The game commenced with a speech. You have been given a card. On that card is either the assignment of Townie, or the assignment of Agent. The agents have been given a task, some aberrant deed to perform in the course of an hour. Some samples:

Your spirit animal is a lobster, communicate this to someone.

Rub someone’s head.

Effect a pouring of drink in memoriam for some dead homie(s).

Toast us all (a little speech).

Get three people to look at the moon [a full moon had just risen] and profess a lie about said moon.

The Agents and Townies alike were on full suspicion to note aberrance and to report it on cards they were given. Bonus point for the precise deed. The point system, no longer even clear to me, was designed to prevent Agents from performing multiple, self-initiated aberrant actions. The Townies were to be only on alert, and to act as normal as possible.

At the end of the hour, we did a go-around. Pass if you were Townie. If you were Agent, describe your deed, so others could rejoice in acknowledging your stealth or, alternatively, in confirming their sound suspicions.

The ballots were counted, but the rules may have been unclear and tallying scores was nearly impossible. (One guest suspected everyone, some wrote down essays on behaviors involving nearly everyone). Deductions for false accusations. Deductions against you for accusations that you were an agent, whether or not it was so.

It worked well. A lot of silly fun was had.

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(concept credit to Daniel Johnston)

because one side is closer to that leaf, the other is half submersed in water. Context of a point of view from which the two sides enjoy their own identity necessitates the separate identities. *We cannot form an orbifrog simply by quotienting the frog by . *At least the quotient doesn’t extend, to the frog’s innards or to the world around.

The total symmetry of the mirror means that the actors in that world have duplicates in our world. The totality allows us to quotient the conceivable double world, and form the orbiverse in which we live.

When we quotient by all symmetries we end up with an essential, and asymmetric space, (or perhaps a point, what symmetries does it enjoy?). Yet there is the mirror, still, the evidence of a universe with symmetry.

My thesis, today: a fundamental group is a symmetry much like the symmetry of a mirror. It is a total symmetry which allows for a redundant, anecdotal universe:

Here, Castor, faced with the humdrum of a thoroughly repetitive life, dreams one day that he comes home not to the home that he left that morning, but to a home with differences ever too subtle to discern (indeed, so subtle they never effect divergent large-scale events). Work the next day has suspicious but subtle differences to work yesterday, and at the end of the second day he is back home to tell his original (or *even*) family about the world he’s sure he inhabits. Here is Castor’s world:

All the while a twin of Castor is both leading and following in his footsteps.

To recap: a fundamental group is a symmetry of a space as a mirror is a symmetry of a space.

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Is this a cringeworthy title? I think so. What images does it conjure in your mind? In its more dignified form, (arguably quite artistic), it’s Escher tessellations and Helaman Ferguson sculptures. Going further down the path you encounter colorful Mathematica tori and fractal art, icosahedra made from marshmallows or macaroni, knitted Möbius strips. The last stop is visual puns on pi=pie and clip art caricatures of skinny white guys with acne and thick glasses, or anthropomorphized cartoon numbers dancing together. The common denominator is a simple formula:

X is visual

and

X references, usually quite literally, an idea, an object, or a stereotype from the world of mathematics.

How does this fail and what right do I have to say most of this is neither math nor art? Good question. I have no authority, except as someone who is not totally blind to the world of art. I see that fine art, in general, has strayed far from the literal, and involves abstract ideas, often challenging ideas, and requires a context, appropriately referred to as a dialog.

(This doesn’t make me an expert. It makes me one of millions of people that pay occasional attention.)

Let me contextualize my complaint. Say, instead of an American Mathematics Society joint meetings art sectional, you had the analogous display at a huge medical technology symposium. What would you find? maybe finger paintings of prosthetic devices. Maybe wood carved hypodermic needles. Paper mache stethoscopes. Maybe some of it would be “cool” like a knitted skeleton with knitted organs (I really liked this one when it made its rounds around facebook). But I guarantee such a display doesn’t exist and wouldn’t, it’s a ridiculous idea. Would lawyers at a law convention humor amateur pencil sketches of handcuffs? No. In part this is because law and medicine have a culture of practicality.

But mathematics, as well as philosophy and physics, concerns itself with abstract ideas. You might expect the intersection of art with these disciplines to capture ideas that had fertile roots in each. Instead, you get a void of any critical standard, except, and this is my point, that works are *expected* to be predictable, to reference math literally and directly and predictably. Some pieces will receive awards. These will demonstrate craftsmanship, precision. The world of art subsumes works which are anti-art, because that has been voiced in dialog. The urinal. The can of shit. It is therefore reasonable to me that you might expect to find antagonism against math in an art show for mathematicians. Generally, sadly, you do not. Not that I need art to be controversial or antagonistic. But math art = vanilla safety pleasant clever clean. The formula is there, repeated to nausea. You know it, and you are bored by it.

I am bored by it, too.

I was asked to draw a cartoon character for our math department T shirt. Initially I was asked to draw a superhero, but realized in the process I do not abide superheros, in principle. This is what I submitted:

As of today, the image has met with disapproval by an invisible committee of people invested in the integrity of this year’s T shirt. All I know is a friend of mine *wants* the eyes to be filled in, and some other people, the identity of whom is unknown to me, have objected that it is “not totally relevant” to the math jokes text it would accompany on the T shirt.

It’s not great art. The face is stolen from Modigliani. The rest, the body, is stolen from 2 other sources. What it is, *what it is to me*, is deliberate. It is what I wanted it to be, and is yet a mystery to me. So let me explain why.

I have a 12 year old daughter who is exceptionally capable at understanding and originating mathematics. She’s receptive to the ideas I introduce to her in graph theory, in set theory, in programming, in logic. She gives back to me solutions and startling reflections on her own thought process. She’s artistic and fluent with analogies. She’s the only 12 year old mathematician I know, because I do not know many kids her age. She also is passionate about style. She has a style that is both conservative and defiant. She has a mother that is an artist and a father figure that is an architect and a father that is a mathematician, and she exhibits an influence of each of these. She has drawn tall, slender women since she began drawing around the age of 3. And so… I had in my mind a role model. An image of a woman that is attractive but not hypersexualized; both fierce and feminine. I think I accomplished that. Why Modigliani? In part, a tribute to the vertical figures my daughter has built on paper and emulated. In part, because of some mysterious attraction I have to this figure. I wouldn’t fill the eyes in. Come forth with a criticism that I have depicted a soulless woman, or that I have shrouded her in a mask, or that she has lost her gaze. I would be interested in any such critique, but not “you should fill in the eyes.” Of course, I will disagree with you. She could be looking down, she could be lost in her thoughts; I really don’t understand the image entirely and that’s exactly why I’ve chosen it.

(a) A committee cannot create a piece of art

(b) Suggesting to the creator a minor change to some piece of art without addressing any of the ideas expressed is, plainly, insulting.

(c) What stands in as the most acceptable stereotype is significantly less genuine than a genuine portrait, which is necessarily idiosyncratic.

(d) I’m bored to vomit by the shit you call the intersection of math and art.

Feel me?

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Can you think of contexts where a particular class of number is inapplicable?

Think about this for a moment. By “class of number” I mean, for example, negative numbers. Negative numbers, we learn, are great for expressing debt. They apply perfectly well, also, to linear orders where there is no smallest element (temperature isn’t the best choice, since Kelvin has a smallest possible value… year is better, since although we estimate a beginning of the universe, we can’t count days since then, so BC years are appropriately negative). What negative numbers don’t seem to apply to are numbers of family members, times you can iterate a process, dimensions of a space, distances. It is no wonder they too were suspect when first conceived.

How about the class of numbers which are not 1? Possibly it makes no sense to talk of multiple universes (I am of this camp, nevermind the fashionable physics which suggests the idea). For if there is a vantage point from which to observe 2 universes, then causally the two are part of your universe, even if they are entirely independent of each other. We might (should) define universe to be all those things which cause change in what we can perceive (to my mind this is possibly an asymmetric relation: a realm can exist which is broadcasted to us, upon which we have no impact, so that it is part of our universe but we are not part of its). Then to any observer there is, by definition, but one universe.

For number of family members and iterations of a process, it doesn’t generally make sense even to apply positive rational numbers. Particularly if the process is instantaneous. Can you wake up 2 and a third times? Well, not without defining what it means to wake up a non-integer number of times, it certainly is not given. Is it okay that certain numbers have limited scopes to the context to which they apply?

The idea of iterating a process, when put in mathematical terms, gives rise to the old subject of solving functional equations. If I have a function f:R->R from the real line to itself, I can define the square of the function, that is f^2, to be the function composed with itself. Then I can ask if there exists a function g so that g^2=f, i.e., does f have a square root? If f is the function of sending x to -x then normally there exists no function which halves this process (for example, sending x to the average of x and -x certainly doesn’t work). For 1×1 matrices, multiplication and composition are the same thing. So for linear maps from R to R, finding half of the inverting map, [-1], is exactly finding a square root of -1. If you extend the real line to be sitting in a plane, then it becomes clear that you can rotate the plane 90 degrees in either direction to give a process which when iterated twice sends x to -x. This mysterious fake number, i, is nothing more than rotation of a plane by 90 degrees. Do rotations by 90 degrees exist? If anything does, they damn well do. Teachers have been heard saying “i is important because it solves x^2=-1, which you want to be able to do for engineering problems” It is a good teacher that says “i is important because it is a compass point for any object with a quality that takes values in the circle, which certainly includes many physics and engineering problems” So, any phenomenon involving waves involves complex numbers. If waves exist, complex numbers do too.

From above: let us know what it is we are doing with language, so that we may deliberately do much more of it.

]]>> says, “mathematical innovations are [my emphasis] linguistic innova-

> tions.” I would rather say that mathematical innovation often entails

> attaching new technical meanings to words, e.g., ’gerb’, and sometimes

> introducing new words, e.g., ’surjection’, but that the mathematical

> innovations are not necessarily (and frequently are not) linguistic as

> perceived by the mathematician. In my experience, mathematical in-

> novation, which constantly accompanies mathematical work, is not

> generally linguistic; the mental imagery is geometric, diagrammatic,

> combinatorial, even kinesthetic (particularly for physicists), as well as

> aural (to use a Buddhist expression, mathematical “monkey chatter”)

> which is that part I would call linguistic. Perhaps Hadamard’s Psy-

> chology of Invention in the Mathematical Field would shed light on

> this issue.

>

Let me clarify.

Whatever the devices used by the individual mathematician, in order for the ideas to become part of a shared theory, some form of written communication is used. This is trivially “linguistic,” and, as I’m using it, includes diagrams and even geometric drawings. But are the choices that go into recording mathematical ideas merely style, more or less independent from content? I’m trying to argue that they are not, that a good deal of the content IS the organization of the (written account of the) collection of ideas.

I’m thinking of developments which dramatically progress

mathematics, not the personal phenomena that come with problem solving. The latter can be written about, but it’s mostly irrelevant to mathematics. Major innovations include the development of the decimal system, certainly this can be seen as both a linguistic and a mathematical innovation. Or Newton’s initial formulation of a differential equation, (just 9 years after calculus had been developed!), or for another example, Eilenberg-Maclane spaces. In both of these examples (and countless not mentioned) you have a heap of theory that goes toward defining objects which become *basic* objects of study. (Included in that theory are theorems, since frequently theorems are necessary to give the context for a definition). Now I can say “suppose I have this K(pi,n) space …” How do I translate that statement into the setting, a few hundred years prior, of Descartes or Newton? How is a differential equation translated into the language of the Greeks? It’s not just that the definitions would be long-winded, but much stronger: that for an adult of moderate intelligence to contemplate articulate questions about 4-dimensional manifolds before Gauss’s time, say, is absurd. In the 21st century it is not. To a great extent this is because us 21st century inhabitants have been given an organized language for asking (and *understanding*) such questions. In summary, language is a lot more than a list of labels, but instead is rich with structure. This structure is crucial to mathematics.

> 2. Isn’t the structural atomism Andrew mentions, “. . . statements and

> derivations in some atomic, axiomatic system of symbols” distinct

> from the structural aspects of the objects of mathematical theory? I

> guess we could take a nominalist position that there are no mathemat-

> ical objects, only the language that constitutes mathematical theories.

> Would a strict nominalist tell us that we are mistaken when we claim

> to be thinking in non-linguistic geometric imagery prior to enunciating

> mathematical statements about those images?

I can’t really address this, as I don’t really understand the

nominalist position. Am I confusing the structural aspects of the

“theory of mathematics” with “structural aspects of the objects of

mathematical theory.” Maybe I am. We can steer the ship back over to that topic.

> 3. Andrew kicks the hornets’ nest (or would have twenty years ago) in

> making points about “A Transformative Hermaneutics of Quantum

> Gravity”, i.e., the Sokol project. A question I would like to study is

> how the unintelligibility of Po Mo writing and the unintelligibility of

> mathematical writing compare. For example, three years ago or so,

> Gayatri Spivak wrote an article in “Art Forum” on Badiou. I couldn’t

> understand much of that article. Compare this with the fact that

> any of us wouldn’t understand much of Categories for the Working

> Mathematician without a great deal of work. Are these two cases

> essentially the same, or is there a signiﬁcant difference lurking here?

I didn’t mean to make a point about that work, but the title serves as a great example of far-reaching abstraction (in natural language). If we know a priori that some work is mathematically sound, albeit requiring, say, a billion pages of difficult math to build up to, then perhaps, a priori, we can claim that the work is fundamentally different than a social philosophy text that requires a billion pages of background, although it might be more

interesting to note the similarities. The distinction between the

math and the philosophy is founded on the idea that math can be

formally correct, i.e., that it can be checked by automation.

Philosophy cannot, both for practical reasons and for the fact that social philosophy (any philosophy, really) requires a point of view (in our case human, of a certain culture, etc.). In math we can claim a conclusion is true even if we don’t understand the proof, provided, say, we understand the algorithm that computed the proof, and we understand the terms in the conclusion. The obvious example is the four colored theorem, or suppose the

Goldbach conjecture had been solved by a program. By contrast, what would it mean to say that a philosophical phrase (e.g., “God is Dead!”) is sound because some billion generated lines of code concluded it. Nonesense! Espousing such a slogan in philosophy is a way of referencing one’s own understanding of the justification, and is not the assertion of a FACT.

Of course, conclusions in mathematics can also be in terms which

require billions of pages of code to parse. So instead of something as simple as the Goldbach conjecture, the program

generates a line relating objects which it has defined, and which are incapable of being understood by a human, yet the human can check the code and conclude that, barring bugs, the conclusion

must be correct. This computer generated conclusion is much more akin to the philosophical assertion which requires billions of pages of justification, since in both cases the conclusion is essentially gibberish.

I won’t say anything more concrete about MacLane vs. Badiou, but would be interested in hearing another’s take on the comparison.

> 4. Meter, millimeter, micrometer, nanometer, . . . . This kind of termi-

> nology is similar to the would-be lengthette, isn’t it? The suffixes

> ’eme’ and ’etic’ seem to be used by philosophers of linguistics. Split

> inﬁnitives are evolving to acceptable status, ’lite’ is understood as a

> variant of ’light’ albeit with a special meaning akin to that of ’je-

> june’. Dennis notes jargon such as texting abbreviations (in an email

> that you might not have received) that arises too quickly for me. I

> have heard that German grammar was reconstructed by grammari-

> ans (18th century?) after Latin grammar and that this constituted a

> major structural change. Ray, you can probably help us here.

The ‘ette’ suffix was an example of what we might call a structural innovation in our language. Indeed it does not exist as I have defined it (again, if I want accuracy but am unable to be precise, I should not use ‘a few nanometers’ to mean a relatively small length). In category theory we have the ‘co’ prefix which is extremely general. The introduction of such modifiers is clearly more substantial than the labeling of some very specific instance of some thing.

I was making a point about how language is only infrequently modified deliberately, to be more structurally robust. It’s mostly aside to the topic here.

> 5. Finally a point that is tangentially related to Andrew’s note. Struc-

> turalism was fashionable in several parts of the liberal arts in the 1950’s

> to 1980’s, but went out of fashion, declared by many to be irrelevant.

> Here is what Peter Caws says (Structuralism, 2000, p.105):

>

> In fact the stress on “formal models” in this statement of

> Chomsky’s points up a general problem for structuralism.

> It is perfectly true that outside some domains of linguistics

> very little progress has been made in developing interesting

> or fruitful formalisms: as we shall see in the next chapter,

> L´vy-Strauss’s attempts, which count among the most am-

> e

> bitious, seem often at once simplistic and contrived. But

> even within linguistics the results have not been much hap-

> pier, and my own view is that the whole attempt to conduct

> the structuralist enterprise in terms of rigorous mathemati-

> cal models is a case of what I call “spurious formalism.” A

> nonspurious formalism has to meet two conditions: what it

> deals with must be precisely speciﬁable in formal language

> (in the ideal case, moreover one frequently encountered in

> the physical sciences, it will be quantiﬁable), and this speciﬁ-

> cation must make possible formulations and operations that

> would not be possible in ordinary language. This last con-

> dition is hardly ever met in structuralist studies; the main

> points can nearly always be conveyed discursively, and the

> introduction of formulas and technical expressions is often

>

> 2

>

> merely ornamental and plays no real role in the argument.

>

> In relation to Andrew’s (2), mathematics and mathematical physics

> could in principle be written in natural language, just as they could

> be written in some ﬁrst-order formal language. It’s not that it is

> impossible, just that it is impractical in the extreme. Caws’s point is

> that the opposite is true of most structuralist efforts in the arts and

> human sciences, that is, natural languages serve better than various

> formalisms of a mathematical kind.

I am, at present, fascinated by what I will call the role of language

in mathematics. In using this phrase I wish to call to my own

attention the similarities between [natural language, and how it is

used to imbue theory with structure] and [language in mathematics]. I

want to make the point that mathematical innovations are linguistic

innovations, and that this observation is important, though frequently

overlooked. Let us draw an analogy between the material reductionist

and the mathematical formalist. In the former, all utterances are

understood to be rough shorthand for statements in terms of atomic

particles or some finer physical stratum. In the latter, all

mathematical theorems and their proofs are understood to be precise

shorthand for statements and derivations in some atomic, axiomatic

system of symbols. While it is sufficient (and often necessary) to

offer a definition of a mathematical term by way of previously defined

terms, (i.e., on a tree of definitions rooted in the formal base language),

one immediate question following such a definition is why this

definition at all? We are lead to believe that “mathematics”

encompasses not only the naming of natural and relevant formal

constructions, but that assigning arbitrary definitions, and remarking

with arbitrary observations, is also mathematics, provided the

definitions are formal and the observations are proved. That is, the

essential in mathematics is its absolute precision, and not in the

particular choices of structure in the theory. And why not? after all,

who can account for particular choices? We call these *tastes*, their

justification resides in a wholly different realm than the

justification for a line in a proof. They depend on the whims of

complex and erudite connoisseurs, not on reason. But when a student

asks why the property of a subgroup H in G, that gHg^{-1}=H, is given

a name (let alone why the particular name), what can they be answered

with? The usual answer would go along the lines of “normal subgroups

are extremely important, ubiquitous, characterize kernels of

homomorphisms, provide for series which classify groups, (etc.)”. That

is, the student is given a structuralist justification, and not “it is

a matter of taste to consider this property important and not, rather,

some other arbitrary property,” which would constitute a very weak answer. I

argue the reason normal subgroups have a name, (versus, say, subgroups

which have order any of {7, 48, 184}) a) is because there are multiple

contexts where normalcy is referenced, and b) has as much to do with

the essence of mathematics as does the precision of the definition of

normal. Indeed, it is the appreciation for concepts such as “normal”

that leads people to mathematics, and not merely the precision, as is

typically perceived.

Another typical perception is that mathematicians invent words and

generate new questions along the path of solving a given problem.

Hence the endless pursuit that is mathematics. Indeed, this perception

is accurate and many a mathematician corroborates it to the outside

world. Related is the myth that generalization=improvement. What is

less often discussed, in the same discussion, is that *any*

philosophical inquiry leads both to new words and questions and to a

generalized view. This is because the vocabulary introduced today

becomes naturalized in tomorrow’s theory. And generalization, when it

happens naturally, is a sign of sophistication. You might work with a

group for some time, because it is the symmetries of some aspect of

this problem you are working on. At some point you identify some

interesting properties of this group. A collection of these properties

fit together nicely, and so you christen the collection of properties.

Now you cannot help but wonder a few things about ****** groups, can

you? The construct of being a ****** group not only has a formal

definition, but it has semantic value to you now. It has an identity,

one that can and should be explored. When I think of generalization I

think of the fact that the Library of Babel has volumes on various

theories (albeit mostly inconsistent and fictional). For every

coherent theory or novel or case study book, there are books which

reference these (although referencing one book from another presents

its own problem, doesn’t it?) so that the theory can be built up to

unimaginable structures (let’s just allow ourselves to adhere labels

to a few of the bindings). Already in the humanities you have a

running gag about names of papers such as the famous “Transgressing

Boundaries, Toward a Transformative Hermaneutics of Quantum Gravity”

(or make up your own). Yet, with enough time, one could very well be

interested in various hermaneutics of quantum gravity, transformative

and otherwise. Why not? But the important thing is not whether some

particular esoteric hodgepodge of theory *has* meaning, but whether it

has interest to you, at your current level of sophistication. As the

levels of sophistication are unbounded, at some point there are

volumes of theory which will be utterly inaccessible to any human. We

have no choice but to call such a text “meaningless.” On a similar

note, one does not begin group theory by studying a subsuming theory,

such as the theory of magmas, not only because it is difficult to

handle such generalization from the beginning, but because groups are

interesting on their own.

Finally, on a somewhat related note, I wish to point out that computer

scientists are frequently fluent in many programming languages and are

frequently at work designing new ones, to improve structural issues of

those existing. I wonder why we do not do this more actively with our

natural language. We evolve our natural language like the brain has

evolved, by packing on new to old, and by tiny discrete changes here

and there. Could our language be more expressive or better organized?

Here is an example: suppose I want to connote a relatively small

amount of some unit, given some context, and I want to be accurate but

not precise (meaning I want to say exactly what I mean, while

remaining vague). It would be ideal if I could attach the same

diminutive suffix to any dimension, so for example, lengthette, timette,

massette, forcette. Not only does no such suffix exist, but if I begin

an essay by “let ette be the suffix which connotes relative smallness

of the linear dimension to which it is appended,” well, let’s just

say this would be unorthodox. I imagine if enough people did this

language would organize itself a little more according to deliberate

design, as is arguably the case in mathematics.

What happens in choice, when alternate outcomes are considered, one is preferred, and then acted out? The universe has encoded itself within itself, in a simplified model which informs events yet to occur. Equipped only with the knowledge of Newtonian mechanics some observer concludes that a car, headed south along highway 81 en route to Asheville, NC, will collide with a tree in approximately 12.03 seconds. Another observer skilled in the art of brain-state reading concludes that the tree will be avoided as the curve in the highway is followed, and that in approximately 6 hours the driver of the car will meet his son in a restaurant, eat and pay the bill. How mysterious is it that the universe encodes past and future in simplified models contained in *mind*?