Cognitive Blending
and the Two Cultures

Examples by Steven H. Cullinane, March 5, 2004

The View from the Sciences:

From John Baez,
http://math.ucr.edu/home/baez/week198.html:

"Analogies are incredibly important in mathematics. Some can be made completely precise and their content fully captured by a theorem, but the 'deep' ones, the truly fruitful ones, are precisely those that resist complete encapsulation and only yield their secrets a bit at a time.  Corfield* quotes André Weil, who describes the phenomenon as only a Frenchman could-- even in translation, this sounds like something straight out of Proust:

     As every mathematician knows,
     nothing is more fruitful than these obscure analogies,
     these indistinct reflections of one theory into another,
     these furtive caresses,
     these inexplicable disagreements;
     also nothing gives the researcher greater pleasure.

I actually doubt that every mathematician gets so turned on by
analogies, but many of the 'architects' of mathematics do, and
Weil was one." 

* David Corfield, Towards a Philosophy of Real Mathematics, Cambridge U. Press, Cambridge, 2003.

The View from the Arts:

"Story is a basic principle of mind. Most of our experience, our knowledge, and our thinking is organized as stories. The mental scope of story is magnified by projection - one story helps us make sense of another. The projection of one story onto another is parable, a basic cognitive principle that shows up everywhere, from simple actions like telling time to complex literary creations like Proust's Ŕ la recherche du temps perdu."

-- Mark Turner, preface to The Literary Mind: The Origins of Thought and Language. This is the same Mark Turner whose work in the journal Cognitive Science is cited below.

It seems that pure mathematics and literature, two apparently unrelated pursuits, may have something in common.

The anecdotal and fragmentary material below illustrates some of my own attempts to do what theorists call "cognitive blending" -- of personal memories (of both reality and fiction) as well as of some mathematical concepts that interest me. Turner's writing, which I have just come across, seems highly relevant to my own efforts along these lines.

This is a sequel to my journal entries of
Feb. 29 through March 5, 2004.

 The Monk's Path 

From a draft of

Mark Turner, Cognitive Dimensions of Social Science: The Way We Think About :Politics, Economics, Law, and Society, Oxford University Press, 2001: 

Consider the riddle of the Buddhist monk, analyzed in Fauconnier and Turner* (1998): "A monk rises at dawn and begins to walk up a mountain path, which he reaches at sunset. He sits and meditates through the night, rises the next morning, and walks down the path, reaching the bottom at sunset. Prove that there is a place on the path that the monk inhabits at the same hour of the day on the two consecutive days."

*Gilles Fauconnier and Mark Turner,
 "Conceptual Integration Networks,"
Cognitive Science, 22(2) 1998, 133-187....

"Abstract:  Conceptual integration—'blending'—is a general cognitive operation on a par with analogy, recursion, mental modeling, conceptual categorization, and framing.  It serves a variety of cognitive purposes." 

---

Through the Looking Glass,
Chapter 7:
The Lion and the Unicorn

Holiday Romance, by Charles Dickens

PART I

Introductory Romance.
From the Pen of
William Tinkling, Esq. (Aged eight.)

This beginning-part is not made out of anybody's head, you know. It's real. You must believe this beginning-part more than what comes after, else you won't understand how what comes after came to be written. You must believe it all; but you must believe this most, please. I am the editor of it....

Through taking the back road, I was so fortunate as to meet nobody, and arrived there uninterrupted.

Road to Nowhere

In memory of
Albert Axelrod

who died on
Mardi Gras, 2004

The above Dickens phrase "taking the back road" suggests both Robert A. Heinlein's Glory Road and the phrase "the road to nowhere."   A search on the latter yields The Road to Nowhere, by John Sealander:

"I ... am the only kid in the third grade ...  trying to read every single Robert Heinlein novel...."

From Glory Road, by Robert A Heinlein, 1963, Ch. XV: "There is a go-for-broke tactic, 'the target,' taught by the best swordmasters, which consists in headlong advance with arm, wrist, and blade in full extension -- all attack and no attempt to parry.  But it works only by perfect timing when you see your opponent slacken up momentarily.   Otherwise it is suicide...." "Lunge and counter and thrust, Sing me the logic of steel!" "Tweedledum and Tweedledee Agreed to rustle cattle. Said Tweedledum to Tweedledee I'll use my nice new saddle."

My own personal back road
to nowhere begins at the
above chapel in
Cuernavaca, Mexico.

For a few details, see

Shining Forth.

In conclusion, a quotation from Fauconnier and Turner -- 

"An organizing frame provides a topology for the space it organizes—that is, it provides a set of organizing relations among the elements in the space.  When two spaces share the same organizing frame, they share the corresponding topology and so can easily be put into correspondence.  Establishing a cross-space mapping between inputs is straightforward when they share the same organizing frame."

-- and two  illustrations.

From Sealander's Road to Nowhere:

From my own
Time Fold:

"This correspondence may serve to illustrate what Eliot called 'a sense of the timeless as well as of the temporal and of the timeless and of the temporal together.' " -- Time Fold

Related websites:

Center for the Cognitive Science of Metaphor Online

Mark Turner's site

Literature, Cognition & the Brain

Many other sites may be found at the above three.

Related literary tools:

Wallace Stevens: Collected Poetry and Prose, The Library of America, 1997 

Thomas F. Walsh, Concordance to the Poetry of Wallace Stevens (University Park, Pennsylvania: Pennsylvania State University Press, 1963)

"Poetry is a satisfying of the desire for resemblance....
If resemblance is described as
a partial similarity between two dissimilar things,
it complements and reinforces
that which the two dissimilar things have in common.
It makes it brilliant."
- Wallace Stevens, "Three Academic Pieces" in The Necessary Angel (1951), page 690 in Collected Poetry and Prose

Page created March 5, 2004