Dinner Party Game, Fête in the Field

September 8, 2015

The guests arrived between 7 and 8.  The sun was setting behind the large barn.  The house was filling with side dishes brought by guests, as the patio filled with guests.  We served Negroni’s and wine, there being a loose Mediterranean theme.  At 9 we instructed the guests, some 24, to each carry a side dish down into the field.  The narrow path is mowed and was lit with lights beneath the wall of brambles.  It led to a large rectangle of mowed lawn, surrounding an apple tree lit with rope lights, and surrounded by a thicket of yellow flowers and an idyllic view of distant rolling acclivity.  A long table to seat 24 was set.

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.

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.

Symmetry

April 15, 2014

Many a child has wondered if the face in the mirror might have a different mind lurking behind her eyes.  The symmetry of a mirror is different than the symmetry of a frog.  It is a total symmetry, a surface where two isometric worlds are glued along a surface.  Why give the mirror world its own identity?  For sake of amusement.  Why take away its identity?  For sake of simplicity, Occam’s razor.  The same cannot be done with the right and left sides of the frog.

(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 $\mathbb Z_2$.  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.

“Intersection of Math and Art”

February 20, 2014

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?

Imaginary Numbers Rant

August 23, 2011

I’m preaching to the choir here, if to anyone at all, I suppose. Still, it should be said. Descartes coined the term “imaginary number” to deride the invention of a number such that squared becomes -1. The name stuck and has done countless harm to the notion, fooling pupil and school teacher alike into implicitly accepting the formalist position: “that which can be defined, precisely, exists. End of story.” It is no wonder that many have been suspicious of the invention of imaginary constructs that can do the impossible. I have a rhetorical question to justify imaginary numbers to my students. The question is:

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.

language, math, structuralism

June 24, 2011

Just a few thoughts.

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.

Icon: dichotomy

December 28, 2010

Really it is quite curious…

May 20, 2010

The universe sings, observes some thing residing on the outside. From a vantage point beyond both space and time, it is an understatement to say the universe appears deterministic. There is no cause and effect from this view, only a single relic with unimaginable beauty and complexity. The minds of animals mimic the happenings of the universe, in memory and foresight, and herein the universe replicates itself like a fugue. The theme of one particularly successful fugue is heard for hundreds of years, from the occasional street musician to the department store radio. The universe favors this fugue.

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?

May 5, 2010

May 1, 2010

1.3(i-iii) Cell World

April 29, 2010

The cellular automata universe offers a universe that’s easy to deal with and yet rich enough to give us all the complexity of our universe. Indeed, I argue that the details are essentially irrelevant at high enough levels of complexity; that no particular material nor specific fine-scale mechanics are necessary for consciousness, and that we might expect quantum mechanics to inform our understanding of pop music or hedge fund management as well as it informs our understanding of the brain. Thus, the cellular automata world is rigid enough to grant the most stringent determinism, it is also rich enough to birth arbitrarily deep levels of complexity, and house any imaginable intelligence. I want to also distinguish my position from those who insist free will (or consciousness) is an illusion. At best this is misleading, more likely it is just plain wrong.

i. Objectness

Long before we need worry about free will and consciousness, we need to worry about what objects are. These will be the nouns of any truth claim we make. Before there is *choice*, there is *person*, and how to deal with the distinction of being a *person* object is not as easy as it may seem. Even in a low resolution cellular automaton (i.e., one with few cells) the problem remains. Is a glider an object? Is every configuration of cells an object? We might suppose that every configuration of cells theoretically has a name, or could be given a name, and even that names in our language must be shorthand for collections of cells (that meaning in language must essentially be built from these building blocks). But then something so simple as ‘glider’ is necessarily shorthand for a list of trillions of configurations. This seems like a faulty way of looking at things. If glider includes not only the same two 5-cell configurations of cells, up to translation, but also larger things which exhibit a gliding property, the problem is harder. Certainly if an object appeared to maintain its shape more or less, as it translated itself through space, perhaps even fizzling out at some time, we’d be tempted to call it a glider. This is especially true if our instruments of detection are unable to detect individual cells, so we cannot discern a glider’s states at that finest level. No one argues that natural language is not fuzzy, as it unarguably is, but then how do we interpret a fuzzy truth claim, in reductionist terms?

ii. Diagonalization

We’ve built in our imaginations a cellular world with trillions of cells, and in this world a creature has been formed. That creature is constantly bombarded with gliders of various sizes and from these collisions, (and internal happenings), the creature processes thought and outputs gliders, as statements. This creature I’m thinking of is essentially a human, or a near approximation. Now, let’s say we agree on an interpretation of its language (i.e., the waves of gliders it sends out of its mouth, each wave differing in shape enough so that a discrete language can be understood, as English is). What can it say in this language? One thing it can say is “the universe is a cellular automaton with the following rule of evolution…” What can it not say? It cannot say “I will now give names to the 10^(10^10) cellular configurations possible in this universe, beginning with ‘aardvark,’…” Indeed, since each utterance takes up space (for the gliders to carry the waves of speech) the utterances are quite limited in the amount of information they can carry (necessarily less than the total number of configurations possible in the same tiny amount of space, let alone the universe). Now, theoretically we can offer the utterer all the time he wants to longwindedly describe each fine detail (indeed each cell) of some object, and terminate after finite time. But who, or what, is his audience that can reassemble the information into a model that contains as much information as the original object? This is one reason computers cannot calculate the evolution of the universe, because you don’t even get to specify the initial condition without generating an infinite descending loop! It almost seems absurd to expect more than fuzziness from meaning in language, but of course our language is not fuzzy, and perhaps this is where some of the confusion lies. Language is ridiculously precise. Unlike facial expressions or performed music, it is exact and codifiable. Yet still we discern subtlety and nuance in our favorite authors, after reading hundreds of thousands of their words (or even a good cadence in a paragraph or sentence).

iii. Emergence

I want to draw attention to the fact that fuzzy terms (as in the referent is fuzzy, such as with natural language) aren’t just ‘fuzzy,’ as opposed to being precise, as a sort of deficit. But instead, that there is meaning in a fuzzy term that is essentially lost with the attempt to make it precise. I am thinking of the cellular human, and imagining her, let’s call her Frida, holding a ball and commenting “it is round.” This roundness property, which seems so elementary, is in reality a reflection of the ball’s resemblance to other objects previously perceived by Frida. So the process goes: an object is in front of Frida, some waves of gliders emanate from it (or rather “bounce” off it), carrying information about the object into Frida’s sensory apparatus, then Frida’s brain momentarily gets a hold of that information. Within a few seconds most of the information is gone, but some faint ghost remains, a ghost which somehow holds information about the object which is general, and connects to yet other things Frida has seen. This object would not be classified under the blanket abstraction ’round’ were it not for the ‘intrinsic’ cellular make up of the ball, but the abstraction cannot be said to be an intrinsic quality of the ball, supported only by the physical state of that ball. I think some would find this distinction too subtle, but it’s an absolutely crucial difference to understanding how meaning works and how the reductionist is wrong.

It is only too easy to imagine ’roundness’ is a concrete quality either enjoyed intrinsically by an object or not. However, terms which are overtly contextual as opposed to physical are readily available and make up the majority of the words we use. Take ‘majority,’ for example, and define it in terms of cells in such a way that nearly all usages found in English literature can be said to reference it. It can’t be done. We can say what a majority of cells being on in a given region means, more or less, but that is not directly referenced by my usage above, nor when we stumble upon it in literature, say in the phrase “tyranny of the majority.”

It is observed, also, that terms which don’t lend themselves easily to reduction, are frequently not necessarily more complex, given a context. This is exactly the point, for if it was necessary to give a description of X with complexity proportional to the extent X resists reduction, then we are in a reductionist framework, and X is just really complex. But the human mind doesn’t work this way. It is frequently possible to communicate great generalities to children, who would have no way of understanding the reduction to finer physical parts. How can one insist “being on one’s best behavior in a restaurant” is really a property or action of physical particles, or even a deep sociological action, when neither of these can be comprehended by the child, while the statement itself is easily understood? Part of the answer is that in building a vast framework of complexity, certain terms become contextually simple, while being intractably complex from a ground-up perspective. You don’t build “one’s best behavior” from scratch. The other part of the answer is that a child begins with such a framework. Knowledge does not stick to an empty slate (not even a blackboard!), but children have a robust way of making sense of generalities from the beginning. My main point here is that meaning is emergent, held together by a framework harder to imagine than a strict partial order. There are more lateral connections in the network of meaning.