@(россия[^1]n accent, unedited) { > _When I was studying in Moscow University, Foundations of Mathematics was really a subject which was extremely unfashionable. We had this different - as we called them "cathedrals" - different groups inside the university, different groups of faculty. So there was the group of algebra, and the group of topology, and then there was also the group of analysis and the group of differential equations. > > And there was also a group of Foundations of Mathematics. > > And at the end of the second year of studies, every student had to choose a specialization. Every student had to choose which of the groups to continue to study with. That was of course a big deal, so we discussed things a lot, and different groups had different reputations. Some groups were considered to be very cool and fashionable, and some groups were considered to be very uncool. At my time, the group of algebra was considered to be cool because there were some very famous people there like Manin, and people about whom various legends were going around in the University. > > The group of topology and geometry was cool, also because there were some very great people there, like Novikov (Sergei Petrovich). > > Some other groups were kind of less cool. But the group of Foundations of Mathematics was probably the least cool group of all. Worse than that was only the History of Mathematics. Nobody really considered going there. At least nobody from "us" who thought about ourselves as good mathematicians considered going there. > > And back then I went to the group of algebra and started to study there, and I still consider myself to be mostly an algebraist, and applying the methods of algebra in various other areas of mathematics. > > So that's in a sense the beginning of the story. Today, I'm working full time on Foundations of Mathematics. So the question is "How did I get here?" and does it mean that I became entirely uncool over all these years, or is there some other reason? > > So I'm going to try to explain why Foundations of Mathematics the way we're doing them now is really a cool subject again._ > > -Vladimir Voevodsky, Fields Medalist 2002, How I became interested in foundations of mathematics. [^1]: Russia. } --- 0: _(ahem)_ _(Narrator: 0 clears 0's throat.)_ At long last! The long awaited categorization. The 32 cultures of the hardest of sciences. The basis vectors of the many nations of stem. (With a few groups of inseparable barbarians mixed in.) This is... # The /etc/groups - [[#α: The Alphas (Abstract Algebra)|α: The Alphas (Abstract Algebra)]] - [[#β: The Betas (Unassigned)|β: The Betas (Unassigned)]] - [[#γ: The Gammas (Geometry)|γ: The Gammas (Geometry)]] - [[#δ: The Deltas (Calculus)|δ: The Deltas (Calculus)]] - [[#ε: The Epsilons (Analysis)|ε: The Epsilons (Analysis)]] - [[#ζ: The Zetas (Number theory)|ζ: The Zetas (Number theory)]] - [[#η: The Etas (The Catedral)|η: The Etas (The Catedral)]] - [[#θ: The Thetas (Topology and Homotopy)|θ: The Thetas (Topology and Homotopy)]] - [[#Θ: The Big Thetas (Finance, institutional)|Θ: The Big Thetas (Finance, institutional)]] - [[#ι: The Iotas (The Institutions)|ι: The Iotas (The Institutions)]] - [[#κ: The Kappas (The Engineers)|κ: The Kappas (The Engineers)]] - [[#λ: The Lambdas (Descendants of Church)|λ: The Lambdas (Descendants of Church)]] - [[#Λ: The Big Lambdas (Foundations)|Λ: The Big Lambdas (Foundations)]] - [[#μ: The Mus (Descendants of Turing)|μ: The Mus (Descendants of Turing)]] - [[#ν: The Nus (Artificial Intelligence)|ν: The Nus (Artificial Intelligence)]] - [[#The Vega (Finance, retail)|The Vega (Finance, retail)]] - [[#Ξ: ???|Ξ: ???]] - [[#ο: The little omicrons|ο: The little omicrons]] - [[#Ο: The Big Omicrons (Complexity)|Ο: The Big Omicrons (Complexity)]] - [[#Π: The Pis (Computability)|Π: The Pis (Computability)]] - [[#Ϻ: The Misunderstood (The Sans)|Ϻ: The Misunderstood (The Sans)]] - [[#Ϙ: ???|Ϙ: ???]] - [[#ρ: "The Rows" (Not a culture)|ρ: "The Rows" (Not a culture)]] - [[#Σ: The Sigmas (Reverse Mathematics)|Σ: The Sigmas (Reverse Mathematics)]] - [[#τ: The Taus (Topoi)|τ: The Taus (Topoi)]] - [[#Υ / υ: The Upsilons (Converts)|Υ / υ: The Upsilons (Converts)]] - [[#φ: The Phis (Physics)|φ: The Phis (Physics)]] - [[#χ: The Chis (Statistics and Probability)|χ: The Chis (Statistics and Probability)]] - [[#ψ: The Psis (Scientific computing)|ψ: The Psis (Scientific computing)]] - [[#ω: The Omegas (Standard foundations)|ω: The Omegas (Standard foundations)]] - [[#Ω: The Big Omegas (Large Cardinals)|Ω: The Big Omegas (Large Cardinals)]] - [[#א: The Alephs|א: The Alephs]] --- The following is a comprehensive list of the various cultures in computing and mathematics that we'll encounter throughout the rest of the story, as we track how computing develops from 1937, where we left off, until the modern era of 20XY. Though our story will follow The Lambdas and The Mus, in order to know how they (i.e. we) fit in to the broader world around us, it is important to gain a better understanding of that world. So here is a brief summary of that world and its inhabitants. Longer descriptions of the other tribes will be provided as the need arises, in `/etc/apache`. --- _(Narrator: No relation to the webserver. I'm assuming he meant [[fsck#The People|this]].)_ --- ## α: The Alphas (Abstract Algebra) > _Abstract Algebra PhD student, to her mentor: What do you picture when you picture a Group?_ > _Abstract Algebra Professor: A capital G._ > > -Anonymous Algebraist The tribe of abstract algebra is generally more syntactically minded and less visually reliant than the other mathematical disciplines, though their introductory textbooks attempt to dispel this notion by including every picture ever taken of any symmetric object on earth. Abstract algebra arises from the examination of mathematical structures in which various generalizations of $+$, $-$, $*$, and $/$ can be defined, leading to a field focused on certain limited forms of symbolic computation over structured sets. We say "limited" because the field is almost exclusively found in practice to be studying functions of the type signature: $op: A \to A \to A$ typically written in infix notation, or maps between two structures that preserve operations of this type signature. This culture rarely studies common data types such as strings (unless the term "free" is mentioned beforehand), integers (unless the term "domain" is mentioned after), floats (though their "fields" can be somewhat floaty (except when finite)), or data structures in the sense of structs, unions, or enums. They approach their objects of study not as concrete objects to be later categorized, but as typeclasses of which instances are subsequently found. Examples of such typeclass structures are the algebraist's groups, rings, fields, modules, lattices, algebras over a field, and of course vector spaces (though there are ongoing territory disputes over precisely to whom the field of linear algebra belongs, the alphas claim some sovereignty over the region, given that "algebra" is in the name.) The field also includes a number of rarely used objects that are largely neglected by the mathematically inclined members of the algebraic tribe, but which members of the functional programming community (see the lambdas) have recognized as valuable data structures for many of their purposes. Many young lambdas, upon first contact with even small quantities of second hand category theory (see the etas) find themselves approaching unsuspecting algebraists with a manic preachy excitement and curiosity about the value of algebraic structures such as monoids, groupoids, semigroups, and other structures that are believed to belong to the abstract algebraists, even by the algebraists themselves, though the algebraists rarely study or speak of these objects, and are usually not sure what the young lambdas are so excited about in these situations. The confusion is typically resolved once the young lambda mentions monads, at which point they are gently redirected toward the etas, where they are guaranteed to be as confused by the etas as the alphas were about the lambdas. ## β: The Betas (Unassigned) It's not polite to call a group the betas. ## γ: The Gammas (Geometry) > _A point is that which has no part._ > _A line is breadthless length._ > > -The Vintage Foundational Gammas. Began with Euclid. Was considered foundations for roughly 2000 years. These days, the genetic makeup of this tribe is almost unrelated to the vintage foundational gammas. The proximity to the alphas and deltas is also not an accident. The culture of the modern gammas is exactly what one would expect if the native gammas were entirely displaced, and a culture was fabricated to fill the gap based purely on stochastic migration from the nearby tribes. If the gammas were to disappear tomorrow, the modern form of their culture could be reconstructed simply by taking any stray alphas and deltas who happened to wander here from above and below respectively, and co-mingling the interests and focus of these two groups. In doing so, one would arrive at at a culture who would often be found found studying differential manifolds via all manner of tensor algebras on them. Their metrics and connections would be both algebraic and differential in nature, and one would find almost no recognizable geometry anywhere. This is, in many ways, the culture of the modern gammas. Scarcely a diagram is found in many of their monographs. What is inevitably found is abstract algebra (α), advanced calculus of the more computational (δ) and less analytic (ε) variety, plus a largely unjustified claim to be the rightful heirs of the subject that was once called geometry. Perhaps they are its heirs. But if the modern gammas disappeared tomorrow, and this area were to be repopulated based purely on migration from adjacent fields, it would be the modern culture and not the ancient culture of geometers that we would find here once again. ## δ: The Deltas (Calculus) The once powerful culture that began with Newton and Leibniz and is commonly known as calculus. This includes differential and integral calculus in one variable, multiple variables, or infinitely many variables in the so-called calculus of variations. Found primarily in physics departments, though it is equally at home in applied mathematics. This culture and its methods are closer than perhaps any other to the language of nature herself. The full name of this culture is not simply calculus, but "The Calculus," the definite article serving as a parabolic homage to this culture's unrivaled ability to simplify computations which without its methods would require infinite resources. This culture is marked by its use of infinitesimals rather than the semi-formal methods of "analysis" found among the epsilons. Unlike the epsilons, many members of the modern deltas evince a sort of ethno-massochism, self-hatred, or compulsive tendency to apologize upon any public use of the native methods of the infinitely small that their people created. Despite being among the most useful of mathematical cultures to other tribes outside themselves, they are perhaps also the least confident and most likely to apologize for their heritage. The deltas are at present a diaspora culture without a diaspora, displaced while at the same time remaining in their original homeland. The deltas are not formally recognized as an official culture by most mathematics departments. Though there was for some time a movement to establish, formally, the existence of the deltas as a separate and distinct people from the epsilons, this movement has largely disappeared. The movement, while it existed, was founded by Abraham Robinson and known as "nonstandard analysis." Nonstandard analysis is now widely acknowledged to have been a psychological operation by the epsilons. Though various theories exist as to the motivation for this operation, many believe nonstandard analysis was designed to undermine and extinguish any sense of ethnic pride among the deltas that might have led the more radical among them to assert the right to use the native infinitesimal techniques of their people directly and without shame. While there is no hard evidence for the subversive motives of the nonstandard analysis movement, no competing theories have yet been proposed that can explain the empirical facts. Those empirical facts are that, in the years between 1961 and 1966, there existed an increasingly large contingent of young mathematicians and physicists who believed that the successes of theoretical physics in the prior half century justified the existence of the deltas and their methods as a separate formally recognized people, descending from Newton and Leibniz, whose methods should be recognized for their empirical success as being every bit as rigorous as the methods of the epsilons, whose far less fruitful constructions were considered unproblematic. Many among the younger and more radical deltas argued that the culture who created the Dedekind reals, measure theory, the Vitali construction, the Banach Tarski paradox and other abominations could hardly claim to be more rigorous or grounded than the deltas, whose supposedly less formal methods undergirded all of modern physics. Though theories vary, the empirical fact is that the inroduction of string theory to the physicists and nonstandard analysis to the mathematicians served to demoralize the young radical deltas, and the group soon fragmented and lost the will to fight. While string theory is still found in mathematics and physics departments as of the year 20XY, nonstandard analysis was a subject that appeared ex nihilo, created as if its purpose was simply to die. All that is known for certain is that the young radical deltas flocked en masse to the nonstandard analysis movement, hearing its claims to make infinitesimal arguments rigorous. And once there, the young radical deltas found themselves faced with a baroque implementation of their subject based on objects called ultrafilters whose very existence required non-constructive methods to prove. The young radical deltas thus found themselves fragmented, demoralized, leaderless, and disillusioned with pure mathematics and formalization. Many turned to string theory, and have never been heard from since. Others went into industry, scientific computing, or left the technical fields entirely. In short, despite being one of the most successful tribes in the history of mathematics, the deltas remain a people without a formally recognized homeland, their positions now occupied all but universally by the epsilons to their south, and the gammas to their north. ## ε: The Epsilons (Analysis) This tribe includes real analysis, complex analysis, functional analysis, and any of the semi-formal methods popularized by Weierstrass, Dedekind, Cauchy, et al. in an attempt to make the arguments of the δs "rigorous." Distinguished by their use of ε-δ arguments. The ε people first arrived in the region during the period of the so called "arithmetization of analysis" in the 1800s. During the arithmetization, the εs colonized and disenfranchised the δs, discouraging the use of the δs native language and techniques, infinitesimal arguments, from being used in early education, and all but outlawing it in academic research. The εs claim to social status rests largely on their claim to have civilized and formalized the δs, and not (as they insist) on having subjugated them or stolen their culture. The Epsilons are unique among the standard mathematical fields in having virtually zero popular books written about them, either covering the field's content, or covering the stories of interesting individuals in the field's history. While many popular books exist on the work of the Alphas (e.g., The equation that couldn't be solved), the Gammas (e.g., The Poincare Conjecture, The Shape of Space), the Deltas (e.g., Infinite Powers, The Calculus Wars), the Zetas (e.g., Fermat's enigma, The Music of the Primes, the Man who Knew Infinity), and the Thetas (Euler's Gem), the Epsilons appear to have virtually no stories to tell that have managed to reach a general audience. ## ζ: The Zetas (Number theory) The macho fraternity of standard mathematics. Home to mathematics olympiad winners and Putnam exam high scorers, they've been recognized as mathematically gifted since birth. However, don't be misled by the name. This field isn't just about numbers. In fact it barely even involves numbers at all these days, except perhaps a prime here and there. This group is named for the Riemann Zeta function. No further elaboration is needed. Natural puzzle solvers, this group is in some ways the analogue of the little omicrons in computing. Unlike the omicrons, however, whose culture is a sort of local minimum of optimizing for well-posed homework problems (and in that sense is a fundamental misunderstanding of the field of computing in which they live) the number theorists, in contrast, understand their field perfectly well, as well as any other mathematical culture, often more. The most widely ranging mathematical polymaths are best categorized as Zetas, given the field's expansive reach across all conventional areas of mathematics. Erdös, for example, while having published wider than anyone in the history of the field except Euler, is best categorized as a Zeta. The title of his biography "The Man Who Loved Only Numbers" bears out this categorization. The polymaths of standard math are culturally Zetas most of all. The Zetas are the home of Ramanujan, and though not one in a million has a mind as mysterious and alien as his, in many ways the field of number theory is the Ramanujan of mathematics, appearing to all the other cultures the way Ramanujan appeared to Hardy. Suffice it to say, this group is among the most advanced, wide ranging, and culturally "mathematical" parts of mathematics. In a sense, it is the capital city of mathematics itself. A number theorist is a mathematician's mathematician, a true team member, not regarded with suspicion by any subfield of mathematics. They belong in the field, and they speak its native language more expertly and fluently than any other group. In this way, the culture of the Zetas is distinct in nearly every way from the "formal number theory" of first order arithmetic commonly seen in foundational cultures. As such, a native ζ who chooses to join the culture of foundations has perhaps the most difficult journey that is possible to make, and the largest cultural transition that any mathematician can undergo in discovering the light of the new foundations. See the Upsilons for an example a zeta who made this heroic journey. ## η: The Etas (The Catedral) > _We didn't invent categories to study functors, we invented them to study natural transformations._ > -Saunders MacLane, founding patriarch of the Etas > _More simply, a monad is just a monoid in the category of endofunctors._ > -The Etas, official motto Category Theory. Completely incomprehensible. Natural transformations are commonly denoted by η, hence the name. Often referred to as The Cathedral, The Catedral, The Cat-holics, The Abstract and Concrete Cats, The Easiest Way to Get Tenure By Doing Nothing, The Hardest Way to Get Tenure By Doing Anything, and various other harmless slurs. Members most commonly refer to themselves as "The Naturals," by the term "Generalized Abstract Nonsense," or simply by drawing diagrams. Don't spend too much time here. It's best to move on quickly. ## θ: The Thetas The Thetas, - aka θ (or "th"), - not to be confused with Θ (or "TH"), - or with τ (or "topotopy") - aka the Topos culture - includes the cultures of - Topology, - Homotopy, - Homology (slightly α) - but does not include - Topos theory - for reasons explained in The Taus. - In other words: | θ is | logy | topy | | ---- | ---------- | -------------- | | homo | homology ✅ | homotopy ✅ | | topo | topology ✅ | ~~topotopy~~ ❌ | While this group claims its field concerns "rubber sheet geometry," the newcomer to this culture will find nothing of the sort in the fundamental definitions of the field. The thetas possess perhaps the most obscure fundamental definitions of any field. The first day of an introductory class among the alphas introduces groups as "sets with a binary operation that's associative and invertible," a definition that can be grasped quickly, even if it feels unfamiliar at first. The first day of an introductory class among the epsilons introduces limits of sequences defined in the typical "For all ε > 0 there exists N such that" style, another case in which the basic definitions, though at first unfamiliar, are quickly understood in terms of the fundamental motivation that might have driven their creators to settle on these definitions in the first place, rather than on others. Similarly for vector spaces, rings and fields. And even for the spaces named for Hilbert and Banach. Though unfamiliar at first, the basic definitions can be understood without a great deal of puzzlement, from first principles, as the abstraction of some idea one might reasonably want to capture. The basic definitions of the thetas, however, are unique. The definition of a topology is perhaps the most poorly motivated definition (in the sense of intuitive justification) in any field of mathematics outside the Catedral and the Topos people. Further, members of the thetas appear blissfully unaware of this fact, and proceed quickly onward to define continuity backwards, in terms of inverse images of sets arbitrary chosen in the codomain happening to come from sets similarly arbitrarily chosen in the domain. Surely this field could use a makeover, in terms of its basic definitions. This is unlikely to occur. The behaviors of the thetas are culturally similar to the epsilons, and they share the dubious claim to be rigorous generalizations of the field known as calculus. While there is overlap with the gammas in the intermediate limbo field of topological-but-not-yet-differentiable manifolds, this field appears not to even know its own name. If it is differential topology, then wherefore the topological manifold that we just assumed had no differential (i.e., smooth) structure? While the subject and its results are solid, the basic definitions of the thetas and of any field touched by theit culture become corrupted with an unrivaled lack of clarity and machinery. A manifold is an atlas of charts? God help us. Give me a metric or give me death. Note: This tribe should not be confused with the Big Thetas, who are a completely unrelated culture, and not part of mathematics. ## Θ: The Big Thetas (Finance, institutional) While vega represents fat tailed upside, the dual financial culture of Theta (capital) represents thin tailed upside and fat tailed downside. This group's homeland is quite different from the vegas. While the vegas homeland Vegas is organized around a "strip," the thetas homeland Wall is organized around a "street." At the Wall and all along the street it flanks, the big Thetas prostrate themselves with much crying, wailing, and tragedy. These are the people of institutional finance. While near in position and name to the theta of topology and homotopy, there is no relationship between these two cultures. ## ι: The Iotas (The Institutions) ``` Ivory ι ι Industry The Two Institutions ``` A transition point in the landscape. The crossroads at which various institutions intersect. The institutions: the ivory tower of academia, and the industry of the private sector. Not strictly a culture of computing, but an important feature in the landscape of computing history. Mirror image of vega, another non-computing culture directly opposite it, flaked by two cultures of aggressive innovation, the kappas and the nus. ## κ: The Kappas (The Engineers) ![[the-most-amazing-coincidence.jpg]] Descendants of Charles Babbage. Often known as the Engineers, the Machine People, or (among their detractors) the Calculator People. Creators of mechanical and electrical calculating machines, and a variety of special purpose devices for specific tasks. An engineering culture, the kappas are in some ways the analogue of the early logicians encountered in lost+found. Wildly innovative, often working alone, with an unclear relationship to what came after them. Includes early pioneers in the creation of physical devices for the automation of computing, such as: - Percy Ludgate (accountancy, developed an analytical engine independently of Babbage, 1909) - Leonardo Torres Quevedo (civil engineering, electromechanical arithmometer, 1920) - Konrad Zuse (creator of the Z1, Z2, Z3, Z4, 1936-45) - Howard Aiken (physics, Harvard Mark I, 1944) - John Presper Eckert (electrical engineering, ENIAC, 1943-46). There is considerable overlap between the kappas and the early mus, and in many cases, it is possible to make a convincing case that the devices they conceived of were nearly or fully Turing complete. For example, although Zuse's Z3 lacked conditional branching, and was thus not universal, it could be imagined to be universal by requiring speculative computation of all possible outcomes of a given calculation. Similarly, Babbage's analytical engine arguably contained the basic primitives required for universality, though this machine was never built. The separation of the Kappas from the Mus is not meant to diminish their contributions to history. Many of the earliest breakthroughs in the field we now call computing were the work of this culture. The individuals in this group are not a monolith. Aiken read Babbage and was inspired by his work. Konrad Zuse was inspired by Hilbert and Ackerman's book _Principles of Mathematical Logic,_ and could therefore be regarded as belonging to the more foundational camp. Nevertheless, this group has been given a distinct name to highlight the differences in culture, mindset, skillset, and approach from the other cultures throughout computing and the related fields of mathematics, physics, and engineering. ...( ## λ: The Lambdas (Descendants of Church) 0: Descendants of Church. Latter day members include McCarthy, Curry, Howard, Milner, SPJ, and Wadler. Well known languages descend from the lambda calculus, and include Lisp, Scheme, ML, Haskell, Nix, and Lean. The last two of these examples, namely the Nix package manager and the Lean theorem prover represent a recent expansion of the lambda culture far beyond the small academic circles which they have historically inhabited since the early 1930s. ## Λ: The Big Lambdas (Foundations) ![[the-big-lambdas-02.jpg]] ![[the-big-lambdas-01.jpg]] The Logos. Logic, Reason, Foundations. No elaboration needed. **Elaboration:** Father of λ and μ. Loves both children equally. Has praise and criticism for both. While λ was the firstborn and inherited the name, Λ is well aware that μ is several orders of magnitude more successful than λ in its career. Then again, λ may be the solution to many of μs problems, if they can find a way to get along. Either way, Λ loves them both. Λ doesn't pick favorites. Λ tries to help you, but Λ doesn't try to change who you are. Λ knows: you are what you are. [Λ is what Λ is](https://en.wikipedia.org/wiki/I_Am_that_I_Am) too. Known to make really bad dad jokes and puns, like spelling "Land" as ΛΛ b/c Λ is L & Λ is And, in Greek Λ Logic respectively. Don't ask. Bible thing. Also a dad thing. Bibles are a big bag of dad jokes. So sayeth the ΛVΔ. ## μ: The Mus (Descendants of Turing) 0: Descendants of Turing with influence from Gödel. Machines, Moore's Law, practical computing, von Neumann architecture, eventually Unix, C, Python, and everything that runs the modern world. Culturally distinct from the lambdas, though the two cultures share a common origin, and have been gradually recognizing their need for a return to a common culture in recent years. Examples include the increasing presence of lambda expressions in μ languages, and the increasing recognition of the need to minimize shared mutable state. )... ## ν: The Nus (Artificial Intelligence) > _Have you heard the good news?_ > -An absolute fanatic. The nus. Cousins of λ & μ, the νs have been around since the beginning of computing, though they are a culturally separate people with distinct norms and traditions from the λs and μs, primarily due to the influence of the δs via backpropagation and the concept of "training" via gradient descent. The culture of the nus has been shaped by repeated mass immigrations and emigrations of outsiders due to hype. The culture therefore consists of an unusual mixture of devoted intellectuals and distracted business types and assorted hype gremlins. In this sense it is not dissimilar from the culture found in cryptocurrency circles. Always focusing on the "new," the nus are the only culture in computing and mathematics known to undergo regular boom-bust cycles, known in the field as "AI winters." The existence of these boom-bust cycles is not due to the nature of the discipline itself, but rather due to the popular appeal of its core goals which has a tendency to attract non-technicals who are interested in anything "nu." Predictably, the non-technicals often grow bored and leave as quickly as they came. As a result, members have grown wary of any mention from outsiders that their field has become "over-hyped," and members were often heard in past eras to utter among themselves the phrase "winter is coming", as they prepared for the funding to dry up and the excitement to abate for another generation. Many believe that the final AI winter has now passed, though the elder members are split among themselves on this issue, both with regard to whether they believe this is in fact the case, and with regard to their individual "p(doom)" -- the probability each person assigns to the proposition that the accelerating advances in their field will literally lead to the collapse of civilization and/or the destruction of all human life. ## The Vega (Finance, retail) The letter "vega" is a non-existent Greek letter, arising from Finance Bros seeing the letter "ν" (nu) used as a parameter to describe the volatility of stock options, and then pretending to know what letter it is. This culture is neither part of mathematics nor computing, though the various other cultures described here have often been found working technical jobs in companies alongside them, with varying degrees of amusement and nihilism. The vegas home is known as Vegas, where they are often known to pilgrimage to engage in the rituals of their people. Represents volatility and the hope of fat tailed upside. ## Ξ: ??? ... UϺ Nobody knows what this is. Sure would be nice to know something. Has variously been called The Xi, The 11, The 3, The 三, The San (sic?), The Word, The Speech, The Call, The LoΓoς (what?), The Yue, The S\*n, The M\*n, The You, The Ni, The 山ξ, Mount S\*n, Mount Sh\*n, Sh\*n S\*n, 山三, The `/*ɢʷad/`, Mount 二二@ (citation needed?), as well as The Γ曰⅂ and The L曰⅂, among others. Seems to be a mixture of schizophrenic gibberish with essentially all of Unicode. Often found in the /boot directory, in kernel space, scrawled in manic graffiti under bridges and overpasses, and in corrupted files in lost+found. Somehow related to the Greek letter [Ξ](https://en.wiktionary.org/wiki/%CE%9E), Chinese character [三](https://en.wiktionary.org/wiki/%E4%B8%89), the Latin letter [L](https://en.wiktionary.org/wiki/L), the Greek letter [Γ](https://en.wiktionary.org/wiki/%CE%93), the Hebrew letter [⅂](https://en.wiktionary.org/wiki/%D7%93), the Chinese characters [日](https://en.wiktionary.org/wiki/%E6%97%A5), [月](https://en.wiktionary.org/wiki/%E6%9C%88), and [曰](https://en.wiktionary.org/wiki/%E6%9B%B0), and sufficiently many others that it's not technically possible to list them all without meeting the DSM-$\forall$[^1]'s diagnostic criteria for mania, and against that background, it's probably best I wrap up. [^1]: A universal quantification over all current and future versions of the Diagnostic and Statistical Manual of Mental Disorders. Pronounced "Dismal." So far, no existing group has claimed responsibility for the Ξ inscriptions. No individual or group appears to possess the ability to write it or read it. Hence, nothing concrete about this writing style is known. It's not clear if it belongs to any past or present culture, or whether's it's simply a fabrication of \\ patterns where none exist. Only the inscriptions appear, without any known equivalent as spoken word. I've got guesses, but nothing I'd be comfortable sharing in a place like this. Only intuitions, not a bit of hard evidence. Or maybe a bit. Not positive. ## ο: The little omicrons The leetcode kids. Puzzle solvers, lovers of structured homework problems, frantic studyers of how to invert a red-black tree on a white board. These may or may not be developers at heart. All we can know for sure is that they're nervous kids doing their best, and studying in precisely the worst way imaginable. There are no interesting polynomial time problems at work, kids. Work involves syscalls and ill-posed problems, often presented in the form of mysteries that no one is even attempting to solve, let alone assign to you as work. It's your job to find out what the interesting problems are, and learn to tell the signal from the noise, because in the real world no one's gonna be there to do it for you. Someone tell these kids they're playing sudoku. They don't know. They're new. I mean Christ, it's not their fault. Academia's dead. ## Ο: The Big Omicrons (Complexity) The culture of Complexity theory and thick books on algorithms. The people who write the leetcode problems consumed exclusively by the little omicrons. This is a largely insular two-sided subculture, and it is nearly but not entirely orthogonal from developer culture at large. Adjacent to big Pi, computability theory. Part of the orthodocs. ## Π: The Pis (Computability) The culture of Computability theory. Part of the orthodocs. The name Π comes from the [Kleene Mostowski hierarchy](https://en.wikipedia.org/wiki/Arithmetical_hierarchy), in which the complexity of the first-order formula that describes a given set is related to its degree of unsolvability. ## Ϻ: The Misunderstood (The Sans) The Constructivists. Named for the deprecated Greek letter [San](https://en.wikipedia.org/wiki/San_(letter)), written Ϻ in uppercase and ϻ in lowercase, and pronounced "S or Sh." Nothing like what one expects at first glance. Often described as a Σ turned sideways, both in its shape, and culturally as an area of mathematics (see Σ below). Similar to Ξ if we allow a cross-language pun, but it's generally assumed that this is an accident, and definitely not some kind of blessing from the gods or an omen about how the ϻeek will inherit the earth. I mean wtf even is San? That's not even Greek. (Linguistic Note: Though San had become largely deprecated and fallen into disuse by the latter half of the 5th century BC, it is often found in abecedaria, particularly those recovered from churches. San became largely obsolete by the second half of the fifth century BC, when it was generally replaced by sigma. Represents no known culture or people in userspace. Despite this, it is often found in enumerations of the /etc/groups, for unknown reasons. ## Ϙ: ??? Deprecated Greek letter Qoppa, written Ϙ in uppercase and ϙ in lowercase. Derived from Phoenician [qopf](https://en.wikipedia.org/wiki/Qoph) meaning "eye of needle." Represents no known culture or people in userspace. Despite this, it is often found in enumerations of the /etc/groups, for unknown reasons. ## ρ: "The Rows" (Not a culture) Common name for the Rectangle Pseudo-culture: A degenerate case of computing, it primarily affects non-developers aiming to accomplish some business or scientific goal, though in more severe cases it may affect those who are culturally developers as well. While object oriented programming may be described by the short-hand "Everything is a noun," and functional programming may be described by saying "Everything is a verb," the Rectangle pseudo-culture can be described by the (never explicitly stated) belief that "Everything is an adjective." In other words, it is _adjectives,_ not nouns or verbs, are taken to be primary in this non-culture. In the "dataframe supremacy" variant of this disorder, all data one recieves is stored in a rectangle with named columns and typically unnamed rows. In this case, it is the adjectives or attributes (namely columns) that recieve names, while the nouns that those attributes belong to (rows) recieve no name, either as types (e.g. Person) or as values (e.g., Dave). In such situations, the fundamental objects on which the program operates are simply represented by unnamed integer indexes into a large rectangle in which they live. Not all uses of rectangles as a data structure are a mistake. For the symptoms to meet the diagnostic criteria for this condition, the use of rectangles must lead to a significant impairment of an individual's decision making ability in technical arenas relative to their overall technical competence. Areas known to be affected by the Rectangle pseudo-culture include: - In modern scientific computing: Matlab, R, Scilab. - In modern business computing: Excel, SQL. - In previous eras: Often found in latter day academic uses of Fortran, but not in its creators. - In the modern era across content domains: The Python scientific computing stack (e.g., Numpy and Pandas) is perhaps the most widely used rectangle toolchain as of the time of this writing. While useful for many problems, overuse of these tools without compensating non-rectangle programming outside of work or school often leads to a progression of the rectangle disorder that undermines the user's ability to achieve their goals. - Exceptions: - APL: Though it is technically an array based language, experts largely agree that APL does not count as an instance of the rectangle disorder. Though consensus is not universal on this matter, the clear majority opinion is that the use or admiration of APL is a distinct phenomenon. Though use of APL comes with its own set of costs and benefits, it should not be confused with the rectangle pseudo-culture or the disorder observed within it. - Tensorflow, Torch, Jax: Many users of these frameworks didn't come for the rectangles, and don't reach for them as the solution to other problems. Sometimes, you just need your variables to be differentiable. I mean seriously, nobody wants to do tensor calculus by hand, except maybe when we're first learning general relativity. But for users of these frameworks, your tensors aren't even real tensors, they're just big obnoxious high dimensional rectangles, and you're just trying to get things done and move on to something nu (see above). Life is about trade-offs. Don't blame these people. Progression of Symptoms: - Early symptoms of the rectangle disorder manifest as a set of _behaviors_, not as a set of explicitly stated beliefs. Such behaviors can be summarized by the phrase "Everything is a rectangle." In early stages of this disorder, every object one represents in any domain is represented as a row of attributes describing the object in question. - Intermediate symptoms commonly involve either proposing or actually using SQL databases of various forms, in cases where a simple text file or directory thereof would be more than sufficient. - In severe cases, as the disorder progresses, affected individuals often acquire a form of Stockholm Syndrome in which they develop explicit beliefs about the superiority of rectangles as a data structure for all or most problems in computing. In such cases, affected individuals are often found using terms or phrases such as "primary key," "foreign key," or "that's just a join," in areas having nothing to do with databases. Such individuals may even brag about their affliction, with language like "I just use Postgres for everything," or "A filesystem is just a database." - The most extreme cases of the rectangle disorder do not lie on a continuum with the other three severity levels. There is disagreement about whether or not the most apparently severe cases of this disorder should be treated as, in fact, more severe at all. Though the symptoms appear on the surface to be more severe than any of the above, the underlying cause among tier 4 cases has been hypothesized to be causally distinct. This is an area of active research, and it is beyond the scope of this file to elaborate on the details. See [[rect|/var/lib/rect]] for a more in-depth discussion of these cases. ## Σ: The Sigmas (Reverse Mathematics) ![[godel-collected-works-092.png]] > > Kurt Godel, collected works Sitting here between two weirdos, others often fail to notice it. Named sigma, due to their common use of the Kleene Mostowski hierarchy, shared with Π. Everything is Σ separation or Σ comprehension. ## τ: The Taus (Topoi) - Claims to be theta. - Speaks like eta. - All introductions and and prose paragraphs sound like Λ, λ, μ or Ϻ (san). - Topos theory claims to be many things, but it is not yet clear (even to Topos theory) what, in fact, it is. For this reason, the group θ described above includes the topologists and topology adjacent mathematics cultures of homotopy theory, but not the Topos people as they are still [sketching their elephant](https://ncatlab.org/nlab/show/Sketches+of+an+Elephant). ## Υ / υ: The Upsilons (Converts) > _It's not easy, but I think it is possible. And such a possibility is emerging for the kind of foundations which are now being developed based on ideas coming from theoretical computer science mostly, at this time. So we need new foundations which will be formulated in a formal system which can be used, despite its inconsistency or possible inconsistency, can be used to construct reliable proofs. So the classical first order logic is not good at it, because of it has an inconsistency then one can prove everything and it stops being informative. However there are other types of formal systems which can be used for the formalization of mathematics. Which react to inconsistency in a much less drastic way in a sense. So inconsistency in such systems doesn't mean the system becomes totally uninformative. And one example of such classes of systems is the so called constructive type theories. And that is a class of formal systems which have been used extensively in the theory of programming languages._ > -Vladimir Voevodsky > _My biggest regret is that I never got to talk to Voevodsky about this, two months after I started playing with Lean, Voevodsky died, so I never got to ask him why he never defined schemes. I think he'd lost interest in Algebraic Geometry, I think he'd become a constructivist, he got interested in Foundations of Mathematics._ > -Kevin Buzzard The Converts. The letter upsilon represents a Fork (Υ) or a U turn (υ). This group represents those who, whether from disillusionment or inspiration, become converted in some way to the new Foundations. > Upsilon is known as Pythagoras' letter, because Pythagoras used it as an emblem of the path of virtue or vice. > > Lactantius, an early Christian author (ca. 240 - ca. 320), refers to this: > > For they say that the course of human life resembles the letter Y, because every one of men, when he has reached the threshold of early youth, and has arrived at the place "where the way divides itself into two parts," is in doubt, and hesitates, and does not know to which side he should rather turn himself. > > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge These individuals begin in the other cultures of /etc/group, primarily mathematics. Sometimes they come from θ, like the Fields medalist Vladimir Voevodsky, who after losing faith in traditional peer review discovered the new foundations, and contributed an entirely new branch to it before dying tragically young. Sometimes they're world class ζs, like Kevin Buzzard, who reached the top of number theory before becoming disillusioned with the state of modern Langlands and peer review, and who has since spent years building a community around the new foundations. Others who come to upsilon have touched so much of mathematics that it would not have been inappropriate to give them an entire letter for themselves, like Terence Tao, though unfortunately τ is already taken. In many ways, these individuals remain members of the cultures of their origin after they discover the new foundations. They were usually never much interested in the old foundations. But somehow through a personal journey they come to see the light of the new foundations and are converted. There tends to follow a period of proselytization, where these new converts travel around the globe giving talks to the youth of their native fields, explaining the need for the new foundations, and the promise that they hold for their home culture. Though no /etc/group is a monolith, the upsilons are especially diverse in their views. They have widely varying beliefs about the importance of constructive foundations, and about the important of The Nus in the future of mathematics. But all of them eventually discover the modern merger of Lambda and Mu, and what they see convinces them that the future of mathematics lies with the new foundations. ## φ: The Phis (Physics) Phi. Physics. Self explanatory. ## χ: The Chis (Statistics and Probability) Names for the Chi squared statistic. Self explanatory. Note: Probability theory ala E. T. Jaynes is Λχ. The rest is statistics. ## ψ: The Psis (Scientific computing) Psi. Scientific computing. Self explanatory. ## ω: The Omegas (Standard foundations) > _Somehow set theory won, for some random reason._ > -Kevin Buzzard Standard Foundations. Set Theory, Model theory, Proof theory. Descended from Cantor and Gödel. ZFC and NBG live here. Culturally part of the Orthodocs. Was believed to be the foundations of mathematics for over a century, until recently beginning to lose ground to computing as the mus and lambdas began to merge again for the first time in a century. ## Ω: The Big Omegas (Large Cardinals) > _The ultimate Large Cardinal axiom is "0 = 1"._ > -Anonymous > _The "0 = 1" axiom is sort of a joke (though it's not wrong). The large cardinal axioms are stronger and stronger in consistency strength meaning they prove more and more. Taking "0 = 1" as an axiom is then the strongest thing you can assume since you can then prove everything._ > -A user named \[deleted\], somewhere on the internet Large Cardinals. The most unhinged subculture in all of human thought. Somehow ended up in "foundations of mathematics," despite doing exactly the same things that led to the original paradoxes, in much more concentrated form. Ingests heroic doses of the most potent substances daily. Part of the orthodocs. Mathematics now loops back around to α, and the cycle continues infinitely. ## א: The Alephs The alephs. The Nameless authors of the books we call Bibles, and of many books in the same genre but not known by the name. Little is known about this group, but whatever is known (e.g., of the J, E, and P sources and the various redactors of the Torah) will be referred to under the broad label of "The Alephs."