Hi all, (01)
This is my first post to this conversation. As the creator / editor of the
Unified Code for Units of Measure (http://unitsofmeasure.org), I am certainly
interested in what this group comes up as a semantic construction of units
of measures, but I can't monitor this thread so closely. However, a few
things stick out at this time: (02)
1. the construction of a unit as more than a representative quantity, e.g.,
as a function that takes a length and maps it to a number, (03)
2. the question of whether certain dimensionless measures (e.g., angular
degrees) should have a dimension instead and the irrationality of the
dimensional reductionism. (04)
Ad 1: WHAT IS A UNIT? (05)
When Peter Hayes suggests that the unit meter "denotes a function
from lengths to numbers" would seem to say that any unit is defined as
a function from its respective kind of quantity to numbers. But that would
mean you contract the entire definition of measurement into the unit,
for measurement is a function from the respective kind of quantity to
measurement results which are numbers given a reference. (06)
That is too much. (07)
A unit is a reference standard for a measurement of a kind of quantity.
As such it is a defined quantity. 1 meter is the length of <fill in the
current agreed metrologic standard procedure>. That reference procedure
may be a factor k times the length of an earth meridian, a particular
stick stored in a museum, or a factor k times the wavelength of some
monochromatic radiation. Interestingly, as these reference procedures are
replaced over the decades and centuries, such replacement is made while
keeping the unit invariant for all practical purposes. That is, the
factor k is chosen to keep unchanged all measurement results expressed
in meter. I.e., if a new reference procedure to realize the meter is
introduced and if that new meter standard now requires anyone to convert
their old results from old-meter to new-meter, then the new reference
standard should be corrected. (08)
Therefore, one would conclude that a unit transcends any of its particular
reference standard that are used at a time. A reference standard is
merely a choice agreed upon for whatever pragmatic motivations, but the
meter does not change. (09)
In the end, one should just define unit as a quantity in its respective
dimension, abstract from the reference procedure, but at any time
approximated as best we can by that procedure. A quantity is expressed
as number times unit, because the unit has that which cancels out
when we divide a to be measured quantity by the unit to leave only
a number. This ontology should not add or remove anything from this
meaning of a unit: i.e., it must remain a quantity that can divide
other quantities to yield a number, or multiplied with a number to
yield a quantity. (010)
Ad 2: THE PROBLEM OF DIMENSIONAL REDUCTIONISM (011)
So what about dimensionless quantities and units which are
defined as dimensionless by any particular system of quantities and
units, such as the angular degree? (012)
One could very well say, and with procedure-defined units which we
encounter a lot in medicine I have been forced to conclude that there
is really an infinite set of dimensions. This is using "dimension"
somewhat loosely meaning the class of quantities that can be converted
into each other. In that sense, (013)
(i) (014)
wave length is in a different dimension from
body height is in a different dimension from
distance between celestial bodies (015)
and (016)
(ii) (017)
amount of carbon atoms is a different dimension from
amount of glucose molecules, (018)
but (019)
mass of glucose molecules is convertible to
amount of glucose molecules, hence is it the same dimension? (020)
and (021)
(iii) (022)
volume concentration of alcohol is in a different dimension from
plane angle (023)
and so forth. (024)
In the (i) the length is used to determine entirely different quantities,
and I cannot divide my body height over the radio station I've tuned into. (025)
In (ii) practitioners often feel the need to annotate units by the specific
thing that is measured of fear that this information is cancelled out
and lost. On the other hand, practitioners are often oblivious of the
difference between amount-of-substance concentration vs. mass concentration
for a particular substance, for you can just convert between them. (026)
In (iii) it is striking that we actually find plane angle in the same
dimension as volume concentration of alcohol. (027)
Having to maintain UCUM and getting many comments and requests by
practitioners, the most difficult thing is to tell people that their
distinction does not matter hence it is reduced into the same dimension.
And at the same time that their unit is not a unit because it cannot
be reduced into any of the 7 dimensions. (028)
Clearly, the fact that there are fewer dimensions than kinds of
quantity is because of a reductionist abstraction, by which we find
that wave length and body height are "just lengths". And where
frequency is "just" number over time. And where many procedure-defined
units, such as "Bioequivalent Allergen Units" are just dimensionless
numbers. It is odd that we try to rescue the dimensional identity of
plane angle when not also rescuing the dimensional identity of
"Allergen Units". What is the rationale for this? Over the 10 years
that I have managed the UCUM system, I have today less a good answer
than I had before. (029)
Fortunately even the hard physical science has seen a major change
in system of base quantities and units, that from CGS to MKS, wherein
the major change is not in these 3 letters but in the approach to
measuring electromagnetic phenomena. This is something to study when
making an ontology of units. (030)
In my view today, just as the meaning of a unit transcends the particular
reference standard chosen for them at different times in history,
the meaning of a quantity must transcend the particular reductionist
construction in terms of base quantities and units, from which the
dimensionality is determined. (031)
The choice of base system of units is entirely arbitrary for a base
system of quantities. The choice of base system of quantities is
somewhat arbitrary, but any such reductionist system seems never
quite sufficient practically. For example, it is entirely irrational
to me that there should be a dimension reserved for "luminous intensity"
in the SI, when there is no dimension for "loudness" and no dimension
for "allergenity". In fact it seems trivial to reduce luminous
intensity to radiation energy flux (or something related), when it
is actually very hard to capture "allergenity". Why should there
be an exception made to photographers but none to allergologists and
pharmacologists? (032)
The plane angle situation is similarly irrational. (033)
And the most irrational thing in the SI is the amount of substance
being admitted as a base quantity. It's amazing if you think that
it is clear to every elementary school kid today that matter is
made up of little particles called atoms and molecules, and in
middle school we learn that each kind of such particles has a
mass which can be computed as the sum of the mass of its constituents.
And that the mass of a block of matter is the sum of the mass of
the particles that make up such matter. Yet, at the same time
scientific society affords itself a dimension that means nothing
more than "number of atoms or molecules". According to this system,
the number of cars and number of people in them are dimensionless,
but the number of carbon atoms and number of Amb-a-1 proteins are
in a common special dimension. (034)
I think this ontology discussion should wrestle with some of these
irrationalities and anomalies. It should construct its concepts
so that they may help to explain what is going on here. (035)
In UCUM, the base is currently defined thus: (036)
* length - meter
* elapsed time - second
* mass - gram (not kilo-gram, because kilo-gram has a prefix)
* charge - because elementary charge is the basis of
electrical phenomena - coulomb as conventional unit
* temperature - kelvin
* candela - luminous intensity (just because the SI has it)
* radian - plane angle (037)
So, you can see, the UCUM is an ongoing experiment to try to uphold
a different base where amount of substance is just a dimensionless
count of particles and plane angle is a dimension different from
length. (038)
Over the past 10 years of running UCUM, there have been 99% of
discussions about procedure-defined units which are no good units
at all, and 1% (if that much) of discussion about the lack of
amount of substance as a base unit. Recently we had to mark a
large (and most rapidly growing) set of units in UCUM with a
marker "arbitrary", i.e., defined by a special procedure so that
there is no conversion relationship to any of the other units
in the system. This is still an ongoing problem. (039)
When I monitor this ontology discussion, I am most interested to
seeing how it will strive to align in its principles as much as
possible with the VIM and metrology convention, yet be able to
avoid its irrationalities and construct the concepts for kinds of
quantities and units as transcending any particular historical
choice of metrologic standard system. (040)
regards,
-Gunther (041)
--
Gunther Schadow, M.D., Ph.D. gschadow@xxxxxxxxxxxxxxx
Associate Professor Indiana University School of Informatics
Regenstrief Institute, Inc. Indiana University School of Medicine
tel:1(317)423-5521 http://aurora.regenstrief.org (042)
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