Originally written in fits and starts from 2015-2018
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
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? I thought about this for months at the time and eventually made a certain conscious choice. I was aware in a child-like way that it would affect my life for decades to come, yet because of the long and careful deliberation, 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 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. Life went on and adolescence continued transforming me as it does everyone, except in regards to this one area.
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, had a lot of fun for the nine years that I was there. (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.) I was insulated from the reality of college life which is itself insulated from the real world. I worked hard at times, especially at the many positions I held for the college newspaper, but when I did I was led by joy, not by any sense of duty or even a compelling sense of future consequences. Still thinking like a child, if I did hard work it was more a matter of doing things correctly than doing things out of an awareness of the future which depended on me doing my best. During and after college, I found myself attracted to the Infinite Sun warehouse which continued to insulate me from growing up (a bunch of twenty-something visionary “Rainbow hippies” living in a warehouse, going on silly adventures ranging from vision quests to political protests, drum-circling, dumpster-diving, hitchhiking around the country, all without paying much in the way of rent thanks to a generous benefactor whose main purpose was keeping his son away from drugs at any cost. Paradoxically for the type of crowd we attracted, for this reason it honestly was a drug-free environment much of the time. Insulated.)
Hence I remembered and kept my teenage decision well until I was about 30 years old. At that point, I finally found myself prey to 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 with all its liberties and responsibilities and had gotten myself into circumstances which absolutely required critical thinking skills I still lacked. In short, the full weight of adolescence finally hit me in the same way it hits everyone else.
The Two Worlds
I won’t go into details here about what brought about the change, but as I studied this confluence of forces which had gotten quite painful because I was living in two different worlds simultaneously and could no longer continue doing so, I made a few observations. One is that there exists an unspoken agreement between all adults, who basically treat children like 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 ones who love children. This was a mystery to me. I wouldn’t be able to put my finger on it for another 15 years, when I made the connection between what I was seeing then and one of the deepest insights of any philosopher I’ve yet encountered (Rene Girard’s scapegoat mechanism) but I could see the undefined border between worlds, an elusive something which silently made all grownups conspire against children. It was particularly clear to me because I was revisiting my long-ago decision, which had shaped my life more than I realized, until circumstances brought me face to face with myself and my future again.
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 conspiracy against children is evolutionary-grade, meaning it’s in all of us. As a working theory, I assumed its purpose is to guard and protect children from certain kinds of dangers, as is the case with many such forces which operate at that scale. There is a layer of this unspoken agreement which is conscious and obvious (e.g. 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 (e.g. all adults assume that a child doesn’t know more than an adult expert on a given subject).
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 effort. I understand it clearly now, and anyone 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 and unbounded imagination. Nowadays I 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 conspiracy of silent forces opposing 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. And for this fragment of my former ability, I am most grateful.
This condition of unbounded imagination, although frowned upon or constrained in many ways by the grownup 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 a non-obvious link, but I recognized it because so many people didn’t understand my innate ability to love and forgive everyone for anything in a way which is rare among grownups… although of course common with children. Children are able to rapidly develop a storyline that allows them to continue loving someone who has harmed them; grownups are much less flexible. 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 enablers and pawns to play roles in their elaborate games. I can recount numerous episodes where I had to get myself out of such circumstances, 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. If you’re paying attention, you’ll see that the childlike behaviors 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 unique 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. There are a number of other problems along these lines, all of which together conspire to create that glass ceiling effect mentioned earlier.
The Former 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 the depression which comes to anyone when they face themselves, I painfully and finally made that conscious decision to grow up which I believe occurs for most people during adolescence. I began doing so, fully aware by this point that I was thereby relinquishing my early teenage commitment. I was entering the world of grumpy grownups.
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, logic finally broke through. 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, in a way. (That’s putting things poetically. The actual decision is too raw to describe in detail here, and it involved accepting a role in corruption, accepting the grim reality that grownups are all corrupt, and 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 currently remaining thread of pain… 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 easy to preserve before. 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.)
Curiously, I now recognize that 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 my 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 read “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. It’s easier to figure out what to do, and people are friendly and encouraging in this area. 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 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 calculus — which still baffles me — but seriously advanced, theoretical math. I had no idea there were two different worlds within mathematics (tedious and amazing) or I might have fallen in love much earlier.
It started by reading 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. At the time, Grigori Perelman had just solved the Poincare Conjecture, and his story is one of the most interesting in all of math history, so it was easy to get drawn in. As I continued reading these biographies, I was exposed to mathematical ideas, usually peripherally, but instead of skimming over those parts, 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. I began to experience the joy of math epiphanies — seeing beautiful structures and patterns and understanding how things connected.
Biography was a great way in, and I had long ago learned how to find good writers out of the larger mix on a subject. As I read casually “for the story” I kept recognizing insights from great mathematicians like Euler, Gauss, and Riemann in ways that drew me in “like a deer panteth for water,” as the Psalmist once wrote. During the early years, I could see these beautiful ideas only intuitively, but the more I studied, the more they began to take 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 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. 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.
Since that kind of math is even still tediously hard for me, I was pleased to discover it was also hard for people like Einstein, an intuitive genius who relied on rational geniuses to make his insights more mathematically sensible (see http://www-history.mcs.st-andrews.ac.uk/Extras/Poincare_Intuition.html). His papers hardly contain any equations and 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 Hidden Theme Within Mathematics
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 like Euler, Gauss, Poincare, and Ramanujan, but it seemed 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. It was as if something about sharing insights with the math community was making people lose a certain sharp edge to their insights. 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 contemplated this pattern deeply because I wondered if I might be able to prevent it from happening to me, similar to how I had figured out how to prevent myself from becoming a “grumpy grownup” for so long. Having been through that process, I eventually determined it was impossible to prevent this inevitable fall from clarity. From that position, 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 the process I’ve described elsewhere (digging a hole and burying my 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. 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 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.
A while back I discovered someone else has written about this same fall, from a slightly different angle, but I recognize it. Therefore I can now frame this idea that was previously too idiosyncratic so others can understand it: At essence, I am seeking to know as much as I can in the pre-rigorous stage of a mathematician’s journey (see https://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/ or at least skim the article because it makes a really important observation relevant to the narrative here).
Alas, The Graceful End 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.
This all happens while pursuing a professional career that has little relationship to mathematics and raising a small family, so the speed is slow, and because I am prolonging the initial joy, I am in no hurry. I am careful to preserve the 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. Sounds crazy, but this kind of crazy is how 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.
Eventually what will be left will be something other mathematicians will be able to understand, with all the magical thinking stripped out or formalized. Along the way, I am keeping a form of math journal that narrates the progress from magical to mundane, which I will use in the future to write about the journey. In fact, this present essay is one of the first which has a beginning, middle, and end, rather than just capturing a beautiful fragment for future purposes. I’ve been writing it for a couple years, even though it isn’t very long. I may end up developing an entire book out of this material, because I am keenly aware of how I have barely skimmed the surface of this story. This book idea is not a pipedream. All of this happens within the larger context that, at heart, I am a writer. Not a mathematician, but a journalist for whom math is a hobby. In the beginning, the reason I write this brief essay is 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 understood. Hence the beginning of this essay with memories of childhood innocence, long before I fell in love with mathematics.
Why Do Children Ask Questions So Much?
This is one of the most exciting components of a unique angle on math that I have come to call Original Math*. There is a certain internal coherence to this discovery, and an obviousness which shocks me when I search and search and find that no one else has written about some of these core ideas previously (revising this essay after it was dormant for a long time, this is no longer true: I have since found others, very intriguing and provocative others, who have similar ideas. Finally.): 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), 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 flaw of the excluded middle of 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 are easily resolved)? And lastly, what does it say of our way of seeing the world if we do not commonly probe the foundations of our own beliefs at this level, or actively ignore those who do, unless they follow a protocol that is as much about protecting egos as it is about seeking the truth?
Even if I fail to sustain the magical-thinking coherence of these insights once I cross over into the rigorous domain, I believe these are the kinds of questions that deserve better answers than we currently accept. Due to the nature of my history, I expect that children, or young people, will see what I’m saying better than adults.
*”origin” referring to the name for zero in XYZ Cartesian coordinates, and also a pun on a new way of seeing mathematics, and for other fun reasons.