Subject: Re: Eight Features of an Ideal Intro Stat Course
         (Response to comments by Ronan M. Conroy)

     To: EdStat E-mail List and Newsgroup

   From: Donald B. Macnaughton <>

   Date: Tuesday February 6, 2001
     Cc: Ronan M Conroy 


In a 99/5/9 post I suggest that relationships between variables 
are more interesting for students than univariate distributions.  
Referring to this point, Ronan Conroy writes (on 99/5/10)

> This is a fundamental point.  Ampere, in a crucial development
> in the philosophy of science, pointed out that science does not
> study things in themselves, but the relationships between their
> attributes. 

The "things" Ronan refers to are what I call "entities".  The 
"attributes" are (essentially) what I call "properties" or "vari-

(I discuss why I recommend the word "entity" over other candidate 
words in a paper [1998, app. E.1].  I recommend the word "prop-
erty" over the word "attribute" because word frequency statistics 
suggest that beginners are more familiar with the former word 
[Breland and Jenkins 1997].  I discuss my view of the relation-
ship between the words "property" and "variable" in a paper 
[1998, app. E.2].)

I fully agree with Ronan's point that science (together with all 
the other branches of empirical research) generally studies rela-
tionships between properties (or relationships between attributes 
or relationships between variables).  I discuss this idea further 
in a paper (1999, sec. 3.4).

André-Marie Ampère (Andre-Marie Ampere, 1775-1836) emphasized the 
concept of a relationship between a certain type of entities -- 
entities he called "phenomena" (Williams 1970, 1989; Hofmann 
1996).  Since both variables and properties can be conceptually 
viewed as "phenomena", Ampère was clearly close to the concept of 
'relationship between variables'.

Probably Ampère viewed "phenomena" as being equivalent to what 
modern speakers of English refer to as "variables".  However, I 
have been unable to confirm this because Ampère wrote in French, 
and my French is weak, and I have not found any English transla-
tions of Ampère's detailed philosophical writing.  So I have been 
unable to directly study his philosophical writing to see how he 
characterized his concept of 'phenomena'.  Do any readers have 
further information about Ampère's characterization of phenomena 
or about translations of his philosophical writing?

                            *   *   *

Ampère brilliantly recognized and described the scientific method  
(Williams 1989).  The method (which is also called the 
hypothetico-deductive method of science) consists of the follow-
ing four steps:

1. A researcher frames (i.e., invents) a new hypothesis about 
   some area of experience.  (The researcher frames the hypothe-
   sis on the basis of knowledge of earlier research, intuition, 
   and logic.)

2. A researcher (perhaps the same researcher) then deduces or in-
   fers an empirically testable implication of the hypothesis. 

3. A researcher (perhaps the same) then performs an empirical re-
   search project to test whether the implication is actually 
   present in the area of experience.  

4. If evidence of the implication is found (and in the absence of 
   a reasonable alternative explanation), the community with 
   prime interest in the area of experience will (by informal 
   consensus) accept (or will be more inclined to accept) the hy-
   pothesis framed in step 1 as being correct.

Ampère's description of the scientific method is important be-
cause most modern scientific research (and most other empirical 
research) proceeds formally according to this method.

(I discuss the formal and informal aspects of the progress of 
science in appendix A.)

Examination of instances of the use of the scientific method sug-
gests that the implication in step 2 can usually be usefully 
viewed as a statement of a relationship between variables in some 
population of entities.  Thus the key statistical concept of 're-
lationship between variables' plays a central role in the scien-
tific method.

                            *   *   *

Ampère's empirical interests were in part in the branch of phys-
ics known as electrodynamics, which he founded (Williams 1989).  
Perhaps his most important contribution in electrodynamics is the 
discovery of the following relationship between variables:

     The magnetic force between small elements of two cur-
     rent-carrying conductors is inversely proportional to 
     the square of the distance between them and is directly 
     proportional to the product of the currents.  (Ampère 
     also described how the three angles that define the 
     relative directions of the two conductors affect the 
     size of the force.)

The force described in this relationship is important because it 
is the force that drives almost all electric motors.  Scientists 
have recognized the importance of Ampère's work in electrodynam-
ics by naming the unit of electric current, the ampere, in his 
                            *   *   *

Ampère's contributions lead to questions about the history of re-
lationships between variables.  Researchers have been formally 
studying such relationships for centuries (with differing degrees 
of emphasis in different eras).  For example, the Pythagorean 
theorem describes a formal relationship between variables that 
was discovered by Pythagoras in the 6th century BCE.

Going back further still, humans (without being aware of it) have 
been informally using relationships between "variables" since 
when we began to reason.  The aphorisms "haste makes waste" and 
"power corrupts" are good examples of informal statements of re-
lationships between variables.

The use of relationships between variables is not limited to hu-
mans and may also exist in animals.  That is, we could build 
plausible models of some animal reasoning in terms of an animal 
(unconsciously) learning about simple relationships between 
"variables" that reflect properties of its habitat.  The animal 
uses the knowledge of the relationships to aid in its survival.  
For example, an animal might learn a relationship between the 
time of day and the likely location of prey.  Such a learning ap-
proach to survival enables an animal to adjust as quickly as pos-
sible (sometimes within seconds or minutes) to unpredictable 
changes in its habitat, such as changes due to a fire, a flood, 
or an epidemic among its prey.  These changes invariably cause 
new relationships between variables to arise.  It is in an ani-
mal's survival interest to learn as quickly as possible about all 
relationships that have response variables that affect its sur-
vival or reproductive capability.

Viewing learning as learning about relationships between vari-
ables is also reflected in the literature of machine learning.  
For example, Vapnik, in the first sentence in the first chapter 
of his book The Nature of Statistical Learning Theory, writes

    In this book we consider the learning problem as a prob-
    lem of finding a desired dependence using a limited num-
    ber of observations (2000).

Vapnik goes on to show that the "dependence" he refers to is a 
dependence (i.e., a relationship) between a single response vari-
able and zero or more predictor variables.

                            *   *   *

Ampère studied relationships between variables in the field of 
physics which (together with chemistry, astronomy, and engineer-
ing) is one of the physical sciences.  As I discuss in a paper, 
most empirical research in the physical sciences can be usefully 
viewed as studying relationships between variables (1999, app. 

When physical scientists study relationships, the standard sta-
tistical concepts of 'population' and 'sample' are often fuzzy or 
not present.  This is because in the physical sciences any "sen-
sible" sample of entities will often perform quite well as a rea-
sonable sample from the (implicit) population.  

For example, Isaac Newton's law of cooling states that the rate 
of cooling of a physical object when it is hotter than its envi-
ronment is proportional to its excess temperature.  The model 
equation for this relationship between variables is

                         r = k * deltaT

        r is the rate of cooling (in degrees per unit of time)

   deltaT is the excess temperature of the physical object above 
          the ambient temperature and

        k is a constant. (k is constant within any given physical 
          object, but varies from one physical object to the 

Newton derived this relationship from empirical research in which 
he measured the excess temperature and computed the rate of cool-
ing at several times in a sample of relatively hot physical ob-
jects.  In all the cases in the sample he found strong evidence 
that this specific relationship between variables was present.  
Thus Newton generalized the findings to his law.  Because the law 
has been repeatedly verified, physicists believe it applies to 
all macroscopic physical objects in the population of physical 
objects at all times.

(It is possible [and may be known] that the constant k in 
Newton's law of cooling is not a perfect constant.  Instead, k 
may be better viewed as varying slightly as a function of tem-
perature, excess temperature, or possibly other variables.  All 
relationships between variables may be refined as measurement 
technology improves.)

Newton's discovery of the law of cooling illustrates how in em-
pirical research in the physical sciences we may not need to pay 
much attention to the population and sample -- the relationship 
between variables under study is exactly the same throughout a 
very broad population.  On the other hand, in most other areas of 
empirical research (including the biological and social sciences) 
the definitions of the population and sample are important be-
cause in these areas a relationship between variables may change 
substantially if we change to a slightly different population or 

                            *   *   *

If we use the instruments and procedures Newton used to investi-
gate the law of cooling, we find that the error variation in the 
relationship in research data reflecting the law is negligible.  
Thus when Newton discovered the relationship he did not need to 
use statistical methods to study the relationship.  Instead, he 
could see strong support for his conclusion that the relationship 
reflected in the law is present by merely scanning his research 
data, or by a few simple mathematical operations on the data, or 
by plotting the data on scatterplots.

More generally, because of the (usually) low error variation in 
relationships between variables studied in the physical sciences, 
physical scientists often do not use statistical methods beyond 
standard graphics to study relationships between variables.  (To 
fit model equations they sometimes use homegrown algebra and 
sometimes statistical methods.)  On the other hand, in other ar-
eas of empirical research the error variation is usually large 
enough that the use of statistical methods greatly simplifies the 
study of relationships between variables.  In addition, statisti-
cal methods can reliably detect subtle phenomena in data that may 
otherwise go unnoticed.

                            *   *   *

Researchers in the physical sciences may have difficulty if they 
omit using statistical methods -- witness the cold fusion contro-
versy.  The key issue in this controversy can be usefully viewed 
as whether certain unexpected relationships between variables are 
present in a cold fusion cell.  

A cold fusion cell is an electrolytic cell with a palladium cath-
ode and often with a platinum anode and containing a solution of 
deuterium (heavy water) and lithium deuteroxide.  The original 
cold fusion researchers reported that, under certain conditions, 
if a cold fusion cell was connected to a source of electric cur-
rent, the heat energy output (response variable) was substan-
tially greater than the electrical energy input (predictor vari-
able).  Such a relationship between variables contradicts ac-
cepted physical theory.  The original researchers believed that 
the (perceived) unexpected extra energy output of the cell was 
due to the nuclear reaction of fusion taking place in the cathode 
-- nuclear fusion gives off heat.  Were it true that nuclear fu-
sion was occurring in these cells, it would have been extremely 
important because it would suggest that we could use cold fusion 
as a new safe and inexpensive source of energy.

Some researchers reported that (using their own cold fusion 
cells) they were able to replicate the relationship between vari-
ables reported by the original researchers.  But many other re-
searchers reported that they were unable to replicate the rela-
tionship (and were unable to demonstrate other implied relation-
ships between variables).  In the end, the weight of evidence led 
most physical scientists to conclude that the claimed new phenom-
ena (which can be usefully viewed as relationships between vari-
ables) are not present (Huizenga 1993).  

The cold fusion controversy lasted several years and consumed a 
substantial amount of physical scientists' resources.  As sug-
gested by the government panel that investigated cold fusion, the 
high resource consumption may have been partly because some re-
searchers involved in the controversy were not using proper sta-
tistical methods to determine whether the claimed new relation-
ships between variables were present (Energy Research Advisory 
Board 1989, app. 2.C).  

If the original cold fusion researchers had used proper statisti-
cal methods for detecting relationships (i.e., statistical tests, 
a proper taking account of negative results, and experimental de-
sign considerations), it seems likely that the repeated high or 
borderline p-values would have quickly alerted them to the fact 
that that the sought-after relationships between the variables 
were probably not present.  Or if the relationships were present, 
they were not properly detectable with the current experimental 
methods, and thus further "more powerful" research must be per-
formed before a responsible positive conclusion could be drawn.  
This might have saved the original researchers considerable em-
barrassment and might have saved the physical science community 
substantial costs.

Omission of the use of statistical methods is not the only possi-
ble cause of the cold fusion errors.  As also discussed by the 
Energy Research Advisory Board panel, some of the mistaken posi-
tive conclusions may have arisen through errors that were made in 
the measurement methods the original researchers used -- errors 
that caused the measured net energy output of some cold fusion 
cells to be over-estimated (1989, sec. II, app. 2.C).  

Cold fusion research is associated with the sub-domain of chemis-
try called "electrochemistry".  But regardless of whether the do-
main is chemistry, or medicine, or psychology, or any other area 
of empirical research, a researcher can marry statistical methods 
with proper measurement methods in the domain.  This marriage, 
when tempered with the proper cautions, yields optimal empirical 
procedures to detect and study relationships between variables.  

> An attribute is the relationship between a measurement process
> and the thing to be measured, 

Ronan gives a definition of the concept of 'attribute' or 'prop-
erty'.  He defines the concept in terms of three other concepts, 
namely, 'relationship', 'measurement process', and 'entity' 
(i.e., "thing to be measured").  

Ronan uses the concept of 'measurement process' to define the 
concept of 'property'.  Thus he appears to view the concept of 
'measurement process' as being more fundamental than the concept 
of 'property'.  Ronan's view does not seem to be contradictory, 
but it is different from mine.  I view the concepts of 'entity' 
and 'property' as being verbally undefined, just like the con-
cepts of 'mass', 'length', 'time', and 'temperature' are usually 
verbally undefined in physics.  

(Mass, length, time, and temperature are all properties of enti-
ties.  Mass and length are properties of physical objects.  Time 
[duration or point in time] can be viewed as a property of events 
or a property of "reality" [the entity that contains all other 
entities].  Temperature can be viewed as a property of environ-
ments.  The ways of measuring these properties are clearly de-
fined in physics, but the concepts themselves are verbally unde-
fined.  I discuss some possible objections to these points in ap-
pendix B.)

The concepts of 'mass', 'length', 'time', and 'temperature' are 
verbally undefined in physics because it is not possible to ver-
bally define every concept in a field without introducing unde-
sirable circularity.  Therefore, physicists have been forced to 
choose certain of their concepts to be verbally undefined.  It 
makes sense to choose the simplest concepts in a field to be ver-
bally undefined and then, when possible, to define the more com-
plicated concepts in terms of the simpler undefined concepts.  
Hence the concepts of 'mass', 'length', 'time', and 'temperature' 
are verbally undefined in physics.

Dictionaries generally give definitions of the four undefined 
concepts because such definitions provide the most benefit for 
dictionary users.  However, examination of the definitions and 
examination of the definitions of the terms used in the definien-
tia invariably reveals (of necessity) that the set of definitions 
associated with any of the four concepts is circular.  Thus with-
out outside help we cannot use these definitions to explain or 
understand the four concepts.  Thus although the four concepts 
have dictionary definitions, they are (due to the circularity of 
the definitions) still effectively undefined.

If no (non-circular) verbal definition is available for a par-
ticular concept, how can we understand it?  We can understand a 
verbally undefined concept through "ostensive" definitions of it.  
An ostensive definition is one that does not rely on words.  In-
stead, the definition works through the defining person physi-
cally pointing at the thing being defined, or it may work through 
some other form of direct experience.  The direct experience al-
lows one to learn the concept without having to use words (except 
for the word or words naming the concept being defined).

For example, although the fundamental properties of 'mass', 
'length', 'time', and 'temperature' have no verbal definitions in 
physics, they have ostensive definitions.  These definitions are 
not given in words and thus do not appear in formal discussion.  
(The definitions are generally taken for granted in formal dis-
cussion.)  Instead, the four definitions are given through direct 
experience -- experience that occurs many times for most people 
before they are eight years old.  

For example, consider the property of the mass of a physical ob-
ject.  (As one learns in high-school physics, this property is 
basically the same as the property of the weight of a physical 
object.  The difference between these concepts, which is some-
times important, is not relevant here.)  We can teach the prop-
erty of mass to an interested child who does not understand it by 
handing them various physical objects and saying "heavy", "very 
heavy", "light", and so on, as appropriate.  Children can quickly 
learn any perceivable property through such ostensive definitions 
if enough opportunities are available for them to properly dis-
criminate different values of the property directly.

As infants we learn our first words (e.g., "mommy", "TV", 
"green") through informal ostensive definitions that occur in the 
speech and actions of the people around us.  Furthermore, most 
people learn almost all the words they use in their day-to-day 
conversation through ostensive definitions, not through verbal 
definitions.  Thus ostensive definitions are more basic than ver-
bal definitions.

In an approach that is identical to the approach taken in phys-
ics, I view the concepts of 'entity' and 'property' as verbally 
undefined fundamental concepts in empirical research (and also 
verbally undefined fundamental concepts in broader day-to-day hu-
man reality).  Humans understand these concepts through the many 
ostensive definitions of them we experience and because the con-
cepts are the basis for a substantial part of our thinking.  

I further discuss the role of the concepts in thinking in the 
1999 paper (sec. 3).

                            *   *   *

Ronan's definition introduces the important concept of a "meas-
urement process" -- a process that we use to measure the values 
of a property.  For any given property of entities, researchers 
may view the property as if there is one specific property, but 
there may be several ways of measuring the value of the property 
in entities.

For example, we can measure a person's weight (or mass) using a 
standard bathroom scale (which may be a spring balance), or we 
can use a pan balance in which we balance the person's weight 
against known weights, or we can use a weight guesser from a car-
nival.  All these methods are valid methods for measuring a per-
son's weight in the sense that they yield estimates that are 
highly correlated with each other.  However, these measures are 
of different quality in the sense that they have different preci-
sion (i.e., high or low random variation about their expected 
value) and different bias (i.e., expected deviation from the 
"true" value).

                            *   *   *

The preceding discussion leads to the philosophical question of 
what it means to speak of the "true" value (at a given time) of a 
property of an entity.  This question breaks into two cases:  the 
case in which a property is represented by a discrete variable 
and the case in which a property is represented by a continuous 

With a discrete variable we can certainly speak of and sometimes 
even know the "true" value of the underlying property.  For exam-
ple, if we wish to know the number of multiple-choice questions a 
student answered correctly on a test, we can, if we are careful 
enough, know the "true" value of this number (which can be use-
fully viewed as a property of the student).

In the more interesting case of a continuous variable, empirical 
researchers usually define the "true" value of a property in 
terms of some commonly-agreed-upon measurement approach because 
this facilitates communication and understanding.  For example, 
researchers in the physical sciences usually define the "true" 
values of the properties they study in terms of the definitions 
and standards provided by the International Bureau of Weights and 

To illustrate measurement standards, suppose we decide to measure 
the mass of physical objects using the kilogram as the unit of 
mass.  The exact formal English version of the definition of the 
kilogram in the International System of Units (SI) is as follows:

    The kilogram is the unit of mass; it is equal to the mass 
    of the international prototype of the kilogram (BIPM 

The international prototype of the kilogram is a platinum-iridium 
cylinder kept under carefully controlled conditions in a labora-
tory of the International Bureau of Weights and Measures in 
Sèvres, France.  The national standards organizations of most 
countries of the world accept the definition that this cylinder 
has a mass of exactly one kilogram.  

Using an accurate balance it is possible to make copies the in-
ternational prototype of the kilogram so that the copies have 
masses that are very close to the mass of the prototype.  Several 
such copies and many copies of these copies have been made.  From 
these copies it is possible to create other physical objects that 
have masses that are known multiples of one kilogram or that are 
known fractions of one kilogram.  

After we have obtained appropriate accurate masses we can use 
them to calibrate other very accurate mass-measuring instruments.  
(Such instruments, when properly calibrated, are more efficient 
than using a pan balance with weights.)  The "true" mass in kilo-
grams of any physical object is the mass we would get if we were 
to measure the object with a calibrated mass-measuring instrument 

1. we had made no errors (no matter how small) in making our sec-
   ondary masses from the international prototype of the kilogram 

2. we had made no errors (no matter how small) in calibrating the 
   instrument with the secondary masses AND 

3. the instrument perfectly interpolates between or extrapolates 
   away from it calibration point(s) AND

4. the instrument has precision (at least) to the level of the 
   mass of subatomic particles.  

Clearly, none of the four requirements can be satisfied.  Thus 
even with mass -- a fundamental concept of physics -- we cannot 
determine the true value of this property for a given physical 
object to perfect precision.  Nor does it seem likely that humans 
ever will determine the "true" mass of an arbitrary physical ob-
ject or the true value of any other continuous property.  

However, if we are careful with our measurement procedures, we 
can estimate the true value of any property to a very high preci-
sion, which is usually all we need.  For example, currently the 
best mass-measuring instruments for small objects can accurately 
resolve ten or more significant digits of mass (BIPM 2001b).  It 
is hard to imagine situations in which we would need more accu-
racy than that.

Although we can generally never know the true value of a continu-
ous property of an entity, defining such "true" values is a use-
ful idealization because it encourages us to strive to accurately 
measure the values.  This leads to more accurate knowledge of re-
lationships between variables, which leads to better ability to 
predict and control. 

Donald B. Macnaughton   MatStat Research Consulting Inc      Toronto, Canada


In the body of this post I note that most empirical research pro-
ceeds formally in terms of the scientific method.  However, most 
empirical research actually proceeds quite differently, with many 
false starts and with frequent serendipity.  But the actual pro-
gression of the work is less important because the community with 
prime interest in a research area usually concentrates its atten-
tion on the formal descriptions of empirical research in the 
area.  (The formal descriptions are generally given in terms of 
the scientific method and often first appear as papers presented 
at meetings and then as journal articles.)  The community concen-
trates on formal descriptions because experience has shown that 
this approach is generally most efficient.

In sharp contrast, at what can be viewed as the highest level of 
scientific thinking, the decision to accept a scientific hypothe-
sis is made informally.  The decision is made by the scientific 
community with prime interest in the area.  (For example, a deci-
sion about a chemical hypothesis is made informally by the commu-
nity of chemists.)  When scientists decide to accept a hypothe-
sis, no formal declaration is ever made that the hypothesis is 
accepted.  Instead, after perhaps a period of debate, scientists 
in the area simply begin speaking as if the hypothesis is an ac-
cepted fact.  Why are scientists so informal about the important 
operation of accepting a scientific hypothesis?  

Scientists are informal because it is risky to say formally that 
"such and such is so".  This is because we invariably find later 
(perhaps years or centuries later) that "such and such" is not 
always "so" the way we had thought, so we must revise our think-
ing, sometimes drastically.  Knowledge of this leads careful 
thinkers to never think that something is "so".  Thus scientists 
generally do not make formal declarations about the truth of hy-
potheses and instead decide things informally.  All decisions are 
open to revision or reinterpretation if a reasonable alternative 
explanation is brought forward.


This appendix discusses some criticisms that might be made of my 
claim that the properties of 'mass', 'length', 'time', and 'tem-
perature' are verbally undefined in physics and also discusses 
some related ideas.

One could argue that the properties of 'mass', 'length', 'time', 
and 'temperature' are defined by the way they are measured.  That 
is, one could argue that defining the way of measuring mass de-
fines the property of 'mass'.  However, humans do not generally 
learn these simple properties through the way they are measured.  
Instead, we learn them through direct experience.

Thus I believe it is useful to distinguish between 

(a) a person sensing the property of 'mass' by lifting a physical 
    object and 

(b) measuring the mass of a physical object with some (non-human) 
    measuring instrument.  

I believe these are best viewed as conceptually different opera-
tions, with (a) being the fundamental form, and with (b) being 
derived from (a) in order to objectify and refine it.

Thus although we could define mass in terms of how it is meas-
ured, this approach appears to yield a "secondary" definition be-
cause the more fundamental concept of mass for humans is in terms 
of "difficulty of lifting".  Thus I believe the concept of mass 
is best introduced in terms of the well-known (but somewhat 
vague) concept of the difficulty of lifting a physical object.  
After this introduction, we can then immediately discuss the 
various systems humans have invented to help us objectively meas-
ure this important property of physical objects.

                            *   *   *

One could also argue that the property of 'mass' can be verbally 
defined as being equivalent to the concept of 'inertia', which is 
the resistance of a physical object to being accelerated by an 
external force.  The inertial definition of 'mass' is not circu-
lar, although it relies on the concepts of 'physical object', 
'length', 'time', 'motion', 'differentiation' (since acceleration 
is usually a derivative), 'force', and 'frame of reference'.  
However, although this definition is used in physics (where the 
physical objects under study cannot always be readily weighed), 
it does not express the way most humans view weight or mass be-
cause the definition requires that we understand mass in terms of 
what is effectively the concept of 'difficulty of acceleration'.  
But most humans do not conceptualize mass in these terms.  In-
stead, as I suggest above, we conceptualize mass in terms of dif-
ficulty of lifting.  

Difficulty of lifting contains a component of acceleration and 
components of static and moving weight, as you can see by di-
rectly sensing the mass of a moderately heavy object, such as a 
book.  In getting a sense of the mass or weight of a book you may 
hold it still in your hand and sense the tension in the muscles 
in your arm, or you may raise it at a steady rate, sensing the 
amount of "work" you have to do to raise it a certain distance, 
or you may swing it up and down, in part sensing the difficulty 
of accelerating it against the gravity of the earth.  

However, sensing mass in terms of difficulty of acceleration is 
difficult because one must sense two different properties (i.e., 
force f and acceleration a, both of which will vary during a 
swing of a book) and then determine the mass by mentally dividing 
according to Newton's second law:

                            m = ---.

The complexity of sensing the two varying properties and then di-
viding the value of one by the other strongly suggests that dis-
criminating differences in the acceleration difficulty of mass is 
the most difficult of the three components for a human to sense.

Since humans appear to understand mass in terms of difficulty of 
lifting and not in terms of difficulty of acceleration, it makes 
sense to view the "difficulty of lifting" sense of mass as being 
fundamental and to view the "difficulty of acceleration" sense as 
being secondary.  This suggests that we should view the concept 
of 'mass' as being verbally undefined instead of viewing it as 
being defined by the inertial definition.  

(The inertial definition of mass is not contradictory, so we are 
free to adopt it, even though it may be "unnatural".  If we do 
adopt this definition, we have not succeeded in eliminating a 
verbally undefined concept because the verbally undefined concept 
of 'mass' has been replaced by the verbally undefined concept of 
'force'.  For me, this replacement is unsatisfactory because 
'force', being often dynamic and somewhat "unattached", is a 
harder concept to understand than 'mass', which is generally 
static, and is always directly "attached" to a physical object.)

Interestingly, empirical research has shown that the mass of our 
everyday experience from lifting physical objects (without accel-
eration) or from weighing physical objects on a standard weigh 
scale (which never involves direct acceleration) has a perfect 
linear relationship with the concept of the inertial mass of a 
physical object.  That is, if object A "weighs" twice as much as 
object B, we will find that the inertial mass of object A is (to 
the best of our measurement capability) exactly twice the iner-
tial mass of object B.  The linear relationship between these two 
variables is an empirical fact, but need not have been -- it is 
not (obviously) a necessary fact.  Because of the existence of 
this (to the best of our knowledge) perfect relationship between 
the two variables in everyday experience, physicists do not dis-
tinguish between the two definitions of the property of "mass" 
when dealing with everyday experience -- both definitions effec-
tively define the "same" property.

                            *   *   *

Although humans learn and understand the properties of 'mass', 
'length', 'time', and 'temperature' through direct experience, 
this approach is usually not possible with more complicated prop-
erties.  In this case, researchers may understand a property in 
terms of how it is measured because observing the measurement of 
the values of the property may be the closest we can come to di-
rectly experiencing different values of the property.  

For example, we cannot easily directly experience the property of 
the electrical resistance of a conductor of electricity.  How-
ever, we can indirectly intuit this property by measuring (or 
simply feeling) the heat dissipation when a known voltage is ap-
plied across the two ends of the conductor.  This is because the 
heat dissipation by the conductor is inversely proportional to 
its electrical resistance -- the higher the resistance, the lower 
the heat dissipation.  Alternatively, we can use an ammeter to 
measure the size of the electric current passing through the con-
ductor when a known voltage is applied across the two ends of the 
conductor.  This is because the current is also inversely propor-
tional to the resistance -- the higher the resistance, the lower 
the current, as given by Ohm's law.

                            *   *   *

When I say in the body of this post that (for example) the 
property of 'mass' is verbally undefined, I am not simply making 
the point that the standard unit of mass (i.e., the kilogram) is 
one of the SI base units (Cardarelli 1999, p. 7).  The fact that 
the unit of mass is an SI base unit is a parallel point to my 
point that the property of 'mass' is verbally undefined.  Both 
points arise due to the basic role of the property of 'mass' or 
'weight' in human reality.

                            *   *   *

The National Institute of Standards and Technology (1991) gives 
an interesting chart showing the relationships among the impor-
tant SI units.  Paradoxically, these relationships between units 
are not relationships between properties or variables in the 
sense I discuss above -- they are perhaps better viewed as rela-
tionships between entities (entities that are measurement units).


BIPM [Bureau International des Poids et Mesures, International 
   Bureau of Weights and Measures]. 2001a. "SI Base Units."  
   Available at

---- 2001b. "Flexure-strip balances."  Available at

Breland, H. M., and Jenkins, L. M., 1997. English word frequency 
   statistics: Analysis of a selected corpus of 14 million to-
   kens. New York: College Entrance Examination Board.

Cardarelli, F. 1999. Scientific unit conversion 2d ed. London: 

Energy Research Advisory Board. 1989. "Cold fusion research".  
   United States Department of Energy report DOE/S--0073.  Avail-
   able at  Search for the phrase 
   "cold fusion research" including the quotation marks.

Hofmann, J. R. 1996. André-Marie Ampère. Cambridge: Cambridge 
   University Press.

Huizenga, J. R. 1993. Cold fusion: The scientific fiasco of the 
   century. Oxford UK: Oxford University Press.

Macnaughton, D. B. 1998. "Eight features of an ideal introductory 
   statistics course." Available at

---- 1999. "The introductory statistics course: The entity-
   property-relationship approach." Available at

National Institute of Standards and Technology. 1991. "Interpre-
   tation of the SI for the United States and federal government 
   metric conversion policy (NIST Special Publication 814)." 
   United States Department of Commerce, National Institute of 
   Standards and Technology. Available at

Vapnik, V. N. 2000. The nature of statistical learning theory 
   2d ed. New York: Springer.

Williams, L. P. 1970. "André-Marie Ampère." in Dictionary of 
   Scientific Biography ed. by C. C. Gillispie.  New York: 
   Charles Scribner's Sons, pp. 139-147.

---- 1989. "André-Marie Ampère."  Scientific American, 260 (1), 

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