Often we begin exploring a topic because we find it interesting, perhaps even mysterious. For example quantum mechanics gets a lot of play in the media these days — probably because it is seen as puzzling, mysterious, even ineffable. There are perfectly coherent ways to understand quantum mechanics without sliding into mysticism, but that is a different post. Whenever a part of the world captures our attention and pulls us in, it is usually because it is represented by a big question mark in our map of things — *here be dragons. *

Other times, however, we have a different experience. Instead of learning about a big ? in our map, we learn that there *is* a big ? in our map, one we didn’t even know existed. We usually stumble onto these haphazardly; this is the nature of unknown unknowns. Although not as immediately interesting as exploring something that has captured us because it was mysterious, I find in my personal experience that these can often have a larger impact on me. It seems to shift more of my world.

There seem to be at least two reasons for this difference. The first is that when I am aware I don’t know something, that uncertainty spreads more obviously to other parts of my web of belief. I know that I don’t know how general relativity works in great detail, therefore other parts of the world and inquiry that rely on general relativity are marked in my mental map with more uncertainty. To the best of my ability, I keep track of how my uncertainty about one thing affects my understanding of other things. However, when I do not know that I do not know something, I cannot do this kind of epistemic bookkeeping. So when I become aware of something that I do not know, the uncertainty ripples through my world, revealing pillars of stone to be sand.

The second difference is more aesthetic: it is beautiful to discover that something mundane, unnoticed, and background is actually complex, nuanced, and puzzling. It reminds us that the world is much richer than we expected. As my undergraduate mentor Darcy Otto taught me (echoing Aristotle), the world presents itself to us as series of puzzles, calling out to be solved. Exploring a puzzle is a great intellectual joy; discovering a new puzzle to be tackled opens up and entire new domain to be explored.

Like many others, I had this kind of discovering-a-puzzle experience when I first read Gödel, Escher, Bach, after my dad gave it to me in high school. The foundations of mathematics, formal systems, undecidability, etc., were things I hadn’t even known existed as question marks in my map. It felt like the world I lived in was much more vast than I had known. I was recently sharing some set theory with a friend of mine, and she was having a similar experience working through how we could build up to the real numbers from the empty set and a few axioms from set theory. Numbers had always just been numbers; now there was a whole new depth to them, and to mathematics as whole.

This has all been a lead up to the concept I want to write about in this post: measurement. Measurement — assigning numbers to things — does not seem like the most attractive topic. Indeed, it can seem trivial, banal. We’ve all measured a piece of wood, or the width of a door, with a measuring tape. Wherein lies the mystery?

Until very recently I had no question marks in my map surrounding measurement. Even though in a lot of ways science is based on measurement, that didn’t seem to be the interesting part. Science to me looked like designing and carrying out an experiment to measure some stuff, an empirical effort managed by scientists, and then making inferences based on those numbers — also managed by scientists, and statisticians. Most of the theoretical action happened in this second half, drawing inferences from the data collected.

Happily, I was very wrong. There is an entire theory of measurement, one that tackles deep and serious problems in the project of measurement. Published in Science, *Measurement Scales on the Continuum* by Luce and Narens (the latter of the two I am lucky enough to learn from personally, since he is at UCI) is itself a kind of summary of field (as it stood in 1987) that provides an overview of how modern measurement theory might be used in the social science. The paper is very technical, and brings together a wide collection of results from decades of work. For this summary, I won’t track the paper very carefully. Rather, I want this post to be more of an introduction to and motivation for measurement theory. But I mention the paper because I think it provides a good starting point for someone who wants a more technical, thorough entry into the field.

***You can find the original paper here.***

When we measure something, we want to assign a number to a thing (actually, many things) that capture some kind of underlying empirical reality. Take length, for example. When we measure the length of different objects we want the numbers we assign to the different objects to reflect the qualitative ordering of the objects. For example, if I have one stick, A, and another stick, B, if A is longer than B, then I want to number I assign to A to be larger than the number I assign to B.

So, in general, there is some kind of qualitative empirical reality (such as the length of objects) that we want to capture using some kind of systematic measurement scheme. One of the central tensions driving measurement theory, then, seems to be this: we want a measurement scheme that is strong enough to capture the empirical situation, but not stronger. In order words, when we use our measurement scheme to describe reality, we don’t want to mistake artifacts of our choice of measurement for features of reality.

For example, consider temperature as described using Celsius or Fahrenheit. The temperature in Irvine now is 20°C. I know that this is hotter than 10°C. Does it make sense to say that 20°C is twice as hot as 10°C?

This makes sense with length. If one ruler is 100 centimeters long, and another, is 10 centimeters long, it makes sense to say that the first ruler is 10 times as long as the first. Does it make sense with temperature?

It does not. We can see why if we represent the temperatures in Fahrenheit.

20°C is 68°F, and 10°C is 50°F. Is 68°F twice as hot as 50°F? Clearly not.

Consider again the meter sticks. Let’s try the same trick, and rewrite the lengths using a different representation — inches. 100 centimeters is about 39.37 inches, and 10 centimeters is about 3.937 inches. In this case, we see that it still seems right to say that the first stick is ten times as long as the second one. What is going on here?

Measurement theory gives us the conceptual tools to understand this difference. It turns out that we represent temperature (not including things like Kelvin) and length using different scale types. In particular, we use a ratio scale to represent length, and an interval scale to represent temperature.

A scale is the set of different representations of an empirical reality. This sounds abstract, so let’s break it down. Let’s take an example. We can think of the the centimeter representation of length as a function that maps physicals objects — such as meter sticks and phones and sticks — to numbers. We usually think of a function as something mathematical, like , but more generally we can use functions to map one set to another — such as the set of physical objects to the set of real numbers. So, the centimeter representation of length is a function that maps the first stick to the real number 100 and the second stick to the real number 10. The inch representation maps the first stick to the real number 39.37, and the second stick to the real number 3.937.

So, the scale for length would include the centimeter representation and the inch representation. It would also include the meter representation, the furlong representation, the light-year representation — every way we have of measuring length (and then some we never use — I’ll get to how that works shortly).

We can then describe the type of scale this is by looking at the kind of transformations we can use to get from one representation to another. Again, this sounds abstract, so let’s break it down. Let’s call the centimeter representation , and the inch representation — remember, they are both functions that map objects to real numbers. So , and , for example. Now, we notice that we can turn into by multiplying the former by 0.393701. So we have . So, multiplying by 0.393701 is a transformation, and it allows us to get from the centimeter representation to the inch representation. In fact, for any two representations of length in the scale, we can get from one to another simply by multiplying by a positive real number. Furthermore, for every positive real number , is also a member of the scale (this is what I meant before when I said there are representations in our scale that we never in fact use). We also don’t allow any other transformations — like adding a constant.

A scale where the only allowable transformation is multiplying by a positive real number is called a ratio scale. For example, length (as we have seen) and mass are measured on ratio scales. They are called ratio scales because on such scales ratios of quantities are *meaningful*. A statement about the objects is meaningful if it is invariant under the transformations allowed by the scale. For example, recall our stick case. Saying “the first stick is twice as long as the second” is a statement about the ratio of the length of the first stick to the second one. We can see this is meaningful by considering the transformations we allow — in particular, we allow us to multiply by any positive number . We can see that under this kind of transformation the ratio is invariant:

This is because the s cancel, and so the ratio is invariant under the scale transformation. Thus we say that ratios are meaningful on a ratio scale.

This is not so with temperature (again, if we choose to measure it with things like Fahrenheit and Celsius), for we (usually) measure it on an *interval scale*. An interval scale is defined by allowing transformations of the form , where is any positive real number, is any real number, and is whatever representation we are transforming. For example, if is the Celsius representation, and is the Fahrenheit representation, we can get from Celsius to Fahrenheit by using the transformation .

Now we can understand in more detail why it is meaningless to say that one thing is twice as hot as another — this kind of statement is not invariant under an interval scale. For example

We see that the ratio is not invariant under transformations of the form . The intuition is that for an interval scale there is no fixed 0 point. Allowing the addition of a constant can allow us to change the 0 point – 0 °C is not the same as 0°F. However, 0 centimeters is the same length as 0 inches — that is, no length at all.

This is why if we use Kelvin, which has an absolute 0, the game changes. Fahrenheit and Celsius are part of a different scale than Kelvin: the first two are part of an interval scale, whereas the latter is part of a ratio scale.

Intervals scales are so called because whilst ratios are not meaningful (they are an artifacts of our choice of representation and do not track an underlying reality), ratios of intervals are meaningful. For example, it makes sense to say “the difference between 20°C and 10°C is twice as big as the difference between 7°C and 2°C.” We can see why here:

Under the kinds of transformations allowed by an interval scale, ratios of intervals are meaningful, and not mere artifacts of our representation.

Besides ratio and interval scales, there are many different scale types, of different strengths (more or fewer meaningful statements). How do we know what kind of scale type to use? That depends on our empirical context. In general, we would proceed in the following way. We would first identify the property in which we are interested in measuring — length, mass, happiness, productivity, anything. Then we would do some theoretical work to figure out what kind of scale seems appropriate. For example, we want want to know if there is a fixed 0, like with length or mass. A good part of measurement theory is trying to understand what kind of properties empirical structures have. Then, given this theoretical starting point, we write down some axioms that characterize this kind of structure. We show that these axioms are sufficient by proving a representation theorem that says that if a qualitative structure obeys these axioms then it can be represented by functions that belong to a scale of a certain scale type. Then we would test these axioms empirically — this is often the most challenging part. Once we have confirmed these axioms, we have our system of measurement up and running.

Notice how much theoretical and mathematical work goes into this. Far from being straightforward and uninteresting, the theoretical foundation of measurement is surprisingly complex and fascinating. Understanding measurement allows us to demarcate the meaningful from the meaningless. Remember all those times when your teachers told you that you needed to look at the units on the graphs in order to understand what the graphs are actually telling you? This is like the super-mega version of that — anytime you hear any statistic, comparison, or summary — for example “doing art makes you three times happier” or “teaching through interpretive dance is twice as effective as teaching with PowerPoint” — you need to ask, is this statement meaningful? Measurement theory can help you answer this.

Indeed, it is foundational for science.