Natural Functorial Categorical Intuition

["Ellis D. Cooper" <xtalv1@netropolis.net>]

\documentclass{article}
\usepackage{amssymb}
\usepackage[all]{xy}
\begin{document}
According to David H. Bailey and Jonathan M. Borwein in an article on
experimental mathematics in the current issue of the Notices of the AMS,
G. Polya quotes J. Hadamard, "The object of mathematical rigor is to
sanction and legitimize the conquests of intuition, and there was
never any other object for it." This predates my motto, "Rigor cleans
the window through which intuition shines."

   I intuit that there is a
Ground beneath all foundations of mathematics. A few details are in
my book, ``Mathematical Mechanics: From Particle to Muscle," from
which the following is derived.

The primary distinction of the Ground of Mathematics is between
Discourse and Surface.
Every expression occurs in a region of the Surface. The Discourse may specify
expressions and regions in the Surface.

A Context is a specified region of the Surface within which smaller
regions may or may not contain
expressions. In any case, the extent of such a Context is clearly
marked, for example, by
Chapter, Section, Subsection, or Paragraph headings. Thus, Contexts
may be nested. Sometimes
a single expression is considered to be a Context.
Care must be taken to observe Context boundaries.

In a specified Context
the choice of a symbol to represent an idea -- including all its
copies in the Context --
may be replaced by some other symbol in all of its occurrences within
the Context, provided
the replacing symbol occurs nowhere else in the Context. In this
sense the replaced symbol is
called bound. For every symbol there is a sufficiently large Context
in which it is bound.

The Ground includes human cognitive ability capable of answering the
following questions:
What is the specified Context of the Surface?
What is the specified region of the Surface?
What is the specified expression?
For a specified region of the Surface is there some expression
occurring the region?
Is a specified expression occurring in a specified region of the Surface?
Of two specified regions is one left, right, above or below the other?
Of two expressions in distinct regions, is one a copy of the other?
What is the total count of expressions in a row, column, or other
specified region?
Is a Context nested within another Context?

The Ground includes human muscle contraction capable of performing
the following actions:
Introduce an expression specified in Discourse into a specified
region of Surface. For example,
to introduce a copy of an expression of Discourse in a blank region
to the right of a specified region.
Repeat this action to yield a list expression on the Surface.
Copy the expression in a specified region into a distinct specified region.
Mark the start of a Context.
Mark the end of a Context.
Delete the expression -- if any -- occurring in a specified region.

These capabilities are called the Ground Rules of Discourse. The book discusses
Symbol \& Expression, Substitution \& Rearrangement, Dot \& Arrow,
and that Diagrams Rule by Diagram Rules.

Natural language locutions such as ``we write," ``we choose to write,"
``we usually write," ``we sometimes simply write," and so on, are
common in mathematical
writing. A declaration in the Discourse that a described diagram
``exists" is equivalent
to asserting the right but not the obligation to
draw the diagram on the Surface.

For example, assertion of the bounded existential quantifier formula
$(\exists x\in A)P(x)$, where

\[
\xymatrix{A\ar[r]^P&\Omega}
\]

\noindent is a diagram,
corresponds to the
existence on the Surface of a commutative diagram

\[
\xymatrix{A\ar[rr]^P&&\Omega\\
&1\ar[ul]^a\ar[ur]_{\top}&\\
}
\]

\noindent such that $a$ does not occur unbound in the Context, and
the assertion of the
bounded universal quantifier formula $(\forall x\in A)(P(x))$
corresponds to the existence on the Surface of
a commutative diagram

\[
\xymatrix{
&1\ar[dr]^{\top}&\\
A\ar[rr]_P\ar[ur]^{\tau_A}&&\Omega\\
}
\]

\noindent In this Ground for foundations of mathematics, everything
is a diagram.

Ellis D. Cooper
\end{document}





[For admin and other information see: http://www.mta.ca/~cat-dist/ ]