>>But this only shows that mathematical objects exist in the sense that chair 
>>as a abstraction from chairs.  So chair isn't identical with any particular 
>>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
> (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:
> Reality Java Code a chair concept:
> 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.

It is an interesting point, but it's not so fundamental as you seem to think.  
We can 
do without 'chair' and 'table' etc.  But we can't do wihtout 'this' and 'that'. 
Without distinguishing objects we couldn't count and we wouldn't have the 
Language, logic, and math are human inventions just as chair is, c.f. William 
Cooper "The Evolution of Reason".  Probably they are nomologically necessary in 
sense that any sentient species that evolves would have to invent them. But 
because mathematics and logic are built into our language and are necessary to 
language that we could recognize, does not show they are "out there" like the 
we call 'that chair' is out there.  That chair would continue to exist even if 
humans were wiped off the Earth - but the concept of 'chairs' wouldn't and 
would '2'.

Ontology is invented too.  Most ontologies put the chair 'out there' and math 
'in our 
heads'.  Some put the chair 'out there' and math in 'Mathematica' (I don't like 
use 'Platonia' because Plato put chair in there too).  Java has it's own 
that we invented to reflect an idea of instances and classes.  There's nothing 
necessary about that as is easily seen from the fact that anything Java can do 
also be done in Fortran or assembly or by a Turing machine.

Brent Meeker
The sciences do not try to explain, they hardly even try to  interpret, they 
make models. By a model is meant a  mathematical construct which, with the 
of certain verbal  interpretations, describes observed phenomena. The 
of  such a mathematical construct is solely and precisely that it is  expected 
to work.
        --—John von Neumann

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