- Were one asked to characterize the life of religion in the broadest and most general terms possible, one might say that it consists of the belief that there is an unseen order, and that our supreme good lies in harmoniously adjusting ourselves thereto. This belief and this adjustment are the religious attitude in the soul. [William James, The Varieties of Religious Experience (1902), Lecture III: The Reality of the Unseen.]
- The awakening reason demands a theory of the universe and ceases to be satisfied with the patchwork schemes of mythology. The moral self coming to partial consciousness of its nature and scope demands a higher rule of life and a deeper understanding of cosmic forces. Instead of inventing stories about the beginning of things and the origin of laws, the mind begins to search for the general truths underlying or permeating experience and giving unity and meaning to human purposes. The forward step achieved by thought in this movement may be described by saying that the imagery of its earlier stage is replaced by defined and reasoned conceptions formed by the analysis and reconstruction of primitive ideas. . . . fail as it may in its attempts at final truth, a deeper religion and a higher ethics are the outcome of each new effort. [L. T. Hobhouse, Morals in Evolution: A Study in Comparative Ethics (1915), Chapter III: “The World and the Spirit”.]
- There is no certainty in sciences where mathematics cannot be applied. [Leonardo da Vinci.]
Virtually every human endeavor can well be seen as an attempt to bring order to life or find order in it. Mathematics, science, governance, law and social systems of all kinds, and even our arts and our everyday pleasures are all histories of the search for order: the attempt to bring life into orderly adjustment with the good.
This is not to suggest that the “unseen order” wills our happiness but only that our lives offer us a chance at happiness. Hard-core secularists observe that most of the universe is cold and incompatible with life. That is true but in our section of the universe, life is abundant, and we are the beneficiaries of the orderliness that made it happen. To miss the opportunities offered by this fortuitous confluence of events and circumstances would be to waste our lives. The search for order is best seen not as the invention of a just-so story about cosmic consciousness but as an attempt to find the principles in nature that will help us lead satisfying and productive lives.
Our challenge is to see order both in close focus and in broad perspective: to achieve both a depth and a breadth of vision. We do not aspire to excessively constraining forms of order represented, for example, by the medieval practice of sleeping in a rigidly defined position so as to align oneself with a certain conception of God; this we see as myopic and unproductive. In our search for order, we recognize that the very idea of the good is a work in progress, and that our success in finding some part of it depends on our devotion to moral and ethical principles, including humility and openness. “Unseen” does not mean magical or supernatural; it means that we have not yet seen it, as people did not see the order in the planets and stars until Copernicus pointed it out, or a modern conception of music’s expressive capacity until Mahler, Shostakovich and Stravinsky demonstrated them through their compositions.
As human beings continue to uncover pieces of nature’s unseen order and the scope of our knowledge expands accordingly, the challenge of ethical living imposes ever greater demands on our time and attention. In these interesting times, with our politics in the United States having devolved into shouting and sloganeering and our mass media having virtually abandoned news in service of entertainment, our survival may depend on our all becoming “Renaissance men” with an appreciation for order both in exquisite detail and on a large scale.
People obsessed with order who made significant contributions in human affairs:
- Joshua Kendall, The Forgotten Founding Father: Noah Webster’s Obsession and the Creation of an American Culture (G.P. Putnam’s Sons, 2011): Webster also “cleaned up” the Bible.
- Joshua Kendall, The Man Who Made Lists: Love, Death, Madness, and the Creation of Roget’s Thesaurus (, 2008): the creator of the great thesaurus, not surprisingly, was an obsessive.
- Alex Danchev, Magritte: A Life (Pantheon, 2021): “He was . . . the embodiment of Flaubert’s famous dictum that artists should live in an orderly fashion and reserve their wildness for their work. His art was brilliantly subversive and based on his belief that everyday life is shrouded in mystery.”
Searching for order, in various ways:
- Jack Lynch, You Could Look It Up: The Reference Shelf From Ancient Babylonia to Wikipedia (Bloomsbury Press, 2016): on “the history of putting words and concepts in accessible order.”
- Cecelia Watson, Semicolon: The Past, Present, and Future of a Misunderstood Mark (Ecco, 2019): “ . . . the semicolon takes the place of a period and yokes together two independent clauses that could function as sentences on their own.”
- Jordan Ellenberg, Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else (Penguin Press, 2021): “A seam in his narrative is a critique of how math, and especially geometry, has been taught. (His strategy for success in teaching is to employ more strategies; multiply approaches so students might find one that works for them.)”
- Kevin Roose, Futureproof: 9 Rules for Humans in the Age of Automation (Random House, 2021): “. . . a concise, insightful and sophisticated guide to maintaining humane values in an age of new machines.”
Technical and Analytical Readings
From science to music, the order in nature is found in mathematics.
- Jan Gullberg, Mathematics: From the Birth of Numbers (W. W. Norton & Co., 1997).
- Victor J. Katz, A History of Mathematics: An Introduction (Addison Wesley, 2008).
- Uta C. Merzbacn and Carl B. Boyer, History of Mathematics (Wiley, 1968).
- Michael G. Gordin, A Well-Ordered Thing: Dmitri Mendeleev and the Shadow of the Periodic Table (Basic Books, 2004).
- American Journal of Mathematics
- International Journal of Mathematics
- New York Journal of Mathematics
- The Quarterly Journal of Mathematics
- The College Mathematics Journal
(Image: a famous fractal: the Mandelbrot set)
We find a magnificent illustration of order in mathematician Benoit B. Mandelbrot’s work on fractals. Mandelbrot saw repeating patterns in nature. This discovery allowed him, and then others, to develop computer models that closely simulated objects in nature, such as mountains or a head of broccoli. Mandelbrot had uncovered a fundamental pattern of nature, hidden from human consciousness for millennia, yet already expressed in art and architecture, and obvious once understood.
- Heinz-Otto Peitgen, Hartmut Jürgens and Dietmar Saupe, Chaos and Fractals: New Frontiers of Science (Springer, 2004).
- Benoit B. Mandelbrot, Fractals and Chaos: The Mandelbrot Set and Beyond (Springer, 2004).
- Benoit B. Mandelbrot, The Fractal Geometry of Nature (W.H. Freeman, 1982).
- Miroslav M. Novak, Thinking in Patterns: Fractals and Related Phenomena in Nature (World Scientific Pub. Co., 2004).
- Manfred Robert Schroeder, Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise (W.H. Freeman & Co., 1991).
- John Briggs, Fractals: The Patterns of Chaos: Discovering a New Aesthetic of Art, Science, and Nature (Simon & Schuster, 1992).
- Michael Fielding Barnsley, Fractals Everywhere: The First Course in Deterministic Fractal Geometry (Academic Press, 1988).
- Kenneth Falconer, Fractal Geometry: Mathematical Foundations and Applications (Wiley, 2003).
- Kenneth Falconer, The Geometry of Fractal Sets (Cambridge University Press, 1985).
- J.M. Li, Li Lü, M.O. Lai and B. Ralph, Image-Based Fractal Description of Microstructures (Springer, 2003).
- V.P. Dimri, ed., Fractal Behaviour of the Earth System (Springer, 2005).
- Gary William Flake, The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation (The MIT Press, 1998).
- Dietrich Stoyan and Helga Stoyan, Fractals, Random Shapes and Point Fields: Methods of Geometrical Statistics (Wiley, 1994).
- Tom G. Blenkinsop, Jörn H. Kruhl and Miriam Kupková, eds., Fractals and Dynamic Systems in Geoscience (Birkhäuser Basel, 2000).
- Jans Feder, Fractals (Physics of Solids and Liquids (Springer, 1988).
- Marc Frantz and Annalisa Crannell, Viewpoints: Mathematical Perspectives and Fractal Geometry in Art (Princeton University Press, 2011).
- Benoit B. Mandelbrot and Richard L. Hudson, The (Mis)ehavior of Markets: A Fractal View of Risk, Ruin and Reward (Basic Books, 2004).
- Fractals (journal)
Foundations of life's order:
- Siddhartha Mukherjee, The Gene: An Intimate History (Scribner, 2016): “ . . . the essential unit of biological information, we call the gene. First the idea of the gene had to be invented. Then the physical entity, present in each cell of our bodies, in every living thing, had to be discovered.”
Documentary and Educational Films
- Jackson Pollock, Pattern (1945)
- Wassily Kandinsky, Red, Yellow, Blue, and Golden Rectangles (1925)
- Piet Mondrian, Composition No. 11, London, with Blue, Red and Yellow (1921)
- Piet Mondrian, Composition: Light Color Planes with Gray Contours (1919)
- Piet Mondrian, Composition in Color A (1917)
- Diego Rivera, The Mathematician (1918)
- Lee Krasner, White Squares (1948)
- Max Ernst, Design in Nature (1947)
- Georgia O’Keeffe, Pattern Leaves
- Pavel Filonov, Formula of the Cosmos (1918-19)
Film and Stage
- My Darling Clementine: a semi-historical Western fable about a town verging on anarchy until Wyatt Earp lays down the law
Music: Composers, artists, and major works
Johann Sebastian Bach’s music will appear frequently on these pages. Widely regarded as history’s greatest composer, he set a standard for composition that others have yet to match. Among many other attributes, the exquisite orderliness of his music is apparent even to the untrained listener. In bringing together an orchresta and chorus in tight compositional structure, and in remaining true to the intent for which Bach composed the works, his Mass in b minor and his “Passions” well illustrate the virtue of order.
- Mass in b minor, bwv 232 (here are links to performances conducted by Gardiner, Bicket and Suzuki)
- Matthew’s Passion, bwv 244 (here are links to performances conducted by Suzuki, Brüggen, Harnoncourt, Mengelberg and Pichon)
- John’s Passion, bwv 245 (here are links to performances conducted by Richter, Suzuki and Harnoncourt)
- Mark’s Passion, bwv 247 (performances conducted by Jedlin, Goodman and Koopman)
- Pärt, Passio (Passio Domini Nostri Jesu Christi secundum Joannem) (1982)
- Ben Johnston, String Quartet No. 3, “Verging” (1966) displays “Johnston’s search for intelligibility within complexity” [Bob Gilmore, program notes for an album of string quartets].
- Berger, Ideas of Order
- Raga Miyan ki Todi, an early morning raag (Todi – Miya ki Todi) (performances by Ravi Shankar, Asad Ali Khan, Joshi, Banerjee and Amonkar)
- Catherine Lamb, divisio spiralis (2019). The composer writes: “the first time I discovered Erv Wilson’s 1965 organization of the overtone series as a logarithmic spiral, the image immediately resonated with me as a lucid means to describe harmonic space as numbers in repetition and interaction, generating/blooming outwards with each new prime and composite. I absorbed this image while working on the piece for JACK, and after applying a 29-limit reductionist omission to the tonal palette and situating the four string instruments inside it as distinct resonating chambers, I utilized this image as an inspiration for the total piece.”
Albums, from the dark/gray side (disorder):
Geometry is harmony. Some fine mansions here and there made magnificent outlines against the picturesque attics of the left bank. The house of Nevers, the house of Rome, the house of Reims, which have disappeared; the Hôtel de Cluny, which still exists, for the consolation of the artist, and whose tower was so stupidly deprived of its crown a few years ago. Close to Cluny, that Roman palace, with fine round arches, were once the hot baths of Julian. There were a great many abbeys, of a beauty more devout, of a grandeur more solemn than the mansions, but not less beautiful, not less grand. Those which first caught the eye were the Bernardins, with their three bell towers; Sainte-Geneviève, whose square tower, which still exists, makes us regret the rest; the Sorbonne, half college, half monastery, of which so admirable a nave survives; the fine quadrilateral cloister of the Mathurins; its neighbor, the cloister of Saint-Benoît, within whose walls they have had time to cobble up a theatre, between the seventh and eighth editions of this book; the Cordeliers, with their three enormous adjacent gables; the Augustins, whose graceful spire formed, after the Tour de Nesle, the second denticulation on this side of Paris, starting from the west. The colleges, which are, in fact, the intermediate ring between the cloister and the world, hold the middle position in the monumental series between the hotels and the abbeys, with a severity full of elegance, sculpture less giddy than the palaces, an architecture less severe than the convents. [Victor Hugo, Notre-Dame de Paris, or, The Hunchback of Notre Dame (1831), Volume I, Book Third, Chapter II, “A Bird’s-Eye View of Paris”.]
Novels from the dark side:
is by admitting
or opening away.
This is the simplest form
of current: Blue
moving through blue;
blue through purple;
the objects of desire
opening upon themselves
without us; the objects of faith.
The way things work
is by solution,
resistance lessened or
increased and taken
The way things work
is that we finally believe
they are there,
common and able
to illustrate themselves.
Wheel, kinetic flow,
rising and falling water,
ingots, levers and keys,
I believe in you,
cylinder lock, pully,
lifting tackle and
crane lift your small head--
I believe in you--
your head is the horizon to
my hand. I believe
forever in the hooks.
The way things work
is that eventually
[Jorie Graham, “The Way Things Work”]