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'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:
Meaning and the Problem of Universals
A highly rated site on Logic and Ontology in the Google Web Directory.
"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
-- G. H. Hardy, section 23, A Mathematician's Apology, Cambridge University Press, 1940
"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 M24 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 M24
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
"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
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.
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