On the Art of Approaching Math and Quantum Physics from a Child's Perspective

Originally written in fits and starts from 2015-2019

In unexperienced infancy
Many a sweet mistake doth lie:
Mistake though false, intending true;
A seeming somewhat more than view;
That doth instruct the mind
In things that lie behind,
And many secrets to us show
Which afterwards we come to know.
-- Thomas Traherne (1637–1674)

A Singular Decision and a Long Childhood

There is something unique in my approach to math and physics which has to do with a curious decision I made in early teens, probably around age twelve. Having reached the height of childhood thinking, I could finally see what previously had been hidden -- although the truth had been dawning for quite a while by then: that grownups were wretched creatures as compared to children. As this dawned on me, I wanted to ensure that I didn't become a "grumpy grownup" like all the others. I thought about this deeply: what invisible thing had ensnared grownups, where did it begin, how did it work?

At the time, I thought about this for months and eventually made a certain conscious choice, which I will describe below. I was aware in a stil-child-like way that it would affect my life for decades to come, yet because of the careful deliberation of several months, I was fairly certain that there was no other reasonable path.

A key element of the decision was in its timing; it's not a choice you can arbitrarily make at any time. I made the choice right at a critical stage in the process of developing an adolescent's understanding of the world which I am now convinced everybody else chooses differently. Without using these particular words, I carefully and intentionally chose to "remain as a child."

Following through on the decision, I began to make certain quiet efforts in my inner world to ensure this happened. The decision was strong enough to be a kind of vow, although that is a grownup concept so it's not how I framed it. It was simply an important decision I made entirely using internal resources, asking no one else's advice, and I began to keep it firmly, at a moment when it was appropriate to do this, whereas later it might not be so easy. Life went on and adolescence continued transforming me as it does everyone, except in regards to this one aspect.

I read a lot of books, visited the library absurdly often, and eventually went to college. Anyone who knew me then can tell you that I did not take higher education seriously and, especially for someone who never had alcohol up to that point, had a lot of fun for the nine years that I was there. In many ways, for reasons which are explored elsewhere, I was insulated from the reality of college life -- which is itself insulated from the real world. I did not even get a bachelors' degree during that time. At one point I discovered I had taken Introductory Logic three times, with a passing grade each time. Apparently, I thought "Hey that looks like a fun class" while reading the syllabus, and took it again. Note that it was with a different professor each time, if that makes any difference.

I worked hard at times, especially at the numerous positions I held for the college newspaper, but when I did I was led more by joy, than by any sense of duty or even a compelling sense of future consequences, although these impulses played some role. I had intellectual awareness of future consequences, but they were not driving motives as they are for some. Still thinking like a child, if I did hard work from time to time, it was more a matter of doing things correctly than doing things out of a sense of the future which depended on me doing my best. The depth of my social immaturity was more obvious to others than to me for a long time yet.

During and after college, I found myself attracted to the Infinite Sun warehouse, which was an environment that continued to insulate me from growing up for several more years. It was a ragtag amorphous group of twenty-something visionary artists, "rainbow hippies," living in a warehouse near downtown, going on silly adventures ranging from individual vision quests to large-group anti-war, pro-environment political protests. There were many drum-circles, regular dumpster-diving adventures, periods of hitchhiking around the country to other intentional communities, endless philosophical discussions, and all without paying much in the way of rent, thanks to a generous benefactor who had his reasons for sponsoring the chaos we lived. Paradoxically for the type of crowd we attracted, it honestly was a drug-free environment much of the time. There are many details I'm leaving out here, which are told elsewhere, the point being: I remained insulated from the "real world."

Hence I remembered and kept my teenage decision firmly, along until I was about 30 years old. At that point, I finally found myself prey to hidden forces tearing me apart -- on the one hand I still fundamentally perceived the world "like a child," and on the other hand I had a fully adult body and intellect with all its liberties and responsibilities and had gotten myself into circumstances which absolutely required critical thinking skills that I still lacked. In short, the full weight of adolescence finally hit me in the same way it hits everyone else, albeit a decade and a half late.

The Two Worlds

By this time, things were rather complicated. I won't go into details here about what brought about the change (it was painful, that's all that's relevant here), but as I studied this confluence of forces which had gotten quite complex because I was living in two different worlds simultaneously and could no longer continue doing so, I made a few observations useful to this narrative. One is that there exists an unspoken agreement between all adults, who nearly unanimously basically treat children like adorable but ignorable idiots in some way or another. From my unique perspective, where an important part of my ability to perceive was still connected to childlike innocence, I could see this was curiously true of even the grownups who love children. This unspoken agreement was a mystery to me, as I could not understand what linked grownups into such unison on this.

I wouldn't be able to put my finger on it for another 15 years, when I finally made the connection between what I was seeing then and one of the deeper insights of any philosopher I've yet encountered (Rene Girard's scapegoat mechanism). Nevertheless, I could see the unspoken border between worlds, an elusive something which silently made all grownups conspire against children. The signalling system was invisible to children, who simply took it for granted and never questioned it, but the border it produced was particularly clear to me because I was revisiting my long-ago decision, which shaped my life more than I realized until circumstances brought me face to face with myself and my future again.

It was gradually becoming clear that I could not continue to remain "as a child." Again, I contemplated for months, again, privately, since no one I knew ever talked about such things. I could see the underlying purpose of this seeming-conspiracy against children as something evolutionary-grade, meaning it's in all of us. As a working theory, I assumed its purpose was to guard and protect children from certain kinds of dangers, as is the case with many other such forces which operate at that scale. There is a layer of this unspoken agreement which is conscious and obvious (for example, all adults "conspire" together to ensure children don't fall into a swimming pool), but there's a subconscious part which was more visible to me than most because I was still living on the "child" side of that agreement (for example, all adults assume that a child doesn't know more than an adult expert on a given subject. That assumption can change, but it's the default starting point).

My experience of this unspoken agreement was similar to encountering a wall, or a kind of glass ceiling; a boundary beyond which I could never cross regardless of intent or effort. I understand it clearly now, and anyone else whose behavior indicates a child-like heart knows exactly what I'm talking about, but most people long ago lost awareness of this boundary, or replaced it with another one which has a similar effect but is no longer oriented around childlike innocence.

Losing awareness of this boundary is an event not noticed; it's just a normal part of growing up, like losing the ability to think of the moon as made of cheese. Who notices the day that happens? I surely did, but I'm getting ahead of the story, so let's continue.

Imagination is an Important Element of True Love

Into my early thirties I was a still a child; a kind of idiot-savant with little common sense, dorkish mannerisms, and unbounded imagination. Nowadays I sometimes miss the immediate access to imagination which was my daily experience, but I do not miss all that comes with it, having since acquired some of the benefits given to those who participate in the more grownup conspiracy of silent forces opposing and suppressing such unbridled imagination. However, even to this day -- when the circumstances are just right -- I am able to be persuaded that the moon is made of cheese, or is home to six-legged sheep tended by shepherds who use flatulence as a way to move themselves across the dusty surface. I really am able to believe such things at times, and do in fact hold some rather peculiar private theories as a consequence. Not everyone can do this, having learned to suppress such crazy ideas in favor of the dominant paradigm. For this fragment of my former unbridled ability, I am most grateful.

This condition of unbounded imagination, although frowned upon or constrained in many ways by the adult world in general, is not entirely a bad thing; in fact, mixed with a healthy awareness of objective reality, it's the best way to be. For example, it turns out that having an unbounded imagination is an essential ingredient in being able to love and forgive people. That's not an obvious link, but I recognize it because so many people didn't understand a natural ability to love and forgive everyone for anything in a way which is common with children but rare among grownups. Children are able to rapidly develop an entirely creative storyline that allows them to continue loving someone who has harmed them; grownups are much less flexible, and tend to adopt well-worn culture-driven storylines shared by many others, inherited through the centuries without much critical inspection. Jt is much harder for grownups to forgive, while for children it is the natural state. I am convinced that true love requires an abundance of imagination, and quick and complete forgiveness.

Sadly, such pure imagination has some equally strong demerits: For example, being so easily persuaded, I was prey to any random narcissist who sought narcissist supply from enablers and pawns to play roles in their elaborate games. I can recount numerous episodes where I had to get myself out of circumstances with narcissists, sometimes -- like Pinocchio -- requiring the intervention of fairies, angels, or some other kind of deux ex machina to do so, because I was in well over my head. Narcissists take a commodity like endless forgiveness and use it as fuel to destroy as much as they can. This is well written about elsewhere, so I won't go into detail.

Such childlike behaviors as quick forgiveness can reflect immaturity, and can easily lead someone into a life of crime or of incarcerated innocence. The fact that I had gotten to this point in my life based on a singular decision in early adolescence -- which separated me from the normal hooligan -- was moot because nobody cares about such things as deeper motivations, or at least extremely rarely. Other than writers like Dostoevsky, people in general react to appearances with a little guesswork toward motivation, thereby confining out-of-the-box thinkers in the box.

There are a number of other problems along these lines, all of which together conspire to create the glass ceiling effect.

The Singular Decision Iterates Again

Circumstances eventually became painful enough that I could not escape my future any longer. I could no longer "live in the moment." After wallowing in pretty serious depression which comes to anyone when they finally face themselves, I painfully made that conscious decision to grow up which occurs for most people during adolescence, when it is easier than in mid-thirties.

I began the journey, sadly and fully aware by this point that I was thereby relinquishing my early teenage commitment. I was at last entering the world of grumpy grownups I had so long avoided.

It was an unbelievably grueling decision to make, moreso then than I think it is for anyone at the normal time in life. I resisted it with everything I could muster, but having taken all those logic classes, it should not be surprising that it was logic which finally broke through and carried me across the first chasms. I was basically letting go of a dear and treasured friendship with a part of my character which had something so sweet as innocence at its core.

Innocence in certain frames is foolishness, though. And that was the crux. Innocence must give way to wisdom.

That's putting things poetically. The actual decision process is written about elsewhere and is far too raw to describe in detail here. It involved accepting a role in the common corruption, accepting the grim reality that grownups are all corrupt, and nevertheless accepting I must become one of them.

Because I thought it through so carefully, I was prepared for all consequences of the decision except the one thread of pain which remains to this day... that I betrayed my own innocence and now no longer have immediate access to childlike innocence. In other words, I have to work hard to preserve what was thoughtlessly easy to preserve before.

I am one of you. I currently believe that once you make this decision, you make it again, over and over, until you're free from mortality, except it's not really a decision, it's just a weary re-acknowledgement that the decision is being made for you by the circumstances you're in because you haven't got the strength to resist the general corruption any more. I think it is from the fountain of this particular sadness that great and noble things are born, kind of like Abraham Lincoln's melancholy being so powerful that all the forces of the Civil War arrayed against him were not enough to break his hard-won commitment to the correct principles. Again, I'm waxing poetic, but it's a way of coping, and if you're going to be a poet, there can hardly be better material than the common melancholy which poets perceive more deeply, but which all share.

Curiously, I now recognize it was at this same time that critical thinking entered my life. Years later, I remember the moment when I formally began to accept the reality that -- because this is a world composed of deception which is layered and subtle on such a deep level I cannot cipher a way out and yet must suffer until I do -- I must consciously choose to accept a cloak for open innocence out of the fabric of deception in the same way as everyone else.

In other words, I needed to consciously create an illusion of who I am and begin convincing others of it, rather than simply being who I am. The only other path I could see was to go live in a desert, away from all people, which was not an option because I had already read the hauntingly well-written Into the Wild by John Krakauer, and knew I wouldn't survive.

When presented with this particular Join or Die, after having successfully postponed the decision for decades, I finally joined, seeing it as the way out of being the fool I knew myself to be into something more palatable to the world around me.

One of the valuable things I gained from having postponed the decision to join the cadre of grumpy grownups was the extremely sharp spotlight shining on the moment I made that decision and began following through. When it happens during adolescence, it can pass without notice because at that stage everyone expects you to be "growing up" and everywhere you look there are cues signaling you to do so, and how to do so. At that age, it's easier to figure out what to do, and people are generally friendly and encouraging in this area, although of course there are some who prey upon this stage. There is no time in life when peer pressure is so exquisite as this time.

However, when it happens in mid-thirties, everyone long has given up on the idea that this will ever happen to you, and you're either in jail, or in a mental ward, or homeless, or some other place where the outcasts and dregs of society, who never "grew up" for one reason or another, survive. Usually you are so deeply embedded in your situation that there is no hope of getting out of this zone permanently, except by that precious everlasting hope offered by mothers and God, who alone hold the kind of love required to bring someone from such a condition into polite society again.

As for me, even my mother had given up on me, so it was up to God.

Now for how Mathematics is Involved in this Story

Almost as if to comfort the loss of innocence, at about this same time I discovered and fell madly in love with advanced mathematics. Not everyday math, or even basics like calculus -- which still baffles me -- but advanced, theoretical math, number theory, prime numbers and quantum mechanics kind of math. I had no idea there were two different worlds within mathematics (tedious and amazing) or I might have fallen in love much earlier. There was no better time for me to fall in love with something that literally ends with infinity, because it took away the sharp edge of that razor which was herding me forward.

It started by reading a fascinating book about the history of zero, followed soon by a number of math biographies. These contained limited actual math and focused more on the mathematician's social relationships with other mathematicians and the process of being discovered, always a fascinating story. Think of the great movie "Beautiful Mind" about John Nash -- a great story with little mention of actual math.

At the time, Grigori Perelman had just solved the legendary Poincare Conjecture, and his story is already one of the more interesting in all of math history. So it was easy to get drawn in. As I continued reading these biographie,s I was exposed to mathematical ideas, usually peripherally. At first, I was skimming over those parts, but soon I realized that I was able to understand things I never thought possible. Quantum physics started to make sense simply because I could finally see enough of it to think about it coherently. I began to experience the joy of math epiphanies -- seeing beautiful structures and patterns and understanding how things connected. In short, I was discovering a way to keep childlike innocence alive and nourished, in a manner which affected no one around me (largely because nobody normal cares about advanced mathematics, so attempted conversations were always short).

Biography was a great way in, and I had long ago learned how to find the better writers out of the larger mix on a subject. As I read casually "for the story" I kept encountering mentions of insights from great mathematicians like Descartes, Euler, Gauss, Riemann, Poincare, Einstein, Gödel, Ramanujan, and more recent ones like Nash, Feynman, Perelman, in ways that drew me in "like a deer panteth for water," as the Psalmist once wrote. Note, you may have already noticed, math encompasses quantum physics -- which is currently only coherent from a mathematical approach since the empirical data is so weird -- so I was reading biographies of people in both fields (and logic) and calling it all mathematics.

During the early years of the journey, I could see these beautiful ideas only intuitively, but the more I studied, the more they began to gradually take their rational form. For example, until only recently, my eyes glazed over in the way non-mathematicians know well whenever I saw a mathematical equation with more than a couple symbols in it. Now, more than a decade into the journey, I am slowly beginning to decipher them, and enjoying it. I take it in tiny pieces at a time, usually buried within a healthy narrative that is compelling me to understand some aspect I've long pondered.

I reckon it will be another decade before I can decipher a string of symbols comfortably; in other words, it'll be a fully twenty year journey for me to get comfortable with what a first-year math nerd does daily. I'm in no hurry. This is my joy. I'm following it.

Since that kind of symbolic math is even still tediously hard for me, I was pleased to discover it was also sometimes hard for people like Einstein, an intuitive genius who relied on the rational geniuses to make his insights more mathematical (see http://www-history.mcs.st-andrews.ac.uk/Extras/Poincare_Intuition.html). His private notes show he was good at the math, but his big papers hardly contain any equations. He famously once said "Since the mathematicians have invaded the theory of relativity I do not understand it myself any more." When I saw what he had done, I realized there was a place for slow-wits like me in mathematics, whereas before I had always thought that math was something at which I would never be any good.

A Repeating Theme Within Mathematical History

While learning math from a biographical perspective first, I began to see a theme repeated over and over throughout mathematical history: a young prodigy makes a powerful insight into math, publishes, and then spends the rest of his career extending or refining that same insight, rather than making any more powerful insights. There are occasional exceptions by prodigies, but it seemed to be a theme repeated often enough to be a general rule. It was, for example, true of Einstein who spent the last half of his life pursuing something he famously never found.

I contemplated this recurring theme. To me, it seemed as if something about sharing insights with the math community was making people lose a certain sharp edge to their early insights. Note this is not a private observation; the Fields Medal, sometimes known as "the Nobel of Mathematics" (which curiously does not award in the category of Math), does not award anyone over 40 for this reason. I pondered causes: Did recognition of an idea by others affect the ability to think with pre-recognition clarity? Is this related to how ego operates? I could see the transformation had to do with intuition becoming cloudy... but why? Was there a way to keep intuition clear? I answered myself with some ideas about humility, likely influenced by some biographical passages about Bernhard Riemann and my surprise at seeing how Einstein wrote in such a humble, readable, fashion.

I contemplated this pattern deeply because I wondered if I might be able to prevent it from happening to me, probably similar to how I had figured out how to prevent myself from becoming a "grumpy grownup" for so long. But how to do it right this time? Having been through that process, I eventually determined it was impossible to prevent this inevitable fall from clarity into the real world. Coming from that unique angle, I began casting about for a path forward which could maximize the underlying power curve if I did things right. I began to aim for a point in the future when I would "lose my mathematical innocence," only this time around, by having prepared to do so for a long time, rather than being forced into it by painful circumstance.

On the Art of Preserving the Joy of Math

I took myself through a process I've described elsewhere (digging a hole and burying my initial math journals for a while, etc) as a way of ensuring that I remembered my own motives and who I was in the earliest stages, already knowing that I was going to be studying math more and more deeply for the rest of my life. This present article is written (over a period of several years) both for the personal therapy of extracting a coherent narrative out of what was at times pure chaos, and also to hopefully help shape the journey of anyone else who enters mathematics through the door of biographical narrative.

My point is simple: "Treasure and protect what innocence you have in the beginning, it's more valuable that it seems at the time."

These days I approach mathematics with a strong desire to harvest everything I possibly can before I cross that inevitable threshold and "fall." It hasn't happened yet, but I believe it will because I want what happens on the other side of that fall, even if only as a simple matter of concluding what I started. Remember how critical thinking began only after I accepted the common corruption of adulthood? That gift alone almost makes up for the loss of innocence.

A while back I discovered someone famous has written about this same fall, from a slightly different angle, but I recognized it immediately. Therefore I can now frame an idea that was previously too idiosyncratic for others to understand. Here is how Terry Tao put it:

One can roughly divide mathematical education into three stages: The “pre-rigorous” stage, in which mathematics is taught in an informal, intuitive manner, based on examples, fuzzy notions, and hand-waving. (For instance, calculus is usually first introduced in terms of slopes, areas, rates of change, and so forth.) The emphasis is more on computation than on theory. This stage generally lasts until the early undergraduate years. The “rigorous” stage, in which one is now taught that in order to do maths “properly”, one needs to work and think in a much more precise and formal manner (e.g. re-doing calculus by using epsilons and deltas all over the place). The emphasis is now primarily on theory; and one is expected to be able to comfortably manipulate abstract mathematical objects without focusing too much on what such objects actually “mean”. This stage usually occupies the later undergraduate and early graduate years. The “post-rigorous” stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory... The emphasis is now on applications, intuition, and the “big picture”. This stage usually occupies the late graduate years and beyond.

See his brief description at https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ for more on this idea. At essence, I am seeking to know as much as I possibly can during the pre-rigorous stage of a mathematician's journey, understanding that doing so carefully will provide a garden with many flowers in my old age.

The Graceful Conclusion of Magical Thinking

I recognize I bring to this stage of the journey a cumbersome reliance on magical thinking -- the native language of intuition -- which is slowly being strained through the sieve of an incrementally growing rigor as I continue to study, a little here, and a little there. I'm no longer sure that I will even recognize when I finally make that transition -- will it happen gradually, or will there be a moment when I turn briefly to finally say goodbye to my "mathematical childhood"? We shall see.

This all happens while pursuing a professional software developing career that has little relationship to mathematics and raising a small family, so the speed is slow, but as I said, I'm okay with that. This is about joy. I am most careful to preserve that beautiful core of the insights which first began to flood my mind while studying the history of zero and realizing how it was connected to infinity. That there is a link between these two may sound crazy at fist, but this kind of crazy is how many math discoveries appear at first, so I'm not worried about it. The more I study, the more this peculiar core is being shaped into something others can understand. I also have a growing sense of the importance of humility in making mathematical discoveries. It's complicated, because there are definitely plenty of egos in the history of mathematics, but I have this compelling hunch that the deepest discoveries come through mathematicians who have a certain purity of thought which has something to do with humility.

Along the way, I am keeping a sort of math journal that narrates the progress from magical to mundane, which I hope to use in the future to write about the journey. In fact, this present essay is one of the first in that series which has an actual beginning, middle, and end, rather than just capturing a single beautiful fragment for future analysis. I've been writing this narrative for several years. I sometimes wonder if I'll eventually end up developing a book out of this material, because even as I write now, I am keenly aware of how I have barely skimmed the surface of this story. All of this happens within the larger context that, at heart, though I have come to love mathematics, I am a writer. Not a mathematician, but a journalist for whom math is but a treasured avocation which helps structure my thoughts in ways useful for other writing.

In the beginning, the reason I began working on this essay was to annotate that I believe I am working on a way of seeing and doing mathematics which is more native to the way children see than what is commonly known. Hence I began with memories of childhood innocence, laying a foundation for falling in love with mathematics as a way to preserve that innocence. I hope I can bring, in a small way, to mathematics, encouragement toward people asking childlike questions about mathematics, which are answered thoughtfully rather than with scorn because we understand that rigor does not necessarily equal grumpy. By doing this, I believe we will make math more fun for all. I'm not alone in this effort, it seems there are many who are waking up to the beauty and joy of math these days.

At this stage of the journey, I have found enough riddles in math to know there is plenty of work to be done once I cross over to the rigorous stage. I currently ask questions like this: Why is it that more people don't question the awkwardness of having both a positive and negative infinity? Why is it that more people haven't realized that math must begin with a singularity if it is to be coherent (not a null for which there is no intuitive image), and that it would be best if the two worlds of pure math and physics could touch in at least one reliable, well-known, spot, instead of being on two parallel paths as if reifying the most controversial of Euclid's axioms? Why is it that more people can't see the fundamental logical flaw of the excluded middle in binary logic and how it affects everything we see, think, feel, and do -- and therefore ought to be fixed or at least understood, instead of deeply hidden and actively ignored? Why is it that more people can't see the countless instances of confusion between Euclid's ancient definition of a point and the very different point required to understand physics (and why therefore famous paradoxes like Zeno's, or certain bizarre assumptions within quantum physics can be easily resolved)?

Even if I fail to sustain the magical-thinking value in these questions once I cross over to the rigorous domain, I believe these are the kinds of questions that deserve better answers than we currently find. Thankfully, the more deeply I study math, the more I am able to find the obscure but rich periphery, where such ideas are entertained with full vigor, although little-known.

Due to the nature of my history, I expect that children, or young people, will "get" what I'm saying better than adults. On that note, I wonder if it's possible to write this same narrative in a way which is more accessible to adolescents and younger. Which is a good thing, it usually means I'm finally done with an essay, if I start seeing the seeds of another one.

Posted in Everything, Mathy Stuff, Postinfinity Tergiversation on Jan 18, 2018