Natural Functorial Categorical Intuition

Natural Functorial Categorical Intuition

[Ellis D. Cooper]

My motto has been "Rigor cleans the window through which intuition
shines." It seems to me that
a great deal more is known about mathematical rigor than about
mathematical intuition.

Economics Nobelist Daniel Kahneman recently published at www.edge.org
a survey of several
men-decades of research on flaws of human statistical intuition. A
paper, "The Intuitive Experience" in
"The View from Within" (Journal of Consciousness Studies, V.6 1999)
discusses in considerable
detail schema and methods of invocation of intuition in
psychotherapy, art, and biology research.

My overall question is whether there really are different kinds of
intuition depending on the research
discipline. In particular, is there some kind of kinetic intuition
specific to category theory that crucially
involves visualization of time-varying diagrams? Do conjectured
adjoint functors arise from distinct algebraic, or
geometric, or logical intuitions? Do categorists deploy special
methods to access their intuition, or
do intuitions just happen to those with a knack for category theory?
Does categorical intuition just
develop with experience, or is there a specialized training to
enhance it? Is categorical intuition
any different from mathematical intuition in general?

Ellis D. Cooper



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Re: Natural Functorial Categorical Intuition

[posina]

Dear All,

My understanding, having studied in some detail the behavioural,
psychological, and cognitive scientific studies, is that a serious study of
mathematics (beginning with Lawvere & Schanuel's Conceptual Mathematics)
can inform cognitive sciences more so than the other way around, with all
due respect to Dan Kahneman and those 'where mathematics comes from' guys.

Thank you,
posina

On Tue, 27 Sep 2011 17:20:41 -0400, "Ellis D. Cooper"
<xtalv1@netropolis.net> wrote:
> My motto has been "Rigor cleans the window through which intuition
> shines." It seems to me that
> a great deal more is known about mathematical rigor than about
> mathematical intuition.
> 
> Economics Nobelist Daniel Kahneman recently published at www.edge.org
> a survey of several
> men-decades of research on flaws of human statistical intuition. A
> paper, "The Intuitive Experience" in
> "The View from Within" (Journal of Consciousness Studies, V.6 1999)
> discusses in considerable
> detail schema and methods of invocation of intuition in
> psychotherapy, art, and biology research.
> 
> My overall question is whether there really are different kinds of
> intuition depending on the research
> discipline. In particular, is there some kind of kinetic intuition
> specific to category theory that crucially
> involves visualization of time-varying diagrams? Do conjectured
> adjoint functors arise from distinct algebraic, or
> geometric, or logical intuitions? Do categorists deploy special
> methods to access their intuition, or
> do intuitions just happen to those with a knack for category theory?
> Does categorical intuition just
> develop with experience, or is there a specialized training to
> enhance it? Is categorical intuition
> any different from mathematical intuition in general?
> 
> Ellis D. Cooper
> 

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Re: Natural Functorial Categorical Intuition

[Dr. Cyrus F Nourani]

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► 


 

On Sep 28, 2011, posina <posina@salk.edu> wrote: 


Dear All,

My understanding, having studied in some detail the behavioural,
psychological, and cognitive scientific studies, is that a serious study of
mathematics (beginning with Lawvere & Schanuel's Conceptual Mathematics)
can inform cognitive sciences more so than the other way around, with all
due respect to Dan Kahneman and those 'where mathematics comes from' guys.

Thank you,
posina


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Re: Natural Functorial Categorical Intuition

[Jocelyn Ireson-Paine]

On Wed, 28 Sep 2011, posina wrote:

> Dear All,
>
> My understanding, having studied in some detail the behavioural,
> psychological, and cognitive scientific studies, is that a serious study of
> mathematics (beginning with Lawvere & Schanuel's Conceptual Mathematics)
> can inform cognitive sciences more so than the other way around, with all
> due respect to Dan Kahneman and those 'where mathematics comes from' guys.
>
Do tell us more. Mathematics has informed cognitive science on, for
example, the structure of natural-language grammars, how neurons compute,
and how the brain uses the geometric constraints on 3D shapes when
understanding images. But you mention Lawvere & Schanuel's "Conceptual
Mathematics". What can category theory contribute? I suggested some
possibilities in
http://www.j-paine.org/why_be_interested_in_categories.html , "What Might
Categories do for AI and Cognitive Science?". There must be lots more.

> Thank you,
> posina
>
Jocelyn


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Natural Functorial Categorical Intuition

[Ellis D. Cooper]

\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}





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Natural Functorial Categorical Intuition

[Ellis D. Cooper]

Many thanks for responses to my initial post on the Subject.
It was partly motivated by the seeming variety of kinds of intuition
in the cited
references.

Also, I am utterly in awe of categorical intuitions codified, for example, by
adjoint pairs of functors in geometry, algebra, and logic. Until recently
I only dreamed of some day having such an intuition. I think I now have one,
and would like to know if you agree that it is specifically a
categorical intuition.
It is a separate question whether there is a rigorous explication and proof.

Define a (two-dimensional) shape to be a smooth injection of the circle
into the plane. By the Jordan Curve Theorem a shape has an exterior.
Given a point in the exterior -- call it a viewpoint -- there exists
a finite set
of intersections of the lines through the viewpoint -- call them
sightlines -- which are either tangent
to or pass through an inflection point (with respect to an orthogonal
coordinate system in which the sightline is one coordinate).

For a sufficiently remote viewpoint (maybe infinitely far away) there
exist exactly
two of these sightlines tangent to the shape between which
the angle is less than pi radians, and such that all other of these sightlines
are contained within the sweep of one of the two to the other.

I am guessing that if a cover relation (as in Hasse diagrams of
finite posets) is
defined to be an acyclic irreflexive relation, then the category of
finite posets
is a reflective subcategory of the category of cover relations, where
the adjoint
to the inclusion is given by taking the reflexive transitive closure.
Is that right?

Given a shape and a viewpoint
construct a cover relation by declaring that among the elements of the
above set of intersections, one element covers another if (1) it is
encountered earlier by a sweeping sightline as above,
and (2) there exists a segment of the shape connecting the two elements
that contains no other intersections. By the aforementioned adjunction this
construction leads to a finite poset for any given shape and remote viewpoint.

It is my intuitive guess that for a given shape there exists an
algebraic structure
comprised of the set of all finite posets corresponding to its
viewpoints, and that
this algebraic structure involves spans of poset maps among those
finite posets. I guess
moreover that there exists a functor from the isotopy category of shapes to
an appropriately defined category of these algebraic structures,
which I like to
call algebraic models of shapes.

Among the aforementioned intersections there are those which are directly
"visible" from a viewpoint in the sense that no points of the shape intervene
between such a point and the viewpoint. Call the set of directly visible points
the partial-view from the viewpoint. It is my intuitive guess that
the set of all partial-views
of a shape also comprise an algebraic structure. Call it the
partial-view model of
the shape -- clearly it forgets information contained in the algebraic model.

It seems to me that the algebraic model of a shape determines the partial-view
model. That is, if multiple intersections lie on the same sightline
in the algebraic model,
then only the one closest to the viewpoint is in its partial-view. So
there exists a functor from a category of
algebraic models to a category of partial-view models. That word
"closest" is what made my intuition click:
this functor has a left adjoint. In other words, I am guessing that
partial-views
of a shape may be "integrated" to form its algebraic model.

If these intuitions can be rigorously worked out maybe there is a
mathematical theory that is to "dents" in shapes
as algebraic topology is to holes in spaces.

Ellis D. Cooper



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Re: Natural Functorial Categorical Intuition

[peasthope]

Ellis,

> ... whether there really are different kinds of
> intuition depending on the research discipline.

I'd say "yes".  For example, most people have a fairly good spacial
perception but individuals appearing unable to imagine a two or three
dimensional image exist.

Isn't this a question of neuroscience rather than of mathematics?
Neuroscience is barely established as a discipline.  Without intending
any offense, the subject has far to go before it can explain phenomena
such as intuition.  The question of how intuition works at present is
perhaps analogous to the question of how planets move in the sky, asked
in the year 1400.  Celestial mechanics required roughly five centuries
to "sort out".  Intuition might take even longer.

Best regards,              ... Peter E.

-- 
Telephone 1 360 450 2132.  bcc: peasthope at shaw.ca
Shop pages http://carnot.yi.org/ accessible as long as the old drives survive.
Personal pages http://members.shaw.ca/peasthope/ .



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Re: Natural Functorial Categorical Intuition

[Fred E.J. Linton]

Hi, Ellis,

Very ambitious questions you're asking here.
I'm not sure there's much in the way of clear-cut answers, though.

To begin with, I'm not nearly as sure as you are that

> a great deal more is known about mathematical rigor than about
> mathematical intuition.

Or perhaps there is -- what particularly did you have in mind that 
"is known about mathematical rigor"?
  
> My overall question is whether there really are different kinds of
> intuition depending on the research discipline. 

Well, "depending on the research discipline"? Who can tell?
But "whether there really are different kinds of intuition"?
Almost surely yes. Both within and across disciplines.
Just as people have different complexions, they'll have
"different kinds of intuition".

> ... In particular, is there some kind of kinetic intuition
> specific to category theory that crucially
> involves visualization of time-varying diagrams? Do conjectured
> adjoint functors arise from distinct algebraic, or
> geometric, or logical intuitions? Do categorists deploy special
> methods to access their intuition, or
> do intuitions just happen to those with a knack for category theory?

Intuitions, I'd say, "just happen to those with a knack for" intuitions.
And no, I'm not just being difficult. I believe you're suffering from a
sort of Aristotelian disease, common to many even two millenia after
Aristotle, that believes any sufficiently well-formed and narrowly 
specific question must have a clear yes-no, or "this-or-that", answer.

How can someone like you, who's really smart enough not to fall for that,
not be smart enough not to fall for that? -- because Aristotelean "yes/no"
black-or-white logic doesn't generally apply to the real world, with all its
gradations and shades of gray :-) .

> Does categorical intuition just
> develop with experience, or is there a specialized training to
> enhance it? 

Why not both? or neither? or other?

> ... Is categorical intuition
> any different from mathematical intuition in general?
  
At the risk of sounding more Zen than I intend, I'd answer
"Of course it's different. And yet it's not different. 
In fact it's both different and not different. As well,
it's neither different, nor not different."

"Do I contradict myself? Very well, then, I contradict myself." :-) .

> Ellis D. Cooper
  
Cheers, -- Fred 
(with a tip o' the ol' hat to Carl Sandburg)



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