Discussion Groups / English Usage / November 2006

Here is a simple example.

Suppose I have 20 apples, but redefine the scale such that 10 is zero.
On my redefined scale, where zero is arbitrary and meaningless, the
count is 10.  Now I add 10 apples to my original number.  On my new
scale, my count is 20, numerically 2X as much.  But that does not
reflect the real change in the number of apples, that is from 20 to 30.
I can say that I have exactly 10 more apples, but not twice as many.
That is the scale is interval, not ratio.

The issue is the relationship with some real and meaningful zero point.
Scales like F for temperature set an arbitrary zero, that is not
related to anything meaningful.  This does not mean they are not useful.

I think I know why the author did that: he/she knows it is an issue
that will bother students over and over again, so wants to deal with it
right away. Before retiring, I taught Operations Research models and
methods for more than 30 years, and every year, year in and year out,
the same problem always arose: how to get students to stop thinking of
a "number of objects" as necessarily being an integer. For example, a
model might involve production quantities, and the variables in the
model might be R = number of red widgets, B =number of blue widgets to
produce per week. So, there would be constraints on manpower, money,
productive capacity, raw materials, etc., and the final model might be
a linear programming problem in variables R and B. Many students would
insist on adding the constrtaints that R and B must be whole numbers;
after all, these quantities are defined as *numbers* of objects per
week. In fact, in a _linear_ model, we let R and B be fractional, and
later deal with the issue of a fractional solution if it arises. The
students were trying to put the cart before the horse, or to run before
they could walk.

Now, you seem to be bothered by several concepts of "number"; you
mentioned zero, negative numbers, etc. Although you didn't do so, you
could have also mentioned fractions (what does it mean to have 2/7 of
an apple pie?), and irrational numbers such as pi or the square root of
2. Well, it all has to do with "extension". We start off with whole
numbers, then generalize to negative integers, then generalize to
fractions, then to real numbers and finally to complex numbers. Why do
we do this? If we assume that many of the concepts and methods arose
from practical problems, it must be the case that we use fractions
because we need them, and use negative numbers because we need them. Of
course, to speak of -2 apples seems nonsensical, but that might be just
because we should, perhaps, have invented a different term than
"number". Or, it may be, as in many business and industrial situations,
that an inventory of -2 apples means we have 2 apples on backorder. If
my bank balance is -$1000 it means that I owe the bank $1000.
Accounting statements indicate this by putting brackets around the
number ($1000) instead of a minus sign, but it really amounts to the
same thing.

To return to your original example: to speak of 15 degrees F as being 3
times as warm as 5 degrees F is wrong, since we first need a "warmness"
measure that makes physical sense. The one that does have physical
significance is the absolute temperature scale, and in those terms, 5
and 15 degrees F are only very slightly different. In casual
conversation we might think about these matters differently, but the
author was, after all, discussing a "technical" subject, in which words
might be used a bit differently than in their everyday occurrences.

R.G. Vickson
Adjunct Professor, University of Waterloo

I think I know why the author did that: he/she knows it is an issue
that will bother students over and over again, so wants to deal with it
right away. Before retiring, I taught Operations Research models and
methods for more than 30 years, and every year, year in and year out,
the same problem always arose: how to get students to stop thinking of
a "number of objects" as necessarily being an integer. For example, a
model might involve production quantities, and the variables in the
model might be R = number of red widgets, B =number of blue widgets to
produce per week. So, there would be constraints on manpower, money,
productive capacity, raw materials, etc., and the final model might be
a linear programming problem in variables R and B. Many students would
insist on adding the constrtaints that R and B must be whole numbers;
after all, these quantities are defined as *numbers* of objects per
week. In fact, in a _linear_ model, we let R and B be fractional, and
later deal with the issue of a fractional solution if it arises. The
students were trying to put the cart before the horse, or to run before
they could walk.

Now, you seem to be bothered by several concepts of "number"; you
mentioned zero, negative numbers, fractions, etc. Although you didn't
do so, you could have also mentioned irrational numbers such as pi or
the square root of 2. How did all these thing come to be? Well, it all
has to do with "extension". We start off with whole numbers, then
generalize to negative integers, then generalize to fractions, then to
real numbers and finally to complex numbers. Why do we do this? If we
assume that many of the concepts and methods arise from practical
problems, it must be the case that we use fractions because we need
them, and use negative numbers because we need them. Ditto for complex
numbers: to an electical engineer, a complex voltage is no more
mysterious than a dozen eggs, and to a physicist, a complex-valued
quantum-mechanical wave function is just as meaningful as a lottery
ticket.

Of course, to speak of -2 apples seems nonsensical, but that might be
just because we should, perhaps, have invented a different term than
"number". Or, it may be, as in many business and industrial situations,
that an inventory of -2 apples means we have 2 apples on backorder. If
my bank balance is -$1000 it means that I owe the bank $1000.
Accounting statements indicate this by putting brackets around the
number ($1000) instead of a minus sign, but it really amounts to the
same thing.

To return to your original example: to speak of 15 degrees F as being 3
times as warm as 5 degrees F is wrong, since we first need a "warmness"
measure that makes physical sense. The one that does have physical
significance is the absolute temperature scale, and in those terms, 5
and 15 degrees F are only very slightly different. In casual
conversation we might think about these matters differently, but the
author was, after all, discussing a "technical" subject, in which words
might be used a bit differently than in their everyday occurrences.

R.G. Vickson
Adjunct Professor, University of Waterloo

Part of the problem is that many people think that mathematics is more
like music than medicine.

One can listen to and appreciate music quite easily and without much
mental effort, though it does require effort to produce it, so that one
gets the notion that one knows a lot more about it than is actually the
case.

Most of us, without having gone though the intensive study of a medical
degree of some sort are quite happy to acknowledge our lack of any deep
understanding of most things medical.

In that sense math is more like medical studies than musical ones.

I think I know why the author did that: he/she knows it is an issue
that will bother students over and over again, so wants to deal with it
right away. Before retiring, I taught Operations Research models and
methods for more than 30 years, and every year, year in and year out,
the same problem always arose: how to get students to stop thinking of
a "number of objects" as necessarily being an integer. For example, a
model might involve production quantities, and the variables in the
model might be R = number of red widgets, B =number of blue widgets to
produce per week. So, there would be constraints on manpower, money,
productive capacity, raw materials, etc., and the final model might be
a linear programming problem in variables R and B. Many students would
insist on adding the constrtaints that R and B must be whole numbers;
after all, these quantities are defined as *numbers* of objects per
week. In fact, in a _linear_ model, we let R and B be fractional, and
later deal with the issue of a fractional solution if it arises. The
students were trying to put the cart before the horse, or to run before
they could walk.

Now, you seem to be bothered by several concepts of "number"; you
mentioned zero, negative numbers, fractions, etc. Although you didn't
do so, you could have also mentioned irrational numbers such as pi or
the square root of 2. Where do these things come from? Well, it all has
to do with "extension". We start off with whole numbers, then
generalize to negative integers, then generalize to fractions, then to
real numbers and finally to complex numbers. Why do we do this? If we
assume that many of the concepts and methods arise
from practical problems, it must be the case that we use fractions
because we need them, and use negative numbers because we need them. Of
course, to speak of -2 apples seems nonsensical, but that might be just
because we should, perhaps, have invented a different term than
"number". Or, it may be, as in many business and industrial situations,
that an inventory of -2 apples means we have 2 apples on backorder. If
my bank balance is -$1000 it means that I owe the bank $1000.
Accounting statements indicate this by putting brackets around the
number ($1000) instead of a minus sign, but it really amounts to the
same thing.

As for complex numbers: while the man on the street might find complex
and/or imaginary numbers to be nonsensical, to the electical engineer a
complex-valued voltage is no more mysterious than a dozen eggs and to
the physicist a complex-valued quantum-mechanical wave function is no
less "real" than a lottery ticket.

To return to your original example: to speak of 15 degrees F as being 3
times as warm as 5 degrees F is wrong, since we first need a "warmness"
measure that makes physical sense. The one that does have physical
significance is the absolute temperature scale, and in those terms, 5
and 15 degrees F are only very slightly different. In casual
conversation we might think about these matters differently, but the
author was, after all, discussing a "technical" subject, in which words
might be used a bit differently than in their everyday occurrences.

Finally, I am not sure who is to blame, but I am not convinced that the
types of "scale" material that is bothering you can be laid at the
feet of mathematicians at all. Many of these things seem to arise in
psychology or sociology and the like. If you Google the term
"interval scale", you will be led to several web pages that discuss
scales of all types (ordinal, nominal, interval and ratio). Different
scales are used in different contexts and for different reasons. The
point being made by the author of the passage bothering you is that
certain quantities are measured in a ratio scale, but others are not.
Standard temperature measures used in meteorology are examples of
quantities measured on an interval, but not ratio, scale. On the other
hand, a chemical engineer would typically use temperature as measured
on a ratio scale by going to degrees Kelvin or their Farenheit
equivalent. Different tools for different purposes!

R.G. Vickson
Adjunct Professor, University of Waterloo

*snip*

*snip*

I don't know if this is relevant to your difficulty, but my experience
as a tutor (not math) has taught me that when a technical point seems
obscure or confusing, there is usually a _historical_ explanation.
People mess things up all the time, usually with good intentions, and
then the resulting mess has enough inertia that it can't be supplanted
by a more logical solution. I don't claim to be an expert, but perhaps
a little story about temperature measurement will make things clearer
to you. It may not; it will only explain why temperature measurement is
messed up, not what to do about it.

The Fahrenheit scale was proposed by Gabriel Fahrenheit in 1724, who
(according to Wikipedia) was annoyed that Ole Roemer's temperature
scale produced negative numbers as measurements in everyday situations.
Fahrenheit wanted to make a new scale which produced 'friendlier'
numbers. Hooray! Good intentions!

Back in those days the thing to do was to pick two useful-seeming
temperatures, call one of them 0 and the other 100. Fahrenheit did
this, and it was called the Fahrenheit scale. The story I heard was
that Fahrenheit picked the freezing point of sea water (useful for
sailors, I guess) and body temperature. Wikipedia has more variations
on the story if you're interested.

In 1742 Anders Celsius did the same kind of thing, with the freezing
point and boiling point of pure water. It was called the Celsius scale.
In 1848 William Thomson, 1st Baron Kelvin wrote a paper complaining
about the lack of a temperature scale rooted at 'absolute zero', the
state where a material has no heat energy (even due to particle motion
and so forth), which is about -273 Celsius. His scale became known as
the kelvin scale. The kelvin scale is still a bit arbitrary; 0 kelvin
is good because you can't get any colder than no heat at all, but the
magnitude of a degree kelvin is still related to the properties of
water.

So, now we have degrees kelvin which starts at absolute zero instead of
the freezing point of water. This kind of scale is much more useful for
physicists because 15 kelvin is three times as hot as 5 kelvin in a
meaningful sense, because 0 kelvin _actually means_ no heat, unlike 0
Fahrenheit or 0 Celsius which just mean 'pretty cold'. But the kelvin
scale never caught on in popular use, perhaps because weather reporters
feel silly saying "it's going to be a chilly 280 degrees today".

Fahrenheit, Celsius and kelvin all measure the same property
(temperature), but they use different scales to do it. They are all an
abstraction of the same thing, and it's the abstraction process which
introduces mathematical problems.

Another way of looking at it is that Fahrenheit is a bad abstraction of
the real world, precisely _because_ 15F != 5F. This unfortunate fact is
because of the way the abstraction was made. I won't try to talk about
things like whether particular abstractions obey associativity rules --
I think that would be a job for someone in sci.logic, and I know just
enough to get myself into trouble.

I apologise if that wasn't helpful for you, or you already knew all
about it (I see some sci.* newsgroups in the reply-to field, I probably
offended someone there with a dangerously over-simplified science
story). The point I'm trying to get at is that 15F is not three times
5F because of a quirk of history, not because of an obscure
mathematical principle.

temperature is a way of measuring "heat".
"Heat" is a way of saying amount of intramolecular motion.
Zero degrees Kelvin is when there is NO intramolecular motion.
Zero degrees centigrade/Celsius is the freezing point of water.

There are different meanings to zero.

It can refer to an abstract notion such as the number line.

It can also refer to counts.

It can also refer to an arbitrary reference point  such as temperature,
longitude, latitude,  any Z-score, etc.  Other arbitrary reference
points are 100 for IQ, 500 for GRE, SAT, and 50 for T-scores used in
psychometrics.
Any of the examples after  Z-scores above use the mean as a reference point.

One way to distinguish ratio level of measurement from interval level of
measurement is to ask whether the zero point has any relation to
nothingness or nonbeing in some way.

Art Kendall
Social Research Consultants

It is 20 degrees warmer, but that  is 15 - (-5), not 15 / (-5).

"What is the value of 0/0? (Is it really undefined or are there an infinite
number of values?)

There's a special word for stuff like this, where you could conceivably
give it any number of values. That word is "indeterminate." It's not the
same as undefined. It essentially means that if it pops up somewhere, you
don't know what its value will be in your case. For instance, if you have
the limit as x->0 of x/x and of 7x/x, the expression will have a value of 1
in the first case and 7 in the second case. Indeterminate."

http://mathforum.org/dr.math/faq/faq.divideby0.html

Plato would have managed just fine in Greek.

What's with the accusation of trolling?  You almost sound like you're
jealous; this is the second time you've mentioned it.  If my problems
don't interest you, please start your own thread.  I don't see why you
should hijack this one and yet accuse me of trolling -- and by
implication accuse the others of being suckers.

Irrelvevant here.

That's a limit value by definition, meaning 5/t can be made larger than
any number anyone would choose just by picking t small enough.  It is
not, however, a number.

-- Lou Pecora  (my views are my own) REMOVE THIS to email me.

On 12 Nov 2006 16:50:36 -0800, "Reef Fish"
<large_nassua_grouper@yahoo.com>

And earlier wrote...

I think between the two of those statements there is a solution for
world hunger.  So we divide a fish by zero people and we end up with an
infinite number of fish.  I take it that infinity divided by
6,556,742,150 (as of 21:30 GMT EST+5 Nov 13, 2006) is also infinity.
The Nobel committee will be contacting you soon, I am certain.  : )

Yes, thanks for that reference.  Of course, as the article points out
you loose some properties of the numbers (e.g. order) which is what this
thread is (partly) about.

Thanks.  I forgot about that use of the word infinity in measure theory.

One can also add infinities to the real number system in the archemedian
(sp?) sense that they are larger than any standard number.  Doing so
consistently produces Nonstandard Analysis, but in that system, too, 1/0
is not defined.

I think one has to be careful since the word infinity (or the symbol oo)
has many meanings in mathematics and quantities that capture behavior
like we imagine of an infinity can be added to number systems to produce
interesting and useful new systems, but they can't get around basic
contradictions.

-- Lou Pecora  (my views are my own) REMOVE THIS to email me.

Hmmm