Science Retrospective

The Pythagorean Paradigm

Phythagoras, founding father of Greek philosophy, logic and science declared that "all things are numbers" (i.e. integers). The Pythagorean paradigm assumes that the behavior of the Universe can be described in the language of mathematics. Since heavenly objects are believed to be controlled through sympathetic harmonies, music, together with mathematics holds the key to the Universe.

In modern terms, musical sounds are produced through sound vibrations having frequencies corresponding to various musical tones associated with different length of strings, tubes, mass of hammers, etc. of musical instruments. The fundamental arithmetic rule of musical harmony (multiple notes sounded simultaneously) is that whereas intervals between tones add, corresponding frequencies multiply. The simplest harmony is the musical octave, related to the simplest  frequency integer ratio 2 to 1. (To the ear, the notes sound much the same.) The term octave results from the division of the musical octave interval into 7 intermediate tones, or notes, called the musical scale. The dominant harmony of the Pythagorean musical scale is the 5th note above the 1st or fundamental note together with the fundamental, which is produced through the next simplest frequency ratio of 3 to 2. Going up (in frequency) by intervals of  repeated fifths should eventually arrive at the same frequency ratio as going up the scale by repeated octaves. Since an interval of a Pythagorean fifth corresponds to a frequency ratio of 3/2, the fundamental formula for commensurably is that after n fifth intervals one should arrive at the same frequency as after k octave intervals, so that (3/2)^n=(2/1)^k, or, taking logarithms of both sides, n/k=Log[2]/Log[3/2]. The ratio n/k is a rational fraction of integers but Log[2]/Log[3/2]= ... is not (in fact it is a never ending irrational decimal fraction). To resolve this discrepancy, one can resort to the continued fraction representation of decimal fractions in terms of rational fractions. Rational fractions have finite continued fraction representations, but irrational fractions have infinite continued fraction representations. Continued fractions have the property that, when truncated, they provide the closest ("best") rational fraction approximations to given decimal fractions. The term Log[2]/Log[3/2] is an irrational number whose smallest truncated continued fraction approximates are equal to the rational fraction ratios n/k = . Thus, possible commensurate musical scales consist of either 1 note (degenerate case), or 2 notes (the higher frequency being times the fundamental), or 5 notes (the pentatonic music scale of the Orient), or 12 notes (the chromatic scale of Western music), etc. (The seven note pythagorean scale is embedded in the modern richer 12-tone scale.) In this way, simple mathematics provides a rational explanation of human invention.  

As far as is known, the music theory of Pythagoras did not provide an explanation of the progression of musical notes in time, called melody. That task fell to his intellectual progeny, who sought rational (mathematical) explanations of motion in time, principally in terms of the newly invented geometry. Hereclitus believed everything to be in a state of flux or change, which seems more in agreement with observation than the rival notion of Parmenides that everything is static, or unchanging. Aristotle quantified the notion of rate of change of some quantity as the ratio of amount of change of the quantity to amount of change in time. Thus velocity is the rate of change in position to change in time. Today, infinitesimal ratios of change are called derivatives. Aristotle preached that objects falling to the Earth fall with constant velocity, but that was before empirical observations blossomed in science during the Middle Ages.

The Pythagorean approach to describing the Universe is based on arithmetic integer ratios describing discrete fixed states. The Pythagorean paradigm is thus a discrete, static mathematical model of behavior.

The Newtonian Paradigm

Descartes unified algebra and geometry with the notion that causal or functional relationships between independent (cause) and dependent (effect) measures of the state of a system, called variables, can be represented equivalently algebraically through equations or geometrically with graphs. The state of a system may be represented with a set of variables, such as position in space, each of which may depend on the time. The number of dependent variables is called the number of degrees of freedom, n.  Galileo, the inventor of modern mathematical dynamics, deduced from careful observations that it is not velocity that is constant for falling objects, but rather the change in velocity with time, or acceleration that is constant. This effectively introduced second derivatives of change into physics since acceleration is the change of the change of position with time.

Newton constructed a general "system of the Universe" in terms of the laws of universal gravitation affecting all known objects, including terrestrial apples and celestial planets. His magnum opus, Principia Mathematica presented the laws of motion (mechanics) in the language of Euclid's geometry, with motion along geometric curves in space. The dynamics of motion was expressed in algebraic terms in his Second Law, f = . where f represents the cause of motion, called force, p is in his words "quantity of motion," known today as momentum, and the dot represents derivative with respect to time. For objects of constant mass moving in a straight line, linear momentum is equivalent to the product of mass (a measure of the amount of matter) and acceleration (a measure of the amount of change in velocity).The gravitational force between two objects is proportional to the product of their respective masses and inversely proportional to the square of their separation. The universal nature of the gravitational law results from the empirical observation that the proportionality constant (symbolized as G) is the same for all objects. Newton did not publish his work on celestial mechanics until accurate values for the masses of the Earth and Moon became available and until he could demonstrate that the gravitational effects of a rigid sphere could be replaced with those of  an equivalent point mass at the center of the sphere having a mass equal to the sum of the masses of the component particles of the sphere (a delay of two decades). Newton also simplified the treatment of several mutually attracting objects to two principal objects perturbed by the presence of other, less influential, objects.

The Newtonian paradigm describes nature in terms of second-order  differential equations, that is, equations involving second derivatives of dependent variables with respect to their independent variables. The multidimensional nature of space suggests a combination of algebraic and geometric descriptions in terms of vectors, or multidimensional scalar quantities representing the states of the system (given by the set of variables describing the system). Modern dynamical theory recognizes that vector equations can be written as sets of scalar equations and, more importantly, that equations involving second derivatives are equivalent to pairs of equations involving first derivatives. For example, the second-order  differential equation = f/m, is entirely equivalent to the pair, or "system" (in the mathematical sense) of first-order  differential equations { = v, = f/m}, where the first equation is the definition of velocity, and the second the definition of acceleration according to Newton's Second Law. Given initial values of the variables, Newton's equations can in principle be integrated to produce system continuous (infinitesimal) motion called trajectories. Since the motion is determined from the equations of motion together with the initial state, the system is said to be deterministic. The Newtonian paradigm is thus a continuous, deterministic dynamical mathematical model of behavior.

The Lagrangian and Hamiltonian Treatments

Historically, first-order  differential equivalents to Newton's equations came about through inquiry into the inverse problem of the motion of a falling object in terms of its time of fall as a function of position rather than its position as a function of time. The first such problem was called the brachistochrone problem in the case of the shortest (brakistos) falling time (kronos), a problem which interested Galileo and Newton, among many others over time. The solution evolved into the field of optimization theory using the calculus of variations, introduced by Johann Bernoulli. This paradigm is based on the concept that the actual path in state space taken by an object is the most economical, or shortest path. In modern mechanics it is known as Hamilton's Principle. In general terms, the extremum of the integral of any quantity f which depends on n independent variables and dependent variables determines a relation between the variables known and the Euler-Lagrange  equation, -() = 0, i = 1, ..n. This formula is entirely equivalent to Newton's Second Law formula when x is identified with time, y with position and f with the difference between functions of velocity and position called, respectively, by Leibnitz vis viva, or today kinetic energy, T() ≡ and the work of force, or today potential energy V(,t). In mechanics, f ≡ L = T - V, is called the Lagrangian. Generalized momenta are obtained from derivatives of the Lagrangian as ≡. Since the Euler-Lagrange  equations result from the very general condition of optimization, they apply generally to arbitrary systems described in terms of arbitrary coordinate or state variables, including physical mechanical and electromagnetic field systems (both classical, relativistic and quantum mechanical), economic systems, biological and chemical systems, and, in short, to all systems known to science. In comparison with Newton's equations, they have the advantages of incorporating constraining conditions in a natural way and produce conserved quantities (constants of the motion) directly from the symmetries of the system When the Lagrangian is independent of a particular variable (said to be cyclic) or its derivative, the Euler-Lagrange  equations simplify to first order differential equations that can be immediately integrated to produce constants "of the motion". Examples in mechanics include the conservation of energy and momentum in time due to time invariance (i.e. no time dependence) of the Lagrangian and the extension from point particles to extended rigid bodies (having constant separation of particles and therefore constant potential energy).

Note that  a set of n differential equations in position and its first derivative with respect to time (Euler-Lagrange)  has replaced a set of n differential equations in second derivatives of position with respect to time (Newton). However, the Euler-Lagrange  equations are second order as well since they involve derivatives of derivatives. By means of a change in variables, however, it is possible to replace the Euler-Lagrange  equation with an equivalent set of 2n first order differential equations in terms of coordinates and momenta, known as Hamilton's equations. In the Lagrange formulation, the velocities are dependent on the positions through time derivatives of the latter. In the Hamiltonian formulation, the equations are strictly first order and the momenta and coordinates are independent variables on equal footing.  It is not surprising that Hamilton's equations are nearly symmetric: = + ,   = - , i = 1, .. n, where H ≡ T+V, called the Hamiltonian,  represents the total energy in a conservative system. This leads to the notion of representing the state of systems in coordinate-momentum "phase" space, instead of coordinate space (motion in both spaces is expressed parametrically in time). Again it is a trivial matter to show the equivalence to Newton's equations since if T ~= T(),  then = - = -  ≡ force. In place of using Newton's equations as the starting point in the description of system behavior, Hamilton's treatment begins with Hamilton's equations. This prescription incorporates cyclic coordinates in a very direct way, since, for example, if H ~= H(), then = 0 and = constant.

Jacobi showed how to find a coordinate transformation that renders all the coordinates ignorable, which, for conservative systems, replaces the 2n first order differential equations of the Hamiltonian formulation with n algebraic equations in the Hamilton-Jacobi  formulation.

The Poincaré Extension

Dynamical system theory, as developed by Poincaré, generalizes mechanics descriptions by treating evolution in general systems as systems of coupled first-order  differential equations (each equation generally involving multiple dependent variables), describing the motion of system state vectors in state and phase space. He also introduced a graphical analyses technique that replaced equations with plots that could be used to determine the ultimate state of a developing system which he used to show that the ultimate stability of the Solar system could not be determined.

Again, very general systems can be modelled in this way. However, Poincaré proved that no general analytic solution can be obtained for more than two interacting objects (the n-body  problem). The development of computational capability has permitted the exploration of new fields and new behavior, including discrete (finite step) as well as continuous motion, chaotic (or indeterminable) as well as deterministic behavior and, through numerical integration techniques, complex as well as simple systems and behaviors.

General Systems

Although the development of mechanics was motivated by the desire to explain the regularities of the behavoir of the Heavens, the resulting tools and concepts have found wide application to many fields, including chemistry, biology, art and economics. The basic idea of using the language and logic of mathematics to describe patterns of behavior has been accelerated with the advent of fast and accurate computational machines (computers).

Summary

Systems and their behavior can be described in a variety of ways, categorized as physical or field, discrete or continuous, deterministic or probabilistic, simple or complex. The search for unification, generalization and simplification has led to alternative, often mathematically equivalent but increasingly powerful descriptions and tools generally based on the fundamental Pythagorean-Newtonian  notion that existence and behavior can be understood and described in the language of mathematics.