On Sep 23, 2009, at 5:19 PM, Ed Barkmeyer wrote: (01)
> David Leal wrote:
>> Dear All,
>>
>> I fear that reaching an understanding of what mass is, is going to
>> be difficult.
>>
>
> I want to leave the "understanding of what mass is" to the SI people.
> For our purposes, mass is a kind of quantity that is one of the "base
> quantities" of the SI system. Whatever the SI definition is, we
> replicate.
>
>> There seems to be agreement that there is an object denoted "1.3
>> kg". In
>> Ed's synthesis, this is:
>> - an equivalence class of members of Q1 ("particular quantity");
>> - a member of Q3 ("magnitude of quantity").
>>
>> There are at least two ideas of what the members of "1.3 kg" are,
>> including:
>> a) the members are mass tropes of different individual quantities
>> of matter;
>> b) the members are different individual quantities of matter.
>>
>
> I don't know what David's (b) means. I would have said the members of
> the equivalence class designated "1.3kg" are either:
> a) 'mass tropes' of different individual things, or
> b) measurements of the 'mass tropes' of individual things. (02)
Wait, wait. I have trouble understanding what a mass trope is, but a
*measurement* of a mass trope really is getting beyond rational
comprehensibility. Can we have examples, or something, to clarify what
we are talking about here? But read down before trying.
>
>> To me approach (b) seems simpler. Using this approach, "1.3 kg" is
>> merely a
>> class of individual quantities of matter which is defined by a single
>> necessary and sufficient condition - equilibrium with the 1 kg in
>> Paris
>> using a 1:1.3 ratio balance (or some practical equivalent).
>>
>
> The problem with this is twofold:
> - There are no "individual quantities of matter". (03)
Of course there are. Any chunk of physical stuff is an individual
quantity of matter. (04)
> There are individual
> material things/phenomena that have a "mass" property.
> - The "practical equivalent" is exactly where the measurement concept
> and notions of uncertainty and tolerance come in. (05)
We are not *obliged* to consider uncertainty and tolerance in the core
of the ontology. The fact that mass can only be *measured* to a given
degree of accuracy does not imply that the *concept* of an exact mass
is incoherent. (06)
> So David' description of 'approach (b)' crosses exactly the two
> notions
> I distinguish above.
>
> I am coming to the conclusion that "1.3 kg" is an equivalence class of
> measurements (07)
Oh no, no. Just ask any physicist. See below. (08)
> that is defined in part by some specification of the
> tolerable variance in those measurements. Put another way, the
> ratio of
> the measurements to the reference kilogram in Paris is 1.3 to the
> accuracy that we decided to express them (which includes both the
> uncertainty in the measurement and the practical tolerance).
> Scientific
> measures have no notion of tolerance, but they express uncertainty;
> business measures tend to assume that the uncertainty is very small
> relative to the tolerance. In both cases, the "1.3 kg" is a 'nominal
> value' that denotes a classification of the measurements.
>
> I must say that I also find this model cumbersome and scary (because I
> lack the intuition to formulate the axioms). So, like the Cowardly
> Lion, "there's just one thing I want you to do -- talk me out of
> it." :-) (09)
Delighted. (010)
First, the basic point above: limitations upon measurements should not
be taken to be necessarily fundamental to our concepts. In the cases
of tolerances and error ranges, indeed, I think we *must* assume that
exact, precise quantities, even though they be unmeasurable, are the
conceptual foundation, since the very idea of an error range or a
tolerance (as usually understood) itself assumes the notions of exact
numbers. 3.52 +/- 0.001 actually *refers to* the exact numbers 3.52
and 0.001, so the ontology must have these concepts in it. (Long ago I
actually tried to build a completely 'topological' ontology of
imprecise quantities based on Zeeman's tolerance space idea, which
replaces classical numbers and ranges by an inherently 'fuzzy' notion.
I did not succeed. It is surprisingly tricky, and its not even clear
that it is more correct in any case, e.g. it is possible to prove that
vernier scales do not work in a tolerance space: but they do work.) (011)
Second, I claim that the notions of a given chunk or piece of matter,
and the mass of that chunk or piece, are well-defined, unambiguous,
simple, and fundamental. By a 'chunk' I mean here a given collection
of molecules, and its (rest) mass is the sum of the masses of those
molecules. As i understand David's point above, this is exactly what
he means by "an individual quantity of matter" and its mass. These are
not metrical ideas, they are basic physical ideas. (012)
Third, we can erect a perfectly satisfactory ontology of mass and mass
measurements upon this basis. Here goes. I will use the predicate
Chunk to be true of such entities and the function massOf to relate
one of them to its mass. We can now say that masses can always be
compared: (013)
(forall ((x Chunk)(y Chunk))(exists ((r Real))(= (massOf x)(times r
(massOf y)) ))) (014)
This assumes that it makes sense to multiply a mass by a real number.
Which it does, of course, so we will simply proceed. We can also say
what zero mass means: (015)
(forall ((x Chunk))(iff (= (massOf x) zero)(Vacuum x)) )) (016)
Note, this says nothing about measurements or errors or any of that.
It is a purely *physical* theory so far. It is basically Dalton's view
of the atomic structure of matter. But given that these ideas are
coherent and well-defined, we can proceed to talk of how to measure
them. We need a way to reliably isolate a set of molecules (such as
having a lump of pure, stable metal and not rubbing it against
anything) and then a way to measure its mass, which we can do with a
balance and a standard weight. Lets call this standard chunk
theKilogram. Cutting through details of how exactly to make a suitable
balance, we can say the following. (017)
(Chunk theKilogram) (018)
Any chunk has a mass which is some number times the mass of the
kilogram. Call this number the massInKg of the chunk: (019)
(forall ((x Chunk))(= (massOf x)(* (massInKg x)(massOf theKilogram)) )) (020)
Note that massInKg is simply a number, not an actual mass. Our earlier
axiom guarantees that this always exists and, assuming basic axioms
about multiplication, is unique. (021)
We could now introduce tolerances and error limits (expressed as real
intervals) and talk of measurements being within them, as required.
Thus for example, 3.25 +/- 0.005 refers to the interval [3.245,
3.255], and its use as a surrogate number asserts the existence of a
number in this interval, eg (022)
(= (massInKg myTable) (Tolerance 0.005 3.25)) (023)
is shorthand for (024)
(exists (x)(and (increasing 3.245 x 3.255)(= (massInKg myTable) x) )) (025)
and more elaborate theories can now be built upon this, of course. (026)
-------- (027)
I claim that this all makes perfect sense and indeed is kind of
obvious; and that it extends in obvious ways to other measurements and
units based on comparisons to some 'standard' mass/length/voltage/
whatever. I also note that nowhere does it require any reference to
'tropes' of any kind, and I claim that this is a distinct advantage.
So far, this ontology refers to: (028)
- chunks of matter (including empty chunks, also called vacuums.)
Think of these as pieces of space enclosing a fixed set of molecules. (029)
- masses. These are 'values', which each chunk has a unique one of,
and can be multiplied by real numbers. (Or; which can be compared as a
ratio, expressed as a real number.) So I might not know what the
numerical value of x is, but it still makes sense to say that y is
2.73928 times it. The exact metaphysical nature of these values is not
specified and is not relevant to anything in the rest of the ontology;
all that matters is that they can be compared. (030)
- real numbers (031)
- closed real intervals (032)
- the Kilogram, a particular named chunk. (033)
and that is all. (034)
I bet I can formalize anything anyone wants to say about mass,
measuring mass and units of mass just using this basic ontological
framework. (035)
>
>> Similarly for length
>> --------------------
>> There is a similar discussion for length (and maybe for all
>> quantities). Two
>> ideas of what the members of "1.3 m" are, include:
>> a) the members are length tropes of different individual physical
>> objects;
>>
> yes.
>
>> b) the members are point pairs.
>>
>
> This is some formal geometric characterization, and I don't want to
> mix
> 'quantities' with geometry, just because 'length' and 'angle' are
> quantity kinds. But in fact, this may just mask a 'measurement'
> concept.
>
>> With approach (a), the length of my_table is a member of "1.3 m".
>> With
>> approach (b), my_point_pair is a member of "1.3 m". There may be a
>> separate
>> statement that the points in my_point_pair are at opposite ends of
>> my_table.
>>
>
> What makes you think your table has "points"? It has "length",
> which is
> how much space it occupies in one dimension. (036)
Its not hard to *prove* that it has points (using some very plausible
assumptions about topology), because you can reverse-engineer the
points from the length intervals (using maximal ideals, a construction
originally due to Whitehead.) (037)
> In one view, you get
> "points" when you make a geometric model of the table (which is not
> the
> table). In another view, you get the "points" when you introduce the
> measuring device and locate it in space relative to the table, which
> gives rise to "points of coincidence" -- places where the two bodies
> touch.
>
> Either way, it is not clear how this kind of conceptualization would
> generalize to a model for quantities that aren't spatial in nature.
> In
> fact, we do the reverse -- we introduce scales, which have an
> intellectual model that is spatial displacement, and use them as the
> model for organizing measurements. (038)
Who are 'we' here? Metric theorists may do this, but most engineers
and physicists (not to mention carpenters, etc.) take the line that
the world exists, has the dimensions that it has, and that we measure
these things with more or less accuracy. But we do not *create*
lengths by measuring them. (039)
Pat (040)
>
>> A question for the ontology experts
>> -----------------------------------
>> Can we agree a definition of the object Q3 ("magnitude of
>> quantity"), whilst
>> at the same time having different understandings of Q1 ("particular
>> quantity")?
>>
>
> Do we have different understandings of 'particular quantity'? It
> seems
> to me that the problem is what Q3 is.
> Surely "1.3 kg" is an "expression of" (a name for) an instance of Q3.
> But we apparently don't agree on what the Q3 is. And this is the crux
> of the ontology, because a 'measurement unit' is either a 'particular
> quantity' (which I doubt) or a Q3.
>
> -Ed
>
> --
> Edward J. Barkmeyer Email: edbark@xxxxxxxx
> National Institute of Standards & Technology
> Manufacturing Systems Integration Division
> 100 Bureau Drive, Stop 8263 Tel: +1 301-975-3528
> Gaithersburg, MD 20899-8263 FAX: +1 301-975-4694
>
> "The opinions expressed above do not reflect consensus of NIST,
> and have not been reviewed by any Government authority."
>
>
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