of Pontius Pilate's unanswered question

-- H. S. M. Coxeter, 1987, introduction to

Richard J. Trudeau's remarks on

the "Story Theory" of truth

as opposed to

the "Diamond Theory" of truth

in

The image at right below shows the cover of a booklet I wrote in 1976. This booklet details the implications of what I call the "diamond theorem," after the diamond figure in Plato's Meno dialogue. For technical details, see The Diamond Theorem.

The site you are now viewing, Math16.com, offers a less formal treatment of philosophical and literary matters related to the diamond theorem.

The following quotation describes, and inspired, the picture on the Diamond Theory cover:

"Adorned with cryptic stones and sliding shines,

An immaculate personage in nothingness,

With the whole spirit sparkling in its cloth,

Generations of the imagination piled

In the manner of its stitchings, of its thread,

In the weaving round the wonder of its need,

And the first flowers upon it, an alphabet

By which to spell out holy doom and end,

A bee for the remembering of happiness."

-- Wallace Stevens, "The Owl in the Sarcophagus"

Another description of this picture may be found in the novel *A Wind in the Door*. A main character in this book is the (singular) cherubim named Proginoskes. A comment from the author:

"Thank you for the diamond theory. It does, indeed, look more like Proginoskes than any of the pictures on the book jackets."

-- Madeleine L'Engle, letter of November 28, 1976

A Mathematician's Aesthetics

The Diamond Archetype

Aesthetics of Parallelism

Geometry of the I Ching

*The Non-Euclidean Revolution*.

This book by Richard J. Trudeau, with a brief introduction by H. S.
M. Coxeter, traces in the recent history of geometry the conflict
between what Trudeau calls the "Diamond Theory of truth" and the "Story
Theory of truth"
-- known to more traditional philosophers as "realism" and
"nominalism."

Plato as a precursor of Gerard Manley Hopkins's "immortal diamond." An illustration shows the prototype of the figure D discussed at the diamond theory site.

Plato's Diamond Revisited

Ivars Peterson's Nov. 27, 2000 column
"Square of the Hypotenuse" which discusses the diamond figure as used
by Pythagoras (perhaps) and Plato. Other references to the use of
Plato's diamond in the proof of the Pythagorean theorem:

'You see,' he said, 'it seemed to me so beautiful....'

I nodded. 'Yes, it's very beautiful,' I said -- 'it's very beautiful indeed.'"

-- Aldous Huxley, "Young Archimedes," in

(See Heath, Sir Thomas Little (1861-1940),

Illustrated legend of the diamond proof

Babylonian version of the diamond proof

This site offers, to those who click on "problem of non-being," a quotation from W. V. Quine regarding this question: If we say of something that it does not exist, then

Meaning and the Problem of Universals

A highly rated site on Logic and Ontology in the Google Web Directory.

-- Simon Blackburn,

"You will all know that in
the Middle Ages there were supposed to be various classes of angels....
these hierarchized celsitudes are but the last traces in a less
philosophical age of the ideas which Plato taught his disciples existed
in the spiritual world."

-- Charles Williams, page 31, Chapter Two, "The Eidola and the Angeli," in *The Place of the Lion* (1933), reprinted in 1991 by Eerdmans Publishing

For Williams's discussion of Divine Universals (i.e., angels), see Chapter Eight of *The Place of the Lion*.

"People have always longed
for truths about the world -- not logical truths, for all their
utility; or even probable truths, without which daily life would be
impossible; but informative, certain truths, the only 'truths' strictly
worthy of the name. Such truths I will call 'diamonds'; they are highly
desirable but hard to find....The happy metaphor is Morris Kline's in *Mathematics in Western Culture* (Oxford, 1953), p. 430."

-- Richard J. Trudeau, *The Non-Euclidean Revolution*, Birkhauser Boston, 1987, pages 114 and 117

"A new epistemology is
emerging to replace the Diamond Theory of truth. I will call it the
'Story Theory' of truth: There are no diamonds. People make up stories
about what they experience. Stories that catch on are called 'true.'
The Story Theory of truth is itself a story that is catching on. It is
being told and retold, with increasing frequency, by thinkers of many
stripes.... My own viewpoint is the Story Theory.... I concluded long
ago that each enterprise contains only stories (which the scientists
call 'models of reality'). I had started by hunting diamonds; I did
find dazzlingly beautiful jewels, but always of human manufacture."

-- Richard J. Trudeau, *The Non-Euclidean Revolution*, Birkhauser Boston, 1987, pages 256 and 259

Trudeau's confusion seems to
stem from the nominalism of W. V. Quine, which in turn stems from
Quine's appalling ignorance of the nature of geometry. Quine thinks
that the geometry of Euclid dealt with "an emphatically empirical
subject matter" --
"surfaces, curves, and points in real space." Quine says that Euclidean
geometry lost "its old status of mathematics with a subject matter"
when Einstein established that space itself, as defined by the paths of
light, is non-Euclidean. Having totally misunderstood the nature of the
subject, Quine concludes that after Einstein, geometry has become
"uninterpreted mathematics," which is "devoid not only of empirical
content but of all question of truth and falsity." (*From Stimulus to Science*, Harvard University Press, 1995, page 55)

-- S. H. Cullinane, December 12, 2000

The *correct* statement of the relation between geometry and the physical universe is as follows:

"The contrast between pure
and applied mathematics stands out most clearly, perhaps, in geometry.
There is the science of pure geometry, in which there are many
geometries: projective geometry, Euclidean geometry, non-Euclidean
geometry, and so forth. Each of these geometries is a *model*, a pattern of ideas, and is to be judged by the interest and beauty of its particular pattern. It is a *map* or *picture*,
the joint product of many hands, a partial and imperfect copy (yet
exact so far as it extends) of a section of mathematical reality. But
the point which is important to us now is this, that there is one thing
at any rate of which pure geometries are *not* pictures, and that
is the spatio-temporal reality of the physical world. It is obvious,
surely, that they cannot be, since earthquakes and eclipses are not
mathematical concepts."

-- G. H. Hardy, section 23, *A Mathematician's Apology*, Cambridge University Press, 1940

-- W. V. Quine in

"It's a thing that nonmathematicians don't realize. Mathematics is actually an aesthetic subject almost entirely."

-- John H. Conway, quoted on page 165, Notices of the American Mathematical Society, February 2001.

"There are almost as many different constructions of M_{24} as there have been mathematicians interested in
that **most remarkable of all finite groups**."

-- John H. Conway in *Sphere Packings, Lattices, and Groups*, third edition, Springer-Verlag, 1999

"**The miraculous enters....**
When we investigate these problems, some fantastic things happen.... At
one point while working on this book we even considered adopting a
special abbreviation for 'It is a remarkable fact that,' since this
phrase seemed to occur so often. But in fact we have tried to avoid
such phrases and to maintain a scholarly decorum of language."

-- John H. Conway and N. J. A. Sloane, *Sphere Packings...*, preface to first edition (1988)

Many actions of the Mathieu group M_{24}
may best be understood by splitting the 24-element set on which it acts
into a "trio" of three interchangeable 8-element sets -- "octads," as
in the "Miracle Octad Generator" of R. T. Curtis.
(See chapters 10 and 11 of the above book by Conway and Sloane.)
It is a remarkable fact that the characteristics of such a **trio** are not wholly unlike those of the more famous structure described below by Saint Bonaventure.

-- S. H. Cullinane, March 1, 2001

"Beware lest you believe that you can comprehend the Incomprehensible, for there are six characteristics (of **the Trinity**) which will lead the eye of the mind to dumbstruck admiration. Thus, there is

- supreme communicability coupled with the individuality of persons;
- supreme consubstantiality with the plurality of hypostases;
- total likeness with separate personalities;
- complete equality with order;
- coeternity with emanations;
- complete intimacy with emission.

All of this is prefigured by

-- Saint Bonaventure,

(translated by Lawrence S. Cunningham, Franciscan Herald Press, Chicago, 1979),

Chapter Six, "On the Contemplation of the Most Blessed Trinity under its Name: Goodness"

"Was there really a **cherubim** waiting at the star-watching rock...?

Was he real?

What is real?"

-- Madeleine L'Engle, *A Wind in the Door*, Farrar, Straus and Giroux, 1973, conclusion of
Chapter Three, "The Man in the Night"

"Oh, **Euclid**, I suppose."

-- Madeleine L'Engle, *A Wrinkle in Time*, Farrar, Straus and Giroux, 1962, conclusion of
Chapter Five, "The **Tesseract**"

For more on philosophy and Quine,

and also theology and angels, see

Is Nothing Sacred? and Midsummer Eve's Dream.

For a small memorial to Quine, see

On Linguistic Creation.

"It is a good light, then, for those

That know the ultimate Plato,

Tranquillizing with this jewel

The torments of confusion."

- Wallace Stevens,

Collected Poetry and Prose, page 21,

The Library of America, 1997

Home-page address of the author is
**http://www.m759.com**.

E-mail address of the author is

URL address of this page is **http://math16.com**.

Page last updated June 11, 2006; created December 10, 2000.