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A Mathematician's Aesthetics

This page is my contribution to the growing genre of "art quotes" web sites. It gives the background for some of my mathematical work. - S. H. Cullinane, August 1, 2000

G. H. Hardy on the Nature of Mathematics:
"A mathematician, like a painter or a poet,
is a maker of patterns.
If his patterns are more permanent than theirs,
it is because they are made with ideas."
-Godfrey Harold Hardy, A Mathematician's Apology (1940),
reprinted 1969, Cambridge U. Press, p. 84
Reviews and ordering information for A Mathematician's Apology
G. H. Hardy Lecture Hall (Western Canon U.)
Hundreds of web sites on G. H. Hardy
C. G. Jung on Archetypes and Visible Reality:
"All the most powerful ideas in history go back to archetypes.
This is particularly true of religious ideas, but the central concepts
of science, philosophy, and ethics are no exception to this rule.
In their present form they are variants of archetypal ideas,
created by consciously applying and adapting these ideas to reality.
For it is the function of consciousness not only to recognize and assimilate
the external world through the gateway of the senses,
but to translate into visible reality the world within us."
- Carl Gustav Jung, "The Structure of the Psyche" (1927), in Collected Works Vol. 8,
Structure and Dynamics of the Psyche, P. 342
Other Jung quotations on archetypes
Paul Klee on Visible Reality:
"Art does not reproduce the visible; rather, it makes visible....
My aim is always to get hold of the magic of reality
and to transfer this reality into painting -
to make the invisible visible through reality.
It may sound paradoxical, but it is, in fact, reality
which forms the mystery of our existence."
- Paul Klee, "Creative Credo" from The Inward Vision:
Watercolors, Drawings, Writings. Abrams, not dated; published c. 1958.
Recommended browsing and reading on Paul Klee
Wallace Stevens on the Visibility of Archetypes:
"These forms are visible to the eye that needs,
Needs out of the whole necessity of sight."
- Wallace Stevens, "The Owl in the Sarcophagus," (first publ. 1947) in
Collected Poetry and Prose, Library of America, 1997
Wallace Stevens and Painting: A Bibliography

Edward Sapir on Linguistics, Mathematics, and Music:
"...linguistics has also that profoundly serene and satisfying quality
which inheres in mathematics and in music and which may be described as
the creation out of simple elements of a self-contained universe of forms.
Linguistics has neither the sweep nor the instrumental power of mathematics,
nor has it the universal aesthetic appeal of music.
But under its crabbed, technical, appearance there lies hidden
the same classical spirit, the same freedom in restraint,
which animates mathematics and music at their purest."
- Edward Sapir, "The Grammarian and his Language," American Mercury 1:149-155, 1924
Biography of Edward Sapir

Heisenberg on Symmetria or Consonantia:
The following refers to an essay by Werner Karl Heisenberg in "Across the Frontier," published by Harper, 1974, p. 183.
"Heisenberg crystallizes the notion remarkably when he notes,
'Beauty is the proper conformity of the parts
to one another and to the whole.'

Like the Sonnets or the Bill of Rights, Heisenberg's 15-word remark smacks of such precision that one could imagine less eloquent thinkers writing entire books without ever arriving at the core truth Heisenberg lighted upon"
- Mark K. Anderson, "Beauty and the Paradigm,"
originally published in the Shakespeare Oxford Newsletter,
Summer 1997 and Fall 1997/Winter 1998
Text of "Beauty and the Paradigm"
Roger Scruton on Symmetria or Consonantia:
"The ancient theory of music
that we owe to the Pythagoreans,
which is endorsed by Plato in the Timaeus
and by Plotinus, St Augustine, and Boethius
in their treatises on music,
and which survives in Al-Farabi, in Aquinas,
and even in such Renaissance theorists as Zarlino,
is centred on the experience of harmony.
Having noticed that the elementary concords
- octave, fifth, and fourth -
are produced by strings whose lengths are proportioned
according to perfect fractions,
those writers concluded that
our experience of music
is an experience of number.
Number, and the relations of number,
provide the hidden order of the universe;
and numbers are known through the intellect,
and known with a certainty that pertains to no other thing.
When understanding mathematics
we have access to
the order of creation,
and this order is eternal,
like the numbers themselves.
In music we know through experience,
and in time,
what is also revealed to the intellect
as outside time and change.
Just as time is, for Plato and Plotinus,
the moving image of eternity,
so is the experience of music
the revelation in time
of the eternal order.
The beauty of music is the beauty of
the world itself,
revealed to the sense of hearing -
a 'point of intersection of the timeless with time.'"
Roger Scruton, The Aesthetics of Music,
Oxford University Press, 1999, pp. 63-64
Vitruvius and Joyce on Symmetria and Consonantia:
The following is an excerpt from Principles of Architecture, by Jorma Manty (online). Manty is disussing terms used by Vitruvius (De Architectura) and by St. Thomas Aquinas and James Joyce (Portrait of the Artist as a Young Man).

Grace (Venustas): When the appearance of the work shall be pleasing and elegant, and the scale of the constituent parts is justly calculated for symmetry.
In addition to venustas, Vitruvius also devises three principal aesthetic conceptions: eurythmia, symmetria, and decor, and explains them as follows:
Eurythmia (Proportion) implies a graceful semblance, the suitable display of details in their context.
Symmetria is the appropriate harmony arising out of the details of the work itself; the correspondence of each given detail among the separate details to the form of the design as a whole.
Decor demands a faultless ensemble of the details of the composed work; a decor "obeys convention."
(Vitruvius, On Architecture)

This triad of aesthetic principles interestingly has something in common with the three principles mentioned in the statement by St. Thomas Aquinas:
James Joyce translates it as: "Three things are needed for beauty: wholeness, harmony and radiance" And he gives a more detailed explanation of these three terms:
Integritas: "An esthetic image is presented to us either in space or in time. What is audible is presented in time, what is visible is presented in space. But, temporal or spatial, the esthetic image is first luminously apprehended as selfbounded and selfcontained upon the immeasurable background of space or time which is not it. You apprehend it as one thing. You see is as one whole. You apprehend its wholeness."
Consonantia: "Having first felt that its is one thing you feel now that it is a thing. You apprehend it as a complex, multiple, divisible, separable, made up of its parts, the result of its parts and their sum, harmonious."
Claritas: "When you have apprehended it as one thing and have then analysed it according to its form and apprehended it as a thing you make the only synthesis which is logically and esthetically permissible. You see that it is that thing which it is and no other thing. The radiance is the scholastic quidditas, the whatness of a thing."
(James Joyce, Portrait of the Artist as a Young Man)

Manty's Principles of Architecture
Aesthetics Chapter (5) in Joyce's Portrait of the Artist
James Joyce Internet Resources
The Brazen Head: A James Joyce Public House

Roger Fry on Formalism in Art and Mathematics:
" is not impossible to draw a fairly sharp dividing line
between our mental disposition in the case of esthetic response
and that of the responses of ordinary life.
A far more difficult question arises if we try to distinguish it
from the responses made by us to certain abstract mental constructions
such as those of pure mathematics.
Here I conceive the emotional states due to the apprehension of relations
may be extremely similar to those aroused by the esthetic apprehension.
Perhaps the distinction lies in this, that in the case of works of art
the whole end and purpose is found in the exact quality of the emotional state,
whereas in the case of mathematics the purpose is the
constatation of the universal validity of the relations
without regard to the quality of the emotion accompanying apprehension.
Still, it would be impossible to deny the close similarity of
the orientation of faculties and attention in the two cases."
- Roger Fry, Transformations (1926), Doubleday Anchor paperback, 1956, p. 8
Ordering information for Art Made Modern: Roger Fry's Vision of Art
Paul Rand's defense of Roger Fry and formalism
Henri Focillon on Formalism in Islamic Art:
"These combinations are produced by mathematical reasoning.
They are based upon cold calculation;
They are reducible to patterns of the utmost aridity.
But deep within them, a sort of fever
seems to goad on and to multiply the shapes;
some mysterious genius of complication
interlocks, enfolds, disorganizes and reorganizes
the entire labyrinth.
Their very immobility sparkles with metamorphoses.
Whether they be read
as voids or solids,
as vertical axes or as diagonals,
each one of them both withholds the secret and exposes the reality
of an immense number of possibilities."
- Henri Focillon, The Life of Forms in Art (1934), translated by George Kubler.
Zone Books, distributed by The MIT Press, 1989, pp. 41-42
Ordering information and reviews for The Life of Forms in Art

Gerard Manley Hopkins - Beauty is a Relation:
"Then the beauty of
the oak and the chestnut-fan and the sky
is a mixture of
likeness and difference or
consistency and variety or
symmetry and change....
And if we did not feel the likeness
we should not think them so beautiful,
or if we did not feel the difference
we should not think them so beautiful.
The beauty we find is from the comparison we make....
Beauty therefore is a relation,
and the apprehension of it a comparison."
- Gerard Manley Hopkins, "On the Origin of Beauty: A Platonic Dialogue" (1865).
Abridged version in Poems and Prose of Gerard Manley Hopkins,
ed. W. H. Gardner (1953), Penguin Classics, 1985, pp. 98 and 103
Gerard Manley Hopkins' Aesthetic Theory, by Marco Graziosi
Wallace Stevens - Poetry is a Resemblance:
"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 Stevens' Collected Poetry and Prose, published by The Library of America, 1997.
Hartford Friends and Enemies of Wallace Stevens

Aesthetics of the Diamond Archetype:
S. H. Cullinane on the diamond archetype
Combinatorial mathematics of the diamond archetype
The Diamond 16 Puzzle
Aesthetics of Parallelism
Block Designs
Affine Geometry of the I Ching hexagrams

Page last modified Feb. 5, 2005; created August 1, 2000.