# [XG W3pm] Quantity and unit

From: David Leal <david.leal@caesarsystems.co.uk>
Date: Fri, 30 May 2008 16:59:05 +0100
Message-Id: <1.5.4.32.20080530155905.00e3ef50@pop3.demon.co.uk>
To: Evan Wallace <ewallace@cme.nist.gov>

```
Dear Evan,

Thank you for drawing my attention to VIM (International Vocabulary of

Terminology
-----------
My initial attempt was:
>Summarising some of the key concepts we have:
>- physical quantity as defined in ISO 31, e.g. length of 10 metre,
thermodynamic temperature of 20 degrees Celsius
>- spaces of related physical quantities, e.g. length, thermodynamic temperature
>- defined points (units) within a space of physical quantities, e.g.
the_metre, the_Kelvin
>- scales which relate spaces of physical quantities to real numbers, e.g.
metre_scale, Kelvin_scale, Celsius

Corresponding to my "physical quantity", "physical quantity space" and
"unit", the VIM has:

1.1
quantity
property of a phenomenon, body, or substance, to which a magnitude can be
assigned

1.2
quantities of the same kind
quantities that can be placed in order of magnitude relative to one another

1.9
unit
scalar quantity, defined and adopted by convention, with which other
quantities of the same kind are compared in order to express their magnitudes

This seems to be a pretty good basis to me.

"Spaces"
--------
The name and definition of "quantities of the same kind" seem to be
deliberately informal, and I understand why - as soon as you try to be
formal you become involved with loads of mathematics.

Being ordered, which is all that the VIM definition requires, is necessary,
but usually we rely upon much more structure. Mohs hardness is just ordered.
Hence there is no concept of a Mohs unit. Instead each Mohs quantity is
separately identified, and is ordered with respect to the others.

The length quantities have more structure. In particular, where a and b are
scalars and x is a length, we assume that:

a.x + b.x = (a + b).x

NOTE Being pedantic, there are two different + operators in the equation
above - one for lengths and the other for reals.

It is this structure which enables us to define a length by reference to a
unit length and a real. I think that the length quantities must be a "1D
vector space over the reals".

Relationship with OWL
---------------------
The VIM document in clause 1.2 has the example
a) All lengths, such as diameters, circumferences and wavelengths, are
generally considered as quantities of the same kind.

Does this imply that the concept "length" is a member of the class
"quantities of the same kind"? I think it does, and so we could write
(apologies for the N3 - but for this simple example it is clear):

iso31:Length  a  vim:QuantitiesOfTheSameKind .

For some purposes, such as the use of lambda calculus to defines scales as
in Gruber and Olsen, it is useful to classify length further as:

iso31:Length  a  maths:1DVectorSpaceOverTheReals .

We then have my "big if" from before - can we classify a length quantity as
being a member of Length. If we can, and I hope we can, we have:

:TheLengthThatIs10Metres  a  vim:Quantity ;
a  iso31:Length .

My feeling that just defining Length as a subClass of Quantity is fine
within OWL-DL, but as soon as we wish to classify Length as a
vim:QuantitiesOfTheSameKind or as a maths:1DVectorSpaceOverTheReals we are
into OWL-FULL. Have I got this right?

Best regards,
David

============================================================
David Leal
CAESAR Systems Limited
29 Somertrees Avenue
Lee London SE12 0BS
tel:      +44 (0)20 8857 1095
mob:      +44 (0)77 0702 6926
e-mail:   david.leal@caesarsystems.co.uk
web site: http://www.caesarsystems.co.uk
============================================================
```
Received on Friday, 30 May 2008 15:59:38 GMT

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