Month: December 2008

Once you get the idea that flipping a coin n times produces 2ⁿ possible results, it’s just a short step to Cantor’s description of exactly how big is the set of real numbers.

Let’s go back to those real numbers between zero and one, but instead of describing them in base ten, let’s describe them in base two – binary – the same way that numbers are stored in a computer. So instead of a 1/10 place, a 1/100 place, a 1/1000 place, and so on, there’s a 1/2 place, a 1/4 place, a 1/8 place, etc. And instead of digits 0 through 9, we just use digits 0 and 1.

Here are some examples of real numbers in base two:

0.100100111001…
0.001010011011…
0.101110101110…

You can use binary digits to express any real number. For example, 1/4 in binary is 0.01, and 1/3 is 0.010101… (repeated forever).

Cantor’s clever idea was to look at each 1 or 0 digit as the result of a coin flip, where 1 means heads and 0 means tails. When you look at it this way, you see that each real number is just one possible outcome of flipping a coin over and over again, and writing down the result after each flip.

Suddenly, just by looking at it this way, it all becomes obvious. Every time you add one digit, you double the possible number of outcomes. If N represents “how many counting numbers there are” (a kind of infinity), and R represents “how many real numbers there are” (another kind of infinity), then it’s easy to show, from the coin flip argument, that R = 2^N.

And that’s just what Cantor showed: That “how many real numbers there are” (ie: any quantity that could ever appear on a number line) is 2 to the power of “how many counting numbers there are” (ie: 1,2,3,…,∞).

In other words, there are a hell of a lot more real numbers than there are counting numbers.

Cantor went on to show that there are even bigger kinds of infinity. In fact, infinities form a kind of nested chinese boxes: You can look at 2^R to get an even bigger infinity. Then you can raise 2 to the power of that infinity to get yet another one bigger than that, and so on and so on.

This sequence of infinities is usually written using the hebrew letter , with a little subscript to indicate which infinity you’re talking about. For example the smallest infinity, which is “how many counting numbers there are”, is written ₀, and the next smallest infinity, which is “how many real numbers there are”, is written ₁.

One question that nags at me, is whether that’s really all you can do to get bigger infinities. I mean, we’re just counting up infinities here one by one: ₀, ₁, ₂,…

Why restrict ourselves to those boring old counting numbers that Euclid was throwing around way back in 300 BC? One thing that Cantor never talked about, and that I can’t seem to find discussed anywhere else, is whether you can go up faster than just counting the infinities one by one. For example, why can’t we talk about ₀ or ₃, or even ₃?

Or maybe I’m just being greedy. 🙂

Oh my, I fear I may be losing my readers. “He’s actually talking about math!” (general panic ensues). Well, maybe it’s ok. Maybe it’s like looking at a lovely but slightly mysterious painting or poem. There’s the surface beauty, but as you keep looking, the inner meanings being to reveal themselves, layer by layer.

When I was a teenager I became fascinated by infinity, and Georg Cantor became one of my heroes. I loved the way he simply took things we already know and understand about the finite world, and showed how you could apply the same principles to the infinite world.

For example, yesterday I described how he showed that the set of real numbers is bigger than the set of counting numbers, by adopting Euclid’s method of making lists.

But he did much more than that. After all, how much bigger is bigger? Just a little bit bigger? A lot bigger? Cantor came up with a very clear way to describe exactly how much bigger. And it turns out that the set of real number is a lot bigger than the set of counting numbers. A whole lot bigger. A humongously stupendous, outrageously phantasmagorical lot bigger.

And he did it with his usual trick: He stole an idea from an earlier mathematician who was trying to describe finite things, and he used the same idea to describe infinite things. As Pablo Picasso once said: “Good artists borrow; great artists steal.”

Let’s go back to 1654 when Blaise Pascal, who was highly religious, nonetheless agreed to help out his friend Antoine Gombaud, who was into gambling, and who wanted to come up with the best answers to what we would now think of as betting odds. To help him out, Pascal invented the entire science of probability (and his letters to Fermat on this subject incidentally led to the development of Calculus, but that’s a different story).

We can take some ideas from Pascal’s innovations to answer a deceptively simple question. The answer will soon lead us back to Cantor’s solution to how much bigger the set of real numbers is than the set of counting numbers.

But first, let’s go back to Pascal, devout Christian yet patron saint of professional gamblers. If you flip a coin three times in a row, what are the odds that it will come up heads every time? Well, one way to figure it out is to list all of the possible outcomes, and then count up how many are all heads:

The first thing you notice is that there are eight possible outcomes. The second thing you notice is that only one of them is all heads. So the odds are 1 out of 8 of getting all heads.

Now it’s easy to see what the answer is for any sequence of coin tosses: Every time you add another toss, there are twice as many possible outcomes. But still, only one of them will produce all heads. So the odds of getting only heads after one toss is 1 out of 2, after two tosses it goes down to 1 out of 4, after three tosses it’s 1 out of 8, and so forth.

One convenient way to say this is to say that after n coin tosses, the odds of getting all heads is 1 out of 2ⁿ (that’s two raised to the nth power).

That’s just a small glimpse into the science of probability, beloved by mathematicians and bookies ever since. Pascal’s brilliant insight was his realization that you could predict how likely it was that something would happen, even if it hadn’t happened yet. We take that for granted now, but before 1654 it hadn’t occurred to anyone that you could reliably predict anything about the future.

That was quite a fundamental shift in human thought, wasn’t it? And these days we think the internet is big news. Compared with something like that, the internet is nothing.

Three hundred years later, Cantor realized that Pascal’s tool for seeing into the future could just as easily be applied to infinite things. But that’s a story for tomorrow.

This evening I was telling my friend Gerry about the recent blog thread here that included Euclid’s proof of the infinity of prime numbers. Gerry said that he was glad it was so easy to show that there are infinitely many prime numbers, but he was still unhappy that he hasn’t been able to explain to anybody, in a simple way, why there are more real numbers (eg: numbers that have an infinite number of decimal places, like 0.3278594…) than there are counting numbers (numbers like 1, 2, 3, …).

I told Gerry that in fact there is a simple way to explain this. By an interesting coincidence, the proof that Georg Cantor came up with of this in 1874 was in some ways the same proof that Euclid had used 2700 years earlier for primes. Actually what is amazing is that it took twenty seven centuries for somebody to realize that Euclid’s method could be applied to this other question.

To make things simpler, Cantor considered only real numbers between zero and one. He asked: How many real numbers are there between zero and one (eg: 0.7, 0.23528315…, etc)? Are there more such real numbers than there are counting numbers? If you can make an infinite list of all the real numbers between zero and one (a first one, a second one, and so forth), then you’ll have shown that no, there aren’t more such real numbers than there are counting numbers.

Cantor proved that yes, there are more real numbers between zero and one than there are counting numbers. He basically adapted the same method that Euclid had used more than two thousand years earlier for primes – the idea that we talked about yesterday – except he upped the ante by letting the list be infinite.

Assume that you have an infinite list of all the real numbers between zero and one. So on your list there’s a first number, a second number, a third number, all the way up to infinity. The question is: Is every possible real number between zero and one contained on your list?

If you can find even one real number between zero and one that’s not on your list, then that means that it’s impossible to list all the real numbers between zero and one. In other words, that there must be more real numbers between zero and one than there are counting numbers.

It’s the same thing that Euclid had done – Euclid had said: “Let’s make a finite list of what we claim are all the prime numbers. Now let’s use that list to show that there must be at least one prime number that’s not on the list. Once we do that, we’ve proven that there is no finite list containing all the prime numbers. So there must be an infinite number of prime numbers.”

Similarly, Cantor said: “Let’s make an infinite, but countable, list of what we claim are all the real numbers between zero and one. Now let’s use that list to show that there must be at least one real number between zero and one that’s not on the list. Once we do that, we’ve proven that there is no countably infinite list containing all the real numbers between zero and one. So there must be an uncountably infinite number of real numbers between zero and one.”

Here was Cantor’s trick to produce a number that’s not on the list: it’s a very simple, elegant trick. Let’s say you have such a list. Each of the numbers on your list can be written in decimal form. Maybe your list looks something like this:

1)     0.73289473682958…
2)     0.32562839727636…
3)     0.27462849637426…
      …

To make things easier to understand, let’s show the list again, but this time highlight the 1/10ths place in the first list entry, the 1/100ths place in the second list entry, the 1/1000ths place in the third list entry, and so on:

1)     0.73289473682958…
2)     0.32562839727636…
3)     0.27462849637426…
      …

To construct a number between zero and one that’s not on the list, just make a new number that has a different digit in the 1/10ths place from your first list entry, a different digit in the 1/100ths place from your second list entry, a different digit in the 1/1000ths place from your third list entry, and so on.

For example, the new number you construct, to three decimal places, might look like: 0.835… (I’ve just added one to each of the highlighted digits – pulling out the “7”, “2”, 4″, etc., and replacing each one with a digit that’s one greater).

So here we have a real number that’s clearly between zero and one, but isn’t on your list. And you can use the same recipe to build such a number for any such list that you try to make.

And so, just as Euclid showed around 300 BC that there are an infinite number of prime numbers, Georg Cantor in 1874 came up with a very simple and elegant way to show that there must be more than one kind of infinity: The real numbers between zero and one form a bigger infinity than do the counting numbers 1,2,3,….

How cool is that?

In particular, as I alluded to yesterday, when I was twelve I had my first real encounter with the infinite – courtesy of Euclid.

Please understand that for me this was a very big deal. I had been wrestling for several years by then with people throwing around words like “infinity” and “God”, and I was getting quite uncomfortable with my growing understanding of the culturo-centrism around me. I found it seriously disturbing that we were being taught about all of the Greek gods to be found in Edith Hamilton’s “Mythology” in terms that made it clear that they were unreal – the product of an immature civilization, whereas (so we were told) our Judeo-Christian God was the real thing.

And yet it was clear even to my pre-adolescent mind that the complexity and subtle humanity to be found in Zeus, Apollo, Athena, Aphrodite and the other supposedly mythological inhabitants of an earlier Pantheon were much more in tune with the human condition that I saw around me every day than was the highly abstracted and invisible God that our society had more recently settled on.

So I guess you could say I was ready for some more tangible encounter with the infinite. And at twelve years old I found it, thanks to old Euclid of Alexandria.

Specifically, I learned that back around 300BC Euclid proved that there were an infinite number of prime numbers, and that he did so in a way that did not in any way resort to belief or doctrine – merely straightforward statements of fact and steps of logic.

I’m sure many of you know the proof – it is simple and elegant, and it goes like this: Suppose there are only a finite number of prime numbers. Well, that means you can write them all down in a list. It might be a very very long list, but it’s a finite list, so eventually – if you write long enough – you can write them all down, every one.

Euclid’s trick was to ask what happens when you multiply together all of the prime numbers on your list. Of course you get some humongously huge number, which you can call N. Then he looked at N+1, the number that is one greater than N.

Here’s where things get interesting. Clearly N+1 is not divisible by any of the prime numbers in your list – if you divide it by any of the numbers in the list, you will get a remainder of 1.

So either N+1 is a prime number, or it must have factors in it that are not in your list. Either way, there must be at least one prime number that you didn’t put in your list.

But that’s a contradiction – your list was supposed to contain all the prime numbers. In other words, it’s impossible to make a list of all the prime numbers. Every time you try, another one will pop up that wasn’t on your list.

And so the prime numbers must go on forever, infinitely.

When I was twelve and saw this, and understood what I was seeing, it was a huge relief. I realized you can talk about things that are not only vastly larger than us but are in fact infinite, bigger than the stars and galaxies, bigger than the universe itself, simply by stating some simple facts and applying a little reason.

Which to me made a lot more sense than trying to decide how the universe works by counting how many people are voting for Vishnu, Zeus, Yahweh, Jesus or the Buddha in any given country or century. Mathematics is beautiful because it is true, in the most powerful possible sense of that word.

Math is, quite specifically, the science that seeks verifiable answers to the question “What is true?”

Even about infinite things.

Wow, this really seems to have struck a chord. I suspect most of us have some childhood war stories about encounters with the tragedy of the way math is taught (or rather, not taught) in secondary schools. In my case, the first hero who came to my rescue was John Schneider, who was at the time a new young teacher at Tappan Zee High School in Rockland County. After the total apathy I had endured on the part of math teachers in middle school, encountering someone like Mr. Schneider was a blast of fresh air for my twelve year old mind as I entered the ninth grade – my first year of high school.

Mr. Schneider truly loved math, and he had none of the “this is just a job” attitude that public schools often beat into their teachers. In addition to a revelatory Geometry class, he also offered, for anyone who cared to take it, an extracurricular course in non-Euclidean Geometry. Five of us signed up, each of us brainy but slightly odd. And all of us were male – I’m sure there were various social forces at work there.

For me, who was still far too young at twelve years old to understand how to fit in socially in high school, our little non-Euclidean group was a dream come true. There were secrets here, mysteries, a sense of power to be had by deliberately breaking the most common sense rules of geometry – such as the rule that says parallel lines exist – and then discovering that things worked anyway. We learned about the spherical world of Reimann, and the beautifully curved hyperbolic world of Lobachevsky. We were amazed to learn that our own universe might be either of these shapes – depending upon how much mass there was out there beyond the stars. But you didn’t even need the physics for it to all make sense – it all stood on its own as pure reason, beautiful and self-consistent. It was my first real encounter with pure thought, the incredible process of building upon a few elegant hypotheses to create an entire world.

And that was the year I first encountered the idea that you can speak truth to infinity without any need to devolve into religious or metaphysical debates. We could all argue in endless circles about whether there is a God, or gods, or whether the world of Edith Hamilton’s “Mythology” was less true than the deity in our prayer books, but five minutes was all it took to understand that there are encounters with the infinite that are indisputable, such as the fact that there are indeed an infinite number of prime numbers – an absolute truth that stands on its own, unassailable, independent of any mere belief system.

Our textbook was a slim dark red hard-cover volume that was blank on the front. The only clue to the treasures within was the phrase “Non-Euclidean Geometry” printed on the binding in plain block letters. Because we felt like rebels, we would pretend it was Mao’s little red book, and we each taped a fake binding onto the edge of our copy that said “Quotations of Chairman Mao John”, in honor of our teacher. I think Mr. Schneider was very bemused by this, but happy nonetheless, because in our quirky adolescent way we had each stumbled upon the beauty of mathematics.

There were so many bad and indifferent teachers in my public school education. I still have one searing memory from when I was eleven years old of bringing a poem I had written – one I was quite proud of – to show an English teacher. He pretended to read it, not bothering to hide his boredom. Never again did I show a poem to a teacher.

But our school was lucky in its math teachers. John Schneider was just the first of a sequence of mostly wonderful math teachers I was to encounter between the ninth and twelfth grades. It was already clear to me, before I graduated high school, that a single good teacher is the most powerful force in the universe for instilling a love of learning in young minds.

That’s how I found my way to math, but it’s also what decided me to end up in a career that involved teaching, and to devote much of my life to turning young minds on to exciting new ideas.

Here in manattan there are various little free newspapers around, supported only by ad revenue. None of them are of very high quality, but one thing they all seem to have in common is a puzzle page. The crossword puzzles aren’t very good (I’m spoiled by The New York Times) but I find the sudoku to be a perfect mindless divertion during subway rides uptown. All I need is a pen in my pocket when I leave home or office, and I’m good to go.

But there is one oddity about the Sudoku page. In many newspapers, it has a little instructions page that reads (and I quote):

How to play:
Fill in the grid so that every row, every column and every 3×3 box contains the digits 1-9. There is no math involved. You solve the puzzle with reasoning and logic.

Today I became curious whether anybody else is bothered by the patently false and absurd claim that “There is no math involved”. So I did a Google search on:

“There is no math involved” sudoku

 

and found 82 hits (Google gives an estimate of over 900 hits, but they only actually find 82). I went through them all, and found that every page simply repeated the nonsensical statement that there is no math in Sudoku, without thinking to question it.

If I were to start printing some equally absurd statement in newspapers, like “There is no gravity, we just all secrete glue out the bottoms of our feet and shoes,” I suspect somebody might complain. So what’s going on here?

My theory is that math education is so fundamentally broken in our society that people actually grow up believing that “math” is a synonym for “arithmetic”. And there is certainly no arithmetic in Sudoku. But in fact Sudoku is a math puzzle. It consists of nothing but math.

Actual mathematics is, quite precisely, any endeavor in which you start with a set of symbols, together with some rules for combining and manipulating those symbols, and then you set about discovering what symbol combinations are provably true or provably false, according to the rules you started with. Sudoku has nine symbols, and a few elegant rules for how you are allowed to combine those symbols. Most combinations produce a provably false result, and one combination produces a provably true result.

Things don’t get much more mathematical than that.

And so I come to the reluctant conclusion that almost nobody in our society has any inkling of what math is. Which is really strange when you consider that kids are required to take math in high school. For example, if you attended high school in this country, the odds are that your state curriculum required you to study Geometry somewhere along the way. That’s an entire year in which you did math pretty much without arithmetic or numbers. You would have encountered a bit of arithmetic, like summing the occasional pair of angles, but that was pretty incidental to what you were actually studying, which was how to prove theorems from a given set of simple initial rules.

It’s no wonder there is so much math phobia in this country. For one thing, people aren’t even aware of what math is. It’s just that scary thing they are supposed to learn from people who apparently also have no idea what it is.

Imagine how bizarre the system must seem to the average high school student. An entire year is spent taking a math course called “Geometry” which – by common consensus – has nothing whatever to do with what most people mean when they say the word “math”, since most people are under the misapprehension that math is the same as arithmetic.

And yet, ironically, Geometry is the one math subject offered in high school which actually is math, as that word is understood by mathematicians. Algebra and Calculus would be math too, if anybody bothered to show you how and why they really work. But in high school that doesn’t seem to happen. Instead, these courses are generally taught as a set of mysterious formulas: Plug in the right formula and the right answer comes out.

I was completely uninterested in math when I was in middle school – it was all taught as a set of formulas by bored teachers who seemed much more focused on stopping us kids from throwing pencils at the backs of each others’ heads. And because I wasn’t interested in math, I wasn’t particularly good at it.

Then in ninth grade I had my first great math teacher, and everything changed. I was finally shown that math (actual math, not what most people mistake for math) is one of the most beautiful things that a human being can encounter in life.

But that’s enough for now. More tomorrow.

I had made a perfectly serious looking computer monitor using my little java modeling system, because I’m working on a project where it’s useful to visualize the relative sizes of various items commonly found on a desk, and I wanted to include a 24″ diagonal PC monitor. I modeled the mug and pencil from objects that happened to be sitting on my real-world desk:

 

But I am also working on a completely different project for which I am building a kind of cartoon world, a world I call the “Jellybean World”. In the Jellybean World everything has that wonderfully crazy retro-future Googie look that was borrowed by Walt Disney for their original Tomorrowland, and by Hannah-Barbara for The Jetsons. It’s a world that says, essentially, “Welcome to the future, and everything really is ok.” Not a very common sentiment in recent times, I’m sad to say.

Since I’d already made the monitor, I simply repurposed it, running it through might be called a “jellybean filter” – a process that transforms real-world objects into Jellybean World objects:

 

In the Jellybean World the computer monitor is no longer required to hang out with mugs and pencils. Instead, as you can see, it finds itself in the company of happy plastic palm trees. The tree, by the way, was modeled after the ones I used to play with when I was a kid – the little plastic trees that came with the cool (and highly influential) set of plastic dinosaurs my brother and I got from our uncle Ned when I was six.

I like the idea of a jellybean filter that can be applied to whatever object you feed into it. There is just something about the sheer loopy hopefulness of this concept that appeals to the six year old in me.

In case you were wondering, in my Jellybean World both the monitor and the trees are perfectly capable of dancing and swaying in time to music, as are all objects. This is a feature not generally found in real-world computer monitors.

Or real-world trees, come to think of it.

When you write a blog entry every day, things tend to get more Proustian. You start to notice all of the little crumbs of madeleines along the way, each dipped in its own lime-flower tea. These crumbs, humble as they are, become precious. Each can lead you to some inner place that connects you to someone you know and love. Or – a tea of a somewhat more bittersweet flavor – to someone you knew and loved in times past.

I tend to notice now when somebody gets that sentiment right. Recently I revisited “Citizen Kane”, and it struck me that the lovely pivotal speech which Herman Mankiewicz (the screenwriter) gives to Bernstein is one of the most perfectly Proustian moments in any Hollywood film. It’s part of the answer Bernstein gives when interviewed by the investigative reporter. At this point Berstein is already quite old, his mind circling around his own memories of Kane, whose precipitous life is now a thing of the past.

I’ll let you savor it for yourself:

“A fellow will remember things you wouldn’t think he’d remember. You take me. One day, back in 1896, I was crossing over to Jersey on a ferry and as we pulled out, there was another ferry pulling in — and on it, there was a girl waiting to get off. A white dress she had on – and she was carrying a white parasol – and I only saw her for one second and she didn’t see me at all – but I’ll bet a month hasn’t gone by since that I haven’t thought of that girl.”

Consider that these words are being spoken by a man who is near the end of a very long life, with the distinct implication that he has never before told anyone of the girl with the white parasol. In a few eloquent sentences it tells you everything you need to know about what truly matters in the film. I think it’s one of the most perfect speeches ever to appear on celluloid.

I went to see a chamber music concert this evening – short performances by ten chamber groups in two hours. The music ranged from Schumann to Shostakovich, and everything in between. My favorite was the “Violin Sonata in A Major” by Cesar Franck, a perfect study in soulful aching lyricism. Tonight, in the hands of a brilliant violinist and pianist, it somehow managed to be deeply sad and ecstatic all at once, like the very best songs by Tom Waits.

Somewhere during the arrangement for flute, clarinet and piano of Debussy’s “Epigraphes Antiques” I had an epiphany. Each of the musicians on stage was focusing on giving the best individual performance possible – each mind on the stage in its own personal zone, a place that had been achieved through countless hours of practice. And yet each player was clearly keeping an ear open to all the others. You could see the occasional glances from one musician to another, the thoughtful pauses between the music as they all tried to sync to the same emotional wave as it surged and ebbed throughout the piece.

And I realized that this is the same experience I have watching a great performance by a dance ensemble, or great actors on a stage, whether the play is by Mamet or Chekhov. In each case there is a dramatic illusion of tension between two opposing players – in the case of the Debussy it was the byplay between the flute and the clarinet. But in fact, they are all aspects of the same mind – and this is what makes it all wonderful.

A good performer, whether in a ballet, a chamber orchestra or a theatrical farce or tragedy, is usually presenting a well thought out creation of a single brilliant mind. The people you see on the stage are not there to act solely as individuals, but rather to illuminate an inner dialog created in this author’s mind. In Ibsen’s “A Doll’s House” Nora and Torvald are not really two people – rather they are artful illusions of two people, perfectly sketched representations of individuals, invented for the purpose of playing out questions that actually arose within the mind of the playwright.

When you see this in a novel, it is all usually tangibly obvious. The reader knows that Darcy, Elizabeth and Collins are merely creations from the single mind of Jane Austen. In fact, Austen reminds us of this fact at every moment in the story – by making her own narrator’s voice the most vivid of all the voices we read.

But in a performance it gets trickier. You can actually see the individual floutist or clarinetist there before you – flesh and blood humans who you know have each spent countless hours to perfect their art. And yet the job of that floutist and that clarinetist is to illuminate the mind of Debussy, to bring you back to the vision of the individual creator.

I think that it is this formal tension – the fact that even in the middle of a knock down, drag out cursing match in the blackest of Mamet plays, or the most devastating psychic wounds inflicted in a play by Conor McPherson, it is still the job of the players to act as an ensemble, to work harmoniously together on the meta-level of illuminating a single author’s story and ideas.

To reveal to us, to the best of their abilities, the creation of an individual mind.

I threw a party this evening. Nothing fancy, nominally a work-related holiday party, but really an excuse to connect with and give thanks to some people I’ve liked and have worked with over the years (and who happen to be in NY at the moment) and to watch them enjoying each others’ company. It was delightful to see people of varying backgrounds and ages, each of them somebody I like, meeting each other and having fun together.

The youngest guests were in the early twenties, the oldest probably pushing seventy, and there was the usual wide ranging mix of ethnicities, nationalities and backgrounds that we get here in Manhattan. Everyone seemed completely engaged by everyone else. Charles and I merged our “very favorite songs” playlists, so an intriguingly eclectic mix of musical sensibilities became our soundtrack for the evening. My old friend Darcy showed up with the gift of various miniature remote control flying toys, and it somehow seemed perfectly natural for guests to take turns throughout the evening, sending these little spinning contraptions sailing overhead.

There didn’t seem any point in telling people that the entire party was vegan, and people didn’t seem to notice – they were just enjoying all the munchies and pies and cupcakes and other goodies. Which in some sense is the point.

It’s strange, isn’t it, how people need an official excuse to simply relax and enjoy life. We spend so much of our time playing serious roles, and then once in a while we have these officially declared things we label “parties”, times set aside when it is ok to be a little silly, to goof around, to smile and have fun with one another.

It would be great if we could figure out a way to do that more often – without needing to put a label on it.