RoR: Forum

My definition: "Measurement is a quantity in terms of real numbers and a standard of magnitude that serves as a unit, with multiples of the unit being additive and subtractive."

Does this allow for measurement systems that are non linier like decibels or the Richter scale?

(I'm asking, not being rhetorical)

Post 1

Thursday, November 24, 2005 - 6:02am

Thank you for the excellent exposition, Merlin! I found it enlightening and edifying. I am smarter for having read it. This day was not going well, but you turned it around for me.

Keith: If I may, I might point out that via logarithms, etc., you can make anything "additive" and basically, the operations of exponentiation and multiplication are defined as kinds of repeated additions. (Likewise, subtraction is just a kind of addition.) But your point is well made and it occured to me, too, as I was reading the article.

Finally, perhaps my favorite example of errant psychology is William Sheldon's somatotyping. Sheldon also invented the bizarre 70-point grading scale loved by American coin collectors. He photographed college freshmen in the nude to develop metrics for psychological profiles. Somewhere are naked pix of Nora Ephron, Judith Martin, and Hillary Rodham. See here for how coins and nudes are intertwined.
http://www.maineantiquedigest.com/articles/shel0298.htm
or here:
http://tafkac.org/collegiate/ivy_league_nude_photos.html

Also, for those who believe that epistemology has moral consequences, Sheldon was posthumously condemned in court as a thief and his stolen property -- subsequently sold via auctions after his death -- was returned the American Numismatic Society.

(Edited by Michael E. Marotta on 11/24, 6:05am)

Post 2

Thursday, November 24, 2005 - 6:34am

Keith Phillips asked if the Richter scale fits my definition of "measurement." Good question and one I expected. Based on what I know about it, I'm inclined to call it a borderline case. It is a ranking or scaling calculated from measurements of amplitudes of seismic waves that do fit my definition. Note that these amplitudes are found. My guess is that Richter decided that using a linear scale would sometimes give amplitude values so large they are inconvenient to use. Therefore, he decided to convert such values to a base-10 logarithmic scale for easier comprehension and expression. It seems he could have chosen scientific notation, e.g. NN.NN x 10^M (N and M numbers and ^ for exponentiation), which would have not masked the linearity. However, he did not choose that route.

You're welcome, Michael M!

Soon I will be preoccupied with Thanksgiving Day festivities for most of the rest of the day, so my replies will be delayed.

Happy Thanksgiving to all!

Post 3

Thursday, November 24, 2005 - 10:40am

Merlin,

Thanks for the nice article. Interesting, I was talking to my wife Karen a couple of weeks ago and she was talking about the different ways they measure larger and larger parcels of land. At some point they run into trouble because of the curvature of the earth. She asked, what makes a measurement anyway? I said a true measurement has to have a reference unit, like feet, pounds, seconds, etc. If there's no reference, then it's not a true measurement.

Regarding decibels: You state "If P claims that X is measurement, then P should be able to name the unit."

This is true of the dB unit. However, dB values only make sense if you name the reference unit and the quantity of reference unit you are comparing a measured value to.

For instance we may define 1 milli Watt of RF power to be 0 dB. If we want to compare some other RF power to our 1 milliWatt reference we do the following calculation: X dB[1mW] = 10 times log X (milliWatts)/1milliWatt. A measurement of greater than 1mW is represented by a positive dB number, a measurement of less than 1mW is represented by a negative number. Many orders of magnitude of powers both less than and greater than our reference can be represented in simple comparable units. Quite convenient. It is not a borderline case at all because dB's are easily converted to real units of power (or voltage or whatever).

Post 4

Thursday, November 24, 2005 - 12:28pm

The points in this article cannot be addressed without asking, what is essential in measurement - and how does measurement differ from other evidence obtained by the senses? As with much else recently, I'll address this when I have time to write about it at adequate length. I am grateful to Merlin for setting the stage.

Post 5

Thursday, November 24, 2005 - 5:59pm

Keith wrote:

"Measurement is a quantity in terms of real numbers and a standard of magnitude that serves as a unit, with multiples of the unit being additive and subtractive.

Does this allow for measurement systems that are non linier like decibels or the Richter scale?"

I would disagree regarding "real" numbers. Imaginary or complex numbers measure a point on a circle.

The "standard of magnitude that serves as a unit" is confusing. I would say measurement is the non-arbitrary magnitude of the units being measured.

There are linear measurements, logarithmic measurements (decibels), even (contrary to the article) statistical measurements - "standard deviations". The more complex logarithmic measurements are transforms of the simpler, but are still true, as long as the relationship is defined.

In general relativity, measurements are in curved space along a path (geodesic) of least-action.

Adam wrote:

what is essential in measurement - and how does measurement differ from other evidence obtained by the senses?

That's the issue - what is essential. Some quantity to measure, and a standard unit to measure with.

Thinking about Goedel's theorem and Turing machines, I'm doubting the reality of "whole" or "integer" numbers!

Yea, sure it looks like you have one or two or three... discrete fingers or toes. But what defines the transition where one begins and the other ends?

What about particles? If you squeeze one or two or three... discrete particles close enough, there starts to become probabilities quantum states could begin blurring - you describe them as waves and probabilities of observing them as discrete or tangling entities.

Even quantum numbers like spin, particle charge or magnetic moment, though discrete for particles, are dictated by the nature of the geometry of interacting fields composing the particle.

So as the quantity of mass-energy increases distance in space and time, approximating the quantities with integers becomes valid.

AFAIK, no ultimate quantized "particle" is known.

Scott

Post 6

Friday, November 25, 2005 - 6:06am

Scott Stephens wrote:

I would disagree regarding "real" numbers. Imaginary or complex numbers measure a point on a circle.

I have no idea what your second sentence means. Please elaborate. If you claim there is such a thing as a measurement in terms of complex numbers, please give a clear concrete example.

The "standard of magnitude that serves as a unit" is confusing.

What exactly do you find confusing? Consider a few concrete examples of measurement units and maybe it will be clearer.

I would say measurement is the non-arbitrary magnitude of the units being measured.

What qualify as units? Is that a definition? If yes, it's circular. Also, "units being measured" sounds strange. One measures an attribute in terms of units.

There are linear measurements, logarithmic measurements (decibels), even (contrary to the article) statistical measurements - "standard deviations". The more complex logarithmic measurements are transforms of the simpler, but are still true, as long as the relationship is defined.

Regarding logarithmic and statistical measurements, you are free to use "measurement" metaphorically. My article concerns literal meaning, not metaphors. About the logarithmic Richter scale, it's discussed above and note that it is called the Richter scale, not the Richter measurement. Suppose you calculate the standard deviation for a series of random numbers. The result is a pure number. My definition of "measurement" also requires a standard unit of magnitude, an analog to inch, gram, mph, etc. That's why I consider statistics quantities but not measurements. Of course, statistics may be about measurements.

Post 7

Friday, November 25, 2005 - 10:05pm

Merlin,

No doubt there is much to analyze and discuss. I'd rather not get pedantic with semantics.

First, regarding examples of measurements of complex vs. real numbers, consider the electrical power grid. "Power Factor" is a measure of reflected power. In the most common application, flourescent or HPS lights in a warehouse. Some have many inductive ballasts that reflect energy, create a phase-shift between voltage and current.

http://en.wikipedia.org/wiki/AC_power

I don't have time to explain. Here:
http://www.google.com/search?sourceid=mozilla&q=%22power%20factor%22%20lighting

http://www.google.com/search?hl=en&lr=&q=%22power+factor%22+lighting+%22complex+numbers%22&btnG=Search
http://en.wikipedia.org/wiki/AC_power

Any further questions regarding complex vs. real power mesurement?

Scott

Post 8

Saturday, November 26, 2005 - 5:49am

Scott, thank you for the links.

I thought my definition might be too restrictive.

1. I included "real number" to exclude ordinal numbers, which don't have arithmetic properties and are often subjective. The electrical power example ("apparent power") seems to warrant not excluding complex numbers. The drawback, of course, is it could much complicate the definition.

2. It could be construed to improperly exclude "derived measurements" like the Richter scale numbers, decibels, and apparent power. I favor including ones like the first two, since the exponent part has arithmetic properties. Also, underlying all three are (non-log, non-derived) measurements which do fit my definition. Again, it could much complicate the definition.

"Apparent power" seems to describe a potential value, in contrast to the real valued units of measure watt, volt and ampere that underlie it. Is that accurate?

Post 9

Saturday, November 26, 2005 - 3:35pm

Merlin,

"Apparent power" seems to describe a potential value, in contrast to the real valued units of measure watt, volt and ampere that underlie it. Is that accurate?

I haven't heard the term "Apparent power" before.

There is another term VAR - volt-amps-reactive. Motor-start capacitors have that rating.

You see, an inductor or capacitor store current or voltage, then discharges it. There isn't anything "apparent" or "imaginary" about it! Its quite real, I assure you!

If you don't think so, try using an under-rated component in a high-power application and you will get a violent object lesson in the reality of reactively-stored energy!

"Real" power is absorbed power, power converted to heat by a resistor, or light, or kinetic energy by a motor. Reactive power is *REAL* electrical power bouncing around; taking the form of magnetic flux in an inductor, or electrostatic potential in a capacitor.

And if you whiz around an inductor or capacitor at relativistic speeds, the electric and magnetic fields change from one to the other, depending on reference frame.

Which demonstrates the real dimension of "charge" is displaced, and is defined according to reference frame.

Scott

Post 10

Saturday, November 26, 2005 - 4:12pm

Scott Stephens wrote:

I haven't heard the term "Apparent power" before.

It was at the URL, http://en.wikipedia.org/wiki/AC_power, you gave in post #7.

There is another term VAR - volt-amps-reactive. Motor-start capacitors have that rating.

Yes, that is the unit for reactive power at the same URL.

You see, an inductor or capacitor store current or voltage, then discharges it. There isn't anything "apparent" or "imaginary" about it! Its quite real, I assure you! If you don't think so, try using an under-rated component in a high-power application and you will get a violent object lesson in the reality of reactively-stored energy!

I didn't say "real power" was "apparent" or "imaginary." The webpage used "apparent power", and the formula for it included an imaginary number. I did use "potential" (in regard to "apparent power"), but that does not mean imaginary or unreal.

Post 11

Wednesday, December 7, 2005 - 11:59am

Merlin defines measurement as "a quantity in terms of real numbers and a standard of magnitude that serves as a unit, with multiples of the unit being additive and subtractive." There is a better definition of measurement than this one.

The standard basic reference for measurement theory today is the monumental 3-volume work Foundations of Measurement. The authors of this work are D. Krantz, R.D. Luce, P. Suppes, and A. Tversky. Volume I appeared in 1971. On the first page, the authors define measurement as: the association of numbers (or other familiar mathematical entities, such as vectors), to the attributes of some class of objects or events "in such a way that the properties of the attribute are faithfully represented as numerical properties."

Volume I treats additive and polynomial representations. Volume II (1989) treats geometrical, threshold, and probablistic representations. Volume III (1990) treats nonadditive representations and foundational issues. One will find also in this last volume the why and wherefore of the modern classification of measurement-scale types, which has supplanted the classification of S.S. Stevens.

Merlin observes that Steven's definition of measurement ---assignment of numerals to objects or events according to a rule---is too loose. Merlin's definition (and that of Joel Michell) is too tight. The definition that has been torqued just right is that of Krantz, Luce, Suppes, and Tversky.

Under the modern classification, in a mathematically sensible order, the (one-dimensional) types of measurement scales are: absolute, ordinal, hyperordinal, interval, and ratio. Correctness of application of a given scale type is determined by the character of the attribute being measured (and not by the level of our understanding of the attribute). Foundations of Measurement teaches us how to apply the right scale type to the attribute at hand.

Rand's thesis that concepts can be cast in terms of (omission of) measurements at the level of the ordinal scale type and above is a meaningful and substantive proposal. To see what Rand's theory can do when modern measurement theory is brought to bear in it, see my essay "Universals and Measurement" in The Journal of Ayn Rand Studies (V5N2). Visit the JARS site for ordering information.

Post 12

Wednesday, December 7, 2005 - 2:56pm

Stephen,

You did not find my article persuasive? :-) Sorry, friend, but I do not find your reply or Foundations of Measurement persuasive either. How you use "measurement" is your choice, and how I use it is mine. So I guess our disagreement will remain just that.

I've read significant chunks of Vol.1 of Foundations of Measurement. Krantz et al are psychologists, so it is no surprise that they define "measurement" in a way acceptable to psychologists. I find their definition very vague. What exactly does "properties faithfully represented as numerical" mean? Does it mean one can arbitrarily label objects or events with numbers and call it "measurement", as long as it's done "in good faith"?

I don't have a copy to cite the page, but early in Foundations of Measurement Krantz et al start addressing "ordinal measurement." The treatment is purely a mathematical one. Not one word is devoted to an analysis and contrast of the differences between it and true measurement qualitatively or philosophically. There is nothing about how the concepts map to concrete reality or how the numbers are obtained. They simply assume that ranking is measurement, which is exactly what my article disputes. Ranking is quantification, but not measurement.

I haven't seen Vol.3, since it's very hard to find. But if the "supplanted classification of S. S. Stevens" is simply Stevens' categories plus hyperordinal, not much was supplanted.

You mention your article in JARS about Rand's claim of "measurement omission" in concept formation. So I will mention mine -- there is a link for it in my profile. Readers can come to their own judgments about the merits or demerits of each.

Post 13

Thursday, December 8, 2005 - 1:58pm

Stephen,

Thanks for the reference and definition. Though I'm inclined to side with Merlin's position. As I understand it, he objects to calling an arbitrary or coincidental mathematical correlation a measurement. As a matter of aesthetics, I immediately agree. But I should identify and define why.

I gather from "Foundations..." the authors have a quite natural abstract/material, analytic/synthetic premise. That mathematic abstractions have non-material origin. Which I suppose is appropriate enough, since the work doesn't address philosophy.

The Objectivist position is numbers, mathematical functions and abstractions, do not exist apart from concretes. And the action preformed by our brains (computers) is to *act* on (external or internal) perceived differences in physical concretes, according to our values and volition.

Our brains have a metaphor, a model, of what we are measuring. We observe a spring and mass bouncing, and in our minds, neurons fire in synchrony and we learn what to expect, how to derive units, how to model, and what ratio of units is appropriate to create measures of analogous phenomena.

The act of measurement is a back-reference, a post-hoc observation, an affirmation of identity, according to similar truths we've already learned. We can measure a volt here or there, and its the same, because an electron here or there is the same, because charge, mass, space and time are the same here or there. What is different is time, location and the flow of energy in its fields.

No doubt physical laws are the universals in existence which enable us to recognize the differences in the fields, particles and energy, which vary through (I should say actually are) space and time.

I will change my definition of measurement, "a ratio between differentiable units". Returning to my thoughts on the nature of integer numbers, is an integer anything different than the observation of the ratio of one complete cycle between different units? Why do you have to use "one" cycle as a unit standard? Why not 1/3, or Pi/4?

We don't have to make life hard, expecting integers to be everywhere just because we don't have webbed fingers and toes!

Thanks for reminding me about apparent power Merlin. Its another rant, but I've always had a problem with imaginary numbers and power.

Power is energy * time. Well, in a loss-less circuit, no "real power" is being dissipated. To an inductor, a capacitor discharging through it does "work" creating flux. And if to a capacitor, an inductor sourcing current is doing "work", giving it energy over time, charging it up. It only from the perspective of a resistive load that power is removed from the system. Transduced.

So "real" or "imaginary" is a kind of arbitrary choice of perspectives. And "i" is only used in the context of complex (two - component) quantities. Saying there is such a thing as a negative scalar or single-valued square-root seems to me context dropping, and a violation of the nature of (single-valued) multiplication. But I'm no mathematician, so I better shut-up now.

Scott

Post 14

Saturday, September 13, 2008 - 1:23pm

Merlin,

In post 12 you said, "Krantz et al are psychologists, so it is no surprise that they define "measurement" in a way acceptable to psychologists."

Hey, I resemble that remark! Please don't lump all psychologists together.

I have my own pet peeves about statistical abuses in my field that go far beyond anything you are describing. For example, what about the psychologist that went into a prison and asked inmates convicted of violent crimes to answer on a scale as to what degree they liked or disliked themselves. Then, based upon the results, "proved" that previous surmises of low self-esteem relating to violent crime were false, and actually claimed that high self-esteem might be a cause of violent crime!

Post 15

Monday, September 15, 2008 - 2:59pm

Merlin,

There's a fair chance I don't understand your article, but...

I think you say that measurement requires a standard unit. The problem is that the unit itself must be "measured", thus creating a circular or regressive problem, for to measure the unit, one must first have a unit by which to measure it, and a unit by which to measure that last unit, etc.

One way around this might be to subjectively stipulate the unit's size, then go from there, but I'm not sure you want this subjectivity, and moreover, I'm not sure you can, under your view, stipulate the unit's size without resorting to some other unit anyway.

What underlies unitization might well underlie ordinality.

Jordan

Post 16

Monday, September 15, 2008 - 7:07pm

Jordan wote:

I think you say that measurement requires a standard unit. The problem is that the unit itself must be "measured", thus creating a circular or regressive problem, for to measure the unit, one must first have a unit by which to measure it, and a unit by which to measure that last unit, etc.
One way around this might be to subjectively stipulate the unit's size, then go from there, but I'm not sure you want this subjectivity, and moreover, I'm not sure you can, under your view, stipulate the unit's size without resorting to some other unit anyway.

No, the unit chosen is not measured. It is what one uses to measure. To some degree the unit chosen is subjective. For example, for lengths in a moderately small range, one could use an inch or centimeter as the standard. In a larger range, one could use a yard or meter. In a still larger range, one could use mile or kilometer. Once the unit is chosen, however, how one uses it is not arbitrary.

Post 17

Tuesday, September 16, 2008 - 6:34am

I think Jordan meant his statements more basically than that (am I right, Jordan?).

I think that he was talking about the length we "choose" to be that length that will, in our future discussions of lengths, shall henceforth be considered an inch, or shall henceforth be what it is that we will all agree will be considered to be a foot (from here on out).

Ed

Post 18

Tuesday, September 16, 2008 - 7:05am

Ed wrote:

I think Jordan meant his statements more basically than that (am I right, Jordan?).

Maybe. But he used "circular or regressive problem" and there is no circularity in my response in post 16.

Post 19

Tuesday, September 16, 2008 - 8:02am