Natural Functorial Categorical Intuition
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ . [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Hi, you might be interested to glance at the enclosed Show more results from fortunecity.comDiscovery and Consiousness Science - netzkollektor // Projektenetzspannung.org/cat/servlet/CatServlet?cmd=netzkollektor...Cached You +1'd this publicly. Undo Intelligent Trees, Consciousness Science, and Discvoery Models Cyrus F. Nourani Revised November 1998. ABSTRACT Multiagent Computing Intelligent Syntax ... Cognition · Cognitive Science · Comparative Psychology · Consciousness Research · Cultural Psychology .... Dr Cyrus F Nourani ... [TeX] Consciousness in Science and Philosophy 1998---“Charleston I” - IJSai.ijs.si/mezi/informatica/Informatica/Vol22/No3/abstract.texYou +1'd this publicly. Undo File Format: TeX/LaTeX - View as HTML The parallel discovery of a unified field of consciousness raises fundamental ...... absnum Cyrus F. Nourani. Intelligent Trees and Consciousness Science. ... Discovery and http://www.aspbs.com/multimedia.html. A cognitive at time spatial intelligent congnitive algebraic glimpse. CyrusFN : cyrusfn@alum.mit.edu Akdmkrd.tripod.com Acdmkrd@gmail.com PS IOS Press Books Online, A Haptic Computing Logic – A haptic logic and computing paradigm is presented with a basis for multiagent visual computing ... Books Online Home · IOS Press Home ... Cyrus F. Nourani ... www.booksonline.iospress.nl/Content/View.aspx?piid=2405 - Cached ► 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Well, you could try looking at Michael Arbib's work: he has, in the past, written a book on category theory, and he's done a huge amount of work on cognitive science. Graham On 17/10/11 12:46, Jocelyn Ireson-Paine wrote:
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
\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/ ]
participants (7)
-
Dr. Cyrus F Nourani -
Ellis D. Cooper -
Fred E.J. Linton -
Graham White -
Jocelyn Ireson-Paine -
peasthope＠shaw.ca -
posina