The Aesthetic Dimension of Education in the Abstract Disciplines

THE JOURNAL OF
Aesthetic Education
REPRINT
VOLUME 4 • NUMBER 3 • JULY 1970

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The Aesthetic Dimension of
Education in the Abstract Disciplines
KENNETH R. CONKL1N
There are two senses in which education can be said to have an aesthetic
dimension: the processes of teaching, learning, and knowing may have
aesthetic aspects as may the subject matter itself. T h e present article
seeks to show that all education has an aesthetic dimension in both
senses and that, indeed, the aesthetic dimension is so essential that no
education is possible without it.
It is fairly obvious that teaching and learning have aesthetic aspects,
although the aesthetic aspect of knowing is quite interesting and highly
controversial. These topics are discussed in section one. Section two ex-
plores the aesthetic aspect of subject matter — especially subject matter
composed of abstract concepts. Section three discusses concomitant
learnings as aesthetic by-products of content and method. Finally, sec-
tion four relates these ideas to certain problems in curriculum develop-
ment, teaching methods, school administration, and teacher education.
1. THE AESTHETICS OF TEACHING, LEARNING, AND KNOWING
Regardless of the subject matter involved, teaching is a performance.
Teachers, of course, should not be judged according to the standards
applied to actors, opera singers, ballet dancers, or artists; yet, it is clear
that teachers do convey moods, use their voices, gesture and move
about, and make drawings on the blackboard and, therefore, aesthetic
criticisms are possible. There may well be disagreement concerning the
KENNETH R. CONKLIN received his Ph.D. in the philosophy of education from
the University of Illinois, Urbana, and taught at Oakland University in Michi-
gan before accepting a position at Emory University. He has published in
several educational journals, including Educational Theory, T h e Educational
Forum, and T h e Record (Teachers College).

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22 KENNETH R. CONKLIN
importance of aesthetic criteria in evaluating teacher performance, and
certainly, it is impossible to specify criteria as either necessary or suffi-
cient for effective teaching. But there is general agreement that good
teaching requires considerably more than knowledge of subject matter.
Subject matter must be presented effectively, and this effectiveness is
primarily determined according to aesthetic considerations.
An effective teacher may have a voice which soothes his students but
on occasion may employ a harsh, rasping voice with equal effectiveness.
But in either case, qualities of voice are significant. Sometimes teacher
enthusiasm will stimulate the productivity of students, while at other
times teacher apathy (perhaps deliberately portrayed) will disturb stu-
dents and thereby encourage their productive thought or action. An
effective teacher, like an effective actor, controls his performance, ad-
justing it to the requirements of changing circumstances in order to
produce intended results. A teacher must appreciate the moods of his
students, but he must also maintain an appropriate psychological dis-
tance in order to provide an intelligently manipulated organization of
his conduct. For best results in handling "the discipline problem," a
teacher must achieve an appropriate balance between empathy and
distance.
While the learner's primary task may be to attend to the content of a
lesson, it is also true that he undergoes an aesthetic experience as a spec-
tator of the teacher's performance. Indeed, as we shall later see, some
of the most important learnings occur as concomitant results of this
aesthetic spectatorship. A behaviorist might well argue that teaching
and learning are entirely analyzable as sequences of empirical stimuli
and responses, but the behaviorist would also analyze art, poetry,
music, and drama in the same way. In short, however one might ana-
lyze the fine arts, the same mode of analysis and criticism will yield
significant observations about the processes of teaching and learning.
In examining teaching and learning as aesthetic processes, we tend to
ignore that these processes serve the purpose of conveying subject matter
and that the conveyance of subject matter may also be subordinated
to the still larger purpose of putting the subject matter to use after the
teaching-learning process has ended. But this argument does not destroy
the validity of regarding teaching and learning as aesthetic perfor-
mances: many works of art (including all that are "representational"),
portray subject matter in the same sense and sometimes have the pur-
pose of providing social or religious commentary. Whether there is
recognizable subject matter and whether there are propositional lessons

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ABSTRACT DISCIPLINES 23
to be learned have nothing to do with the fact that criticism of a per-
formance may be made on aesthetic grounds. Indeed, it has been as-
serted that a primary purpose of education is the sheer enjoyment of
undergoing it.
Those who most strongly defend education for self-realization usually
draw a distinction between learning a n d knowing. Learning enables
one to cope with sensory phenomena, while knowing transcends the
senses and has no purpose beyond itself. To use Plato's way of speak-
ing: learning can provide right (or wrong) opinion about the world
of appearances, while knowing provides certainty and wisdom pertain-
ing to the world of Forms. Knowing is considered infinitely more val-
uable than learning; knowing is both a cause and a result of intense
personal involvement and commitment; knowing is the highest kind of
aesthetic experience.
T h e personal involvement and commitment to be found in the act
of knowing have been explored at length by Michael Polanyi. He shows
that knowing requires the creative integration of whatever evidence or
propositions are available. Knowing always goes beyond the data. No
proof ever forces the acceptance of its conclusion; rather, a successful
proof expresses a truth in such a convincing way that whoever wrestles
with the proof comes to agree with its conclusion. Knowing is a per-
sonal commitment to t h a t which is known, as indicated by the tenacity
and fervor with which knowledge is held and proclaimed.1 T h e aes-
thetics of mystery are involved in a problematic situation; a drama un-
folds as evidence is organized and partially understood; tension resolu-
tion, emotional release, and psychological closure occur as knowledge is
finally discovered.
An excellent review of the history of the claim that knowing is an
aesthetic activity is given by Frederic Will in his book Intelligible Beauty
in Aesthetic Thought.2 Will notes that Plotinus supplied a monistic,
mystical completion to Plato's doctrine of ideas, and subsequent aes-
theticians have used this tradition as a basis for the notion of intelligible
beauty. According to Will, the notion of intelligible beauty is "the belief
that man's higher cognitive faculties are deeply and appropriately en-
gaged in aesthetic experience."3 Will shows how the notion of intelli-
gible beauty functions in the philosophies of a number of thinkers. He
1 Michael Polanyi, Personal Knowledge (London : Routledge an d Kega n
Paul, 1958).
2 Frederic Will, Intelligible Beauty in Aesthetic Thought (Tubingen : Ma x
Niemeyer Verlag, 1958).
'Ibid., p. 16.

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24 KENNETH R. CONKLIN
claims, " T h e most general agreement of Plotinus with Hegel, and with
the post-Kantians in general, is on the tenet that reality is essentially
thought, or intelligibility, and that the end toward which reality strives
is total intelligibility."4
Plato's sun, cave, and divided line allegories in the Republic provide
metaphysical explanations of what is being asserted in the claim that
knowing is essentially an aesthetic activity. It will be recalled that as
a potential philosopher-king acquires greater knowledge at higher levels
of reality, he approaches knowledge of the supreme Form of the Good.
When he finally does achieve this highest kind of knowledge, he under-
goes a mystical conversion experience which alters his personality char-
acteristics. There is thus profound personal involvement in the struggle
for wisdom. Furthermore, Goodness, T r u t h , and Beauty are regarded
as three aspects of the unified Form of the Good, so that knowing and
aesthetic experience are identical at the highest level.
In speaking about the rise of the soul into the world of the Absolute,
Plato says in the Phaedrus,
It is there that Reality lives, without shape or color, intangible, visible only
to reason, the soul's pilot; and all true knowledge is knowledge of her . . .
when the soul has at long last beheld Reality, it rejoices, finding sustenance
in its direct contemplation of the truth and in the immediate experience
of it 5
In the Symposium Plato is even more explicit:
This is the right way of approaching or being initiated into the mysteries
of love, to begin with examples of beauty in this world, and using them
as steps to ascend continually with that absolute beauty as one's aim, from
one instance of physical beauty to two and from two to all, then from physi-
cal beauty to moral beauty, and from moral beauty to the beauty of knowl-
edge, until from knowledge of various kinds one arrives at the supreme
knowledge whose sole object is that absolute beauty, and knows at last
what absolute beauty is."
T h e following points made by Plato deserve special emphasis here:
particular phenomenal instances of beauty are inferior to a n d derive
their beauty from more general, abstract kinds of beauty; both physical
and moral beauty are particular embodiments of the beauty of knowl-
edge; ultimate Reality (and hence ultimate beauty and ultimate knowl-
edge) is "without shape or color, intangible, visible only to reason."
4 Ibid., p. 205.
5 Plato, Phaedrus, trans. W. C. Helmbold and W. G. Rabinowitz (New York:
The Liberal Arts Press, 1956), p. 30.
"Plato, Symposium, trans. W. Hamilton (Baltimore: Penguin Books, 1956),
p. 94.

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ABSTRACT DISCIPLINES 25
O n e of the most controversial points here is the claim that pure
cognition, without any immediately antecedent sense perception, can be
an aesthetic experience. It is customary to speak of aesthetic experience
in connection with the perceptions of the five physical senses; yet, as
Hospers points out,7 aesthetic experience is actually concerned with
meanings, associations, and emotions, whether these come to us through
the senses or otherwise. This assertion is especially true of literature,
where the actual sound (if any) is not important. Of course it may be
claimed that reading produces mental images, so that some kind of
sensory-like basis exists for the aesthetic experience in literature. But
Hospers refutes this claim:
. . . many readers can read appreciatively and intelligently without having
any visual or other images evoked in their minds. . . . The inclusion of
literature in the category of the perceptual by means of some image evoca-
tion theory constitutes a desperate attempt to make the facts fit a theory.
However, the dismissal of literature as not being the object of aesthetic
attention because of its nonperceptual character would seem to be a prime
case of throwing the baby out with the bathwater.8
We shall see in the next section of this paper that abstract mathematics
has important aesthetic aspects, although it is completely nonperceptual.
As Hospers says,
When we enjoy or appreciate the elegance of a mathematical proof, it
would surely seem that our enjoyment is aesthetic, although the object of
that enjoyment is not perceptual at all: it is the complex relation among
abstract ideas or propositions, not the marks on paper or the blackboard,
that we are apprehending aesthetically. It would seem that the appreciation
of neatness, elegance, or economy of means is aesthetic whether it occurs
in a perceptual object (such as a sonata) or in an abstract entity (such as
a logical proof), and if this is so, the range of the aesthetic cannot be
limited to the perceptual.'
Indeed, according to Plato the best aesthetic experiences are the most
abstract and least perceptual.
Any attempt to provide a definitive characterization of what is meant
by "aesthetic" or "aesthetic experience" is beyond the scope of this
paper. O u r purpose in die present section has been to indicate impor-
tant similarities between activities generally acknowledged to be aesthetic
and the activities of teaching, learning, and knowing. T h e significance
for education of those similarities will be explored in section four. We
7 John Hospers, "Problems of Aesthetics" in The Encyclopedia of Philosophy,
ed. Paul Edwards (New York: Macmillan, 1967), Vol. I, pp. 38-39.
8 Ibid., p. 39.
8 Ibid., p. 38.

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26 KENNETH R. CONKLIN
have emphasized the aesthetics of abstract knowing in preparation for
the remainder of this paper. T h e author has elsewhere provided a more
extensive analysis of what is meant by "aesthetic" and how the aesthetic
aspects of teaching, learning, and knowing are interrelated.10
2. THE AESTHETICS OF ABSTRACT SUBJECT MATTER (ESPECIALLY MATHEMATICS)
Let us imagine that the subjects in a school curriculum have been
arranged according to the relative "aestheticness" of their subject matter
as normally conceived. Surely the arts would be close to one end of the
continuum, while the abstract, logical disciplines such as mathematics
and theoretical physics would be at the opposite end. Yet we shall see
that the arts have a mathematics-like aspect, and that the subject matter
of mathematics has an essential aesthetic aspect. T h e continuum sug-
gested above is therefore really a continuum rather than a multichotomy.
But what is most significant for our purposes here is that abstract subject
matter has an aesthetic aspect and that, therefore, teaching mathematics
and other abstract subjects for appreciation is as reasonable and, indeed,
as necessary as teaching art for appreciation.
In claiming that mathematics possesses an essential aesthetic aspect,
we must distinguish between mathematics as it is written and mathe-
matics as it is held in the mind. Mathematicians may have poor hand-
writing, small in size and hard to read. Mathematical symbols might be
considered ugly. Although it is true that configurations of symbols are
manipulated by the mathematician in the process of proving theorems,
and that seeing the configurations is usually helpful and sometime?
apparently necessary in making discoveries. Manipulating symbols on
paper strongly resembles moving furniture in a room or assembling a
jigsaw puzzle: we often must try out a configuration before deciding
whether it fits together and is pleasing. But the way die symbols appear
on paper is obviously not crucial to the mathematician, who can adopt
alternative systems of notation with no effect on his mathematical results.
What really matters is the fittingness and pleasingness of the configura-
tions of abstract concepts in the mind of the mathematician.
Mathematical discovery is a species of knowing, and as such has
already been discussed in section one. Plato, for example, regarded
mathematical objects as "shadows" of the Forms in his divided line
allegory. He recommended a ten-year program of abstract mathematics
in the curriculum of prospective philosopher-kings to accustom their
10 Kennet h Robert Conklin, " T h e Aesthetics of Knowing and Teaching, " The
(Teachers College) Record, forthcoming.

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ABSTRACT DISCIPLINES 27
minds to the abstract beauty of the Forms and to develop the power of
intuitive, aesthetic appreciation of nonsensuous entities in order to
prepare them for their "vision" of the Form of the Good.
T h e role of intuition in mathematics has been widely discussed. On
the one hand, proofs must be based on strictly logical reasoning and
must avoid overt dependence on intuitive or heuristic appeal. On the
other hand, mathematical discovery seems to draw heavily upon intu-
ition, and the most profound discoveries were often the products of
the most profound intuitions. While it is true that intuition sometimes
leads mathematicians to false conclusions, it is also true that mathema-
ticians accept numerous theorems as intuitively obvious even though
all efforts at proving them have failed.
K u r t Godel, who has been among the most successful mathematicians
in using rigorous techniques of logic, is also one of the strongest de-
fenders of the role of intuition. He argues that mathematical intuition
is very much like sense perception, and that the question of the objec-
tive existence of the objects of mathematical intuition "is an exact
replica of the question of the objective existence of the outer world."11
I don't see any reason why we should have less confidence in this kind of
perception, i.e., in mathematical intuition, than in sense perception, which
induces us to build up physical theories and to expect that future sense
perceptions will agree with them and, moreover, to believe that a question
not decidable now has meaning and may be decided in the future. The
set-theoretical paradoxes are hardly any more troublesome for mathema-
ticians than deceptions of the senses are for physics."
What, however, perhaps more than anything else, justifies the acceptance
of this criterion of truth in set theory [clarity of intuition] is the fact that
continued appeals to mathematical intuition are necessary not only for obtain-
ing unambiguous answers to the questions of transfinite set theory, but also
for the solution of the problems of finitary number theory (of the type of
Goldbach's conjecture), where the meaningfulness and unambiguity of the
concepts entering into them can hardly be doubted. This follows from the
fact that for every axiomatic system there are infinitely many undecidable
propositions of this type.1*
Intuition in cognitive discovery functions much like sensation in
artistic appreciation. Perhaps Plato (doctrine of reminiscence) and
Jung (racial memories) would argue that intuition in cognitive dis-
covery is more like memory than sensation, but they would also say that
"Kurt Godel, "What Is Cantor's Continuum Problem?" in Philosophy of
Mathematics, ed. Paul Benacerraf and Hilary Putnam (Englewood Cliffs, N.J.:
Prentice-Hall, 1964), p. 272.
"Ibid., p. 271.
13 Ibid., p. 272.

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28 KENNETH R. CONKtIN
the "aestheticness" of sense experience comes from memory as well.
In any case, the role of intuition in mathematical discovery is closely
similar to the role of aesthetic sensitivity in artistic creation or appreci-
ation. Henri Poincare uses vivid language to describe the drama,
beauty, and joy of mathematical discovery in his own experience,1*
and Jacques H a d a m a r d has made a major study of the psychology of
mathematical discovery.15 There can be no doubt that mathematical
discovery is an aesthetic experience of the most profound kind.
We distinguished earlier between mathematics as it is written and
mathematics as it is held in the mind, and we noted that only the latter
kind of mathematics has a genuine aesthetic aspect. W h a t should be
compared with the arrangement of pigments on canvas is not the ar-
rangement of symbols on paper but the arrangement of concepts in the
mind as suggested by the written symbols, a kind of perception not far
removed from the artistic appreciation of our perception of the colored
shapes rather than an appreciation of the colored shapes themselves.
Thus far we have viewed the aesthetics of mathematics from the per-
spective of someone who discovers important mathematical truths, and
the discussion to this point may seem relevant only to creative mathe-
maticians engaged in original research. But this is not the case at all.
T h e aesthetics of mathematical discovery is valid whether the discovery
adds something new to the stock of mathematical knowledge or whether
it is merely a rediscovery by a child of a truth commonly known to all
who are not mathematically illiterate. Each discovery is original for the
person who makes it.
But there is clearly a distinction between making a discovery inde-
pendently and following a given argument, just as there is a distinction
between creating a symphony and listening to one created by someone
else. Yet, some of the aesthetic experience of discovery is surely present
even for the person who only appreciates the work of another. Following
a single proof or studying a whole branch of mathematics provides an
aesthetic experience closely similar to that of reading a novel or seeing
a play: there is a plot with sometimes unexpected twists and turns,
there is a buildup of suspense, and in the end things either get resolved
or stimulate us to look for a sequel. Even though a mathematician
may have read a particular proof several times, if it is a significant or
14 Henri Poincare, "Mathematical Creation" in The World of Mathematics,
ed. James R. Newman (New York: Simon and Schuster, 1956), Vol. IV, pp.
2041-50.
15 Jacques Hadamard, The Psychology of Invention in the Mathematical Field
(New York: Dover Publications, 1954).

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ABSTRACT DISCIPLINES 29
beautiful proof he enjoys reading it again. Mathematicians enjoy cre-
ating or reading many different proofs for the same theorem, just as
music and drama lovers enjoy variations on a common theme. Just as
there are standards of aesthetic judgment for works of art, music, and
drama, so we should expect there to be standards of aesthetic judgment
for mathematical products — and indeed there are such standards.
Poincare talks in general terms about the feeling of mathematical
beauty, the harmony of numbers and forms, and elegance.
Now, what are the mathematic entities to which we attribute the char-
acter of beauty and elegance, and which are capable of developing in us
a sort of esthetic emotion? They are those whose elements are harmoniously
disposed so that the mind without effort can embrace their totality while
realizing the details. This harmony is at once a satisfaction of our esthetic
needs and an aid to the mind, sustaining and guiding. At the same time,
in putting under our eyes a well-ordered whole, it makes us foresee a mathe-
matical law. . . . The useful combinations are precisely the most beautiful,
I mean those best able to charm this special sensibility that all mathema-
ticians know, but of which the profane are so ignorant as often to be tempted
to smile at it."
G. H. Hardy, a professional mathematician, wrote a book celebrating
the aesthetic aspect of mathematics as a justification for devoting his
life to the subject. He provided a lengthy and precise description of
standards for the aesthetic criticism of mathematics. Hardy notes that
the mathematician is a maker of patterns with ideas. Creating or read-
ing mathematics has aesthetic qualities like those found in playing or
watching a chess game, except that mathematics is superior to chess in
many ways.17
According to Hardy, the beauty of a mathematical theorem depends
greatly on its seriousness, and the seriousness of a theorem is determined
according to the theoretical significance of the mathematical ideas which
the theorem connects.18 Significant mathematical ideas are those having
generality and depth.19 A theorem is general if it summarizes a host of
concrete facts or lower-level generalities,20 a n d it is deep if it is some-
how essential to a number of important truths.21 In addition to serious-
ness, a beautiful theorem or proof also has unexpectedness (either in its
16 Henri Poincare, op. cit., pp. 2047-48.
" G. H. Hardy, A Mathematician's Apology (Cambridge: University Press,
1967), pp. 84-85.
18 Ibid., pp. 89-98. Hardy gives two examples of theorems with proofs which
he considers beautiful.
"Ibid., p. 103.
10 Ibid., pp. 104-09.
21 Ibid., pp. 109-12.

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