Over the years I have come to form a vague but persistent conviction that there's a distinct analogy between reactionary ideologies and certain mathematical constructs.

To the true believer of (say) "ethnonationalism", their ideology must surely make sense in their head, feeling as sound and whole and true as a mathematician's appreciation of a complicated proof. It's been a long time since I've worked complex mathematical problems but I'm aware of the clear cold feeling of truth that comes with successfully working out a difficult exercise. It's like a sudden gust of wind from a high snowy mountaintop, or like the pleasurable sting of being hit in the face with cold water on a hot day. One feels, for a moment, as if illuminated by starlight. I have no doubt that all the fascist ideologues I've seen scurrying around on Twitter, with names like "TheSensibleCentrist" and "Apollonius of Fresno", or the ones who proudly advertise a long string of memberships in "non-political" think tanks, all believe they've experienced this same brisk, bracing feeling of uncovering a deep truth. They must feel like their ideas make sense.

And I think perhaps they do make sense, but in the same way that a Klein bottle makes sense, or the way that one can find solutions for otherwise impossible mathematical problems among the complex numbers, i.e. numbers that have no physical representation. "Imaginary numbers" excited my imagination (hah) as few other intellectual concepts did, when I was a child learning mathematics. Aha! negative numbers do have square roots! So they do, but a "real" square root may be constructed with a compass and straightedge, while an imaginary root can only be represented symbolically. The real root of a function is a dot on a graph; the imaginary root isn't to be seen on the graph at all and must be obtained through symbolic manipulation. Not all mathematical entities, even if they can be logically defined and manipulated as readily as ordinary numbers, have no concrete form. One can't assemble -5 + i√2 beans in one place.

Hence I conjecture that extremist ideologies of all sorts, not just right-wing ones, must have some equivalence to such virtual constructs in mathematics. They can be defined and discussed, but they can't ever be realized in the material world, any more than one can construct a real Klein bottle, only an approximation or lookalike to one. In the heads of the "gender critics", for example, their ideas must make perfect sense and seem like the only logical ones. They measure the rationality of the world against their bizarrely contorted (but logically consistent) hypotheses, find that the world doesn't measure up, and blame the world for not being real enough. They would like to codify their ideas in a fully rational and concrete form, because then it would be persuasive. They can't ever get it quite right but no matter, they'll keep trying the way that people are still trying to "trisect the angle".

The example of the Klein bottle is especially useful here, for the difficulty with building Klein bottles and many other such mathematical surfaces is that they require "self-intersection" in three dimensions. If we could somehow imagine two wholly separate pieces of matter occupying the same space at once, as if they could pass harmlessly through each other, then you could make a genuine Klein bottle, but real-life matter does not behave that way. One sees something like "self-intersection" happening with fascist ideologies. Consider the "doublethink" involved in the infamous TERF slogan, "Sex is Real", which seems like mere tautology. The slogan works for them because of "self-intersection". The word "Sex" is carrying a heavy load of multiple meanings. They wish outsiders to see "Sex" and think of real-world things—sexual things. But they also wish "Sex" to mean something subtle and insidious and indeed metaphysical, referring to their absurd desire to enforce the "laws of biology" through coercion and violence.

Can this be formalized somehow? The analogy between harmful ideology and abstract mathematical constructs seems so natural that I'm persuaded to believe that it's valid and can be thrashed out in some practical form.

~Chara of Pnictogen

An excellent response, and I'm trying to figure out how to respond, and explain further where I was coming from.

It's something like this. I don't think that a fascıst ideologue has actually worked out their reasoning in advance. I don't think they've consciously reasoned out their beliefs—indeed, I think it's likely that many of their core beliefs are things that they don't even recognize as beliefs, because in order to maintain themselves in public they need to avoid articulating everything that's on their mind. However, I've come to think that they must FEEL, in a wordless way, that all their ideas hang together into a coherent pattern. They must feel the sort of certainty that comes from thinking they've got everything worked out in their heads. Hence I conjecture that, internally, their minds must contain some sort of coherent system—some construct in which every concept links up, in a locally rational manner, with other concepts. I don't think the kind of certainty that these people think they have, their indomitable sense of rightness and certitude and purity of logic, can possibly exist merely through bluff. In some way, their beliefs must make sense to them and all link up with each other in a logical way.

Hence I conjecture that they're able to do so only through constructing logical systems that are mathematically equivalent to "impossible surfaces". I don't think they're doing this consciously. I don't think they're even thinking about mathematics at all! But I think they've constructed chains of reasoning to defend their beliefs that are "impossible", and if you could map their interconnections among concepts in some 3D way, I think you'd see the equivalent of an "impossible" or self-intersecting surface.

~Chara of Pnictogen

I'm sure lots of writers have made the obvious analogy between American "process cheese" and the general homogenization and flattening processes behind the generation of mass-market American culture. One can't deny there's a certain American genius for processing things. British culture has its processed foods but it's like they want to stay stuck in a Victorian time warp in which it's still novel to put ground-up beef into a tin or a bottle. I like certain British processed foodstuffs but they all have a sort of "we're still working out the kinks" taste. Things taste strong without necessarily tasting good.

American processing, though, has become a fine art. More inventive cuisines have figured out new things to do with American foodstuffs, too, e.g. Korean Army Base stew, which is delicious. One can sneer at the poor imitation of respectable foods, but the processed foods have unique properties and serve somewhat different purposes. The gluey quality of American cheese is perfect for burgers, but not much fun on a cracker. American processed entertainment must seem like that to outside viewers. You get a whiff of that from the way tokusatsu occasionally gets its American moods, and suddenly there's burgers everywhere and Rangers in cowboy hats, all in good fun.

I have spent so much time being merely pissed at being an American, like my mother was pissed. (She was more than peeved, I'm sure...more like "enraged to the point of paralysis".) Fortunately I evaded the H. P. Lovecraft trap and escaped becoming a wannabe British gentleman, sniffy about the colonials, but it was a near thing and it's left traces on our plurality. My mother was always watching PBS, which during the 1970s and 1980s (and onward) was relentlessly British. All the really good programming was imported. My mom detested Britain too but she still preferred them to anything American. She read a lot of British mystery writers and took a subscription to the Guardian because she couldn't stand American newspapers. So, as a kid, I quickly acquired the idea that Britishness was better.

Uh, er, em. I got better!

I...almost feel proud to be an American all of a sudden.

~Chara of Pnictogen

Over the years I have come to form a vague but persistent conviction that there's a distinct analogy between reactionary ideologies and certain mathematical constructs.

To the true believer of (say) "ethnonationalism", their ideology must surely make sense in their head, feeling as sound and whole and true as a mathematician's appreciation of a complicated proof. It's been a long time since I've worked complex mathematical problems but I'm aware of the clear cold feeling of truth that comes with successfully working out a difficult exercise. It's like a sudden gust of wind from a high snowy mountaintop, or like the pleasurable sting of being hit in the face with cold water on a hot day. One feels, for a moment, as if illuminated by starlight. I have no doubt that all the fascist ideologues I've seen scurrying around on Twitter, with names like "TheSensibleCentrist" and "Apollonius of Fresno", or the ones who proudly advertise a long string of memberships in "non-political" think tanks, all believe they've experienced this same brisk, bracing feeling of uncovering a deep truth. They must feel like their ideas make sense.

And I think perhaps they do make sense, but in the same way that a Klein bottle makes sense, or the way that one can find solutions for otherwise impossible mathematical problems among the complex numbers, i.e. numbers that have no physical representation. "Imaginary numbers" excited my imagination (hah) as few other intellectual concepts did, when I was a child learning mathematics. Aha! negative numbers do have square roots! So they do, but a "real" square root may be constructed with a compass and straightedge, while an imaginary root can only be represented symbolically. The real root of a function is a dot on a graph; the imaginary root isn't to be seen on the graph at all and must be obtained through symbolic manipulation. Not all mathematical entities, even if they can be logically defined and manipulated as readily as ordinary numbers, have no concrete form. One can't assemble -5 + i√2 beans in one place.

Hence I conjecture that extremist ideologies of all sorts, not just right-wing ones, must have some equivalence to such virtual constructs in mathematics. They can be defined and discussed, but they can't ever be realized in the material world, any more than one can construct a real Klein bottle, only an approximation or lookalike to one. In the heads of the "gender critics", for example, their ideas must make perfect sense and seem like the only logical ones. They measure the rationality of the world against their bizarrely contorted (but logically consistent) hypotheses, find that the world doesn't measure up, and blame the world for not being real enough. They would like to codify their ideas in a fully rational and concrete form, because then it would be persuasive. They can't ever get it quite right but no matter, they'll keep trying the way that people are still trying to "trisect the angle".

The example of the Klein bottle is especially useful here, for the difficulty with building Klein bottles and many other such mathematical surfaces is that they require "self-intersection" in three dimensions. If we could somehow imagine two wholly separate pieces of matter occupying the same space at once, as if they could pass harmlessly through each other, then you could make a genuine Klein bottle, but real-life matter does not behave that way. One sees something like "self-intersection" happening with fascist ideologies. Consider the "doublethink" involved in the infamous TERF slogan, "Sex is Real", which seems like mere tautology. The slogan works for them because of "self-intersection". The word "Sex" is carrying a heavy load of multiple meanings. They wish outsiders to see "Sex" and think of real-world things—sexual things. But they also wish "Sex" to mean something subtle and insidious and indeed metaphysical, referring to their absurd desire to enforce the "laws of biology" through coercion and violence.

Can this be formalized somehow? The analogy between harmful ideology and abstract mathematical constructs seems so natural that I'm persuaded to believe that it's valid and can be thrashed out in some practical form.

~Chara of Pnictogen

It may be conceded to the mathematicians that four is twice two. But two is not twice one; two is two thousand times one.

That's a line from G. K. Chesterton's The Man Who Was Thursday, and it's a great line. Then he follows it up with "That is why, in spite of a hundred disadvantages, the world will always return to monogamy," because that's my lovely friend Gilbert for you, always bringing it back round to Catholic orthodoxy somehow.

The context? The protagonist, Gabriel Syme, who has felt himself to be alone as an undercover detective amidst a den of anarchists, realizes that there's another like him, and he's overwhelmed with a feeling of camaraderie and strength. One person, alone, needs to be a tremendous fighter indeed to face an entire mob. Two persons, back to back, just might be able to cover the whole field. Chesterton is right: two is much, much stronger than one.

It's observations such as this one which prompt me to think that there's something genuine about numerology, which as a youthful science nerd I was taught to flee with horror, the same way astronomers are taught to despise astrology and chemists are taught that alchemy is obsolete mediaeval junk. Every major science, I suppose, has its twilight counterpart with evil connotations. I'm sympathetic to astrology these days but I'm also forced to admit that it's a tool of bıgotry. The shadow companion to biology is truly frightening—Lysenko, "race science", and eugenicism live with biology's sinister twin. Numerology conjures up other horrors, like people irrationally frightened of the number 13 or who gamble away entire fortunes because they have feelings about odd integers. It's tempting to think, as the science nerds think, that it's all nonsense.

Yet I've come to think there's value in numerology, though I'm still uncertain about how best to express it. Observations like Syme's remind us that there's a subtle difference in quality and mood to numbers and how they interrelate. Two is not merely one more than one. And three is not merely one more than two. x² + y² = z² has infinitely many solutions among the integers; x³ + y³ = z³ has none, and proving this was the work of centuries. Ordinary mortals can't even understand the proof—I certainly don't. The "two-body problem" is an undergraduate textbook example; the "three-body problem" has no general analytical solution, though many special cases are soluble. Once again we sense that going from 2 to 3 is a fearfully complicated business in reality. It's not simply counting.

There's an opposing extreme to numerological complication, I daresay, and it's to be seen in the bean-counters and the computer programmers who seem to regard numbers as dead objects, meaning nothing more than quantities to be shuffled about with calculators or computers and plugged into simple formulae. How many programmers realize that the sequence 1, 2, 3 is not actually simple? It's not merely incrementing an object. In reality, altering numbers (even by a simple increment of one) produces drastic changes sometimes. The computer code, however, makes it seem quite dull. It's merely...advanced bean-counting. It's not surprising that the programming world has virtually merged with the business and financial worlds, and commits all the same sins.

I—well, the Pnictogen Wing—have daydreamed about attempting to formalize astrology and even alchemy in some way, establishing a more solid bridge between these disreputable domains and the world of empirical science. Perhaps structure can be applied to numerology as well.

~Chara of Pnictogen