>But this only shows that mathematical objects exist in the sense that chair 
>exists;
>as a abstraction from chairs.  So chair isn't identical with any particular 
>chair.
>
>Brent Meeker


What follows is actually a very important and profound metaphysical
point, absolutely fundamental for understanding platonism and reality
theory.

Both the *concept* of a chair and mathematical concepts are *abstract*
things.  But there's a big difference.  In the case of the chair
concept, it's simply a human creation - it's simply a word we humans
use to summarize high-level properties of physical arrangements of
matter.  There are no 'chairs' in reality, only in our heads.  We can
see this by noting the fact that we can easily dispense with the 'chair
concept' and simply use physics descriptions instead.  So in the case
of the 'chair' concept, we're obviously dealing with a human construct.


Critical point:  The 'chair' concept is only a (human) cognitive
category NOT a metaphysical or ontological categories.

Mathematical concepts are quite different.  The key difference is that
we *cannot* in fact dispense with mathematical descriptions and replace
them with something else.  We cannot *eliminate* mathematical concepts
from our theories like we can with say 'chair' concepts.  And this is
the argument for regarding mathematical concepts as existing 'out
there' and not just in our heads.  There are two steps to the argument
for thinking that mathematical entities are real:

(1)  A general mathematical category is not the same as any specific
physical thing
AND
(2)  Mathematical entities cannot be removed from our descriptions and
replaced with something else ( the argument from indispensibility).

It's true that both 'chair' concepts (for example) and math concepts
are *abstract*, but the big difference is that for a 'chair' concept,
(1) is true, but not (2).  For mathematical concepts both (1) AND (2)
are true.

There's another way of clarifying the difference between the 'chair'
concept and math concepts.  Math concepts are *universal* in scope
(applicable everywhere - we cannot remove them from our theories) where
as the 'chair' concept is a cultural construct applicable only in human
domains.

To make this even clearer, pretend that all of reality is Java Code.
It's true that both a 'chair' *concept* and a 'math' concept is an
abstraction, and therfore a *class* , but the difference between a
'chair' concept and a 'math' concept is this:  'Math' is a *public
class* (an abstract category which can be applied everywhere in
reality), where as a 'chair' concept is *private* class, applicable
only in specific locations inside reality (in this case inside human
heads).

Reality Java Code for a math concept:
PUBLIC CLASS MATH  ()

Reality Java Code a chair concept:
PRIVATE CLASS CHAIR ()

Big difference!

The critical and profound point if we accept this argument, is this:

*There is NO difference between *epistemological* and *metaphysical*
categories in the cases where we are dealing with cognitive categories
which are universal in scope.  Math concepts of universal applicability
are BOTH epistemological tools AND metaphysical or ontological
categories.  One needs to think about this carefully to realize just
how important this is.


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