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Math Guy

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  1. This pattern holds for Japanese emperors, Roman emperors, Pharoahs, Popes, Chinese emperors, and on and on. It holds even for very short dynasties. If a dynasty only lasted long enough to make one handover of power -- from one king to his son -- nonetheless the son's reign has tended to be shorter than the father's. I am sure some of you will be tempted, as I was, to hypothesize that this has something to do with limited human lifespans, and the vigor of youth, intergenerational dynamics, and the inherent limitations of hereditary rule. We might call it the "King Lear" effect. A young, ambitious man seizes power and founds a dynasty. During his lifetime he reforms the law, expands the territory of the kingdom, wins new allies, and so on, and so on. But being self-confident and ambitious, he is reluctant even in old age to hand over control. He delays until the last possible moment, in most cases until he actually dies. This means that he is succeeded by his adult son, or perhaps several sons, who have spent a lifetime waiting in their father's shadow. Perhaps there is conflict. Perhaps external enemies sense weakness once the old man is gone. In any case, the fortunes of the kingdom go downhill after the promising beginning, not for reasons of probability, but because of human nature and the limitations of hereditary rule. This is a good theory, but ultimately wrong. It is relatively easy to test. In a number of countries, one dynasty has followed another for thousands of years. Egypt, China, Rome, Byzantium, and other great states outlasted even their strongest dynasties by an order of magnitude. As many as 30 dynasties have succeeded one another in governing the same geographic region. So we can perform the same analysis as we did for the individual rulers, but on a wider scale. We find that here too, there is decline. Later dynasties do not last as long as the early ones. The shape of the curve is the same; the average decline of 20 percent from the first half of the list to the last half is again evident. Obviously, as these dynasties tend to last much longer than a human lifetime, and the states they govern last thousands of years, any sort of simple father-son effect cannot possibly be responsible. This is an impasse I encountered again and again in my studies: we start with a strange power-law phenomenon, which we explain using some sort of local, anecdotal cause. Then we find a very similar power law operating nearby for which our anecdotal explanation is useless. Eventually, given enough examples, we are forced to the conclusion that this is a very broad, universal rule, one that transcends specific causes. It has, if you will, a meta-cause or mathematical cause. It is a rule relating to randomness as such rather than a rule relating to fathers and sons, or political dynasties. The next stage is even stranger. I will be back later tonight with another post.
  2. This first example concerns the life (and death) of political regimes -- starting with hereditary monarchies. This, by the way, is a pattern I discovered on my own, not one of the ad hoc laws named for esteemed but long-dead scientists above. We'll start with a really elementary question. Suppose we were to pull out a reference book and choose some random dynasty out of history -- the Tang dynasty, who ruled China from 618 AD to 907 AD, or the Valentinian dynasty that ruled Byzantium in the 4th century. Each ruler managed to stay in power for some span of years, before dying, passing on the throne to an heir, or being overthrown in some fashion. Suppose we render all those spans as integers, rounding to the nearest year, and look at them in sequence, like this: 37 12 9 9 8 10 10 11 4 15 11 8 15 10 18 5 8 5 1 14 3 1 3 1 10 6 9 13 8 1 2 3 22 1 15 14 18 15 2 16 ... This particular sequence, by the way, is the first 40 Popes of the Catholic Church. Now, the question I want to ask is: If out of N rulers, I took an average of the first N/2, and compared it to the average for the last N/2, what would I see? There are three possibilities: -- Rising trend (rulers stay in power longer) -- Level trend (rulers experience no change in stability) -- Falling trend (rulers are deposed or killed sooner) The standard assumption in classical statistics is the "indifference principle," which says basically that if we don't know anything specific about the situation, we should expect to see no difference between two sides of an arbitarily chosen line. We expect a level trend, a stationary or unchanging mean. Later rulers in a dynasty should hold power as long as early ones. In Jaynes' system we replace this assumption with a general rule: in any large, complex system composed of many moving parts, entropy will increase. What this rule means will vary from case to case, but here it is simple to apply. Since long spans in power are rare, and short spans are more common, for entropy to increase requires that the rare item (long spans) become rarer. The average span in office should therefore decline. In the example above, it does decline, by about 20 percent. Later Popes varied enormously in their terms of office, from just 1 year to 22 years. But overall they were in office for a shorter time. The same is true -- much to my surprise, and yours too, I imagine -- for virtually every dynasty I have examined. Some decline only a little (say 5 percent). Others decline steeply (50 percent or more). The mean decline is about 20 percent. The longer the dynasty, the more likely it is to converge on the 20 percent figure. The form of the decline is scale invariant, meaning that if we compute cumulative decline for each of the N rulers, we get a power-law curve depending on N. By the time we get to the 10th ruler, the average time in office is half that of the first ruler. By the time we get to the 100th ruler, the average time has fallen to one-quarter that of the first. And so on. This pattern, taken by itself, would not excite much comment. It is strange, anomalous, intriguing, but it hardly suggests that the whole framework of classical probability needs to be rethought. However, we are just getting started. . . .
  3. Thank you Jake (and thank you to everyone else who replied). Your question is a very good one. I've had some very fruitful discussions with professional mathematicians over the years, in fact some of the inspiration for the work came from editing another man's book on a related subject. I suppose technically I am myself a professional mathematician, although I don't work at a university. I have talked with other professionals as well about specific applications. I haven't lacked for people to talk to. I haven't tried to put my work through the usual peer review process for several reasons. Leaving aside all the standard complaints about academia, something like this, which sprawls across so many disciplines, would not be suitable for formal review. I am convinced the way to present it is to put all the arguments and all the footnotes in one place (675 pages, 220 graphs and illustrations), and just let laymen and scientists alike make up their own minds. However, this group does have at least two unique qualifications that I am interested in. First, I take it as given that you are all engaged on the question of philosophical fundamentals -- how we know what we know, how do we define concepts, what is free will, and so on. That level of engagement is rare, even among academics. Second, the book was inspired in part by Rand's theory of integration, and the critique of modern science that she and Leonard Peikoff put forward decades ago. Peikoff referred in The Ominous Parallels to science being divided into two camps, those who generated theories without any facts to support them, and those who accumulated facts but neglected to explain them by means of coherent theories. Rand of course has said similar things on a number of occasions. This is precisely the situation I found with regard to power-law distributions. Rand's philosophy of science, at least on this point, has proved far superior to that of Popper, or Kuhn, or Feyerabend (to name a few). Science has gradually accumulated a mountain of discrete, ad hoc "laws," each one a power-law curve, each describing some very specific empirical relationship. They are not merely present, but central to current thinking, in biology, economics, military history, criminology, and so on. Somehow, no one has noticed (or bothered to do anything about) this mass of similar yet unintegrated facts. All these laws, I contend, are actually expressions of one basic principle: — Pareto’s law of elite incomes — Zipf’s law of word frequencies — Lotka’s law of scientific publications — Kleiber’s law of metabolic rates — the Clausewitz-Dupuy law of combat friction — Moore’s law of computing costs — the ‘paradox of higher demand’ in religion — the Wright-Henderson cost law — Weibull’s law of electronics failures — the Flynn Effect in IQ scores — Benford’s law of digit frequencies — Farr’s law of epidemics — Hubbell’s neutral theory of biodiversity — Bass’s law of innovation diffusion — Rogers’ law of innovation classes — Wilson’s law of island biogeography — Smeed’s law of traffic fatalities — the Cobb-Douglas factor elasticity curve At the same time, for the past 50 years, people working in these fields have for the most part ignored or under-used the maximum-entropy framework laid down by Jaynes, an idea with phenomenal potential and sweeping implications. The theory needed to explain their "orphan" facts has been lying in plain sight for decades. What I found, over 20 years of investigation, was that integration was in fact possible, and more than that, necessary. Rather than dozens of discrete "floating" laws, each unrelated to the next, we actually have a small number of standard curves, which can be constructed using an even smaller number of Jaynesian postulates. And of course, having established a new "universal," I could then explore numerous applications for which no ad hoc rules had even been proposed. Some of these curves are very weird and disturbing. At least, I find them so, and so do all the people I have shown them to, Objectivists or not. But the method by which I arrived at these findings owes a great deal to Rand, and so I think it would be particularly interesting to see what you all make of them. I've already submitted a draft of my book to my editor, and in the next few months I will need to commit to a final, authoritative edit. Perhaps something will come out of this discussion worth including. If not, I can at least get a better sense of how audiences respond to my arguments. I'll present my first case in my next post.
  4. Hello everyone, I have a series of implausible and shocking empirical discoveries that I would like to share with this forum. They will be published next spring in my book The Decline Effect: The law behind diminishing returns and wildly varying outcomes in markets, politics, culture, religion, disease, and war. Before committing myself to a final edit, I would very much like to have a discussion of their philosophical implications with a group of Objectivists. My work builds on a theoretical foundation of probability and inference using the principle of maximum entropy, that was laid down by Edwin Jaynes starting in 1957. Essentially what I have found is that a huge range of phenomena all obey the same simple statistical rules -- NOT the normal or bell-shaped curve, but a power-law curve with very different implications. My findings raise questions about the nature of markets, the power and spread of ideas, the evolution of political and religious institutions, even the nature of free will. I'd like to present a summary of several key discoveries and see what people say about them. Given the posted rules of this forum, I need to offer some information about my background and motivations. I was involved in the Objectivist movement for many years, having for example attended "Debate 84" in Toronto between Leonard Peikoff, John Ridpath, and a pair of socialists. I wrote for several Objectivist publications in the 1990's, and attended discussion groups in Kingston, Toronto, Seattle, Vancouver, and Victoria. I was also very active in the AOL Objectivist forum 15 years ago. However, I gradually stopped participating in Objectivist activities, in part due to lack of time, in part because the arguments in 2004 were not very different from 1984, and in part because my research findings raised awkward implications. I would describe my discoveries to Objectivist friends, and the reaction would often be a mix of skepticism (that I was just wrong) and suspicion (that I was somehow being immoral in even pursuing the question). There were plenty of people who found my arguments interesting and were prepared to listen, but as it turned out, none from the "closed" school of Objectivism. Although we have not corresponded in several years, I remain on friendly terms with David Kelley. I am on even friendlier terms with Chris Sciabarra, in part because some years ago, I wrote a review of The Facets of Ayn Rand for the Journal of Ayn Rand Studies. I hope, sometime after the book is out, to draft an essay on the implications of my findings for Objectivism. Chris Sciabarra has expressed interest in having such an essay appear in JARS. Let me stress that I am not trying to re-ignite the "open" versus "closed" debate. Twenty years of such debates are more than sufficient for me. What I want is for a group of intelligent adherents to the "closed" view to look at my findings. My hope is that in the course of the discussion, I may hear arguments that shed new light on the problem, a perspective that my "open" friends are not able to supply. Is this something that would interest people here? Should I proceed?
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