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## Introduction

1 “Music” is not musical notation and the theory of chord progressions. It is possible to play music without reading music, and you don’t need some minimal level of training to hear and understand music.

2 “Math” is not mathematical notation and rigorous proof. It is thinking about something, and then bringing it up in conversation to examine it. To define it further. To think about what properties it must have. To come up with examples to make it more specific, or to come up with generalizations, sometimes in more than one way.

3 Suppose you are enjoying thinking about triangles. You might start by drawing a triangle on a piece of flat paper and figuring out what is true about it. You might then try and figure out what is true about every triangle that can be drawn on flat paper—that would be one way to generalize. Or you might decide to see how triangles drawn on paper differ from a triangle that you could draw on a globe. Or you might decide to generalize by thinking about four-sided figures, and five-sided figures, and so on—generalizing in the direction of ‘true for multi-sided figures.’ Or you might generalize by thinking about what triangles look like in three dimensions—you’d have to decide whether to think about flat, two-dimensional triangles in three-dimensional space, or maybe you’d prefer to think about tetrahedrons, since those are what you get when you add a point outside of the plane of a triangle.

4 Some people love the notation and the rigor. I’m one of them. But it’s not necessarily the thing that has to come first, or even at all. We can play with abstract ideas without having to write them all down in a careful scrimshaw of Greek letters, because everyone can talk about ideas.

5 You are a human.1 You can use words, which are abstract, to refer to things which are concrete. That is your birthright as a creature of language. So you also get to talk about things that do not “exist” in the same way that that rock over there and this air over here exist. And because you can talk about those things in quasi-existence, you can imagine a world that is finer than the one you live in at this moment.

6 Which brings me to my second point about the philosophy of punk and mathematics: the role of destruction in social criticism. Why are things so fucked up? Some people would say, “Because of capitalism.” Others would say, “Because of not enough capitalism.” Either group can agree that colorful bits of paper—or electronic signals that represent what colorful bits of paper have historically represented—which have no inherent value except what we collectively assign to them, influence and control our lives and how our lives interact, and influence and control how matter mined in Africa is refined and assembled in China to the specifications of Californians and then shipped across the world to become a prized attention-seizing device in Poughkeepsie.

7 Matter governed by physics, moved about by global shipping networks, rearranged to align with patterns of hard- and software engineering, becoming “owned” by a consumer, going obsolete, becoming a lump of fairly expensive and somewhat toxic matter again.

8 Why does the world have to be this way? Will it be this way for much longer? Why is there poverty in some places and abundance in others? Is the system failing or working exactly as designed? Do we smash it or try to perfect it? Is there No Future? Am I part of the next-to-last generation and what do I do about it?

9 Well, I don’t know about you, but I’m going to apply math to it.

10 You may not believe me, but math is really an elaborate and infinite game. It is important to note that math is more than simple arithmetic. I can do some mildly impressive computations in my head, because I know the commutative and associative laws (we’ll meet them soon enough) to transform a computation I find difficult into several computations that are easier. But, more times than I’d care to count, I will say something like: “Six times three is twenty-four” and totally screw everyone up when we’re divvying up a dinner check.

11 Math is also not statistics, although statistics is a division of math. A knowledge of statistics would help folks understand that climate change is real, and that inequity is a problem. If someone writes “Punk Statistics” I’ll be the first in line to buy it. But this is not a book on statistics. The real world is messy and uncertain. Statisticians can not only tell you how likely it is that their conclusions are true, but how frequently it is that they will incorrectly tell you that their conclusions are true.

12 If arithmetic is about how to combine numbers, and statistics is the relationship between numbers in the world, then mathematics in the broader sense is the study of patterns. It is the way things relate, the patterns appearing in those relations without caring much about what those things are. In brief, math = real facts about imaginary things.

## The Problem

13 Now, numbers may be imaginary concepts—have you ever met “3” in a dark alley?—but that’s not stopping them from kicking our asses right now. Consider certain measurements of the natural world: the number of parts-per-million of greenhouse gases in the atmosphere, and the decreasing area of the arctic ice. The number of people on the planet increasing exponentially, and the volumes of water and dirt staying stubbornly constant.2 And then there’s the volume of earth’s petroleum, burning slowly toward zero.

14 And those are just measurements of the physical world. There are social measurements, too, though their relation to the real number system is a bit fuzzier. For example, the 85 richest people in the world control as much wealth as the bottom 3.5 billion people—or half the population of the planet.

15 Putting off for the moment what “the total wealth” even means as a concept,3 consider that a dollar is a sort of vote on how to spend our world’s limited resources and our population’s limited time. (How money flows and how prices vary, after all, are supposed to determine how capitalists and workers decide where to apply their capital and labor. If there isn’t a lot of something, the price is supposed to go up, so that if you buy it anyway, you’re saying, “It is so important that this thing gets produced that I vote that more people should get on producing it.”) Doing a very rough back-of-the-envelope calculation, the richest people in the world have 40 million votes to the single vote of, say, an average Liberian. You may not reject this on its face, but consider the sentiments of Georgina Rinehart, the fourth richest woman in the world, lamenting that Australians are not willing to work for $2 a day. While we may empathize with the loss that bequeathed her this inheritance, we may wish to think about the implications of Ms. Rinehart’s dollars-as-votes for constructing society. 16 There are many other numbers with political and social implications. The exponents describing the debts of post-Econopocalyptic Iceland and Ireland. Returns on investment from Congressional lobbying on the order of 22,000%.4 Probabilities of risk selectively assessed in whatever way ensures that the deepwater drilling goes forward and that the nuclear power plant gets built, and intellectual property decisions that lead to ten-dollar medications being sold for$1500.

17 Beyond simple numbers, there are other, weirder abstract structures manipulating us and the world, and they don’t seem to care that they are imaginary. When I trade inedible pieces of paper to a total stranger in exchange for food, I am participating in a mathematical structure we have decided to believe in for convenience. On a larger scale, note that American houses are made of the same atoms no matter who owns them, and yet, perturbations in the imaginary structures of prices, debts, and ownership have caused great economic suffering, and produced the peculiar state of affairs in which the number of American houses laying empty in 2010 outnumbered the homeless by a factor of five-to-one.5

18 We’re also finding that the computing revolution is supercharging the ability of certain well-studied mathematical structures to screw up our lives in a variety of entertaining and dangerous ways. Trading firms use high frequency trading algorithms to buy and sell the ownership of stocks and commodities on a scale of nanoseconds, making staggering profits without doing a joule of actual work. Our worldwide computer network facilitates both political protest and state repression, and within the net’s ecosystem, services like Facebook exist for free by selling the users themselves—or at least, social graph data stratifying us into precision-marketable demographics.

19 Well, fine. Invisible mathematical monsters surround and coerce us. But don’t we have nerds to handle these problems? Modern culture has grown comfortable offloading hard computational problems to professional geeks, and expecting the supercomputers on their desks and in their pockets to handle distasteful numbers if they come up. People who are otherwise intelligent and curious will cheerfully admit to giving up on learning mathematical thinking. I have yet to come across someone who says, “Meh, I never really got ‘reading.'”

20 Innumeracy is not a problem because it hurts nerds’ feelings. It is a problem because the innumerate are easy to mislead.

21 People use the average Joe’s poor mathematical understanding as a way to control, exploit, and numerically fuck him over. When the American political sphere debates cutting heating assistance to the poor, we don’t talk much about how the cut is equivalent to about five days of American warfare costs, because the public doesn’t understand exponents.

22 By the way, the best method I’ve found to help my dumb ape brain understand the relative enormity of huge numbers—millions, billions, trillions—is by considering how many seconds there are in each. A thousand seconds is about 15 minutes. A million seconds is about 12 days. It’s possible to imagine counting to these numbers, although you’d have to skip sleeping so I wouldn’t recommend it. A billion seconds would take about 32 years—it is inconceivable to imagine actually counting to it, but, it is a length of time that is within the comprehension of a human life: I’ve lived for more than 1 billion seconds (=1 gigasecond), but I’m unlikely to live for 4 gigaseconds. A trillion, though? Nearly 32,000 years. That’s not merely prehistory: that’s before the Ice Age, before the first domestic dogs, and before the extinction of the Neanderthals.

23 The moral is: In matters of State and the Economy, don’t pay so much attention to the digits in the number, as to the number of digits the number has. In trying to conceive of enormous numbers, the magnitude must be considered first, before the particulars of the digits. It’s the difference between a sign on a locked-up store that reads, “Back in 30 minutes” versus one that says, “Back in half an Ice Age.”

24 This rule-of-thumb is closely related to scientific notation which, once you get used to it, allows you to easily compare the order-of-magnitude difference of numbers. When I write 9 million as $9×10^6$ and 2 billion as $2×10^9$, it reminds me that the larger leading digit of the millions number is nowhere near as important as the order-of-magnitude difference: nine minus six is three, and so the billions number is around a thousand ($10^3$) times as big. But I digress.

25 The problem is: people who believe we can offload mathematics to computers let themselves to be pushed around by programmers and financiers, lobbyists and lawyers (and don’t get me started on casino operators). (Gosh, can you imagine… what if there were segments of society that somehow benefited from us not thinking too hard about these things? “Remain calm. Keep shopping. Everything’s fine.”)

## School Sucks Math Rules

26 Since the aforementioned Joe certainly received a basic math education in school, and yet Joe’s eyes dart around for the nearest exit which I start explaining how important it is to distinguish 10-to-the-7th policy discussions from 10-to-the-10th policy discussions, it seems clear that basic math education is failing us.

27 When I taught, many students made it clear that before they got into my classroom, before they experience me shouting and waving my arms and jumping up and down preaching the beauty and frustration of math, their teachers emphasized algorithms and answers over concepts and questions. To them, math appeared to be a series of cargo cult pencil-and-paper algorithms (ritually performed to avoid the Mark of the Red Pen), proofs by intimidation (if there were proofs at all), and meaningless algebraic rituals for finding answers to contrived questions. Math classes frequently lack real-world motivation, or if motivation is present, it is financial—textbook authors seem to believe that nothing fires students’ imaginations like amortization and compound interest.

28 Since students’ interests are not respected, they cannot find intrinsic motivation—the perfectly rational response to a question like, If Billy is twice as old as Sally was two years ago, and Sally will be Billy’s age now in ten years, how old is Billy? is “Go fuck yourself.” Instead, they are motivated by the extrinsic reward of grades, which we know to be less effective for learning.

29 Furthermore, emphasizing hand computations when your typical college student could solve it in seconds using a five-dollar smartphone app makes the traditional math class seem downright Kafkaesque—if the only thing that matters is “the answer.”

30 Students exposed to this kind of math education conclude that math is not relevant to their interests or their world, and it’s hard to fault their reasoning.

### The Solution: Punk Mathematics

31 “Breaking down the distance between performer and audience is central to the punk ethic. Fan participation at concerts is thus important; during the movement’s first heyday, it was often provoked in an adversarial manner—apparently perverse, but appropriately ‘punk.'”—unknown, Wikipedia

32 Too many people see math as a set of arcane rules—delivered from on high—that one manipulates to pass a final exam and then eject from memory. But to me, mathematics is a deep and playful game that demands the courage to fail spectacularly, to make noisy mistakes, to ask inconvenient questions, to call bullshit when you see it, and to believe nothing until you’ve proven it for yourself. In short, math is punk as fuck.

33 In punk rock, both social and political commentary is wrapped in chaos and feedback. Taste and beauty are rejected for technical accessibility and energy. The secret ingredient is relevance, interest, and accessibility. Passion, anger, and the intent to see structures as they are, not as we wish them to be.

34 What I find personally maddening about the common misperceptions about mathematics is that in reality, math is probably the least authority-dependent subject of study. Whereas the sciences have theories, explanations of data which have been observed (but probably not by you), mathematics has theorems: we make as few assumptions as we can, then prove universal truths outward, carefully stepping on the stones of logic, heeding intuition, but remembering that obvious “truths” are frequently falsehoods. Any skeptic with enough time and frustration-tolerance can in principle verify any mathematical claim with only a pencil, some paper, and some friends with whom to work out ideas (or their written words, if these friends are dead). Math is cheaper than chemistry, cleaner than biology, and the most dangerous thing you can do is divide by zero to see what happens. No government-funded particle accelerators or tweedy experts are necessary.

35 Mathematics is made of words and symbols, numbers and diagrams, agreed-upon definitions and rules of transformation. As is the case in poetry, the language does not merely describe the thing, it is the thing. Therefore, people doing mathematics together are on even ground—what is right is determined by the mathematics that you’ve written down so far. In this sense, mathematics is anarchistic. (Note: the etymology of “anarchy” suggests its original meaning: “no rulers!” [“an-,” without; “arkhos,” the archons, or chiefs] which is quite different from the common conception that “anarchy” means “no rules!”)

36 If you are playing chess with world-class chess player Judit Polgar and you were to somehow discover a flaw in an opening she thought was good, you would only need to say: “But… look!” and upon examining it, she could determine if she was in error. If you are playing “Let’s create a 5-year economic plan” with Josef Stalin and you identify a flaw (e.g., millions starving, general suffering, bunches of death), saying: “But… look!” will not get you anywhere but the gulag.

37 It is likely that a mathematician or chess player would actually be excited to be proven wrong; the new understanding of truth would radiate outward in their concept map and update their general knowledge of how their discipline works. There is no board of trustees determining mathematical truth—we are free to explore the logical entailments of whatever mathematical constraints we choose. We are not even bound to talk only of worlds that are known to exist in our own physical, biological, or social reality. If we’re careful about our definitions, we can venture beyond the looking glass into realms of alien logic as well.

38 People don’t generally associate mathematics with breaking shit, with questioning authority, with auto-critique and medium as message, with photocopies and safety pins. That’s only because most people don’t know about the insolubility of the quintic, the nature of the axiomatic method, with Godel’s theorem and Wiener’s information theory, about logical algebras and industrial optimization.

39 The reason that punk mathematics is a thing worth thinking about is that punk’s naturally aggressive drive to obliterate bullshit in all forms can be the guidance system for translating what many of us “believe” are “facts” about the “world” into mathematical language. Mathematical language is a magical tongue, good as any shaman’s words-that are-things, a lawyer’s pro-ergo-procto-cum-coc, a hacker’s source code. The worst problem for a mathematician is to have no problems. Punk identifies problems loudly and brooking no bullshit; math solves problems.6 The specific magic of mathematical language is that it un-utters falsity: statements that are not true drop out of the universe of consideration.7 When you are working a mathematical problem, there are ways of keeping straight when you are assuming, when you are using, and when you are contradicting.

40 Both punks and mathematicians have an instinct for calling bullshit. Both punks and mathematicians will insist on verifying claims for themselves. Both punks and mathematicians create far more bad noises and wrong paths than successes, and this is why they succeed. Both punks and mathematicians recognize that things worth doing may well be frustrating, or terrifying, or beyond their capabilities, and then they try to do them anyway. In a punk mathematics, there is no discipline but self-discipline, no truth but proven truth, and it is better to be wrong than quiet.

41 And everyone gets to play—regardless of their skill.

42 At the beginning of a term, students will come up to me and say, apologetically, “I’m sorry, I’m really bad at math.” They’ve accepted this, internalized it as a true fact about them—as an external fact over which they have no control. They’ve decided that they were passed over for some “math gene” in their brains. Fuck. That. Shit. I explain to them that humans are bad at math: logical chains of inference are no good for escaping saber-toothed cats, and thinking in that way is hard work.

43 Grades don’t account for the wonder of failure.

44 When I taught college mathematics, my job was to create a function from the set of facts about, say, calculus to a performance of duties. Then my other job was to create a function from the set of infinite learning experiences in their infinite combinations and infinite diversity to the tiny finite set: A, A-, B+, … and F.

45 But my students made such interesting mistakes! This system of grading leaves out “Fail better.” And THAT IS how you find out what works and what doesn’t.

46 Aside: How many functions are there from a class of size N to A B C D F? Five raised to the nth power. Is there a linear order of the N, of how good they are at whatever’s being taught? No. It’s the same flaw as IQ tests, assuming that there is a ranking of intelligence instead of a multiplicity of partially-ordered, fascinating multitypes, with emotional intelligence, logical, verbal and visual all playing their parts in the aforementioned mathematical lenses.

47 Math should occur on both sides of the brain. Dance is subdividing four with the body. And math should occur above and below the eyebrows. Why be ashamed of counting on your fingers? If you’re willing to teach the muscles of your hands binary, you can count to 1023 on your fingers.

48 Your eyes are attuned to symmetry

49 Fig. Scott McCloud’s symmetry images from Understanding (or was it Making?) Comics.

50 and your ears are attuned to modular arithmetic, through your perception of rhythm in music. If you are a musician, you might further be intrigued by the world of harmonics, which govern musical instruments (though not bells or drums, those are a different mathematical beast), or L-systems, which can be used to algorithmically generate self-similar structures like chord progressions or melodies.

51 “Here’s a chord, here’s another, here’s a third, now form a band.”
{image}

52 This felt-pen-and-photocopy drawing from 1977 says so much to me about punk rock. It presents three chords without prescribing an order for them. It introduces tablature notation in a way that ought to make sense once you figure out what end of the guitar points up. But most importantly, it says: Find an instrument, find some friends, and join us, right now. This is what I aim to do for the mathematically downtrodden—the mathematically discontent.

53 I will grant it takes a lot of so-called ‘mathematical maturity’ to explore these fields in detail. And it takes rigor and precision—skill with the instruments of mathematics—to advance them.

54 But who says we need to advance them? Why not screw around with them? What’s the worst that can happen? Maybe you’ll reinforce a cognitive bias; maybe you’ll incorrectly reject a true hypothesis; maybe you’ll monkey around with a system you think you understand but don’t know.

55 There are notions and concepts at the ‘ground level’ that should be open to everyone, that require a solid explanation and not a million complex calculations. Concepts that use language, or if-then statements, or counting, or a bit of simple algebra that you can be re-taught with a few short examples. These concepts, unpacked, can lead to new patterns of thoughts about how the world might work. You, dear readers, and my dear students from semesters past, I want you to help disambiguate what is, from what must be.

## very axiom / so constraint / many generative / amaze

56 When you ask a mathematical question, there often is an answer OUT THERE because it predicts many things. The rate of falling objects. How to make a hologram out of crazy “coherent” light. Yet it is also derived a priori—based on theoretical deduction rather than empirical observation—out of the machinations of the mind. Mathematics is therefore imaginary, abstract, invented. Because it is about the world AND entirely imaginary, mathematics is “a set of patterns that fit the world and the mind at the same time.”

57 You might, therefore, think of mathematics as being a sort of property of the universe, or of all possible universes, and if you like you can take this as proof of the existence of God. Or, if you are of the opposite disposition, you can take it as proof against, since math pops for free out of any universe and so He (or She) has been made “unnecessary.” If you are Lakoff, you think of mathematics as a set of metaphors that people came up with, and so mathematics is “only” those kinds of thoughts that people can think.8 Whereas Kronecker said that God gave us the integers, and all else was the work of mathematicians.

58 What concepts and definitions we choose to begin with determines the mathematical universe we are working in. The fundamental concepts that are assumed in order to begin are called axioms.

59 Now, no one knows why the Internet does what it does, but one of the things it does very well is create micro cultures of imaginary grammatical rules of adorable animals. One of these sets of grammatical rules is a fine example of the beautiful paradox of axioms: they constrain the choices we can make in our mathematical or linguistic universe, yet simultaneously create an enormous and generative space in which to play.

60 I would like to tell you about the grammar of doge. In 2010, a Japanese woman put a picture of her shiba inu dog, Kabosu, online. Something ineffable about this dog’s expression and round face struck an etheric resonance field on the internet, or something, and people began photoshopping this picture and adding text.

61 Much like the predecessor meme, Lolcats, the appropriate text for a doge meme has a broken grammar—as though we expect that if our pets could talk, then they would break grammatical rules, like young children do. What makes doge unique compared with Lolcats is that the grammar is very regular.9

62 A “doge phrase” can be one or two words long. The one-word phrases are chiefly “wow,” “amaze,” “excite”; since it’s questionable to claim a list of single words has any grammar at all,10 we will concentrate on the two-word phrases.

63 Here are some examples:

64 {LIST OF AWESOME DOGE PHRASES}
{Let us gather unique examples from the tubes of the Internet.}

65 The first word of a doge phrase comes from a very small set, chiefly “very,” “such,” “so,” “much,” and “many.” The second word can be just about anything, but it must violate the usual “selectional restrictions” of these words so it produces an effect that sounds “off.” In order to speak fluent doge, you must possess a fluency in English that permits you to make word choices that are deliberately less coherent.11 Much strange. Very word. The grammar of doge is ridiculous, and highly restricted in how you can say things. But it does not restrict what you can say. As in mathematics or in chess, the rules are inviolable but the expressions within those rules are nearly infinite.

66 On Twitter, @SecurityDoge tweets on the fight for the open internet (“very surveillance.” “much corporate.” “so regime.” “scare.”). And people all over the internet use doge to comment on how they are feeling about their everyday lives. Occasionally, to make oneself clear, one violates the rules of doge grammar in order to express yourself clearly. But such rule-breaking is more the domain of poetry than that of mathematics. Consider Shakespeare, who coined over 1700 now-common words like “road,” “addiction,” “laughable,” “hint,” “gossip,” “invulnerable,” “bedazzled,” “majestic.” He wrote sonnets, which have restrictive metric and rhyme rules. What might he have done with the grammar of doge?:

67 “What light. So breaks. Very sun. Wow, Juliet.

68 What Romeo. Such why. Very rose. Still rose.

69 Very balcony. Such climb.

70 Much love. So Propose. Wow, marriage.

71 Very Tubalt. Much stab. What do?

72 Such exile. Very Mantua. Much said.

73 So, priest? Much sleeping. Wow, tomb.

74 Such poison. What dagger. Very dead. Wow, end.” 12

75 Well… maybe not.

## The Nuts and Bolts of Mathematics

76 The mathematicians who have come before have bequeathed a huge library of abstractions they found particularly rich, beautiful, generative of surprising results, or applicable to real non-dog-grammar problems, in much the same way the English language is enriched by Shakespeare and other speakers generating new and useful terms for expression that are picked up (or not) by other speakers. Through a long process of consensus across centuries, mathematicians have come to consensus on what definitions seem to lead to interesting results. Similarly, the consensus of English speakers establishes a grammar that is particularly precise and generative of meaning. But so long as we’re clear, we can choose axioms to give us a new mathematical or linguistic universe to play in.

77 On occasion, choosing a new axiom leads to new universes that are very different but nonetheless equally valid and consistent as the one you’ve been in before. (See: Euclidean and Non-Euclidean Geometries in Chapter X)

78 On very rare occasions, we drop an axiom and discover something true about our own universe. When asked how he came up with Special Relativity, Einstein said he had “ignored an axiom”—specifically, the previously assumed axiom that time is universal, as though there were an Absolute Clock ticking away the seconds for us all. It is obvious that two observers, no matter where they are or how fast they are moving, will agree whether event $A$ preceded event $B$ or not; however, this “obvious” thing is also wrong.

79 More frequently, messing with these axioms leads to contradictory or degenerate results. For example, an editorial error in a linear algebra text led Stephen M. Walk at St. Cloud State University into an examination of a structure13 which “looks like a vector space, walks like a vector space, and quacks like a vector space,” but instead of having a single unique zero element, has many zero elements: one for every other element in the space. “[T]his fragmenting is antithetical to the foundation of anything that we call ‘algebra,'” he points out, “that foundation being the ability to solve equations by undoing operations.” With non-unique zero elements, we can no longer, say, subtract three from both sides of $x + 3 = 7$, because when we do so, we have no guarantee that the zero we get from $3 + (-3)$ is the same zero that makes $x + 0 = x$ a true equation. Or in more metaphorical terms, try to imagine a space of modular furniture pieces — let’s call it an ikeaspace — in which screwing together two pieces could never be undone if you’d made a mistake. Doing algebra in a degenerate space is nowhere near as fun as being a degenerate.

80 In practice, there are some axioms that most mathematicians take up, like the Axiom of Choice, because not taking them up seems really weird.14 If there is a statement which cannot be proven from the axioms, and cannot be disproven from the axioms, the statement is called undecidable. And there are many. These are the weird edges that bound separate mathematical universes from one another, and unless you want only to prove theorems that do not require the use of those statements15 you will have to make a choice of what universe you want to live in. At least, if you want to live on these weird frontiers.

81 Having made our choice of assumptions, many consequences occur. For obvious reasons, in theoretical physics we choose assumptions whose consequences match up with experiment and observation in our own universe. In math, we don’t have to do that, but there are still a lot of necessities that come from our choices, and many choices explode in contradiction quickly. It is the nature of the mathematical game that not all sets of rules are equal.

82 We might choose a smooth universe of waves and fluid dynamics, or a choppy one of counting problems and prime number factorizations. We might choose to explore the set of all possible relations that can be drawn among 7 billion entities, or the 219 repetitive patterns that crystals in 3-space can form (it’s 230, when you can tell the difference between right and left). Or we might select some axioms for rationality, choosing to study how we choose. You can choose to explore exciting forks of mathematical reality—nonstandard analysis, doxastic logic, and topics even further away from the K-12 curriculum. If you do go off to explore new frontiers of mathematics, don’t forget to write.

## Definitions, Propositions, and Theorems

83 Math is formed in the interplay between choice, constraint, and freedom. You can choose axioms to give you a new mathematical universe to play in, and once we’ve chosen, we get to derive propositions by way of logical inference. You can define new terms as you like, to wrap up a particular concept you want to explore.

84 Without the ability to define new concepts, you wouldn’t get very far (we wouldn’t want to go through proving every theorem directly from axioms and rules of inference anyway because it would take forever). To do this, we list out some properties, and say that any object $X$ that satisfies those properties gets named… a whatever. The name is just a shorthand to say, “things that satisfy these properties.” This way of defining mathematical objects may seem silly and tautological but that’s how definitions work: “a bachelor is an unmarried male”—that sort of thing. We permit ourselves to create new abstractions out of the old ones.

85 If mathematics really is a language—as they say—this is what makes it so. By wrapping up new concepts in new definitions, the language becomes more powerful: it can express more concepts, more precisely and more eloquently. The speaker’s thoughts are denser, and finer distinctions can be made. There are languages that lack distinct names for green and blue, and people who grew up speaking those languages have some difficulty picking out the difference visually.16 When a name is given to an unfamiliar concept, you can see that concept where before it was invisible.

86 The longer our little mathematical program—the longer we think through the implications of our initial axioms and develop new concepts with names—the more concepts we can express, and express them in simpler terms.

87 Anything we prove about such abstract objects will be true for anything that has that list of properties. But, this does not work in reverse: if we pick an example object, it may have additional qualities that are not true for every object in the domain of the abstraction. Every square is a rectangle, but not every rectangle is a square; Hitler was a vegetarian, but not every vegetarian is Hitler. This is why we can use examples to hone our intuition and to look for counterexamples, but we can’t use examples as a substitute for proof, if we’re doing serious mathematics (so we’re off the hook there).

88 A mathematical proposition is something like, “The square of an even number is itself even.” It is a clear statement that has been proven true. A proposition that is proven in order to be used in the proof of something more important is called a lemma: they are the pitons of mathematical mountain climbing, pounded into the problem surface to step on but not important in their own right.

89 An important proposition is called a theorem. And if a theorem is very important, it might be called a fundamental theorem.17

90 Because a theorem has two parts (the hypothesis, and the conclusion), we are always beginning sentences with “Suppose Y” or “Given an X such that P is true” or similar constructions. I’m claiming that if it is ever the case that the hypothesis statement is true, then the conclusion statement will also be true. If you only say things that start with suppositions, then you are not responsible for your hypothesis being misconstrued as a statement about everything.

91 You may not believe this yet, but ultimately, mathematics is extremely simple. Not easy: simple. If you permit yourself only to speak undeniable truths, you’re going to have to keep the questions simple.

## An Illustrative Story in Mathematics

92 I’d like to take the time to give you a hint of what doing mathematics feels like. This story is based on actual historical accounts (possibly embellished), and it is my favorite story about slowing down to think deeply and playfully about the properties of the imaginary things numbers are.

93 A little over two hundred years ago, there was a young schoolboy who attending lessons in Germany. One day he was in math classroom, and the teacher instructed all the pupils to add the numbers from 1 to 100 into one big sum and tell him what it was. This was perhaps because he felt this was an important exercise in addition, or possibly he simply wanted all the kids to shut up for 20 or 30 minutes.18

94 Now this was at a time when paper was still handmade, and presumably pencils were, although I don’t know if they had pencils yet, but at any rate, paper was way too expensive to treat as cavalierly and disposably as we do now. So the students did their work on little rectangles of slate, writing on them with chalk. Each boy got his own.19

95 The teacher is presumably rooting around in his desk for his flask or rosary beads or whatever, and the kids start furiously calculating. One plus two is three, plus four is seven, plus five is, uh, twelve, plus six is eighteen plus 7 is… um, twenty-something…. You can imagine that they probably started out blazing fast and then started to slow down as they had to really think about each sum.

96 Meanwhile, as his classmates are cranking away, our boy is just sitting there, thinking. There are a lot of tellings of this little classroom story, but in mine, he’s leaning back, with his hands in his lap or on his desk, just staring off with a faraway look. And after a moment, he noticed something funny.

97 The funny thing came from imagining all the numbers, from one to one hundred, all laid out in a line. I see them as going from left to right. You can see them as starting with one on the left, and then, glossing over some numbers, turning your head to the right and “seeing” 100 at the end of this line of numbers on the right. (I actually turn my head sometimes when I think like this.) The curious thing that our schoolboy noticed was that the two ends of his line of numbers (1 and 100) added to 101—but so did 2 and 99, the next-to-leftmost and next-to-rightmost numbers. And the third-to-leftmost number and the third-to-rightmost number would be 98 and 3, and those also sum to 101. And the pattern continues, all the way to the two central numbers, 50 and 51, which sum to 101.

98 So he folds the row of numbers in half, as though there was a hinge between 50 and 51. Try to imagine it in your mind: a row of numbers from 1 to 50 going left to right, and folded under, a row of numbers from 51 to 100 going from right to left. Each column adds to 101.

99 Can you see it? Try to see it. You’re holding a hundred numbers in your head all at once, but they’re all sequential so it’s no big deal. Each column adds to 101. So how many 101’s is that? The numbers here are polite enough to count themselves: that top row ends in 50, which tells us we have fifty 101’s. So the folding trick turned 99 increasingly difficult addition operations into a single multiplication problem: 50 \times 101. One hundred fifties makes five thousand, plus one more fifty makes 5050. Voilà! There’s our answer.

100 Now remember, this schoolboy is just sitting there, thinking. Looking for all the world like he’s blown off the problem as too difficult or too boring. Then suddenly he leans forward, writes the correct answer, 5050, on his slate, walks to the front of the room, tosses it on the desk, and says, “Ligget se”: There it lies.

101 Not bad for an eight-year-old little boy. Fuckin’ A.

102 Now, I want you to look at what our hero did at an abstract level. He didn’t apply massive computational horsepower to his problem. And trust me, he could have; this was a child who was correcting his father’s accounting at the age of three. Instead, he stopped, thought, imagined, created a picture, and observed what the inside of his mind looked like. Instead of going off half-cocked in a frenzy of computation, he slowed down to walk around his problem and view it from all angles.20

103 The teacher, to his credit, realized that while this kid may have been a smart-ass, he was a smart-ass with incredible potential. With his own money, he bought him the best mathematics textbook he could, reportedly saying, “He is beyond me—I can teach him no more.”

104 The smart-ass schoolboy grew up to be the finest mathematician of his place and time—Karl Friedrich Gauss. {Give a shout out to the internet meme of Gauss Facts here? “Erdos believed God had a book of all perfect mathematical proofs. God believes Gauss has such a book.”} Probably he had a natural, inborn talent for doing mathematics, but the important part of the story is how Gauss demonstrates what, to me, is the essence of mathematical thought. By taking his time to reframe the problem, the solution was not merely possible—it was obvious.

### The Problems Today Belong to All of Us

105 Our world, by definition of unsustainable, is coming off of hyperconsumerism, US industry dominance, and the post-bullshit era (with new ones to come, don’t worry). Seventy years or more of military-industrial-advertising complex, hundreds of years of colonialism, and hundreds of years of centralized currency, and ape power games dating back millions of years. I have a lot of concerns about how global hypercapitalism seems to be organizing our world for maximum ecological destruction and minimum compassion for humans who are disempowered, whether that disempowerment is because of where they were born, what they look like, what genitals they have, or a whole host of factors that lead our society to ignore their beliefs, their safety, and their suffering. Our problems are, in part, medieval institutions, paleolithic emotions, and newly godlike powers. So many smart people working to get people to give up the names of everyone they know, and to click on ads.

106 The misuse of power is the most bullshit of all. But “power” is an overloaded and ill-defined concept. In physics, it’s clear enough: power is defined as a rate of work done per second. If we can both lift a heavy weight over our heads, but I can do it twice as fast as you can, I am more powerful.

107 What is power in the societal sense? My working definition is: power is the ability to constrain another’s actions. I posit this as my working definition because, at least in principle, this ought to have a measurement: compare what an actor is able to do alone, and then compare their possible actions with a new element. If the collection of actions that can be taken expands under a new condition, the actor has been empowered. If the collection of actions that can be taken contracts under a new condition, they are disempowered.

108 Consider the following power structures that are in a state of flux. Our information flows around. When it flows around through a 3rd-party, the NSA grabs it, and the courts don’t protect it because it’s not on paper. 3-D printing is about to disrupt the material flows in ways we can fantasize about but not predict.

109 Media power influences the information we take in and the conversations we engage in. When there were just three networks, there was the “news of the day”; Walter Cronkite signing off his nightly newscast with, “And that’s the way it is.” This brought about a mindset in which there are “Two sides to a story” instead of a different story depending on whether you’re unmonied, colored, genderfucked, or otherwise marginalized. Because there were only so many minutes to be devoted to news (on television or radio) or only so many column inches (in a newspaper), the number of possible conversations around “the news of the day” were severely constrained by whoever was making the editorial decisions. This metanarrative—the story that there were a small number of stories—wasn’t so much an intentional conspiracy to construct societal discussion as an effect of the constraints of the media of the day. But to satisfy those physical constraints, a small number of demographically similar editors — mostly white, mostly dudes — filtered the complexity of the world through their own biases and instincts. In retrospect, “All the news that’s fit to print” should probably have been “All the news that fits in print.”

110 Now, there has been an explosion in the number of points of view. A Cambrian explosion of stories, from cheap media production platforms and cheaper distribution networks. These new points of view are not necessarily controlled by white men with advertisers to please; these new points of view can help illuminate the sexist racist classist nationalist ableist ageist poverty-blaming slut-shaming working-class-dividing Muslim-surveilling black-person-shooting health-care-denying mental-illness-stigmatizing water-despoiling toxin-spewing nature-fracking bias-reinforcing woman-silencing school-defunding child-droning terrorist-producing austerity-forcing species-extinguishing vaccine-backpedaling elder-threatening dissent-criminalizing infrastructure-corroding discourse-oversimplifying kyriarchal system of oppression that, don’t forget, is also killing the bees.

111 Human knowledge and communication, and in particular communication across the internet, is big. Like, very big. We have moved from an information-scarce to an information-abundant environment. The main corporate video hosting site, YouTube, claims to receive 100 hours of uploaded video every minute.21 Put another way, it would take you over a year of continuous viewing to catch up with the video that’s been uploaded in the last hour and a half.

112 The explosion of mobile technology means that the first connection some people are having to the media are mediated through their phones. There are a couple billion people that will be added to the media network in the coming couple of decades.22 Media power is falling apart, fragmenting, losing the metanarrative, going weird and flowy. Information and media melting into internet meme ooze. 23

113 Monetary power on the other hand, controls the way we dispose of our time. (And our resources, if we had any.) The need to accumulate wages constrains many, many hours of our time. It makes us not do things that are beneficial to society. I’ve been paid as a teacher, and the pay is shit. (If only I could just get it in my head that, as spoken by the Investment VC Market, the most pressing challenge issue of our time is to get people to click on ads.)

114 That power is not falling apart. Rather, it seems like it’s using the increased fluidity of the information we generate to insert itself in between every human transaction it can 24 But it is going weird. The taxicab and hospitality industries are in a flux of a “sharing economy” brought on by services like “Uber,” “Lyft,” and “Air B&B” that allow people to “network” these services peer-to-peer, rather than going through conventional industries. But market forces and agents are going weirder than just being ever-present. Bots that operate by algorithmically trading at a nanosecond level read the online rumor mills to attempt to make pricing predictions; as a result, it has been observed that when actress Anne Hathaway has a good news day, the shares of Warren Buffett’s Berkshire-Hathaway corporation experience a small but noticeable rise in value, a phenomenon known as the “Hathaway effect.” Elsewhere on the Weird Money Internet, cryptocurrencies—forms of money created through sort of cooperative, sort of competitive algorithm for keeping a distributed ledger—experience surges and crashes in value; these currencies include but are not limited to Bitcoin, the alpha software of distributed finance, and Dogecoin, a currency created as a joke that derives from a meme that arose from a slightly funny photograph of a round-faced dog, that at the time of this writing trades for N American dollars.

## Math as a Powerful Tool

115 There is a huge amount of bullshit out there that could really use at least a half-assed mathematical analysis. The oldest bullshit aspects of society, like war and oppression, and relative newcomers like consumerist sleepwalking. The idea that poverty and environmental degradation are necessary simply in order to have an economy. The systems of control, the structures of power, and the aftereffects of the de-organization of society. Is it coincidence that American wages entered their long stagnation just as large organized labor strikes ceased? That might be a question that can be answered. After all, collective bargaining is a game theory problem—so why not do some game theory, rather than declare that unions rule or drool based on unexamined assumptions, political tribalism, and family lore?

116 Mathematical truths are not dependent on The System (whatever that is), and so math is a uniquely powerful tool for examining our biases and social constructs, whether your beliefs are rooted in science, religion, or neither. We need a toolkit to understand the scale of the problems we face as a species, identify the structures of society that can aid or interfere with possible solutions, and contemplate what truth is in a complicated universe full of primates trying to control one another with various forms of bullshit. Punk mathematics is one such toolkit. In particular, we can use mathematics to assess the validity of 20th century structures as petro-capitalism spins down, to make independent personal choices in an Age of Bullshit, and to imagine new ways to use the emerging network structures to tackle the hard problems of our endangered civilization.

117 At first, I assumed the phrase “the purpose of a system is what it does” was some punk anarchist description of The System, when in fact it was the phrase of a cyberneticist.

118 The Firm and the State do not have eyes or hands; they operate by algorithmic methods. They always have: censuses, surveys, records, charters, debts—all are finite categorical or numerical expressions of the will of one-dimensional beings that do not live, yet act. They attempt make us discrete, orderly, and finite. Predictable—so much the better. Classify and measure us and our relations to people and things to make us finite—the reduce variety of our expression. You know they’re not afraid of getting their hands dirty to figure out the angles and profit opportunities.

119 So ask questions. Stupid questions, uncomfortable questions.

120 For gods’ sake, ask uncomfortable questions.

121 Lincoln said that if he had four hours to chop down a tree, he’s spend the first three sharpening his axe. Einstein said that if he had an hour to save the world, he’d spend the first 55 minutes defining the problem. The challenges that we face in trying to avert Total Global Clusterfuck are not going to yield to “Let’s do what we’ve always done, but faster.” Instead, we need to think like Gauss: sacrificing mindless computation time to find clarity, generality, and elegance. We need to see through the fear, uncertainty, and doubt down to the heart of simplicity.

122 Change is possible; a new future is possible. We need tools to dissect old structures and pull out what is good and worthwhile, and relegate the out-dated, aggressive and nasty zero-sum ideas to well-deserved oblivion.

1. Or cyborg, or uploaded brain state vector, or text-mining robot, or some new consciousness I have no name for. I don’t judge.
2. It’s actually worse than that—apparently soil depletion is a thing. Fucking hell.
3. This is surprisingly slippery, and computing a reasonable estimate probably requires summing an infinite series. Why on earth should that be the case? Consider this: if I deposit money in the bank, it doesn’t just sit in a vault gathering dust; it is used to give out loans. Suppose my bank gives you a small business loan with the money I’ve deposited. Do you have my money? Maybe kinda, but you will keep the money not in your pockets but in a bank account, which is loaned out to another small business, ad infinitum. *Literally*, ad infinitum: if you want to try and compute how much money there is you’ve got to sum an infinite series.
4. Measuring Rates of Return for Lobbying Expenditures: An Empirical Case Study of Tax Breaks for Multinational Corporations, Raquel M. Alexander, Stephen W. Mazza and Susan Scholz, Journal of Law and Politics, Vol. 25, No. 401, 2009.
5. http://www.truthdig.com/eartotheground/item/more_vacant_homes_than_homeless_in_us_20111231#. Data from 2010. The number of empty houses in Europe outweigh their homeless population by a factor of two-to-one: “Deutsche Bank has warned that it will take 43 years to fill the oversupply of empty homes in Ireland at the current low population growth rate.” http://www.theguardian.com/society/2014/feb/23/europe-11m-empty-properties-enough-house-homeless-continent-twice)
6. Or, at the very least, precisely describes the conditions under which solutions may exist.
7. I suspect those false statements actually do kind of “go” somewhere, but that it would take more logic and type theory for me to know for sure.
8. That’s pretty amazing to contemplate: the set of all thinkable thoughts!
9. The comment on pets expected grammar, and the grammatical deconstruction, is from Gretchen McCulloch, “A Linguist Explains the Grammar of Doge. Wow” on the-toast.net.
10. Although, McCulloch observes that the shortest possible form of a word is used in these: “scare” makes a better addition to one-word doge phrases than “scary” or “excitement.”
11. For those who are anxious to get into a bit of mathematics right away, this is related to the combinatorial problem of finding *derangements*: if you give me a set in some order, the number of derangements is the number of ways I can give the set back to you in a new order, one in which none of the elements are where they started. For instance, consider the problem of the angry hatcheck girl: if she ignores the number on the customers’ tickets, giving a random hat to each customer and telling them to get out, then what is the probability that no one gets his or her own hat back?
12. Originally posted by queerqueerspawn, November 2013, republished in MuCulloch, cited above.
13. Which he toys with calling a “wector” space, “deflector” space, or “Hannibal Lecter” space, before finally settling on just $W$.
14. Even though taking them up also leads to incredibly weird results, like the freaky Banach-Tarsky paradox, which allows one to cut a finite sphere into a finite number of pieces that can be reassembled into two solid spheres the same size as the original.The question of whether the Axiom of Choice should be assumed when dividing loaves and/or fishes is a better question for theologians than mathematicians.
15. Which you can of course do, but you will only get so far.
16. http://web.ics.purdue.edu/~felluga/sf/NewYorkTimes.html
17. The fundamental theorem of calculus roughly says that differentiating and integrating are opposites, in some sense. The fundamental theorem of arithmetic says that any integer can be broken into prime factors. The fundamental theorem of algebra says that given any polynomial function of degree $d$, there are $d$ roots (possibly repeated) somewhere in the complex numbers.
18. In some versions of this story it was a special punishment given only to our little protagonist. The real details have been lost in the retellings over the centuries.
19. I’m deliberately not saying “boy or girl” because I’m pretty sure we’re in the “No girls allowed” stage of public education here. {Can I get a fact check from historians of education here?}
20. This is why computers do not replace mathematics. They aid it, but if you ask your computer from 2015 to do 1000000000000 operations it will choke. Find a creative way to reduce it by 4 orders of magnitude? Back in business. Of course, by 2020, \$1000 personal computers are expected to surpass human computational abilities, which is on the order of 10^24 Floating-point Operations Per Second (or “yottaFLOPS”)—a number much larger number than the scale of operations suggested above—but the principle will still be the same, and mathematics will still have a role to play in making the process more efficient.