The Deeper Roles of Mathematics in Physical Laws

Abstract

Many have wondered how mathematics, which appears to be the result of both human creativity and human discovery, can possibly exhibit the degree of success and seemingly-universal applicability to quantifying the physical world as exemplified by the laws of physics. In this essay, I claim that much of the utility of mathematics arises from our choice of description of the physical world coupled with our desire to quantify it. This will be demonstrated in a practical sense by considering one of the most fundamental concepts of mathematics: additivity. This example will be used to show how many physical laws can be derived as constraint equations enforcing relevant symmetries in a sense that is far more fundamental than commonly appreciated.

Familiarity breeds the illusion of understanding

–Anonymous

Technical Endnotes

This technical section provides the formal definitions of a partially-ordered set (poset) and a lattice as well as short derivation of the generalized sum rule for non-exclusive join.

A partially-ordered set (poset) P is a set of elements S along with a binary ordering relation, generically denoted $\le$, postulated to have the following properties for elements $x, y, z \subseteq P$

$$\begin{aligned} \begin{array}{lll} &{}\mathrm{For}\,\mathrm{all}\,\, x, x\le x &{}\qquad \qquad \qquad \mathrm{Reflexivity}\\ &{}\mathrm{If}\,\, x \le y \,\, \mathrm{and} \,\, y \le x, \,\, x = y &{}\qquad \qquad \qquad \mathrm{Antisymmetry}\\ &{}\mathrm{If}\,\, x \le y \,\, \mathrm{and}\,\, y \le z, \,\, x\le z &{}\qquad \qquad \qquad \mathrm{Transitivity} \end{array} \end{aligned}$$

A poset $P = (S,\le )$ is referred to as a partially ordered set since not all elements are assumed to be comparable. That is, there may exist elements $x, y \subseteq P$ where it is neither true that $x \le y$ nor that $y \le x$. In these situations, we say that x and y are incomparable, which is denoted $x\, \Vert \, y$.

A lattice L is a poset where each pair of elements has a supremum or least upper bound (LUB) called the join, and an infimum or greatest lower bound (GLB) called the meet. The meet of two elements $x,y \in L$ is denoted $x \wedge y$ and the join is denoted $x\vee y$. The meet and the join can be thought of as algebraic operators that take two lattice elements to a third lattice element. It is in this sense that every lattice is an algebra. The meet and join are assumed to obey the following relations

$$\begin{aligned} \begin{array}{lll} &{}x \vee y = y \vee x &{}\qquad \qquad \qquad {\mathrm{Commutativity}}\\ &{}x \wedge y = y \wedge x \\ &{}x \vee (y \vee z) = (x \vee y) \vee z &{}\qquad \qquad \qquad {\mathrm{Associativity}} \\ &{}x \wedge (y \wedge z) = (x \wedge y) \wedge z \\ &{}x \vee (x \wedge y) = x &{}\qquad \qquad \qquad \mathrm{Absorption}\\ &{}x \wedge (x \vee y) = x \end{array} \end{aligned}$$

Lattice elements (and poset elements in general) can be quantified by assigning a real number (or more generally a set of real numbers) to each element. This is performed via a function v called a valuation, which takes each lattice element to a real number: $v:x \in L \rightarrow \mathbb {R}$.

Fig. 3

(Left) An illustration of the join, $x \vee y$, of two disjoint elements x and y resulting in an additive measure. Disjoint elements have a null meet so that technically this structure is known as a join semi-lattice. (Right) The situation considered in the derivation of the Sum Rule, which applies to general cases

Valuations are meant to encode the ordering of elements in the lattice and this is accomplished by insisting that for $x\le y$ we have that $v(x) \le v(y)$. Furthermore, if the valuation is to encode the relationships among the elements, it must be that the valuation $v(x \vee y)$ assigned to the join of two disjoint elements x and y (Fig. 3, left) can be expressed as a function of the valuations v(x) and v(y) assigned to those two elements. We write this as

$$\begin{aligned} v(x \vee y) = v(x) \oplus v(y). \end{aligned}$$

where the operator $\oplus$ is to be determined. The concept of a generally-useful quantification by valuation (non-degenerate ranking) implies that the operator $\oplus$ obeys a cancellativity property for disjoint elements x, y, and z, where for $v(x) \le v(y) \le v(z)$ we have $v(x) \oplus v(z) \le v(y) \oplus v(z)$. Commutativity of the lattice join requires that the operator $\oplus$ is commutative, and associativity of the lattice join requires that the operator $\oplus$ is associative. The associative relationship represents a functional equation, known as the Associativity Equation, for the operator $\oplus$, whose solution is known to be an invertible transform of additivity [1, 4, 10], which can be written as

$$\begin{aligned} a\oplus b = f^{-1}(f(a) + f(b)), \end{aligned}$$

where the function f is an arbitrary invertible function. In terms of the valuations this is

$$\begin{aligned} f(v(x) \oplus v(y)) = f(v(x)) + f(v(y)) \end{aligned}$$

which is

$$\begin{aligned} f(v(x \vee y)) = f(v(x)) + f(v(y)). \end{aligned}$$

This suggests that one can always choose a simpler quantification than the valuations v by instead assigning values u(x) defined by $u(x) = f(v(x))$ so that $\oplus$ transforms to simple addition for disjoint x and y:

$$\begin{aligned} u(x \vee y) = u(x) + u(y). \end{aligned}$$

Recall that this result holds only for disjoint elements (in a join semi-lattice). We now derive the result for two lattice elements in general. Consider the elements $x \wedge y$ and z illustrated in Fig. 3 (right). Since their join is y, we have that

$$\begin{aligned} u(y) = u(x \wedge y) + u(z). \qquad (\mathrm{A1}) \end{aligned}$$

Next consider that elements x and z are disjoint and their join is $x \vee y$. This allows us to write

$$\begin{aligned} u(x \vee y) = u(x) + u(z). \qquad (\mathrm{A2}) \end{aligned}$$

Solving (A1) for u(z) and substituting into (A2) we have the Sum Rule

$$\begin{aligned} u(x \vee y) = u(x) + u(y) - u(x \wedge y), \qquad {\mathrm{Sum}\, \mathrm{Rule}} \end{aligned}$$

which holds for general elements x and y [8–10].