William Kent

August 1994

> PROLOGUE

> 1 INTRODUCTION . . . 3

> 2 GOALS OF MEASUREMENT SYSTEMS . . . 5

>> 2.1 Symbolic Representations . . . 6

>> 2.2 Operations . . . 6

> 3 FUNDAMENTAL PRINCIPLES FOR MEASUREMENT DOMAINS .
. . 7

>> 3.1 Distinction Between Physical Quantities
and Representations . . . 7

>> 3.2 Types of Physical Quantities . . . 7

>> 3.3 Instances of Physical Quantities . . . 9

>> 3.4 Equality . . . 9

> 4 FUNDAMENTAL PRINCIPLES FOR SCALAR MEASUREMENT
DOMAINS . . . 10

>> 4.1 The Goal: Units . . . 10

>> 4.2 Combinational Closure . . . 11

>> 4.3 Commutativity and Associativity . . . 12

>> 4.4 Nil . . . 13

>> 4.5 Multiplication . . . 13

>> 4.6 Ratios . . . 13

>> 4.7 Rational Quantities . . . 14

>> 4.8 Coverage . . . 14

>> 4.9 Other Properties . . . 16

>>> 4.9.1 Order . . . 16

>>> 4.9.2 Negatives and Subtraction . . .
16

>>> 4.9.3 Division and Rational Closure .
. . 17

>> 4.10 Coercion . . . 18

> 5 APPLICATION TO VARIOUS DOMAINS . . . 18

>> 5.1 Points and Intervals . . . 18

>>> 5.1.1 Fixed and Movable Points and
Intervals . . . 19

>>> 5.1.2 Sameness . . . 19

>>> 5.1.3 Length . . . 19

>>> 5.1.4 Reference Intervals . . . 20

>>> 5.1.5 Points . . . 20

>>> 5.1.6 Modulo Combination . . . 20

>> 5.2 Distance . . . 21

>> 5.3 Volume . . . 21

>> 5.4 Time . . . 21

>> 5.5 Speed . . . 21

>> 5.6 Angles . . . 22

>>> 5.6.1 Unusual Properties of Angles . .
. 23

>> 5.7 Color . . . 23

>> 5.8 Sounds and Music . . . 24

>> 5.9 Temperature . . . 24

>> 5.10 Counts, Frequencies, and Molarity . .
. 24

> 6 ALGEBRAIC RELATIONSHIPS AMONG DOMAINS . . . 25

> 7 SUMMARY OF PRINCIPLES . . . 25

> 8 ACKNOWLEDGMENTS . . . 26

> 9 TO BE COMPLETED . . . 26

> 10 RESTART (8/10/94) . . . 26

>> 10.1 Measurement . . . 26

>> 10.2 Unit-Based Measurement . . . 27

>> 10.3 Other Measurement Paradigms . . . 28

>> 10.4 Operations on Measurement Data . . .
28

>>> 10.4.1 Operand Descriptions . . . 28

>>>> 10.4.1.1 Forms of Data Terms .
. . 28

>>> 10.4.2 Operation Validity . . . 29

>>> 10.4.3 Operation Execution . . . 29

>>> 10.4.4 Result Descriptions . . . 29

>>> 10.4.5 Specific Operations . . . 29

>>>> 10.4.5.1 Addition . . . 29

>>>> 10.4.5.2 Difference and
Subtraction . . . 30

>>>> 10.4.5.3 Multiplication . . .
30

>>>> 10.4.5.4 Division . . . 30

>>>> 10.4.5.5 Assignment . . . 30

>>>> 10.4.5.6 Display . . . 31

>>>> 10.4.5.7 Equality . . . 31

>>>> 10.4.5.8 Order . . . 31

>> 10.5 Relationships Among Domains . . . 31

>>> 10.5.1 Domain and Delta . . . 31

>>> 10.5.2 Canonical Form Expressions . .
. 32

>>> 10.5.3 Similar Domains . . . 32

>>> 10.5.4 Quasi-dimensions . . . 32

>>> 10.5.5 Specialization . . . 32

>> 10.6 Notes . . . 32

> 11 ANOTHER RESTART (8/28/94) . . . 32

>> 11.1 Basic Distinctions . . . 32

>> 11.2 Distinguishing Domains . . . 33

>>> 11.2.1 Clarity of Concept . . . 33

>>> 11.2.2 Equality and Distinctness . .
. 34

>>> 11.2.3 Number . . . 34

>>> 11.2.4 Behavior Under Operations . .
. 34

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.

The paper still rambles a bit, and could use further tightening.

*[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?]*

Section 2 elaborates some of the general goals. Section 3 identifies a few principles which apply to all domains. Section 4 goes on to describe principles which apply to those domains which exhibit familiar sorts of numeric behavior. Section 5 then provides some illustrations in various specific domains.

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.

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?

In this section we focus primarily on well-defined scalar measurement systems. The other issues should be dealt with in other sections of the main paper.

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.

*[Try to characterize which operations are supported in which domains. Tie in with
Bruce's stuff.]*

*[Depending on what we want to admit as domains in the first place, some of these
principles might not be universal. The problem with some domains such as intelligence or
beauty might be precisely that they don't have well-defined instances or a well-defined
equality operator. We're not likely to have consensus that two people are equally
intelligent, or equally beautiful. Does such analysis extend to more significant domains,
such as usability or productivity?]*

Just as an athlete's performance is a different thing from a gold medal, so is the weight of a rock a different thing from an expression of the form "3 grams". In most of what follows, we focus more on the physical quantities themselves than on their symbolic representations.

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.

Types (domains) are characterized by their populations, i.e., their instances. It is sometimes necessary to distinguish domains which are very similar yet differ in their populations. For example, there are several such domains all commonly called "angle" [Section 5.6]:

- Angle of rotation is purely cumulative, so that 0°, 360°, and 720° are all distinct. It is modeled as lengths on an infinite line.
- There is the familiar circular measure, where 0°= 360°. Here the combining operation
is more like the residue arithmetic for lengths [Section
5.1.6], where combination of angles is modulo 2
*pi*. - Heading is like circular measure but with respect to a fixed reference direction on Earth. There is also, of course, the distinction between true and magnetic north.
- The dimensional analysis notion of angle as a ratio of distances reduces angles to those in the first quadrant, having the same absolute value of their tangent.
- There is also a notion of angles being equal under inversion, as in the assertion that
the base angles of an isosceles triangle are equal. Here x=|
*pi*-x\2*pi*|.

Populations which do or don't include negative quantities [Section 4.9.2] constitute different types. Weight in a vacuum has no negative quantities, while weight in air does. Weight in air is really a measure of buoyancy. A helium balloon would pull upward on a scale, and lighten the weight of a person holding one.

It may also be useful to clarify whether a domain need be limited to quantities which
can exist in reality. Dimensional systems are used not only for measuring real phenomena,
but also for describing hypothetical or fantastic situations. Does the speed domain
include speeds greater than the speed of light? Should 2c be an invalid speed expression?
Does the energy domain include half a quantum? Then perhaps it might also be meaningful in
some contexts to contemplate negative masses, and negative temperatures. *[Illustrates
the point that users of a measurement system need to establish consensus on the nature of
the domains involved.] *

Some domains are difficult to measure precisely because we don't have a clear idea of their populations. We all have a reasonably clear sense of what is meant by a set of distinct weights, or lengths. We don't have an equally clear sense of a set of distinct beauties, or intelligences. This is intimately connected with the question of equality [Section 3.4].

Language doesn't always help. Sometimes natural language fuses different concepts into a single term (as with the several domains called "angle"), and sometimes makes artificial distinctions within the same concept. Compare the notions of time and distance. They are quite similar, both being isomorphic to the one-dimensional real-number line. They become even more nearly identical if we imagine ourselves to be moving at constant speed in a straight line. There is a sense of place in both domains; "here" and "now" are isomorphic concepts. Both domains also have a notion of interval.

Yet there are strange asymmetries in the language we use in the two areas. With
distance, we use distinct terms for the notions of point and interval, one being
"location" and the other being "distance". Where I am is a location;
how far I am from you is a distance. However, we use the term "time" for both
point and interval: when I was born, and how long I've lived, are both considered to be
notions of time. If we go by linguistic conventions, then point and interval seem to be
different physical quantities in relation to distance but the same physical quantity in
relation to time. (We do have the word "duration" for a time interval, but we
don't have a corresponding word that makes a single concept of the phrase "point in
time". *[Jim suggested "instant", but it doesn't have exactly the right
connotation. It may have something to do with an implied granularity. "In my
chair" and "in California" are both location concepts, despite the
disparity in granularity. In contrast, it seems comfortable to call "3 PM today
" an instant, but not "today", even though both may be a description of
when something occurs. To be resolved.])*

On the other hand, distance has one set of terminology for all granularities, while time uses two. Whether we say that I'm in my chair or in California, both are considered to be "location". However, if we say that I was born at 3 PM or on February 19, one is considered to be "time" while the other is "date".

There is another difference. For times we have distinct names for periodically repeating points and intervals, such as Tuesday, 3 PM, January, and January 15. We don't seem to have a natural counterpart for distance. It may just be an accident of astrophysics, in that our notions of time derive from the behavior of the spinning and rotating body we live on. If planet Earth moved in a straight line through space, without rotating around the sun or spinning on its axis, our notions of time and distance might be more isomorphic, with no natural notion of cycles in either.

The bottom line is that we can't always rely in our intuitions, common sense, or
linguistics to guide an objective identification and differentiation of the types of
physical quantities. We need some other basis for a systematic formal model of physical
quantities and measurement. *[Do we ever discuss that further?]*

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°?

Such ambiguity will be ignored for most of the present discussion. We assume that a reference to a physical quantity refers to a very specific one.

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.

Inequality will be denoted as x ¬EQ y.

A *scalar measurement domain* is defined as a domain in which a measurement can
be represented by a single number.

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.

This simple scenario implies a number of distinct elementary requirements, as described in subsequent sections.

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.

(However, we do use the opposite reasoning for particle charges, where we do think of them as being positive and negative because they cancel each other out. The real message is that this is at least a point of potential ambiguity which needs clarification in any particular measurement system before we can agree on what sorts of operations are valid, and what their results are.)

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).

This is a non-trivial requirement which might not be satisfied, for example, when mixing chemicals.

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.

The importance of such a nil is not clear yet. It may become relevant in connection with negative quantities, or with natural origins for units.

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.

Note that different values of n can yield the same physical quantity, as illustrated for angles in circular measure [Section 5.6]. Combining n one-degree angles yields the same result as combining n+360 one-degree angles.

If there is a nil physical quantity Ø, then we should have 0*x EQ Ø. *[Does that
have to be postulated as an independent axiom?]*

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.

Note that we still haven't measured anything. Though we might multiply or divide by a number, we aren't multiplying or dividing a number. Five times the weight of a certain rock is a bigger weight, not a number. Half the weight of a rock is again a weight, not a number. If we say one rock is 0.6 times as heavy as another, this is a ratio of weights ("heavinesses"), not of numbers. It simply means that five rocks as heavy as the first weigh the same as three rocks as heavy as the second.

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.]*

Hypothesis: any non-nil physical quantity within such a rational set of quantities could serve as the unit for the set. I.e., given two such quantities x and y, there exist non-negative integers k1 and k2 such that x EQ (k1/k2)y.

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 adeltaIN D such that d EQ kr#deltaanddelta<epsilonr for some rational k.

The steps:

- 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. - 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. - 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.

Continuity may not be an especially important axiom to worry about, until we find some domain in which its absence creates a problem. We do point out that continuity is a prerequisite for approximation, which is essential in most domains. Continuity cannot simply be taken for granted, though in most domains we would simply assert that the axiom holds, without further proof.

We mention some other possibly interesting properties that don't seem essential. Have we overlooked something? Is there some need for them?

We don't seem to have a need for order as an independent requirement in a scalar measurement domain, except in connection with continuity [Section 4.8].

Note that it's not sufficient to simply define x>y to mean there exists a non-nil z such that x EQ y#z. This does not induce an order relation, which must be anti-symmetric. Angles in circular measure provide a counter-example [Section 5.6.1]. For any two angles x and y, there exist angles z1 and z2 such that x EQ y#z1 and y EQ x#z2.

*[It has occasionally been suggested that we can use between rather than order in
certain contexts. I have trouble understanding a precise useful characterization of
between in the absence of order. The situation usually seems to arise in circular systems.
Thus I don't know exactly what it means to say that Monday is between Sunday and Saturday
but not between Saturday and Sunday. Similarly, though my intuition does say that NE is
between N and E but not between N and W, I don't know how to formalize that intuition. *

*What are the axioms of betweenness?]*

The *difference* x*Delta *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 x*Delta *y ¬EQ y*Delta *x.

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

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

We require *Delta *to at least be "semi-total". That is, for any x,y
IN D, either x*Delta *y EQ z or y*Delta *x EQ z for some z IN D. *[Caution.
Does that presume some sort of coverage? Maybe this is analogous to **Section 4.9.3.]*

*Delta *being *totally* defined on a domain D would mean that x*Delta *y
EQ z IN D for any x,y IN D. This leads to the following sorts of properties *[need to
sort out which are primitive axioms and which follow from others]:*

- The domain has a nil Ø.
- (x
*Delta*y)#(y*Delta*x) EQ Ø. - A domain can be partitioned into Ø, positive quantities, and negative quantities.
- If x ¬EQ y, then one of {x
*Delta*y, y*Delta*x} is positive while the other is negative. - If x#y EQ Ø, then y is denoted as x¯. I.e., x#x¯ EQ Ø. By convention, positive quantities are denoted without the overline, negative quantities with.
- x
*Delta*y EQ x#y¯. - (x
*Delta*y) EQ (y*Delta*x)¯.

The definition of a domain should specify whether or not it includes negative quantities. If it doesn't, then the difference operator is not totally defined on the domain, and subtraction of measurements does not always yield a valid measurement. If negative quantities are valid in some contexts but not in others, then it may be appropriate to define two different domains for these purposes.

Domains with negative quantities often involve a sense of direction, e.g., yardage
gained or lost in a football play, or heat going into or out of a chemical reaction. This
suggests an isomorphism with velocity vectors having a trivial two-valued direction
component. Positive quantities have one value of the direction component, while negative
quantities have the opposite value. *[Elaborate on the nature of the combinational
operations.] *

There is also a strong parallel between adding a negative quantity and subtracting a positive quantity, but these don't provide exactly the same capabilities. In order to express the net yardage gained by a player in a football game as a summation, it is necessary to be able to express positive and negative yardages for individual plays.

The preceding axioms don't guarantee closure under division. What we have established
is that, given a reference weight x0, *if* there is a certain weight x1, then it
can be expressed as (or approximated by, if continuity holds) kx0 for some k. However,
nothing in the axioms seems to imply that there must exist a weight equal to kx0 for
arbitrary x0 and k. Closure is guaranteed for an existing x0 and an arbitrary *integer*
k, due to the closure property of # [Section 4.5],
but not for arbitrary rational k.

Thus closure under division seems to be an independent axiom. This also doesn't seem especially important, until we find some domain where such closure doesn't hold. We simply make the observation in passing; in most cases it will simply be postulated that this axiom holds.

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.] *

[We might say more about coercion in connection with "soft" measurements. Tests and polls are examples of techniques for coercing abstract quantities into manageable numbers.]

*[We might say that gauges represent a form of physical coercion. Gauges convert
quantities such as heat or speed or weight into other quantities such as length or angle
of rotation in order to be displayed on a thermometer, speedometer, or scale.]*

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.

*[Much room for cleanup and expansion here. Probably much redundancy.]*

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.

In particular, when we speak of point and interval quantities, we should be careful that we are not making an analogy with a fixed interval on a line, but with an equivalence class of intervals having the same length. That's what we mean when we say that the interval between 30° and 40° is the same as the interval between 40° and 50°.

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.

The other kinds of points and intervals could be called *movable*. The location
of a movable point is a fixed point. It is possible for two distinct movable points to
have the same fixed point as their location.

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.

If x==y, then any expression, operation, etc. in which x and y are evaluated should behave the same. In particular, x==y=>x=y and x==y=>x EQ y, regardless of what particular (reasonable, i.e.,non-quoting) interpretations we put on = and EQ.

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).

Any partitioning induces an equivalence class, and a corresponding equivalence operation. Would any partitioning serve our purpose? In a sense, that's one of the things we're trying to isolate. What exactly are the requirements? Minimally, repeatability of # [Section 4.2] has to be preserved. That is, if we combine any two members of two given equivalence classes, the result should always be in the same equivalence class:

length(i1) EQ length(i2) AND length(j1) EQ length(j2) => length(i1)#length(j1) EQ length(i2)#length(j2).

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.

We could also be more specific and pick a set sharing a common end point which serves as an origin, with all the intervals lying on the same side of the shared point. We could do that by introducing either direction or inside vs. outside. In the latter case, we could postulate that for any two non-zero intervals, one end point of one of the intervals is strictly inside the other interval.

We can define points as intervals having no length. We did not define intervals in terms of points. We can postulate that there is a unique interval associated with any pair of points, but we don't need that as a definition of interval. The set of points is one of the equivalence classes induced by length equality.

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 <---------------------------------------------->

Note that we've been able to do this without introducing other assumptions such as
order. *[Do we have to prove the existence and/or uniqueness of such a set Dm? Do we
need additional postulates, such as a nil element, or maximality over all such sets?]*

As mentioned earlier, the combinational operator is described in terms of laying straight sticks end-to-end. Describing the straight-line equivalent of a curved edge, or a coastline, is non-trivial.

As mentioned earlier, physical mixing is not the right model for combining volumes. A more accurate model for combining the volumes of two shapes is to imagine we can get volumes of water equivalent to each, and then combining the volumes of water. Simple immersion won't do, unless we postulate porosity for enclosed spaces. We might think in terms of melting ice sculptures which faithfully reproduce the two shapes involved (ignoring the contraction which occurs during melting).

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?

Time intervals probably need a bit more care. I believe that we have been primarily
talking about them in the sense of *duration*, being directly analogous to length.
This should be distinguished from things which are analogous to line intervals, in the
sense that several distinct time intervals might have the same duration. And we can
further distinguish fixed, movable, and repeating intervals. ("Cyclic" or
"periodic" might be an imprecise term, since the periods are not always
uniform.)

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.

Thus speed is directly proportional to distance and inversely proportional to time, justifying the expression of speed as the quotient of distance over time.

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

- Angle of rotation is purely cumulative, so that 0°, 360°, and 720° are all distinct. It is modeled as lengths on an infinite line.
- There is the familiar circular measure, where 0° EQ 360°. Here the combining operation
is more like the residue arithmetic for lengths [Section
5.1.6], where combination of angles is modulo 2
*pi*. - Heading is like circular measure but with respect to a fixed reference direction on Earth. There is also, of course, the distinction between true and magnetic north. While circular measure is analogous to length, heading is more like a reference set of intervals anchored at an origin [Section 5.1].
- The dimensional analysis notion of angle as a ratio of distances reduces angles to those in the first quadrant, having the same absolute value of their tangent.
- There is also a notion of angles being equal under inversion, as in the assertion that
the base angles of an isosceles triangle are equal. Here x=|
*pi*-x\2*pi*|.

This is a good illustration of the ambiguities that this sort of precise axiomatic analysis can expose.

In general, these domains have different populations, which can be illustrated in terms of the number of distinct angles in the domain whose measure is an integer number of degrees. The numbers for those five cases are infinity, 360, 360, 90, 180. Do you agree?

The combinational and equality operators are probably best described via mappings into and out of the domain of rotational angles. Note that the combinational operators assume rotation in the same direction; reversing direction would give rise to subtraction, or negative quantities.

Are there other variants on the notion of angle?

There are a couple of questions that might be trivial or interesting in these domains.
Are the endpoints equal? I.e., is 0° the same thing as 360°? If we visualize an angle as
defined by two line segments meeting in a point, does it matter which side we are on? Does
that configuration define one angle or two? (Not counting rotations modulo 2*pi *.)

Angles of circular measure, i.e, modulo 2*pi *, 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 *=3*pi
*=5*pi *...

Order is funny. For any two angles x and y, there exist angles z1 and z2 such that x=y#z1 and y=x#z2.

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?] *

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.

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.

This definition also yields a basis for defining a combinational operator. Suppose ti=ksi². Then t1#t2=t3 would be defined to mean s1²#s2²=s3² - provided we could explain what it meant to square a speed, or to combine such things. (Tedious reminder: we are working with physical quantities, not numbers.)

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.

Molarity is a form of concentration which counts the number of particles (molecules, atoms, ions) in a volume. Here count is serving as another variation of amount.

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?

The expression of speed as a quotient of time over distance was justified in Section 5.5.

Area is a little trickier. As Bruce pointed out, we can imagine that a certain area is "swept out" by a stick of length l1 moving sideways a distance of l2. It takes just a little more work, probably not too difficult, to justify that multiplication is the right operation between the measures of l1 and l2.

We might want to apply similar analyses to other algebraic relationships.

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:

I'd like to add an acknowledgment to Jim Davis for his insightful observations and incisive questions.

*Things left to be done:
*

- Resolve all editor's notes in the text.
- Cluster the principles into significant subsets.
- Might want to refer to the "ideal" case as one which satisfies all principles, including non-residue arithmetic. The intuition is that such domains are the ones which behave most like numbers.
- Analyze and classify various domains according the principles.
- Possibly integrated into Section 5.
- Possibly in the form of a matrix of domains vs principles.

*Describe various levels of programming language support based on the principles.*

Another repackaging of old and new stuff. Maybe this will be more coherent.

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.

There are fundamental axioms and other principles which describe measurement domains and characterize differences between them. Some will be described in connection with the measurement paradigm [Section 10.2], others in the context of operations on measurement data [Section 10.4].

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)

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?

The first and last of these require some form of description of the operands. For now we postulate the following descriptive elements: form, domain (dimension), units, value type, value. Others that may be relevant include precision, accuracy, measurement method and context, etc.

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.

Validity of operations depends very much on the operator and the domains of the operands. This will be discussed in detail in Section 10.4.5.

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.

There may also be other sorts of auxiliary operations.

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.

This is where dimensional analysis is important, and where its limitations also show up. Other mechanisms to supplement dimensional analysis may be needed.

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

For some domains, residue arithmetic applies, in which case addition and subtraction would not be order-preserving. [Illustrate.]

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

Discuss negatives, closure under subtraction.

Other than domain and delta. Are difference or subtraction ever valid in this case?

Treat multiplication by numbers as a special case.

This is where ratios come in. Potential problems in dimensional analysis.

Treat division by numbers as a special case. Implications for closure, continuity?

Is this where coercions arise?

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.

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.]

[Temperatures are funny. T and T¯ both seem to have zeros, but they don't mean the same thing.]

[Highlight the non-uniqueness.]

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 2*pi *.

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.

Managing measurement data involves much more than units and units conversion. Let's explore a few ideas without worrying about their usefulness just yet.

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.

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?]

[What sort of generic criterion distinguishes weight and mass?]

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.]

Many other domains involve complex combinations of factors on which there may or may not be general consensus. These include such domains as intelligence, beauty, performance in various academic, artistic, and athletic endeavors, productivity, efficiency, usability, etc. While there may be schemes for assigning numeric or letter values to such things, there is no real clarity as to what such symbols actually denote. They may or may not even be valid for simple comparisons.

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: