Cathedral Builders: The Sublime in Mathematics

Vladislav Shaposhnikov

Faculty of Philosophy Lomonosov Moscow State University

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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.

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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).

THE HYPOTHESIS OF THE RITUAL ORIGIN OF MATHEMATICS

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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.

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)

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)

A falcon-shaped altar

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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)

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GOD AS ARCHITECT & GEOMETER IN MEDIEVAL MINIATURES

God the Architect and the Geometer (13th century)

Holkham Bible, 14th century British Museum, Add. 47682, fol. 2

For extended discussion see:

Friedman, J.B. (1974). The Architect’s Compass in Creation Miniatures of the Later Middle Ages, Traditio, 30, 419-429.

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

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THEOS

ANTHROPOS COSMOS

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THEO-COSMO-ANTHROPOLOGICAL TRIANGLE (TCA-triangle)

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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

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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/TradesCrafts/indexByTrade.htm)

Drucker, P.F. (1993/1954). The Practice of Management, New York, NY: HarperCollins.

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).

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AIMS & GOALS of PURE MATH NEEDS → MOTIVATION → AIMS & GOALS

EXTERNAL vs. INTERNAL GOALS EPISTEMIC vs. NON-EPISTEMIC or PRACTICAL GOALS

What are the ultimate non-epistemic aims and goals of the mathematical activity?

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.

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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)

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THE NUMINOUS & THE SUBLIME RUDOLF OTTO, 1917, Das Heilige (The Idea of the Holly)

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).

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)

WHAT IS THE SUBLIME?

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“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).

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).

Caspar David Friedrich, Wanderer above the Sea of Fog, 1817, Kunsthalle, Hamburg

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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)

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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.

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Das höchste Leben ist Mathematik. - Es kann Mathematiker der ersten Größe geben, die nicht rechnen können. - Man kann ein großer Rechner sein, ohne die Mathematik zu ahnden. - Der echte Mathematiker ist Enthusiast per se. Ohne Enthusiasmus keine Mathematik. - Das Leben der Götter ist Mathematik. - Alle göttlichen Gesandten müssen Mathematiker sein. - Reine Mathematik ist Religion. - Zur Mathematik gelangt man nur durch eine Theophanie. - Die Mathematiker sind die einzig Glücklichen. Der Mathematiker weiß alles. Er könnte es, wenn er es nicht wusste. Alle Tätigkeit hört aus, wenn das Wissen eintritt. Der Zustand des Wissens ist Eudämonie, selige Ruhe der Beschauung, himmlischer Quietismus. - Im Morgenlande ist die echte Mathematik zu Hause. In Europa ist sie zur bloßen Technik ausgeartet. - Wer ein mathematisches Buch nicht mit Andacht ergreift, und es wie Gottes Wort liest, der versteht es nicht. – (Novalis, 1837, pp. 147-148)

The highest life is mathematics. - There can be supremely ranked mathematicians who cannot calculate. - One could be a great calculator without having an inkling of mathematics. - The true mathematician is an enthusiast per se. Without enthusiasm there is no mathematics. - The life of the Gods is mathematics. - All divine messengers must be mathematicians. - Pure mathematics is religion. - One only advances to mathematics through a theophany. - Mathematicians alone are fortunate. The mathematician knows all. He could know it, even if he did not already. All activity ceases when knowledge enters. The state of knowledge is eudemony, the blessed peace of contemplation – heavenly quietism. - True mathematics is at home in the orient. In Europe it has degenerated into a purely technical science. - Whoever does not take hold of a mathematical book with devotion, and read it as the word of God, fails to understand it. – (Translated by David W. Wood)

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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.

VEHICLES FOR THE MATHEMATICAL SUBLIME

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1.Mathematical infinity or high complexity. 2.Mathematical perfection or optimality. 3.Mathematical certainty.

THE SUBLIME & THE INFINITE

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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)

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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)

THE INFINITE IN THE FINITE

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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).

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).

A hyperbolic tessellation in Poincaré’s disc model using regular heptagons

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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)

THE STARS OF THE HEAVEN & THE SAND OF THE SEA

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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)

MODERN, POSTMODERN & THE SUBLIME

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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)

MATHEMATICAL MONSTERS

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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.

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)

MATHEMATICAL PERFECTION: GREEK GEOMETRY

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1. Pythagorean tilings

5. Platonic solids

4. Circle & Sphere

2. Hemitetragonon tiling

3. Hemitrigonon tiling + hexagram & pentagram

MATHEMATICAL CERTAINTY & THE SUBLIME

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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.

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MATHEMATICAL BEAUTY is well-established as a term in the philosophy of mathematics.

MATHEMATICAL SUBLIME

and MATHEMATICAL NUMINOUS

are candidates.

THANK YOU!