Foundations of a Theory of Measurement

PROLOGUE

This material is intended to eventually be incorporated into our main paper. It is presently in an early stage of personal rumination. It still has to reach two more stages: consensus among ourselves, followed by appropriate presentation for our audience. The material mainly focuses on fundamental concepts and axioms for scalar measurement domains. It mentions problems associated with other domains, but does not address them here. That should be done elsewhere in the main paper.

1 INTRODUCTION

[Some of this stuff can be moved out to the general introduction.]

Dimensioned data is important in many computer applications, and will become even more important in emerging application domains such as health care and the environment. The most familiar computational aspects involve units conversion and dimensional analysis. Behind these deceptively simple aspects lies a remarkably confusing set of issues concerning the appropriateness and meaning of various operations. Sorting out these issues is essential to the design of appropriate facilities in programming languages and information modeling.

Even the scope and boundaries of the topic are unclear. Measurement generally involves assignment of a symbolic representation to a physical quantity in a way that communicates certain information and supports certain operations. "Physical quantity" is a very loose term, covering obviously physical things like length and weight as well as not so physical things like intelligence, efficiency, and beauty. Sometimes we'll refer to these more vaguely as "phenomena".

Several aspects of dimensioned data need to be distinguished:

• The abstract phenomenon being characterized.
• The corresponding symbolic representations.
• The abstract paradigm for assigning a representation to the phenomenon, e.g., judging, testing, polling, measuring, etc.
• The real process for assigning a representation, involving such aspects as accuracy, precision, and repeatability, as well as such auxiliary mechanisms such as averaging of multiple readings.

The symbolic representation is often numeric, but letter grades in school and letter compass headings such as NNE may also be considered to be dimensioned data.

The representation may simply convey a relative order, such as first, second, and third place, or gold, silver, and bronze medals. Very often the information has the sense of how big or how much, but not always. Some phenomena don't involve a sense of order or bigness at all, such as locations or directions.

There are various paradigms for assigning symbolic representations to phenomena, which may or may not all be considered "measurement". At one extreme are purely subjective judgments, as in grading essays, or in judging Olympic events, beauty contests, and livestock competitions at the county fair. A bit less subjective are polling and testing procedures which simply count things, like people's preferences or the number of correct answers on a test. Some paradigms are based on more complicated computations, as in batting averages and quarterback ratings.

In its narrowest sense, the term "measurement" involves a unit, which is a particular instance of the measurement domain. Any thing in the domain is then expressible as being equal to k of the unit things. The value of k is clearly different for different choices of the unit thing.

Symbolic representations have behaviors of their own, such as ordering among letters and numbers, and arithmetic among numbers. Such behaviors may or may not correctly model the behaviors of the underlying phenomena. Ordering often makes sense: it's common sense that a rock whose weight is a bigger number is a heavier rock (assuming common units). On the other hand, compass headings don't have order: NE is not bigger or smaller than SE, nor is a direction of 30° bigger or smaller than a direction of 60°. And it is at least arguable whether a higher IQ number implies a smarter person, or a higher letter grade implies a better performance, or a higher price implies a greater value.

Arithmetic can be similarly misleading. Just because numbers can be added and subtracted doesn't mean that these operations are meaningful for measurements. Weights are well-behaved: putting two rocks on a scale yields a measurement which is the sum of the individual weights, so it is sensible to add weight measurements (in consistent units, of course). But it doesn't make sense to add temperatures, or numeric compass headings, even though they are numbers. If the direction of travel of two ships is 30° and 60°, what does it mean to add those two numbers? (Curious coincidence that temperatures and headings are both measured in degrees.)

Other arithmetic behaviors may also fail to correspond to the underlying phenomena. Adding a positive number to a positive number yields a larger number, but such ordering doesn't hold for circular angles: adding 200° and 200° yields an angle of 40°. Adding the same angle to two angles can invert the order of the results: 50°<70° but 300°+50°>300°+70°, since 350°>10°.

Subtraction doesn't always make sense, either. While numbers are closed under subtraction, sometimes yielding negative numbers, it doesn't follow that subtracting a large weight from a small weight yields a weight.

Even simple counting can be anomalous. With angles, counting does not lead to a purely monotonic sequence. As we keep accumulating one-degree increments, we eventually come back to angles we've encountered before. Among other things, this implies that the "measure" of an angle is not unique: x=x+360i for any integer i.

Which operations with dimensioned expressions are valid, and what their results signify, depends on the type of underlying phenomena involved. Weights, temperatures, angles, velocities, and intelligence all behave differently. Even though measurements are expressed as numbers, we can't assume that all properties of numbers automatically apply in these domains. In fact, it's necessary to reason the other way. We need to independently establish whether certain principles hold in the underlying domain in order to justify applying various arithmetic procedures to the measurements. In other words, we need to understand the extent to which we can or cannot reason about a measurement domain by reasoning about the measurement numbers.

Simply put, you can get into trouble if you just assume that properties of numbers automatically imply corresponding behaviors in the underlying domain.

Subsequent sections identify a set of principles (sometimes in the form of axioms) which may or may not hold in various measurement domains. If an axiom doesn't hold in a physical domain, then arithmetic operations which depend on that axiom may not be meaningfully performed on measurements of that domain.

Such principles serve several purposes. First of all, they can be grouped into subsets to characterize different sorts of measurement systems. Such sets of principles precisely and generically describe different levels of support which can be provided in a language.

Further, the principles provide an analytical tool, helping to determine precisely whether and how a given domain can be subjected to measurement. The analysis can expose ambiguity and fuzziness of concept, leading sometimes to differentiation of similar domains, sometimes to clarification of exactly how measurement principles can be applied. The principles also illuminate why certain domains such as color or beauty are difficult to measure.

[I seem to be switching to the term "domain" instead of "dimension". Can we use them both?]

2 GOALS OF MEASUREMENT SYSTEMS

In general we think of a measurement system as something involving units, providing the ability to do units conversion and to validate and evaluate expressions involving units. In broadest terms, a measurement domain contains a set of physical quantities we wish to map into symbolic representations which convey information and support certain operations, in a manner consistent with our intuitive understanding of behaviors in the domain. The term "measurement" is used to mean either the mapping process or the resulting symbolic representation. A measurement system thus provides:

• Symbolic notation: a mapping from the abstractions in some domain (such as weights or performance in a course) into some symbolic representation.
• Operations: operations on the symbolic representations which should mimic behaviors of the abstractions being represented. For example, addition of numbers representing weights should preserve the physical behavior of weights being combined.

2.1 Symbolic Representations

The simple business of assigning symbolic representations to the abstractions in some domain is sometimes a major challenge in itself.

We usually think of measurement systems as dealing with numbers, but non-numeric representations are sometimes used. School grades and letter compass points (e.g., SSW) are examples.

Some measurements require a vector of representations rather than a simple number.Velocity is the most familiar example, requiring two values to express the speed and heading. Other examples include location in n-dimensional space, and colors expressed in terms of hue, saturation, and value [explore the behavior of those three as scalar measurements]. This category doesn't include "additive" notations such as "five pounds, three ounces" which can trivially be transformed into a single number. True vector measurements don't have a single-number equivalent.

Even scalar (single-number) measurements can be difficult to quantify. Thus the "soft" sciences have problems trying to assign measures in such domains as intelligence, performance in various senses, usability, quality, productivity, efficiency, wealth, net worth, cost of ownership, and various other economic indicators. Similar issues also arise with beauty contests, movie and restaurant ratings, and judging of Olympic events. Symbolic representation can often be misleading, implying greater precision than actually exists, and sometimes even giving rise to incorrect comparisons. Does higher IQ really imply greater intelligence?

2.2 Operations

Various combinations of the following operations with symbolic representations are supported in various measurement domains:

• Communication. This simple facility should not be overlooked, particularly in non-numeric representations. Letter grades and compass headings convey information. So do all the other measurement notations. In the case of letter grades or economic status indicators, this may not always be self-evident. More generally, we expect numeric representations to convey some intuitive sense of magnitude, i. e., how big, how much.
• Equality comparison. We generally expect that if the symbolic representations match, then the abstractions being represented are the same in some sense. (The converse needn't hold; units conversion presumes that equal quantities may have different representations.) Even this could be problematic, as with school grades. Exactly what does it mean to give the same grade to two students in two courses at two schools?
• Order comparison. If one symbolic representation is "larger" than another, we expect the corresponding underlying abstraction to be larger than the other. Problems again, as noted with IQ's and school grades.
• Combination within a domain. We very often wish to add the measurements of two weights, or of two lengths. This is intended to mimic the behavior of combining the underlying abstractions. Adding the measurements of two weights should yield a result that's consistent with the effect of putting two rocks on one pan of a balance scale. Other operations such as subtraction, multiplication, and division should make similar sense.
• Combinations across domains, as in dividing distance by time to get speed.

3 FUNDAMENTAL PRINCIPLES FOR MEASUREMENT DOMAINS

3.1 Distinction Between Physical Quantities and Representations

3.2 Types of Physical Quantities

It is necessary to be able to identify, refer to, and differentiate various types of physical quantities (domains) such as weight, mass, length, area, volume, time points and intervals, speed, velocity, acceleration, temperature, frequency, etc.

Identifying and differentiating types of physical quantities is a non-trivial task. Some, like length and weight, are reasonably simple concepts to identify and differentiate. Even these, though, can be troublesome if we want to be very careful, such as the distinction between weight and mass, or the distinction between weight in vacuum and weight in air.

Others, like locations in space or points in time, might raise a question as to whether they should even be considered physical quantities, since they don't seem to have a sense of magnitude inherently associated with them. A place is just a place, not anything which has a bigness to be measured.

The well-behaved ones are relatively precise concepts, often expressible in algebraic relationships such as area being the square of length, or speed being the ratio of distance to time. Others, like amount, while being a reasonably clear concept, are more like a family of closely related physical concepts. An amount of stuff might be expressed in terms of weight (iron), volume (water), area (carpet), length (rope), or even a simple numeric count (people, atoms). The amount of a given chunk of stuff is often expressible in several such terms. Is "amount" a single domain or several?

Other such problematic domains also involve the notion of amount. Concentration, for example, is essentially the ratio of two amounts, each expressible in any of the forms that amounts can take. Productivity is a ratio of amount to time, fuel consumption might be a ratio of amount to distance or time, and so on.

We need to be clear as to whether we consider amount to be a single type of physical quantity or a family of such types. If it's a family, does the single family behave in some ways like a single type? E.g., units which measure the same type of physical quantity can be converted to one another. Can the same be said of units which measure the same family of physical quantities? Can a weight be compared with a volume?

Amounts illustrate concepts which are not algebraically equivalent (e.g., weight and volume) yet seem to be the same in some sense. Conversely, some things which are algebraically equivalent don't seem to be the same. Are length and width the same type of physical quantity? What about the length of an object vs. the distance something travels? Is the production rate of rope or licorice in feet per hour the same concept as speed? Dimensional analysis treats angles as being algebraically equivalent to a ratio of two lengths, yielding a dimensionless quantity if we do cancellation. Does that make angle the same as a pure number? Then what do degrees and radians measure? Torque and work are both algebraically equivalent to force times distance, but these are clearly not the same physical quantity.

3.3 Instances of Physical Quantities

It is necessary to be able to identify, refer to, and differentiate various instances of physical quantities. This often takes the form of some property of some "subject" under certain circumstances, such as the temperature at which water boils at a specified pressure, the weight of Bill Kent on the planet Earth at 8 AM on May 19, 1994, and the distance between New York and San Francisco.

Such references are often somewhat ambiguous, leading to issues of precision and accuracy. Bill's weight may or may not include his clothing. The distance between New York and San Francisco may be measured between various precise points. It might not even be straight-line distance; it might be airline miles or driving miles.

We sometimes visualize an angle in terms of two lines meeting at a point. Are we always sure that a picture like < means the inside angle? How do we then refer to an angle greater than 180°?

3.4 Equality

This is perhaps the most crucial principle. It must be meaningful to speak of the equality of two physical quantities in the same domain. This equality has the sense of "the same magnitude", not "the same thing". Thus we must be able to say whether Bill's height is equal to Bruce's height. We often carelessly use the word "same", but we don't mean to force the issue that Bill's height is the very same thing as Bruce's height. They are different notions.

Equality of magnitude is significant. Being "the very same thing" is not. There are many issues of identity that we don't need to get involved with, all having the flavor of "how many instances". Is the distance between New York and San Francisco the same thing as the distance between San Francisco and New York? If you and I are boiling water in different kettles, does the temperature at which each kettle boils constitute one temperature or two? It doesn't matter. We only need a notion of equality.

Accuracy and precision of the measurement process are not the big problems here. Preciseness of concepts is. The time domain is really problematic. When is a month a month? January and February are both months; are they equal? How many days in a month? Are any two years equal? What does it mean to add a month or a year to a given interval? And let's not forget about the mythical 30-day month and 360-day year. [Such concerns should be addressed elsewhere. The point here is simply that a well-defined notion of equality is required as a foundation for this algebra.]

Lots of domains have equality: weights, temperatures, spatial locations, colors. Lots of domains don't, such as beauty and intelligence. (Don't confuse equality of measurement with equality of physical quantities. Equal IQ doesn't really mean equal intelligence.)

Thus there must be a well-defined equality comparison operator, which we will denote as x EQ y, which must be true or false for any two instances of a domain. This operator must be totally defined:

• Totality: x EQ y is true or false for any x, y in D,

and satisfy the usual axioms of equivalence relations:

• Reflexivity: x EQ x.
• Symmetry: x EQ y => y EQ x.
• Transitivity: x EQ y, y EQ z => x EQ z.

The equality operator must be chosen with care to satisfy these axioms. Real-world measurements are not perfectly transitive if there is any imprecision in the measurement. Suppose that y weighs a tiny bit more than x and a tiny bit less than z. A real scale might not detect any difference in weight between x and y or between y and z, yet detect a difference between x and z.

4 FUNDAMENTAL PRINCIPLES FOR SCALAR MEASUREMENT DOMAINS

4.1 The Goal: Units

Symbolic representations of phenomena in a scalar measurement domain D are expressed in terms of units. A unit is defined as some instance u of D such that any instance of D is equal to a combination of some number of occurrences of u.

For example, we can pick a certain rock as the Reference Rock. We can find a lot of rocks each of whose weight is equal to the weight of the Reference Rock. For an arbitrary rock, we might find that its weight is equal to the weight of k rocks, each of whose weights is equal to the weight of the Reference Rock. We then say that this rock weighs the same as k Reference Rocks, or that its weight is k Reference Rocks.

If the weight of a rock isn't an integral multiple of the weight of the Reference Rock, we can generalize to rational numbers by saying that the weight of a rock equals the weight of n/m Reference Rocks if the weight of m of the rocks equals the weight of n Reference Rocks. If the weight of a rock isn't a rational multiple of the weight of the Reference Rock, it can be approximated by the weight of another rock that weighs nearly the same as the rock we want to weigh.

Thus the weight of any rock is equal to or approximated by the weight of k Reference Rocks, where k is some rational number.

That's what measurement is. It is rooted in counting: how many reference units are equal to the physical quantity being measured? Notice the close association between two meanings of the word "unit". The unit of measure is the size of the things we are counting; it corresponds to a count of one, i.e., a unit.

Different units are trivially illustrated by introducing reference rocks whose weights are not equal to each other. Let's say, though, that all Blue Reference Rocks weigh the same, and that all Green Reference Rocks weigh the same. Then a given rock might weigh the same as x BRR's or y GRR's. If we know how many Blue Reference Rocks weigh the same as one Green Reference Rock (or vice versa), then we know how to convert between x and y. Given either, we can compute the other.

4.2 Combinational Closure

The measurement scenario is based on combining unit quantities. Does it in fact make sense to combine the quantities? Is the result an instance of the same type? Does it satisfy certain algebraic properties? Under what conditions is this meaningful?

This is where we can start to differentiate the behaviors of various physical quantities. Weight, length, time intervals, and frequencies seem well-behaved, but temperature, location, and time points don't. Remember, we are talking about the physical quantities themselves, not numbers associated by measurement.

The underlying phenomenon seems to involve equality as well. Thus we can imagine combining the weights of two rocks and judging the combination to be a weight which is equal to the weight of some third rock. We can imagine combining the lengths of two sticks and judging the combination to be a length equal to the length of a third stick.

This breaks down for temperature and location. I don't get any good intuition as to what it means to combine the temperatures of two kettles of water and judging that to be equal to the temperature of a third kettle. Nor do I get a good intuition of what it means to combine the locations of two points in space and judging that to be equal to the location of some other point, in the absence of a superimposed coordinate system.

Thus there must be a well-defined combinational operator, which we will denote as x#y. This operator is something in the real world which combines physical quantities, not an operation on numbers. However, under the right conditions, this operator will be isomorphic to addition, making the measurement of a combination of quantities equal to the sum of the measurements of the components. The intent is that

measure(x#y,u) = measure(x,u) + measure(y,u),

where x and y are quantities in some domain and u is a unit of measurement in that domain.

The combinational operator must be totally defined and closed on the domain. That is, the result of x#y must exist and be an instance of the domain for any instances x and y of the domain.

The operator must be repeatable. Combining the same two quantities should always yield the same result, i.e., x1 EQ x2 AND y1 EQ y2 => x1#y1 EQ x2#y2. This sounds trivial with respect to mathematically defined operations, but it is not automatically true for operations on physical quantities. Mixing two cups of water does not yield the same volume as mixing a cup of water and a cup of salt. It is thus important to carefully define the combinational operator which provides the basis of measurement as some operation which is repeatable. Physical mixing is not an appropriate combinational operator for measuring volumes.

We thus have:

• Totality: x#y exists for any x, y in D.
• Closure: x#y is itself an instance of D.
• Repeatability: x1 EQ x2 AND y1 EQ y2 => x1#y1 EQ x2#y2.

Formal description of the combinational operator for a measurable quantity can be complex. Combining lengths, for example, involves a notion of laying end to end. Things which are not straight (perimeters of circles, coastlines of countries) must first be mapped into something straight before being combined.

Colors have such a combinational operator, but spatial locations don't.

Weights (masses) combine by inert juxtaposition. Just because anti-matter might annihilate matter when they come in physical contact doesn't necessarily mean that anti-matter has negative weight (or mass). This is analogous to saying that the sum of masses of some atomic particles is not given by the mass of the results of their fusion, since some mass might vanish into energy.

4.3 Commutativity and Associativity

There is no sense of order in the combination process. Our linear notation requires that we write one participant before the other, but x#y means the same thing as y#x. We thus have

• Commutativity: x#y EQ y#x.

Order-independence is more than notational. If we first combine x and y, and then combine the result with z, we must get the same end result as combining x with the result of combining y with z. We thus have

• Associativity: (x#y)#z EQ x#(y#z).

4.4 Nil

Some physical quantities can be combined with others without yielding something different.

If combining a particular physical quantity Ø with any arbitrary physical quantity x yields a physical quantity equal to x, i.e., x# Ø EQ x for all x in the domain D, then Ø is a nil quantity. (We use the term "nil" rather than "zero" to underscore the fact that we are not talking about numbers but about physical quantities.)

We could also postulate uniqueness of nil: if Ø1 and Ø2 have this property, then Ø1 EQ Ø2.

Many domains have a nil. I can be holding no weight of rocks, or no length of stick. Adding my rocks or sticks to yours then doesn't change the weight of rocks or length of sticks you have.

4.5 Multiplication

Associativity and commutativity insure that combining a set of occurrences of physical quantities yields the same resulting physical quantity regardless of how the subsets are grouped. I.e., we could combine them sequentially one at a time, or we could combine them pairwise and then combine the pairs, and so on. No matter how we do it, the result is the same physical quantity. Thus, the result of x#x#...#x involving n occurrences of x is a unique physical quantity, which we can denote as nx or n*x or xn or x*n. Note carefully that n is a number but x is not; x is a physical quantity. Furthermore, nx is a physical quantity, not a number.

4.6 Ratios

The ratio of two physical quantities x1 and x2 is k1/k2 if k2*x1 EQ k1*x2, where k1 and k2 are non-negative integers. Let r1 and r2 be rocks weighing x1 and x2. We are saying that the weights of the two rocks are in the ratio 3/5 if five rocks as heavy as r1 weigh the same as three rocks as heavy as r2. We don't yet assume that the ratio of any two physical quantities in a domain can be so expressed.

We use this as a basis for introducing a notion of fractional physical quantities. Under the given conditions of the example, we can say what 3/5 of the weight of r2 means: it means a weight the same as the weight of r1, i.e., (3/5)x2 EQ x1.

4.7 Rational Quantities

We have to account for the distinction between real and rational numbers in our measurements. Whether measurement maps a physical quantity into a rational or irrational number depends on the choice of unit. Angles measured in degrees and radians provide a familiar example. The conversion factor is 180/pi . Angles which are a rational number of radians are an irrational number of degrees, and conversely. The angles which can be precisely measured as a rational number of degrees or a rational number of radians constitute disjoint sets.

We can talk about a rational set of physical quantities relative to a given physical quantity x0, namely the set of physical quantities xi such that xi EQ (k1/k2)x0 for some non-negative integers k1 and k2. The number k1/k2 is a "measure" of xi relative to the "unit" x0.

[Other numbers can also be measures, so long as there is some mapping to k1/k2. That's how Bruce's ß gets into the act. If the measure is simply k1/k2, then the measure of the unit quantity is 1 and the measure of the nil quantity is 0.]

4.8 Coverage

In order for something to be a unit, every instance of the domain must be "reachable", i.e., expressible in terms of the unit. For a unit u, any instance x must equal ku for some number k.

Consider the rational set defined by some unit quantity u, i.e., the set of quantities equal to ku for some rational k. Does there always exist a unit u such that its rational set covers the entire domain? I.e., can any quantity in the domain be so measured with this choice of unit?

Coverage fails whenever there is no quantity which can be repeatedly combined with itself to generate "enough" other quantities in the domain.

Coverage obviously fails when there always exist quantities which can only be expressed as xu where x is some irrational number.

Coverage also obviously fails for velocities: repeated combination of a given velocity can only yield velocities in the same direction. The same is generally true for any quantity which must inherently be expressed as a vector. [Could that help to actually define the essential nature of a vector quantity?] Colors may illustrate the extreme failure of coverage. It's hard to imagine a combining operation under which repeated combination of a color with itself yields anything other than itself (except for some sort of subtraction, which would only yield the nil color).

For a different sort of example, consider the possibility of a certain pebble together with a certain boulder which is heavier than any number of these pebbles, i.e., whose measure relative to this pebble is infinity. Rocks in such a domain could be stratified into numbered "sorts", and we can define pebbles to be rocks of sort 0. The difference in weight between rocks of the same sort equals the weight of a pebble. Combining the weights of rocks of sorts x and y equals the weight a rock of sort x+y. Such a domain doesn't seem to violate any of the previously proposed axioms. This domain doesn't have a unit.

A numeric analog to this domain can be modeled in terms of a combining operation # which behaves differently on the integer and fractional portions of a number. It involves a special operator f which is closed in the interval [0,1). That is, 0=<x<1 and 0=<y<1 => 0=<f(x,y)<1. Thus no combination of fractional values can ever equal or exceed 1. f might simply be defined as addition modulo 1, or there might be a more sophisticated formula such as f(x,y)=1/(1/x + 1/y). If xi and xf are respectively the integer and fractional portions of x, then we define # as x#y=xi+yi+f(xf,yf). In this domain, numbers smaller than 1 correspond to pebbles.

While this is a purely academic exercise, it does suggest an interesting mapping. Breaking up the quantities into integer and fractional parts creates an isomorphism with vector quantities. In effect, these parts constitute distinct vector components, each with its own combination rule.

[Closure and/or transition needed here.]

There are several possible approaches to the matter of rational and irrational quantities.

It might simply be true that a rational set does cover the domain. The domain might involve some intrinsic quantum quantity, as with molarity. [Does quantum mechanics provide any help here?] Or, the domain might be defined as consisting only of quantities which can actually measured by certain devices. Audio and visual phenomena might be restricted to some digitizable set of states.

Another approach would be to arbitrarily extend our model to let k be any real number, without any further explanation. This would account for irrational weights and angles, but not for pebbles and boulders.

Approximation is a common approach, in which any quantity in the domain can be replaced by some "acceptable" quantity in the rational set. Acceptability usually involves some notion of nearness, expressed in terms of some difference being small. Smallness requires a notion of order, which hasn't yet been required for any other reason.

The assurance that there always exists a rational quantity sufficiently close to any arbitrary quantity is usually expressed in terms of a continuity axiom:

Let R be a rational subset in some domain D. For any d IN D and any r in R, and for any rational number epsilon >0, there exists a delta IN D such that d EQ kr#delta and delta <epsilon r for some rational k.

The steps:

1. You pick a boulder d that you want to weigh, a pebble r that you want to use as a unit, and a nice small number epsilon as the margin of error.
2. The axiom guarantees the existence of a small rock (pebble or boulder; it doesn't matter) delta such that it weighs less than epsilon r.
3. The axiom further guarantees that we can find such a delta rock with the additional property that kr#delta is exactly the weight of the boulder d, for some rational number k. That is, if we combine k reference rocks of weight r, the result will be so close to the weight of d that the difference is less than delta .

[Could we use that as a basis for introducing "real" weights, in contrast to rational weights?]

The pebbles and boulders example does not satisfy the continuity axiom.

4.9 Other Properties

4.9.1 Order

4.9.2 Negatives and Subtraction

The difference xDelta y between two quantities x and y can be defined as a quantity z such that z#y EQ x.

[Do we know that the difference is unique? I'm not sure that the repeatability axiom assures that.]

Difference is generally not symmetric, allowing xDelta y ¬EQ yDelta x.

[Do we need/want the stronger axiom x ¬EQ y => xDelta y ¬EQ yDelta x ?]

If the domain has a nil Ø, it seems desirable to postulate that x=y => xDelta y=Ø.

4.9.3 Division and Rational Closure

4.10 Coercion

Certain phenomena which don't exhibit additive closure can be described via an isomorphism with some phenomenon which does. More precisely, there is often a family of such isomorphisms, differentiated by some parameter. This typically applies to point quantities being coerced into interval quantities.

Thus, while locations of points in a one-dimensional space don't exhibit additive closure, their distance from some fixed reference point does. [Elaborate as needed. The parameter is the choice of reference point.] Locations of points can thus be "measured" so long as some common reference point is understood. [Multiple reference points work, so long as their relative locations to each other are known.]

This implies a form of coercion. The locations of points are "coerced" into the corresponding distances from the reference point, appropriate operations are performed on such distances, and the result then "coerced" back to some location. [Illustrate.]

Coordinate systems provide two essential ingredients: an origin to establish the reference point, and a scale for measurement. [In n-dimensional systems, there is also an "orientation" provided by the position of the axes. Is that significant here?]

[Extend to n-dimensional space, then to other things analogous to such spaces. E.g., time, temperature.]

5 APPLICATION TO VARIOUS DOMAINS

Some examples of how the theory might apply in various domains. One main theme is to examine the underlying combinational operator that makes measurement possible in each domain. In general, it must be an operation such that we believe combining n occurrences of a unit quantity using this operation yields something equal to a quantity we want to measure.

5.1 Points and Intervals

Much of what we have to say about physical quantities is often mapped into an analogy with points and intervals on a line. I'm going to use this section to collect a number of observations and clarifications about points and intervals. It may or may not all be relevant in the end, and it may belong somewhere else. Some of it may seem pedantically precise, but I do want to sidestep some troublesome but subtle ambiguities.

5.1.1 Fixed and Movable Points and Intervals

To begin with, I'm talking about fixed points and intervals on a straight line, infinite in both directions. Thus some corner or edge of a rectangle which can move around in space is not what I mean here by a point or interval. It doesn't make sense to speak of two points being in the same place. The point is the place, and there is only one point there.

5.1.2 Sameness

Let's also get clear about "sameness", which I use here in the sense of "one and the same". The predicate Same(x,y), which I denote as x==y, means that x and y refer to the same thing. Anything which attributes plurality to x and y is misleading, as in "the points x and y are the same point". The terms x and y are two things only in the sense that they are two expressions (variables, designations) which evaluate to (refer to, designate) the same thing. If the butler is the murderer, we wouldn't say they were two people who were the same.

5.1.3 Length

In this context we take length to be an undefined primitive property of intervals, for which an EQ operator is defined. Thus length(i) is defined for any interval i, and the expression length(i1) EQ length(i2) is true or false for any pair of intervals i1 and i2. The EQ operator is required to be an equivalence operator, i.e., to be reflexive, symmetric, and transitive.

Length is not the same thing as an interval. A length corresponds to an equivalence class of intervals induced by length equality. By definition, length(i1) EQ length(i2) if and only if i1 and i2 are in the same equivalence class.

The # combinational operator on lengths is also an undefined primitive. It is required to be total and closed in the sense that length(i1)#length(i2) exists for any two intervals i1 and i2, and the result is equal to length(i3) for some interval i3: length(i1)#length(i2) EQ length(i3).

5.1.4 Reference Intervals

It might be useful to define a set of reference intervals, isomorphic to the set of lengths, which contains one interval of each length. We could simply define this as a set containing one member of each equivalence class.

5.1.5 Points

5.1.6 Modulo Combination

For modeling some domains, such as angles, it will also be useful to have a different combining operation corresponding to residue arithmetic.

Residue arithmetic can be expressed in terms of a "residue" operator \ on numbers such that x\y yields the remainder of x/y. [Is that an adequate definition?] One could then define a residue-based summing operator such that s(x,y,z)=(x+y)\z. To get back to binary operations, we could define a family of such s operators, one for each value of z, such that sz(x,y)=(x+y)\z.

We can define an analogous sort of combining operation on physical quantities which does not depend on numbers or arithmetic.

The modulo operator µ will be described as a parameterized family of operators, one for each distinct quantity m (the modulus). To avoid clutter, though, we will simply write µ rather than µm. For a given domain D, we define Dm to be a subdomain containing a subset of instances such that, for any x,y IN Dm, either x#y IN Dm or there is a distinct z IN Dm such that x#y EQ z#m. Then we define xµy as follows for any x,y IN Dm:

xµy EQ x#y if x#y IN Dm,
else xµy EQ z where x#y EQ z#m and z IN Dm.

Here Dm itself is a domain, with combining operator µ, though they are defined in terms of a larger domain D with combining operator #. The operator EQ is the same for both.

The modulo operator µ is illustrated in a nearby figure, which shows what it means to combine lengths x and y modulo length m to obtain a length z.

                       x#y
<---------------------------------------------->

:                       :                      :
:            x          :         y            :
................................................
:               m             :       z        :
:                             :                :

                                     xµy
                              <---------------->

                             m#z
<---------------------------------------------->

5.2 Distance

5.3 Volume

5.4 Time

We have probably been thinking of time points as fixed. Do we need to say much about the notion of movable time points, such as the next time I go to New York?

5.5 Speed

The basic relationship between speed, distance, and time is that an object traveling in a straight line at a constant speed s travels the same distance d in any time interval of duration t.

Combining speeds s1 and s2, i.e., s1#s2, can be defined in terms of combining distances. If objects traveling at constant speeds s1 and s2 travel distances d1 and d2 during some time interval of duration t, then an object traveling at a constant speed equal to s1#s2 will travel a distance equal to d1#d2 during a time interval of duration t. This definition involves two meanings (overloading) of #, one for speed and one for distance. The same is true for equality.

So far we could have simply treated speed as a pair s={d,t}, with the combining operation defined as {d1,t}#{d2,t} EQ {d1#d2,t}. How do we justify the arithmetic operation of division as the appropriate connection between distance and time? By showing that speed is directly proportional to distance and inversely proportional to time.

Doubling the distance doubles the speed, for constant time. If an object is traveling at a speed s1 EQ {d,t}, what is the speed s2 of another object which covers twice the distance in the same time?

s2 EQ {2d,t} EQ {d#d,t} EQ {d,t}#{d,t} EQ 2{d,t} EQ 2s1.

Each step is justified from earlier results concerning distances and speeds, not from properties of numbers. The result is readily extended to any multiplier k.

Now consider a third object traveling at speed s3 which covers the same distance as the first object in twice the time, i.e., s3 EQ {d,2t}. Since the first object is traveling at constant speed, it would cover a distance of d#d in time 2t, so that

s1 EQ {d#d,2t} EQ {d,2t}#{d,2t} EQ 2{d,2t} EQ 2s3,

i.e.,

s3 EQ s1/2.

5.6 Angles

There are several distinct domains, all informally called "angle":

5.6.1 Unusual Properties of Angles

Angles of circular measure, i.e, modulo 2pi , have some noteworthy characteristics which we should factor into our analysis. Most of it follows from the behavior of residue arithmetic. Some of this material should find its way into other sections.

The combination of two non-nil, non-negative quantities can be nil, e.g., pi +pi =0.

The integer k which measures an angle relative to the unit angle is not unique: pi =3pi =5pi ...

5.7 Color

The kinds of combining operations we might imagine for colors include mixing crayons on paper, mixing lights, or mixing filters. I can't image any sort of combining operation and any sort of color such that repeatedly combining that color with itself will eventually reach any arbitrary color. Is there an inherent reason for this? Is that because it is a vector quantity? (No, that's probably not the whole reason.)

Furthermore, I find it hard to even imagine a combining operation under which combining a color with itself yields anything other than the same color, except for an operation that annihilates the color to yield black.

[Explore the behavior of color as a vector of hue, saturation, value (HSV). Does it still satisfy our intuition that all vector quantities can be combined by vector addition? Also explore the behavior of each of those components as a scalar measurement domain.] [Actually, is color an HSV vector, or a vector of such vectors, one for each primary color?]

5.8 Sounds and Music

Note names in music constitute nomenclature for cyclic values, just like days of the week or months.

We may want to distinguish between combinations of sounds (directly experienced physical phenomena) and combination of frequencies (attributes of sounds). There are several analogies here with light (color), including the distinction between pure and mixed frequencies. Combining sounds, i.e., the result of playing two sounds together, is not simply described as a sum of frequencies.

5.9 Temperature

The more we say about this stuff, the less I understand how temperature works. Little of the preceding analysis seems to apply to temperature.

Perhaps the only way to rationalize temperature is to be very simplistic and consider the coercions that go on in the measurement process. What is a degree of temperature? It's a distance! What we look at on a thermometer is how far something has moved up or down in the tube. The coercion is via the thermal expansion of the substance in the tube. We don't really know anything about the linearity of heating, or expansion. We simply mark the tube in uniform distance increments. We don't know if it takes twice as much heat to make the fluid move up twice as many notches. [There seems to be some dependence here on the uniformity of a substance's specific heat at different temperatures.]

So, temperature makes sense if we consider it a coercion to distance. Our naive intuitions about temperature are really based on the behavior of distance.

As with volumes, mixing of substances doesn't give the right metaphor for combining temperatures. Ignore chemical reactions, i.e., let's assume we are dealing with kettles of water. They act somewhat like the self-nils of color: mixing two kettles at the same temperature yields something of the same temperature again. More generally, mixing two kettles does not yield something bigger than either one separately, but rather something in between: x<y => x<x#y<y.

A better metaphor for combining temperatures would be expressed in terms of taking heat out of one kettle and putting it into the other. But the parallel between heat and temperature is still a difficult concept to grapple with [cite that reference].

On the other hand...

Temperature can be defined in a way that makes it proportional to the square of the speed of something, perhaps an average particle. This yields at least a natural nil concept, corresponding to a speed of nil for an average particle. The interesting observation here is that the zero number for the common measurement scales does not correspond to the natural nil of the physical quantity.

5.10 Counts, Frequencies, and Molarity

In some domains the measure consists of simply counting things, as in populations, committee or team sizes, and inventories. The unit is intrinsically defined as one thing, and we are simply counting how many things. The combinational operator is intrinsically defined as addition.

Combination must be done carefully, in terms of distinct things. Combining the sizes of two committees does not necessarily correspond to combining the committees, if some people are serving on both.

The unit notion has to be carefully distinguished from other properties. Bigger populations do not assure larger weight or volume. Walls can be measured in bricks, even if the bricks are of different sizes. Walls whose sizes in bricks are equal may have different weight, height, area, or volume. (Distinctness matters here, too. If two walls are connected - i.e., share bricks - then their combined size is not the same as the number of bricks in the two walls.)

Sometimes the things being counted are events, like births, or arrivals, or a cyclic phenomenon reaching a maximum or minimum. Frequencies or rates then correspond to counting the number of such things in some interval of time. This is the meaning of birth rates, arrival rates, and frequencies as expressed, for example, in cycles per second. Combining such things sometimes needs to be carefully distinguished from mixing together the underlying phenomena. Combining frequencies for measurement purposes does not correspond to composition of frequencies, as might arise from mixing tones.

6 ALGEBRAIC RELATIONSHIPS AMONG DOMAINS

We are familiar with such notions as area being the square of length, or speed being a ratio of distance to time. Can we justify those semantics without reference to the arithmetic of their measurements?

7 SUMMARY OF PRINCIPLES

Principles applicable to all domains:

• A physical quantity is a different thing from its representation.
• Types (domains) must be carefully identified and distinguished.
• A type (domain) is characterized by its population (instances) and by the behavior of the equality (EQ) and combinational (#) operators.
• A domain has an equality operator (EQ) which is totally defined:

  Totality: x EQ y is true or false for any x, y in D.

• and is an equivalence relation:

  Reflexivity: x EQ x.

  Symmetry: x EQ y => y EQ x.

  Transitivity: x EQ y, y EQ z => x EQ z.

Principles which apply to some domains, and which can be used to characterize or differentiate domains:

8 ACKNOWLEDGMENTS

9 TO BE COMPLETED

Things left to be done:

10 RESTART (8/10/94)

10.1 Measurement

In its narrow, familiar sense, measurement is a paradigm for assigning numeric values to physical quantities. The term also has broader usage, sometimes hard to distinguish from naming. The things that get measured aren't always physical quantities, but might also be things like intelligence, academic performance, efficiency, performance (in various senses), various economic conditions, populations, etc. The results of measurement are sometimes alphabetic rather than numeric, as with school grades, shoe sizes, or compass headings like NNE.

There are also different sorts of measurement paradigms. We often think first of the units-based paradigm for measurement data, but measurements can also be determined by tests, polls, judging, complex combinations of factors, and perhaps other means as well.

After measurements are recorded, computational systems support operations on the measurement data. Familiar computational operations on numbers and characters can't be applied directly to measurement data; some form of adaptation is needed. Measurements of 3 and 4 can't simply be added unless they have the same units. If the units are different but compatible (measuring the same "dimension", such as length), then the numbers can be added after first performing an appropriate units conversion. If they are incompatible, then addition is meaningless. Thus 3 inches and 4 inches can be added directly; 3 inches and 4 feet can be added after converting to the same units; and 3 inches and 4 grams can't be added at all.

The results of such operations also need to be interpreted in order to know how they should be displayed or whether they may be assigned to a data item. The product of two measurements may be assigned to a data item representing area if the two measurements are lengths, but not otherwise.

10.2 Unit-Based Measurement

In unit-based measurement, any instance of some domain D is equivalent to a combination of some number of occurrences of a particular instance of the domain, which serves as a unit. For example, if we pick a particular weight W to be a unit, than any weight is equivalent to kW for some number k.

That innocuous formulation hides a remarkable number of subtle assumptions. Articulating these assumptions can help explain how measurement works, how measurement data should be treated, why some domains behave differently from others, and why some domains can't be measured at all using units.

Summary of assumptions:

• The domain is well-defined, having a well-defined population with clear concepts of distinctness and equality.

  We can immediately identify domains that don't satisfy such criteria...

  We can also observe ambiguities in the definitions of some domains, leading us to distinguish several closely related domains. [Elaborate: angles.]

  Also observe that there are some domains which are distinct but may share the same units (not just unit names) and/or the same canonical forms. [torque and work, speed and linear productivity, paint coverage and length]

  Also introduce the notion of "delta domain". Generally have the same canonical form, same units.

• Carry on. (include singularity of number)

10.3 Other Measurement Paradigms

10.4 Operations on Measurement Data

Computers operate on numbers and characters. Under what conditions and in what way are such operations meaningful for measurement data?

We focus on these operations:

• Comparison (equality, order).
• Basic arithmetic (addition, subtraction, multiplication, division).
• Assignment.
• Display.

Given that a requested operation takes the form of an operator with one or more operands, the following questions are relevant:

• What are the characteristics of the operands?
• Is the operator valid for such operands? I.e., is it a valid operation?
• How should the operations be performed?
• What are the characteristics of the results?

10.4.1 Operand Descriptions

10.4.1.1 Forms of Data Terms

Operands (data terms) may take the following forms:

• Literals, which are specific data values such as 3 or "abc". For measurement data, this usually has to include units (e.g., pounds) and dimension (e.g., currency), though there are various syntactic techniques allowing these to occur either explicitly or implicitly. We use constructs such as "3 inches" or "3 pounds/currency" as example literals in this paper, but other syntaxes are possible.
• Variables, whose values are usually mutable via assignment operations. These come in two forms:
  • Simple variables, as commonly encountered in programming languages, often denoted by simple names such as x or h2.
  • Properties or attributes, as commonly stored in databases, such as the heights of persons. A reference to a particular value may take a form such as height(Bill). A property such as "height" is sometimes called an instance variable, behaving like a variable which has different values for different instances, e.g., for different people.
• Expressions, which take the general form of an operator applied to one or more data terms. This is a recursive definition.

10.4.2 Operation Validity

10.4.3 Operation Execution

For measurement data, auxiliary operations are often required in addition to the normal behavior of the specified operator on numbers or character strings. The most common of these is units conversion [discussed elsewhere].

In some cases, as with certain angle domains, residue arithmetic is involved, mapping results outside the domain back into the domain. For example, adding 100° to 300° might yield an angle of 40°.

Subtraction may require special handling in case of negative results.

10.4.4 Result Descriptions

This is an important and difficult topic. Some notes...

This is the mechanism by which the meaning of results is "interpreted". It determines how the results will be treated on assignment and display, as well as in other operations in nested expressions.

10.4.5 Specific Operations

10.4.5.1 Addition

10.4.5.1.1 Same Domain

For units-based measurements, combinational closure is required: both a combinational operator, and total closure.

10.4.5.1.2 Domain and Delta Domain

10.4.5.1.3 Different Domains

10.4.5.2 Difference and Subtraction

10.4.5.2.1 Same Domain

Define "difference" to be absolute value. Domains closed under difference have a zero. Domains not closed under difference have a different delta domain.

10.4.5.2.2 Domain and Delta Domain

10.4.5.2.3 Different Domains

10.4.5.3 Multiplication

10.4.5.3.1 Same Domain

10.4.5.3.2 Different Domains

10.4.5.4 Division

10.4.5.4.1 Same Domain

10.4.5.4.2 Different Domains

10.4.5.5 Assignment

10.4.5.5.1 Same Domain

10.4.5.5.2 Different Domains

10.4.5.6 Display

10.4.5.7 Equality

10.4.5.8 Order

10.5 Relationships Among Domains

Domains can relate to each other in various ways:

• Domain and delta.
• Canonical form expressions.
• Sets of "similar" domains. [This may not be a crisp concept. It's intended to deal with domains that are often ambiguously confused with each other, e.g., various angle domains, weight and mass, domains and their deltas, etc.] [Often share units and canonical forms.]
• Quasi-dimensions.
• Specialization.

10.5.1 Domain and Delta

The elements in a delta domain D¯ correspond to differences between elements in a related domain D. For example, duration is the delta domain for the domain of time points. Some domains are their own delta domains: the difference between two weights is a weight.

For di IN D and dj¯ IN D¯, the following familiar rules apply:

• |d1-d2| IN D¯
• d1+d2¯ IN D

If a domain is not its own delta domain, then it is not closed under difference (absolute subtraction), and probably not even closed under addition. [Can that be proved?] Without addition, there are no units. Such a domain is often "measured" by mapping an element into the delta domain relative to some fixed element of the delta domain serving as the origin:

µ(d) ::= µ¯(d-do),

i.e., the measure µ of an element d IN D is defined to be the measure µ¯ in D¯ of the difference d-do relative to an origin element do IN D. [Illustrate!!]

[Account for sign or directionality, i.e., absolute values.]

10.5.2 Canonical Form Expressions

10.5.3 Similar Domains

10.5.4 Quasi-dimensions

10.5.5 Specialization

10.6 Notes

Letter compass headings: a finite domain?

Different measurement paradigms can apply to the same domain: compass headings in letters and degrees. Well, strictly speaking, one domain is a subset of the other.

Non-singular measurements: angles modulo 2pi .

Operations on mixed operands.

Assuming that the result of an operation corresponds to a domain element relies on continuity, closure.

The role of dimensional analysis.

Dimension expressions occurring in dimension declarations, e.g., declaring x to be force/angle.

Where do zeros come in? Relate to subtraction. Relate to origins.

11 ANOTHER RESTART (8/28/94)

11.1 Basic Distinctions

At the very start, we need to understand the distinction between the things being measured and the measurements themselves. If a certain rock weighs two pounds, we have several distinct concepts: the rock itself, a measurement represented as "two pounds", and the weight denoted by that measurement, which is the "heaviness" of the rock. It's like talking about a red apple. There's the apple, there's the word "red", and there's the color itself which the eye sees and which the word "red" denotes, but which is a different thing from that three-letter word. The heaviness of the rock is what we shall mean by "weight". It might be denoted by various measurements such as two pounds or 32 ounces.

The things being measured are grouped into various domains such as weight, distance, time, speed, velocity, area, angles, money, color, beauty, intelligence, and so on. One of the important topics we'll return to is making sure that we know exactly which domains we are dealing with. Ambiguity and confusion often arises from a failure to distinguish closely related but distinct domains, such as weight and mass, or torque and force, or circular measure and angle of rotation.

[What is color temperature?]

We also need to sort out some terminology in connection with units. Units associated with totally unrelated domains may have the same name, such as pounds of weight and money, or degrees of angle and temperature. Let's agree to say that degrees of angle and degrees of temperature are different units because they apply to unrelated domains, though they are homonyms. Let's further agree that degrees Fahrenheit and degrees Celsius are different units, even though they apply to the same domain. We'll see later that these units are based in different elements of the same domain (one degree Celsius is a different temperature from one degree Fahrenheit).

On the other hand, the same units may be used to measure related domains. Thus angle of rotation and circular measure are different domains (0°is the same as 360°in one domain but not the other), but they can be measured in the same units. That is, degrees of rotation and degrees of circular angle are the same unit. Of course, we'll have to clarify what we mean by "related domains". Often the elements of one are a subset of the elements of the other.

[Are pounds of weight and pounds of mass the same units? If so, these related domains are not subsets of one another.]

To summarize:

• Degrees of temperature and degrees of angle are different units which happen to have the same name. They measure unrelated domains.
• Degrees Fahrenheit and degrees Celsius are different units even though they measure the same domain. They are based on different elements of the domain.
• Degrees of circular measure and degrees of angular rotation are the same units even though they are measuring different domains. These domains are closely related.

11.2 Distinguishing Domains

A measurement domain contains a set of similar but distinguishable elements. Various criteria determine how well domains are defined, and how they relate to or are distinguished from one another.

[Are we essentially trying to distinguish their populations?]

11.2.1 Clarity of Concept

Sometimes the underlying concept is simply not clearly defined. Without getting into subtleties of precise scientific definitions, we all have a clear enough idea of what constitutes a distinct weight, or a distinct length, or a distinct speed, etc.

Color is a bit more problematic. [Get into formulations involving hue, etc.]

11.2.2 Equality and Distinctness

Closely related to clarity of concept are the dual properties of equality and distinctness. Without clarity of underlying concept, it's hard to say whether two people have the same intelligence, or the same beauty. That's not the same as determining whether they scored the same on an IQ test or had the same rating in a beauty contest.

Even when there is clarity of underlying concept, domains with similar elements may be distinguished by different notions of equality. As we have seen, circular and rotational measure both involve angles, but they have different populations. Rotational measure has no upper bound, and a rotation of 360°is different from a rotation of 0°. Circular measure has an upper bound, and the angles measured as 0°and 360°are the same angle.

In fact, we could differentiate the following similar domains, all involving angles, based on differences in populations of the domains:

11.2.3 Number

11.2.4 Behavior Under Operations