The other day, a friend of mine asked if I could tell him what light is.

A loaded question. One I can answer in two distinctly different ways.

Observationally, light is what you sense with your eyes. No further explanation required (at least so long as your eyes are functioning well): it is something you can directly experience. Even a caveman knows exactly what light is.

Of course my friend asked me not in my capacity as a caveman, but as the amateur physicist that I fancy myself to be. And physics, theoretical physics especially, concerns itself with one goal: finding the mathematical language that best matches the physical world around us.

Mathematics has many branches. Not everyone knows that we can derive all of mathematics: everything from number theory, algebra, calculus, geometry, you name it, from a very small number of basic concepts. One possible starting point is set theory.

When we start from set theory, we really begin with the following building blocks:

- The concept of a set
- The concept of "being a member of a set"
- The rules of formal logic, in particular being able to formulate "compound statements" such as "
*A*and*B*" where*A*and*B*are both statements (e.g., "*the sky is blue*and*it is raining*") and determine their truth value from the truth values of the individual statements.

With these basics, we can proceed as follows:

- We can identify members of one set with members of another set.
- We can make this identification one-on-one, i.e., match a member in one set with exactly one member in the other set and vice versa.
- If there are no "leftovers", we say that the
*cardinality*of the two sets is*equal*. - If there are leftovers in one set, we say that its cardinality is
*greater than*that of the other set. - We call the cardinality of the empty set (the set with no members)
*zero*. - We put all other sets into
*equivalence classes*: sets of the same cardinality belong to the same equivalence class. We call these equivalence classes*natural numbers*. - We find that there's an equivalence class with cardinality that is greater than zero, but not greater than the cardinality of any other set. We call this
*one*. - Similarly, we can define cardinalities of
*two*,*three*, and so on. (We now have number theory.) - We take two sets with no common elements. We call the cardinality of their union set the
*sum*of their individual cardinalities. We call the operation that identifies the cardinalities of the individual sets with the cardinality of the union*addition*. - We find that addition satisfies certain properties: it's
*commutative*(*x*+*y*=*y*+*x*), it has a*zero element*(*x*+ 0 =*x*), and it's sometimes*invertible*(?*z*:*x*+*z*=*y*). - When addition is not invertible among the natural numbers (what number do you add to 5 in order to get 3?) we define new numbers: the
*negative numbers*. Specifically, we define –*x*to be the number that satisfies the equation,*x*+ –*x*= 0. We find that there's a unique way to do this, while preserving all the nice properties of addition. Addition is now fully invertible: we have*subtraction*. - We define, through repeated addition,
*multiplication*. - Once again we find that the operation is not always invertible, but we can uniquely define new numbers, the
*rational numbers*, and now we have*division*. (And we now have high-school algebra.) - We find that the rational numbers form a
*dense set*(between any two rational numbers there are more rational numbers) but that there are sequences of rational numbers whose limit is not rational. We add these new numbers and call them*irrational numbers*. Together, the rational and irrational numbers form the*continuum*of the*real numbers*. - We call a set that can be subdivided into parts that can be mapped by one or more real numbers (i.e., you can cover the set with one or more
*coordinate charts*) a*manifold*. (And we now have coordinate geometry.) - On a coordinate chart, we call an arrow with a length and a direction a
*vector*. We find ways to attach vectors to points in any manifold, not just to the coordinate charts. When we attach a vector (or a number, or some other quantity) to each point in a manifold, we call this a (number, vector, etc.)*field*. (Not to be confused with the algebraic concept of a field.) - Whatever quantities (numbers, vectors, etc.) form a field, we can find ways to measure their "rate of change" as we move from point to point: i.e.,
*differentiation*. (We now have calculus.) We cannot always perform differentiation, but we can sometimes even differentiate the result of a differentiation a second time. When a field can be differentiated an arbitrary number of times, it is called*smooth*. - There are many operations with vectors. In particular, we have one that's called the
*exterior product*. The result of the exterior product is an antisymmetric*tensor*. We can also form the exterior product of a tensor with a vector, or with another tensor. - We can also compute the exterior product of a vector with the
*differential operator*. We find that computing the exterior product with the differential operator twice yields a quantity that is identically zero. - We can define another operation between vectors and tensors, called the
*interior product*.

So far, this is mathematics: specifically, all this stuff is constructed from the concept of a set, being a member of a set, and formal logic. (Never mind the English-language scaffolding I used in this brief overview, this can actually be done without resorting to any English-language explanations. For instance, I could build all this using a very simple LISP-like language that models set theory.)

Now comes the punch line when we turn towards physics:

- We assume that the world we live in has the geometry of a four-dimensional manifold.
- We assume that on this manifold, there exists a smooth four-dimensional vector field. Computing the exterior product of the differential operator and this vector field, we can form a tensor field whose components we label
*E*and*B*. We find that these will accurately correspond to what we call the*electric and magnetic fields*in our observations. - Computing the interior product of the differential operator and this tensor gets us two equations that define two quantities, which we call
*charge density*and*current*. Further differentiation shows that these form a conserved quantity: charge is conserved. - Computing the exterior product of the differential operator and our tensor produces zero; this gives us two more equations. Together, these four equations are known as
*Maxwell's equations*. - One particular class of solutions to these equations is plane waves moving with unit velocity. Experiments show that these waves accurately correspond to what we know as
*electromagnetic radiation*, specifically light.

In other words, what we call light is just the geometric properties of a 4-vector field over a 4-dimensional spacetime manifold.

Well, that should serve as an outline. If you wanted to carry out this derivation in a formal way, it'd probably fill up a thick volume or two. That said, it might be an interesting exercise to actually construct a LISP-like language to do this. Then again, for all I know, perhaps it has already been done.