The Impossibility Theorem
(a) the moral and spiritual qualities of the present personality's behaviour determine the quality of life of the new personality in the next incarnation. Expressed simply: actions determine karma.
(b) The quality of life of the present personality determines the moral and spiritual qualities of the personality's behaviour in the next incarnation. Expressed simply: karma determines actions.
Let R be the probability that reincarnation is a fact, ε be a small error in the operation of karma, and N be the total number of series of personalities, then
RANDOM VARIATIONS ABOUT THE MAXIMAL PATH
Although chaotic, trendless swings of karma – as illustrated in the diagrams above – make little sense from an evolutionary standpoint there is some merit in the argument that karma might move randomly about the maximal path. In other words, although the long term trend in karma will be positive in order to reflect the upward spiritual ascent of the personality, changes in karma from one incarnation to the next could be negative. Thus, an evil personality could succeed a good one! Given the propensity of human beings to behave in erratic and unpredictable ways, such a scenario is not merely possible but highly likely. Under this assumption, the karmic path would look as follows:
The actual karmic path Kr(x) - drawn in black - moves randomly about an upward path of spiritual improvement. This upward path is the maximal path K(x) and is drawn in red. It will be seen that, whereas the maximal path “hits” the incarnation axis only once, the actual path “hits” it fifteen times. Hence, it might be thought that the chances of eliminating karma are increased and that the Impossibility Theorem is invalidated. Actually, this is not the case. In fact the chances of eliminating karma are decreased. [The proof of this assertion will be given in a revised and expanded version of [Refuted ]. The reader will need to study Chapter 4 of Refuted which develops the mathematics of karma and the notion of a karmic probability density function. For those who wish to familiarize themselves with the elements of probability theory and mathematical statistics, I would recommend John Freund’sMathematical Statistics (Prentice-Hall) by Irwin Miller and Marylees Miller; and Paul G. Hoel’s Introduction to Mathematical Statistics (John Wiley & Sons). For more advanced reading I would suggest Introduction to Mathematical Statistics (Collier-Macmillan) by Robert V. Hogg and Allen T. Craig.
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The system of karma here presented incorporates two basic features:
(a) the upward ascent of the personality in the long run expressed as the maximal path.
(b) the erratic and unpredictable behaviour of the personality in the short run expressed as random fluctuations about the maximal path.
A karmic path can, therefore, be envisaged as the sum of two components: an evolutionary, maximal component associated with the personality’s behaviour in the long run, and a random component associated with the personality’s behaviour in the short run.
The error in the operation of karma does not, of course, refer to errors in the workings of nature. We are here concerned only with the analysis of a doctrine and its mathematical representation. The concept arises because it’s unreasonable to insist that karma must be exactly zero for liberation to be possible. Suppose, for example, the magnitude of bad karma is:
If this had to be counter-balanced by an equal quantity of good karma, then liberation would not be possible if it was
The error in the operation of karma permits small differences in good and bad karma to be disregarded.
It is important to note that, since the karmic graph “hits” the incarnation axis at a point between two whole numbers, the error in the operation of karma cannot exceed 0.5. Hence, unless it “hits” the axis at the midpoint, it will always be nearer one number than the other. The nearest whole number defines the incarnation of liberation (because karma will be smaller). If the error relative to one incarnation number is increased to a figure above 0.5, it will ipso facto be decreased to a figure below 0.5 relative to the other incarnation number. This latter incarnation will then define the incarnation of liberation and the error will accordingly be less than 0.5. The probability of the graph “hitting” the incarnation axis at the midpoint, or an integral value of x, is precisely zero. Although 0.5 is the upper bound for ε, the error will, of course, need to be much smaller than 0.5 to ensure that residual karma is negligible.
On the question of the so-called "Noble Eightfold Path," it is a mistake to assert that the personality can control and eliminate karma by “right” living, meditation and general renunciation of the world. The reasons are twofold:
(1) the character traits needed to engage in such activities are themselves determined by karma and can be traced back to the actions of the first
incarnation. Since the actions of the first incarnation are the product of chance, the character traits needed to eliminate karma are also the product
(2) the actions of the personality can have unintended consequences and, therefore, unintended karma. If karma is eliminated, it will be due to a
chance combination of factors and not to the efforts of the devotee. As we have seen, the probability of this happening can be calculated.
In the case of point (2) unintended consequences of actions will add to the fluctuations of karma about the maximal path and, accordingly, reduce the probability of liberation. In many ways point (2) is typical of the undeveloped personality. If A shoots and kills B but didn’t mean to kill him, A’s actions will have karmic consequences over and above the intended action.
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Random motion about a maximal path – which includes the special case of motion with a random disturbance of zero - describes karma in its most general setting. As we have seen, other descriptions are possible but are grossly inconsistent with karma’s function as a facilitator of human spiritual evolution. The inability of karma to determine the personality’s actions in the first incarnation coupled with the erratic and unpredictable behaviour of the personality in the short run means that liberation from the cycle of rebirths is a chance event – an event that is beyond the control of the personality. It is quite remarkable that, whatever the response to karma, the probability of liberation can never exceed ε – the error in the operation of karma. An immediate consequence of this result is that the smaller the error, the smaller the probability of liberation - approaching zero in the limit. It is the universalist nature of karma that is its undoing because it permits of no exceptions: every series of personalities without exception must eliminate karma in order to escape the cycle of rebirths. As we have seen, the probability of this happening is so small it can be equated to zero. Consequently, some series of personalities will be infinite in extent; and a series of personalities that is infinite in extent has no final personality. In short, karma cannot describe anything that happens in nature. Without karma, the notion of a “stream of consciousness” linking a group of non-contemporaneous personalities is entirely gratuitous. One might just as well speak of a stream of consciousness between a person and his ancestors; or a stream of consciousness between a group of trees; or a stream of consciousness between a group of any living things – non-contemporaneous or otherwise. If a group of non-contemporaneous personalities have nothing in common other than the fact that they are personalities dwelling on the same planet, then it is meaningless to apply the term ‘rebirth.’ Nothing is reborn because every personality is unique unto itself.
Conflicting ideas about the self and the nature of the reincarnating entity are the source of much confusion in the reincarnationist world. In Hinduism, for example, the reincarnating entity is the eternal Self. In Buddhism, no such entity exists; instead, a chain of personalities are linked together by a "stream of consciousness." Between these two extremes are many variations reflecting different historical and cultural traditions. This disparate state of affairs has led some people to argue that refutations of reincarnation are essentially meaningless because it's unclear what is being refuted. In reply to this - using the Four Postulates - it can be proven that the self does not incarnate more than once which is to say it does not reincarnate. Discussion of reincarnationist belief is, therefore, unnecessary. For those who don't accept Postulate III, such as Buddhist monks, the analysis of karma, alone, is sufficient to show that single incarnation is the only viable alternative. Once again, discussion of reincarnationist belief is unnecessary.
In the analysis of karma, the concept of a reincarnating entity is dispensed with entirely. Instead, one considers a series on non-contemporaneous personalities whose purpose is the elimination of karma. The underlying principle linking these personalities may vary from one system of thought to another but the concept of a temporal series of connected personalities does not. Crucially, there must be a reason for these personalities to exist - they do not appear by chance - and this reason is the motive force for reincarnation. This reason may be grounded in choice or compulsion but, ultimately, the distinction is irrelevant because all reasons are functionally equivalent to karma - only the language varies. Thus, a series of non-contemporaneous personalities exist to eliminate karma or the need to learn new lessons.
One of the most surprising features of karma is its ability to be quantified and analyzed with the tools of mathematics. What is hidden beneath a jungle of metaphysics can be dug up, dissected and demolished by mathematical methods well within the grasp of any first year university student in the physical or social sciences. (Mathematically inclined readers may consult Refuted for details.) In the first instance it can be shown that liberation from the cycle of rebirths is a chance event, and this leads to a remarkable theorem - the Impossibility Theorem - which states that the probability of every series of personalities eliminating karma and achieving liberation from the cycle of rebirths is zero. Consequently, at least one series of personalities will be infinite. Since this is impossible reincarnation cannot be a fact of nature. This demonstrates that the doctrine of karma is fundamentally incoherent i.e. internally self-contradictory. The contradiction arises because karma is supposed, ultimately, to liberate everyone from the cycle of rebirths through a process of learning and spiritual evolution but liberation for everyone is impossible. This incoherence shows up in another way: the more evolved a personality becomes, the less likely the personality will achieve liberation - the reverse of what is supposed to happen. This latter proposition is proven in Refuted.
The formula in the panel above is based on the assumption that every series of personalities follows a slow, consistent path of moral and spiritual improvement. Given any linked series of non-contemporaneous personalities, adjacent personalities will not be very different from one another in terms of their moral and spiritual qualities. The particular roles in life may vary considerably e.g. one personality may be a king and the next a janitor, but their moral and spiritual qualities will not be very different. However - and this is very important - the later personality will be slightly better (morally and spiritually speaking) than the earlier one. Of course, this scenario does not cover all possibilities. It is conceivable for an evil personality to succeed a good one, or even an animal incarnation to succeed a human one. As we shall see, however, large, irregular and possibly inconsistent changes in the moral and spiritual qualities of personalities between adjacent incarnations, reflected as large, irregular and inconsistent swings of karma, will decrease the chances of attaining liberation. To illustrate, a man carrying a rifle moving around quickly and haphazardly on rough, uneven terrain is less likely to hit a given target than a man walking slowly and progressively uphill. Any departure from this slow, upward path will decrease his chances of hitting the target. We shall refer to a series of personalities that is following a slow, consistent path of moral and spiritual improvement as a maximal series of personalities and the path followed a maximal path. This terminology is used because a path of slow, consistent improvement maximizes the chances of eliminating karma - an assertion we will justify below. Suppose the probability of a maximal series of personalities attaining liberation is 1 chance in 100, then the probability of a non-maximal series attaining liberation would be smaller e.g. 1 chance in 150, or 1 chance in 200 etc. depending on the degree of variation from one incarnation to the next. The difference between a maximal and non-maximal path can be illustrated graphically:
The question is: what determines the karmic path? Before we consider this question let's clarify the meaning of the term 'actions'. In the doctrine of karma actions relate not so much to the outward behaviour of the personality but to the moral and spiritual qualities of the personality's behaviour. For example, firing a gun at a wall is very different from firing a gun at a person. The outward behaviour may be identical but the motive is vastly different. Actions can, therefore, be good or bad. Karma, on the other hand, refers to the quality of the personality's life and can also be good or bad. If, for example, a child is murdered or born handicapped, or someone's house is repeatedly burgled or burns down - that's bad karma. If someone wins the national lottery, or lives a life of ease and luxury, or is highly intelligent - that's good karma. Actions and karma interact with one another from one incarnation to the next in the following way:
There is no circularity here because present actions determine future karma, and present karma determines future actions. For example, someone who murders (bad actions) in the present incarnation will experience a poor quality of life (bad karma) in the next incarnation. Actions in the following incarnation will then change for the better. This means that future personalities will learn from the mistakes of previous personalities. If this were not the case then reincarnation would be pointless because no lessons would ever be learned. Different modes of interaction are possible but anything more complicated is likely to produce a non-maximal path. (The reader may refer to Refuted for further discussion of this point.) Now, if the actions of one incarnation determine the karma of the next incarnation, and the karma of the next incarnation determines the actions of the incarnation after that, what determines the actions of the first incarnation? They cannot be determined by the karma of the previous incarnation because there is no incarnation previous to the first one. This means that initial actions cannot be determined within the system of karma, and if this is the case then they must be the product of chance. The subsequent path of actions and the corresponding path of karma must, then, also be the product of chance. What is supposed to be a system of justice turns out to be a rather arbitrary system of punishments and rewards.
Those familiar with engineering systems will recognize the above scheme as an input-output system with a feed-back loop:
To transform the input-output scheme into a mathematical model, it is necessary to quantify actions and karma, and in Refuted I spend some time showing how this can be done. Since the present discussion is expository we shall take quantification as given. Now, a mathematical model in which the quantity of something in the present time period depends on the quantity of something else in a previous time period is known as a difference equation, and anyone familiar with difference equations will recognize that the scheme outlined above can be so represented. Now, the solution of a difference equation can, in many cases, be represented graphically, and this turns out to be possible in the case of a karmic path. For illustrative purposes, a karmic path can be shown as a smooth or jagged line depending on whether it is maximal or non-maximal but the true graph will be a series of points - each point corresponding to a particular incarnation. (The gaps between the points correspond to the gaps between incarnations). Karmic paths will, therefore, look like the ones below. The points have been joined to show the general trend.
A maximal karmic path can be thought of as a single entity that is randomly determined by some hypothetical experiment like the throw of dice. The dice are thrown in the first incarnation, the actions are set and the entire karmic path is determined. (Precisely how the path is determined depends on the response to karma). If the actions had been different, the path would have been different. The following diagrams show maximal paths corresponding to different initial actions (diagram on the left) and maximal paths for the same initial actions but different responses to karma (diagram on the right). For ease of viewing, points on the graphs are not shown. Note than karma begins in the second incarnation - hence the gap between the graphs and the vertical axis. No significance should be attached to the numbers; the scale is shown to indicate that the graphs are mathematically correct representations of karma (drawn in MathCad).
In mathematics, smooth, continuous curves are easier to handle than discontinuous points, and in Refuted I introduce a "smooth" approximation to the series of points that define the karmic path. This replaces the path of points with a smooth, continuous curve like the ones shown in the diagrams above. Given that the maximal path can be represented by such a curve, we need to find the probability that the curve will "hit" a point on the incarnation axis close to a number that defines an incarnation. When this happens, the karma in that incarnation will be negligible and liberation from the cycle of births will be possible. A direct "hit" occurs if the curve crosses the incarnation axis at integral value (i.e. whole number); karma will then be zero. Note that the graph of a maximal path never "hits" the incarnation axis more than once. This means that if liberation is achieved, the incarnation of liberation is uniquely determined.
In the following discussion, we shall refer to the distance between two incarnation numbers as the standard unit. The Greek symbol, ε (epsilon) shown in the diagrams above is defined as the error in the operation of karma and sets the limit to the permissible deviation of the curve from the number of the incarnation. The error is indeterminate but needs to be fairly small to ensure that residual karma is negligible e.g. one hundredth of the standard unit. In the diagrams above, the error is one-fifth. If the error wasn't small, liberation would be possible with non-negligible karma. In the following diagram the error is two-fifths of the standard unit and residual karma is much larger.
Let n be the number of the incarnation, then liberation is possible if the curve "hits" the incarnation axis anywhere from n - ε to n + ε where n can be any integer greater than 1. (n must be greater than one for reincarnation to happen.) Using standard techniques in mathematical statistics it's possible to calculate the probability that this will happen and it turns out to be equal to ε - the error in the operation of karma. This probability is the same for every series of maximal personalities.
Let's suppose that the error is 0.1. Then, statistically speaking 1 series of personalities in 10 will eliminate karma - where eliminating karma means reducing it to zero or some negligible quantity - and achieve liberation. But, the doctrine of reincarnation requires that every series of personalities eliminate karma otherwise reincarnation for some would be infinite. Since all karmic paths are determined independently - they are the product of a hypothetical random experiment in the first incarnation - to find the probability of, say, two series of personalities eliminating karma one simply multiplies the individual probabilities together i.e. (0.1) x (0.1) = 0.01 - one chance in a hundred. The probability of three series of personalities would be (0.1) x (0.1) x (0.1) = 0.001 - one chance in a thousand. It is evident that the probability of eliminating karma diminishes rapidly with the number of series of personalities. In mathematics multiplying a number by itself is called "raising" it to the power of 2; multiplying it by itself three times is raising it to the power of three. Multiplying it by itself any number of times is raising it to the power of that number. Thus,
0.1 raised to the power of 2 = (0.1) x (0.1) = 0.01
0.1 raised to the power of 3 = (0.1) x (0.1) x (0.1) = 0.001
0.1 raised to the power of 4 = (0.1) x (0.1) x (0.1) x (0.1) = 0.0001
The probability of a million series of personalities eliminating karma would be:
0.1 raised to the power of one million = (0.1) x . . . . . . . . . . . . . . . . . . . . (0.1) ≈ 0
one million times
The wavy equal sign means almost equal to. But, of course the number of series of personalities must be counted in billions, say 10 billion. We should, therefore, write:
0.1 raised to the power of 10 billion = (0.1) x . . . . . . . . . . . . . . . . . . . . (0.1) = 0
10 billion times
which can be equated to zero because anything with that probability will definitely not happen! Generalizing, if ε is the error in the operation of karma and N the number of series of personalities, the probability of every series eliminating karma is ε raised to the power of N.
Since ε is small in relation to the standard unit and N is a very large number, this quantity is, for all practical purposes, equal to zero. This means that it's virtually certain that at least one series of personalities - in practice it could be billions - will not eliminate karma and, consequently, reincarnate an infinite number of times. Since this is impossible the doctrine of reincarnation cannot describe anything that happens in nature. This is the essence of the Impossibility Theorem.
On the question of non-maximal paths - though conceivable - they make little sense from an evolutionary point of view. In biological evolution organisms progress from simple to more complex forms producing an orderly, progressive series - not a random, haphazard one. The same must apply to any theory of spiritual evolution - including the doctrine of reincarnation - because a series of personalities that swings endlessly from one extreme to another in a random, purposeless way is anything but evolutionary. Indeed, any departures from the maximal path may be considered counter-evolutionary. Therefore, it is highly reasonable to assume that personalities follow a path of slow, consistent and progressive improvement, each personality learning from the mistakes of previous ones. (However, random variations about the maximal path itself, in which one personality may or may not progress from the previous one, is probably a more reasonable assumption. Though counter-evolutionary in the short-term, such variations are consistent with evolutionary change in the long-term. See the discussion below for the analysis of karma in this scenario.)
It is easy to show how large departures from the maximal path decrease the chances of eliminating karma. In the diagram below the non-maximal path "hits" the incarnation axis more than once and, in one case, comes within a distance ε of an incarnation number. (The karma for this "hit" is the vertical distance from the incarnation number to the point b.) But, because the variation is relatively large, the corresponding karma is relatively large and is, therefore, inadmissible. For the maximal path, the karma is the distance from the incarnation number to the point a, which is relatively small. It will be noticed that at high levels of positive karma - where the graph becomes steep - the maximal path will also produce "hits" that are inadmissible, and explains why liberation becomes more difficult as the personality develops.
Of course, one may design a non-maximal path that scores direct "hits" or close "hits" such as the one below. But because this path is only one of an infinite number of non-maximal paths, the probability of this happening by chance is precisely zero.