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Vagueness in Mathematical Terms (reworked and more accessible)

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High Up (On Vagueness and Mathematics)

Imagine walking down Olympus Mons — the largest volcano in the solar system, on Mars. Its slopes are so gradual that you might walk for hours barely noticing the descent. At the top, you are clearly “high up.” At the bottom, you are clearly not. But somewhere in between, the description falters. You become unsure whether “high up” applies or not. That fuzzy middle zone is what philosophers call vagueness.

We can make this precise — or try to. Map the mountain onto a number line: 0 at the summit, 1 at the base. The set of points where you count as “high up” has a boundary. Call it sup — the least upper bound, the last point before “high up” runs out. The set of points where you count as “not high up” also has a boundary: inf, the greatest lower bound, the first point where “not high up” begins.

Now suppose — reasonably — that sup itself counts as “high up” (since every point below it does), and that inf counts as “not high up.” What happens when we ask how sup and inf relate to each other?

There are only three possibilities, and each one produces a contradiction:

  • If sup = inf, then that single point is both “high up” and “not high up.”
  • If sup > inf, then the real number line — which is dense, meaning there’s always another number between any two — guarantees a point z sitting between them. That point would be both “high up” (since it’s below sup) and “not high up” (since it’s above inf).
  • If inf > sup, the same logic applies in reverse.

The standard response is to blame the vague word. “High up” is imprecise — a folk term, not a technical one. Strip it away and mathematics, supposedly, is safe.

But stripping the vague term doesn’t solve the problem. It moves it.

Consider the wave theory of light. Its mathematical core — the equation governing refraction:

sin(α) / sin(β) = μ

— looks clean and precise. But the philosopher Mary Hesse pointed out that the equation, on its own, is ambiguous: it can be interpreted in multiple, entirely different ways. The symbols don’t come pre-labeled. Perhaps α and β aren’t angles of light at all — perhaps they’re the angles between the Pole Star and two planets at midnight. The mathematics would fit. Which interpretation is correct? The equation doesn’t say. Meaning doesn’t live in the symbols alone.

Vagueness and ambiguity are usually treated as distinct problems. Ambiguity means a word or expression has more than one possible meaning. Vagueness means a word has unclear edges — cases where it’s genuinely uncertain whether it applies. But consider: what if a word were both at once?

Thai has a word, krup, that technically means “yes” but functions more like a polite acknowledgment — because outright agreement can feel presumptuous, as if you’re confirming what the listener already knows. It occupies a middle space between assertion and non-assertion.

Now invent a word: snook. It means “tall” in some contexts and “not tall” in others. When applied to someone of borderline height — someone exactly at the edge of where “tall” is uncertain — is snook ambiguous, vague, or somehow both? Is there a vagueness between vagueness and ambiguity? If so, what does that do to the apparent clarity of mathematical symbols?

Even pure mathematics — mathematics with no interest in mountains or light — is soaked in vagueness. The discipline’s foundational concepts carry it: continuity, completeness, integral, limit. These are not casual approximations. They are the load-bearing terms of analysis, the branch of mathematics that underlies calculus.

And they are vague. Any careful textbook in real analysis will show you functions that slip through the formal definition of continuity — technically satisfying the definition while still behaving in ways the definition was meant to exclude. The definition doesn’t quite capture the intuition. The intuition doesn’t quite surrender to the definition. The gap between them is not a failure waiting to be fixed. It is where thinking happens.

Without words like “continuity” and “completeness” — words that mean something intuitively before they mean something formally, and that keep some of that intuitive life even afterward — mathematics would be unlearnable. Students would have nothing to grab onto. The vagueness isn’t what mathematics tolerates in spite of itself. It’s part of what mathematics thinks with.

The fantasy of a perfectly precise formal world, unsullied by the messiness of natural language, is just that — a fantasy. Vagueness goes all the way down.

The Title of the Song (reworked)

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Has justice become sense-making? The many senses of the over-worked concept of justice allow justice to generally sound like a good idea to the atomized American. Even our best politicians repeat 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, and they crave agency in what makes sense to them. 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 neoliberal “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, because we are 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 compelling force in politics. And it should be persuading people, not compelling us the way we got used to in our math classes.

The problem isn’t that Americans lack meaning, it’s that they’ve been told which meanings are permitted. Media outlets employ Elite People like Anand Giridharadas, a fixture of elite media commentary who wrote for the New York Times to argue that we shouldn’t listen to just anybody.

[Giridharadas]: “114 percent of Americans now having their own podcast, … 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.”

That desire to regroup atomized communities to the tune of the New York Times was visible then. Not that the New York Times wanted 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 attempted to rhetorically close that space, which is unhealthy politically. We need synthesis but not to the tune of the elite who brought us more neoliberal presidential candidates which were, 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 Deirdre agrees, read her wonderfully brief book on writing!). Here Deirdre writes “they are egg and yolk in a scrambled egg.” or “their differential equations are nonseparable.” Sweet language, such as poetry, expresses the truth best (not mathematical or statistical language). There is still freedom where there is truth. The freedom to be found is in our own creative interpretations. This freedom should not be limited to poetry. The more elastic concept that also allows freedom but is less categorically artistic is Rhetoric.

Related: https://questionsarepower.org/2014/09/08/the-valid-logical-argument/

The House Builder (Revision from June 2015)

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“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” bbb, perhaps each inferential step discounts it by a factor k≤1k\le 1k≤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.

The Spider and the Whatever: A reconstructed conversation

02 Friday Jan 2026

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Preface

My father died before I could argue with him enough. That is probably the most honest thing I can say about grief. This conversation didn’t happen exactly this way — but the ideas did, across calls and emails and the margins of things he sent me. I’ve reconstructed it as faithfully as I can, which means I’ve invented the connective tissue and preserved the bones. He would have pushed back harder. That’s what I miss most.

Kevin: The title’s the spider. That’s the whole book, really — Pulitzer as a builder of webs. Not just a media mogul. A spider.

Andrew: Why not a fly?

Kevin: Because a fly just — lands on things. Randomly. The spider makes something.

Andrew: That’s what I mean. The fly is more honest about what journalism actually does, most of the time. Who, what, when — and then whatever. The “why” is usually retrofitted.

Kevin: That’s cynical.

Andrew: It’s Pyrrhonist. There’s a difference.

Kevin: (a pause) Say more about the whatever.

Andrew: A fly doesn’t have a theory of the room. It just moves through it according to hunger and reflex. “Whatever” isn’t stupidity — it’s a refusal of teleology. It says: I won’t pretend there’s a plan here. And a lot of what gets called news is exactly that — contingency dressed up as narrative. The fly is more honest than the spider because the spider implies a design that may not be there.

Kevin: But the web is there. That’s the whole point. The web is the design — not in the editor’s head, but in the structure. Power doesn’t need a planner. It just needs a pattern.

Andrew: (slowly) Okay. That’s actually — yes. Power as topology. Not as intention.

Kevin: Exactly. The web isn’t about the spider’s motives. It’s about what gets caught and what passes through. What becomes a story and what doesn’t. The web is what makes some contingencies into facts and lets others dissolve.

Andrew: The holes.

Kevin: What?

Andrew: The web is mostly holes. That’s what makes it a better model than force. Force has no holes. But power — real power — is structured emptiness. It’s not what it catches, it’s also what it allows to pass, what it makes unthinkable. The whatever doesn’t get caught. It gets — dissolved. Made invisible. That’s more frightening than capture.

Kevin: That’s Foucault.

Andrew: That’s Arachne.

Kevin: (laughing a little) You and your myths.

Andrew: I’m serious. Arachne isn’t just a craftsperson. She’s a rival weaver. She weaves a different world — she depicts the gods doing exactly what they actually do, all the abuse and deception — and her work is technically flawless. Even Envy can’t find a flaw. And Minerva destroys it anyway.

Kevin: Because it was a political threat, not an aesthetic one.

Andrew: Right. And that’s the moment that breaks the fly/spider binary for me. Arachne was weaving her own web. She wasn’t a fly — she wasn’t just drifting, all whatever. She had a design, a structure, a rival topology. And the response wasn’t rebuttal. It was censorship. Which means — what? That the original web knew it was being threatened by another web, not by chaos.

Kevin: Power responds to counter-power. Not to noise.

Andrew: Yes. Noise gets ignored. The whatever gets ignored. It’s only when someone builds a competing structure — an alternative account of what’s real — that the original web has to act violently. Because you can’t argue with a web. You can only build another one or destroy the competition.

Kevin: So the spider in my title —

Andrew: Is more afraid of Arachne than of the fly. The fly just buzzes. Arachne weaves. That’s the actual threat.

Kevin: (quiet for a moment) I’m not sure I had that in the book.

Andrew: You had the web. I’m just adding the rival weaver.

Kevin: Well. Maybe that’s the paperback edition.

Andrew: Or the conversation you and I keep having without quite finishing.

(Neither of us said anything after that. Some arguments are better left open — not because they’re unsolvable, but because the opening is where the thinking lives.)

The Monk Who Looked for Space Final Version

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The Monk Who Looked for Space

by Andrew Nightingale

Adapted from the Dhamma for Children

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.

Vagueness is Essential

10 Wednesday Jan 2024

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Vagueness is not a matter of semantics. It is a problem that troubles the basic assertion that, as our old friend Berty Russell asserted as an axiom in the Principlia Mathematica, "Everything that is, is." The basics are a very interesting place to stay. We would like to say this is a basic and acceptable assertion, and it turns out to not be basic at all. In fact, no subject is elementary, and also every subject is elementary. 
 
Channell and Rowland argue that vagueness has pragmatic usefulness: "For language to be fully useful, therefore, in the sense of being able to describe all of human beings' experience, it must incorporate built-in flexibility. This flexibility resides, in part, in its capacity for vagueness" (p201 Channell 1994) Dr. Channell outlines various views of where vagueness comes from, from the difference between the "same idea" in different minds (Fodor 1977 in Channell 1994), to language (Peirce 1902 in Channell 1994), to physical reality (Russell 1923). Vagueness is found discussed in logic (Lakoff 1972 in Channell 1994) where it is argued (along with Russell) that "true" and "false" are vague, and so classical logic could be modified..." (p66 Nightingale 2019) 

"[i]t is perfectly obvious, since colours form a continuum, that there are shades of colour concerning which we shall be in doubt whether to call them red or not, not because we are ignorant of the meaning of the word "red," but because it is a word the extent of whose application is essentially doubtful." (1923 Russell as quoted in Nightingale 2019, p66). 

"The word "red" is vague in this respect because there are borderline cases where it is not clear whether or not we should call the case "red". Russell says "essentially doubtful" because this uncertainty is essential, in the sense of being a part of the nature of red. One deception here is in asserting that the "continuum" is a perfectly precise reality that can be expressed numerically. This renders vagueness a kind of error; without a perfectly known continuum underneath our words, vagueness is not error but has a reality of its own. Does the continuum suffer from vagueness?...
Peirce claimed that another way to describe generality is where the Law of Excluded Middle ("A or ~A is always true") does not hold. This makes sense because normally, the LEM decides which of "A or ~A" is true (even if we don't know which is decided, it asserts that "out there" it is decided.) When the LEM does not apply "A or ~A" is left undecided, which allows for a generalization on "A or ~A", you can choose which. However the claim that something can be essentially uncertain is directly against the LEM." (p66-68 Nightingale 2019)

I mean to say that reifying vagueness proves the LEM is false, in general. (The ideas of general and of vague are intimately connected) Russell asserted "everything that is, is" in order to "prove" the LEM. And here I am arguing against the LEM, which would also be against "everything that is, is" What makes red red? In this question i mean to be vague between term red and the actual red. If everything that is depends on other things to be, there is a certain spaciousness to Being, an undefined vagueness between Being and Space.

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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.

The Two Fools

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The first story is about a girl who loses her family to illness and cannot keep her house because of debt. She leaves her house with only her winter clothes and a loaf of bread, and begins walking down the road with no-where to go. As she walks the day goes by, and a beggar comes and asks for some food. The girl gives half her loaf of bread and keeps walking. But then she meets another beggar on the road who asks for food, so she gives her other half of the loaf. Again she meets a beggar on the road who says he is cold, and she gives him her coat. And again, a beggar with bare feet asks for her shoes, and she gives. Finally it is night on the road, and she has only a little cloth to wear. As the stars come out, they fall down to her feet. She picks them up and realizes they are gold. So many stars fall that night, that she lives the rest of her life without a worry for food or things.

The second story is about a man who works for his employer five years and after completing his work is paid a lump of gold as big as his head. The man takes his wages, the lump of gold, and begins making the day-long trip down the road to his mothers house. As he is walking he meets a man on a horse. The man on the horse explains that a horse can make a lot of money because it can do a lot of work on a farm, and can carry him home to his mother as well. So the Fool agrees to give his 5 years wages to the man with the horse, in exchange for the horse. As the Fool rides down the road on his new horse, he meets a man with a goat. The man with the goat explains that a goat can give milk, and can be shorn for its coat of hair to make clothes. The Fool agrees to exchange his horse and takes the goat, walking on down the road. The Fool meets another man, who is standing next two millstones. This man explains that millstones can do great work grinding the wheat for bread, and so the Fool agrees to exchange his goat for the millstones. Now the Fool is heavily burdened by the stones and grows tired. He finds a well, and thanks God that he can drop the millstones into the well to make water come up so he can have a drink. Finally he reaches his mother with nothing for his 5 years of work.

These two stories at first seem to have two very different ideas about what a Fool is. In one, the girl is not at fault, but continues to give everything away. The idea being that there is an accountancy in an invisible world of magic that repays her for her selflessness. It is rather easy to see that the second Fool is not being selfless, because he is being duped into making bad exchanges for his wealth. The second Fool makes no gain in any invisible world, or so the story goes. There is a strange part of the second story, taken from Grimm’s Fairy Tales, where he thanks God for easily being able to give up his millstones at a well in exchange for a drink of water. Here the logic of the story admits that there is an invisible world, and even though he has failed in this world, as well as the material world, God cares for him anyway, and relieves him of the burden of the stones out of mercy. The girl getting gold coins from magic is not mercy, I think, but the universe paying the debt incurred in the invisible world in the form of gold coins.

We can see clearly here that the world of the invisible and the material world is getting confused. If the girl knew the universe would repay her, she would not be repaid, of course. And if the Fool in the second story knew his exchanges were not savvy in the material world, he would have gained some repayment from the invisible world, beyond mercy. If you use knowledge, you had better get it right, because even though the trades for the second Fool were done with knowledge of the merits of millstones, etc, the loss incurred from this knowledge was complete. The Fool who plays the game of exchange to lose, wins; and the Fool who plays the game of exchange to win, loses. It is clear that if you know you are a Fool, you should play to lose. And, well, it is rather easier to be a Fool and play to lose than to be wise and play to win… isn’t it?

Speaking as a diagnosed Fool, I have to attest that the game of losing is a lot harder than it seems. Eventually you will find that you are wise in spite of yourself, because if you do things for people expecting to gain in the invisible world, you can gain nothing. Eventually you have to learn to control your mind and do things without expectation, and what do you end up doing then? Forest-wandering, or anything else practiced by both involuntary fools and the deliberate ascetic.

I will end with Whitehead’s quote in his masterwork on process philosophy. “God is the Fool of the World.” The World card is the end of the Tarot, and the Fool is the beginning.