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Math
My wife didn't want me to include this chapter in the book, and it is probably safe to assume that most of the people interested in this book are not interested in mathematics, as it is usually conceived. Professional mathematics is, admittedly, the worst of the barbaric specialties, so fragmented by jargon and bristling technicalities, that even mathematicians do not understand one another.
Humanist math is nothing like that. It is a fully adequate and complete form of mathematics, yet it is simple and anyone can understand it.
But why have anything to do with mathematics? Humanists need have nothing to do with professional mathematics; humanist math is an important part of the humanist synthesis. Humanist math is part of the attack on the prestige of specialists and part of the claim that the humanist synthesis can include man's technical and scientific knowledge as well as humanities and religion.
The assumptions of humanist math are these: mathematics is a language. Becoming fluent in this language has no more to do with formal definitions and proofs that does becoming fluent in Russian or Polish. You couldn't learn Polish by studying Polish logic.
The second assumption is a corollary of the first. The only reason for learning mathematics is to master the literature in the language, which mainly consists in the half-dozen great equations of physics. For our purposes, mathematics is not defined by what professional mathematicians do, but is solely defined by the users of the language, again, mainly the physicists. Concepts which are not part of the working language of physicists are not, therefore, part of math. This means we can dispense with the endless abstractions (the n-dimensional spaces, for instance) which have no real application.
Our last assumption is that only methods which are both practical and universal will be used. Since all the great equations can be cast into a single form, that of the variational equation, that is the only one we will develop. Since the only method of solving this equation (or any other) for any possible case (including most practical problems)is trial and error, that will be the single method of humanist math.
We have already simplified things considerably. We only need to study one form of equation, with 5 or 6 specific variants of it, and only one method of solving the equation. Unlike the methods of solving things in professional (formal) mathematics, skills in humanist math are cumulative; skills in solving one problem carry over to the next one.
A philosophical point: Thales regards all the sciences as empirical, including the conceptual sciences such as logic and mathematics. The subject matter of such sciences is the natural language, which is a living system, not at all like the formal models of it developed by formalists. Fastening a formal definition on such systems only inhibits their growth, as does the rejection of a well-used concept (like differentials) because it does not fit the already established formal system. In humanist math, we avoid the trappings of 'rigor' and formality altogether.
Math is the language of counting and measuring. It follows that all numbers in humanist math will have a finite number of decimal places or significant numbers (rarely more than 5 or 6), and they will also have definite units and a coordinate system or space. It also follows that complex numbers need not appear.
There are two distinct styles of humanist math, one developed with the small electronic calculator, and the other developed on a conversational programming system such as the telenet. In this brief introduction to the subject we will not get far enough to describe the salient differences.
The literature we are trying to make accessible and part of the general understanding of the humanists is stated in the form of equations. The elements of an equation are numbers, functions, operators, possibly arrays, and a test or desideratum. Physical systems are described in terms of functions or arrays of functions. Some of the buttons on an electronic calculator are functions, such as X-to-the-Y or e-to-the-x or 1-over-x or ln(x) or sin(x). If you enter a number, then press a function button, and look at the display, and do this for many different entered numbers, you will find a distinct pattern or picture for each function. When plotted, each function has a distinctive shape. The variational equation of humanist math takes your guesses as to the behavior of the system, operates on them according to the rules for the particular equation, and produces a number. If you make a variety of guesses in function space, and plot the numbers produced, the point or points in function space which correspond to allowable reality are those which make the desideratum or test number an extremum, that is, a maximum, a minimum, or an inflexion point on the graph.
The variational equation is called that, because if you vary the path followed by the real system and vary it only a little, the test number will not change (unless the system is chaotic). The paths of the system which occur in linear systems are thus the ones which are stable.
To the professional mathematician, the trial and error method would not seem a method of solution at all, sincere there is no formal procedure which will lead to solutions. It may therefore seem as if all the advantage lay with the formalist. In reality, this is not so, for two reasons: formal methods only work for a few special cases rarely found in nature, and even for those cases, learning the formal methods takes longer than solving the problem by trial and error.
In any equation, there are certain things you know (laws of varying degree of generality) and things you don't (the precise description of the system). You use the things you know to form the functions and perform the operations necessary to evaluate your guesses as to the behavior of the system.
There are two kinds of operation on functions which the humanist mathematician needs to master. One is symbolic differentiation of functions, which is easily learned, and numerical integration of functions (using the hand electronic calculator or the computer) and this is also easy to learn. Symbolic differentiation of a function produces another function, in general. For instance the derivative of sin(x) is cos(x), of x**2 is 2*x and so forth. There are only about 6 distinct rules for differentiation, given in any mathematical handbook. Numerical integration of a function produces a number. You can find a description of the methods of numerical integration (e.g. Simpson's rule) in books on numerical analysis.
The way the variational equation works to codify mathematical laws of physics is something like this: The final step of the equation is a numerical integral of a function which sums up the salient physical characteristics of the system. In about half the physical equations, the characteristic function is L, the Lagrangian, which is a function describing the total kinetic energy of the system (T) minus the total potential energy of the system (V). T and V are themselves functions...functions of other functions which describe the motion of each free variable as it varies with time. In the other half of the cases, the characteristic function is N, the index of refraction. N is usually a function of other things as well, possibly of potential energy or position. In this type of equation, you guess as to the behavior of the system takes the form of functions describing the path of a ray of the wave in question. These functions are substituted for the space variables in N and you integrate along the path length. This sounds complicated, but with practice turns out to be extremely simple. As in learning Russian, things look hopelessly difficult until you've used them and become familiar with terms.
The traditional way of writing these two forms is the variational integral of the Lagrangian function, and the variational integral of the index of refraction function. In linear systems, the variation of the integral around the point in function space which describes the real system approaches zero. In non-linear, chaotic system, this is still true around the semi-stable regions of phase space. There may be more than one such semi-stable region. If small variations produce large changes in the integral, you know you are in an unstable region of phase space.
It is possible to form a Lagrangian for electro-magnetic problems just as well as mechanical ones. Your guesses are arrays of functions of the time, each component of the array describing a component of the electric field, or the magnetic field, or the current, or the magnetic potential fields. The Lagrangian function is defined in terms of an array product, which reduces each of them to a single term.
The laws of special relativity are satisfied by making a transformation of space and time variables (using the Lorentz) transformation) before forming L.
The laws of general relativity require a transformation of the differential dS.
The laws of quantum mechanics (de Broglie theory) can be formulated in terms of the laws of the index of refraction of the de Broglie wave.
All of the basic mathematical laws can be written on a single page. This would probably be no more intelligible to most than a precise poetic statement of Russian wisdom in a single great poem. As in the case of any other foreign language, what is needed to become fluent is not a lot of formal definitions, but rich involvement in examples, and that will have to be left to another book. I hope I have said enough to show that the humanist can learn the great mathematical laws of the universe as well as the specialist---without spending years wandering among the bewildering jargon and technicalities of the formalist.