>But why can't it be reduced to classes of specific physical things? How
>can you show that it is necessary for anything corresponding to this
>description to 'exist' apart from its instantiations as documented
>procedures and actual occurrences of their application?

I did point out in my last post that there appears to be no simple way
to make such reductions (between math concepts and classes of specific
things).  For instance no one has yet succeeded in showing how math
concepts such as infinite sets and transfinite sets (which are precise
math concepts) could be converted into physical notions.  A also
pointed to David Deutsch's excellent 'Criteria For Reality':

 'If according to the simplest explanation, an entity is complex and
autonomous, then that entity is real.' ('The Fabric Of Reality', Pg 91)

As Detusch points out, mathematical entities do appear to match the
criteria for reality: 'Abstract entities that are complex and
autonomous exist objectively and are part of the fabric of reality.
There exist logically necessary truths about these entities, and these
comprise the subject-matter of mathematics.'

>Language, logic, and math are human inventions just as chair is, c.f. William 
Cooper "The Evolution of Reason".
>That chair would continue to exist even if all
humans were wiped off the Earth - but the concept of 'chairs' wouldn't
and neither
would '2'.
>Ontology is invented too.
>Brent Meeker

I distinguish between two kinds of abstract concepts - abstract
concepts of universal applicability, which I think are objectively real
and abstract concepts of limited applicability, which are clearly human
inventions.  You don't accept the distinction.  But I pointed out that
for abstract concepts of universal applicability, there appears to be
no difference between cognitive and ontological categories, where as
for abstract concepts of limited applicability, there clearly is a
difference between cognitive and ontologic categories.

So I would tend to say that the concept of '2' is clearly 'out there',
where as the concept of 'chair' is 'in our heads' and quite possibly
even the concrete instances of a 'chair' is 'in our heads' as well!
After all, is it really the case that a chair is an object 'out there'
with definite objective physical dimensions like length?  Isn't it
actually the case that all that's 'out there' is a 4-dimensional
'chair' world-time? - which I point out to you as really a
*mathematical construct* ;)

>Actually, it's an arguement against doing so. If mathematical
>terms referred to particular things, they would not be universally
>They are universally applicable because they don't refer to anything.


Math concepts are super-classes or abstract classes being used to
classify *other* astract classes.  I pointed out three different
ontological catgories:

(1)  Abstract entities of universal applicability (like math concepts)
(2)  Abstract entities of limited applicability (human constructs like
alphabets or a chair concept)
(3) Concrete instances (like a particular example of a chair)

I'd say you can make a good case that the entities in (1) are the only
real objective reality.  It's (2) and (3) that are actually 'in our

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