Mandelbrot’s The Fractalist, Chapter 14: First Kepler Moment: The Zipf-Mandelbrot Distribution of Word Frequencies, 1951

Our scientist-hero has identified a goal – creating a ‘Keplerian’ field of study which he can pioneer. Mandelbrot has found a mentor, Kastler, who understood the need to drift between worlds and excelled at it. He has a method to pursue this need – surrounding himself with brilliant peers, and then finding out how to outshine them and realizing that their bureaucratic games of hierarchy hold no attraction for him.

Best Quote(s)

Path dependency matters. Many had noted the trends and methods that Mandelbrot would make into a formal field, but under ‘Important’ areas of study, it was hard to take the intellectual risks necessary to openly discuss their use. Failure would be too painful. By starting in an atypical area that threatened no one, he was allowed room to develop and grow. (On a personal note – this feels like my time in nanofibers.)

“My luck was to begin with the distribution of word frequencies—a thoroughly atypical example without any important consequences, and uniquely easy to handle.” Mandelbrot, Chapter 14, Location 2387

Mandelbrot, Chapter 14, Location 2387

Scaling – ‘being fractal’ – in your activities is one of my personal big take aways from reading this book. Mandelbrot declares that if something is not smooth, then it is rough – and things that are rough have common traits in how they behave. Apply this to life – if things are not smooth, then they must be rough. Do things that scale up and scale down. Much in business is focused on ‘can it scale up’ – but in many ways this avoids the hard challenges of finding ways for activities to be worthwhile if they must scale down. To win in a world of roughness, scaling – both up and down – must be part of the strategy.

“The language—English, French, Latin, whatever—does not matter. Neither—quite oddly—does the writer’s degree of literacy. This is an example of what physicists were soon to call a universal relationship. Another notion in physics, called scaling, is one that underlies fractals.”

Mandelbrot, Chapter 14, Location 2408

If ever a movie is made of Mandelbrot’s life – then the scene where he reads this paper, a gift from Uncle Szolem, and hits this Eureka moment would be a highlight.  

“In one of the very few clear-cut eureka moments of my life, I saw that it might be deeply linked to information theory and hence to statistical thermodynamics—and became hooked on power law distributions for life.”

Mandelbrot, Chapter 14, Location 2425

Zipf’s paper was not new – it had first been published 16 years earlier. Mandelbrot would be in a tenuous position – he believed that his mathematical capabilities provided him a unique perspective, but he was alone with this view point. Only as he was able to add to this perspective with advances in topics far afield from word theory would the strength of Mandelbrot’s position improve.

“My good fortune resided in an unfair advantage. I was to be the first—and for an interminable time, the only—trained mathematical scientist to take Zipf’s law seriously.” Mandelbrot, Chapter 14, Location 2425

Mandelbrot, Chapter 14, Location 2425

Page by Page, Swipe by Swipe, Location by Location

2349

“TAKE THIS REPRINT. That’s the kind of silly stuff only you can like.”

2353

“Oddly but almost ineluctably, that string, that reprint, ended up directing me to some of the main themes of my scientific life: unevenness, inequality, roughness, and the concept of (as well as the word) fractality.”

2364

“…George Kingsley Zipf (1902–50). Independently wealthy, this academic character was a university-wide lecturer at Harvard in a self-invented field he called statistical human ecology.”

2381

“Inequality and Unevenness Are Everywhere:

How long does a book on the best-seller list remain there?”

2387

“Extreme inequality is a familiar pattern in nature and in the works of humans. Such distributions are called long-tailed distributions. For them, no value is typical, and the contrast between short and long tails came to play a central role in my work.”

“My luck was to begin with the distribution of word frequencies—a thoroughly atypical example without any important consequences, and uniquely easy to handle.”

2392

“Statisticians rarely use this method, but there is nothing wrong with it.”

2398

“An odd and hard-to-read pattern emerges. The curve does not fall gradually from most common to least common word.”

“By the very definition of rank, frequency varies inversely with rank.”

2403

“To compare such curves, it is best to replot them more legibly by replacing both the rank and the frequency with their logarithms.”

2408

“The language—English, French, Latin, whatever—does not matter. Neither—quite oddly—does the writer’s degree of literacy. This is an example of what physicists were soon to call a universal relationship. Another notion in physics, called scaling, is one that underlies fractals.”

2425

“In one of the very few clear-cut eureka moments of my life, I saw that it might be deeply linked to information theory and hence to statistical thermodynamics—and became hooked on power law distributions for life.”

“My good fortune resided in an unfair advantage. I was to be the first—and for an interminable time, the only—trained mathematical scientist to take Zipf’s law seriously.”

2436

“A difficulty: a well-defined probability may exist for common words, but what about rare words, especially in multiauthor works or composite files of newspaper articles?”

2441

“Worse, experimentalists try to help by simplifying what they see, and key facts are often unwittingly overlooked.”

2451

“Early on, a shadow was present—the example I worked on was devoid of important consequences. No one could predict that I was to be called the “Kepler of word frequencies,” then more generally the “father of long tails.”

2455

“Had I approached it from a seemingly more “worthy” angle, I am convinced I would have failed.”

2460

“On December 19, 1952, the die was cast. My Ph.D. dissertation loudly affirmed a Keplerian determination to become a solo scientist—the kind my world thought had vanished.”

2477

“So, no, I was not acting spoiled. I did not want to hide—I wanted to find the best conditions to fulfill my Keplerian dream.”

2483

“Had I sought their advice, I am sure I would not have taken it.”

About flybrand1976

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2 Responses to Mandelbrot’s The Fractalist, Chapter 14: First Kepler Moment: The Zipf-Mandelbrot Distribution of Word Frequencies, 1951

  1. Pingback: Mandelbrot Part 2: Chapters 8 – 20, “My Long and Meandering Education in Science and in Life” | Fred Lybrand

  2. Pingback: Mandelbrot’s The Fractalist, Chapter 22; At Harvard: Firebrand Newcomer to Finance Advances a Revolutionary Development, 1962–63 | Fred Lybrand

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