There is a story, universally known to mathematicians, about Leonhard Euler, Denis Diderot, Catherine the Great, and the epistemological authority of mathematics. It apparently first appeared in English in Augustus De Morgan‘s book A Budget of Paradoxes:

Diderot paid a visit to the Russian Court at the invitation of [Catherine the Great]. He conversed very freely, and gave the younger members of the Court circle a good deal of lively atheism. The Empress was much amused, but some of her councillors suggested that it might be desirable to check these expositions of doctrine. The Empress did not like to put a direct muzzle on her guest’s tongue, so the following plot was contrived. Diderot was informed that a learned mathematician was in possession of an algebraical demonstration of the existence of God, and would give it him before all the Court, if he desired to hear it. Diderot gladly consented: though the name of the mathematician is not given, it was Euler. He advanced towards Diderot, and said gravely, and in a tone of perfect conviction: Monsieur, $\frac{(a + b^n)}{n} = x$, donc Dieu existe; repondez!¹ Diderot, to whom algebra was Hebrew, was embarrassed and disconcerted; while peals of laughter rose on all sides. He asked permission to return to France at once, which was granted.²

One interesting thing about this is De Morgan’s misrepresentation of the point of the anecdote he quotes. He attributes it to Dieudonné Thiébault, whose version ends quite differently:

…Diderot wanted to explain that this alleged proof was nonsense, but was unable to escape the embarrassment of realizing he was being fooled with and would not be able to escape the jokes with which they were ready to assail him…³

De Morgan’s version, contra Thiébault, paints Diderot as an actual clown, too ignorant of algebra to even understand the mockery. Mathematicians have been fervently repeating De Morgan’s version since he published it. It’s as foundational a myth for mathematicians as the story of George Washington and his dentures made of slaves’ teeth is for the United States. What lessons does it teach its audience? How does it comfort those who share it? Myths reinforce and reproduce social systems — what’s being reinforced here? How does this myth help reproduce the community that relies on it?

There’s a clue in De Morgan’s weirdest and least plausible addition — the claim that to Diderot “algebra was Hebrew.” Diderot, a famously brilliant guy, wrote an entire monograph on mathematics. If algebra to him was Hebrew it would only have been because he was fluent in Hebrew.⁴ The folk process really latched onto this detail, which shows its significance. One recounter changed “Hebrew” to “Chinese,” while another used “Arabic.”⁵ Thiebault wrote in 1804, De Morgan in 1872, and the other two in the early 20th Century. What had changed?

For one thing, capitalist domination of the world, still not a done deal in the 18th Century, was essentially complete by the turn of the 20th. This happened over the course of De Morgan’s lifespan.⁶ By then capital had discovered many of its modern uses for mathematics, and consequently mathematicians were integrated into social power structures in many of the same ways they are now.⁷ Which means that they were in the market for worldviews that justified their privilege in contrast to the violence necessary to create and maintain it.

This need is reflected in the evolution of the story. On first telling the point was obscure, perhaps because it was close to a description of something that actually happened. De Morgan had an axe that needed grinding, so he punched it up a little by beclowning an old-school renaissance man and natural philosopher of precisely the type that supplied the theoretical foundations of power in the 18th Century, of precisely the type whose role was now filled in part by mathematicians. But by the first decades of the 20th Century much graver threats to peace of mind had surfaced, which therefore called for new mythmaking.

The replacement of Hebrew, in the relevant time a dead language, with Arabic and Chinese, in their relevant times languages spoken by stereotypically subhuman colonised peoples, served to pearl-coat the jagged violence of colonialism, much more apparent in elite white circles by the early 20th Century than had previously been the case. The story justifies white Europe’s colonial violence by framing its intellectual architects as rightfully, due to their unassailable knowledge authority, advising Catherine the Great’s world-ruling counterparts.

Another essential element of De Morgan’s version of the story is the stature of the characters, including his own personal stature. Euler, the greatest mathematician of his time, was responsible for much of the framework of modern mathematics. Carl Gauss, himself the greatest mathematician of his time, said that “The study of Euler’s works will remain the best school for the different fields of mathematics, and nothing else can replace it.”

Diderot’s fame is also essential to the story’s point. Titans clashed and the one with the recognizably, at least to De Morgan, modern worldview won out. Part of the pleasure, part of the community, that the anecdote provides to mathematicians in its recounting lies in vicarious identification with Euler’s triumph and its parallel to their own triumphs. Another part lies in vicariously identifying oneself with a cause championed by the also-eminent De Morgan. Without archetypally famous characters reading the lines the scene misses its mark.

Something all versions from De Morgan’s on have in common is that they’re told by mathematicians to mathematicians. By definition then members of the audience have already claimed a share of power and are looking for a way to feel better about that choice. The narrations must build community, and aggression works against that so the tellers take a respectful tone. It’s different when they’re defending against the non-mathematical world. Here’s De Morgan’s description, from the very same book that has the Euler-Diderot story, of an amateur mathematician who had the nerve not only to disagree publicly with famous mathematicians but doubled down on being told he was wrong:

The behavior of this singular character induces me to pay him the compliment which Achilles paid Hector, to drag him round the walls again and again. He was treated with unusual notice and in the most gentle manner. The unnamed mathematician, E. M. bestowed a volume of mild correspondence upon him; Rowan Hamilton quietly proved him wrong in a way accessible to an ordinary schoolboy; Whewell, as we shall see, gave him the means of seeing himself wrong, even more easily than by Hamilton’s method. Nothing would do ; it was small kick and silly fling at all; and he exposed his conceit by alleging that he, James Smith, had placed Whewell in the stocks. He will therefore be universally pronounced a proper object of the severest literary punishment: but the opinion of all who can put two propositions together will be that of the many strokes I have given, the hardest and most telling are my republications of his own attempts to reason.⁸

This De Morgan doesn’t gently invite his peers to self-soothe with shared humor and community reaffirmation — this De Morgan repeatedly announces his intention to physically assault a human being. Like many whose continued existence relies on brutal violence directed by others on their behalf De Morgan needs to justify his intention by showing how many have tried nonviolently to convince the heretic. He was given many chances to conform! Clearly he has willingly forfeited the protection of civilization! He is a semi-civilized subhuman! Beat his ass! De Morgan goes so far as to describe his criticisms as “strokes,” a punishment administered to slaves by their masters. The metaphor is striking and striking is the metaphor. Intellect righteously serves power by wielding a pen as an implement of physical torture and subaltern correction.

Which brings us to the next subject of this essay, a self-proclaimed anarchist known as William Gillis.⁹ Gillis, who seems to be some kind of minor cult figure among left-wing libertarians,¹⁰ is a dude with a website, some thoughts about things, and a tsunami of untempered rage. Apparently there are people to whom his work provides comfort and community given that it’s pretty widely published and discussed in certain circles. These people are to Gillis as the mathematicians in his audience were to De Morgan. He supplies conscience-soothing, community-creating myths to his followers. But Gillis lacks an essential tool that De Morgan had a surfeit of — knowledge authority.

Knowledge authority is the ability to have one’s assertions accepted as fact by virtue of one’s social position without adequate supporting argument.¹¹ It’s not a function of the truth or falsity of the claims, which is irrelevant to the authority. Knowledge authority is a relation between knowers and consumers of knowledge rather than a property of the particular knowledge in question. It’s a social fact. Social facts are backed by society, and therefore in a coercive society ultimately backed by violence rather than reason. This situation is colorfully evoked by De Morgan’s putatively hyperbolic threats against a man who refused to accept his putatively rational arguments.

Scientists in modern American society have a great deal of knowledge authority because their work is essential to capitalism, the ultimate source of violence. We know physicists are telling the truth because their bombs explode. Their conclusions are justified by three unassailable social facts: They know stuff we don’t know, if we did know it we’d agree with them, and we will never know it like they do.

Ultimately this kind of authority doesn’t come from the truth values of its claims but from the utility of those claims to people with the power to impose worldviews that serve their purposes. Without power there’s no knowledge authority, but only authoritative knowledge. I am not saying that scientific claims are false, but rather that, true or false, they’re authoritative because they’re backed by force.¹²

So Gillis has a problem. Anarchists have zero knowledge authority in the world and less than that in anarchist circles. They can’t wield state violence for obvious reasons, and they can’t get other anarchists, hardened skeptics who’ve already rejected violence-backed state agendas, to accept their dogmas through mere amateur violence. Many anarchist writers resolve this dilemma by explaining their ideas clearly and respectfully to their audience to allow them to come to their own conclusions. Gillis chose another path: cosplay. See, for instance, his website bio, where he proclaims publicly for all the world to see that he “…is a second generation anarchist activist who studies high energy theoretical physics.”

What an interesting phrase! He “studies high energy theoretical physics.” In the ordinary denotational meaning of the words this is probably a completely true statement, but connotations can be tricky. The connotational meaning of the verb “to study” conjugated this way, in some kind of ongoing present tense,¹³ and followed by an academic discipline, changes radically depending on how the discipline is described. If it’s something general, like “physics,” the word “studies” retains its ordinary meaning. If someone asks what my kid does at college I could plausibly answer that she studies physics. But if the discipline is highly qualified, narrow, esoteric, a different meaning becomes available.

It still makes sense to say that a college kid “studies high energy physics”¹⁴ if they’re taking a series of classes in that subject, but it also makes sense to say of a working scientist engaged in original research that she or he “studies high energy physics.” It’s a kind of modesty, a way of saying that even though the subject knows more about high energy physics than all but a few hundred or even a few dozen people in all of human history, they’re still humble students. For instance, “Allison Hall studies high energy physics using the Large Hadron Collider at CERN in Geneva, Switzerland.” It’s easy to find plenty of similar examples.

So what is up with Gillis? He doesn’t seem ever to have published anything in any area of physics. He doesn’t have an advanced degree in the subject as far as I can tell, and nothing on his website suggests he’s done original scientific research.¹⁵ He may dip into Physics for Dummies to kill time in airports but equivocating on different connotations of the verb “studies” is dishonest — stolen humility rather than stolen valor. On his Twitter bio he calls himself a “lapsed physicist” and on Mastodon unqualifiedly a “physicist.” Without access to genuine authority the guy needs knowledge authority and he’s not shy about grabbing it.

The three social facts that scientific knowledge authority relies on are these: they know stuff we don’t, if we knew it we’d agree with them, and we can never know it. Gillis stakes his claim to such authority in a remarkable essay, the general tenor of which is well-represented by this passage, a sort of Platonic ideal of the first and third pillars:

The qualia of physics and math, the richness, the crystal clarity, the complex humor of someone’s proof, the overwhelming resonance of the revealed relations and their potency at further exploration make sad jokes of all the cheap fragmentary poetic or neural associations one can momentarily garner and perhaps struggle to hold onto from drugs and religions. Trying to explain this kind of experiential depth to those who have never even glimpsed mathematics beyond arithmetic isn’t like explaining sex to a preschooler, it’s like trying to explain the subjectivity of other individuals’ knowledge to a toddler or self-awareness to an newborn. The doors it opens to experiencing reality and the remarkable solidity of the whole affair are not even fathomable beforehand.¹⁶

And just like De Morgan, when Gillis is talking to his followers, rather than aggression he uses explanations, examples, arguments, discussion. And just like it did with De Morgan, Gillis’s knowledge authority plays a more tacit role in this context than it would for outsiders. He’s careful to project respect for his followers, a necessary element of any discursive style appropriate for intra-group community strengthening and reaffirmation.¹⁷

Although intragroup aggression destroys community rather than building it, extragroup aggression may build community. It’s a dangerous tool, though, and must be handled carefully. Thus the contextually reasonable Gillis, like the contextually reasonable De Morgan before him, can become quite unreasonable, even aggressive, when dealing with external threats to his community:¹⁸

We agree to leave you that stupid house you bought in the surburbs, with firm social norms against violating such. You can operate on the market, collect food and basic needs from post-state social services, and we’ll retrain anyone to work in professions without power. But the moment someone organizes a hierarchy or fields an ex-cop gang to spread terror again that gang gets exterminated by every surrounding watchful civilian. We have to be willing to, at the drop of a hat, race out of our houses and confront and stop with violence the predatory gangs the ex-cops will try to form.¹⁹

Like I said, though, such talk can be dangerous if not handled carefully. Remember that De Morgan, when fantasizing about the violence with which he’d like to meet the challenge posed to his power by an amateur mathematician, took care to demonstrate to his peers that the aggression was justified, that its target deserved its fate. Such moves supports social stability in the sense that they reassure spectators that as long as they meet relevant community norms they won’t be subject to a violent fate.

Without this reassurance the aggression can’t build community because the intended audience is too anxious to attend to the performance. Gillis is cheerfully planning vigilance committees and community lynchings and at least some of his followers, the ones with any sense, will wonder if they’ll end up lynched. It’s not hard to see proposed impromptu communitarian death squads settling personal scores for Gillis or other wannabe lefty-lib Robespierres. If leaders propose violence to build community they must have effective ways to reassure the community that they themselves won’t become victims.

To this end the criteria for outlawry must be as visible as possible and one way to do this is to perform the determination process as publicly as possible. Insiders must be reassured that they’ll get a fair trial if they’re suspected of being potential targets.²⁰ A tentative approach might be useful, a sort of hand extended to someone who may be a transgressor but who could conceivably still redeem themselves by conforming if offered a chance. If they turn out to be the first Gillis can make an example of them. If the second he may gain a follower.

De Morgan’s litany of evidence justifying his proposed violence is part of this process and we can see the same decision process unfolding in this recent toot thread, which began with Mastodon user Ben Chambers posting the following claim:²¹

knowledge and economic calculation problems are solved by relational egalitarianism, democracy, and usufructuary commons, not by market transaction, property, and commercial enclosure

This makes a lot of sense to me, but it’s a fairly dense aphorism. I can see how people might fail to understand it if they haven’t been thinking along these lines or aren’t willing to put some thought into deciphering it. But regardless of what you think about the truth or the content of the claim it’s clear that it’s plausible, to be taken seriously, and that it has something to do with left libertarian concerns. Gillis, as some kind of left libertarian leader, must be alert to social-capital-building opportunities. Inducing a heretic to recant is such an opportunity, and violently attacking a heretic, whether metaphorically or literally, is another. Gillis, needing to decide which is appropriate in this case, a few hours later, chimed in with a who-goes-there challenge: “Cool assertion. Now let’s see the proof!”²²

There is so much aggression imbedded in just these few small words! The word “cool” followed by a communicational noun, “assertion,” evokes the tagline “cool story, bro,” which, as Wiktionary tells us, is “[u]sed to dismiss a comment perceived as boring or pointless, or refute an anecdote that one considers difficult to believe.” The next word, “assertion,” does more than just connotationally influence the word “cool.” First, it’s an unusual word choice, at least superficially. Compare the n-gram viewer results for it compared to “claim,” “statement,” and “idea,” all of which seem much more natural in the sentence.

This result suggests that the choice was deliberate, and as such likely intended at least to add an air of esoteric technical knowledge to Gillis’s challenge. It’s entirely plausible that Gillis meant to evoke the spirit of proof by assertion, which is an informal fallacy. We’ll see below that Gillis relies heavily on this kind of connotational innuendo, each individual instance of which might be a coincidence but the aggregate weight of the instances is hard to explain other than by intention. So the first two words of Gillis’s toot constitute a rhetorically complex and ideologically loaded challege to Ben Chambers’s aphorism.²³

The next sentence, wherein Gillis demands “a proof” of Chambers’s claim, is more straightforward. It’s also an archetypal invocation of scientific knowledge authority, by the way. The word “proof” sounds technical, like there are technical standards that make an argument so good, so foolproof, that it can be called “a proof.” Self-proclaimed physicist Gillis understands this, do you?! Mathematics is the paradigmatic example of a discipline whose claims admit of “proof,” and when people hear the word in even mildly technical contexts they tend to visualize 9th grade geometry and its associated feelings of ignorance and shame. But mathematical ideas of proof, whatever they might be, can’t apply outside of mathematics, so this must not be what Gillis means.²⁴

The general consensus, shared by such mainstream figures as Albert Einstein and Karl Popper, is that scientific truths don’t admit of definitive proof, but can only be definitively disproved. Popper, quoted in Wikipedia’s article on Scientific Evidence, said:

In the empirical sciences, which alone can furnish us with information about the world we live in, proofs do not occur, if we mean by ‘proof’ an argument which establishes once and for ever the truth of a theory.

Gillis spends a lot of energy and time pretending to be a scientist, so maybe he’s referring to some kind of scientific but not mathematical proof, something that would establish the truth of the claim not necessarily mathematically but in accordance with the epistemological standards of some accepted scientific community. The consensus is that each scientific discipline is a community of knowers with its own community standards of proof. If someone demands a proof of a non-mathematical statement, then, it’s appropriate, if only for the sake of efficiency, to ask what standards of proof they’ll accept, which I did in my response to Gillis:

What kind of proof do you think statements like that admit? This is a purely good faith question, because it seems like an important issue, and one I don’t know the answer to even after a lot of thought.

Such questions are a staple of trolls, but I wasn’t trolling, so I asserted my good faith. Of course trolls assert their good faith too, so it’s also respectful and pragmatic to offer some evidence that one’s question is thoughtful as a token of commitment to the discussion, which I did. My complete response is here and in the footnote.²⁵ I don’t want to rehash the argument in this essay, but some parts of my response are necessary to understand Gillis’s tactics.

First, I outlined how I use the word “proof,” which is essentially as it’s used in mathematics. Second, to give a motive to my question other than trolling I explained why I don’t think that the word applies in that sense to anything outside of mathematics, which makes it reasonable to ask what he means by the word. Third, I used the word “worldviews” to describe the epistemological frameworks in which proof-admitting truths reside. I mean by this essentially what Thomas Kuhn meant by “paradigms” in a purely scientific context, although the concept has much broader application.

Finally, I proposed that however the problems mentioned by Chambers have been solved in the past it couldn’t have to do with private property, an institution created and maintained at great cost by coercive states, which therefore has only existed for a few centuries. Like I said, I’m not arguing here for the truth of these claims, although I believe they are true, but just describing them enough to make Gillis’s response intelligible:

Naw. You can’t negate the constraints of gravity (() or complexity classes ) by changing worldview.

As to your historical appeal. 1) Almost every hunter-gatherer society recognizes some form of property, indeed the Kung San can even trade land titles! 2) No one is saying you can’t live inefficiently, but pursuit of material freedom involves pursuit of some measure of economic efficiency.

The first sentence is a single word, a negation, denotationally simple but connotationally complex. This word suggests weariness, a speaker worn down by refuting the same tired arguments against the rationality that he, as a physicist and member of the knowledge elite, is required by all he considers sacred to defend. “Sigh,” it says. “Here we go again!” This move is intended to put me in my place, to let me know that I’m not even wrong, as physicists, even the self-proclaimed variety, love to say. My thoughts are long-ago-refuted cliches, and so on. The next sentence is more complex, containing shorthand versions of two purported arguments against my claim that propositional knowledge is only possible inside a worldview. He didn’t address standards of proof at all.²⁶

The rest of his response, about the !Kung, is based on an equivocation between the standard meaning of the phrase “private property” in economic discourse and whatever it means to hunter-gatherers who, whatever they’re doing with land they control, aren’t fencing people out of it in order to enslave them. The existence of private property in this sense relies on the existence of a state. No state, no private property. The !Kung don’t have a state, so they don’t have private property in the only sense the phrase could have meant in context.²⁷

So what does it all mean? De Morgan, writing on behalf of capital’s world-spanning power, to justify his position in it and that of his peers, had to rationalize capital’s actually-existing violence. He was writing after-the-fact justifications for capital’s victory and his role in its maintenance and growth, his reliance on violence. De Morgan is defending the existing order, so has no need to court followers. His socially granted knowledge authority is an essential element of this process.

But Gillis is in a different position. He has no access to state power, no way to impose his individual will on the world by physical force, ultimately the only reliable way for one person to control multiple people. He lacks De Morgan’s legitimized knowledge authority but by hook or by crook has cooked up a functional substitute. It’s backed neither by the socially granted knowledge authority or potential state violence at De Morgan’s disposal, but it works in Gillis’s context. In short, what Gillis lacks is De Morgan’s state-associated political power.

If he wants to get anything done, then, he has to build political power outside the state. One way to do this, as I said, is through respectful discourse. Another is by dividing the world up into the chosen and the unchosen, the lynchers and the lynched. Promising followers a share in the impending violence, whether vicariously as violence-theorizers or directly as members of community lynch squads is a way to build power in the present, and that’s what Gillis is essentially up to. In particular, his unprovoked and untenable attack on an aphorism he didn’t take the time to understand can be fruitfully seen as an abortive political move.

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Postscript: Because it’s not clear to me how much of my mental space a minor and fairly inconsequential figure like Gillis is worth it’s also not clear to me if I’ll ever write the second part of this essay, but if I did it would be about the kinds of worlds that might result from Gillis’s tactics. I was joking when I called him a wannabe Robespierre, but it’s not entirely a joke. His utopian visions terrify me, which will be the subject if it ever feels worth writing.

1. Therefore God exists! Respond!
2. Augustus De Morgan. A Budget of Paradoxes. Pp. 250-1.
3. DIDEROT, voulant prouver la nullite et l’ineptie de cette pretendue preuve, mais ressentant malgre lui, l’embarras oil l’on est d’abord lorsqu’on decouvre chez les autres, le dessein de nous jouer, n’avoit pu echapper aux plaisanteries dont on etoit pret a l’assaillir; que cette aventure lui en faisant craindre d’autres encore, il avoit temoigne peu de temps apres le desir de retourner en France… Dieudonne Thiebault. Mes souvenirs de vingt ans de sejour a Berlin ou Frederic Le Grand, sa famille, sa cour, son gouvernement, son academie, ses ecoles, et ses amis litterateurs et philosophes par Dieudonne Thiebault. De Morgan also, less substantially, changes the denominator of the lefthand side of the equation to $n$ where Dieudonne has $z$, but that’s not important. I apologize for my lousy translation, which is a superficial rewrite of Google translate output.
4. I have no idea if he was or not.
5. For Chinese see The Mathematical Writings of Diderot. Krakeur and Krueger. Isis
   Volume 33(2). June, 1941. For Arabic see The So-Called Euler-Diderot Incident. R. J. Gillings. Amer. Math. Monthly. 61(1954). 77-80.
6. 1806–1871.
7. Capital’s modern uses for mathematics are so various, so technical, and so obscured that it’s hard to imagine describing them comprehensively. Piecewise methods like this essay are easier. That being said, mathematics in the service of engineering, e.g. weapons, infrastructure, strength of materials, and so on, is well-understood. For some less familiar aspects, Accounting for Slavery by Caitlin Rosenthal is astonishing. Without mathematical abstractions, she argues — among many, many other things — absentee ownership of plantations wouldn’t have been feasible. Think of how Atlantic history might have changed without this capability.
8. In A Budget of Paradoxes pp. 104ff.
9. I’m not omitting a link to Gillis’s Wikipedia page out of spite. At press time he didh’t have one. I am, however, noting that fact out of spite.
10. I can’t keep track of what these folks like to call themselves. They act like libertarians and say they’re left-wing, so that’s what I call them. Read the first few paragraphs of their Wiki page for examples of their interminable terminologizing.
11. This may not be a standard definition. I’m treating this as a technical term here, and defining it only for this discussion.
12. Is this controversial? I feel like it might be, but it’s too much to argue for in detail here. The idea is that scientific theories will sometimes conclude that the natural order of the world requires people to allow themselves to starve in the face of abundant food, to die of exposure in the face of abundant housing. No one actually believes this when it’s themselves or their family that has to starve and die, so it can’t actually be true for anyone. Anyone would do anything to feed themselves and their children, and only the threat of a more violent fate than death by starvation can make them stop themselves. So the antihuman conclusions of science must be backed up by violence rather than by reason. But scientists don’t distinguish between kinds of conclusions. They’re all just scientific truths, value-free, they are so quick to remind us. They use the same justificatory tools on all of them, so their authority is backed by violence. Without violence they’d have to convince, and no one can convince someone that it’s not just pragmatic to starve themselves, but actually right and good. This is part of the reason mathematicians and presumably other scientists repeat myths like De Morgan’s. They dampen the contradictions rather than heightening them, they soothe the conscience rather than inspiring right action.
13. I’m sorry I don’t know the technical term for this.
14. I’m dropping the qualifier “theoretical” because it’s too easy to make fun of.
15. I’m not at all claiming that possession of an advanced degree is required for someone to be reasonably called a physicist. All I’m saying is that I’m willing to treat possession of an advanced degree as sufficient to establish one’s status as a scientist. This is clearly overly generous, but in a direction that cuts against my conclusion, so it’s reasonable. Publishing scientific work is also sufficient but not necessary on this account, again being generous.
16. Isn’t it fricking convenient that no one can experience whatever it is he’s talking about? How can we deny that it’s like whatever it is he says it’s like? I’ve been a working mathematician for thirty years and I have no idea how to explain how deeply false, how deeply deceptive, this statement is. Gillis comes off as a clueless undergraduate suck-up wannabe grade booster trying to buddy-buddy a professor by falsely claiming to share their ultrarefined aesthetic perceptions. Making new science is hard work, much of it incredibly tedious, a fact missing from every one of Gillis’s descriptions. He doesn’t even know enough about science to know how revelatorily wrong his statements are. Gillis also inexplicably ignores the fact that many scientists also teach and almost all of us on the truth-and-beauty side of the business are teachers. One of our major activities is explaining whatever it is Gillis is going on about to a bunch of college kids, many of whom may or may not have glimpsed “mathematics beyond arithmetic,” but they don’t necessarily remember it. We teach those kids something about how to think about mathematics and science also. You shouldn’t take my word for it, though. If you know an actual working scientist ask them what they think of this passage and see if they don’t agree that it’s pure bullshit.
17. Examples of the overtly reasonable Gillis abound, and here’s just one:

    … under many systems of property-titles if the legal experts cannot reach consensus on who is the legitimate owner of an object nothing is done with the object in the meantime. Those involved in contending differing uses for an object in a propertyless society are directly capable of far more diverse means of negotiation, but so to, if they can’t reach consensus, then nothing is done with the object. Because literally everyone in the world has the capacity to veto.

    William Gillis. From Whence do Property Titles Arise?. Appears in Markets Not Capitalism. Chartier and Johnson eds. Minor Compositions, 2011.

18. De Morgan was defending the existing community of mathematicians. Gillis, as a revolutionary, defends at least two different communities. The first is the ideal community he and his comrades work towards and the second is the existing community made up of him and the comrades. This may complicate the analysis but I’m not really considering it in this essay.
19. Quoted in The Superior Race of Good People —
    On William Gillis’ “Bad people”.
20. Which is as good an explanation for public criminal court trials as anything else I’ve heard.
21. This thread, which drew Gillis to my attention, is also a necessary condition for this essay’s existence.
22. One possible objection to my line of reasoning here is a claim that Gillis wasn’t actually engaged in either of these activities but was just participating as an equal in the conversation. That he had no ulterior motives for participating, but just wanted to talk. To me this theory is utterly inconsistent with his over-the-top level of aggression. People whose only motive is the pleasure of the conversation aren’t generally so angry.
23. It was at this point in the interaction, by the way, that I started to think about responding to Gillis. I had no idea who he was or why he so aggressively inserted himself into the conversation but I had been working intermittently on an essay about De Morgan’s version of the Euler Diderot incident and recognized the same dynamic at play. Learning more about Gillis and his career only solidified the picture.
24. I’m applying the principle of charity here. If Gillis actually does mean mathematical proof his request is incoherent and not worthy of a response, so I don’t interpret it that way. I’m not arguing in favor of my claim that mathematical standards of proof can’t apply outside of mathematics, but only because it’s too tangential to this essay. The basic idea is that only problems which admit of acceptably mathematical solutions are part of mathematics. If a problem can’t be solved mathematically it’s not a mathematical problem. Likewise if a truth doesn’t admit of mathematical proof it’s not a mathematical truth. Does this strike you as a tautology? It is, but so is every other true statement. Change my mind about that if you can!
25. I tend to think of proving statements as something that’s only possible in a system of propositional knowledge, where truth is established deductively, or at least synthetically in line with community standards. I’d go further and say that proof is only possible in systems of knowledge that have been artificially restricted to the kinds of truths that do admit of proof. Whatever the uses of proof-based knowledge, I don’t see how either of the two positions involved in the assertion could plausibly be seen as living in such an epistemological space.

    Instead I think they’re more embedded in worldviews. If you see things one way one version is obviously true and if the other then the other is obviously true. But worldviews aren’t established propositionally. Propositional knowledge exists inside worldviews rather than the other way round.

    Or maybe the answer is much more simple than that. In 200,000 years of human history private property and commercial enclosure have only existed for a few hundred. Knowledge and economic calculation problems were solved before private property existed or we wouldn’t be here today. Can they be solved after private property is smashed along with the state? No one knows the future, but obviously I think they can or why do I fight?

26. The first argument, that “[y]ou can’t negate the constraints of gravity … by changing worldview,” seems to go like this:

    1. By way of contradiction, assume propositional knowledge is only possible inside worldviews.
    2. “[T]jhe constraints of gravity” are an example of propositional knowledge.
    3. Therefore “the negation of the constraints of gravity” is an example of propositional knowledge.
    4. Contradictory propositions can’t exist inside the same worldview.
    5. Therefore it’s possible to establish the truth of “the negation of the constraints of gravity” by adopting an appropriate worldview.
    6. Statement 5 is false, therefore statement 1 is false.

    That’s the strongest version of the argument I could come up with, although it’s not strong. I can’t even see how to refute it because it’s not clear what Gillis means by “the constraints of gravity.” If he means that I’ll still die if I jump off the roof no matter how I look at things, well, I agree. If he’s talking about the kind of thing that admits of proof, or does in his mind, maybe he means constraints of gravity as expressed in natural laws? If so, he’s going to have to explain a lot to overcome the historical fact that the constraints of gravity in that sense changed as physicists’ worldview changed from Newtonian to relativistic. Not only that, but he’s guilty of the fallacy of the converse here. I asserted that propositional knowledge exists inside worldviews but certainly not that given any piece of propositional knowledge there’s a worldview inside which it exists. Without this obviously false claim he doesn’t have an argument at all. There’s more to say about this, but not here.

27. There’s no way to be sure about what Gillis means. He might not even know himself. But given his left-wing libertarian connections it’s possible he means something like Benjamin Tucker‘s idea of every household having community-granted control over 10 acres, or some fixed amount of land, which is supposed to be as I understand it determined by how much they can use and how much over which the community is willing to cede control. Like most of Tucker’s ideas this one is self-contradictory, false, and uninteresting.