Well, not that it really matters, but... (01)
On Jan 14, 2010, at 9:56 PM, John F. Sowa wrote: (02)
> David and Pat,
>
> DL> Presumably you also suggest two mappings: numbers <-> measures,
>> and measures <-> quantity values.
>
> PH> I would actually just have measures and values...
>
> DL> I do not see the utility of the concept of "measure".
>> It does not seem to be used in practice.
>
> I agree with Pat. The concept of 'measure' is very widely used.
> For example, "1 inch" and "2.54 cm" represent exactly the same
> measure, but they use different numbers and units.
>
> DL> Trivially there is a difference between a total function and
>> a partial function -- how can this be if a function is just
>> a set of pairs?
>
> I agree that Pat should have mentioned the two sets, domain and
> range (AKA codomain), from which the elements of the pairs were
> taken.
>
> I would also quibble with the claim that a function *is* a set
> of pairs. I would prefer to say that a function is a mapping
> between two sets. (03)
But then we have to say what a mapping is... :-) (04)
> That mapping could be specified by a set of
> pairs (extensionally) or by a rule (intensionally), which for
> any element of the domain determines the corresponding element
> of the range. For infinite sets, all specifications must be
> intensional. (05)
What you say here could be interpreted as a claim that infinite sets
cannot be specified (since sets are extensional and extensional must
be finite?) which is of course not true. The specification is one
thing, and the thing specified is another. Sets are not a way to
specify, they are the stuff of mathematics itself. Or at least, they
are in one way of looking at the foundations of mathematics (the way I
was brought up :-) (06)
Pat (07)
>
> For the distinction between extensions and intensions, see the
> following excerpt by Alonzo Church:
>
> http://jfsowa.com/logic/alonzo.htm
>
> John Sowa
>
>
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> (08)
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