Now we reach a value that pulls all values together, in a sense. Every value is part of an order. We need other values to decide what that order should be but the idea of an order is essential to any values system: ethics, morality, law, spirituality and religion.
- 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 an “unseen order” wills our happiness but only that life offers us a chance at happiness. Most of the universe is cold and incompatible with life. Still, in our part 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 politics having devolved into shouting, sloganeering and disinformation, 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.
Real
True Narratives
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 (G.P. Putnam’s Sons, 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.”
- James Vincent, Beyond Measure: The Hidden History of Measurement, from Cubits to Quantum Constants (W.W. Norton & Company, 2022): “The story of humans measuring things is no less than the story of civilization — a claim that sounds like irritating hyperbole but in this case turns out to be true. Vincent conveys how measurement developed as a “scaffold for knowledge,” encouraging us to categorize and make comparisons. It is also extraordinarily powerful, ‘a tool of social cohesion and control.’ When people agree on a standard of measurement, they can coordinate their actions.
- Rebecca Boyle, Our Moon: How Earth’s Celestial Companion Transformed the Planet, Guided Evolution, and Made Us Who We Are (Random House, 2024): “. . . Boyle walks the reader through a history of both Earth and humanity, from the formation of our planet and the evolution of life to the development of civilization, religion, philosophy and, eventually, science. And through it all, she argues, the moon has played a starring role in how we came to be, and who we are.”
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, A History of Mathematics (Wiley, 1968).
- Michael G. Gordin, A Well-Ordered Thing: Dmitri Mendeleev and the Shadow of the Periodic Table (Basic Books, 2004).
Mathematics journals:
- American Journal of Mathematics
- International Journal of Mathematics
- New York Journal of Mathematics
- The Quarterly Journal of Mathematics
- The College Mathematics Journal
The Mandelbrot set within a continuously colored environment
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 in Space (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, ed., Thinking in Patterns: Fractals and Related Phenomena in Nature (World Scientific Pub. Co., 2004).
- Albert-Laszlo Barabasi & Harry Eugene Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, 1995).
- Tamas, Fractal Growth Phenomena (World Scientific Publishing, 2nd edition, 1999).
- John Fauvel, Raymond Flood & Robin Wilson, eds., Music and Mathematics: From Pythagoras to Fractals (Oxford University Press, 2003).
- Michael Frame & Amelia Urry, Fractal Worlds: Grown, Built, and Imagined (Yale University Press, 2016).
- 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 Behavior 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 & 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, Fratctals (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.”
Photographs
Documentary and Educational Films
Imaginary
Fictional Narratives
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:
- Joan Didion, A Book of Common Prayer: A Novel (1977): “A Taut Novel of Disorder”
Poetry
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
advantage of.
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
something catches.
[Jorie Graham, “The Way Things Work”]
Music: Composers, artists, and major works
Widely regarded as history’s greatest composer, Johann Sebastian Bach “was a musical master of mathematical manipulation. . . He was fond of using geometric operations to explore melody — techniques like transposition, inversion, and retrograde inversion all have analogs in the world of classical geometry”. The exquisite orderliness of his music is apparent even to the untrained listener. “. . . Bach was . . . a mathematician in a . . . general sense, as a composer whose works are replete with patterns, structures, recursions and other precisely crafted features. There are even hints of Fibonacci numbers and the golden ratio in Bach’s music . . .” Bach has been famously linked with the mathematician Kurt Gödel, and the Dutch graphic artist M.C. Escher. Ruth Tatlow, Ruth Tatlow again, and Alan Shepherd, have authored highly-respected scholarly books on the mathematical underpinnings of Bach’s music, a subject also addressed in scholarly books by Christoph Wolff, Robin A. Leaver (ed.), and Bettina Varwig (ed.). Virtually all of his music is an expression of order, in a deliberate and forward way. In bringing together an orchestra, a chorus and a story 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” add another dimension to the deeply religious Bach as a composer of orderly intent and structure. The narrative in each work is from standard Christian theology; the musical composition is what most clearly illustrates the value of order.
- Mass in B Minor, BWV 232 (1749) (110-135’) (libretto): top recorded performances are conducted by Klemperer in 1967, Parrott in 1985, Suzuki in 2007, and Gardiner in 2015.
- St. Matthew Passion, BWV 244 (1727) (approx. 160-180’) (libretto): top recorded performances are by Klemperer in 1962, Gardiner in 1988, Suzuki in 1999, Harnoncourt in 2000, McCreesh in 2002, Jacobs in 2013, and Pichon in 2022.
- St. John Passion, BWV 245 (1724) (approx. 105-130’): top recorded performances are conducted by Herreweghe in 2001, Butt in 2011, Herreweghe in 2018, and Gardiner in 2022.
- St. Mark Passion, BWV 247 (1731) (approx. 110-120’), is reconstructed, the original having been lost. Top recorded performances are conducted by Willens in 2009, and Savall in 2019. Here is a link to some his releases.
Other works:
- Arvo Pärt, Passio (Passio Domini Nostri Jesu Christi secundum Joannem) (1982) (approx. 60-68’)
- Ben Johnston, String Quartet No. 3, “Verging” (1966) (approx. 11’), displays “Johnston’s search for intelligibility within complexity” [Bob Gilmore, program notes for an album of string quartets].
- Arthur Berger, Ideas of Order (1952) (approx. 12’)
- Catherine Lamb, divisio spiralis (2019) (approx. 59’): 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:
- Peter Brötzmann, “For Adolphe Sax” (2006) (50’)
- Franz Koglmann, “Orte der Geometrie” (1989) (64’)
Albums, from the dark/gray side (disorder):
- Thomas Heberer, “X Marks the Spot” (2019) (40’): “Concise and cosmopolitan, the eight selections here offer a slice of contemporary New York improvisation . . .”
- James Brandon Lewis, “An UnRuly Manifesto” (2019) (45’): “It's always refreshing to find a young artist who professes keen links back to the tradition, but is determinedly taking it to new places. With his blend of adventurous funk, electric free-jazz and indomitable spirit, Lewis seems set fair for the future.”
Music: songs and other short pieces
- The Fixx, “One Thing Leads to Another” (lyrics)
Visual Arts
- 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
.jpg){kind=link}
- 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