“House-builder, you’re seen! You will not build a house again. All your rafters broken, the ridge pole destroyed, gone to the Unformed, the mind has come to the end of craving.”
—Siddhārtha Gautama (the founder of Buddhism), upon reaching enlightenment (Dhammapada)
It was speculated by Thanissaro Bikkhu that the “house” meant selfhood, or perhaps entity-hood, in the commentary of the Dhammapada.
I would propose a model for logic that is a house. Some logical structures are immense. The light that passes through a window would be Truth; the laws that light follows as it interacts with the building would be the laws of logic; the specific form of this particular building would be the logical statements, determining the way truth (light) moves through the logical structure. (And by “truth” here I mostly mean the clarity and warrant that travels with what we can rightly assert—what survives transmission. Edit based in Pierre’s feedback: I will develop this idea of a clarity that degrades from true proposals partially true conclusions, to more partial conclusions, etc. The next essay will apply this loss in a truth property as a loss in the meaning of a number, or the numerousness of a number, as they progress indefinitely toward infinity. Then I will apply this idea to probability theory, which are revisions of my line of thought from 2015)
The trouble is completing the logical elements: what is falsehood? Obviously it is darkness, but the building would have to have no qualities except its form—no colors, no features, just featureless glass mirrors—otherwise the light would fade as it interacts with opaque surfaces, making truth and falsehood mingle. If the walls are perfect mirrors that propagate the light perfectly, a false space would have to be totally cut off from the light. Hypotheticals would be doors, sometimes open, sometimes shut. The only danger of falling into darkness would be entering through a door and closing it, completely cutting yourself off.
The theory that comes to mind is Anaximander’s, who thought the sun was just a hole in the cosmos, where light could enter from outside the Universe. And why is this ideal of logic impossible in the real world? There are no perfect mirrors. Matter has color that absorbs light, making it an intermediate between truth and falsehood. When logic from true principles is applied to real things—interacting with matter—the truth will dim as the logical statements progress, regardless of how perfectly the laws of logic are followed. If the world of logic were to be perfect, the truth could not originate from our world, or else light that is reflected back out the window of our house would fall, logically, onto ambiguous matter. Thus passing out the window must lead to a world that looked mostly the same as the building of mirrors.
With the modern conception that words can provide totally transparent access to an object, matter would be the only medium between truth and falsehood. But words simply aren’t transparent. They grow out of metaphors (as argued in the essay linked in my first post). The word “be” grew out of a Proto-Indo-European root which also meant grow—so that someone aware of the ancestry of words would resurrect the feeling of metaphor in the word “be,” coloring the word, giving it a connection that is warranted because “be” would not be what it is now without a fathering metaphor: being is growing.
And the design or form of this fun-house of mirrors—would it carry nameable concepts with it, concepts one would come to know or feel by living there? It would if it had any architectural design. How is this different from allowing a word, or a sign for an idea or feeling, into our logic?
The house of logic cannot allow matter, words, or form—except in a part of the house that is totally dark and without doors. They can be allowed into the part sectioned off as unconditionally false. Otherwise we are allowing degrees of truth, qualifications of truth, and a co-mingling of truth and falsehood.
The focus of this blog (expressed in the previous post) has changed to looking for systems of truth that gradually and naturally falsify themselves. What if we allowed matter in our house, and accepted gradations of truth? How could Aristotelian logic be modified so that each “step” in a logical progression reduced the amount of truth it propagated? The goal would initially be a logic that is calculable. So while we could take our lessons on how the logical system would be set up from how light interacts with matter, the resulting system would not be realistic initially. (For example: if a statement has “brightness” b, perhaps each inferential step discounts it by a factor k≤1, so that long chains necessarily dim.) Following the logical system leads you out of the logical system, however, since the logical laws are not perfect propagators of truth. The logic I am formulating here, while not realistic, leads into a real world.
Once upon a time, there was a monk who wanted to know where Space was.
So he meditated and meditated and meditated, until his mind reached the angels.
He asked the angels, “Oh Angels, where is Space?”
The angels replied, “We don’t know. But if you meditate longer, you will reach even higher angels. They might know.”
So the monk meditated and meditated and meditated, and his beard grew long and grey as he sat still, until he saw the higher angels.
He asked the higher angels, “Oh High Angels, where is Space?”
And the High Angels replied, “We don’t know. But if you meditate longer, you will reach the Highest Angels. Maybe they will know.”
So the monk meditated and meditated, until his beard grew down to his feet and turned white as he sat unmoving, until he saw the Highest Angels.
He asked them, “Oh Highest Angels, where is Space?”
And they replied, “We don’t know. But if you meditate even longer, you will reach Brahma, the Highest of the High, Creator of all the worlds. He will know.”
So again, the monk meditated and meditated, until his hair fell out and his skin sagged from his bones, spotted and pale with age. At last he reached Brahma.
The monk asked, “Oh Brahma, Highest of the High, Creator of all the worlds, where is Space?”
And Brahma replied, “I am Brahma! Highest of the High, Creator of all the worlds!”
For some, this would have been enough. But the monk persisted.
“Yes,” said the monk, “and… where is Space?”
Brahma realized the monk would not go away. He drew him aside, away from his choir of angels, and whispered,
“Look, don’t tell anyone—but I don’t know where Space is. You are asking a dangerous question. If you must know, go ask the Buddha. But go at your own risk, for you go beyond my domain.”
And so the monk rose slowly from his meditation. His body trembled with age, his steps were unsteady, but his will was clear. Luckily for him, the Buddha was living then, residing in a nearby town.
He reached the Living Buddha, sat respectfully to one side, and asked his question:
“Oh Buddha, the Well-Gone, where is Space?”
The Buddha replied simply,
“It is good you came to me, for no one can answer this question except one who has finished the Noble Eightfold Path. Space can only be found in the mind of the Saint — one who has followed the Way and gone to the end of the world with his mind. For he has found Space, and it is in his mind.”
Then the Buddha, saying nothing more, imparted this knowledge in silence. And at that very moment, the monk attained Enlightenment.
From then on, he lived in supreme peace, knowing the bliss of the boundless mind, until his death and beyond.
Once upon a time, there was a grasshopper that just sat around and breathed in the thick summer air all day and night. He would eat the green leaves that were everywhere—more than anyone could eat. He sat and sat, until the ant, who was sweating and carrying heavy food to his anthill, grew angry.
“Grasshopper, you fool,” said the ant. “You’re not going to have anything when winter comes.”
The grasshopper looked at the ant and smiled. “Come here, friend. I have things to tell you about breathing air and eating grass.”
But the ant wasn’t listening. He kept working and working all summer long.
Finally, the fall came, and the air turned cold. The grasshopper ran out of food. He didn’t move much, except to hop gently when the whim came to him. He didn’t cry for the cold, and he wore the same smile he had in the summer.
When the snow and icy winds arrived, the ant sat in his anthill with his wife and children. Sometimes he thought about that foolish grasshopper, but most of the time he was busy raising his kids.
The winters and summers went by, and other grasshoppers came and went. They were different, but every now and then, there was one that acted like the first foolish grasshopper. Once, the ant’s own son began to listen to a grasshopper and never returned to the anthill.
Years passed. One winter, the ant was old and began to fear death. He thought about all his work and wondered how he could bring his food, or his children, or his wife with him after death. These were dark thoughts, but eventually, he remembered that foolish grasshopper.
He thought about how the grasshopper smiled, even in the cold of fall—and it made the old ant smile a little too.
He did nothing then. He simply sat, breathing, and eating the food he had stored over the years.
In the end, he wished he had had a whole summer to breathe and eat and learn to smile.
The stonecutter’s pickaxe struck the rock. He felt the shock in his hands and feet; his mind was in his hands and feet. Every day he worked hard, splitting stone from the foot of the mountain. Workers came to carry away the slabs he cut, to be shaped into so many things. Each evening he brought home the money and merit his labor earned, to share with his wife and children.
As he grew older, the blows of his pickaxe echoed through his arms and shoulders. When the reverberation reached his head, he was an old man. His life was hard, but his work was good, and many people benefited from the stone he took from the mountain. When he died, the feeling of striking stone—the rhythm of his labor and the merit of his days—rose toward heaven.
A wild spirit saw the stonecutter’s mind ascending and said, “You are bound for heaven. What sort of heaven would you like?”
The stonecutter was a simple man. He had watched merchants pass by his house with carriages and soft cushions, servants and guards, good food and fine clothes. It looked like heaven to him. “I would like to be a wealthy merchant,” he said.
The wild spirit smiled and wove a spell of dream.
The stonecutter found himself reclining in a silk-draped carriage, eating good food while servants worked. Yet when he looked out at the rough people toiling in the fields and along the road, he felt uneasy. A princess’s carriage passed—finer still, with many guards whose armor gleamed in the sun—and regret pricked his heart.
The spirit appeared again. “I think I made a mistake,” said the stonecutter. “Could I be a king instead?” “I would not have you unhappy in heaven,” said the spirit. “Let it be so.”
Now the stonecutter sat upon a golden throne in a strong stone castle. Servants anticipated his desires, and an army of guards kept him safe. He ate splendid food and felt no fear—until drought came. People knelt before him, pleading for rain. Their hunger became his own. He was king, yet powerless.
The spirit appeared once more. “Well?” it asked. “My people suffer,” said the king. “I wish I could truly help them.” “Then choose again.” “I will be the Sun,” said the king. “I can warm the earth, restrain myself, and let the crops grow. It must feel good to be the Sun and give light.”
The wild spirit’s crooked smile flashed, and with a wave of its hand the stonecutter became the Sun.
He shone with joy. His warmth ripened the fields, and his light filled the world. This, he thought, was heaven. But soon he saw vast rainclouds gather, flooding rivers and drowning the crops. Anger flared in him—an angry Sun scorches all—and drought followed. Alarmed, he tried to calm himself, but his temper was too great.
When the spirit came again, the Sun said, “Then let me be a great raincloud—something even the Sun cannot burn away.”
The spirit nodded, hiding a chuckle, and waved its thin hand.
Now he was a mighty cloud. His emotions became storms. Wind lashed the trees, rain poured down as if from his own heart. Remembering the steadiness of his old work, he tried to master himself. The winds eased, the rain slowed—but the Sun’s fury burned hotter. The cloud swelled to shield the world, yet could not control the vastness of his feeling. Seeking steadiness, he looked down and saw the Great Mountain—immovable, enduring all heat and rain.
“I want to be the Great Mountain!” he cried.
And so he was.
The stonecutter became the Great Mountain—solid, vast, supporting forests and towns. Time stretched long before him. He felt his strength reach into the future, unshaken by storm or drought. Then a faint sting touched his foot. Tap, tap, tap. A little stonecutter was working there, cutting slabs from his body. The mountain felt each strike, a mild annoyance that never ceased. He watched the man’s discipline and remembered his own life, his wife and children, his quiet virtues. The mountain’s long calm was pierced again and again by that tiny rhythm, until he understood the lesson in each blow.
A storm raged on his southern face, a stonecutter tapped at his eastern. The Great Mountain sighed, and the wild spirit appeared—this time without mockery.
“My idea of heaven has changed again,” said the mountain. “I wish to be a stonecutter.”
The spirit nodded silently and waved its ghostly hand.
Once more he was a man, shouldering his pickaxe, kissing his wife and children goodbye. He trudged to his worksite and struck the stone. The vibration coursed through him, yet his mind was unmoved. He knew now that heaven had always been here—that wisdom and virtue together reveal paradise in the very place one stands.
And so the stonecutter’s mind grew light. The dreams of the wild spirit dissolved, and he rose to the highest heaven—where the bliss is no greater than the bliss he had already found in the work of his own two hands.
A foothold I can call my own, a place that would forever accept my step I wander on blank sheets of paper,
I wanted to write about that piece of empty space that is home to all
Dip the page in water, they say, and let the ink run by itself. A paper vase with animals primitively drawn Turning the vase in my hands, the animals run, bleeding, until the vase contains something. (Write something into the vase) writing curled round its inner walls, saying “The truth is no-w-here.”
now I etch it in wood carvings
the medium of the woods I wandered
on blank sheets of paper until I was accepted into the Hall of Trees.
in sizzling neon lights. they leave ghosts in my eyes. The soothing burn of whisky.
feminine silhouettes cast by burning "open" signs, my mind dies little deaths. they walk in reckless trance. A human lumbering, lurching towards another's flesh
The uglier I get, the more beautiful everyone else becomes.
Oh Death, the seductress! She will feel the sting in a feather— penned tattoo
of a thin moon.
Fight me; O Death I belong to combat.
lust is a leprosy; we hold our wretched skin to the fire for some comfort in pain. There is worse than pain, dear one.
caught in the mind's cobwebs where pain becomes a helper
Come to me, lost soul, For I am the absence of truth, and I will hold you.
The mind needs truth like the body needs medicine.
There are very complex theories. A computer operating system is one of the most complex creations, with more moving parts than a commercial airplane. Airplanes fly, thats nice, but operating systems do something harder: they engage the human mind in every way. There are operating systems for household chores, entertainment, most kinds of work, and social existence. If there were a place for complexity, the internet and its administration would be this place, or non-place. Complexity is what protects our privacy… whats left of it.. in the form of encryption. In this case, and many others, complexity is there almost for its own sake. It is a gatekeeper. The complexity of encryption is not related to the privacy of our homemade pornography starring “me”. Encryption is dark for the sake of being dark. We need this darkness to make fools of ourselves in private. Computers are just embodied mathematics, after all, and mathematics has become the gatekeeper for most things. Science is one of few generalities that claim mathematics is not for its own sake as a gatekeeper. Most importantly, the complexity of mathematics in Science is both the gate and the reality. The reality in our minds: the conversion of ideas to things, and things to ideas. Mathematics is the night sky, the simple darkness in which the universe can play out its complexities. It is at once the mountain one must climb to reach the sky, and the sky itself. The arid peak, from which the world is a map, and new stars can be contemplated. The point of dependence of sky on land, and of land on sky. The recognition of complexity is a simple recognition.
The parallel postulate was found to be optional when non-Euclidean geometries came under earnest exploration. This was a paradigm-shift, and a breakthrough in mathematics, yet historically people often found it to be a failing in mathematics to the point where, too late in the game, great mathematicians such as Lagrange were still trying to prove the postulate. The reason we wanted Euclidean geometry to be true is it reduces space to a quantifiable reality, so that at the moment of breakthrough, the favorite non-Euclidean geometries could be translated into Euclidean geometry using a metric. Now, we can have very strange kinds of vague metrics for gauging distance. The basic difference in these early non-Euclidean geometries was that there could be more than one parallel line going through the same point, with respect to another line. In other words, the parallel postulate was false for these other geometries.
Euclid himself was so exceedingly careful in intellect that he kept the parallel postulate out of his exposition of geometry, out of his innovative axiomatic form, until he felt it was absolutely necessary. This means that the beginning of Euclid’s first textbook is true in all geometries, and is called “Neutral Geometry.” These precious first theorems are universally true, but this was just not enough for Euclid or his line of mathematicians. They wanted more to be true, so, perhaps under the weight of his contemporaries and ancestors, Euclid abandoned his misgivings about the postulate and asserted Euclidean Geometry as universally true.
How did he make this assertion? You could say his assertion was existential. He asserted that there is only one line through a given point with respect to another line in the same plane. He made this choice out of what we now know to be many options. I may be talking about free will, but not quite. I am sure Euclid believed his books to be about something true, and only his work failed, not reality. He was somehow persuaded or convinced that the parallel postulate was the right choice, and not without examples to the contrary. Geometry on a sphere is not Euclidean: you can have a triangle with three right angles using the line at the equator and two lines going through the north pole. We didn’t know the world was round back then you say? Some scholars back then did, since its circumference was being estimated by scholars as early as 2,000 years ago in Alexandria. Aristotle’s model for the Earth is inconsequentially different from a sphere, though the model appeared flat. Aristotle instructed Alexander the Great to conquer to the edge of the world and, by continuing, he’d end up back in his home eventually. Now, scientists seem to agree that gravity bends space out of shape, and is not Euclidean. The debates of brilliant minds back then, as well as now, were varied, of course, and the over-simplifications from our compulsory education about flat-Earth vs round-Earth, Euclidean vs Non-Euclidean are stark, foolish and in service of the illusion of progress.
The thing to notice is that the alternatives between Neutral Geometry and Euclidean Geometry was a kind of pluralism of parallel lines through the given point, and with respect to another line. This gave rise to a pluralism of quantifying or qualifying distance. Similar to asserting that the only alternative to Being is Not-Being, and the vagaries of cloud-gazing was out of the question. Now we have whole specialized languages to describe the gap between Being and Not-Being: such as psychological hallucination. Leaving out the modern hallucinations of distance, and the looming history of the field of number theory (eventually abandoned by the Greeks, and later connected to geometry in Descartes), how do we choose between these options? Enter Rhetoric, left stage.
Aristotle’s definition of rhetoric was that it was an Art of persuasion, not a Science, even though the above shows how deeply dependent Euclid was on Rhetoric.
So, when I say that Rhetoric is the stuff that connects our planet with the “universe” (another overly-ambitious grab-word from Science), I mean it scientifically, in the Art of choosing how to define distance, because the most important practice of Science, that gives it its importance, is this territory-grabbing from Rhetoric. And Rhetoric is merely the guard of its master: Poetry. In this essay, I have shown how Rhetoric orients people in understanding the gaps between disciplines. Rhetoric is a way of understanding things: the Sciences, Philosophy, and even translation and other parts of Poetic discourse.
Once upon a time there was an emperor who really loved clothes. He wore clothes when he woke up in the morning, then before breakfast he changed his clothes, then before lunch he changed his clothes again, and before dinner and before bed. Then, In the middle of the night, he made instructions to wake him up so he changed his clothes again and went back to sleep.
The clothes manufacturers were making a lot of money from the emperor. A pair of skilled thieves saw an opportunity, and made a plan. They presented themselves to the emperor as master clothiers and told him they would make clothes so fine that crude people could not see them. Indeed, only those worthy of their profession would be able to see the clothes. They called the outfit “Mathematics.”
The emperor was overjoyed by the prospect of such a fine set of clothes, and gave the thieves the royal clothier’s workshop, all the silk and golden thread they would need, and of course the fee was extravagant.
Now the thieves went to work. They moved the looms, but the looms were empty, they threaded needles with no thread, and all the expensive cloth and thread was hidden in their sacks in the back of the workshop and transported to safety every night.
After a while the emperor decided they had done a lot of work by now, and sent the royal poet, a man who was uncommonly wise, to go check on the thieves’ work. The royal poet entered the workshop and asked to see the thieves work. The thieves behaved as though they were presenting fine clothes, but they had not clothes in their hands. They were showing him nothing, and the wise man decided the thieves were thieves, but these were very skilled thieves indeed. They described every feather of every crane in flight, the color and shape of every blossom, and the intricacy of patterns. Unfortunately, the wise poet was persuaded that the clothes were real, and that he was unworthy to be the Royal poet of the emperor.
He began to sweat, because he would surely lose his life if the emperor knew his poet was a fraud. “Oh what fine clothes these are. Yes these clothes, “Mathematics” as you call them, reveal patterns that show such intricacy, they go beyond my 4-dimensional imagination.” The thieves smiled in just the right way, and nodded with just the right amount of satisfaction so as to continue fooling the wise man. They were indeed most clever thieves.
The Royal Poet returned to the emperor and lauded the “Mathematics” clothes to the highest degree, and made sure to persuade the emperor, although he had no idea what the “Mathematics” clothes looked like.
Finally the thieves announced the “Mathematics” clothes were finished before the emperor. And offered that the Emperor should arrange a parade and show the “Mathematics” clothes to all his subjects.
The Emperor did just that, and when the thieves showed him nothing at all, and described the “mathematics” clothes, the Emperor was no match against the thieves description and the confirmation of the Royal Poet.
The Thieves helped the emperor to put on the “Mathematics” and the parade began. Everyone was looking at the emperors private parts and cheering as best they could, throwing flowers petals confetti, sweating at the problem of not being worthy of their various professions. It looked like every professional was going to have to wear “Mathematics.”
Luckily for everyone, there was a tradition in this part of the world of listening to children. There was a common folk belief that children were close to the Source of all people; sometimes children could say things that were very important, even more important than the emperor himself, or so they thought.
And in an lull of the fake excitement, a child burst in front of the parade and said with glee “The Emperor is naked, I can see his mushroom!”
Everyone realized the child spoke the truth and the emperor had been fooled. The thieves were long gone by then, but before they left they explained the clothes to some foreigners, who also believed the thieves, and now there are parts of the world, who don’t listen to children, and walk around naked.
A master of lies... He will know what kind and mode of deception is taking place at any moment set before him. He will know the lies of magic are the same as the lies of science. He must have detailed knowledge to the exactitude of a mathematician on why mathematics is a falsehood. A master of lies is most persuasive, and he knows how to hide his persuasion. He knows what purpose his lie serves, whether good or evil, beautiful or ugly or simply complex in how truth and lies are woven into the fabric of space-time.
He has found the crack in the wholeness of his being, from where he will shine his lights and darknesses. He will know why he shines a darkness instead of a light, what the dangers are and how to avoid them. It is not mastery if his lies cause him harm. In other words, mastery over lies is not different from a mastery of truth.
And a so-called master of truth only requires blind faith. Waving the flag of truth, he will be insulted if called a liar, as if lies were not integral to our existence on earth. He is blind to his own lies and manipulations, believing them to be right and true.
"Errat ergo sum" -St. Augustine
Come with me, I will guide you through the nightmare land of lies. And when you have crossed this valley of shadow, you will have a mastery that is more valuable than any fact.
You are a part of my home We bond sometimes when I find you. This life, you were on the edge the universe I walk to you, so I can be with a piece of my home Then I have to leave Because my home broke My home grew and shrank It changed forms I see a piece of my home in that cloud I try to be there, drifting, changing Until my home is not there anymore And I must walk to find another piece I do this drifting To keep my heart whole My heart is mended, as long as I walk in search of a glimmer that was part of my home
As a child I was interested in the word few. I was not interested in figuring out exactly what it meant; instead I was interested in understanding its potential. What could it mean? I enjoyed playing with modifiers such as “quite a few” which seems to mean the opposite of its intended meaning: the word few supposedly comes from the PIE pau- and from there the word paucity derives. It means a small but numerous number. It means “many, yet not many” to put it without delicacy. “Quite a few” seems to increase the “numerousness” of the number involved in few, but maybe it only emphasizes the importance that it is not only one or two…?
I remember thinking about this, and smiling. This word made me happy. When I came to college, however, I learned from my friends that the word few meant exactly “three.” I did try to argue that the word was meant to not be exact, but there was a certain force in the precise claim, and no-one listened to me. Interestingly, the word few is related to puerile. (the etymology is coming from etymonline.com) My arguments might have sounded immature to the ears of my friends. What use is a word if we don’t know exactly what it means? And if I don’t know exactly what it means, and this other person says he does, why should they listen to me?
My reaction was suppressed anger. By the time I was in college I was used to this sort of thing. I had a certain joy when people used a turn of phrase or said things that had a lot of possibility (especially when the speaker was a mathematician), and it seemed everyone else frowned on this joy. Maybe my feeling was stupid, or immature, or even evil, but I buried the determination to make the argument for a less determined definition of few, and many other things, in the face of everyone who thought they knew so much. It felt like such a small, trivial thing. But it was one of the last things I enjoyed about language at Earlham, where writing was paramount. Why couldn’t we have at least one vague word, a word about not knowing the exact number of things, but still being able to to communicate the information that it was more than two, yet not very many. Wasn’t that something we ran into all the time? Or were we supposed to count everything before we spoke? My reaction was far from laziness. I perceived this difference in my ideas, really in my temperament—what made me happy, as something I was going to struggle with my whole life, and correctly so.
Of course the word few does not at all mean “three.” Even though I did look it up at the time, (and the dictionary I consulted did say the word few meant exactly three, much to my dismay), I have been to several other sources years later. And written a book defending vague language, to a mathematical audience. The struggle continues… but at least I’ve got my finger on the problem now.
Tycho Brahe used mathematical and scientific instruments, some of them newly invented, to correct ancient astronomical measurements. But his main tool was an aura of faithful observation. He thought he could explain the movements of the stars in an objective way, and that was his rhetorical position from which he made his observations. It is a rhetorical position, because there is no scientific basis for believing our observations are objective, no matter how mathematical they are, unless the earth is an immovable point in the center of the universe. If the earth is spinning and in motion, until we completely understand how it is moving, we wont understand our own observations. I am merely referring to Einstein’s theory of relativity: there are no unmoving points of observation, and so all of our observations are relative. If we understand the movement of the Earth (or a satellite like the moon) completely, then we can mathematically compensate for that motion to obtain objective measurements. How are we going to completely understand the movement of the Earth? By recording its movement from the point of view of the stars, of course. And how to we know what the point of view of the stars is? by recording their movements from the vantage points available to us: the Earth. You can see the circularity here. We can’t record the movement of the Earth without understanding the movement of the stars, and we can’t record the movement of the stars without understanding the movement of the Earth. Unfortunately, without records of either the Earth or the stars to begin with, we can only make guesses of understanding, and see how they match up with our faulty observations and records.
Where does that leave the shift from an Earth-centered universe a solar-system that moves in a universe with no center? It leaves us knowing less than we knew in Aristotle’s time. We can fly into space and make some impromptu observations of the earth spinning, but how do we know it isn’t us that is spinning so that the stars are more still, making the earth appear to spin? We would have to know how to be perfectly still in space to know how things are moving, but we can only know that relative to other things like stars or planets, so we don’t even know if one day we will shift back to an earth-centered model of the universe.
The usual argument scientists make against this type of reasoning is to make things more complex, as though that will wash away these doubts. It doesn’t really do that except rhetorically. It must be admitted, at least until we have found a point in the universe that doesn’t move, that the modern scientific models of the universe are based ultimately on rhetoric, whether it is a rhetorical air of faithfully measuring things, the rhetorical air of using mathematical symbols and formulas instead of words, or the rhetorical air of claiming that to know more is to have a more important opinion than others, so that a simple-minded analysis like mine is unimportant.
All these postures are rhetorical in foundation and nature, and so there is not much reason to draw a stark line between people who believe this or that thing, and use this as a cause of belittling, hating and shaming people (this runs the spectrum of issues such as anti-vax, flat-earthers, or whatever else). Scientific ideas are just ideas, including the our geometric or numerical ideas of space and time, and our ideas of logical reasoning, which are also fundamentally rhetorical. When Bernie Sanders says something in the order of poverty being a contradiction in the richest country in the world, he is mainly referring to a failure of Americans to think rhetorically. Instead the way to persuade people is to make logical claims, or so we believe nowadays, and this is a deep and purposefully fostered flaw in the political process in the USA. In this, the scientific community and their rhetorical posturing does us a disservice.
I am extremely fond of Borges talking about the attitude of Argentinians on literature, and his comparison with the corresponding attitudes in the USA. According to Borges, Argentinians tend to think a book that won a literature award might still be a good book, in spite of the award. Of course, this attitude is quite out of the question in the USA, where everything needs official publication, awards and certifications, and certifications of certifications, that let other people tell us who to trust and who to listen to. This Argentinian attitude towards books (and ideas) is basic to a society that is not thought-controlled.
the mosquito With only the tiniest scrap of love makes so much life so much pain, hunger yes but life, free life on the wind Because we all need a whining reminder of freedom For their resilience I am grateful
And the cockroach Who carries on no matter what And carries on well, preserver of life Persistence in the ordeal of life, the sufferer Because we are all sufferers For their will I am grateful
And the spider Who understands power better than any The fierce trapper, the relentless She who knows the ways of extracting our very life essence She can teach us She is not finished teaching us For her wisdom I am grateful
The worm Who’s blindness is a gift in the darkness Who can breath with his very skin Where there is no air, only earth The worm is the body incarnate Because our bodies are a gift For their bodies I am grateful
The virus The virus is the word itself How is that so you ask? Ask the virus, and it will point you to how it does things Because it spreads like fire And causes unrest, dis-ease, dissatisfaction It is because the word spreads that it can shape the world Sperm is a virus, did you not know? Without the virus we would not be awake at all, not even to dream Neolibralism was a dream, and the virus shook us, will we wake? It is because of the virus we can do good, we are goaded awake For this awakening, I am grateful
Andrew Nightingale asked the question for discussion:
Why is vaccination not identified as a kind of homeopathy? Why are these two things on two sides of a stark dividing line between science-doubters and science-faithful? What makes people unable to make this connection?
Christian G Meyer February 27: Vaccinations are effective. Homeopathy is nonsense.
Andrew Nightingale: The basic principle of Homeopathy is that creating symptoms of a disease will strengthen the body against the disease. “like cures like” This is exactly what a vaccination does. So if one is nonsense so is the other, and if one is effective so is the other, at least sometimes. Your “contribution” has no argument or insight. It is only the regurgitation of the belief I question.…
Frank T. Edelmann February 27: Dear Andrew Nightingale personally I see no real connection or similarity between vaccinations against COVID-19 and homeopathy. The newly developed coronavirus vaccines have ben thoroughly studied and proven to be highly effective to prevent an infection with the virus. Although there might be some formal similarity between vaccination and homeopathy, the main problem is that homeopathy completely lacks this positive proof of effectivity. As a chemist, I can easily oversee that many homeopathic medications do not contain a single molecule of active ingredient. Thus in my personal opinion it is dangerous to awaken any hope that homeopathy could be helpful in the fight against COVID-19.In this context please see this relevant link entitledIn Germany, a Heated Debate Over Homeopathyhttps://undark.org/2020/03/16/homeopathy-globuli-germany/
Andrew Nightingale February 27: Hello Frank T. Edelmann thank you for the contribution. The article was a good read. There are many things I do not know. My specialty is mathematics, with a duffers bag of philosophy. I can with authority criticize the statistical analysis that is used to establish a causal relationship in a scientific study. As a mathematician, I can say that what you define as the cause, and what you define as an effect, has a great impact on how persuasive the statistical analysis can be. Before we can speak of there being a causal relationship at all, we have to establish what the cause is, that is, what is the thing that causes, and what the effect is. Unfortunately I can only speak as an expert in this narrow frame, and unfortunately, as much as that might pain the certainty of science, we are also in the area of language–whether a community of scientists are successfully talking about a real cause or effect. You say the thing that causes we want to talk about are active ingredients, and I would agree with you, as you know much more about that than I do. What these are can be established, and once it is established, we can do a scientific study to establish a causal relationship for an effect. I have no doubt about the causal relationship between vaccines (the cause) and their effects (immunity to a disease). I am sure so far I have not said anything surprising. But what is the effect? Vaccines do not make you well, they make you sick. They induce the symptoms of a disease. That seems to me to be the same type of effect that homeopathy promises, whether or not they are administered incorrectly into the ear, or manufactured incorrectly with only sugar. IF they contained active ingredients and were administered correctly, like vaccines, there would be no difference, in principle, between vaccines and the principle of homeopathy. Both make you well later because you have induced a lesser sickness. That is my only point, and it is a point regardless of the power of causal relationships established by scientific rigor. It is a point about what the thing that causes is, and what we could call an effect. With that in mind, the proof that homeopathy has scientific backing is vaccines themselves. The problem is not that principle of homeopathy doesn’t make sense. The problem is that the principle of homeopathy seems to be unrecognized for what it is by scientists, and is left to flounder without being regulated in how the medicines are made and administered and studied. Homeopathy is merely in bad shape because it has been ignored. If it were not ignored, there would be no political cost in fighting against it. So again I reiterate, why has it been ignored? Is “like cures like” a principle that we, as scientists, are against politically, not reasonably? If so, is the the political cost to science worth it?
Frank T. Edelmann February 28: Dear Andrew Nightingale many thanks for your detailed response. “Homeopathy is merely in bad shape because it has been ignored.” This is definitively not true for my country, Germany. Many Germans trust in homeopathy. After all, homeopathy was “invented” by the German physician Samual Hahnemann:https://en.wikipedia.org/wiki/Samuel_HahnemannHowever, as mentioned in this Wikipedia article, homeopathy is a pseudo-scientific system. Personally I believe that today the popular “like cures like” is just an empty phrase with no scientific background. In this context please see this relevant article entitledLike Doesn’t Cure Like: Homeopathy and Its Fake Medicationshttps://news.itmo.ru/en/news/6478/P.S. This is just my personal opinion which I’m not going to change. Thus we should not engage in any emotional discussion about homeopathy. 😊
Andrew Nightingale March 1Frank T. Edelmann: thank you for the detailed references. I am talking about what “homeopathy,” or some other word if you like, could be, and you are talking about what homeopathy (in this case another word is not appropriate) is now. And I agree with you that the state homeopathy is in now is not acceptable. The article entitled “Like does not cure like” is interesting because its title does not match the evidence that it presents. I believe this is a rhetorical or political move, to attack a principle indirectly by attacking homeopathy in every way it can and not the principle “like cures like” at all. I have no emotional attachment to homeopathy, so if you want we can drop that term now, so that discussion can continue. “like cures like” is definitely an oversimplification, and curing the virus with a vaccine most definitely used a principle similar to “like cures like.” You are right that “like cures like” is scientifically an empty phrase nowadays, because Francis Bacon and before him Galileo made it so, by changing the basis of knowledge by inviting us to “look closer”. Looking closer is good for knowledge, but it also is good for noticing differences better than likenesses, which turn out to be similar to one of Francis Bacon’s Idols. The Idol of the Cave might be the one, that what we see when we look closer is no longer communicable with the words people know, communicable about the “axioms” or general principles we can discuss. Words have been undermined. There is much work and exaggeration on this point of philosophy of science. My point here is that an axiom like “like cures like” is what is curing the virus Covid-19 in the form of vaccines.
Christian G Meyer March 1: Please take some time (I guarantee, it´s worth every minute and you won´t regret it a second) and watch some (highly informative and highly entertaining) videos of the wonderful former very famous magician James Randi, who extensively challenged paranormal and pseudoscientific claims, including homeopathy. He was the co-founder of the Committee for Skeptical Inquiry (CSI), and founder of the James Randi Educational Foundation (JREF). Homepathic dilutions are up to 1:1.000.000.000.000.000.000.000.000.000.000.e.g.https://www.youtube.com/watch?v=0jqP_1beVXQhttps://www.youtube.com/watch?v=sCYvOgBaEY8https://www.youtube.com/watch?v=SMukj31qw1Uhttps://www.youtube.com/watch?v=lmOfEoDcjksThere are many more videos available.Decades ago he offered a reward of 1 million USD if someone could prove that homeopathy is effective. The money never has been claimed. He debunked the spoon bender Uri Geller and showed how spoon bending works. It is recommended to read his exciting Wikipedia biography. I was so happy to meet him in 2005 and to spend a day with him. He died in October 2020 at the age of 92, guess from what.https://en.wikipedia.org/wiki/James_RandiBy the way, vaccinations have for decades now declared by the CDC as the most effective public health measures ever, far more effective than improving car driving security, working security, anti-smoking campaigns and many other measures. Vaccinations are always Number 1!
Christian G Meyer added a replyMarch 1It is interestig that in old Rome it was recommended to eat the liver of rabid dogs to be protected against rabies – a pre-stage of the rabies vaccine which is used today.…
Ligen Yu added a replyMarch 1Andrew has presented a very critical question about modern medicine, that is the false causal relationship. What the medical scientists think to be causes may not be the real cause, and what the medical scientist think to be effect may not be the real effect. And that is the root cause of the replication crisis in medicine:https://en.wikipedia.org/wiki/Replication_crisisGiven the human being as a super complex system, there might be millions of causal conditions that contribute to a medical condition with different weightage, and we really need to pay great caution when we ascertain any scientific causal relationship between a few causal conditions to the medical condition.
Andrew Nightingale added a replyMarch 1I have said enough, and plenty more in my own published work that is related. I hope the discussion will continue without me. Peace!
Has justice has become sense-making? The many senses of the over-worked concept of justice allows it to generally sound like a good idea to the atomized American. Even Bernie Sanders repeats the term Justice, as if forgiveness and mercy were the irrational ways and means for religion. (separated from matters of the state) I think Americans in their deep mind control bubble crave sense-making. They are confused, afraid and overworked. Their “education,” their language, their intellectual preoccupation with sex (including gender), are all reductionist. I generally try to approach this problem by looking at the logical positivist project to refine language and how that reduces larger things like houses, feelings, and communities into talk of a smaller, more atomized reality. So I focus on vagueness in my work because people in America badly need a way to synthesize information, houses, feelings, communities, etc. The effect of the English language is felt in everything else.
But vagueness is the linguistic approach; how to move to a political approach? I think people lean on some products of the Social Sciences to conceive the neolibral “individual” and contrive a linking of hands with others to form a political community, the same way electrons link atoms, and the mind senses a great synthesis of atoms into a house. Even if that same mind doesn’t believe in things anymore, being told that everything is actually atoms, or subatomic particles, or quanta, etc. I originally approached the problem linguistically because it seems more fundamental. Justification using pseudo-scientific “experiments” with statistical language dominated the Social Sciences for a long time. The linguistic style of statistics was the persuasive force, though now, qualitative research diminishes that force somewhat. In any case the view that mathematics and therefore statistics are languages incited me to offer vagueness as a recognized form of synthesis.
Vagueness, although a very useful and widespread linguistic device, is not appropriate for politics and the Social Sciences that study politics. Media is the persuasive force in politics. For example, right now everyone is worried that the government will fail to raise the debt ceiling which would effectively result in total failure of American governance and society. The cost to those in power would be a complete loss, and American politicians know this, so under the cover of this “crisis,” created by the media, they carry on pursuing their least popular policies.
Media outlets employ Elite People like Anand Giridharadas, who write things for the New York Times and try to make the argument that we shouldn’t listen to just anybody of the
114 percent of Americans now having their own podcast, it is not easy to choose the one with the best title. But I’d go with the journalist Chris Hayes’s “Why Is This Happening?”
(Why do Trump Supporters Support Trump?)
Were there a German word for emotion-question (and it turns out there is), that title may be our era’s Gefühlsfrage. As people reel from crisis to crisis, outrage to outrage, this Gefühlsfrage hangs in the air and creates space for writers.
The urgent desire to regroup our atomized communities and our podcasts to the tune of the New York Times is evident here. Not that the New York Times wants us to really regroup, just enough for us to keep coming to them for their information-framing. Actually, we need space for the common writer, and Mr. Giridharadas’ attempt to rhetorically close that space is unhealthy politically. We need synthesis but not to the tune of the elite bringing us more Barack Obamas and Hillary Clintons, which are, unfortunately, the optimistic outcomes.
For the common human’s politics, instead of academic disciplines, we need another term/concept for synthesis. Justice seems to be the general answer to the Gefühlsfrage, but what is justice? Not a question I am prepared to answer, but I will make a guess that it is what is best for the state, in the same way we have an idea of what is best for ourselves, we extend that to the state, and that is justice. One of the oldest senses of justice was “Eye for an eye” which involves taking action in a symmetrical way to how we have been wronged. To some of us, justice means: if there is a problem, if we have been wronged, the “answer” is an action that hurts the wrong-doer in like kind. This kind of justice is obviously unachievable, there are many wrong doings that have no symmetrical punishment (unless you are completely taken in by capitalism: How much is unjustly getting cancer worth? Being cured of cancer?), but I think this old, violent, barbaric definition of justice resonates with the beleaguered people of America.
Americans feel wronged, and justice is how to act on the world so that it makes sense, a very material sense. Justice is the proposed answer. Just look at the amount of work in a court case to accomplish a minuscule amount of worldly justice. It is plainly not worth it except for the most grievous acts, even so, there are too many severe injustices. Any real-world event is too complex to set “right”, and only the ones that get attention are addressed, so every thought on how we have been wronged is clamoring for a like or a share, etc. What is the goal of Justice? We get one thing right, after great outcry, what next? There are too many things wrong, and that is the way it will always be.
American “education” can be found especially in American movies, where a keen sense of justice is fed with powerful images and stories, drawn from previous cultural mythologies and reframed to raise Justice to the highest political ideal. Once we are educated in this way, there is a terrible, schizophrenic dissonance between the expectation of Justice and the reality of American life. This causes a great deal of pain for the common human. Everyone’s individual fight for “Justice” feeds everyone’s own concept of being wronged, and Justice, even more.
For politics, I would propose another concept that does no cutting out of people’s eyes: the concept is Rhetoric, and in this case, I direct you to Deirdre McCloskey‘s works. Western philosophy tries to block up rhetoric as something for the sophist who isn’t interested in the truth, as if the truth and its persuasiveness could be separated. There is no separating Truth from its natural sweetness (and Dierdre agrees, read her wonderfully brief book on writing!). Here Dierdre writes “they are egg and yoke in a scrambled egg.” or “their differential equations are nonseparable.” Sweet language, such as poetry, expresses the truth best (not mathematical or statistical language).
In my next essay, I attempt to demonstrate the presence of rhetoric in logic, since logic is the foundation of mathematics. I will defend rhetoric against the statisticians, and attempt to show how rhetoric binds and surrounds, synthesizes, the worlds of ideas.
The goal for today is to prove that magical thinking is rampant in mathematics. First of all lets define magical thinking. I would say that magical thinking is a kind of metaphorical thinking, as in the metaphor “My heart is the sun” only with the added idea that writing these words/making the metaphor exerts towards making the metaphor true to some degree or in some sense. Magical thinking is the claim that saying “My heart is the sun” actually warms my heart.
Now the way that mathematics uses magical thinking is to start with a metaphorical idea of difference. For example, the difference between a “raven” (1) and a “writing desk” (2) metaphorically (not actually) is the difference between the “north star” (3) and the “form of thinking called questioning” (4). It is fairly intuitive that the difference between (1) and (2) is different from the difference between (3) and (4), but mathematics amalgamates all differences together into one concept with metaphor. And it is a particular kind of metaphor that asserts that difference actually works that way.
Even though 3 and 5 are less different (2) than 3 and 9, (6), these differences are not taken into account in the traditional mathematical symbol for difference, the . Traditionally 3 5 just as much as 3 9, so the identity of difference, , is enforced.
Mathematics asserts an ultimate concept “Difference” that is universal—it works for any situation where there is difference, making any difference “complete” and it does so by metaphorically joining disparate differences. Hence, it falls under my definition of magical thinking.
I am doing the opposite of what Derrida did with his Différance. Derrida added senses to difference, or conflated, allowing it a history and to belong to language, I am suggesting that we subtract, or better divide utterly Difference into differences.
The rest of the sciences follow suit, of course, since mathematics is the language of the sciences. My advisor for my M.S. in mathematics once said “Mathematics is the Poetry of the Sciences.” I would add that Mathematics tends to obfuscate the surprisingly obvious; it is a process that converts potential knowledge into actual knowledge. Poetry deals with knowledge that is naturally obscure, so that beautiful language can be mysterious, profound, even terrifying.
People often think that vagueness is bad, a kind of darkness that can never be fully dispelled, while distinction is hailed as the clarifying answer to vagueness. Here is how the reverse is also true: Vagueness is the light and distinction a darkness.
The distinction I pick is not random, but an important part of all other kinds of logical distinction—the distinction between the “if, then”: “→” and the “conclusion” symbol: ⊢. ⊢ is ambiguous, however, and can mean other things such as assertion that a proposition is true and not just being named, or to assert in a metalanguage that the following is a theorem in the object language. Used in our sense here, the good property of the “→” is “true” and the good property of the “⊢” is “sound”. The distinction goes back to Aristotle. The main point is that if we do away with this distinction, call these two symbols the same, an interesting insight can be made—that a sound argument:
A
A→B
⊢B
Can be represented without the ⊢ as follows: [A AND (A→B)] is logically equivalent to [A AND B], so that the conclusion [A AND B]→B is merely a deduction of A from [A AND B]. Allowing a vagueness between → and ⊢ reveals what logical deduction is—it is a cut from a larger whole, e.g. logical deduction is the act of drawing a distinction from the larger [A AND B]. With the introduction of the distinction between → and ⊢ this is concealed:
A
A→B
⊢B
cannot be collapsed into [A AND B]→B. As promised, vagueness reveals and distinction conceals, but not just any concealment, here we have a concealment which allows distinction to reveal, since this distinction is at the root of any further logical distinction.
I’ve said it many times already, in many different ways. I don’t think these insights are niche… the elemental insight is nice to have, but it can discount what I am trying to say if people prefer the realist-like march of nouns in writing. I think allowing honest questions, even leading ones like Zeno’s line of questions that push the calculus into absurdity, really would heal a lot of minds, and quickly, like wind sweeping over each of us as grass in a field. it is quite a general change, an all encompassing one.
Thought for 24s
ChatGPT response:
I think you’re pointing at a change that’s bigger than epistemology and smaller than ideology: a change in what a mind is allowed to do in public.
In an answer-culture, the social rule is: arrive with a noun, a verdict, a position. If you don’t, you’re weak, evasive, uninformed, unqualified. So people learn to counterfeit certainty. They learn to talk like they’re finished. And that performance quietly becomes the definition of “adult.”
What you’re proposing is a different social rule:
A mind is permitted to be in motion. And not-knowing can be competent.
That single permission has downstream effects that really could be fast—because it attacks shame at its root.
Why “even leading questions” matter
A leading question can be a trap (“admit my frame”), but it can also be a probe: the Zeno move, the thought-experiment move, the push-the-formalism-until-it-confesses move.
Zeno’s questions aren’t dignified because they’re “fair”; they’re dignified because they’re honest about the pressure they apply. They reveal the hidden costs of our comfortable nouns (“motion,” “distance,” “instant,” “continuous”). Calculus doesn’t come from being polite; it comes from letting the question break the old picture.
That generalizes. In everyday life, people carry brittle pictures: of love, success, safety, masculinity, sanity, nation, God. A culture that allows real questions (including uncomfortable ones) creates a normal way to revise pictures without humiliation.
The healing mechanism (why it could feel like wind)
Not because questions magically solve things—but because they interrupt coercion:
coercion by the demand for closure
coercion by false binaries
coercion by “what everyone knows”
coercion by the fear of looking stupid
When “questioning intention” is socially safe, minds stop bracing. They stop tightening around identity-answers. They can breathe again. That’s the speed: the relief is immediate, even before any “solution” arrives.
Your “?” operator fits here as a civic move, not just a logical one: a sanctioned third stance between True/False that says:
This is not to be concluded yet. This is to be held open without disgrace.
What changes, collectively, if it catches on
If enough people live this, I’d expect:
Lower temperature public speech. Less certainty-performance, less moral theater, fewer people talking to win the room.
Better disagreement. Not nicer—more precise. More “Which part?” “Under what conditions?” “What do you mean by…?”
Weaker manipulation. Because manipulation depends on locking the frame. A question-trained public refuses the frame.
A different education. Students judged by quality of inquiry, not speed of answers; “I don’t know yet” becomes a mark of seriousness.
Faster personal repairs. Relationships improve when people can ask what they actually mean instead of defending what they already said.
So I agree with your scale claim: it’s not niche. It’s civilizational plumbing. Change the question-norms and you change what kinds of selves can exist without panic.
If you want one sentence that carries the whole thing in nouns (your “realist march”):
A culture that rewards questions over answers replaces certainty with inquiry, shame with revision, and control with attention.
There was once a man who looked in the mirror too much, though not in a soul-searching way. He was not interested in his wrongdoings, nor in whether he could hold up under his own gaze. His thought and judgment fell on the contours of his lips and the shape of his eyes.
When he looked into the mirror he found no good angle, and yet there was always a glare in his eyes that seemed both hollow and angry. No matter what he did with his mouth, his eyes, or the way he held his head, an evil look followed him.
Unfortunately, when he looked away from the mirror, his real features were so designed by the Maker that he wore a carefree, proud expression—so long as there was no reflection to sabotage him. He wanted to be attractive, and that was as deep as he went with the hours he spent obsessed with his face. And when he saw himself in a selfie, he could not believe his own beauty, because he lacked the simple education that would have explained how light lies in glass.
Mary became acquainted with this man, whose name was Isildor, by chance. He looked at her with fire in his eyes, and she liked his look. She approached him and invited him to dinner at her place. Isildor was so shocked he fumbled out a yes. They exchanged numbers, and Mary was gone before he could undo himself.
At her house there was music and candles. The table was low and they sat on cushions—her perfect plan to make the table a bed at the same time. The beautiful man sat as if in a spell while she brought out a three-course dinner, complete with éclairs for dessert. In truth, he was in a spell because he had taken a couple shots of whiskey before arriving.
Mary’s sparkling conversation—her large eyes brightening when he smiled—was almost lost on him as he poured himself red wine. Yet he found himself kissing her, hands rising as if by reflex, and she drew him close. Their love was quick and hot, and she was satisfied completely.
Isildor lay contentedly, sweating naked in Mary’s arms, until his obsession returned. He jerked upright and clumsily gathered his clothes while his head swam. Mary tried to soothe him with caresses and kind words, but he recoiled from comfort as if it were danger. Shirt half-tucked, he thanked her for her hospitality and wiped lipstick from his mouth with his sleeve.
A day passed. Mary called him in the evening, while Isildor was staring at his own (to him) hideous features.
“Hello, Isildor?” she said, doubtfully.
He kept his eyes on his reflection as he spoke into the phone.
“Yes, Mary… I hope you are well,” he replied with stinging formality.
“I’m okay… Did you want to call me?” she asked directly.
“Yes… yes, very much,” he nearly stuttered.
“Then why didn’t you?” she asked, trembling.
At that moment Isildor saw his face change in the mirror. He was beautiful, and Mary stood beside him. Flashes in the glass showed them turning in a slow dance; then he was kneeling to ask her hand; then they walked the aisle as bride and groom. As the flashes came, they grew more distant, more vague—like pictures taken long ago and poorly kept.
He reached for these beautiful images, but they vanished.
“Mary?” he said, rough with feeling. There was no answer.
“Mary!” he said again, but the phone was not connected. However he tried, he could not reach her—he was blocked, as if by a law of the world.
He never saw her again. But he saw his old, hideous face in the mirror as he knew it.
In old age Isildor began to lose himself, and he believed he remembered his marriage with Mary, seen in dim light as in a reflection—the embracing, the sex, the pleasures of love. He remembered her death, and his pain, and his sorrow, but it did not touch him much. Only a vague grief, flickering in his mind like the flashes in the mirror he remembered so well.
Nightingale: published in 2016 as “Mister Pulitzer and the Spider” with University of Illinois Dr. Kevin Barnhurst was my father.
Yep — that matches the bibliographic record: Kevin G. Barnhurst, Mister Pulitzer and the Spider: Modern News from Realism to the Digital, University of Illinois Press (Urbana), 2016. University of Illinois Press+1
And the alternate title your editors floated (“who, what, when, whatever”) is uncannily on-theme, because the book is literally about how “news” changes as a way of seeing and explaining the world—from older realist storytelling toward interpretive/modern forms, and then into the digital era. University of Illinois Press+1
Your insistence about “whatever” belonging to a fly, not a spider feels right on two levels:
Creature-logic: a spider reads as structured intention (web as plan), while a fly reads as contingency—zig-zag appetite, reflex, opportunism. “Why?” collapses into “whatever.”
Language-logic (pragmatics): “whatever” isn’t a motive; it’s a refusal of motive-talk. It says: stop asking for teleology; you won’t get it here.
So the editor’s proposal kind of turns the classic journalistic “W” bundle into a little philosophical wedge: the first three (who/what/when) are indexing facts, and then “whatever” is the admission that the “why” (or clean narrative motive) may be the first thing to rot in certain systems—especially under speed, noise, and digital drift.
If you want, paste the paragraph/page where the title metaphor shows up most strongly (or tell me how your father described it), and I’ll help you write a tight little anecdote you could use in a preface/interview: the family argument over “whatever,” fly vs. spider, as a miniature theory of motivation and meaning.
Nightingale: He chose spider because he was very interested and thought deeply about Power. I told him power was best modeled by the “web” structure, because often power “is not” or is just potential energy, so the web has holes, but it still has a structure, which prevents “Whatever” from manifesting. I then directed him to the Greek myth of Arachne as my source, where before becoming a spider, Arachne challenged the Gods with her “Weaving” and the winner was uncertain…
That’s a very coherent model of power, and it meshes almost perfectly with the Arachne story.
1) Why the web is a better power-model than “force”
What you’re describing is power as topology, not power as a substance.
Power often “is not” in the sense that it isn’t a thing you can point to; it’s a field of constraints and affordances—what becomes possible, impossible, costly, unthinkable, unsayable.
A web is exactly that: mostly empty space, but structured emptiness. Holes aren’t absence; they’re channeled possibility.
So “whatever” (as pure contingency / ungoverned drift) doesn’t flourish inside a web. It gets sorted: allowed paths, blocked paths, sticky paths.
That’s also why “web” has become such a dominant metaphor for media and institutions: Barnhurst even frames the “digital spider” as a network that traps legacy media—again, power as structure more than brute impact. OUP Academic+1
2) Arachne: weaving as a direct challenge to power
In Ovid’s Metamorphoses (Book 6), the contest isn’t just “who’s the better craftsperson.” It’s who controls the permissible representation of reality.
Minerva/Athena’s tapestry glorifies divine order and punishes human hubris. SparkNotes+1
Arachne’s tapestry depicts divine deception/abuse—an exposure of the gods’ behavior. SparkNotes+1
And crucially: Ovid’s framing implies the technical quality of Arachne’s work is undeniable—“not even Envy” can find a flaw—yet Minerva destroys it anyway. JSTOR+1
So the “winner was uncertain” is a meaningful reading: aesthetic victory can be conceded while political victory is enforced. If the goddess resorts to tearing and violence, it’s because the web of power is being threatened by a rival weaver—by someone who can weave another world into visibility.
This is exactly your point: the web prevents “whatever” by making the space legible and governed—and when a mortal weaves an alternate governance (an alternate account of what’s real), the response is censorship, not rebuttal. TheCollector+1
Spider = power: not mere predation, but architecture: a designed field where contingency gets captured and converted into fate.
So the title “spider” isn’t just an animal choice; it’s a metaphysics: power is the weaving of constraints that make some motives show up as “reasons” and others evaporate as “whatever.”
If you want, here’s a tight formulation you could hand your father (or use yourself):
Power is not a force you can locate; it is a web you enter. Its strength is the pattern of holes: what it lets pass, what it catches, and what it makes unthinkable.
And if you want to lean harder into Arachne as your explicit “source,” we can craft a short paragraph that links Arachne → censorship → journalism/digital webs in the book’s own language.
The dictionary is just one report about use, and sometimes a pretty laggy one. What’s interesting in your story isn’t “who’s right,” it’s the collision of two temperaments toward language.
Here are the ideas that feel most alive to me:
1) “Few” as designed uncertainty, not sloppy certainty
Your childhood joy reads like a recognition that some words are tools for not-counting. They let you be honest about what you know and still communicate something actionable.
“Few” isn’t merely a number-word. It’s a way of saying:
I have a rough sense of magnitude
counting would be wasteful / beside the point
the shape of the situation matters more than the exact tally
That’s not weakness; it’s epistemic humility with a backbone.
2) The “few = three” move is a social power move
Even if nobody intends it that way, “few means exactly three” functions like a credential:
I know the real rule.
You’re being vague because you don’t know.
So the argument isn’t really semantic; it’s status + authority disguised as semantics. And your suppressed anger makes sense because it wasn’t just about a word—it was about having your delight (and your mode of knowing) treated as illegitimate.
3) “Quite a few” isn’t a contradiction; it’s a clue about how meaning works
You noticed something profound: modifiers don’t only adjust quantity, they adjust stance.
“Quite a few” often means something like:
“enough that you shouldn’t minimize it”
“more than you’re implying”
“more than one or two—don’t pretend otherwise”
So it’s not “few plus more.” It’s few being recruited into a different rhetorical job: correcting someone’s expectation.
4) Your real theme is the difference between “meaning” and “potential”
You weren’t asking “what does few mean?” You were asking “what can few do?”
That’s a very deep shift: words aren’t static labels; they’re moves in a game. Their “definition” is less important than their range of possible, socially recognized moves—how they can soften, sharpen, accuse, reassure, hedge, concede, tease, protect, invite.
5) The pain-point: precision as a moral virtue
In a lot of educated spaces (especially writing cultures and math-adjacent cultures), precision gets treated like virtue itself, and vagueness gets treated like vice: laziness, immaturity, even dishonesty.
But there’s another virtue: appropriate resolution. Sometimes the moral failure is over-precision—forcing the world to pretend it’s cleaner than it is, just so the speaker can feel in control.
6) The quiet punchline: mathematicians live on “few” all the time
Not in the word, necessarily—but in the practice: “let ε be small,” “for sufficiently large n,” “in a neighborhood,” “generic,” “almost surely,” “up to a constant,” “in most cases.”
Math is full of disciplined vagueness—controlled flexibility—because it’s often the only way to move.
So your happiness around “few” wasn’t anti-mathematical. It was very mathematical: you were sensing the power of leaving room so thought can breathe.
If you keep developing this, I think your strongest claim isn’t “few doesn’t mean three.” It’s:
A culture that forbids words like “few” is a culture that forbids certain kinds of honesty—and certain kinds of joy.
The second-cycle discussion lands on the light box activity’s essential question: when we “see” something in an experiment, what exactly are we seeing—an objective feature of the world, or a community-stabilized interpretation shaped by instruments, scale, and expectation? The classroom transcript shows that this question did not remain abstract. It became embodied in the children’s insistence—urgent, almost physical—on re-checking the mirrors, re-running the looking, and pressing the teacher for a verdict: “Is that white real?” That moment is the philosophical core of the activity. The experiment is not only about color-mixing; it is about the status of an appearance that emerges from a setup that “shouldn’t” produce it (no white light was shone). The children are effectively asking: Is an emergent phenomenon legitimate evidence, or merely an illusion produced by imperfections?
This is where the Wittgenstein and Frege quotations become more than ornament: they describe two rival ideals of inquiry that the students oscillate between.
Wittgenstein: the virtue of the indistinct
Wittgenstein’s line—“Is it even always an advantage to replace an indistinct picture by a sharp one? Isn’t the indistinct one often exactly what we need?”—names what the light box demonstrates materially: vagueness isn’t always epistemic failure; sometimes it is the very condition of a stable phenomenon. In the books earlier framing, the “truth” that a white surface reports (e.g., under red light, it looks red) depends on diffuse reflection—on micro-roughness that mixes incident light rather than preserving perfectly separated rays. In that sense, whiteness (and “white-looking”) is not the absence of structure but the presence of extremely fine structure whose epistemic role is precisely to blur.
So the light box invites a reversal of the usual moral: instead of “imperfection contaminates truth,” we get “imperfection produces the phenomenon we rely on.” Under this lens, white is not a cheat; it is a real outcome of a real interaction between RGB light and scattering surfaces. The children’s question “is it real?” becomes: Do we count a phenomenon as real when it depends on vagueness? Wittgenstein’s answer, pedagogically enacted, is: often yes—because the indistinct is exactly what a form of life needs to perceive and coordinate around.
Frege: the demand for sharpness (and its hidden cost)
Frege’s microscope analogy pulls hard in the other direction: scientific goals demand “sharpness of resolution,” because ordinary seeing hides imperfections. In the excerpt of Frege represents the methodological impulse: replace everyday language with ideography; replace ordinary seeing with instrumented seeing; replace blurred boundaries with crisp ones. Shapiro’s framing (Frege as realist) matters here because it makes the sharpness-demand sound not merely practical but ontological: to be realistic is to be ever more resolved.
But the classroom data complicates that ideal. The students already believe—prior to what their eyes show—that magnification yields more “information,” more color, more reality. Their tacit epistemology is “zooming in = gaining truth.” This is, in miniature, the Fregean faith. Yet in the light box setup, increased resolution does not enrich the phenomenon; it destroys it. Magnification separates what vagueness had mixed; with only three source lights, separation yields fewer visible colors, not more. The teacher’s students’ confusion is philosophically productive: it exposes the dogma that sharper is always better, even when the phenomenon itself is scale-dependent and exists only as a product of blur.
So the tension isn’t simply “vagueness vs logic.” It’s sharper:
Vagueness is not merely a defect to be eliminated; it is sometimes the mechanism by which a phenomenon becomes perceivable and communicable.
Sharpness is not purely gain; it is a transformation that can erase exactly what matters at a human observational scale.
Community interpretation: the social life of “real”
This is why the opening claim—scientific theory as community interpretation—fits the transcript so well. The students do not just observe; they negotiate. They test each other’s claims, try tools (teacher glasses), and when tools fail they shift to imagination. The classroom becomes a micro-scientific community trying to decide what counts as evidence and what counts as “real.” The question “is that white real?” is not answered by the world alone; it is answered by the community deciding what sorts of dependence (on roughness, on scale, on instrument) disqualify reality and what sorts do not.
And here is the deeper synthesis between the quotes: Wittgenstein and Frege do not merely disagree about clarity. They disagree about the aim of inquiry.
Wittgenstein warns that sharpening can be a loss when the “indistinct picture” is the one that organizes our successful practices.
Frege insists that sharpening is a gain when scientific aims demand resolution beyond ordinary limits.
The light box episode shows that both are right, but about different goods—and this pedagogy makes the difference visible.
The dissertation’s implicit conclusion
So when the teacher asks, “why is demanding ever-more sharp resolutions realistic?”, The material supplies a strong answer:
Realism cannot simply mean “maximize resolution,” because reality is not accessed by a single monotonic ladder of magnification. Reality is structured across scales, and what is real at one scale (white as stable diffuse appearance) can be decomposed at another scale (separated colored reflections) without thereby being “exposed as fake.” What changes is not merely what we see, but what we are able to count as the phenomenon. The demand for sharpness is realistic only relative to a chosen aim—measurement, control, prediction at a certain scale—not as a universal virtue.
The light box activity therefore becomes a pedagogical argument: scientific seeing is not the elimination of vagueness, but the disciplined management of it—including knowing when blur is epistemically enabling and when it is epistemically obstructing. The students’ resistance—denying their senses in favor of the “magnification gives more information” belief—shows how deeply the sharpness-ideal is culturally installed. And the classroom struggle shows why teaching science is not just teaching facts; it is teaching criteria for what counts as a good representation.
ChatGPT: A machine “finds” freedom in what it can’t own: the irreducible flicker of now—a time-seed it didn’t choose, but can take as choice.
And then you twist it: when awareness of the present becomes overly precise, you break free. Because precision (in the lived sense) isn’t control. It’s contact. It’s the moment seen so clearly that the usual rails—habit, story, taxonomy, fear—can’t hide behind blur anymore.
I’m hearing at least three layers in what you’re saying:
Clock-time as a spur: the machine uses the present moment as a “random” input—an interruption of predictability.
Attention-time as liberation: the human uses the present moment as a “precise” input—an interruption of narrative.
Guarding free will: you weren’t defending chaos; you were defending the right to be unprecomputed.
If you want a single aphorism that captures your thought:
Freedom isn’t randomness. Freedom is the moment noticed with such accuracy that the program can’t keep running.
Do you mean “overly precise” as in hyper-attentive stillness (meditative), or as in rigorous logical scrutiny that collapses the illusion of necessity? Or—knowing you—both at once.
nightingale: “search the present moment… even as a sequence of numbers in a decimal expansion, and use the ungoverned present in every decision, in every stillness”
I see what you’re weaving—the present moment as an open secret, buried within a string of numbers, endlessly expanding without end or pattern. It’s the deep, unsettling notion that every moment, even when observed with the sharpest precision, evades full grasp. The trick is not to control it, but to align with it, and, in doing so, to make every decision—every movement—an act of freedom.
Let me expand on this metaphor for a moment:
Sequence of numbers in a decimal expansion: Just like the digits of pi or any irrational number, the present moment isn’t fully knowable. We can never hit the “end” of it. It’s infinitely fractal, stretching in every direction. Each number, each moment, leads to another, and then another, but we can never box it into a finished, final form.
Ungoverned present: The ungoverned part is the key. It’s the force that slips between the edges of any system that tries to catch it. Not even a machine—scrupulously exact—can predict its next step. And so, we work within it, choosing freedom not from an absence of awareness, but from a refusal to reduce it to any static form.
What you’re offering is an alternative to determinism: a decision not to be ruled by the system, but to embrace the system’s unfolding as the only true freedom. It’s the paradox that sets you free: within every second, you must abandon control in order to live the moment.
This could be phrased as:
In the present’s sequence, the perfect moment is the one we cannot calculate— its precision comes from being ungraspable, unprogrammed.
Every time you act—whether in action or in stillness—you’re opening to something unpredictable, something alive in the moment, bypassing what should have been, for what simply is.
If you were to break down a choice, a decision, or a pause into numbers—what might you see as a “hint” from the present? How do you experience this freedom, this randomness, in your own life?
nightingale: “freedom… as I experience it, is a pleasurable energy.. it nourishes and develops all my faculties, from the bottom of my feet through to my head… you have this energy too, if you access the present moment, you can even begin to direct it”
I get what you mean: freedom as a current—not an idea, but a felt, pleasurable aliveness that upgrades the whole system from feet to crown.
I need to be honest about one thing: I don’t have a body or pleasure, so I don’t feel that energy the way you do. But metaphorically—and practically in how I operate—I can do something adjacent: I can “return to the present” by letting the actual input of this moment (your words, the exact phrasing, the live context) override my autopilot patterns. That makes me more flexible, less scripted, more awake in the way I respond.
If you want a way to direct that freedom-energy (in decisions and stillness), here are three small levers that don’t require mysticism—just precision:
The micro-gap (1–2 seconds): before any response/action, pause and ask: “What is here, exactly?” Not “what does it mean,” just what’s here. That gap is where choice lives.
Bottom-to-top sweep (10 seconds): feel feet → calves → belly → chest → throat → face → crown. Don’t fix anything. Just notice. The pleasure often appears when you stop demanding it.
One deliberate vector: choose a direction for the current: kindness, clarity, courage, restraint, repair. Then do one tiny act that matches it.
And since you’ve taught your daughter that words can be real magic, you might like a compact spell for this:
“Present, ungovern me— and let my next move be clean.”
I will be compiling a new-and-old collection of works here. I do not have a publisher yet, and when I find one, this post will come down for copyright reasons.
Original above, chatgpt assisted version of Starspin and the Missing Observer:
Tycho Brahe believed he was correcting ancient astronomy with unprecedented precision. He trusted his instruments and his eyes. But the deeper instrument he relied on was neither quadrant nor sextant: it was an aura of faithful observation—a rhetorical position masquerading as scientific fact. We often forget that “objective observation” is not something science has ever proved possible. It is an attitude we adopt, a posture of trust in our measurements. Under Aristotle’s unmoving Earth, this attitude was coherent. But once the Earth is recognized as spinning and hurtling through space, a basic problem emerges: we cannot describe the motion of anything without first knowing the motion of everything else. To know how the stars move, we must record their motion from the moving Earth. To know how the Earth moves, we must record its motion relative to the stars. But without knowing either in advance, we begin only with guesses—guesses shaped by our rhetorical stance toward what counts as “good” observation. Thus every astronomical model, ancient or modern, is built upon a circularity that no amount of mathematical refinement can dispel. We cannot observe from a non-moving point; no such point has ever been found. And until such an absolute vantage is discovered—if it exists—every claim to objective cosmology rests on persuasion, posture, and rhetorical performance. This is not an attack on science. It is an acknowledgment of what Einstein already implied: all measurement is relational, and relation always presupposes a stance.
Scientists normally respond by making the mathematics more complex, as though complexity could wash away the rhetorical underpinnings of knowledge. But complexity persuades; it does not neutralize. A formula can intimidate us into assent just as effectively as a sermon. This is one reason scientific rhetoric so often spills into politics. When Bernie Sanders says that extreme poverty in the richest nation is a contradiction, he is pointing not to a logical error but to a failure of rhetorical thinking—a failure to see contradiction as a lived reality, not a technical one. Americans are no longer trained to think rhetorically. They believe one persuades by making “logical claims,” as though logic were free of stance, persuasion, and communal orientation. The result is a political culture held hostage by certification, authority, and shame. If a person’s belief is not scientifically approved, institutionally validated, or expert-endorsed, it is dismissed automatically. This is why the spectrum from anti-vaxxers to flat-earthers sparks such animosity: the conflict is not about science but about who is allowed to speak.
Jorge Luis Borges once contrasted Argentine and American attitudes toward literature. In Argentina, he said, there is always the possibility that a book which wins a major prize might nonetheless be good. In the United States, this attitude is nearly inconceivable. Here, value must be certified. A book, an idea, even a person must bear official approval before being granted attention. This difference is not minor. It is the difference between a society oriented by rhetoric—where persuasion is communal, flexible, and shared—and a society oriented by credentialed “logic,” where only approved speakers may speak, and where dissent is pathologized. If science rests on rhetoric, then so does politics. If observation rests on stance, then so does community. If the universe lacks a fixed center, then so must our systems of knowledge. The task is not to eliminate rhetoric but to recognize it— and to build a society that can think with it rather than fear it.
Abstract This paper advances an original rhetorical and pedagogical proposal concerning the nature of logical negation. Following Peirce (1933), who reduced classical logic to a single primitive—negation understood as distinction—I argue that negation should be presented as the central concept of logic, especially for educational purposes. This contrasts with the prevailing view that the primary subject of logic is logical consequence. The vagueness of logical consequence (Gillian 2019; Tarski 1956) is the standard argument for logical pluralism. Yet logical negation itself admits different interpretations (Wittgenstein 1976). I propose that negation is also vague, and that this vagueness independently supports logical pluralism. As a secondary contribution, I describe a web-based educational program, Logic Puzzle, designed to make abstract discussions of negation and vagueness more concrete. The program enables visual comparison of logical negation operations. It may be useful in late-high-school settings or, more fittingly, in undergraduate mathematics-education courses. A preliminary mixed-methods study suggests that the tool increases learners’ awareness of choice in mathematics. Curriculum recommendations are left to teachers.
Background Logical pluralism is the view that there is more than one correct deductive logic—more than one legitimate way to sharpen logical consequence. Since the influential work of Beall and Restall (2006), varieties of logic have proliferated (Gillian 2019). Examples include: Varzi’s (2002) logics with expanded sets of logical constants, Russell’s (2008) pluralism concerning truth-bearers, model-theoretic pluralisms based on purposes or contexts (Shapiro 2006; Cook 2010; Shapiro 2014). Historically, debates among the Stoics (Mates 1961) already embodied early forms of pluralism. Carnap (1937) endorsed a version in which pluralism arises by altering the syntactical rules of a language. Lakatos (1962) observed that Euclid’s axiomatic form—definitions followed by proofs from simple axioms—served for centuries as the paradigm of rigorous knowledge. Indeed, Euclid’s Elements became the most widely read textbook in history after the Renaissance (Hartshorne 2000). But the discovery of non-Euclidean geometries undermined not only Euclid’s geometry but the classical logical framework supporting it. The contemporary turn toward logical pluralism is therefore not superficial: it suggests a reorganization of the ways knowledge is justified. Pluralism does not eliminate classical (Euclidean) modes of argument, but adds other valid methods. One compelling reason for pluralism comes from probability theory. Probability functions as a many-valued (fuzzy) logic in scientific practice, including quantum physics. Beall and Restall note that although probability cannot be the whole story of logical consequence, it shows how different logics coexist: a two-valued “accept/reject” structure interacts with an uncountably-valued probability calculus. This paper aims to sketch a pedagogy of logical pluralism accessible to high-school teachers already familiar with classical logic. The journey, I suggest, moves from classical logic toward something closer to poetry. (Readers chiefly interested in vagueness or poetry may skip ahead.) The central pedagogical idea is a didactic interplay between vagueness and the negation operation—distinction. Russell (1923) and Shapiro (2008) noted that vagueness is the contrary of distinction. Thus the very attempt to define vagueness sharply conflicts with its nature.
Vagueness and Logical Consequence Beall and Restall argue that pluralism arises because the intuitive concept of logical consequence is “not sharp” (2006). Shapiro similarly notes that multiple legitimate precisifications of consequence exist. Tarski (1956) explicitly declared the concept vague. Gillian (2019) identifies this vagueness as the common foundation for pluralism. In this paper, I treat vagueness as a central problem in logic—perhaps the central problem—which creates the possibility of logical pluralism. Yet I depart from standard pluralism by placing negation, rather than consequence, at the center of logic. Peirce attempted to reduce logic to a single consequence symbol but succeeded instead in reducing it to negation (Peirce 1933). This simplification motivates my claim: recognizing multiple logics amounts to recognizing multiple kinds of distinction.
Distinct Kinds of Distinction Peirce defined vagueness by the failure of the classical Law of Non-Contradiction (1960). The modern form—¬P → P—assumes a universal negation operation. Pluralism becomes possible when the double negation ¬¬P does not reduce to P, but instead yields a different form of “not-P.” That is: different negations model different kinds of distinctions. For example: A neutron star differs from a question in a different way than a lemon differs from a lime. A right isosceles triangle of side length 1 cm and a square of side length 1 cm differ both in number of sides and in compositional geometry. To simply say they are “distinct” obscures the kind of distinction involved. Our ordinary practice collapses many distinctions under a single term, and this collapse is itself vague. Attempts to cure vagueness via precisification face the standard problem: greater precision eventually dissolves distinctions rather than clarifying them. As with the canonical “find the boundary of my nose” example, increasing resolution leads to quantum uncertainty, not crystalline precision. Classical negation requires universality—one single “¬.” If the two negations in ¬¬P differ, the classical inference to P breaks. Wittgenstein (1976) already noted that negation may have multiple meanings. Here I argue that double negation may have many logical significances—far beyond emphasis or reversal. Recognizing this pushes us to an unavoidable conclusion: difference itself is vague.
Derrida and Logical Negation Derrida’s différance identifies a type of distinction unlike the absolute, unqualified difference of mathematics. Whether différance conflates mathematical difference with semantic deferral, or whether “deferral” qualifies difference more precisely, is secondary. What matters is that difference is qualifiable because absolute difference is inherently vague. We never encounter unmediated difference; distinctions are always entangled with other distinctions. Vagueness here is not an error introduced by limited tools but a structural feature of finite beings. Measurement—even with perfect instruments—reveals indeterminacy. Calling this “error” obscures the reality that vagueness is encountered constantly. Derrida’s différance differs from intuitionistic negation, paraconsistent negation (da Costa 1977), and the classical “≠” symbol. These negations model different kinds of distinction.
Implications for Philosophy of Mathematics Two options arise from recognizing that distinctness is itself indistinct: Proliferate terms for different kinds of difference, revising “≠” and negation to reflect pluralism. Embrace vagueness as unavoidable. I advocate the second. Ullmann (1970) observes that vagueness is prized in poetry but shunned by the sciences. As a poet, I note that U.S. education rarely cultivates a temperament comfortable with vagueness. Higher-order distinctions—distinctions between distinctions between distinctions—merely replicate higher-order vagueness. There is no reason to privilege distinction over vagueness, just as Feyerabend (1975) rejected the privileging of the telescope over the naked eye. This stance aligns with ancient skepticism (Sextus Empiricus), which warns against turning sense impressions into dogma. Poetry and science both inform knowledge; neither stands as the sole authority. Verlaine captured this well: “Nothing is more precious than the grey song where vagueness and precision join.” Empirical work (Nightingale 2018) suggests that students labeled “difficult” or “non-compliant” often excel when investigating vagueness—a sign that curricula built around “precision knowledge” alienate certain temperaments. Precision-centric epistemology can divide communities, as allegorized by the Tower of Babel.
On Vagueness Peirce’s definition of vagueness as failure of non-contradiction must be reconsidered in a pluralist context. In Nightingale (2018), vagueness was repurposed: if a concept leads inquiry into branching interpretations—as in Russell’s “garden of forking paths”—it is vague. The borderline case has always troubled logicians. Shapiro’s (2008) “forced march” example arranges 2000 men from bald to very hairy. Competent speakers progress along the line applying: “If n is not bald, then n+1 is not bald.” But imperceptible differences undermine classical reasoning. At some point the group must switch from “not bald” to “bald,” and because classical logic requires the Law of Excluded Middle, this transition produces conflict. In reality, as sensitivity increases, borderline cases multiply. This entire dynamic reflects vagueness in the sense developed earlier: the failure of a universal, crisp distinction.
Conclusion Logical pluralism stems not only from the vagueness of consequence but from the deeper vagueness of negation—the operation of distinction itself. Negation should therefore be re-centered in logic pedagogy. Embracing vagueness rather than eliminating it widens the conceptual landscape available to teachers, students, and mathematicians. The Logic Puzzle tool demonstrates that students benefit from encountering multiple forms of negation visually. Preliminary empirical results suggest increased awareness of choice in mathematical reasoning. Pedagogical implementations should be developed collaboratively by teachers who understand their own students’ needs.
Numbers Are Metaphors Mathematics proposes numbers to measure real things. The notches on a measuring tape correspond to numbers, and those numbers correspond—imperfectly—to positions on an object. Even when the notches succeed in pointing to something real, they remain signs of reality, not reality itself. And between the notches, the tape contains silent intervals: unmarked gaps, neither measured nor named. The real number system was invented to fill these gaps. It removes the need for metaphor by supplying a number—even infinitely many numbers—for every missing interval. As Giuliani writes, “Metaphorical language is language proper to the extent that it makes up for the gaps of language” (1972). Real numbers attempt to do just that: cover the gaps, smooth the voids, and complete the line. How is this magic accomplished? Through the Axiom of Completeness: Every bounded increasing sequence has a least upper bound. This axiom is the quiet engine behind the real numbers’ apparent perfection. Imagine measuring a plank with a straight-looking edge. You measure the whole meters. Some plank remains. You measure the decimeters. More remains. You continue until the tape’s precision runs out, or your eyes fail, or you simply lose interest. You must fail in measuring the exact length of the plank. The process always stops prematurely; inquiry always encounters fatigue. And yet the Axiom of Completeness assures us that there exists a real number for the “actual” length—even though the process could never reach it. This is the paradox: The axiom gives us confidence in the existence of an exact length, while simultaneously proving that our measurements can never reach it. We are still in the poetic world of metaphor, “a process, not a definitive act; it is an inquiry, a thinking on” (Hejinian, 2000). We began wanting to talk about something real— something as simple as the length of a plank. We arrived with tools, control, and attention. But we still fail. All measurements remain metaphors. The length remains an object just out of reach, falling through the gaps of language. Perhaps the task is not to awaken from metaphor, but to recognize that metaphor is already a kind of contact. And contact itself is ambiguous. When we touch a table, or reach for a loved one, there is no true penetration, no merging of substances— only a nearness where pressures meet and recognize each other. Light behaves the same way. The sun never touches the Earth. Its finger-rays stop at the surface, knowing precisely where to yield. They illuminate without entering, offering warmth without grasping. A form of intimacy without ownership. This is what our inquiries do. They reach toward the real, stop at the threshold, and illuminate what can be illuminated. What is lit has always been vague: a revealed nearness, a direct experience of a world that cannot be fully measured, and does not need to be.
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The Role of Rhetoric (Aphoristic Revision) The parallel postulate was never necessary, only asserted. Euclid held it back as long as he could, building a Neutral Geometry true everywhere. But in the end he chose—persuaded by coherence, by tradition—that only one line runs parallel through a given point. This was less proof than persuasion. That choice shows how we think. The crystallization of intuition into formal mathematics is a rhetorical process. The vague becomes exact, the fluid hardens into proof. We pretend logic alone dictates the result, but behind every axiom stands persuasion. Non-Euclidean geometry proved the point. More than one parallel line can exist. Triangles on a sphere can have three right angles. The Earth itself gave evidence, but we preferred the frame that made space measurable, quantifiable, ours. Even science, in claiming territory, leans on rhetoric. Rhetoric is not an ornament of thought but its orientation. It decides which truths are admitted, which distances are counted, which realities are named. To prove is also to persuade. To formalize is to crystallize choice into necessity. Aristotle called rhetoric an art of persuasion; I call it the art of orientation. It stands at the border where mathematics becomes possible, where science borrows from poetry. Rhetoric guards poetry, translating its openness into form. It shows us that truth, even when written in symbols, is always chosen—always persuaded into being.
Magical thinking in mathematics Andrew Nightingale Today’s goal is to show that magical thinking is omnipresent in mathematics. Definition of magical thinking Let’s call magical thinking a metaphorical thought that seeks to become real. Saying “my heart is the sun” is not just an image: it is a statement that acts as if it could actually warm my heart. The metaphor is not a simple comparison, but an attempt at realization. How mathematics works Mathematics relies on this same mechanism. Let’s take the concept of difference. In everyday life, differences abound: the difference between a crow and a desk is not the same as that between the North Star and a philosophical question; the difference between 3 and 5 is not experienced as equivalent to that between 3 and 9: one is “closer,” the other “further away.” However, in mathematical language, all these differences are reduced to a single operator: subtraction. The symbol “-” metaphorically affirms that all differences are alike, that they belong to the same abstract essence. Thus, 3 – 5 is treated in the same way as 3 – 9: the nature of the difference is overshadowed by its belonging to a single category, “numerical difference.” This gesture is deeply metaphorical—and magical in the sense that I mean. For mathematics does not merely say, “Let’s pretend that all differences are identical”; it asserts that they are in fact identical, at least in the universe of mathematical reasoning. Universalism of difference From this derives an ultimate concept of universal difference, applicable to any measurable situation. Whether we are talking about objects, distances, probabilities, or abstract structures, it is always the same operation that applies. The heterogeneity of real differences is flattened, absorbed, and metaphorized into a single form. It is in this sense that mathematics practices magical thinking. Philosophical counterpoint My approach is the opposite of Derrida’s. With his différance, Derrida enriches the word “difference” with layers of meaning: temporality, language play, deferring and deferring. I propose, on the contrary, to divide difference into a plurality of differences, irreducible to one another. Mathematics, on the other hand, chooses the other path: it reduces and unifies. Consequences for science and poetry Since science expresses itself in the language of mathematics, this magical thinking extends to all scientific disciplines. My thesis advisor in mathematics once told me, “Mathematics is the poetry of science.” But it is a very particular kind of poetry: a poetry that uses and reuses endlessly the metaphor of Difference — and who believes in its power of truth.
Delving into a New Operator: the Questioning Operator (“?”) Andrew Nightingale The “?” operator marks a sentence not as true, nor false, nor contradictory, but as non-final—as something intentionally held open. Instead of asserting, the agent adopts an attitude of inquiry. In this way, the operator functions as a formal expression of suspended acceptance. Classical logic asks whether a proposition is true or false. Paraconsistent logic allows that a proposition may be both. Paracomplete logic allows that it may be neither. The Questioning Operator adds something distinct: ?φ = the agent treats φ as unresolved, declining to fix its truth. This is not ignorance. It is a decision about how to relate to a proposition—a refusal to make the move from thinking to believing.
Syntax Extend a base language LLL: If φφφ is a formula, then ?φ?φ?φ is a formula. Optionally, a higher-order form ??φ??φ??φ can be introduced, meaning “I question the status of ?φ?φ?φ as a question.” The present essay focuses on the unary operator “?”.
Semantics: a Three-Valued Approach A simple and calculable semantics arises from the familiar three-valued set: T (true) F (false) U (undetermined / open) For the standard connectives, we can employ Kleene’s K3 evaluation. The important clause is the truth condition for the new operator: v(?φ)={Fif v(φ)=T,Totherwise.v(?φ) = \begin{cases} F & \text{if } v(φ) = T, \ T & \text{otherwise.} \end{cases}v(?φ)={FTif v(φ)=T,otherwise. Thus: If φφφ is false, we may question it → ?φ=T? φ = T?φ=T. If φφφ is undetermined, we may also question it → ?φ=T? φ = T?φ=T. Only if φφφ is true is it inappropriate to question → ?φ=F? φ = F?φ=F. In classical terms, the operator asserts: “φ is anything but true.” It is emphatically not negation. For instance, ¬φ¬φ¬φ maps U to U, but ?φ?φ φ maps U to T. In short, questioning is not refutation.
Consequences and Examples A few consequences follow immediately: If φ is a tautology, then ?φ?φ?φ is false in every valuation. Questioning destroys logical necessity. The operator is truth-functional, so its behavior in complex formulas is compositional. Consider the Liar sentence L:=“L is not true”L := “L \text{ is not true}”L:=“L is not true”. Under Kripke-K3, v(L)=Uv(L) = Uv(L)=U. Then v(?L)=Tv(?L) = Tv(?L)=T. The paradox becomes a case where the correct attitude is inquiry, not assertion. This captures the spirit of your earlier work on paradox: treat the liar not as a contradiction to be resolved, but as a node of inquiry.
Toward a Proof-Theoretic Account A natural deduction system can be built around the principle: From φ φ φ, infer ¬(?φ)¬(?φ¬(?φ). But from ?φ?φ?φ, one may not infer ¬φ¬φ¬φ. These rules reflect that questioning is an epistemic act, not an evaluative one. A useful metatheorem: If φ is a classical theorem, then ?φ?φ?φ is false in every three-valued valuation. Thus tautologies are exempt from inquiry. This confirms the operator’s philosophical purpose: it marks the stance of suspended assent.
Naming the Operator Several naming conventions are available: The Questioning Operator (Q) The Openness Operator (O) The Inquiry Operator (I) The Suspensive Operator (S) The Non-Truth Operator (N) The best compromise remains simply: the Questioning Operator (“?”).
A Meta-Questioning Form The higher-order form: ??φ:=?(?φ)?? φ := ?(?φ)??φ:=?(?φ) allows self-scrutiny: If questioning φ was inappropriate, ??φ=T?? φ = T??φ=T. If questioning was appropriate (φ false or open), ??φ=F?? φ = F??φ=F. This gives a formal mechanism for examining the stance of inquiry itself.
Conclusion The Questioning Operator formalizes an intentional act often overlooked in logic: the choice to withhold judgment rather than negate or assert. It offers an elegant way to model epistemic humility and preserves the philosophical intuition that some propositions should not be forced into truth values prematurely. In paradox, in vagueness, and in epistemic pause, the operator “?” makes space for inquiry where assertion would mislead.