Written Monday, May 25, 2020

There are, periodically, moments in my life where I feel an uncontainable surplus of words - a sort of incessant need to say things. It's taken a lot of life experience (and more than a few "well now I feel silly" moments) to realize that this is generally a sign that I need to write.

Most of the things I feel I have to say are smallish in magnitude: a quick journal entry, a blog post, maybe a several-hour conversation with a dear friend. Lately, though, the journal entries, the conversations, all the quiet contemplation... it just seems to suggest that there is even more I need to say than usual.

This will be the first attempt of mine at starting to distill a larger cloud of ideas that has been forming for many, many years. Most of these ideas are not my own, and very few are new. The thing about ideas - especially in large groups - is that with enough sheer quantity of stuff to think about, it can often be the case that previously-undetected patterns become startlingly clear.

I'm not sure, yet, if that is what has happened for me, but it sure feels that way.


This essay is going to lean a little heavier on the mathematics than usual. I've spent a lot of time thinking about things in rigorous, logical, and analytical terms. These are my strongest "mental muscles" as it were.

I hope that isn't a deterrent, but I certainly understand if what follows ends up being obtuse or opaque. However, I want to express it this way first, so that eventually I can work to say these things in perhaps a more comprehensible way.

After all, to shamelessly mangle one of my favorite quotes, "nobody sits by the shore at sunset to watch the abstract geometry frolicking on the lake."

There is so much more to life, to the way we experience existence, than mathematical or scientific approaches can possibly describe. But this is the best set of expressions I have available for what I want to say, and perhaps this paves the way to richer words down the line.


I have been fascinated for many years by a concept called metastability. I learned of it while reading about chaos mechanics (a deeply important field it its own right) but it turns out - much like fractal mathematics - to appear in virtually everything in the known world.

In very simplistic terms, metastability occurs when you have a balance that is not perfectly at one extreme or another, and yet it is not necessarily motionless. The term "stable" in English tends to carry a connotation of the immovable, of solidity, of certainty.

But there is a very real phenomena called dynamic equilibrium - the ability of something to stay balanced even while constantly moving. One of the most familiar and immediately recognizable examples of this is riding a bicycle.

If you put a bicycle upright on its wheels and let go, it is extremely hard to keep it from falling over. It is not stable. Of course there are some truly impressive feats of static balance performed by people around the world (such as balancing stones) which rely on careful and precise arrangement of things to maintain balance without movement.

But of course people do also ride bicycles (and unicycles!) which means that there is a way to keep a balance while moving.

One of the easiest examples of dynamic equilibrium to overlook is simply standing on two legs. This is, believe it or not, a feat unto itself even on solid ground. Human anatomy is phenomenally complex and requires literal constant, tiny adjustments of dozens of muscles, ligaments, joints, and bones just to stay standing upright. We are not still - we simply have been trained to ignore our bodies making those subtle movements.

This formed, I think, a crucial sort of realization for me; dynamic equilibrium is all around us, all the time, and yet we are trained (more or less accidentally) not to see it.

A key element of chaos mechanics is the idea that any set of moving or changing things can find multiple different equilibria. For instance, consider a pencil. If it is well made, you can balance it on the tip, but that is not a very reliable balance - a tiny breeze will disrupt it. A much more reliable balance is to leave the pencil lying flat on a surface. It takes much more effort (comparatively) to disrupt a pencil that is simply flat on a table. So even this trivial object has multiple ways to balance - some more precarious than others.

I remember being fascinated by dynamic balancing toys when I was a kid - the egg-shaped dolls, the plastic birds that could somehow remain balanced on the tip of their beaks even when you poked them from different angles. There is far more to learn from these toys than I would have imagined.

Constancy and Change

Another fascination of mine is the field of mathematics typically known today as "calculus." More specifically, I've long been enthralled with the idea of studying the behavior of infinite combinations of infinitely tiny changes.

For a while, it seemed to me like the simultaneous inventions of Leibniz and Newton held so many resolutions to so many paradoxes. Zeno of Elea was finally given a proper answer. Calculus gives us the tools to understand how change works.

And yet, at this point in my life, I wonder how much wisdom we have sacrificed by deeming math to be the answer to those questions. Certainly, there is tremendous elegance and even power in the mathematics. The fundamental theorems of calculus are beautiful in a soul-stirring way to me. The technological and scientific brilliance and accomplishments unleashed by that mathematics are certainly testament to its effectiveness.

So why is it that, even with centuries of expertise on studying change, the world still seems so desperately unchanged?

The more things change, the more they stay the same. There is nothing new under the sun. All things old are new again. The only constant is change. You will never stand by the bank of the same river twice. Pick an era - someone had something deep to say about this fact.

The twenty-first century reaction to this idea is strange to me. We must be better than this, now. We have learned too much to think there is value in those silly, antiquated ideas. Life isn't all about cycles and repeating patterns, we're too powerful, us humans, to be imprisoned by that nonsense anymore. We don't need anything but intellectual prowess to live our best lives.

Ten years ago I would not have imagined myself disagreeing with that attitude. I was a staunch believer in objective, rational, rigorous study of the world. People's brains would solve everything, given enough time.

But first I discovered the work of Benoit Mandelbrot; and then I discovered the work of Kurt Gödel.

Fractals and Paradoxes

My journey into the mathematics and logic of these two thinkers started with the work of a genuinely phenomenal author, Douglas Hofstadter. But it was not the mathematics that I think captured me so fully - it was his connection between the rigorous thought and the undeniable artistry of J.S. Bach.

For years I found Hofstadter's slowly-evolving thesis compelling. It is certainly poignant and almost perfect for me - a lifelong nerd with fascinations in music, mathematics, logic, analogy, and computing. I devoured his writings whenever I could.

And yet, after endless contemplation of his work, and following up on half a dozen other brilliant thinkers via his various bibliographies, I came up empty. Something did not quite fit yet. I saw in his work - and that of many others - a sort of decades-long flailing, a late-career despair and exhaustion that reminded me of philosophers from many other approaches to thinking about life. Nobody found answers they were completely happy with. Nobody ever solved "everything." A few lucky souls found contentment in chasing new questions, revising old answers, pursuing thought as far as they could.

I walked away, time and again, with an uncomfortable question about it all. The question was simple, and the answer even simpler - except I'd already decided that I didn't like the answer, so I stopped asking the question:

"What if we can't simply think our way to solving everything?"

Problem Solving With Intuition

Part of what I do for my career involves working with very sophisticated and often unintuitive computing systems. Thinking about these things is, ostensibly, a vital part of the job.

I learned early on, however, that "thinking" is actually not a very good way to create software. Even now this feels like a slightly heretical thing to say (snark about the quality of computer software notwithstanding) but I think I finally have a way to articulate the intuition that I started gathering even as I dabbled with programming at home as a school child.

What most people would describe as the English word "thinking" involves a ponderous amount of words. Many people describe thinking in images, or sounds, or even emotions; and yet, for the sake of communicating with others, we all eventually need to turn out thoughts into words on some level.

I remember learning about mathematicians who made massive discoveries because they did not follow the rules. Young Mandelbrot was certainly famous for not obeying the processes by which most students were expected to answer exam questions, and yet he scored brilliantly and went on to pioneer some of the most important mathematics of human history. The entire idea of non-Euclidean geometry (which almost immediately revolutionized cartography, and eventually made it possible to derive Einstein's General Theory of Relativity, along with an endless list of other revelations) came about because someone dared to break the rules.

Even calculus was a testament to this: the rules of algebraic mathematics could not answer certain questions. They were considered deep mysteries that the learned experts of the age despaired of ever answering. Today, high school students solve them routinely as part of their course work, often without ever knowing how terrifying those questions were a few hundred years ago.

The common thread is actually very well documented and any good math teacher will explain it as part of the history of the field: every revolution of thought, in mathematics, has come about because someone changed the rules.

What is less-often discussed is how people found the inspiration for how to change the rules to begin with. It is not haphazard, it is not random. There is something that people are capable of that provides an insight - an intuition, perhaps - that on those lucky occasions gives someone the right set of tweaks to make, and a new discovery happens.

Certainly it is possible to just slap together arbitrary sets of made-up rules and see if they work; most often, they do not. Ask anyone who has constructed a nontrivial system of axiomatic logic. It is frustratingly easy to end up with nonsense. Also, thank Gödel for finally revealing that it simply isn't possible to make a logical system of any significant level of interest that does not eventually give way to either nonsense or simply leaves some questions permanently unsolvable.

At some point, trying to understand things through systems of rigorous, rule-based thought simply cannot be enough.

It is a great (and tragic) irony, to me, that mathematics - arguably the most pure distillation of thinking as a practice - has conclusively proven its own inability to completely answer its own questions and yet people widely remain convinced that thinking will solve everything.

If so many great advancements in thinking originate in something closer to intuition or imagination, why do we prize the thinking so highly, at a time when our problems feel unsolvable?

My job, as I mentioned, often includes things that could rightfully be called "unintuitive" - and yet I think this is actually a mistruth.

Distributed network systems, parallel (or concurrent) computer programs, and realtime systems have a very strange set of behaviors if you train your intuitions with a certain type of thought and form certain expectations.

What I have discovered, over two decades in the field, is that it is possible to have an intuition that perfectly understands these systems. As a younger programmer, I often felt amazed and profoundly awed at people who could look at what seemed like an intractable problem and simply know what was wrong as if by magic.

The more time I spend in the field, the more I realize that I have developed that magic - because, now, much of my job involves teaching other people the things that I can simply intuit about software on account of having been immersed in it for so long.

How did she know that? is a common question I overhear at work. These are not unintelligent people. Many have been programmers for nearly as long as I have, and some have far more time in the field than I do. All are excellent thinkers in their own right. And yet I can often join in with solving a challenge and land on an answer almost immediately; often the answer I start with is not a good one, but after a couple of (what must seem like random) attempts, it inevitably gets people un-stuck.

It's an evolved form of a pattern I first noticed a long time ago, one that almost anyone who does a lot of thinking can recognize: if your thinking is getting nowhere, take a break. Do something else.

There is something of a trope in math and software problem-solving: sleep on it. More people solve fascinatingly complex problems overnight than you might believe.

Call it the power of the subconscious, or the refreshing effect of relaxing and coming back with a fresh mind; either way, anyone who has struggled on hard thought-problems knows the experience of getting smarter by not thinking.

Even my first teacher of the electric bass guitar was fond of reminding me to not play for a few days, every now and then - and, much to my confusion at the time (and delight now), I often come back to the instrument after a few days of not playing and find myself a better musician for it.

The World

It's no secret the world has problems: inequality, injustice, violence, hate, failing systems of economy and government, prejudice, ignorance, sickness, climate destruction.

Many people spend a lot of time thinking about how to solve these things. And, in some areas, there is a lot of success. The past century has seen serious advancements in our global recognition of human rights. We have a scary long way to go on that front, but we are moving forward. For all the progress and improvement, though, it feels like other things have gotten worse - certainly the climate of the planet is not trending towards "better" over the past century. Economic and governmental problems are thrown into sharp and terrifying clarity by events like the the current global pandemic.

Worse, it seems like every time we try to solve one problem, some other issue takes its place.

This is not just a perception. It is real. I believe that the reason for it is simple to express but intensely difficult to understand... if one is trying to think about it.

In Buddhist traditions, it is often said that everything is interconnected. Up cannot exist without down. Nothing is isolated. Theoretical quantum mechanics tells us the same thing: a tiny change here can affect something seemingly unrelated in a distant galaxy. Chaos mathematics has the exact same lesson in its own words, distilled into the popular "Butterfly Effect" idea.

All the problems we see around us - those I have named, those I have not, those which I personally am unaware of - are interconnected.

In the language of the mathematics, the problems we see form a metastable dynamic equilibrium. Pull on one thread, and you don't unravel the knot - you simply create room for a different thread to join the tangled mess. It may seem to change, to get better, to get worse again, to evolve; but the core problems are not gone.

Politics is maddeningly cyclical in the United States: every few years we see a pendulum-swing of partisan nonsense, each side struggling to undo the efforts of the other. Anyone who looks at complex systems recognizes the trends here over the past century: this is a system that is thrashing, whose pendulum is increasingly erratic and wild. It is a system that any engineer would recognize as on the edge of complete overload and breakdown.

But it does not break - it seems bizarrely resistant to change. Nearly 100 years after the advent of women's suffrage in this nation, we still struggle to be permitted the legal right to reproductive health-care. After decades of overt civil rights struggle and some unquestionable improvements in those areas, we still see racism and bigotry literally embodied in the highest offices of the country. Why does this not change?

Put simply, the political system is not in isolation. It cannot be separated from economic power. It cannot be separated from cultural inertia. It cannot be separated from the collective unspoken experience of what technology has done to our lives - both the good and the bad.

More importantly, despite the deeply-ingrained nationalist attitudes of many influencing people in this country, it cannot be separated from the situation of the rest of the world. Many of the most catalyzing and profound changes in the way life unfolds in the US have been the result (directly or indirectly) of things happening on the rest of the planet.

Any one single problem we see can probably be solved in isolation, given enough thought and effort. But we do not have that luxury. Our problems are not in isolation, our problems are interconnected, and they reinforce each other.

At this moment, it may be tempting to feel a sense of hopelessness, a sense that we can't possibly think our way out of this.

I believe that conclusion, as painful as it is, to be correct. But it is not the whole story.

Slow Down and Live

People have lamented for decades the way technology has affected the "pace of life." We all feel busy, rushed, constantly in a hurry, always need to do something. Of course, culturally this is reinforced by dismissing this attitude as that of "Luddites" - don't fight the pace of change, stop living in the past, forget that old-fashioned silly stuff.

It should give us pause that the youngest generation, who have never experienced that "old fashioned" life first-hand, tend to be the most vocal opponents of the frenzied need to Do Things - and the most disillusioned by the incessant expectation to have an education, a career, a family, a retirement. They have never known any other way of life and they still recognize, on some level, that it doesn't work and cannot possibly work. Anyone fortunate enough to remember the world before the Internet - or, even rarer, life pre-Internet with no TV in the house - would do well to take notice of this. We're all telling each other to do more; the emperor is not the one with no clothes. We are all taught to compliment each other's figurative wardrobes to distract from the reality that we're all roaming around naked. The world has created a set of patterns that make it feel impossible to escape - and layered on top of this, we're taught to believe that we should not want to escape.

For most of us, trying to "do nothing" is an instant recipe for unrest. It feels wrong, feels bad, feels lazy. Others are quick to reinforce this emotional response: I don't want to pick up your slack. For me, I spent years trying to be more efficient. Do more with less. Work smarter, not harder. A perfect mantra to ensnare a computer engineer.

And it snared me - but it didn't work. I never could do "enough."

What has finally gotten me to begin to relax, to feel like I am not simply "doing enough" but in fact doing phenomenally well, has been slowing down.

It is incredibly tough. Nothing about the world I live in suggests that this is OK to do. So I started out by undermining the my system of internalized stigma against laziness using the tools of accomplishment as a scalpel.

I set myself a goal of spending at least an hour doing something "not productive" every single day: watch a show, read a fiction book, awkwardly try to learn to juggle, anything. If it did not smell like productivity, I wanted to do one hour of it a day. At first I did not even try to do a solid hour: ten minutes here, five minutes there. Add up to an hour and I can reward myself with something at the end of the day. Any link of the "dopamine hit" to inactivity would work.

It didn't take long to get hooked. Watching short videos gave way to longer shows, then movies. Reading a few pages turned into "just one more chapter." Even playing games became a way to undermine the need to do things: "how long can I take just messing around instead of finishing this part of the story/this level/etc.?" I no longer tried to play games to finish them or get "achievements" - I started playing just to time how long I could spend not playing by the rules but still having fun.

Within a week or two, I no longer felt bad about not being productive. More tellingly, I actually got more done - as corroborated by my long habit of journaling and writing down lists of things I have done every day, as well as observations from my colleagues at work.

And then the next phase: spend ten seconds, eyes closed, just staying still. Once a day, maybe twice. It was also intensely hard at first. Ten seconds of silence and motionlessness was almost impossible - but after a couple of days, I couldn't resist the urge to get better. The need to "do things" transformed into a need to break a personal record.

Sit still for one minute. Two minutes. Five minutes. Just stare out into the yard and watch the sunlight in the trees for one minute. Two minutes. Five minutes.

Listen to music and do nothing. Ten minutes. One album. Build on it.

Yesterday I fell asleep, curled up in my beanbag chair, listening to music. I did nothing meaningful, productive, or "significant" in the eyes of the world for nearly two hours. And I want more.

Relinquishing the urge to do has started to create room, in me, for an utterly unexpected but massively welcome change in motivation. It isn't a problem to spend two minutes "doing nothing" - so now I can spend two minutes simply drinking a glass of water. Two minutes "doing nothing" while also trimming my nails. The time isn't relevant as much anymore, so I can use that time more richly.

I still have difficulty not falling into old habits, but tiny slices of time are manageable, and I can already see the spans getting longer and longer as I practice. The shift is hard to describe, but unmistakable: from the outside, I have not stopped doing things, but from my own perspective, I have started living.

When I get carried away by the need to solve things, to think, to accomplish, I get tired faster. I get cranky. I get bored. I get upset by the state of the world.

But somehow, when I slow down and can manage - even if it is just a few times a day - to spend a minute or five doing "nothing", or just pouring my limited and flighty attention into a single activity for as long as I can... I do more, and I enjoy what I do. I have better ideas, clearer insights, more energy, and more of a sense that things can change, somehow. I sat down early this morning to spend five minutes recording some bullet-point notes in preparation for writing this entry, and the rest just spilled out as if by sheer momentum; two hours later, I'm ready to post a completed set of thoughts.

We can solve the inter-tangled mess of dynamic equilibria that underlie the problems around us, no matter how big, how small, or how knotted together they may be. We won't fix them with thinking.

We'll fix them by living.

Slow down, and live. Create the space for something different to happen. The most radical act of self-care we can perform, in the world as we occupy it right now, is to be alive - especially if we've been told we're not doing anything in the process.