Pat and Chris, (01)
I agree with Pat's note, but I'd like to add another comment. (02)
PH> ... a point I should have made is that the 'pure' set
> theories without ur-elements are purged of non-mathematical
> things but also of mathematical things. They are *entirely*
> about sets. So it is somewhat misleading to characterize them
> as being about mathematical topics. Now, to be fair to John,
> he did say that they were *applied* to mathematical topics,
> and in a sense this is correct... (03)
The important distinction here is between *pure* mathematics
and *applied* mathematics. A lot of the confusion in this
forum results from a failure to observe this distinction. (04)
In pure mathematics, the primitives (ur elements) are totally
unspecified. They could be people, numbers, or the empty set.
The purpose of pure mathematics is to define an abstract
formal system, which could be studied for its own aesthetic
beauty, or it could be applied to something else. (05)
But as soon as the primitives of a pure mathematical system
are identified with something outside that system, it becomes
applied mathematics. One can apply a pure mathematical theory
to anything -- physical, imaginary, or even mathematical. (06)
For example, group theory could be applied to geometry in order
to analyze the group of rotations of a geometrical object that
preserve symmetry. That is an example of one pure mathematical
system applied to another pure mathematical system. (07)
In some earlier notes in this thread, we have insisted that
certain terms in the UoM axioms should be assumed as primitive.
But other people have objected that more explanation is needed
to avoid misunderstanding about how measurement is done. (08)
Those two points are not in conflict, and I agree with both.
The pure mathematical system should designate certain terms
as primitives. But the standard should also specify exactly
how the mathematical primitives are applied to any subject. (09)
John (010)
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