When I was 13, I developed a love for stock trading. I spent many hours learning how to analyze the fundamentals of companies, as well as the technical patterns found within their price charts. Whenever I went to Barnes & Noble with my family, I would head to the finance section and pick out a few titles to look through while sitting in the carpeted lanes between bookshelves. (Back then they didn’t stop people from doing this.)
One day, I found a book which changed the direction of my life. It wasn’t the same as the other books I had been reading. And, in fact, it seemed to be saying that much of what I had learned was flawed or outright wrong.
The Tails
The book was The (Mis)Behavior of Markets by Benoit Mandelbrot and collaborator Richard Hudson. Mandelbrot, I learned, was considered the “Father of Fractals” for his work in the field of fractal geometry. He had been using fractal analysis to study nature—finding that they are a universal presence in infinitesimal organisms, gargantuan galaxies, and everything in between.
“My life’s work has been to develop a new mathematical tool to add to man’s small survival kit. I call it fractal and multifractal geometry. It is the study of roughness, of the irregular and jagged. I coined its name in 1975. Fractal is from fractus, past participle of frangere, to break... It is used today for an improbably diverse set of tasks: compressing digital images over the Internet, measuring metal fractures, analyzing brain waves in an EEG machine, designing ultra-small radio antennae, making better optical cables, and studying the anatomy of lung bronchia.” - Benoit Mandelbrot
The book I picked up that day was about the failures of our traditional methods for understanding, analyzing, and predicting stock market behavior. And it explains the relative success of fractal-based methods in those same areas. It was an application of a powerful, universal phenomenon—fractal self-similarity—to the complex system of “the market”.
I found many beautiful and persuasive concepts in this book. One idea which stayed in my mind ever since is what Mandelbrot called “tail events”. To understand what this means, we should start by thinking of a “bell curve”—a visual depiction of a probability distribution. The most frequently invoked bell curve is presumptuously called the Normal distribution—otherwise known as a Gaussian distribution.
As you can see, the bell curve is sectioned up into “standard deviations” with vertical lines. Each piece of the bell curve contains a set of possible events, and the entire thing represents a complete state-space of possible events. This delineates what is likely, unlikely, and impossible within a given system. A financial market might experience a small price fluctuation, or an extremely large one, but the market will never turn into a cat.
In a Gaussian distribution of these events and their probabilities, about 68% of events fall within one (+/-) standard deviation of the median, 95% within two, and 99.7% within three. This means that tail events of four standard deviations or more are exceptionally infrequent, and trend toward impossibility. And there was a longstanding trend of using Gaussian distributions to represent/predict the price movements of stocks. Mandelbrot, however, employed fractals to show why this has been an extremely costly mistake.
“By the conventional wisdom, [the market crash of] August 1998 simply should never have happened; it was, according to the standard models of the financial industry, so improbable a sequence of events as to have been impossible. The standard theories, as taught in business schools around the world, would estimate the odds of that final, August 31, collapse at one in 20 million—an event that, if you traded daily for nearly 100,000 years, you would not expect to see even once. The odds of getting three such declines in the same month were even more minute: about one in 500 more minute: about one in 500 billion. Surely, August had been supremely bad luck, a freak accident, an ‘act of God’ no one could have predicted. In the language of statistics, it was an ‘outlier’ far, far, far from the normal expectation of stock trading. Or was it? The seemingly improbable happens all the time in financial markets.” - Benoit Mandelbrot
Mandelbrot suggests that bell curves with “fat” or “long” tails are needed to accurately represent the stock market. The extra “tail space” in these bell curves capture the fact that in many complex systems, the “extreme” and almost-impossible events of a Gaussian probability distribution occur far more often.
And this is a concept which I believe also reveals hidden features of love.
The Love
The middle of the bell curve accounts for much of our daily lives. But the greatest treasures and dangers alike are frequently found in the tails of love. This “tail space” contains, among other things, certain quantitative dimensions of what I’ve been calling “extreme love”. I discussed different flavors of this phenomenon in an effort to get to the essence of extreme love, and provided a playlist whose songs evoke these themes.
This essay follows from these thoughts, and my hope is to categorize some possible dimensions of extremity. As with Mandelbrot’s fractal-inspired view of financial systems, where certain tail events account for more change than many months or years of trading activity combined, these dimensions of extreme love may radically and suddenly transform a person, relationship, or world.
Rarity
There is a natural temptation to see phenomena as adjacent—related to each other in a straightforward, linear fashion. We have been trained to look for “normal” distributions, even where they don’t apply. So the first dimension of extremity I suggest we consider is rarity. We can begin by imagining a range of average “middle of the bell curve” relationships—some a little worse and some a little better than the median. We would find many healthy, happy, fulfilling relationships within two or three standard deviations on the positive side of this probability distribution. At three standard deviations from average, we would already find some pretty excellent relationships—ones in which most people would be happy to spend a lifetime.
A Gaussian view tells us that we shouldn’t waste time thinking of tail events beyond this point. A relationship that falls in that part of the bell curve would be so rare that you are almost assured of never meeting that other person. In fact, given the assumptions of this model, they likely don’t exist.
However, I suspect that love has longer tails than many realize. Extreme love is by no means common, but where a Gaussian distribution tells us that there might be only 1/100th of one person (based on a population of eight billion people) in the “tail space” beyond three standard deviations, long-tailed distributions say that there is a real chance of meeting someone five or more standard deviations above love’s median. In my essay, Hircocervus, you can read letters from lovers who felt that they had relationships so rare that ones of equal status would never be found again in their lifetimes. Maybe they were right. These relationships could have been of the type we are discussing here—which push the boundaries of what we consider normal or even possible.
Intensity
In the example given earlier, of a huge market boom or bust, the assumptions of a Gaussian distribution are shattered. It does not account for events of that intensity—or at least radically underestimates their scale and frequency. Could the same be true of extreme love?
In this dimension of extremity, we might place certain constructs like Obsessive Love Disorder, limerence, ecstasy, or amigeist along a spectrum of extreme love. Something like OLD would be on the negative end of extreme intensity, while amigeist might be on the extremely positive end. As I wrote in a previous essay, “extreme” does not necessarily mean extremely good.
Consider Martin from A Severed Head, or Catherine and Heathcliff in Wuthering Heights for examples of intensity which seems too extreme to be explained by a Gaussian model of possible events. Additional insight about occurrences of extremely intense love events (such as the two examples above) can be found via the idea of turbulence.
“We see turbulence almost any day, just looking up at the billows upon billows of a cumulus cloud... Though studied for more than a century, turbulence remains only partly understood by either theoreticians or aircraft designers. Wire a wind tunnel at Boeing or Airbus with appropriate instruments; and you can detect the complex motion of the water vapor, dust, or luminescent markers blowing inside it. When the rotor at the tunnel’s head spins slowly, the wind inside blows nice and smoothly. Its currents glide in unison in long, steady lines, planes and curves like parallel sheets of supple, laminated plywood. This kind of flow is called laminar. Then, as the rotor accelerates, the wind inside the tunnel picks up speed and energy. Here and there, it suddenly breaks into gusts—sharp, intermittent... Eddies form; and on those eddies yet more, smaller eddies form. A cascade of whirlpools, scaled from great to small, spontaneously appears... [But] can you seriously compare the wind to a financial market, a gale to a rally, a hurricane to a crash? In terms of the underlying causes, certainly not. But mathematically, yes.” - Benoit Mandelbrot
In other words, extreme events often do not occur in isolation; we should expect one strong “gust” to be followed by a cascade of smaller “eddies”. And, because of the extremity of the initial event, even its relatively small “aftershocks” could be too extreme to be modeled with the Normal distribution.
The key point to carry forward is that the bell curve which represents love’s intensity likely has longer tails than the Normal probability distribution. Compared to average experiences, extreme love is not twice as intense as typical love; it is many times more intense.
Valence
In psychology, valence refers to how pleasant/unpleasant we perceive something to be. Thus, all experiences of love will have a valence which is either more positive or negative from an individual’s perspective.
Within the central areas of a bell curve, we could expect events with a mildly positive valence, like receiving flowers from one’s lover, or a mildly negative valence, like getting into an argument.
However, extremity in this dimension may not be contained by the boundaries imagined by a Gaussian bell curve. We should expect there to be events which have an extreme level of positive or negative valence. On the negative end, we might find destructive and obsessive forms of love, infidelity, or abuse. On the positive end, perhaps we might find life events like getting married, parenthood, or the ecstatic sexual states accessed through tantra.
The way we should interpret this is similar to how one extreme day in the stock market can account for more price movement than decades of other days combined. As in, the extreme valence of certain experiences might profoundly alter the love of two individuals in a way which overshadows almost everything else. At that point, one might feel (rightly) that Nothing Compares 2 U.
Growth
One key theme which emerged in my previous explorations of this topic is that love has a lot to do with growth and the orientation of how we change. This sets up an ideal to which lovers may aspire: to be in a relationship which leads to the optimal growth of each individual.
This idea is central to constructs like twin flames, the nemesis bond, and amigeist. In these situations, extremity shows its face as a principle of change. Towards the median space of love, growth occurs in small, linear steps. But that is hardly the whole story. Along with that steady drumbeat, there is an occasional crescendo of activity. And we should expect that the latter could change us more extremely, abruptly, and profoundly than the former. Perhaps this is one reason that love has often been compared with head over heels madness. This mania, as I wrote, can be a great gift. The greater the disorientation is, the greater is the potential for discovery.
Love in general can be understood as mutual psychagogia. So we should expect extreme love to strengthen, magnify, and accelerate this process.
Necessity
I hold a nonstandard view of love in which it is not judged solely from the perspective of an isolated relationship. The love of two individuals is much like a song—whose notes are a relative instantiation of absolute patterns of harmony. The first line of my first book is: “Value and Action are lovers, and this is a book about their relationship.” Our own love stories are microcosms of this original macrocosmic bond, and I believe we can (and should) judge the former by the part it plays in the latter.
By this, I mean to say that we can personally judge the valence of our love, but so can our love be judged from a kind of universal valence and morality. Our love can be “good” only if, in some way, this goodness is universally coherent with the Good. Such non-anthropocentric judgement is based on the degree to which one instance of love affects the entire system in which it is situated. The best of love, in that view, leads to the most perfect actualization of possible perfection.
Thus, the universe might be said to deem individual relationships more or less necessary. And the necessity of some of these might be extremely great compared to a large set of others.
Concluding thought
I believe that quantitative and qualitative views can be complimentary. For example, the valence of extreme love will be subjectively different from individual perspectives. But all experiences of this type may than be analyzed and compared based on some kind of quantitative scales of extremity—such as the ones in this essay. By focusing on extreme love from a quantitative perspective and the viewpoint of tail events, I hope I’ve added to the qualitative portrait of love I’ve been painting in my recent writing.
Go forth and love!
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