Perhaps 'tolerance' is a concept within a uom ontology? (01)
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On 2009-09-23, at 10:05 PM, "Pat Hayes" <phayes@xxxxxxx> wrote: (03)
>
> On Sep 23, 2009, at 5:19 PM, Ed Barkmeyer wrote:
>
>> 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.
>
> 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".
>
> Of course there are. Any chunk of physical stuff is an individual
> quantity of matter.
>
>> 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.
>
> 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.
>
>> 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
>
> Oh no, no. Just ask any physicist. See below.
>
>> 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." :-)
>
> Delighted.
>
> 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.)
>
> 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.
>
> 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:
>
> (forall ((x Chunk)(y Chunk))(exists ((r Real))(= (massOf x)(times r
> (massOf y)) )))
>
> 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:
>
> (forall ((x Chunk))(iff (= (massOf x) zero)(Vacuum x)) ))
>
> 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.
>
> (Chunk theKilogram)
>
> Any chunk has a mass which is some number times the mass of the
> kilogram. Call this number the massInKg of the chunk:
>
> (forall ((x Chunk))(= (massOf x)(* (massInKg x)(massOf
> theKilogram)) ))
>
> 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.
>
> 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
>
> (= (massInKg myTable) (Tolerance 0.005 3.25))
>
> is shorthand for
>
> (exists (x)(and (increasing 3.245 x 3.255)(= (massInKg myTable) x) ))
>
> and more elaborate theories can now be built upon this, of course.
>
> --------
>
> 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:
>
> - chunks of matter (including empty chunks, also called vacuums.)
> Think of these as pieces of space enclosing a fixed set of molecules.
>
> - 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.
>
> - real numbers
>
> - closed real intervals
>
> - the Kilogram, a particular named chunk.
>
> and that is all.
>
> I bet I can formalize anything anyone wants to say about mass,
> measuring mass and units of mass just using this basic ontological
> framework.
>
>>
>>> 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.
>
> 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.)
>
>> 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.
>
> 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.
>
> Pat
>
>
>>
>>> 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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