Cathedral Builders:
The Sublime in Mathematics
Vladislav Shaposhnikov
Faculty of Philosophy
Lomonosov Moscow State University
AGENDA
1. Mathematics and religion through the ages.
Secularization and quasi-religious phenomena.
2. The ‘cathedral builders’ parable. A need for
self-transcendence and doing mathematics.
3. The numinous and the sublime.
4. Are there such things as the mathematical
numinous and the mathematical sublime?
5. The mathematical sublime and its vehicles.
2
MATHEMATICS & RELIGION
THROUGHOUT THE AGES
(1) Abraham Seidenberg on the ritual origins of mathematics in the pre-Greek period.
(2) Mathematics and theology in Platonism and Neo-Platonism: mathematics as initiation
to the mysteries.
(3) Mathematics and theology in the medieval Christianity: God as mathematician.
(4) Mathematics and theology in the Scientific Revolution: TCA-triangle.
(5) Mathematics as a chief successful rival of theology and metaphysics in the 19th-20th
centuries.
I believe that in my generation, the belief in a platonic
mathematics has often been a substitute religion for
people who have abandoned or even rejected traditional
religions. Where can certainty be found in a chaotic
universe that often seems meaningless? Mathematics has
often been claimed to be the sole source of absolute
certainty.
Philip J. Davis (2004, p. 35).
3
THE HYPOTHESIS OF THE RITUAL ORIGIN OF MATHEMATICS
Abraham Seidenberg (1916-1988) – a mathematician and historian of
mathematics at the University of California, Berkeley, who proposed
two interconnected hypotheses (late 1950s – 1980s):
1) the ritual origin of mathematics;
2) a common origin of mathematics of Ancient Civilizations (which
dates back to the Neolithic period, ca. 3,000 - 2,000 BC)
A falcon-shaped altar
Until quite recently, we all thought that the history of mathematics
begins with Babylonian and Egyptian arithmetic, algebra, and
geometry. However, three recent discoveries have changed the picture
entirely. The first of these discoveries was made by A Seidenberg. He
studied the altar constructions in the Indian Śulvasūtras and found that
in these relatively ancient texts the "Theorem of Pythagoras" was used
to construct a square equal in area to a given rectangle, and that this
construction is just that of Euclid. From this and other facts he
concluded that Babylonian algebra and geometry and Greek
"geometrical algebra" and Hindu geometry are all derived from a
common origin, in which altar constructions and the "Theorem of
Pythagoras" played a central rôle. (van der Waerden, 1983, p. XI)
Seidenberg, A. (1961). The Ritual Origin of Geometry, Archive for History of Exact Sciences, 1, 488-527.
Seidenberg, A. (1962). The Ritual Origin of Counting, Archive for History of Exact Sciences, 2, 488-527.
Seidenberg, A. & Casey, J. (1980). The Ritual Origin of the Balance, Archive for History of Exact Sciences, 23, 179-226.
van der Waerden, B.L. (1983). Geometry and Algebra in Ancient Civilizations. New York: Springer-Verlag.
4
MATHEMATICS & INITIATION TO THE MYSTERIES
We can again compare philosophy to the initiation into things truly holy, and to the revelation of the authentic
mysteries [Phaedo 69d]. There are five parts in initiation: the first is the preliminary purification, because
participation in the mysteries must not be indiscriminately given to all those who desire it, but there are some
aspirants whom the harbinger of the path separates out, such as those of impure hands, or whose speech lacks
prudence; but even those who are not rejected must be subjected to certain purifications. After this purification
comes the tradition of sacred things (which is initiation proper). In the third place comes the ceremony which is
called the full vision (the highest degree of the initiation). The fourth stage, which is the end and the goal of the
full vision, is the binding of the head and the placement of the crowns, in order that he who has received the
sacred things, becomes capable in his turn of transmitting the tradition to others, either through the dadouchos
(the torch bearing ceremonies), or through hierophantism (interpretation of sacred things), or by some other
priestly work. Finally the fifth stage, which is the crowning of all that has preceded it, is to be a friend of the Deity,
and to enjoy the felicity which consists of living in a familiar commerce with him. It is in absolutely the same
manner that the tradition of Platonic reason follows. Indeed one begins from childhood with a certain consistent
purification in the study of appropriate mathematical theories. According to Empedocles, “it is necessary that he
who wishes to submerge himself in the pure wave of the five fountains begins by purifying himself of his
defilements.” And Plato also said one must seek purification in the five mathematical sciences, which are
arithmetic, geometry, stereometry, music and astronomy. The tradition of philosophical, logical, political and
natural principles corresponds to initiation. He calls full vision [Phaedrus 250c] the occupation of the spirit with
intelligible things, with true existence and with ideas. Finally, he says that the binding and the crowning of the
head must be understood as the faculty which is given to the adept by those who have taught him, to lead others
to the same contemplation. The fifth stage is that consummate felicity which they begin to enjoy, and which,
according to Plato, “ identifies them with the Deity, in so far as that is possible.”
(Theon of Smyrna, 1979, Mathematics Useful for Understanding Plato, San Diego: Wizards Bookshelf, pp. 8-9)
5
GOD AS ARCHITECT & GEOMETER
IN MEDIEVAL MINIATURES
God the Architect and the Geometer (13th century)
For extended discussion see:
Friedman, J.B. (1974). The Architect’s Compass
in Creation Miniatures of the Later Middle Ages,
Traditio, 30, 419-429.
Holkham Bible, 14th century
British Museum, Add. 47682, fol. 2
6
DIVINE MATHEMATICS
& HUMAN MATHEMATICS
Geometry, which before the origin of things was coeternal with the divine
mind and is God himself (for what could there be in God which would not
be God himself?), supplied God with patterns for the creation of the
world, and passed over to Man along with the image of God; […]
Johannes Kepler, Harmonices Mundi (The Harmony of the World), 1619
Tr. by E.J. Aiton, A.M. Duncan and J.V. Field
The mathematical truths which you call eternal have been laid down by
God and depend on him entirely no less than the rest of his creatures.
[…] Please do not hesitate to assert and proclaim everywhere that it is
God who has laid down these laws in nature just as a king lays down
laws in his kingdom. There is no single one that we cannot understand if
our mind turns to consider it. They are all inborn in our minds just as a
king would imprint his laws on the hearts of all his subjects if he had
enough power to do so.
René Descartes in his letter to Marin Mersenne, April 15, 1630
Tr. by A. Kenny
7
THEO-COSMO-ANTHROPOLOGICAL TRIANGLE
(TCA-triangle)
THEOS
ANTHROPOS
COSMOS
8
Mathematics, rightly viewed, possesses not only truth, but
supreme beauty – a beauty cold and austere, like that of
sculpture, without appeal to any part of our weaker nature,
without the gorgeous trappings of painting or music, yet
sublimely pure, and capable of a stern perfection such as only
the greatest art can show. The true spirit of delight, the
exaltation, the sense of being more than man, which is the
touchstone of the highest excellence, is to be found in
mathematics as surely as in poetry.
Bertrand Russell, The Study of Mathematics, 1902
He had no thought of beauties, but had already run
beyond beauty […], like a man who enters into the
sanctuary and leaves behind the statues in the outer
shrine; these become again the first things he looks at
when he comes out of the sanctuary, after his
contemplation within and intercourse there, not with a
statue or image but with the Divine itself; they are
secondary objects of contemplation.
Plotinus, Ennead VI.9.11
9
THE ‘CATHEDRAL BULDERS’ PARABLE
One may recall an old parable, the parable of
the three stonecutters working at the
construction of a cathedral (e.g. Drucker,
1993/1954, p. 122). In my country the story is
usually associated with the builders of
Chartres Cathedral in the 13th century.
Their external goal was exactly the same: to
cut stones giving them the shape required; but
being asked what they were doing they gave
different answers. The first one said: “I am
making a living”. The second one: “I am doing
the best job of stonecutting in the entire
county”. The third one: “I am building a
cathedral!”
Chartres Trades and Crafts in Stained Glass:
Stonecutters
(Retrieved from
http://snapageno.free.fr/Churches/Chartres/Trade
sCrafts/indexByTrade.htm)
Drucker, P.F. (1993/1954). The Practice of Management, New York, NY: HarperCollins.
10
MASLOW’S HIERARCHY OF NEEDS
1. PHYSIOLOGICAL needs: air, food, drink, warmth, shelter, sleep,
sex, etc.
2. SAFETY needs: security, order, law, limits, stability, etc.
3. BELONGINGNESS & LOVE needs: to affiliate with others, to be
accepted and belong, intimate relationships, friends, etc.
4. ESTEEM needs: to achieve, be competent and responsible, gain approval and recognition,
etc.
5. COGNITIVE needs: to know, understand, explore, be self-aware, etc.
6. AESTHETIC needs: form, symmetry, order, balance, beauty, etc.
7. SELF-ACTUALIZATION needs: to find fulfillment, realize one’s potential, creative activities,
etc.
8. SELF-TRANSCENDENCE needs: service to others; devotion to an ideal or a cause, including
the pursuit of science and a religious faith; a desire to be united with what is perceived as
transcendent or divine; a communion beyond the boundaries of the self through peak
experience, may involve mystical experiences, etc.
This rectified version of Abraham Maslow’s hierarchy is based on Atkinson (1993) and Koltko-Rivera (2006).
AIMS & GOALS of PURE MATH
NEEDS → MOTIVATION → AIMS & GOALS
EXTERNAL vs. INTERNAL GOALS
EPISTEMIC vs. NON-EPISTEMIC or PRACTICAL GOALS
1. EXTERNAL EPISTEMIC GOALS:
- problem setting and problem-solving;
- development of a new technique and/or
notation;
- classification;
- generalization;
- proving;
etc.
2. INTERNAL EPISTEMIC GOALS:
- systematization;
- unification;
- simplification;
- explanation;
- justification;
etc.
3. INTERNAL NON-EPISTEMIC AIMS & GOALS:
- esteem;
- cognitive;
- aesthetic;
- self-actualization;
- self-transcendence.
What are the ultimate non-epistemic aims and goals of the mathematical activity?
12
A rose window in Chartres cathedral
Σχᾶμα καὶ βᾶμα, ἀλλʾ οὐ σχᾶμα καὶ τριώβολον.
A figure and a stepping-stone, not a figure and three obols.
Proclus, In Euclidem, 84.17 (Proclus, 1970/1992, p. 69)
13
THE NUMINOUS & THE SUBLIME
RUDOLF OTTO, 1917, Das Heilige (The Idea of the Holly)
1. R. Otto argued that religious experience had a non-reducible
core for which he coined the term ‘numinous’.
2. He distinguished numinous from aesthetic categories but
closely associated the former with one of the latter – the
sublime: “‘the sublime’ […] is an authentic ‘scheme’ of ‘the
holy’” (Otto, 1936/1923, p. 47).
In the arts nearly everywhere the most effective means of
representing the numinous is ‘the sublime’. This is especially true
of architecture, in which it would appear to have first been
realized. One can hardly escape the idea that this feeling for
expression must have begun to awaken far back in the remote
Megalithic Age (Otto, 1936/1923, p. 68)
Oxford Dictionary of English:
‘NUMINOUS’ means “having a strong religious or spiritual quality;
indicating or suggesting the presence of a divinity” (Stevenson,
2010, p. 1219).
14
WHAT IS THE SUBLIME?
“A quality of awesome grandeur in art or nature, which
some 18th-century writers distinguished from the merely
beautiful” (Baldick, C. 2008, The Oxford Dictionary of
Literary Terms, 3rd ed., Oxford UP, p. 321).
“An idea associated with religious awe, vastness, natural
magnificence, and strong emotion which fascinated
18th‐century literary critics and aestheticians. Its
development marks the movement away from the clarity
of neo-classicism towards Romanticism, with its emphasis
on feeling and imagination; […]” (Drabble, M., Stringer, J.,
& Hahn, D. (eds.), 2007, The Concise Oxford Companion to
English Literature, 3rd ed., Oxford UP).
Caspar David Friedrich, Wanderer above the Sea of
Fog, 1817, Kunsthalle, Hamburg
The sublime is one of the central aesthetic categories that refers to “an experience, that
of transcendence, which has its origins in religious belief and practice”; even nowadays it
remains “an experience with mystical-religious resonances” (Doran, R., 2015. The Theory
of the Sublime from Longinus to Kant. Cambridge, UK: Cambridge University Press, p. 1).
15
MATHEMATICAL NUMINOUS?
The adjective ‘numinous’ is sometimes applied to mathematics in the context
of Pythagorean and Platonic tradition.
Describing Plato’s view Richard Tarnas says that mathematical objects
are numinous and transcendent entities, existing independently of both the
phenomena they order and the human mind that perceives them.
(Tarnas, R., 1991. The Passion of the Western Mind: Understanding the Ideas that Have
Shaped Our World View. New York, NY: Harmony Books, p. 11)
Marsha Keith Schuchard is speaking of “numinous mathematics” in the context of
medieval Jewish tradition that was inherited in the European Middle ages:
It is perhaps one of the strangest ironies of history that this originally Jewish
yearning for transmundane and numinous mathematics would find its
greatest architectural expression in the towering Gothic cathedrals built by
Christian stonemasons.
(Schuchard, M.K., 2002. Restoring the Temple of Vision: Cabalistic Freemasonry and
Stuart Culture. Leiden: Brill, p. 24)
16
MATHEMATICAL NUMINOUS?
One can find a striking appeal for the
recognition of mathematical numinous in
Novalis’s “Mathematische Fragmente”
(1799/1800). Wilhelm Dilthey named
them “die Hymnen auf die Mathematik” .
Shall we take Novalis’s hymns to mathematics seriously? Is there some sense
in his project aimed at “a fusion of mathematics and religion” (Dyck, 1960, pp.
80-81) or rather recognition of their initial intimate connection?
Novalis’s romantic enthusiasm about mathematics has a serious historical
background. The idea to view mathematics as “numinous” (that is
“transmundane” and “transcendent”) is closely associated with the idea of
God as mathematician.
17
Das höchste Leben ist Mathematik. The highest life is mathematics. Es kann Mathematiker der ersten Größe geben, die There can be supremely ranked mathematicians
who cannot calculate. nicht rechnen können. One could be a great calculator without having an
Man kann ein großer Rechner sein, ohne die
inkling of mathematics. Mathematik zu ahnden. The true mathematician is an enthusiast per se.
Der echte Mathematiker ist Enthusiast per se.
Ohne Enthusiasmus keine Mathematik. Without enthusiasm there is no mathematics. Das Leben der Götter ist Mathematik. The life of the Gods is mathematics. Alle göttlichen Gesandten müssen Mathematiker All divine messengers must be mathematicians. Pure mathematics is religion. sein. Reine Mathematik ist Religion. One only advances to mathematics through a
Zur Mathematik gelangt man nur durch eine
theophany. Mathematicians alone are fortunate. The
Theophanie. Die Mathematiker sind die einzig Glücklichen. Der mathematician knows all. He could know it, even if
Mathematiker weiß alles. Er könnte es, wenn er es he did not already.
All activity ceases when knowledge enters. The
nicht wusste.
Alle Tätigkeit hört aus, wenn das Wissen eintritt. state of knowledge is eudemony, the blessed
peace of contemplation – heavenly quietism. Der Zustand des Wissens ist Eudämonie, selige
Ruhe der Beschauung, himmlischer Quietismus. - True mathematics is at home in the orient. In
Europe it has degenerated into a purely technical
Im Morgenlande ist die echte Mathematik zu
science. Hause. In Europa ist sie zur bloßen Technik
Whoever does not take hold of a mathematical
ausgeartet. Wer ein mathematisches Buch nicht mit Andacht book with devotion, and read it as the word of
ergreift, und es wie Gottes Wort liest, der versteht God, fails to understand it. –
(Translated by David W. Wood)
es nicht. –
(Novalis, 1837, pp. 147-148)
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The MATHEMATICAL NUMINOUS
refers to an experience which we
interpret as a meeting with the Divine
through doing mathematics. It meets
our need for self-transcendence.
The MATHEMATICAL SUBLIME refers to
the same feelings but transferred from
the religious to the aesthetic sphere in
their interpretation.
19
VEHICLES FOR THE MATHEMATICAL SUBLIME
1.Mathematical infinity or high complexity.
2.Mathematical perfection or optimality.
3.Mathematical certainty.
20
THE SUBLIME & THE INFINITE
Immanuel Kant in his Kritik der Urteilskraft (1790)
introduced the concept of “the mathematically sublime
(das mathematish-Erhabene)” in the sense of “a
mathematical disposition of the imagination (eine
mathematische Stimmung der Einbildungskraft)”. Kant
decisively connected the sublime with the infinite.
I am profoundly grateful that understanding infinity does not deprive it of its majesty. If
the infinite were only interesting because of the paradoxes it generates, and the
absorbing academic issues raised by the need to resolve them, then it would not be
studied any more than self-reference, a prolific but more pedestrian engine of paradox.
But the infinite is also majestic, one might say infinitely majestic. An hour under a clear
sky at night, looking up, gives some sense of this. The depth of space is a wild blue
yonder, not a true, perceived infinity. But it inspires contemplation of the true infinite,
and the slightest brush with that idea is breath-taking, invigorating, expanding, lifting,
calming, but also agitating, alluring, but also distant and magnificently indifferent. One
reason to study mathematics is that you can get these feelings in broad daylight or
indoors. There are many ways to become precise about these feelings, and many ways to
praise and honor the infinite. I'd like to use Kant's term: it is sublime. (Peter Suber, 1998,
Infinite Reflections, St. John's Review, XLIV(2), 1-59)
21
Nature is […] sublime in those of its appearances the intuition of which brings with them
the idea of its infinity. Now the latter cannot happen except through the inadequacy of
even the greatest effort of our imagination in the estimation of the magnitude of an object.
Now, however, the imagination is adequate for the mathematical estimation of every
object, that is, for giving an adequate measure for it, because the numerical concepts of the
understanding, by means of progression, can make any measure adequate for any given
magnitude. Thus it must be the aesthetic estimation of magnitude in which is felt the effort
at comprehension which exceeds the capacity of the imagination to comprehend the
progressive apprehension in one whole of intuition, and in which is at the same time
perceived the inadequacy of this faculty, which is unbounded in its progression, for grasping
a basic measure that is suitable for the estimation of magnitude with the least effort of the
understanding and for using it for the estimation of magnitude. Now the proper unalterable
basic measure of nature is its absolute whole, which, in the case of nature as appearance, is
infinity comprehended. But since this basic measure is a self-contradictory concept (on
account of the impossibility of the absolute totality of an endless progression), that
magnitude of a natural object on which the imagination fruitlessly expends its entire
capacity for comprehension must lead the concept of nature to a supersensible substratum
(which grounds both it and at the same time our faculty for thinking), which is great beyond
any standard of sense and hence allows not so much the object as rather the disposition of
the mind in estimating it to be judged sublime.
Critique of the Power of Judgment, § 26 (Kant, 2000, pp. 138-139)
22
John Horgan gives the following as a
formulation of mysticism: “awestruck
perception of the infinite in the finite”
(2003, p. 215).
This very idea was familiar to European
Romantic Movement. William Blake
expressed it in famous verses: “To see a
World in a Grain of Sand, / And a
Heaven in a Wild Flower, / Hold
Infinity in the palm of your hand, /
And Eternity in an hour” (Auguries of
Innocence, the Pickering Manuscript,
c.1801-1803, 1947/1905, p. 288).
THE INFINITE IN THE FINITE
A hyperbolic tessellation in Poincaré’s disc model
using regular heptagons
According to F.W.J. Schelling, sublimity is constituted by “the informing of the infinite
into the finite”. Then he continues: “wherever we encounter the infinite being taken up
into the finite as such — whenever we distinguish the infinite within the finite — we judge
that the object in which this takes place is sublime” (The Philosophy of Art, § 65,
1859/1989, pp. 85-86). In this case, the finite turns into “a symbol of the infinite”
(Schelling, 1859/1989, pp. 62-69, 79, 87-90).
23
THE STARS OF THE HEAVEN & THE SAND OF THE SEA
God promised to Abraham: “I will greatly multiply your descendants so that they will be as
countless as the stars in the sky or the grains of sand on the seashore” (Genesis 22:17; cf.
Genesis 32:12; Hosea 1:10; Jeremiah 33:22).
To be able to handle the infinite (or something very big) well means to obtain
divine powers.
This idea was made perfectly explicit in the apocryphal Greek Apocalypse of Ezra:
And God said: Number the stars and the sand of the sea; and if thou shalt be able to
number this, thou art also able to plead with me. And the prophet said: Lord, Thou
knowest that I wear human flesh; and how can I count the stars of the heaven, and the
sand of the sea? (Roberts, Donaldson, & Coxe, 1886, p. 572)
24
THE STARS OF THE HEAVEN & THE SAND OF THE SEA
A mathematician pretends to fulfill the job rejected
by Ezra as Archimedes famously shown in his
Psammites (The Sand Reckoner). He obtains a
divine power over very big numbers (about 1063
grains of sand in the case of Archimedes) and even
infinity.
According to Scott J. Aaronson, a theoretical
computer scientist at MIT, “one could define science
as reason’s attempt to compensate for our inability
to perceive big numbers”.
See Aaronson, S.J. (1999). Who Can Name the
Bigger Number?
(Retrieved from
http://www.scottaaronson.com/writings/bignumbers.pdf)
25
MODERN, POSTMODERN & THE SUBLIME
Jean-François Lyotard in his Answering the Question: What is Postmodernism?
(1982) wrote:
Modernity, in whatever age it appears, cannot exist without a shattering of belief and without
discovery of the "lack of reality“ of reality, together with the invention of other realities. What does
this "lack of reality" signify if one tries to free it from a narrowly historicized interpretation? The
phrase is of course akin to what Nietzsche calls nihilism. But I see a much earlier modulation of
Nietzschean perspectivism in the Kantian theme of the sublime. I think in particular that it is in the
aesthetic of the sublime that modern art (including literature) finds its impetus and the logic of
avant-gardes finds its axioms (Lyotard, p. 77).
The sublime […] takes place […] when the imagination fails to present
an object which might, if only in principle, come to match a concept.
[…] I shall call modern the art which devotes its "little technical
expertise" […] to present the fact that the unpresentable exists. To
make visible that there is something which can be conceived and
which can neither be seen nor made visible; this is what is at stake in
modern painting. […] One recognizes in those instructions the axioms
of avant-gardes in painting, inasmuch as they devote themselves to
making an allusion to the unpresentable by means of visible
presentations. […] The postmodern would be that which, in the
modern, puts forward the unpresentable in presentation itself; […]
(Lyotard, pp. 78, 81)
26
MATHEMATICAL MONSTERS
WEIERSTRASS’S FUNCTION (1872)
“I turn with fear and
horror from the
lamentable plague of
continuous functions
which do not have
derivatives”
(Charles Hermite in his
letter to Thomas Stieltjes
dated May 20, 1893)
where a is a real number with 0 < a < 1 while b is an odd integer with ab > 1+3π/2.
It was the first published example of a function which is continuous everywhere, but is
differentiable nowhere.
27
MATHEMATICAL PERFECTION: GREEK GEOMETRY
1. Pythagorean tilings
3. Hemitrigonon tiling + hexagram & pentagram
2. Hemitetragonon tiling
4. Circle &
Sphere
5. Platonic
solids
28
MATHEMATICAL CERTAINTY & THE SUBLIME
A sense of the sublime is well known to mathematicians.
Reviel Netz drops a general remark on the subject:
[M]ost mathematicians feel that there are aesthetic qualities
to the mathematical pursuit itself. The states of mind
accompanying the search for mathematical results are often
felt as sublime; an aesthetic study seems warranted (2005,
p. 254).
He specifies his use of “sublime” later on in the same paper:
the genre of Greek mathematical texts, “as a whole,
possesses beauty in its sublime impersonality”, that is in its
claim to possess absolute objectivity and truth (2005, p.
261).
Netz, R. (2005). The Aesthetics of Mathematics: A Study. In P. Mancosu, K.F. Jørgensen, & S.A. Pedersen (Eds.),
Visualization, Explanation and Reasoning Styles in Mathematics (pp. 251-293). Dordrecht: Springer.
29
MATHEMATICAL BEAUTY is wellestablished as a term in the
philosophy of mathematics.
MATHEMATICAL SUBLIME
and
MATHEMATICAL NUMINOUS
are candidates.
30
THANK YOU!