My Incomplete Measurement Theory

Based on the equivalence relation and the homomorphism

Figure 1. Two armed balance scale(taken by the author in Ho Chi Minh City)

More than 15 years ago, I was asked to give a talk to an audience of CEOs of calibration service companies. The subject was to be general to help them understand measurement from a fundamental point of view. Having been involved in metrological research on radiation thermometry, photometry and radiometry, I was unaware of a measurement theory. Although I understood measurement only as assigning a number to a quantity, I wanted to explain more fundamentally how we can construct the scale of a quantity such as mass, length and time. Based on my previous article, my understanding of the theory is that of the 1900s. Although I am ashamed that I was brave enough to give the lecture, some issues are still worth considering. I want to describe the incomplete theory in this article. Nevertheless, I will first describe the essential mathematical background and then apply the mathematics to a mass problem. Finally, I will explain what makes my theory incomplete and how to improve it.

Basic mathematics

At that time, I looked at some basic mathematics that could be a key to my problem. The math ranges from the equivalence relation to algebraic homomorphism.

Binary relation
Given sets X and Y, the Cartesian product X ☓ Y is defined as {(x, y): x ∈ X and y ∈ Y}, and its elements are called ordered pairs. A binary relation R over sets X and Y is a subset of X ☓ Y, where the set X is called the domain or departure set of R, and the set Y is called the codomain or destination set of R. The statement "(x, y) ∈ R" reads "x is R-related to y" and is denoted by xRy. The domain of definition or active domain of R is the set of all x such that xRy for at least one y. The codomain of definition, active codomain, image, or range of R is the set of all y such that xRy for at least one x. The field of R is the union of its domain of definition and its codomain of definition. A binary relation is called a homogeneous relation when X is equal to Y. A binary relation is also called a heterogeneous one to emphasise that X and Y may differ.

An example of a binary relation is the “divide” relation over the set of prime numbers P and the set of integers Z, where each prime p is related to each integer z that is a multiple of p. For instance, the prime number 2 is related to numbers such as −4, 0, 6, 10, but not to 1 or 9, just as the prime number 3 is related to 0, 6, and 9, but not to 4 or 13. Note that the order of the elements in a binary relation is important; if x ≠ y, then yRx can be true or false independently of xRy, just as 3 divides 9, but 9 does not divide 3.

Equivalence relation

A binary relation ~ on a set X is considered an equivalence relation if and only if reflexive, symmetric and transitive. That is, for all
a ~ a (reflexive)
a ~ b if and only if b ~ a (symmetry)
if a ~ b, and b ~ c, then a ~ c (transitive)

Examples of binary relations that satisfy equivalence relations:

- “is equal to” for the ratios of any two elements in the set of integers: for instance, 1/3 ~ 2/6.

- “Has the same birthday as” on the set of all people.

— Given a natural number N, “is congruent to, modulo n” on the integers: for instance, 38 ~ 14 (mod 12)

Example relations that are not equivalences:

- The relation “≥” between real numbers is reflexive and transitive, but not symmetric. For example, 7 ≥ 6 but not 6≥ 7.

- The relation “is approximately equal to” between real numbers, even if more precisely defined, is not an equivalence relation, because although reflexive and symmetric, it is not transitive, since multiple small changes can accumulate to become a big change.

- The relation “has a common factor greater than 1 with” between natural numbers > 1, is reflexive and symmetric, but not transitive. For example, (2, 6) have a common factor 2 > 1, and (6, 3) have a common factor 3 >1, but (2, 3) have no common factor > 1.

Equivalence class
Given an equivalence relation ~ on a set X, a subset of X satisfying the equivalence relation ~ for an arbitrary a in X is defined as an equivalence class under ~, denoted [a], is defined as [a] = {x∈X: x ~ a}. The equivalence class have properties as follows:

• For each a ∈ X, a ∈ [a]
• For each a, b ∈ X, a ~ b iff [a] = [b]
• For each a, b ∈ X, [a] = [b] or [a] ∩ [b] = { }

The first property listed above means that each element belongs to its class. The second means that if and only if the classes of two elements are the same, the two elements are in an equivalence relation. The last means that any two classes are equal or their intersection is empty. Therefore, the union of all classes gives the original set X.

For example, on the set X = {a, b, c}, the relation R = {(a, a), (b, b), (c, c), (b, c), (c, b)} is an equivalence relation. The following sets are equivalence classes of this relation: [a] = {a}, [b] = [c] = {b, c}. So the set of all equivalence classes for R is {{a}, {b, c}}.

Partition
If X is a non-empty and P is a family of sets that is a subset of X, then P is defined to be a partition of X if it satisfies the conditions as follows:

• For each V ∈ P, V ≠ { }
• For each x ∈ X, there exists a V ∈ P such that x ∈ X
• For each x ∈ X, there exists a V ∈ P such that x ∈ X.
• For every V, W ∈ P, V = W or V ∩ W = { }

Expressing the conditions listed above in our language, the first is that a family of sets does not contain a empty set. The second is that there is at least one subset of a family of set containing an element in the family of sets. The third is that two subsets of a family of sets are either the same or their intersection is a empty set.

Partition and equivalence relations
Without going into detailed proof, the three properties of the equivalence class mentioned above satisfy the three conditions needed to define a partition. It means that a family of sets obtained by applying some equivalence relation ~ to a non-empty set X partitions the set X.

For example, considering a binary operation ‘x mod 3’ (the remainder of x divided by 3) on an element x of the set of integers Z, the elements in {0, 3, 6, 9, …} are in an equivalence relation such that all remainders are 0. Similarly, {1, 4, 7, 10, …} are in an equivalence relation such that all remainders are 1, and {2, 5, 8, 11, …} are in an equivalence relation such that all remainders are 2. They can be written in equivalence class notation as follows:

[0] = {x ∈ Z; x ~ 0 (mod 3)} = {0, 3, 6, 9, …}
[1] = {x ∈ Z; x ~ 1 (mod 3)} = {1, 4, 7, 10, …}
[2] = {x ∈ Z; x ~ 2 (mod 3)} = {2, 5, 8, 11, …}

For simplicity, although we have only considered the set of 0 and positive integers, we can see that this set is partitioned by three non-overlapping equivalence classes. It is interesting that establishing an equivalence relation on a set can partition the set without overlapping.

Homomorphism

A homomorphism is a map between two algebraic structures of the same type, preserving the operations of the structures. It means a map f: A → B between two sets A, B with the same structure, such that if ◉ is an operation of the structure (for simplicity, assume a binary operation), then

f (x ◉ y) = f(x) ◉ f(y) for any pair x, y of elements of A.

There are several types of homomorphism: a group and ring homomorphism, a linear map (homomorphism of vector spaces), an algebraic homomorphism, and so on. But we will concentrate on an algebraic homomorphism, which preserves the algebraic operations. Here the set A is a real physical domain, while the set B is real numbers. We should define an operation on the physical domain and a corresponding operation on the real numbers.

How to construct a mass-scale

I thought that all measurement begins with comparison and that comparison can be achieved by different methods in different fields of measurement. For example, a two-armed balance is used to compare two weights and a galvanometer in a resistance bridge is used to compare two resistors. Because the comparison requires two objects and they must exist in a physical domain, it is a homogeneous binary relation. Suppose we have a perfect balance that can tell whether two weights are equal in mass. Then, we define an equilibrium state with two weights as an equivalence relation within a physical domain, set A, which has an infinite number of elements whose material, shape, or colour may differ but whose mass is the same or different.

Let's consider, one by one, how the equivalence relation can be satisfied with the balance. First, the condition 'reflexivity' seems embarrassing at a glance cause it must be too obvious. But my interpretation, after a closer look, is that 'reflexivity a ~ a' requires the existence in set A of at least two weights that maintain equilibrium on balance. However, they may differ in surface colour and material. They can maintain equilibrium in the air with varying masses due to buoyancy if made of different materials. This condition requires the balance scale to be operated in a vacuum.

Secondly, it is easy to guess what the symmetry condition means. It requires a perfect balance in terms of symmetry, where if we exchange the positions of the two blocks, both arms should maintain their equilibrium. In practice, it requires that the two arms and the pans of the balance are symmetric about the central pivot and that gravity is uniform in the place where the balance is located. Without gravity, the balance will not work. Because gravity is a vector quantity, it requires any presence of gravity and any gravity direction that satisfies symmetry. If 'reflexivity' is a logical requirement for the existence of two blocks of equal mass, then a balance that satisfies symmetry could give a practical solution for how to create such blocks of equal mass. In this context, symmetry and reflexivity are not independent properties. It makes sense if we remember that one form of symmetry is reflexivity.

Thirdly, if b = a and c = a, the 'transitivity' condition requires a third element in the set A, which is in an equivalence relation with two elements of equal mass. By mathematical induction, this 'transitivity', together with 'reflexivity', allows for the existence of n elements of equal mass. Furthermore, because the balance for comparison between (a, b) and the balance for the comparison between (b, c) do not have to be the same, this condition requires a second balance. Interestingly, the last condition allows us to have an additional balance scale, which can be used to disseminate the mass scale after constructing it. Now, because we have all elements in set A to meet the equivalence relation conditions, set A can be partitioned by equivalence classes.

Figure 2. Set A can be divided into equivalence classes, where any pair of elements maintains equilibrium in the balance. Note that each equivalence class has an infinite number of elements.

Now select two elements from one class of the equivalence classes and place them together on the left arm. Each element will act as a unit mass in the mass scale we will build. Then, we can find the third element from a second class by placing it on the right arm, which is in equilibrium with the two elements from the first class. If we put three elements from the first class together, we can find the fourth element from the other class, which is in equilibrium with the three from the first class. Repeating this process to reform set A allows us to construct a mass scale, as shown on the left side of Figure 3. This figure shows that the empirical operation in the physical domain A has a map to a numerical domain B where the algebraic structures in set A (empirical addition) and within set B (arithmetic addition) are preserved. We say that they are in a homomorphism. Now the set B is an integer set.

Figure 3. A homomorphism is a map between two algebraic structures of the same type, preserving the operations of the structures. How to make a physical operation within a set A of a physical domain ⊕ should be defined, and what conditions should be satisfied ⇔. For example, we can define the operator ⊕ as putting mass blocks together on a pan of a balance. The balanced condition of both arms can be defined as ⇔.

So far, I have explained the scale's so-called scale-up process. However, applying this similar concept to the scale-down process to obtain m/n times the unit mass, where m and n are integers, is relatively easy. This scaled-down process extends set B from an integer set to a rational number set. Depending on which set the unit mass is in, set B can be extended to a set of real numbers.

What was an incompleteness in my theory?

When I gave the talk, many people in the audience were nodding their heads, so I was happy with that. However, last year, while on sabbatical at a university, I had the opportunity to look at measurement theory, and I looked at my theory again. I realised that the problem was not the measurement itself but the uncertainty. In practice, when we compare two masses with a balance, we can only say whether or not two masses are approximately equal within the resolution of the balance. Oh my God! My theory is against the transitivity in the abovementioned equivalence relation in 'Examples of relations that are not equivalences'. I had to abandon my primitive theory of measurement.

I have confessed my mistake to my shame. Nevertheless, I would like to draw a lesson from it. Even if I have to give up the equivalence relation, my interpretations of the condition of the equivalence relation are still meaningful. And the homomorphism still holds. After all, any theory without a consideration of uncertainty becomes useless. My embarrassment has led me to study more advanced measurement theory, suggesting that I must start with a weak-order relation rather than an equivalence one. I hope to write more about this in the future.

All the maths here are well explained on Wikipedia.