I wrote: (01)
>> I would say that it is logically impossible to have a unit without a
>> value. (02)
Pat Hayes answered: (03)
> Its clearly logically possible. Conceptually they seem distinct. The
> idea of the meter, for example, is not the individual measure 'one
> meter', though the two are obviously closely related. I would say that
> 'meter' denotes a function from lengths to numbers, and one meter is
> the length, such that the value of that function when applied to it,
> is the number 1:
>
> (the (x) (= (meter x) 1) ) (04)
Perhaps from a purely logical perspective such a definition of 'meter' is
fruitful, but this is not the definition used in the metrological context
that I took for granted. Here units are measurement units. VIM 2008 says
(1.9): (05)
"measurement unit = real scalar quantity, defined and adopted by
convention, with which any other quantity of the same kind can be compared
to express the ratio of the two quantities as a number." (06)
The last 8th edition of the SI Brochure says (1.1): (07)
"The value of a quantity is generally expressed as the product of a number
and a unit. The unit is simply a particular example of the quantity
concerned which is used as a reference, and the number is the ratio of the
value of the quantity to the unit." (08)
In the SI system, most of the base units are nowadays theoretically
defined. The meter, for instance, is defined as the length of the path
travelled by light in vacuum during a certain time interval. But the
kilogram is still defined as being equal to the international concrete
prototype of the kilogram. (09)
Pat H repeated:
>> Its clearly logically possible. Conceptually they seem distinct. The
>> idea of the meter, for example, is not the individual measure 'one
>> meter', though the two are obviously closely related. I would say that
>> 'meter' denotes a function from lengths to numbers, and one meter is
>> the length, such that the value of that function when applied to it,
>> is the number 1: (010)
Chris Partridge responded: (011)
> It seems a little odd to think of the unit meter and not think of one
> metre
> - especially when historically the unit were actual physical objects.
>
> Isn't this the difference here between a unit and a scale - and PatH's
> function a kind of scale. (012)
I think you are right, but note then that to say that a scale is a
function from equivalence classes (of a certain dimension such as length)
to numbers is to say too little. The scales implicitly talked about in the
above quotations from VIM and the SI system are ratio scales, and this
fact puts quite specific restrictions on the function in question. (013)
> Ingvar, doesn't Ellis argue that a unit is insufficient in some way, so
> one
> needs to uses scales. (014)
I am sorry Chris, but I can't answer. I don't know by heart, and I don't
have his books in my library. Also, I know for sure that over the years
Ellis has changed some of his views on measurements and scales. My own
view, however, is that a measurement unit makes no sense when isolated
from a linear ordering of equivalences classes (of a certain dimension). (015)
Best, Ingvar (016)
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