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Kuhn, 1/f noise, and the fractal nature of scientific revolutions

Bill Harris writes:

I was re-reading your and Shalizi’s “Philosophy and the practice of Bayesian statistics” [see also the rejoinder] and noticed a statement near the end of section 6 about paradigm shifts coming in different magnitudes over different time spans. That reminded me of the almost-mystical ideas surrounding 1/f (f being frequency”) noise in some areas — the notion that almost everything exhibits that effect, and that effect extends to arbitrarily low f. (I sense the idea only gets mystical when f gets low enough so that the event that may happen stochastically is really big—say, you model the height of waves in the Atlantic as 1/f and discover that, at some low frequency, Bermuda becomes submerged. In other words, does the same mechanism that accounts for physical vibrations in the range of Hertz also account for the creation and destruction of islands that may occur in the range of reciprocal centuries?)

When I first encountered 1/f noise in the area of electronic noise measurement, I was intrigued, but I’ve not read seriously in that field for perhaps 25 years or more. Still, I thought you might find it of interest. Perhaps it could help smooth some of the challenges you point out in Kuhnian philosopy and contradict the so-called “law of the excluded middle,” in which (in Kuhnian terms) everything is either normal science or a paradigm shift, or so I presume.

Here are three links on 1/f noise:

– “1/f Noise.” Marvin S. Keshner.

– Scholarpedia.

– “1/f noise: a pedagogical review”. Edoardo Milotti.

Does 1/f noise make any sense in the Kuhnian paradigm context?

My reply: I don’t know, but I think it’s safe to say that Cosma will have some strong opinions on the matter. I did take a look at the Scholarpedia article, which seems excellent.

I’m guessing Cosma will express skepticism about all this. Our article does have that bit about the fractal nature of scientific revolutions, but that’s something that I wrote that Cosma didn’t really like very much, he was just nice enough to let me keep it in. It’s one of my pet ideas (and, as a commenter noted, perhaps related to the fractal nature of scientific taxonomies) but I’ve never had any idea of how to study it formally.


  1. Especially since I co-created a course for non-science-major undergrads that involves both fractals and scientific revolutions (, this idea of a fractal structure of science keeps coming to mind. I think it’s a nice illustration, and even a satisfying mental framework, but I doubt there’s anything substantive to be done with it. The whole notion of what is and isn’t a revolution, or how to quantify the magnitude of a revolution, is too fuzzy. That said, I have no doubt that someone will develop some sort of citation-measuring, word-counting metric for revolution-ness and show that its distribution has a power-law structure. And: the paper on this will be highly discussed, will be useless, and will spawn lots of trivial extensions of its algorithm — a larger-scale version of the h-index! (I’m surprised this hasn’t happened already.)

  2. Mayo says:

    I am fairly sure this isn’t what you’re after, but here’s something on the philosophical side. I think it’s generally accepted that Kuhn’s normal-revolutionary science dichotomy fails to reflect actual science. In my analysis of Kuhn’s three stages: normal science-crisis-revolutionary science, the dichotomy simply stems from an analytic definition of his—not from a historically sensitive description of actual science at all (despite what he claims to be doing):

    That is, science is defined as an activity within a paradigm (made up of large scale theories, aims, and methods). This is sometimes described as an ontology, an axiology, and a methodology. Each field X (disciplinary matrix, he later calls them) can have it’s own paradigm—but just one at a time. Science = science-within-a paradigm, according to what Kuhn says. Its practitioners accept the paradigm and busy themselves with little tasks, fleshing out the paradigm, deriving predicted quantities, and mopping up activities (e.g., papering over any anomalies and unsolved problems). If a paradigm X, e.g., its core theories, is seriously challenged, practitioners of X are no longer doing science. There is a move to crisis science. Science (or the scientific field in question) is suspended, all is in limbo, like in Egypt today. Science is only restored within a new normal science that results from a revolution which involves changing the entire paradigm–theories, aims, methods– by some non-rational means. (It is non-rational because, allegedly, the methods are suspended. There are shared methods alright, but they are so vague, and so capable of being interpreted to support one’s favorite paradigm, that the revolution cannot be directed by them. According to the Kuhnian fable, which even he tended to distance himself from.)
    In my chapter, I have a lot of fun doing a deconstruction of Kuhn (and Popper). On my (revolutionary) reading, what Kuhn says about normal science makes perfect sense, and is the basis for distinguishing science/pseudoscience (i.e., normal science allows learning from stringent testing). (But what he says about revolutionary science is kooky).

  3. jrc says:

    This is a nice chance for me to brush off a pet theory from my old Philosophy undergrad days (so you know it’s not a very well-thought theory):

    I always thought that there was a natural fit between some sort of Kuhnian historicism not unlike the version you describe (something like a hierarchy of paradigms within which normal science occurs) and Quine’s ideas about the holistic structure of scientific knowledge. We can chop off little branches of sub-fields of knowledge and re-arrange them basically at will, but re-arranging the “interior” (say, the metaphysics of sub-atomic particles or the biology of infection) is much more difficult (in part because of sociological reasons and in part because you have to be able to explain everything the old theory explained plus something new).

    I’m sure Mayo can correct and/or crush this thinking, but it always seems strange to me that people take Kuhnian epistemology to understand any particular field of science as some sort of unified thing resting on a particular, singular foundation (one particular paradigm), and not as this sort of complex web with certain core theoretical and metaphysical positions and extending out through decreasingly fundamental assumptions and theories. Maybe that’s Kuhn’s fault, but my memory of reading him was that his thinking was more nuanced than just One Paradigm, Anomalies, ????, REVOLUTION! There’s that whole bit about trying to save the paradigm by tweaking little assumptions, right?

  4. Bill Harris says:

    To Deborah’s point about normal science (I’m partway through the chapter she linked to), I recall a prof in grad school saying that Claude Shannon’s ideas on information theory were pretty much all correct, but his proofs were not, giving many grad students the opportunity to do research by creating corrected proofs of his theorems. Is that part of what you mean by normal science? BTW, the statement about Shannon’s proofs is simply my memory of a statement made in a class a few decades back; I don’t know if it has any basis in fact.

  5. Bill Harris says:

    I happened to receive a link to today, which led to Stuart Firestein’s Ignorance: How It Drives Science at Do you know him? Does this add to the discussion?

    If you like graphs :-), Richard Hake’s “Early History of High Field Superconductivity: 1930-1967 A Tragicomedy in Twelve Acts” at has a number of old-school ones as it explores the development of high-field superconductivity over almost four decades.

  6. Entsophy says:

    […] be more misguided. They’re more akin to numerology than science. (the 1/f noise described here as almost mystical, is a similar example. The giveaway is that this noise occurs in many physically […]