why finite-number versus infinite-number cannot be a primitive concept as point or line #319 ; Correcting Math [Logic]

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From: Archimedes Plutonium on 20 Jan 2010 03:44

pnyikos wrote:
> On Jan 16, 11:38 am, Archimedes Plutonium
(snipped)
> > Why not examine your own
> > definition of Finite versus Infinite.
>
> My own, inexpressible-to-others concept, has been under severe
> examination since 1978, when I made a searching study of the
> foundations of mathematics and realized that this concept is not
> definable in terms of other concepts; it is a primitive concept, like
> "time".
>

Peter, lets raise the stakes here. Instead of the physics "time or
space"
as intuition. Let us talk about Primitive Concepts and the two most
famous
in mathematics are in geometry, not numbers, and are the primitive
concepts
of "point and line" in geometry.

Now let us ask a question. Are these the only two primitive concepts
in all of mathematics?
I would say no, from a physics standpoint of duality. That Geometry is
the dual of Numbers,
and since Geometry gets by with having two primitive concepts, it is
likely that Numbers
must have two and only two primitive concepts that relates to "points
and lines".

So are there any primitive concepts in Numbers? I can think of two
primitive concepts,
set and membership.

So I think that in the whole of mathematics, there are two and only
two primitive concepts
and all the other things have to be either axiomatized or defined.

Now looking at "point and line" as primitive concepts of Geometry, do
they reflect in some
meaningful way "membership and set" for Numbers? Is not a line a set
of points and is not
a point a member of a line? Likewise, the reverse, is not a set a line
of members and is not
a member a point of a set?

So I think where Peter is trying to escape from having to well-define
or precisely define a
Finite-number versus an Infinite-number by shrugging the task off as a
intuit or primitive concept, is not allowed. That mathematics already
has all the allowable primitive concepts and that everything else in
mathematics has to be precision defined. If Geometry can have
only two primitive concepts and all the rest are precision defined
then Numbers have to
have a precision defined Finite-number and Infinite-number.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

From: A on 20 Jan 2010 11:28

From: pnyikos on 28 Jan 2010 23:23

On Jan 20, 3:44 am, Archimedes Plutonium
<plutonium.archime...(a)gmail.com> wrote:
> pnyikos wrote:
> > On Jan 16, 11:38 am, Archimedes Plutonium
> (snipped)
> > > Why not examine your own
> > > definition of Finite versus Infinite.
>
> > My own, inexpressible-to-others concept, has been under severe
> > examination since 1978, when I made a searching study of the
> > foundations of mathematics and realized that this concept is not
> > definable in terms of other concepts; it is a primitive concept, like
> > "time".
>
> Peter, lets raise the stakes here. Instead of the physics "time or
> space" as intuition.

Isn't that lowering the stakes from your POV? Don't you want to base
everything on physics?

> Let us talk about Primitive Concepts and the two most
> famous
> in mathematics are in geometry, not numbers, and are the primitive
> concepts
> of "point and line" in geometry.

More famous, but not the most primitive. They can be defined in terms
of simpler concepts, and you will probably protest that the
definitions are arbitrary, but how can they possibly be more arbitrary
than saying that (10^500 -1)+ 1 = 10^499? Here you have a number with
two different immediate predecessors, so when you subtract 1 from
10^499 you get two different numbers, the usual one and the unusual
one. Why do you like this setup?

> Now let us ask a question. Are these the only two primitive concepts
> in all of mathematics?
> I would say no, from a physics standpoint of duality. That Geometry is
> the dual of Numbers,
> and since Geometry gets by with having two primitive concepts, it is
> likely that Numbers
> must have two and only two primitive concepts that relates to "points
> and lines".

I take it the connection is to arbitrarily pick one point on a line
and call it 0, then another and call it 1.

Euclid said lines can be extended indefinitely in both directions.
Evidently you are jettisoning this axiom and having the real line loop
back on itself in both directions. Have you figured out other
idiosyncrasies of your system?

> So are there any primitive concepts in Numbers? I can think of two
> primitive concepts,
> set and membership.

Yes. And I can define the real numbers just using these two
concepts.

> So I think that in the whole of mathematics, there are two and only
> two primitive concepts
> and all the other things have to be either axiomatized or defined.

Yes.

> Now looking at "point and line" as primitive concepts of Geometry, do
> they reflect in some
> meaningful way "membership and set" for Numbers? Is not a line a set
> of points and is not
> a point a member of a line? Likewise, the reverse, is not a set a line
> of members

Not unless you impose a total order on it. Consider the set of points
in the plane, for example. What total order do you want to impose on
that?

> and is not
> a member a point of a set?

It does no harm to call it that.

> So I think where Peter is trying to escape from having to well-define
> or precisely define a
> Finite-number versus an Infinite-number by shrugging the task off as a
> intuit or primitive concept, is not allowed.

You don't want to allow it, but I do.

> That mathematics already
> has all the allowable primitive concepts and that everything else in
> mathematics has to be precision defined.

If that's the way you want it, you have to content yourself with the
fact that there are many different models of the Peano axioms, some
with infinite integers in addition to the usual finite ones.

> If Geometry can have
> only two primitive concepts and all the rest are precision defined
> then Numbers have to
> have a precision defined Finite-number and Infinite-number.

Numbers are not the primitive concept for mathematicians working in
foundations. Sets are. We define finite and infinite sets in a
certain way, then conjure up numbers (like aleph-nought and aleph-one)
to describe their cardinality.

All natural numbers are finite, but the set of natural numbers is
infinite.

Peter Nyikos

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