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Topic: LOGIC & MATHEMATICS
Replies: 96   Last Post: Jun 6, 2013 5:19 AM

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 fom Posts: 1,968 Registered: 12/4/12
Re: LOGIC & MATHEMATICS
Posted: Jun 2, 2013 9:52 PM

On 6/1/2013 8:21 AM, Julio Di Egidio wrote:
>
> By the way, that link does not work for me, this one does:
>

Forgive my laziness. I attempted to direct to a link
with extraneous information rather than simply restate
myself.

The only necessary background is that I am highly motivated
by understanding the nature of identity and equivalence
classes in mathematical foundations. That has led to
discerning certain structures for which distributive laws
do not necessarily hold. Once recognized, Lawvere's
discussion of the category of pointed sets as an example
of a category in which the distributive laws fail helped
me to crystallize some of my views.

In that context, one sometimes sees a comparison between
Fregean number classes and von Neumann ordinals as a
definition of number class from which the von Neumann ordinal
is a representative. Although this had never been my
motivation, it reflects the sense of 'number-as-object'
as 'number-as-pointed-class'.

What follows is the relevant excerpt from that link

---------------------------------------------------

been the idea of an "equivalence attachment".
My description at that time had simply been that
every equivalence class must have distinguished
element. So, if

{a,b,c,d,...}

was an equivalence class, it would have an
internal organization corresponding with

a*=*b=c=d=...

or, more accurately,

a*=*b
a*=*c
a*=*d
:
:

a=b=c=d=...

where '*=*' is intended to reflect the
relation with the distinguished element
within the class.

Before commenting on this further, I want to
observe that an underlying "claim" in
metamathematical research is that standard
mathematical practice has been "represented".
It is not that metamathematics does not stand
on its own as an interesting and relevant
discipline. Rather, its subject matter is
often interpreted relative to beliefs
concerning mathematics that are not
mathematical. So, what is or is not
being "represented" will come into play
here.

As obnoxious as my participation on these
newsgroups may have been, it is because
of that participation that I have been able
to work out that the idea concerning equivalence
classes mentioned above involves mathematics
in which the distributive law does not necessarily
hold or,at least, is not two-sided.

My first indication of this had been when
I found the paper,

http://arxiv.org/pdf/quant-ph/9906101v3.pdf

In this paper, Pavicic and Megill determine
that the canonical algebraic model for
the classical logic of Principia Mathematica
is a model in which the usual distributive
laws are not necessary. They did so by
analyzing equivalences in algebraic models.
In my mind, it supported the idea that there
were, in fact, foundational issues associated
with identity, although it would be occurring
at an even lower level than I had originally
considering the eliminability of negation
using the complete connectives, it
redirected some of my efforts to
considering logical equivalence (the
biconditional) more closely.

Another indication came recently when my
geometric constructions involving finite
projective geometries led me to consider
91-point projective planes. There are
four of these planes. One is Desarguesian
and three are not Desarguesian. These
planes are generated from near-fields
on nine elements. Although one of the
planes is generated from a Galois field,
the others correspond with an algebraic
system called miniquaternions.

http://en.wikipedia.org/wiki/Near-field_%28mathematics%29

http://en.wikipedia.org/wiki/Hughes_plane

http://en.wikipedia.org/wiki/Hall_plane

These systems only have one-sided
distributivity.

The paper by Pavicic and Megill is
motivated by investigations into quantum
logic. My reasons for investigating
91-point planes is precisely because of
a construction intended to relate the
free Boolean lattice on 2 generators
to the free orthmodular lattice on 2
generators in order to understand
how the "benzene ring" (the ortholattice
O_6) is central to both models. This
structures closely associated with
studies of quantum physics. What I
believe accounts for this is another
instance where distributive laws are not
necessarily two-sided.

This instance comes from automata
theory. One of the more recent issues
is mobile communication. In his book
describing the pi calculus,

http://en.wikipedia.org/wiki/Pi-calculus

http://en.wikipedia.org/wiki/Robin_Milner

Robin Milner shows how classical
language equivalence with regard to
regular sets (where regular sets are
built up by concatenation -- the system
obeys semigroup associativity and has
two-sided distributivity) fails to
distinguish between deterministic and
non-deterministic automata. He uses
this to motivate his definitions
of simulation and bisimulation.

In that illustration, it is precisely
because of an application of one of
the distributive laws that the two
different automata become identified.

So, while my investigations have
led toward mathematics which is of
interest to those who investigate
quantum logics, it seems that this
arises from a relationship between
non-deterministic automata and the
distributive laws.

Returning to the discussion of equivalence
relations with a distinguished element, such
equivalence classes would be pointed sets.

Session 28 of "Conceptual Mathematics"
by Lawvere and Schanuel is included
precisely for the purpose of giving
an example of a non-distributive
category. That category is the category
of pointed sets.

Wikipedia does not have much to say, but
in the article on pointed spaces, you will
see that quotients of pointed spaces map
distinguished elements to distinguished
elements as would any maps in the
category of pointed sets

http://en.wikipedia.org/wiki/Pointed_set

http://en.wikipedia.org/wiki/Pointed_space

Thus, at least I know the relationship
of my original idea to various pieces
of standard mathematics. I did not
know that before.

But why should equivalence classes
be thought of as pointed sets?

Since I do not feel bound to logicism
or the metamathematics that is used to
enforce that philosophy on others, I
would immediately direct attention to
the non-Boolean partition lattices and
topological quotient maps as the general
intuition grounding this claim.

However, that would not be acceptable.

The "acceptable" answer is that metamathematics
has chosen to ignore a standard practice in
its "representations".

When we teach the arithmetic of fractions,
we speak of one particular fraction being
in lowest terms.

When we speak of the operation inverse
to the squaring of real numbers,
we speak of a principal square root.

When we speak of periodic functions,
we speak of principal branches.

When we speak of well-formed formulas,
we speak of normal forms.

When we speak of equations for a line
in analytical geometry, we speak of
a general equation of a line.

In other words, it is standard mathematical
practice to deal with a multiplicity
by trying to find a criterion for selecting
a distinguished element in order to
have a system with single-valued reference.
It is true that these are just stipulations.
But,these stipulations are done with specific
intent as a matter of dealing with
multply-valued collections that need to
be given a single-valued reference.

Constructions like Dedekind cuts make that
practice a difficult thing to visualize.
But, like the axiom of choice, one does not
need to visualize it if one recognizes that
it is a matter of practice that should be
"represented" (For those of us who respect
model theory, the axiom of choice is necessary
in order that the sign of equality can be
represented relationally).

Perhaps it is not necessary that every
equivalence class be such. However, in
the case of logic and quotient models, one
forms equivalence classes of terms in order
to form term models. It is certainly
not unreasonable to suggest that the
application of logical systems depends
on the presupposition of a system of
canonical names that would then characterize
those equivalence classes as pointed sets.

Does this make me right? Right and
wrong have no place here. It is just
a different viewpoint that leads to a
different opinion concerning the foundation
of mathematics.

As for "private language" or "this is
not mathematics", there are practical
difficulties as to what is good mathematics
when one is investigating a problem
that is independent of known axiom
systems. One cannot be a formalist, and,
one cannot simply comply with 'received
knowledge'. For that, however, I am
sorry.

In any case, it has been difficult. I was a
good student. I had good prospects.
This took that all away. And, in fact,
much more than that.

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5/27/13 Charlie-Boo
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