The Wave Theory of Morality
Copyright © 2006, David A. Epstein.
All Rights Reserved.

June 18, 2006

Truth, Morality, and Provability

While the existence of truth might appear to be self-evident or intuitive to most people, it remains elusive from even the brightest of thinkers to prove that it exists. Neither absolute nor relative brands of truth are suited to withstand the rigors of provability or scientific inquiry. This simple logical proof demonstrates the futility of proving truth:

1) Suppose that truth doesn't exist.
2) Then we can make the statement that truth doesn't exist.
3) But this statement itself is a truth.
4) Contradiction. Therefore, truth exists.

But the problem with this proof is that every step (or statement) of the proof presupposes the existence of truth to judge its veracity. When we make Statement #1, #2, or #3, we must determine whether each one is either true or false. This leads to another contradiction: the thing we set out to prove (truth) must be a pre-existing condition in order to establish its existence! Hence, while the proof is valid, it's not sound. The proof has two unresolveable problems: one with language (these Statements can't encapsulate its linguistic content without inducing contradictions), the other with provability (we can't prove the conclusion that truth exists without first presupposing the existence of truth utilized to formulate the proof).

In theory, we can resort to the use of metalogic, the logical framework which determines the soundness of the given proof, to resolve this problem. But the metalogic pointing to this problem is itself problematic. It can't escape the wrath of Godel's Incompleteness Theorem! There will be some rule that is known to exist, but can't be proven or disproven; or the pointer to a statement (e.g. Statement #2) will get entangled with its content; hence the metalogical spotlight is turned back upon itself.

But here's the real rub: the converse condition:

1) Suppose that everything is true.
2) Then we can make the statement that nothing is false.
3) But then there would be no such thing as a lie, falsehood, or untruth.
4) This would mean that contradictory realities exist: the world is both flat and round, gravity pulls objects down and pushes them upward, 1 + 1 = 2 and 1 + 1 = 3, etc.
5a) Either our perception of the world is all messed up (contradictory truths and falsehoods really do exist), OR
5b) Statement #1 is false.
6) Since a falsehood is really a truth by Statement #2, BOTH 5a and 5b are true, meaning that contradictory realities exist AND falsehoods are true.
7) But this leads to another contradiction: Statements 5a AND 5b are true. It's not an OR condition.
8) Therefore, falsehoods must exist.

The problem with this proof is that the rules of proving an assertion must determine whether each step is EITHER true or false. And again, it must presuppose the existence of truth and falseness before commencing with the proof. But by Statement #1, everything is true, meaning that every step of the proof must be TRUE, regardless of whether it is actually false! Hence, this is really not a proof at all, but rather a Rubber Stamping of a series of assertions. So neither the existence of truth nor falseness can be proved here. We can't prove the existence of falseness without first presupposing the existence of falseness in determining the validity and soundness of the proof.

Hence, taking the two proofs together leads us to this startling conclusion: truth may or may not exist, but we can't prove either condition. Yet if truth doesn't exist, everything in the universe falls apart. That is the unfortunate quandry that is slapping humanity in the face!

Now, morality is the set of guidelines which humans subscribe to in their day-to-day lives; this code is essential for human coexistence and interactions. The code itself may warrant flexibility to include new moral precepts that will improve human living and remove old ones that actually harm it. We can't prove that the contents of the code are absolute and deductively conceived; they are subject to the forces of evolution and experimentation. Some moral precepts, which were earlier determined to be logically sound, might in practice turn out to be untenable, or if actualized might prove to be more detrimental than beneficial to human affairs; but the existence of the code and the necessity to implement it must be absolute. The meta-ethics are absolute, otherwise society can't exist.

The existence of Truth is the major pillar which Morality rests upon; but if we can't prove that Truth exists, can morality exist in any shape or form? That is the question I'll attempt to answer in this short paper as I propose my wave theory of morality.

Paired Concepts and Moral Fibers

Every pair of related concepts can be spectrally represented along a line, whether that representation is for an individual or society. For example, along a line, the far left side could represent total truth while the far right side represents total falseness. The points lying between these extremes will represent different degrees of truth and falseness. Another line could feature total goodness (far left) and total evil (far right), with varying degrees of good and evil lying between the polar opposites. In theory, an infinite number of lines can exist; the only limiting factor is the enumeration of conceived paired concepts. An intelligent mind is capable of conceiving of many such pairings, and then discover the intricate relations between them.

Each line has a length representing the distance between the polar opposites. Some lines will naturally be longer than others; the distance between absolute truth and falseness might be much longer than the distance between total goodness and evil. Each line might have a curvature representing the degrees of difficulty reaching certain points along the line. For example, it will be much easier to reach the midpoint between truth and falseness, and far more difficult to obtain total goodness. This line might look like a giant "U" shaped object, with the right-side measurably shorter than the left-side since it will be easier to achieve total evil if one allows oneself to fall pray to the darker side of life. Finally, over time, the line will change its length and shape. As one becomes older, it might be easier for one to learn from experience and practice goodness more readily that when one was much younger. The curved line will in fact transform into a wave formation or a family of waves (more on this later).

One obvious question is whether the lines will start off with equal length and curvature for all people, or will they vary throughout the population. Alternatively, perhaps there are no lines when one is born, that they form during the early development stages of life. Learning and behaviorism would be the forces driving the creation of these moral fibers. Fibers, in mathematical terminology, aren't merely constitutional "lines", but are preimages of inverse functions, meaning that they linearly map a range of elements back to its originating set. Do these moral fibers then represent the return values of moral actions that are functionally directed towards a range of social and personal values? These types of normative questions don't have definitive answers, but give us something to think about while we are formulating our theory.

Relations Between Paired Concepts

Once all of the lines are conceived, we can turn our attention to their relations. An example of a paired relation is between the "good-evil" and "true-false" paired concepts. This is plotted in 2-dimension space, in the x-y Cartesian plane; each point will represent the degree of goodness or evil (along the x axis, where a positive value is "good" and a negative value is "evil") in relation to the degree of truth or falseness (along the y axis, where a positive value is "true" and a negative value is "false"). A simplistic formula could stipulate that for a given person, the greater the amount of goodness the person possesses, the more truth will be obtained in her or his life (of course, a good person could be a sloppy thinker and continually live by falsehoods). Likewise, there will be trios of paired concepts represented in 3-dimensional space, quartets in 4-dimensions, and so forth. For example, in a three dimensional representation, "good-evil", "true-false", and "positive-negative" could be contrasted.

Let us suppose that n equals the total number of conceived paired concepts, represented by the set of  lines {L1, L2, ... Ln}. Then the total number of potential relations (say T) is expressed as:

T = "L1 x L2" (the number of pairs of lines) + "L1 x L2 x L3" (the number of trios of lines) + ...
       "L1 x L2 x ... x Ln" (all n lines are related)
   =  "n taken 2 at a time" + "n taken 3 at a time" + ... + "n taken n at a time"
   =  (n 2) + (n 3) + ... (n n)
To solve for T, we first need to solve for:
S = (n i)  = (n 0) + (n 1) + (n 2) + ... + (n n).
      i = 0
Then T(n) = S - (n 0) - (n 1). It's easy to show that S = 2 ^ n ("2 to the power of n"); i.e. if n = 3, S = 2 ^ 3 = 8.
Then T(n) = (2 ^ n) - 1 - n

We say "potential" because it's conceivable that some paired concepts won't be related to other ones. Only the lines that intersect will be related (mathematically, we use the term correlated). There will be 2-pair relations graphically plotted in 2 dimensional space; 3-pair relations plotted in 3 dimensional space; all the way up to n-pair concepts plotted in n-dimensional space. These graphs represent all the related pairs. Hence, the upper bound on the number of relations equals 2^n - n - 1.

Now, it's important to realize that the lines could be curved, and hence two pairs of lines, representing two different people, could conceivably intersect at more than one point. The "Truth-Falseness" and "Good-Evil" measures of the two people might coincide at several points, where the two people have equal "truth-falseness" and "good-evil" values. Between these points lies an intersection area. That area represents the comparative worth of the two people with respect to these qualitative measures, and the intersection point(s) can be illustrated with the colloquial phrase "with all things being equal". To continue with the simplistic example above, a positive area indicates that the "benchmarked" person is better than the other in the "truth-goodness" categorization within that intersection area, while a negative area would indicate the person is worse. If the curved lines had mathematical equations, the intersection area, call it 'A', equals the integral of the difference between the curves, integrated from one intersection point to the next:
A = (P1(x) - P2(x))dx,

where P1(x) represents the moral worth function of Person #1 (the "benchmarked" person), and P2(x) represents the moral worth function of Person #2. x0 and x1 are the points where the two curves intersect.

So for example, if Person #1 (P1(x)) is represented by y = x^2, and Person #2 (P2(x)) is simply y = 1, then the curves will intersect at y = x^2 = 1 => x = -1 or 1. This point represents where the two people have equal "truth-falseness" values.
Then A = (Area under Person #1's curve) - (Area under Person #2's curve)
            = (x^2 -1)dx
            = x^3/3 - x  (from -1 to 1)
            = ((1^3)/3 - 1) - ((-1^3)/3 - (-1))
            = (1/3 - 1) -(-1/3 + 1)
            = 2/3 - 2 = -4/3

If we interpret these results, Person #1 is worse than Person #2 because since we're starting with Person #1's curve, the result is a negative number. For a given interval of "goodness", Person #2 is more truthful than Person #1. It's possible that for a given pair of equations P1(x) and P2(x), P2(x) will often have a larger y-value for a given x-value than P1(x), but that the integral of (P1-P2) will still be positive. This is the reason the integral is calculated, for that will give us a truer value of comparative worth between the two people rather than a simple comparison of y-values for a set of x-values.

A more illustrative example of relationship between paired-concepts is the pairing of "good-evil" with "success-failure". Here, it's not so clear-cut, for many "evil" people could be quite successful while "good" people are utter failures.

In general, while this method divulges some useful information about comparative worth, it is problematic for a number of reasons:
1) It's difficult to derive these equations for individuals. A lot of data has to be collected about a person's behavior, actions, decisions, and preferences to formulate these equations.
2) The curves might not be continuous, for there might be regions where data can't be collected. For example, at some points, the amount of goodness doesn't translate to any degree of success or failure.
3) For a given person, the y-values might vary with respect to the x-values. This would mean that for any level of goodness, the person will assume a certain degree of success at one point in time, but later on will achieve less success even though s(he) is just as good.
4) The variables (e.g. goodness, truth) are considered to be absolutes, meaning that they can be objectively benchmarked; but as we have illustrated at the beginning of this piece, this is inherently problematic.

We'll deal with these issues in the next few sections.

Measuring Social and Individual Morality

Up to this point, we have introduced a few factors and equations regarding moral behavior (i.e. comparative moral worth between two people). We have not, however, provided any analysis about what constitutes moral worthiness and if it can be measurable. This section will be devoted to providing an initial argument in favor of the measurability of an individual's moral worthiness.

One can argue that the overall worthiness of a society is greater than the mere summation of its individuals. After all, when people purposely work together to pursue a common goal, their collective output is greater than the contributions of the individual members. This is not to say that this greater output is not measurable, only that some "external" factor is at work to generate the surplus output in addition to the sum of the individual outputs. In fact, if this external factor and these output levels have observable measures at any given point in time, then the collective output would also be measurable. This argument extends into the realms of individual and societal morality.

Even if we accept the uncertainty of measuring social output (whether that be represented in terms of economic productivity, or for our subject matter collective morality), we can establish some measurability of an individual's moral worthiness by analyzing the actions performed during her or his lifetime. An action has a measurable effect upon the individual's behavior or society at large, even if there are perhaps unmeasurable effects that can't be determined at that point in time (they might be determined at a later time).

Let us consider an individual's empirical moral worthiness and assume that it's measurable. Then if we wish to calculate the total moral worthiness of a person's actions, it would exhibit this formula:

T = mi * (wi - di), where
      i = 0

mi = intensity of a specific moral action
wi = the moral weight of that action
di = dampening factor from that action

The intensity of a specific action (mi) is the degree of effort in pursuing and realizing this action. The moral weight of an action (wi) is the impact and influence of that action upon one's life and surroundings. We could further decompose wi into personal and social impact components. For example, the action could directly affect the individual's personal life; or alternatively, if performed in the public arena, it would impact many people. The dampening factor (di) is the counter force exerted upon wi. Again, we could further decompose this into personal and social components. For example, this dampening factor could be the actualization of an individual immoral action. Or it could be the result of socialization of similar moral actions (i.e. wi has a significant impact in isolation, but in a societal context has a limited impact due to widespread moral crystallization).

Now, suppose we treat wi - di, say oi, as a normalized individual moral value with respect to a societal context. And let us further make the connection that the intensity of a specific action (mi) is functionally related to the normalized moral weight of an action (oi), such that oi = f(mi). Then

T = mi * f(ni) = m1*f(n1) + m2*f(n2) + ...
      i = 0

There are three possibilities here:

1) If f is a continuously increasing function, then                                        
                     mi                   wi
mi * f(ni)f(m)dm + f-1(m)dm
                     0                      0

        n                                    n          mi                       wi
T = mi * f(ni) =  ( f(m)dm + f-1(m)dm)
       i = 0                                i = 0      0               0
2) If f is a continuously decreasing function, then                                        
                     mi                  wi
mi * f(ni)f(m)dm - f-1(m)dm
                     0                    0

        n                                    n         mi                        wi
T = mi * f(ni) =  ( f(m)dm - f-1(m)dm)
       i = 0                                i = 0      0               0

3) If f is neither continuously increasing nor continuously decreasing function, then                                        
                     mi                     wi
mi * f(ni)f(m)dm +/- f-1(m)dm
                     0                       0

        n                                    n       mi                              wi
T = mi * f(ni) =  ( f(m)dm +/- f-1(m)dm)
       i = 0                                i = 0      0                  0

Without delving into a deep analysis at this time, we can observe that this diffusion of moral actions and counter forces ostensibly results in wave-like behavior. Think of the moral actions as stones thrown into a pond. They will create concentric circles of waves in the water. Now, think of the dampening factors as large rocks jutting out in the pond. They will obstruct the waves and diffract them into different directions and alter their speeds and potential impact.

Morality and Wave Behavior

One theme that continually appears in human behavior is that consistency in moral uprightness varies from person to person. Some people adhere to a high standard of ethics throughout their lives, seldomly deviating from exhibiting the highest virtues in everything they do. Others live their lives confined to a morally gray area, while still others lapse into outright immorality. The variance of morality levels throughout society is a reflection of the values that the people embrace; they are represented by a family of equations that reflect the behaviors of the collection of individuals.

Few individuals, if any, experience a constant level of morality throughout their lives. Each individual, in all likelihood, exhibits a variation of moral levels. Such variations might appear to be conspicuously missing from the most virtuous of people; but even at these ethereal levels of human existence, there are nevertheless fluctuations of ethical behavior that are even characterized with occassional moral lapses. Furthermore, at any given point of time, a person can assume different levels of morality depending upon the courses of action they choose to pursue. 

What appears to be transpiring throughout each individual's life are wave patterns of moral behavior. Perhaps this can be represented by a singular wave function or superimposed wave functions. The variable inputs into these wave functions must be closely related, for a disparate collection of variables will not possibly form an equation much less exhibit wave behavior. For example, a wave equation could be generated diagramming a motion between truth and goodness, which are closely related attributes; but one could not be formed between patience and fear, which have little if anything in common. The challenge in any proposed alignment of related attributes is to demonstrate it actually exhibits wave behavior.

Wave Equations

Infusion of Morality Equation: One Dimension wave

First, let us look at the formulation of a simple wave equation in one dimension. In our example where the "x" represents the degree of goodness, and "y" the degree of truth (we'll use truth in this example), the simple wave equation is:

2y/∂x2  = (1 / v2) * ∂2y/∂t2

2y/∂x2 = second partial derivative of y with respect to x = "acceleration" of truth with respect to goodness;
2y/∂t= second partial derivative of y with respect to time t = "acceleration" of truth with respect to time;
and v = phase velocity of the wave motion = frequency * wavelength. The frequency represents the rate of periodicity in the wave's cycle. A higher frequency translates into a higher amount of truth obtained for a given amount of goodness. That amount of goodness is represented by the wave's wavelength. But there is a tradeoff between the two, for a higher frequency results in a shorter wavelength. Hence, there is a natural upper limit of the wave's velocity.

Visually, the wave moves from left to right along the x-axis at the velocity v. This constant "speed" represents the rate of moral infusion into the human being in question. A higher velocity illustrates that the person takes decisive moral action faster than those with lower velocities, obtaining a greater amount of truth per expended amount of goodness; and since velocity entails directional speed, a higher velocity also demonstrates that the person is moving in the right direction (along the x-axis left to right, towards greater goodness). A negative velocity would indicate the person is moving right to left, or in the reverse direction (from goodness to evil).

Though it would appear that a faster velocity is desirable, there is a price to pay. In order to preserve the equation, ∂2y/∂x2 must increase by the square of the velocity, assuming that ∂2y/∂t2 remains constant (and this in itself isn't a certainty, for a change in  ∂2y/∂x2 might result in a corresponding change in  ∂2y/∂t2, depending upon the wave equation) . For example, if the velocity doubles, ∂2y/∂x2 (the acceleration of truth with respect to goodness) must quadruple. In other words, for an increase in moral velocity to occur, the amount of truth must double to increase the goodness that will be obtained from that increased velocity. Given the uncertainties in acquiring truthful information, knowledge, and commitments, there will be a natural dampening factor upon the moral velocity that can be achieved. As mentioned above, this dampener will be contingent upon the limits of the wave's frequency and wavelength.  Moral righteous people must be careful not to break that speed limit!

This wave equation is a simple linear partial differential equation that can be solved by a variety of methods taking the boundary conditions into account. The simplest form was discovered by d'Alembert using substituted variables and functions f and g:

y(x-vt, x+vt) = f(x + vt) + g(x - vt)
where f is a waveform traveling in the negative direction (right to left) and g is a waveform traveling in the positive direction (left to right).

In our example, this is a solution for the level of truth; however, the truth function is not merely dependent upon the level of goodness (x) and the amount of elapsed time (t), but is a more complex functional transformation of truth that is also dependent upon the velocity of the wave function (v). This truth solution entails a "goodness" function g countered by the "evil" function f, representing the classic struggle between good and evil! In other words, to obtain a positive amount of truth, goodness has to outweigh evil, meaning that a family of g functions (represented by goodness) must be found that consistently move faster than the family of f functions (represented by evil)! The greater the net movement of goodness in the world, the greater amount of truth is to be discovered. This will undoubtedly counter the claims that truth can be discovered by adopting or analyzing evil in the world.

However, a more realistic wave equation will involve a varying velocity that is a function of the y-variable: v(y). The resulting wave equation will be non-linear:

2y/∂x2  = (1 / v(y)2) * ∂2y/∂t2

This is a more difficult equation to solve. It involves a periodic velocity (the movement of moral action) that is functionally dependent upon the level of truth. This adds complexity to the equation because the velocity oscillates between different levels of truth. Thus, changes in the moral velocity results in changes in the level of truth, and visa versa. So it's more complicated to calculate the corresponding change in  ∂2y/∂x2 for a given change in v, because modifying the velocity will directly affect the level of truth (the y-value), which in turn will alter ∂2y/∂x2 . For a simplistic example, if the velocity is doubled, it might itself require a doubling of truth plus the initial velocity V0 from the linear example (i.e. v(y) = V0+ 2*y) to manage the increased wave motion speed. Since ∂2y/∂x2 is inversely proportional to the velocity-squared, assuming ∂2y/∂t2 remains constant, ∂2y/∂x2 must now increase by (V0+ 2*y)2, certainly requiring more than a quadrupling of ∂2y/∂x2 to preserve the equation.

In any case, the wave equation solves some of the problems stated in the previous section, namely the second and third problems: the wave is continuous, and for a given level of goodness (the x-value), there are multiple values of truth (the y-value) that are assumed as the wave moves directionally from left to right. The other problems will be addressed in other sections.

Infusion of Morality Equation: Two Dimension wave

If we introduce a "third dimension" into the picture, say "success vs failure" along the z-axis, the wave equation would be moving across a planar surface of the 3D object, along combinations of the x and y axes. The linear two-dimensional wave equation is:

2z/∂x2 + ∂2z/∂y2  = (1 / v2) * ∂2z/∂t2

The considerations discussed above apply here as well, but the complexity level is greater in this case. For example, a doubling of the velocity will require a quadrupling of the Laplacian term ∂2z/∂x2 + ∂2z/∂y2, assuming ∂2z/∂t2 remains constant. This is a more difficult problem to solve because there are now two mutually-dependent terms that need to be calculated.

The problem is solved using separation of variables with various substitutions and transformations. After all of the brunt work is performed, the solution for the rectangular membrane is:
                   ¥     ¥
z(x, y, t) = [Apq*cos(wpq*t) +Bpq*sin(wpq*t) ] *sin(ppx/Lx)*sin(qpy/Ly)
                        p=1     q=1

where  p* v* sqrt( (p/Lx)2 + (q/Ly)2 ), p and q are real integer values,   
and Lx & Ly  are boundary conditions for z(Lx, y,t) = 0 and z(x,Ly,t) = 0 respectively.

Solutions for Apq and Bpq are:

                                Ly     Lx
Apq  = (4/LxLy) *    [  z(x,y,0)*sin(ppx/Lx)dx] * sin(qpy/Ly)dy
                               0        0
                                      Ly     Lx
Bpq  = (4/wpqLxLy) *    [  ∂z/∂t(x,y,0)*sin(ppx/Lx)dx] * sin(qpy/Ly)dy
                                      0        0

Immediately, we can see this is a far more complex solution, though far more representative of a real-world situation. From the presence of "sines" and "cosines" we can infer some periodic motion between the operating variables; but we can't simply plug in values for truth and goodness as we did in the one-dimensional case. Here, there are summations and integrals to compute. The only exception to this complexity is to consider the singularities when there is no truth (x = 0) or no goodness(y = 0). With respect to our self-defined variables, this is distinct from evil which assumes a negative x-value and falsehood which assumes a negative y-value. I interpret x = 0 and y = 0 to be values expressing ambiquity or uncertainty in truth or goodness. In any case, if x = 0 or y = 0, then since sin(0) = 0, z(0,0,t) = 0. This means that if there is no goodness or truth, there will be no success! This appears to defy what we experience in the real world because even a lack of truth or goodness, or the ambiguous values they may assume, would still lead to some success or failure. If this is the case, then the wave motion approaches the values x = 0 and y = 0 asymptotically without actually assuming those values.

Likewise, if *both*
Apq & Bpq = 0, then z(x,y,t) = 0. This could occur if z(x,y,0) = 0 and ∂z/∂t(x,y,0) = 0, meaning that there is no initial success or rate-of-change of success. Since there are built-in factors to prevent both of these conditions from occurring (i.e. there might not be any initial success at time t = 0, but there will be an initial rate-of-change as soon as some moral decision is made), z(x,y,t) will be non-zero.

The general relationship between truth, goodness, and success is therefore quite complex, described by the periodic wave motion along the two-dimensional surface. There is undoubtedly some stable region obtained from the interplay of these participating factors; this would translate into some vibrational frequency or resonancy pattern that would be applicable to various behavioral and decision-making processes. We would have to carry out several integrations and summations to analyze this further. Equally difficult would be to calculate the optimal value of success (z). We would need to take the partial derivatives ∂z/∂x, ∂z/∂y, and ∂z/∂t and set them all to zero. Then we could solve for z(x,y,t) for those imposed conditions.

Higher Dimensionality and Wave Propagation

In the  "Truth, Morality and Provability" section, I demonstrated the futility of proving or disproving the existence of truth. Simply put, logic and mathematics, while powerful tools, can't sufficiently establish the objective reality of this vital attribute in life. Truth be told, so to speak, if truth doesn't exist in our world, then our world would fall apart and cease to exist. We desperately need truth and without it we're doomed; and there doesn't appear to be a substitute for it. Thus we are stuck in a quagmire, entrapped in a paradox that permeates right to the core of cosmological and human existence: we depend upon but can't prove the existence of truth.

So how is this resolved? What may be at work is that truth is a metaphysical phenomenon (perhaps a circular or vibratory form) existing in a higher dimensional universe. If so then we are receiving the reverberations of it in our 3D world, but with our limited logic and reasoning powers, we can't circumvent the paradox that it exists yet can't be proven; but in the higher world, there is no paradox. One of the ideas I wish to explore, fully consistent with and a comcombitant part of my wave theory of morality, is that higher dimensional wave forms propagate into our 3D world. If this can be established, then it will be relatively a simple exercise to demonstrate that moral waves follow this pattern of propagation.

Let us look at a multi-dimensional wave equation with boundary conditions. Let D be a domain in a space existing in x dimensions, and let B be its boundary. Then along B, for any b contained in B and an x-dimensional vector u, the wave equation will have this form:

∂u/∂n + a(b)*u = 0

where n is the unit normal vector to B, and a is a non-negative function defined on B. The initial conditions are

u(0,x) = f(x)ut = g(x),

where f and g are defined in D.

As I interpret it, the u vector represents the levels of morality in x-dimensional space. The boundary value a(b), visually the (x-1)-dimensional surface of the x-dimensional space, represents the constraints upon that moral space such as social and personal challenges to living a virtuous life. Here, it becomes clear that for any boundary value (a(b)), as u increases, then ∂u/∂n  must decrease; conversely, as ∂u/∂n increases, u must decrease. Hence, for any level of morality (u), the rate of change of that morality must decrease. This demonstrates the phenomenon that at high levels of morality, the tendency for moral decline is greater, and at lower morality levels, the tendency for moral ascendancy is greater. The key to equalize this equation is with the boundary conditions a(b). For a higher a(b), every increase in morality u is offset with a greater decrease in ∂u/∂n. This means that the greater the social and personal pressures (e.g. perils to personal safety, stress, temptations, etc) any courageous move to increase one's morality level will be countered with a severe dampening factor; it will become more difficult to further increase one's level of morality. Finally, at the extreme, when the boundary condition is increasing to infinity, u(x) will tend to zero as will ∂u/∂n. Unlimited social and personal pressures will result in the total collapse of morality (i.e. a total lack of scruples and ethics).

This general problem will be solved by expanding the functions f(x) and g(x) in the eigenfunctions of the Laplacian in D, which satisfy the boundary conditions. Thus the eigenfunction v satisfies the equation in the domain D.
\nabla \cdot \nabla v + \lambda v = 0, \, and
∂v/∂n + a(b)*v = 0
For this solution of the v eigenfunction, there will be a stable level of morality in the higher x-dimensional space. The question is how this translates in our 3D world.  

Morality-Generating Equation

Moral Probabilistic Wave Equation

Last update: February 13, 2007