the nut screwed onto the bolt", this does not work since we cannot go on to write c+d - there is no symmetry associated with this interpretation - the product of the two elements is different to either of them - there is no symmetry in the substrate available here, to permit algebra to be done.
But there is a refinement we need to make.....the symmetry that our phenomenological navel gazing perceives in the fabric of any algebraic sentence, may not necessarily be a global symmetry, preserving the same substrate no matter which operators are involved and allowing us to compose any two elements to obtain a third - it may only be a local symmetry allowing us to compose some terms. When we can combine any two terms , then these terms are truly operators, and there is a global symmetry ; by contrast when only some compositions of terms makes sense, then these terms are what we might call morphisms, and we are not doing algebra, though we may use formulations that look algebraic.
Consider for example the morphisms depicted by the diagram :
* -- a --> *
| \ b
e c |
| \ | |
| - . v
*<-- d -- *
- we may write as above
c = a + b
(i.e. c is the directed diagonal from upper left to lower right, and gives the same result as "a" followed by "b")
e = c + d
but we cannot write a + d - these two morphisms cannot be combined.
(And we cannot write a + a - a morphism cannot in general be composed with itself - unless it is the loop-back morphism that goes back to its starting point.)
This is because the symmetry inherent in these sentences is now just a local symmetry - the transformation a + b takes us to a place that looks locally the same as the place we started from - a point with incoming arrow and outgoing arrows - and from which we can embark on some further morphisms, but not any morphism. This is because there is not a global symmetry inherent in these sentences - when we zoom out and look at the whole picture, a + b has moved us to somewhere that looks different to the place we started from.
So there is this fundamental thing about operators , that distinguishes them from morphisms, and it is that thing that gives groups their distinctive nature, and that explicates the connection of groups with symmetry : an operator does not care where it starts from or where it sends something to , whereas a morphism does. Another way of putting it is that in a system of operators, such as a group, any substrate that one may provide for these operators to operate on - e.g. a square under rotation and reflection, a number system under addition and subtraction - must be mapped identically back to itself in some sense by any and all operators, simply because the result of any operation must be indistinguishably suitable as a jumping off point for further operations by any other operator.
"Where algebraic sentences are written then operators are not far behind"
"Where operators appear then symmetry and groups are extremely close"
I feel that the relationship between groups and symmetry is a little deeper and more interesting - less accidental - than is suggested by a cursory reading of the usual group axiomatics. And something of the flavour of the idea of a group is lost when we insist on the fully abstract set-based axiomatisation, and refuse to call the elements of the set "operators".
A few interesting (to me) questions come up from this :
* Is a system of morphisms almost a system of operators ? - can we incrementally augment a system of morphisms - i.e. , a directed graph - attaining a series of intermediate structures , which concludes in a fully globally symmetric system of operators forming a group ?
* Is there any graph-like diagram that can be used to depict a group ?
* What sorts of systems of operators are there, that are not groups ?
* Can we develop in logic an operator-based system of predication ?
...Since - I claim that the symmetry inherent in writing down algebraic composition of operators, consists essentially in the way the results of the composition may in turn be composed with other operators - as though there is some substrate being operated on that remains invariant under the operation, and so is identically available for further operation.
...and one way to think of this would be , as though the operators were predicates - "blue", "heavy", "sharp", and then the invariant substrate is the subject being described.
...and so an operator-based formulation of predication then would look something like
Blue(Heavy(Sharp(x))), rather than the predicate calculus formulation
Blue(x) and Heavy(x) and Sharp(x)
...and we may compose operators to obtain other operators in an interesting way - for example (maybe)
Heavy(Light(x)) = Does Not Exist(x)
* Physics tends to involve systems of operators; biology systems of morphisms ? (I'm thinking of networks of gene regulation etc). That is why Biology is "less symmetrical" than Physics ?
* Quantum physics tends to involve systems of operators, groups and much symmetry ; General Relativity systems of morphisms, without a group structure, and little symmetry ?
* Computer programming languages typically allow us to write sentences in various algebras, i.e. operator based ( - e.g. all languages support Boolean algebra ; most linear algebra , over various fields - real , and sometimes complex). Could we conceive of languages that are augmented to allow us to calculate with morphisms rather than operators ? Would this be useful for biology and bioinformatics ? (for example for doing graph transformations and calculations)
* Appetites and the self are fundamentally morphism-like things ; ethics and universal values are fundamentally operator-like things - "good", "evil" - that may be composed and studied independently of the self ? Can fundamental mathematical concepts inform moral philosophy - for example is there an ethics group, that summarises the pattern of interaction between fundamental operator-like moral terms ?