By a mathematical approach to art it is hardly necessary to say I do
not mean any fanciful ideas for turning out art by some ingenious
system of ready reckoning with the aid of mathematical formulas. So far
as composition is concerned every former school of art can be said to
have had a more or less mathematical basis. There are also many trends
in modern art which rely on the same sort of empirical calculations.
These, together with the artist individual scale of value, are just
part of the ordinary elementary principles of design for establishing
the proper relationship between component volumes; that is to say for
imparting harmony to the whole. Yet it cannot be denied that the same
methods have suffered considerable deterioration since the time when
mathematics was the fundation of all forms of artistic expression and
the covert link between cult and cosmos. Nor have they seen any
progressive development from the days of the ancient Egyptians until
quite recently, if we except the discovery of perspective during the
Renaissance. This is a system which, by means of pure calculation and
artificial reconstruction, enables objects to be reproduced in what is
called "true-to-life" facsimile by setting them in an
illusory field of space. Perspective certainly represented an entirely
new aspect of reality to human consciousness, but one of its
consequences was that the artist's primal image was debased into mere
replica of his subject. There with the decadence of painting, both as a
symbolic art and an art of free construction, may be said to have begun.
Impressionism, and still more Cubism, brought painting and sculpture
much closer to what were the original elements of each: painting as
surface design in colours; sculpture as the shaping of bodies to be
informed by space. It was probably Kandinsky who gave the immediate
impulse towards an entirely fresh conception of art. As early as 1912,
in his book on The Spiritual Harmony in Art, Kandinsky had
indicated the possibility of a new direction which, if followed to its
logical conclusion, would lead to the substitution of a mathematical
approach for improvisations of the artist's imagination. But as he
found other ways of liberating painting from romantic and literary
associations he did not adopt this particular line in his own work.
If we examine a picture of Klee or one of Brancusi's sculptures we
shall soon discover that, though the "subject" may be an
indeterminable echo of something or other in the actual world about us,
it is an echo which has been transmuted in a form that is original in
the sense of being elemental. Kandinsky confronted us with objects and
phenomena which have no existence in ordinary life, but which might
well have meaning or be portents on some unknown planet; a planet where
we should be quite unable to gauge their purpose or relevance. Yet it
was undoubtedly Mondrian who went furthest in breaking away from
everything that had hitherto been regarded as art. If the technique of
structural design may seem to have inspired his rhythms the resembance
is fortuitous and one which was not present in his own intention or
consciousness, Although the specific content of his work is constricted
with the utmost discipline, the horizontal-vertical emphasis represents
a purely emotional factor in his composition. It is not for any
whimsical reasons that he called his latest pictures "Broadway
Boogie-Woogies" and "Victory Boogie-Woogies", but simply
to stress their affinity with jazz rhythms,
If we can agree that Mondrian realized the ultimate possibilities of
painting in one direction - that is by his success in eliminating most
of the remaining elements which are alien to it - two others still lie
open to us: either we can return to traditionalism (in its wider
sense), or else we can continue the quest for subjects with a content
of a new and altogether different nature.
Let me take this opportunity to explain why it is impossible for many
artists to go back to the old type of subjects. In the vast field of
pictorial and plastic expression there are a large number of trends and
tendencies which have more or less originated in our own age. Different
people look at modern painting and sculpture with different eyes
because what they severally recognize as significant of our age is
necessarily various. Clergymen have a different idea of art from
scientists. Peasants and factory-hands live under radically different
conditions. There are inevitable variations in standards of living and
levels of culture. Similar differences can be found among artists.
They, too, come from different walks of life, and their work reflects
different emotional and intellectual undercurrents. There is another
attitude to modern art which must not be overlooked as its now numerous
followers can always be relied upon to take their stand against every
disinterestedly progressive movement. I mean the much-boosted school
which demands that, if art itself cannot perhaps solve social and
political problems, these shall at least be made dramatically
"actual" and suitably glorified through its medium. We have
good reasons to be skeptical about any "Political Art" -
regardless of whether it emanates from right or left; especially when,
under the cloak of antagonism to the prevailing social order, its aim
is to bring about a new, but in all essentials, almost identical
structure of society - because this is not art at all but simply
propaganda.
After this digression into the potential alternatives which may be said
to have existed prior to somewhere about 1910, let me try to make clear
why some of us were unable to rest content with what had then been
achieved. That would have meant going on marking time over the same
ground during the last forty years and painting in one or the other of
those manners which may be called "a la Klee", "a la
Kandinsky", "a la Mondrian", or, what was more usual
"a la Picasso", "a la Braque", and "a la
Matisse". A great many.jpgted and intelligent artists are still
wearing out their talents in ringing the changes on these modern
masters. In fact this sort of painting has now become something in the
nature of a substitute for the masters themselves, and "a la"
pictures begin to rank as interesting variants of their originals. To
acquiesce in this state of stagnation was impossible because we have no
right to allow a halt to be called in any genuinely creative field of
human activity.
We can safely assume that all the various forms of expression open to
painting and sculpture at the present day are now sufficiently known,
and that the techniques they postulate have been sufficiently
demonstrated and clarified in the work of their respective pioneers
(except perhaps for a very few which can be already anticipated, but
which have not so far been realized). What, then, it may be asked, are
the possibilities of future development? But there are two other
important points which must be dealt with before that question can be
answered: namely, whether the several idioms just referred to can claim
general validity in the plastic arts; and whether there is reason to
enlarge the existing limits of their content. Careful study of those
forms of expression has led me to the conclusion that all of them were
the discoveries of individual artists, either born of their will to
overcome particular prolems or else expedients called forth by
exceptional circumstances; and that therefore they cannot be considered
universally applicable or appropriate. As regards content, most of the
modern work which is often held to have been largely inspired by
mathematical principles cannot, in point of fact, be identified with
that entirely new orientation I have called the Mathematical Approach
to Art. And as this needs to be more nearly defined, I will now
endeavour to elucidate it, and at the same time answer the question I
have left in suspense.
I am convinced it is possible to evolve a new form of art in which the
artist's work could be founded to quite a substantial degree on a
mathematical line of approach to its content. This proposal has, of
course, aroused the most vehement opposition. It is objected that art
has nothing to do with mathematics; that mathematics, besides being by
its very nature as dry as dust and as unemotional, is a branch of
speculative thought and as such in direct antithesis to those emotive
values inherent in aesthetics; and finally that anything approaching
ratiocination is repugnant, indeed positively injurious to art, which
is purely a matter of feeling. Yet art plainly calls for both feeling
and reasoning. In support of this assertion the example of Johann
Sebastian Bach may be credited; for Bach employed mathematical formulas
to fashion the raw material known to us as sound into the exquisite
harmonies of his sublime fugues. And it is worth mentioning that,
although mathematics had by then fallen into disuse for composition in
both his own and the other arts, mathematical and theological books
stood side by side on the shelves of his library.
It is mankind's ability to reason which makes it possible to coordinate
emotional values in such a way that what we call art ensues. Now in
every picture the basis of its composition is geometry or in other
words the means of determining the mutual relationship of its component
parts either on plane or in space. Thus, just as mathematics provides
us with a primary method of cognition, and can therefore enable us to
apprehend our physical surroundings, so, too, some of its basic
elements will furnish us with laws to appraise the interactions of
separate objects, or groups of objects, one to another. And again,
since it is mathematics which lends significance to these
relationships, it is only a natural step from having perceived them to
desiring to portray them. This, in brief, is the genesis of a picture.
Pictorial representations of that kind have been known since antiquity,
and, like those models at the Musee Poincare in Paris where conceptions
of space have been embodied in plastic shapes or made manifest by
colored diagrams, they undoubtedly, provoke an aesthetic reaction in
the beholder. In the search for new formal idioms expressive of the
technical sensibilities of our age these borderline examplars had much
the same order of importance as the "discovery" of native
West African sculptures by the Cubists; though they were equally inapt
for direct assimilation into modern European art. The first result of
their influence was the phase known as Constructivism. This, together
with the use of new materials such as engineering blueprints, aerial
photographs, and the like, furnished the necessary incentive for
further developments along mathematical lines. At about the same time
mathematics itself had arrived at a stage of evolution on which the
proof of many apparently logical deductions ceased to be demonstrable
and theorems were presented that the imagination proved incapable of
grasping. Though mankind's power of reasoning had not reached the end
of its tether, it was clearly beginning to require the assistance of
some visualizing agency. Aids of this kind can often be provided by the
intervention of art.
As the artist has to forge his concept into unity his vision vouchsafes
him a synthesis of what he sees which, though essential to his art, may
not be necessarily mathematically accurate. This leads to the shifting
or blurring of boundaries where clear lines of division would be
supposed. Hence abstract conceptions assume concrete and visible shape,
and so become perceptible to our emotions. Unknown fields od space,
almost unimaginable hypotheses, are boldly bodied forth. We seem to be
wandering through a firmament that has had no prior existence; and in
the process of attuning ourself to its strangeness our sensibility is
being actively prepared to anticipate still further and, as it were, as
yet inconceivable expanses of the infinite.
It must not be supposed that an art based upon the principles of
mathematics, such as I have just adumbrated, is in any sense the same
thing as a plastic or pictorial interpretation of the latter. Indeed,
it employs virtually none of the resources implicit in the term
"Pure Mathematics". The art in question can, perhaps, best be
defined as the building up of significant patterns from the
everchanging relations, rhythms and proportions of abstract forms, each
one of which, having its own causalty, is tantamount to a law in
itself. As such, it presents some analogy to mathematics itself where
every fresh advance had its immaculate conception in the brain of one
or other of the great pioneers, This Euclidian geometry no longer
posseses more than a limited validity in modern science, and it has an
equally restricted utility in modern art. The concept of a Finite
Infinity offers yet another parallel. For this essential guide to the
speculations of contemporary physicists has likewise become an
essential factor in the thinking of contemporary artists. These, then,
are the general lines on which art is daily creating new symbols:
symbols that may have their source in antiquity but which meet the
aesthetic-emotional needs of our time in a way hardly any other form of
expression can hope to realize.
Things having no apparent connection with mankind's daily needs - the
mystery enveloping all mathematical problems; the inexplicability of
space - space that can stagger us by beginning on one side and ending
in a completely changed aspect on the other, which somehow manages to
remain that selfsame side; the remoteness or nearness of infinity -
infinity which may be found doubling back from the far horizon to
present itself to us as immediately as at hand; limitations without
boundaries; disjunctive and disparate multiplicities constituting
coherent and unified entities; identical shapes rendered wholly diverse
by the merest inflection; fields of attraction that fluctuate in
strength; or again, the square in all its robust solidity; parallels
that intersect; straight lines untroubled by any relativity and
ellipses which form straight lines at every point of their curves - can
yet be fraught with the greates moments. For though these evocations
might seem only the phantasmagorial figments of the artist's inward
vision they are, notwithstanding, the projections of latent forces;
forces that may be active or inert, in part revealed, inchoate or still
unfathomed, which we are unconsciously at grips with every day of our
lives; in fact that music of the spheres which underlies each man-made
system and every law of nature it is our power to discern.
Hence all such visionary elements help to furnish art with a fresh
content. Far from creating a new formalism, as is often erroneously
asserted, what these can yield us is something far transcending face
values since they not only embody form as beauty, but also form in
which intuitions or ideas have taken visible substance. The primordial
forces contained in those elements call for intimations of the occult
controls which govern the cosmic structure; and these can be made to
reflect a semblance of the universe as we have learned to picture it
today: an image that is no mere transcript of this invisible world but
a systemization of it ideographically conveyed to our senses.
It may, perhaps, be contented that the result of this would be to
reduce art to a branch of metaphysical philosophy. But I see no
likelihood of that for philosophy is speculative thought of a special
kind which can only be made intelligible through the use of words.
Mental concepts are not as yet directly communicable to our
apprehension without the medium of language; though they might
ultimately become so by the medium of art. Hence I assume that art
could be made a unique vehicle for the direct transmission of ideas,
because if these were expressed by pictures or plastically there would
be no danger of their original meaning being perverted (as happens in
literature, for instance, through printer's errors, or thank to the
whim of some prominent executant in music) by whatever fallacious
interpretations particular individuals chance to put on them. Thus the
more succinctly a train of thought was expounded and the more
comprehensive the unity of its basic idea, the closer it would
approximate to the prerequisites of the Mathematical Approach to Art.
So the nearer we can attain to the first cause or primal core of things
by these means, the more universal will the scope of art become - more
universal, that is, by being free to express itself directly and
without ambivalence; and likewise forthright and immediate in its
impact on our sensibility.
To which, no doubt, a further objection will be raised that this is no
longer art; though it could equally well be maintained that this alone
was art. Such a structure would be like saying that Euclid's was the
only geometry, and that the new conception of geometry associated with
the names of Lobachevsky and Riemann was not geometry at all. One claim
would stand against the other and that would be that!
Although this new ideology of art is focused on a spectral field of
vision this is one where the mind can still find access. It is a field
in which some degree of stability may be found, but in which, too,
unknown quantities, undefinable factors will inevitably be encountered.
In the ever-shifting frontier zones of this nebular realm new
perspectives are continually opening up to invite the artist's creative
analysis. The difference between the traditional conception of art and
that just defined is much the same as exists between the laws of
Archimedes and those we owe Einstein and other outstanding modern
physicists. Archimedes remains our authority in a good many
contingencies though no longer in all of them. Phidias, Raphael, and
Seurat produced works of art that characterize their several epochs for
us because each made full use of the means of expression as his own age
afforded him. But since their days the orbit of human vision has
widened and art has annexed fresh territories which were formerly
denied to it. In one of these recently conquered domains the artist is
now free to exploit the untapped resources of that vast new field of
inspiration I have described with the means our age vouchsafes him and
in a spirit proper to its genius. And despite the fact the basis of
this Mathematical Approach to Art is in reason, its dynamic content is
able to launch us on astral flights which soar into unknown and still
uncharted regions of the imagination.
This essay was first printed by the review Werk (Nr.3, 1949, Winthertur). It appeared afterward in another version in the catalogue of the exhibition Pevsner, Vantongerloo, Bill in the Zurich Kunsthaus in 1949 and finally in a monograph, Max Bill, published in Spanish, French, English and German, edited by Thomas Maldonado (Buenos Aires, 1955). This essay has been reprinted in several countries and languages. This translation by Morton Shand first appeared in Arts and Architecture (Los Angles), No,8, 1954. It was reprinted in Max Bill (The Buffalo Fine Arts Academy & The Albright-Knox Art Gallery, 1974).
Article found at http://hebert.kitp.ucsb.edu/studio/a-m/mb-maica.html