I finally published my compressible flow notes online. They are under classes – comp. flow. They are the result of teaching the course over nine years at University of Florida. I compressed the file a bit to conserve my website bandwidth. Enjoy!
Order, Error, and the Eye
Greek culture did not claim that the world was wholly beautiful. Its myths include monstrosities, mistakes, and bodies that fall outside any ideal. Plato sharpened that tension into philosophy. Sensible reality, as he saw it, was a diminished copy of a more perfect world of forms. The visible world was not the final standard. It was an imitation, and often a poor one.
At the same time, Greek artists treated the Gods as the model of supreme beauty. Olympus became a reference point. Statuary aimed at perfection through proportion, balance, and restraint. Beauty was not assumed. It was pursued.
In Christianity, key parts of this relationship were inverted. From a theological and metaphysical point-of-view, the entire universe is beautiful because it is divine workmanship. Creation is beautiful because it is made by God and with intention. In this framework, even ugliness and evil are pulled into a larger account of meaning. They are not celebrated, but are no longer outside the order. Christ, the human expression of divinity, is often portrayed at the moment of humiliation and suffering. The central image of redemption is not a serene body but wounded. Early Christian writers insisted on the goodness of all beings, and they anchored that insistence in scripture. Genesis states that at the end of creation God saw all that he had made and it was good. The Book of Wisdom adds a striking claim: the world was created according to number, weight, and measure, that is, according to criteria that resemble mathematical perfection.
Philosophy reinforced an aesthetic view of the cosmos. The idea that the world reflects ideal beauty has Platonic roots. Calcidius, in his late antique commentary on Timaeus, speaks of the splendid world of beings and its peerless beauty. The world, in this view, is not only functional. It is radiant. It is meant to be seen as an image of a higher form.
Pseudo Dionysius the Areopagite intensifies the metaphor of radiance. Beauty is not merely a property that objects possess. Beauty is a cause that pours itself out. It distributes fairness to each being according to its measure, and it produces harmony and splendor across creation. The language is luminous: beauty appears as light that makes things beautiful.
Similar and dissimilar things, species and forms, and the many layers of causes and appearances, are gathered into a unity that exceeds any single part. Medieval authors return repeatedly to this theme, sometimes described as the beauty of the whole, or the all beauty of the universe.
But the claim that the universe is beautiful confronts an obvious problem. Evil exists. Deformity exists. Corruption exists. If beauty and goodness are traditionally linked, then saying the universe is beautiful can also imply that it is good. How can that be reconciled with what the world plainly contains?
Augustine provides one answer. In De ordine, he acknowledges that we often perceive disharmony, even an insult to sight, when parts are arranged incorrectly. Yet he argues that error itself belongs within a larger order. In the Confessions, he insists that evil and ugliness do not exist as positive realities in the divine plan.
A related approach justifies ugliness by analogy with art. Deformity and evil function like shadow in a painting. Chiaroscuro makes light intelligible. Contrast reveals structure. The presence of darkness can disclose the coherence of the whole, even when the darkness is not desirable in itself. Others push the argument further. Sin disrupts order, but punishment restores it, so even the damned can be presented as evidence of an underlying law of harmony.
What about attention from the object to the observer. What is judged ugly may appear so because of flawed perception: poor light, the wrong distance, an oblique angle, or air that blurs contours. The judgment of ugliness, then, can be a report about conditions of viewing, not only about the thing itself.
During and after the Renaissance, the reevaluation of the human body produces a new problem: excessive beautification. Distressing events are sometimes rendered with an emphasis on noble strength or gentle sweetness rather than raw brutality. The torture becomes secondary to the aesthetic of endurance. In some cases, the result is openly eroticized, as in many depictions of Saint Sebastian, where the wounded body is presented as an object of desire as much as a symbol of faith.
One subject resists this softening more often, that is the hermit. The hermit is marked by deprivation, exposure, and time in harsh places. The desert is not flattering. Baroque spirituality uses hermits to celebrate disdain for bodily comfort, fasting, flagellation, and discipline. The stylites, who lived on columns, intensify the image. Weather, insects, illness, and relentless interior struggle become the visible cost of their spiritual stance. Their ugliness is treated as testimony and not failure.
Sartre’s No Exit presents an inferno without flames. Three people are confined in a room with the light always on and the door closed. In ordinary life, people can evade the gaze of others, or at least pretend that they are not fully seen. In Sartre’s room, there is no escape from judgment. The scorn of others becomes the punishment. One character begs for physical torture instead, because pain seems simpler than exposure, but the point remains: there are no instruments needed. Hell is other people.
This long arc yields a critical question – what to do with ugliness in a world we want to call meaningful? Greek culture showed that the world was not uniformly beautiful and treated beauty as an achievement. Christianity, in many of its dominant strands, insisted that the whole is beautiful because it is created, and then built philosophies to account for what harms the eye.
One can use the beauty of the whole to minimize the suffering of the part. A better use of the tradition is more demanding. Ask whether beauty is fragile, or whether it is robust enough to include shadow without denying it. Ask whether number, weight, and measure describe a world that is easy, or a world that holds together even when it hurts.
References
- Stanford Encyclopedia of Philosophy, “Plato’s Timaeus,” online reference article, Accessed February 12, 2026.
- Pseudo Dionysius the Areopagite, On the Divine Names, online translation at Christian Classics Ethereal Library.
- Augustine of Hippo, Confessions, Book VII, online translation at New Advent.
- Stanford Encyclopedia of Philosophy, “Medieval Theories of the Transcendentals,” online reference article.
- Tertullian, “To the Martyrs,” online translation at New Advent.
- The Metropolitan Museum of Art, online essay and collection materials on Saint Sebastian iconography.
- Oxford University, Cult of Saints in Late Antiquity database, record and materials on Simeon Stylites and stylite traditions, Accessed February 12, 2026.
- Jean Paul Sartre, No Exit, bibliographic record and synopsis in major online reference sources.
Our Nation – Lincoln
Whence shall we expect the approach of danger? Shall some transatlantic giant step the earth and crush us at a blow? Never. All the armies of Europe and Asia could not, by force, take a drink from the Ohio river, or make a track on the Blue Ridge in the trial of a thousand years. If destruction be our lot, we must ourselves be its author and finisher. As a nation of free men, we will live forever or die by suicide.
Abraham Lincoln, 1837.
Remembering Family – Photographer Paul Liebhardt
Recently my cousin and family member Paul Liebhardt passed away. He worked at NASA for a time and then became a photographer. While a photographer, he traveled internationally. His work as a photographer is well known. The remembrance celebration (link below) shows nice videos and writing of his students coming together from various paths of life sharing his teachings. Archive.org hopefully forever stores snapshots of his website and some of his photos.
A news article of remembrance celebration
He use to photograph for Semester at Sea, which gave a nice memorial tribute
https://www.semesteratsea.org/a-love-letter-to-what-it-means-to-be-human-remembering-paul-liebhardt
The achive.org of his personal website
https://web.archive.org/web/20190127182433/http://www.paulliebhardt.com/about
Eros Barefoot, Beauty Unfinished
Walk through a museum and the first signal is chromatic. Marble reads as white, and white reads as pure. Aphrodite and Apollo become an argument before they become a figure: ideal beauty, ideal calm, ideal proportion. Neoclassicism did not merely preserve the antique, it manufactured an antique that could serve as a stable reference for modern taste. The whiteness of the stone is not neutral. It is an aesthetic claim and a moral one.
The formal claim is older. In the fourth century BC, Polyclitus is said to have produced a Canon, a body that functions as a rule set. Later, Vitruvius writes proportion as arithmetic. The face is one tenth of total height, the head one eighth, the torso one quarter, and so on. The body becomes a fraction, and the fraction becomes a norm. From there the inference is almost automatic: what departs from the norm becomes ugly.
That inference is natural only if one forgets the scope of Greek visual and literary inheritance. The Greeks idealized beauty, but later Europe idealized the Greeks, and then forgot that Greek culture also trafficked in the disproportionate, the monstrous, the comic, and the grotesque. The tradition did not only leave us the calibrated athlete. It also left us Sileni, satyrs, gorgons, hybrid beings, old bodies, wounded bodies, the deliberate violation of canon. Sometimes figures carry traces of Eastern Mediterranean and Near Eastern motifs, which means that even the supposed purity of the Greek line is already a composite line. The West inherited both the rule, and the refusal of the rule.
Greek perfection is often summarized by kalokagathia, the fusion of kalos and agathos, beautiful and good, but also capable, dignified, courageous, disciplined, and worthy of honor. It maps cleanly onto a later English ideal of the gentleman, style plus virtue plus competence. Once you install that ideal, you can generate an entire literature of correlation: physical ugliness signals moral ugliness. The body becomes evidence. The face becomes a verdict.
Yet the Greek record does not permit a simple equivalence. The most obvious counterexample is Helen. The expedition to Troy is motivated by her extraordinary beauty, and rhetoric can even produce praise of her, as Gorgias does in his encomium. But Helen as a moral figure is unstable, at best. Beauty moves nations, and beauty also breaks vows. If beauty is merely what pleases the eye, then it is powerful, and morally indifferent. If beauty is a quality of the soul, then Helen is a problem, because body and life do not align.
This ambiguity becomes explicit in Plato. If reality is the realm of Forms, and sensible things are copies, then ugliness looks like failure, a slide toward nonbeing. Plato even rejects in the Parmenides,the need for forms of base things such as mud, hair, or stains. The foul has no ideal model. It exists only as defect in the sensible order, a local imperfection against a perfect template.
Then the symposium complicates the picture further by relocating beauty inside desire. The dialogue distinguishes kinds of love, including loves aimed at women, and loves aimed at boys, and it stages competing images of Eros. Agathon presents Eros as eternally young and handsome, a familiar Greek pairing: youth with beauty, age with ugliness. It is the aesthetic of the unmarked body, the body before decay, before contingency becomes visible.
Socrates refuses the easy portrait. Speaking through the figure of Diotima, he argues from lack: we desire what we do not have. If so, Eros cannot be beautiful or good in possession, because he is desire for beauty and goodness, not their completed state. Eros is an in between. Plato calls him a daimon, not a god, not a brute, but an intermediary, a striving. His parentage encodes the concept. He is the child of Penia, Lack, and Poros, Resource. From Penia, he inherits a wretched appearance, shaggy, barefoot, homeless. From Poros he inherits the capacity to hunt, to scheme, to pursue what he lacks.
This is not a sentimental move. It is a structural one. If the beautiful is what we lack, then the lover is defined by incompleteness, not by perfection. Desire is not the halo around an already ideal body. Desire is the engine that reveals the gap between the body we have and the value we want.
Plato then connects Eros to immortality. Human beings want to persist. Physical procreation is one route, but …
Diotima insists on a second route: the production of spiritual values. Poetry, laws, philosophy, and the formation of character are also modes of reproduction. They create a different kind of afterlife, immortality by glory, and by transmission. Ordinary people produce children. Those who cultivate the aristocracy of the spirit produce wisdom, and beauty, as durable forms.
From this comes the ladder of love. A person begins by admiring one body. Then many bodies. Then the beauty of minds, practices, and institutions. Eventually, the lover aims at Beauty itself, not a local instance, but Beauty as Idea, the hyperuranian object. Here kalokagathia shifts meaning. The good and beautiful person is no longer the one whose body matches canon. He is the one who sees beyond canon, who prefers the beauty of the soul, and who can care for a young person of promise even if the body is not exemplary. The body becomes a starting point, not a stopping point.
Alcibiades then enters the dialogue as a kind of stress test. Handsome, drunk, charismatic, he offers a vivid confession: he tried to trade his body for Socrates’ wisdom. Socrates refused. He lay beside him chastely. The scene is not a prudish anecdote. It is a demonstration of priority. Carnal beauty does not purchase moral and intellectual beauty. The exchange rate is not defined.
Alcibiades’ praise of Socrates turns on appearance. Socrates looks like a Silenus, an ugly exterior, a comic face, a body that does not satisfy the classical smoothness. Yet Alcibiades insists that inside this exterior is a profound beauty. The dialogue forces a reversal: ugliness can be the shell of wisdom, and beauty can be the mask of chaos. The simplistic opposition between ugliness and kalokagathia collapses. Greek culture knew this collapse, and returned to it repeatedly, as shown by later admiration for Aesop, another figure marked as ugly, but granted noble soul, and sharp intelligence.
Even within Plato, the line is not stable. In the Republic he treats ugliness, understood as disharmony, as a threat to the soul’s formation, and he recommends sparing the young from ugly portrayals. Aesthetic diet becomes moral diet. And yet he also concedes that every thing has a kind of beauty proper to its function, its fit to its own Form. A pot can be beautiful as a pot. A mare can be beautiful as a mare. The hierarchy remains, because a pot is not beautiful compared to a girl, but within its category, it can still satisfy an internal standard. Beauty becomes relational, and contextual, not only absolute.
Aristotle, more empirical, sanctions a principle with long afterlife: it is possible to make beautiful imitations of ugly things. Representation can redeem what it depicts. People can admire a skilled portrayal of the unattractive, whether the unattractiveness is physical, moral, or both. Homer’s Thersites becomes a canonical example, a figure whose ugliness is part of his narrative function, and whose depiction becomes an aesthetic success.
Then the Stoics widen the frame further. Marcus Aurelius can look at imperfections, like cracks in a loaf of bread, and see them as contributing to the agreeability of the whole. The defect is not merely tolerated, it is integrated. It becomes part of the texture that makes the object real, and therefore fitting. Ugliness is redeemed by context, not by denial.
If you follow this arc, the museum myth looks thin. The Greeks did not hand us a single doctrine of beauty. They handed us a contested field: proportion and its violation, body and soul, youth and decay, desire as lack, representation as transfiguration, context as redemption. Neoclassicism took one slice, polished it, and called it the whole. The older material is messier, and more useful. It does not let beauty settle into a static measurement. It forces beauty to argue with virtue, and it forces ugliness to appear as more than a simple opposite. It is a reminder that canon is not nature. Canon is a decision, and Greece, like us, argued about what that decision should mean.
2025 in Review
2025 was a unique year. I lived in four different states and was a resident of those states. I moved across the country three times, with professional movers, packing and unpacking, registering my car, updating bank accounts, IDs, insurance, and permanent addresses. I leased apartments, sold my beloved house in Florida, and navigated multiple career changes quickly.
Along the way, I said goodbye to long term friends, familiar rhythms, favorite restaurants, coffee shops, and the commutes that had become quiet anchors, the route to the university, the small towns, the streams, and the art museums that made a place feel peaceful. In the middle of so much change, I found myself thinking of what I have done, where I have been, and what I still carry. When I look back at the people I knew, and the work, and the ordinary days that now feel distant, I do not see loss. I see what I learned, what it built in me, and the good that came from it.
Socially Sanctioned Incompetence
In America, there is a funny, and revealing, version of mathematics.
One popular joke is that, at a dinner party, the only socially acceptable thing to be bad at is mathematics. Someone recalls high school and says they are terrible at math, another says, me too, and everybody laughs. This is not so common in other countries. America is not unique, but this is a recognizable piece of day to day American culture. Americans are rarely proud of being bad at anything, yet mathematics is the one domain where the common American can sound almost pleased to be incompetent.
Americans often connect academic effort with outcome. A scholarship, a job, a credential, a title, a salary. So how does one convince American families that mathematics matters for a child’s education if the payoff is not immediate, or even legible? This is part of the motivation behind the Clay Mathematics Institute putting serious money behind difficult, century class problems. There is an old story, I cannot remember whether it comes from von Neumann or someone adjacent. The idea is simple: attach a financial reward to hard mathematics so American parents can tell their kids to study it because there is a concrete outcome.
This is very different from what drives professional mathematicians at the top of the field, like Andrew Wiles, who solved one of the great century class problems. I recall watching a documentary on him, and in one interview he cried while talking about the moment it finally worked. He was not crying because of money, status, or career positioning. He was crying because he had touched mathematical beauty. Many engineers, and many students in engineering courses, do not fully register how powerful that can be, the lived experience of beauty in a symbolic language that does not need a practical pretext.
Note on Early Medieval Universities, the Scholastic Method, and the Formation of Critical Thought
Early medieval universities formed a distinct intellectual system built around structure, discipline, and the controlled expansion of reasoning. These institutions emerged from cathedral schools and monastic centers and evolved into formal environments where knowledge was not only preserved but interrogated. Their core mechanism was the scholastic method, a systematic approach that treated ideas as analytical objects. The process was not ornamental. It was engineered to extract clarity from ambiguity through ordered reasoning.
The foundation of the scholastic method began with the authoritative text. Masters performed the lectio, reading the text line by line, isolating definitions, identifying structure, and parsing each conceptual element. This was not a passive reading but an initial decomposition step. The goal was to understand the internal architecture of the argument and to expose points where logical tension might develop.
From this, the quaestio emerged. The method treated contradictions as formal problems to be resolved. A question was constructed with precise logical framing that forced the students to examine the boundary where an argument failed to align with another authority or with its own implications. The university classroom became a controlled environment for isolating assumptions, identifying conflicts, and preparing for systematic resolution.
Disputation was the operational phase. In this setting, objections were raised, defended, and countered with ordered rigor. The disputation structure required the student to articulate objections with precision, evaluate their strength, and identify where an argument carried or collapsed. The master responded with the determinatio, a final synthesis that reconciled the competing claims and provided a coherent resolution grounded in logic rather than force of assertion. This served as a model for integrative reasoning.
The outcome was a unique form of critical thinking. Students were trained to interrogate any claim, including those from revered authorities, through systematic analysis. The method normalized structured critique. The reliance on logical form eliminated vague reasoning and required explicit premises. Students learned to perform rapid evaluation during oral defense, to track argument flow, and to maintain internal consistency. The result was a disciplined intellectual framework that enabled scholars to construct large, stable bodies of knowledge, such as the theological and legal systems that later defined medieval and early modern thought.
The scholastic method represents an early creation of cognition. It imposed constraints that produced clarity and forced internal coherence. While the topics were often theological, the underlying process was analytical, and it shaped the evolution of Western intellectual life. The early universities did not invent critical thinking, but they built the first institutional architecture that required it, refined it, and transmitted it across generations.
Remember a Perfect Day
On a spring day at University of Florida, I walked from Professor Haftka’s office, now my own. Up the small rise from aerospace, near the student union and under the trees, I made my way to class. Years of developing the Navier–Stokes equations for aerospace students, joined by mathematics students. The room held thirty, yet forty arrived, ten seated on the floor. A failed attempt at seeing mathematical beauty, but a perfect walk. Then Covid arrived.
Leaves
In the late summer of that year we lived in a house in a village that looked across the river and the plain to the mountains. In the bed of the river there were pebbles and boulders, dry and white in the sun, and the water was clear and swiftly moving and blue in the channels. Troops went by the house and down the road and the dust they raised powdered the leaves of the trees. The trunks of the trees too were dusty and the leaves fell early that year and we saw the troops marching along the road and the dust rising and leaves, stirred by the breeze, falling and the soldiers marching and afterward the road bare and white except for the leaves.
Ernest Hemingway, A Farewell to Arms