A Mathematical Proof of Reincarnation

[Guest guest]

Reference http://www.hypatia-lovers.com/experiments/Section01.html

 

 

Khan's Infinite Binomial Probability Experiment

 

A Mathematical Proof of Reincarnation

 

 

 

 

 

 

 

— INTRODUCTION —

 

 

 

For time beyond measure, you were dead. No, more than just dead:

Nonexistent. You didn't mind, though — after all, there wasn't a you to

mind. Then, one day, just when you had really gotten the hang of

nonexistence, you were wrenched into existence, quite against your

will. Mercifully,

the sudden sensory bombardment and profound shock attending your birth would

quickly be forgotten, but it was a trauma you'd never quite get over — a

trauma you'd always have to block from conscious memory. The surprise,

outrage, and sheer terror of this violent transitional experience would

reverberate within your subconscious for the rest of your life.

 

 

 

 

 

As you grew older, you came to realize that you were really just an animal,

made of flesh, and that like all animals, you would one day die. Curiously,

this was not a comforting thought. Somehow, the certainty that you would

one day be going back home to nonexistence evoked the same visceral terror

that the birth experience had. Just as you had no say in whether or not you

would be born, so you will have no say in whether or not you will die.

 

 

 

So much for the preliminaries. The obvious questions that arise at this

point are:

 

1) Where were you before your conception (if anywhere)?

 

2) Where will you be after your death (again, if anywhere)?

 

And 3) Is life strictly a one-shot event, or is it a recurrent complaint?

 

 

 

 

 

The first two questions don't seem amenable to any sort of mathematical

sleight-of-mind so we will defer these questions to the realm of theology. Let

each person come to grips with these two questions in his/her own way:

either 1) By ignoring them, 2) By selecting the most comforting answers, or

3) By parroting the official dogma of one religion or another. We simply

can't know for sure where we were before our conception or where we will be

after our death (if anywhere). We can believe whatever we like, but

believing is quite different from knowing.

 

 

 

 

 

The question of whether or not life is a one-shot event is another matter,

however. This question does not enjoy the same immunity from mathematical

mischief as do the other questions. As one who finds it difficult to resist

the temptation to engage in the whimsical misuse of mathematics, and other

forms of tortured logic, I hereby propose a thought experiment which I am

calling Khan's Infinite Binomial Probability Experiment.

 

 

 

 

 

— THE PREMISES —

 

 

 

First, let us assume, for the sake of the argument, that before your

conception you did not exist. Let us further assume that events such as

your birth are governed by chance — this is to say, that events such as your

transition from nonexistence to existence are random events. If (and only

if) we make this assumption, we are allowed to use the theorems of

probability in our reasoning. If you find it difficult to swallow the

assumption that your birth was a " random event, " please bear with me, even

if only to follow this assumption (and it is only an assumption) to its

logical conclusion. Keep in mind, though, that entire mathematical fields

(such as probability and statistics) are based on the assumption that random

events can and do occur.

 

 

 

 

 

Beyond the assumption that your transition from nonexistence to existence

was a random event, we need only one further premise before embarking upon

the thought experiment: We must assume that Time is infinite. If correct

reasoning based upon these two premises leads to an obviously erroneous

conclusion, then we may reject at least one of the premises, although we may

not know which one. Our mathematical reasoning will consist of setting up a

Binomial Probability Experiment in an effort to determine precisely what the

odds are, of being born only once — and then dying, to stay dead forever.

 

 

 

 

 

 

 

— THE CRITERIA OF A BINOMIAL PROBABILITY EXPERIMENT —

 

 

 

 

 

1) There are n repeated trials of an event governed by the rules of

probability. In this probability experiment, we define the event as your

passage from nonexistence to existence.

 

 

 

2) Each trial has two and only two possible outcomes, usually

referred to as " success " or " failure " . In this probability experiment, we

will call your transition from nonexistence to existence a " success. "

 

 

 

3) The n repeated trials must be independent, i.e., the result of one

trial does not affect the probabilities of any other. In this probability

experiment, we will let n, the number of trials, equal infinity since time

is (for the moment) assumed to be infinite and circumstances might favor

your birth at any given instant within an infinite time continuum.

 

 

 

4) The probability of success on each trial (assumed to be constant)

is p, and the probability of failure on each trial is q = (1 - p). In this

thought experiment, we can't possibly know the probability of success on

each trial, but we do know that p is greater than zero, since you are

presumably alive as you read this, which implies that you have already made

the transition from nonexistence to existence once that we know of. Anything

that has already happened is possible, and therefore has a non-zero

probability of occurring. Therefore, we can say with complete confidence

that p > 0.

 

 

 

 

 

 

 

— THE " URN OF NONEXISTENCE " ANALOGY —

 

 

 

 

 

The Binomial Probability Experiment we have set up here is roughly analogous

to drawing balls at random from a very large urn. If a black ball is drawn,

this is analogous to a " success " (i.e., you make the transition from

nonexistence to existence.) If a white ball is drawn, this is analogous to

a " failure " (i.e., you do not come into existence.) After being drawn, each

ball is placed back into the " Urn of Nonexistence, " analogous to returning,

at death, to the nothingness whence you came.

 

 

 

 

 

Bear in mind that this analogy is a rather rough one, for these " drawings "

do not require the existence of an actual being that makes these drawings,

and who then violates conservation laws by making beings come into existence

out of nothingness. The underlying assumption here is that things, even

apparently orderly things, happen at random in the Universe by themselves,

controlled only by the laws of probability. Take, for instance, the

spontaneous decay of a Thorium-228 atom — nobody needs to make it decay, it

just decays all by itself when it's good and ready. In fact, nobody can

even predict when a given Thorium atom will decay. The best we can do is to

give a probability that it will decay within a given time-frame — a

probability contingent upon the number of other Thorium atoms nearby at the

time. In view of this " self-motivated " behavior of matter, it will not at

this point be necessary to infer the existence of a " higher being " whose job

it is to draw balls from the urn.

 

 

 

 

 

— THE QUESTION —

 

 

 

 

 

Given that the urn is known to contain at least one black ball among the

multitude of white balls, (and this is known because a black ball has

already been drawn from the urn) what is the probability of drawing a black

ball again if you keep drawing balls forever? It should be intuitively

clear that, as long as each ball is returned to the " Urn of Nonexistence, "

(perhaps a life-time after being drawn) the probability of drawing a black

ball again is a dead certainty, no matter how infinitesimal its chances of

being picked on a given draw may be. It would seem that, if a black ball

was drawn once and then put back in the urn, a black ball will certainly be

drawn again if you keep drawing long enough.

 

 

 

 

 

Although this conclusion may seem to be intuitively clear, if we employ the

Binomial Probability Function we can determine the precise numerical

probabilities associated with the black ball's being drawn any given number

of times. The Binomial Probability Function tells us the probability

*P(x)*that there will be exactly

*x* successes in n trials, given *p*, the probability of success in each

trial.

 

 

 

 

 

As a formula, the Binomial Probability Function can be written as:

 

 

 

[image: General Binomial Probability Function Equation, and Definition of

Variables.]

 

 

 

Although it is easy to use in more common problems, the Binomial Probability

Function becomes quite difficult to work with when the number of trials

becomes large, as it does in our probability experiment. Try evaluating the

limit of the Binomial Probability Function as n approaches infinity (holding

x constant) and you'll see the problems that arise as the number of trials,*n

*, goes to infinity. The fact that the mean of our " Infinite " Binomial

Probability Function lies at infinity serves to corroborate the conclusion

that the black ball would not be drawn from the " Urn of Nonexistence " only

once, but for further numerical analysis of this probability experiment, we

can escape the intractability of the Binomial Probability Function if we

turn instead to the Poisson Probability Function.

 

 

 

 

 

The Poisson Distribution is a discrete probability distribution which is, in

essence, the " rare event " special case of the Binomial Distribution, where

the probability (*p*) of an event is very small, but the number of trials (*

n*) is very large. As in Binomial Probability experiments, every trial in a

Poisson Probability experiment is an independent event with two possible

outcomes: " success " or " failure. " It is assumed that the product of *n* and

*p* is a constant, symbolized by the Greek letter, lambda. Thus, the

Poisson Distribution is the limiting case of the Binomial Distribution as n

approaches infinity and p approaches zero.

 

 

 

As a formula, the Poisson Probability Function can be written as:

 

[image: General Poisson Probability Function Equation, and Definition of

Variables.]

 

Recapitulating, we know that if time is infinite, then* n *(the number of

trials) will approach infinity; and if an event has happened, then it has a

non-zero probability of occurrence; and if n approaches infinity

while* p*(the probability of success on a given trial) is greater than

zero, then

lambda (the product of *n* and* p*) will also approach infinity.

 

 

 

 

 

Now let's see what the Poisson Probability Function tells us about your

chances of ever having come into existence. The probability of your making

the transition from nonexistence to existence zero times is:

 

 

 

[image: Evaluation of the Limit of Poisson Probability Function as the

Number of Trials Approaches Infinity, With a Non-Zero Probability of Success

on Each Trial. This Turns Out to be the Odds of Never Being Born.]

 

 

 

This would seem to signify that, if time is infinite, then there is a zero

probability that you would never be born (assuming* p* > 0). In other

words, the fact that it has already happened in effect guarantees that your

birth was inevitable — in other words, predestined by the Rules of

Probability.

 

 

 

 

 

Now let's see what the probability is, of your making the transition from

nonexistence to existence once and only once:

 

 

 

 

 

[image: Evaluation of the Poisson Probability Function as the Number of

Trials Goes to Infinity, With a Non-Zero Probability of Success on Each

Trial. This Turns Out to Be the Odds of Living Only Once, and Then Remaining

Dead Forevermore.]

 

 

 

Uh-oh, this seems to numerically confirm our intuitive reasoning from the

" Urn of Nonexistence " analogy. The Kafkaesque implication of this result

would seem to be that there is no chance of living only once, then dying and

having the decency to stay dead, forever. If time is infinite, it would

seem that life cannot possibly be a one-shot event.

 

 

 

 

 

If we continue evaluating the limits of the Poisson Probability Function

(for x = 2, 3, ... etc.) we find, to our horror, that no matter what finite

value of x is used in the Poisson Probability Function, as long as lambda

(the product of n and p) approaches positive infinity, the probability of

the event occurring a finite number of times appears to turn out to be zero!

In a way, this makes sense in the context of our " Urn of Nonexistence "

Analogy: if you keep drawing from the urn (with replacement) forever you're

going to end up drawing a black ball an unlimited number of times, even if

there's only one black ball per zillion white balls in the urn.

 

 

 

 

 

By the rules of probability, the summation of the probabilities of all

possible mutually exclusive outcomes must equal unity (i.e., 1), therefore,

since the probability of coming into existence any finite number of times

appears to be zero, we might reasonably conclude that the probability of

coming into existence an infinite number of times must be 1 (certainty). This

conclusion seems to be at least vaguely corroborated by the fact that the

means of both the Binomial and Poisson Probability distributions used for

our experiment are located at infinity.

 

 

 

 

 

Under this scenario, once you die you simply return to the " Urn of

Nonexistence " for an unknowable — perhaps unimaginable — amount of time

before becoming existent again. Since we can assume that while you're in

the nonexistent state you don't experience the passage of time (as there is

no you to experience this), your next apparition as an extant being must

seem to occur instantaneously after your last (i.e., most recent) death. From

your point of view, anyway.

 

Although this conclusion seems to be in keeping with the Law of Conservation

of Absurdity, it is nonetheless a disquieting thought. What is worse, this

line of reasoning can be applied with similar results to the Universe's

transition from nonexistence to existence, or (by a somewhat circular

argument) to the origin of Time itself.

 

 

 

 

 

Having thus used statistics (the " Mathematics of Ignorance " ) to rush in

where wise men fear to tread, we can add one final outrage to the list of

logical atrocities by summarizing the result of Khan's Infinite Binomial

Probability Experiment with *KHAN'S LAW: " Anything that did happen,

canhappen, and anything that can

happen, will happen, given enough time. " *

 

* *

 

* *

 

— CLOSING NOTE —

 

 

The preceding use of mathematics in an effort to search for the answers to

ontological questions was first published in the April 1990 Issue (# 108) of

Vidya, The Journal of the Triple Nine Society. A refutation is yet to

emerge. It is submitted here for the amusement, bewilderment, or annoyance

of the intelligentsia in the hopes that some better mathematician than its

originator (Khan Amore) will be able to put a hole below the water-line of

this unfortunate line of reasoning. Won't somebody please disprove this

argument? One existence is more than enough, thank you!

 

 

--

Haridev S V

 

 

इनà¥à¤¦à¥à¤°à¤‚ मितà¥à¤°à¤‚

वरà¥à¤£à¤®à¤—à¥à¤¨à¤¿à¤®à¤¾à¤¹à¥à¤°à¤¥à¥‹ दिवà¥à¤¯à¤ƒ स

सà¥à¤ªà¤°à¥à¤£à¥‹ गरà¥à¤¤à¥à¤®à¤¾à¤¨ |

à¤à¤•à¤‚ सद विपà¥à¤°à¤¾ बहà¥à¤§à¤¾

वदनà¥à¤¤à¥à¤¯à¤—à¥à¤¨à¤¿à¤‚ यमं

मातरिशà¥à¤µà¤¾à¤¨à¤®à¤¾à¤¹à¥à¤ƒ ||