The last two issues of C3M ranged pretty far afield from what most people think of as cybernetics. I got an unusually high number of positive comments back, but I also got my first unsubscribe request. So this month I am returning to the mainstream, and plunging into some heavy-duty mathematics as well. Here goes... I am sometimes amazed to encounter people who say they are very interested in cybernetics and/or systems theory but they don't like math. It seems to me that without math these pursuits quickly devolve into just word games, or at best the kind of paying attention to externalities and relationships that brought us group therapy in psychology and man-in-the-loop testing in engineering. Math is vital to the study of systems because it helps us understand all of the things systems are CAPABLE of doing, that is, all possible behavioral modes of a given set of assumptions. In the 1930s Lewis Fry Richardson was a pioneer of the application of mathematical models to social science. His "Generalized Foreign Policy" (1939) offered a mathematical model of arms races, and generated a firestorm of criticism from the "experts" in foreign policy and diplomacy. His next book was "Arms and insecurity: a Mathematical Study of the Causes and Origins of War" (1949). ( www.amazon.com/exec/obidos/ASIN/0835703789/hip-20 ) In it he devoted quite a few pages to rebutting his critics. They had claimed that you couldn't reduce the complexities of international relations to mathematical equations, because it required human judgment and intuition to analyze these types of problems. Richardson's main argument was that when people propose a hypothesis in this (or any) field they often state the "obvious" conclusions from that hypothesis incorrectly; their conclusions do not follow "logically" from their premises. Only the rigor of math can verify such conclusions. Likewise, more recently, misguided critics have attacked the methodology of computer modeling as if working things out "in your head" represented a superior and more reliable way to test if conclusions follow from premises. For example, in his essay "Understanding the counterintuitive behavior of social systems" (1971) Jay W. Forrester described how government funded low income housing projects would usually aggravate the lack of affordable housing, by attracting more people to an area than the projects provide. His models showed that somehow creating more jobs instead would cause private investors to overbuild new housing, increasing supply and reducing cost. The essay is reprinted in "Collected Papers of J. W. Forrester" (1975). ( www.amazon.com/exec/obidos/ASIN/1563271923/hip-20 ) So, if math is so vital to cybernetics and systems theory, why do many of the fans of these fields hate and avoid math? I think it is because it is so badly taught to our children. Kids are very aware of rules, and have high standards of integrity. Our standard math curriculum lies to them, and this turns a lot of them off. We could tell them, "We're going to make up some rules this year and follow them and next year we're going to change the rules." But no, we say, "You can't subtract a larger number from a smaller one," and then the next year we say, "Surprise! You can after all, and the answer is a new kind of number, called a negative number." That scrapes a few of them off. Then we pull the same trick with division, and surprise them with fractions. Then we do it again with square roots, telling them you "can't" take the square root of a negative number, only to go back on our word and introduce "i" the "imaginary" quantity. By this time we've lost almost all of them. Even the people who TEACH math to grade schoolers are pretty nervous about i in my experience. And yet "complex" numbers (formed adding "real" and "imaginary" numbers) are among the most powerful and elegant tools in mathematics. (This goes along with my theory that the purpose of public education is to inoculate people against knowledge so they don't catch it later in life.) I was fortunate in that I had some very good teachers, including my father who taught me at home, and the inoculation never "took" with me. I hung in there through the lies and "got it" time and time again. By 9th grade I had noticed a pattern: We were taught to count. Then counting was generalized to addition (which was "closed" over the counting numbers, or positive integers, i.e., add any two counting numbers and you get another counting number). Then we learned the inverse of addition, subtraction: which was not "closed" over the counting numbers. They had to introduce negative numbers to make a complete set of numbers. Then addition was generalized to multiplication (which was "closed" over the integers), and we learned the inverse of multiplication: division, which was not "closed" over the integers. They had to introduce fractions to make a complete set of numbers (and we still couldn't divide by zero). Then multiplication was generalized to powers (which was "closed" over the rational numbers -- as long as the exponents were whole), and we learned the inverse of powers: roots, which were not "closed" over the rationals. They had to introduce irrationals to make a complete set of numbers (and we still couldn't take the square root of minus one). At this point I came up with what I call Scrivener's Conjecture: Every time we generalize an operator and then take its inverse we will have to invent a new kind of number. So I tried it out. I invented an operator I called "gorp" (for no particular reason) which generalized powers. I defined gorp(2, n) to be n^n. (I am using BASIC's notation for exponents here since this plain text format doesn't allow much else.) Then I defined gorp(3, n) to be n^(n^n) since putting the parentheses the other way, (n^n)^n would reduce to n^(n*n) which didn't seem as interesting. Of course gorp(4, n) would be n^(n^(n^n)), and so on. The I defined the inverse, "prog" (gorp spelled backwards) so that if m = gorp(p, n) then n = prog(p, m). Here is a table of some values: ======================================================================= Cybernetics in the 3rd Millennium (C3M) -- Volume 2 Number 9, Sep. 2003 Alan B. Scrivener --- www.well.com/~abs --- mailto:abs@well.com =======================================================================## Do Nothing, Oscillate, or Blow Up:

An Exploration of the Laplace Transformn p gorp(p, n) -- -- ----------- 0 2 [undefined] 1 2 1 2 2 4 3 2 27 0 3 [undefined] 1 3 1 2 3 16 3 2 3^9 = 19,683Clearly this function rises much faster than anything else I knew of; also, clearly, prog(2, 0) had to be a new kind of number, since there is no n such that n^n = 0. (Though 0^0 is undefined, its limit is 1.) But the math I was being taught took a different turn. Powers were not generalized to gorp, but to the EXPONENTIAL function, b^x where b was a constant, and its inverse was the LOGARITHM, where if a = b^x then log(a) = x (to the base b). Logs are very useful (and they are undefined at 0, with a limit of minus infinity) but they weren't the same as prog, nor did they yield any new types of numbers. For a while it looked like the imaginary quantity i, and complex number z = a + bi, were the last new types of numbers to be defined in western math. It did seem very curious to me that a funky new irrational number called "e" was introduced out of nowhere, and used as the base of the so-called "natural" log, but since e was approximately 2.7182818284 and not defined in terms of any roots or other irrationals I knew about, it didn't seem very "natural" to me. That is, until I learned that if you take the area under the curve of the inverse function, y = 1/x, evaluated from the vertical line x=1 to the vertical line x=k for some number k, the resulting function is log(k) to the base e! Huh? Where did that come from? (In calculus notation, the definite integral from 1 to k of 1/x dx is log(k) to the base e.) Then one day in about 11th grade an older student told me that I wasn't supposed to know this yet, but e^(Pi*i) = -1, or as he glibly said, "e to the Pi i is minus one!" Boy was I confused. First of all, what did it mean to take a real number to an imaginary power? How could you multiply e times itself "i times" anyway? And secondly, e was from logs, Pi was from circles, and i was from square roots. How could these unrelated numbers combine in such a goofy way to make something simple like minus one? I didn't figure that one out for years. Every now and then I'd ask a mathematician about my conjecture, and my ideas for gorp and prog. One told me it sounded a little like Ackermann's function, a super- quickly growing function which has been studied since 1928. Interesting, but not helpful (at least to me). ( www.nist.gov/dads/HTML/ackermann.html ) My friend Bill Moulton alerted me to the work on "hypernumbers" by Charles Muses. He co-edited a book called "Consciousness and Reality: The New Pivot Point" (1972) which included his own essay, "Working With the Hypernumber Idea." ( www.amazon.com/exec/obidos/ASIN/038001114X/hip-20 ) Muses claimed that Hamilton's discovery (or was it an invention?) of quaternions in 1843 represented a new form of imaginary number, which did not obey the commutative law: a*b did not equal b*a. (A quaternion contains 4 components, much like a complex number contains two. They've never made much sense to me, and they fell out of favor in mathematics early in the 20th century.) He goes on to describe a series of such new numbers, ultimately reaching seven of them, counting real and imaginaries as types one and two. Each new type breaks another law, such as the associative law, until the seventh does not even obey identity, i.e., a=a no longer holds. Muses draws spiritual lessons from all of this, equating the seven types of hypernumbers with seven stages of the evolution of human consciousness. (I am reminded of the alchemists, who made a similar association with the stages of transmuting base metals into gold.) I have studied this paper extensively over more than a decade but I've never "gotten" it. By coincidence (or maybe not) Muses has written extensively on cybernetics. He passed away in 2002, and "Kybernetes: The International Journal of Systems & Cybernetics" (which he contributed to frequently) devoted a special issue to him, Volume 31 Number 7/8 2002, "Special Issue: Charles Muses - in Memoriam." ( matilde.emeraldinsight.com/vl=3034422/cl=67/nw=1/rpsv/cw/www/mcb/0368492x/v31n7/contp1-1.htm ) This work is also related to "Surreal Numbers" (1974) by Donald Knuth. ( www.amazon.com/exec/obidos/ASIN/0201038129/hip-20 ) Both Muses and Knuth introduce the idea that there are positive and negative forms of zero (!), each of which satisfies the equation x^2 = 0, which also relates to Kurt Godel's famous Incompleteness Theorem -- Godel proved that these forms of roots of zero cannot be definitely proved to exist or not to exist. But all of these revelations inspired me to go back again and look at my conjecture. After learning calculus I was able to determine that for positive x the derivative of y = x^x was y' = log(x) * x^x, that is equal to 1 when x = 1 (obviously) and has a limit of 1 when x = 0, and between the two values it forms an asymmetric "dip" whose minimum is 1/e of all things! When I learned to manipulate complex numbers I was able to compute the real and imaginary parts of z^z where z = a + b*i. When I learned computer programming I was able to draw graphs of the function, and later do 3D surface plots of the real and imaginary parts over the complex plane. These are beautiful -- I will share them in a future C3M if I can find or rewrite the code I used to generate them -- but I never was able to use these steps to reach a definition of a new kind of number. Along the way, though I was able to learn why the mysterious e^(Pi*i) = -1 is true. It is a specific result of the general form of Euler's Formula: e^(i*theta) = cos(theta) + i*sin(theta) When theta equals Pi, cos(theta) is zero and sin(theta) is one. This "magic" formula, which like e^(i*Pi) = -1 I learned from an older student before I was supposed to know about it, is massively useful. For example, you can use it to find the so-called "trig identities" which I had to memorize in high school, such as sin(2*x) = 2*sin(x)*cos(x). ( www.math2.org/math/trig/identities.htm ) I was shown how to derive these "the hard way" but it never stuck with me. Using Euler's Formula makes it a breeze, and I never had to memorize another trig identity again. But where did Euler get this amazing equality? How do you take the imaginary power of something? The answer is to be found in the tool known as the Taylor Series. I studied this in college calculus class, and was able to pass the test, but never had a clue what the symbols meant. It fell to a physics professor, David Dorfan, to provide an intuitive understanding. ( scipp.ucsc.edu/personnel/profiles/dorfan.html ) I still remember the day in electromagnetics class that he asked if any of us could explain the concept. There was silence. We'd all studied it; calculus was a prerequisite for his class. I remember him muttering in his charming, clipped accent (British? South African?) about "what are they teaching you in the math department," before going to the board and drawing a few figures and explaining it all in about ten minutes. "How do you think they compute sines and cosines for the tables?" he implored us. (This was before affordable scientific calculators, and we used books of tables of numbers to find the values of trig functions.) "Do you think they draw giant circles and measure them?" Here, then, is a brief explanation of the concept which I wrote for a book I'm currently working on, "A Survival Guide for the Traveling Techie" (more on that another time): The Taylor Series allows you to approximate certain well-behaved functions with simple arithmetic. For a given function of x -- f(x) -- you create a series of terms using x, x squared, x cubed, and so on, based on knowing the function's value for some single value of x (often called a) along with it's derivatives at x = a. You need to able to find its first derivative, second derivative, third derivative, etc., or in other words: rate of change, rate of change of rate of change, rate of change of rate of change of rate of change, etc., or in still other words: slope of the graph, slope of the graph of the slope of the graph, slope of the graph of the slope of the graph of the slope of the graph, and so on. So when you go to predict the function, you use a potentially infinite expression, but you only add as many terms as you feel like doing the arithmetic for; if you add up N terms, we say the result is an Nth order approximation. Let's look at a simplified example that uses discrete data. A say we want the value of the function where x = a + 1. A zeroth order approximation of the function would be zero. No matter what the value of f(a) and its derivatives are, who cares, the result will be zero. And in some cases this is not a bad approximation. It's like assuming nothing will happen. Sometimes you're right. A first order approximation would be whatever the function was at x = a. It will just stay the same. This is true for all constant functions. It's like assuming the same thing will keep happening again. Sometime it does. A second order approximation involves looking at how the function's value has been changing, say over the interval from a - 1 to a. Call that difference delta (it's not really calculus without a Greek letter here and there) and say that the prediction at x = a + 1 is equal to the value at x = a with delta added. The actual definition of the series is a summation of an infinite series involving all the infinite derivatives of f(x).