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Notice!
This paper is the second in a series. In order to understand this paper, the reader should read the
first paper: Parapsychology? Or is it Paraphysics?, www. paraphysicstoday .org. – R.F.R.
.
Pythia, Troi and the Arrow of Time
Two Realms
Commander Data and Mister Spock can try forever, but they will never be able to develop a
bullet-proof model that will ‘explain’ the results obtained at the game of psychicards. To
understand why this is so, we must realize that the realm that Data and Spock can access is not
the same realm that Pythia and Troi can access.
The realm that Data can access is a sequence of actual countable events. By countability, we
mean that the sensations that Data experiences can, in theory, be stored at discrete,
distinguishable addresses in Data’s memory bank. But to be countable, the events must actually
exist. Data cannot count events that have not occurred. Even if a visual event reaches Data’s
sensorium from a phenomenon one-foot away, it will have occurred about 1-nanosecond ago.
Auditory phenomena 1-foot away will have occurred about 1-millisecond ago. Smells will have
much longer delays. To these delays we must add the delay times required to process the
information in Data’s sensorium and then to relay the information on to Mister Spock in a
meaningful format. Thus we see that Commander Data can only access the past, never the
future. Past events are actua and are countable. Future events are potentia and are uncountable.
To better understand the distinction between actua and potentia, consider a game of coin tossing.
To evaluate the Data-Spock team’s skill in prognostication, Data has been ordered by Captain
Kirk to toss an honest coin a large number of times and for Spock to predict each outcome as
either a Head or a Tail. Before each toss, the potential of a Head is one; the potential of a Tail is
also one; the potential of both a Head and a Tail is zero; and the potential of neither a Head nor
a Tail is also zero. In order to test their skill at prognostication, Spock has been ordered to
register the outcomes of his predictions in his memory bank -- the successful Hits as ones, and
the failed Misses as zeroes. After each toss there is only one outcome -- either a success or a
failure; the four potentia have engendered one actua, one bit of information. The actual bit has
become a fait accompli that cannot be undone.<1>
It is intuitively obvious that the potential outcomes of future tosses of the coin have not been
altered by previous outcomes. The next toss of Data’s coin will still be either a Head or a Tail
regardless of past outcomes. Thus the potentia are uncountable while the actua are
countable.<2>
You might wonder whether Commander Data can be allowed to count events that Spock ‘knows
for certain’ will be observed in future tosses. The answer is No. Even if Data has achieved 20successful hits in succession, Spock cannot deduce that Data’s next gamble will result in a hit.
Unless Data’s sensations are filtered somehow, his performance as a prognosticator will
approach a dead heat between 50% Hits and 50% Misses. An outside observer might be able to
say that a faithfully repeated ‘cause-effect’ relationship in past observations lends a degree of
confidence that the same cause-effect relationship will be observed in future observations, but
neither Spock nor Data can ever be confident enough to place the mantle of certainty on it. The
same applies to Spock’s logical deductions since the premises of any calculus must be absolutely
True if the conclusions are to be absolutely True.
Assuming that Commander Data is the sole source of information, what can Mister Spock do
with it? Like Data, Spock has no access to the future. Mister Spock’s brain is actually a
computer that strictly applies the rules of deductive (‘Boolean’) logic. If Spock has been fed
garbage by Data, Spock can only transmit garbage to Kirk. But if Spock has been suitably
programmed, there are many items that Spock can deduce from Data’s information that may
prove useful to Captain Kirk -- items such as the likelihood that Commander Data has detected
an enemy intruder, the likelihood that the enemy has detected the presence of the Starship
Enterprise, the optimal strategy of attack or retreat, and so on.
Without a useful program, Spock’s brain serves no useful function on the Starship Enterprise. In
order to write a program for Spock’s brain there must be some perception of a past cause
producing a future effect. Even if Spock’s brain can be programmed to note and report oftrepeated sequences, there is no getting around the fact that inductive reasoning is, at best, only
the statistics of past events projected onto future expectations, and never certain at predicting the
future. <3>
Pocketless pool
After their failure to impress Captain Kirk at predicting the outcomes at coin tossing, Data and
Spock have devised a new game that they have named “pocketless pool.” Pocketless pool is
different from pocket billiards in that it is to be played on a standard billiard table without
pockets. The challenge of this game is for Data and Spock to skillfully utilize the physical laws
of mechanical dynamics to achieve a correctly ordered set of strikes of the fifteen numbered
balls. Data will operate the cue stick while Mister Spock will compute the optimal strategy of
Data’s plays. Mister Spock has been programmed with the well established physical laws of
mechanical dynamics, and, since Data is a machine that also obeys the laws of mechanical
dynamics, Spock ‘knows’ ahead of time what Data’s optimal play will be in the future positions
of the balls. Spock’s initial prediction to Captain Kirk is that Data will strike every one of the
15-balls in correct sequence in exactly15-plays. (Spock is an optimist!)
The game starts out with the fifteen numbered balls carefully ordered and precisely placed in a
triangular array at one end of the game table with the cue ball precisely placed at the other end of
the table. The first strike of Data’s cue ball hits the 1-ball dead on with a perfect break. The 2ball and the 3-ball stop at symmetrical positions on the left and right sides of the table. So far, so
good. Data is able to strike the 2-ball and the 3-ball without difficulty. But then something
unforseen by Spock begins to happen. The beautiful symmetry of Data’s early strikes begins to
erode. Soon the balls roll to unpredicted locations on the table making Data’s shots more and
more difficult. Then Data finds the 5-ball behind the 8-ball without a possible bank shot. Data
soon encounters other impossible shots. The game of pocketless pool degenerates into chaos.
Since Data and Spock are incompetent at predicting the future, Kirk must turn to his two female
counselors, Pythia and Troi to do the job. The only way that Pythia and Troi can do a better job
than Data and Spock is to access the realm of potentia. This leads to another question: “How can
Pythia and Troi do it?”
A Model of Pythia and Troi
All models created by human beings have been, and always will be, incomplete.<4> This is
because humanly devised models are described in terms that Mister Spock can express. Spock’s
models are designed to be valid in the realm of countable actua, while the realm of potentia is
uncountable and inaccessible to Spock. Models of Pythia and Troi will fail for this same reason,
and the following model will fail to represent all the realities that Pythia and Troi can experience.
But it is a start, hopefully in the right direction, that can be improved upon by future explorers
into the unknowable
We begin by resorting to quantum physics where it is a well established fact that potentia behave
as waves, very much like the waves in a swimming pool. Physicists and engineers perform
calculations on these waves using the rules of wave mechanics to determine the probable
outcomes of specific hardware arrangements. For example, electrons passing through slots on
their way to a flourescent screen are first computed as if they were interfering waves of potentia.
The resulting potentia are then converted into waves of probability before they impact the screen.
Once they hit the screen, they become actual particles and become countable. The outcomes of
these experiments, in turn, are used by scientists and engineers to design improved models for
describing quantum behavior and thereby enhancing the realm of potentia.
Now even though the distinction between potentia and actua is evident at the quantum level, one
must not assume that the same distinction fails at the human level. Take sound, for instance.
Sound waves behave as potentia in the atmosphere but are sensed as actua by Commander Data.
Once they impinge on his ear drum they become particles of action called phonons. If the
phonons are mathematically related they are declared ‘harmonic’ by Mister Spock. If the various
harmonics are determined to be mathematically related by Mister Spock, they become ‘chords.’
If Counselor Pythia perceives a ‘meaning’ in the precession and timing of the chords, she
declares it to be ‘music.’ If Troi likes the feeling conveyed by the music, she declares it to be
‘good’ and relays her opinion on to Captain Kirk. The feedback of what ‘sounds good’ to Troi,
results in better musical instruments, better composers, better players and singers. Thereby the
realm of potentia is enhanced by the process.
Indeed, we are surrounded by a realm of potentia that impacts the realm of actua, and vice
versa. The DNA that we carry with us from the womb to the grave is another example of
potentia made actua. Our parents walked and danced around on the earth carrying the potentia of
our existence as viable human beings. The DNA resulting from their sexual union was the
progenitor of our proteins and enzymes that interface the realm of actua. In turn, the experience
of actua determines which DNA codes are capable of survival in the college of hard knocks. Its
survival determines the potentia of the next generation. The mutual feedback between the two
realms of potentia and actua is the driving force of the upward evolutionary trajectory of life.
The ubiquity and importance of potentia and actua at all levels of the human experience cries out
for us to give it a special place in our lexicon. Many names are possible for denoting this
universality, but in order to stress its antiquity and influence throughout all the cultural systems
and ages of the evolutionary process, let us label the realm of all possible potentia the Realm of
Yin, and let us label the realm of all possible actua the Realm of Yang. By using these ancient
labels we intend that the two Realms are not to be restricted to any particular domain of human
enquiry, whether it be the domain of physics, of biology, of psychology, of sociology, of history,
of psychicards, whatever.
In the game of psychicards, we have decided that Pythia must somehow have access to the
Realm of Yin if she is to be of any help on the bridge of the Enterprise. From the discoveries of
quantum physics we infer that potentia are wave-like and Pythia can only ‘understand’ potentia
by dealing with them as waves.
If we are correct in our inference that the potentia are wavelike, we must infer that there is a
distinct waveform associated with each of the four suits. It may be impossible to fully express
these waveforms in mathematical terms usable by Spock, but it is not too farfetched to assume
that Pythia perceives her reality in terms of waveforms. She must ascribe a specific waveform to
the color black, and another to the color red. Similarly she must ascribe a specific waveform to
the shape of spades, another to hearts, and so on. Combinations of the two waveforms of color
and shape define the four suits. Pythia must have been permitted to ascribe these distinctions
before the game is played in order to distinguish between the suits.
After the deck has been well shuffled, Captain Kirk picks up the first card, face down. His
handling of the card results in an effect known in physics as quantum entanglement or
‘quantanglement.’ This effect, though bizarre and counterintuitive, is known to be real and has
been verified experimentally with microscopic quantum objects. Basically, the phenomenon of
quantanglement is manifested when two quantum objects share the same event in time and space.
When this happens, both objects share the same information of that interaction forever. The
knowledge of the state of one of the objects guarantees that the state of the other object is also
known, instantaneously and independent of the distance between the two quantum objects.
Another strange aspect of quantanglement is that until the state of either object is observed, both
objects appear to be in random states quite independently of each other. The fact that mere
observation of the state of one of the quantum objects seems to ‘fix’ the state of the other, lends
an almost ‘mystical’ aspect to the act of ‘observation.’ This is one of the great problems in
quantum physics that has not been resolved by quantum theorists. Quantanglement is sometimes
called the “twin-state” because of its analogy with the observed parallel behavior patterns of
human identical twins. The analogy is not quite accurate, but the term “twin-state” relays some
of the ‘mystical’ attributes of quantanglement.
Another perplexing question in quantum physics is: “Does quantum physics operate at all levels
of the observable universe?.” Quantum theorists believe that quantum physics is operational at
all levels, but at the macroscopic (= large object) level, some of the effects observable at the
microscopic (= small object) level become “decohered” and cease to be observable. We will not
discuss this perplexing problem further, but in this paper we will agree with the quantum
theorists that quantum physics does operate at all levels. We also agree that the human being is a
quantum object, albeit a very complex one. We go one step further and propose that quantum
physics is operable at all levels both in the Realm of Yin and in the Realm of Yang. In the YinRealm, we propose that potentia are ‘invisible’, continuous waveforms, while in the YangRealm, we propose that actua are ‘visible’, countable events.
Getting back to Kirk’s first draw in psychicards, we see that its suit has both a Yin aspect and a
Yang aspect. But since Data cannot physically observe the other side of the card and Spock
cannot deduce the outcome of a random draw, Kirk must now rely on Pythia and Troi to tell him
which suit it is. Kirk first creates a potential choice and asks Pythia and Troi: “If I were to
actually place this card on the Ace of Spades, what would be the actual outcome?”
Counselor Troi feels that Captain Kirk has bonded with the card, and therefore the card has
become quantangled with Kirk. If Kirk decides to place the card on the correct Ace, its actua
will be called a successful hit. If he decides to place the card on the wrong suit, its actua will be
called a failed miss. It is up to Pythia and Troi to advise Kirk what the actual outcome will be if
he actually places the card on the Ace of Spades.
Pythia is next to be consulted at the conference table. Pythia lives in the Realm of Yin and
perceives the card’s suit as a waveform, invisible to Data or Spock. She first combines the
waveform she calls ‘Spades’ with the waveform of the card that Kirk holds in his hand. The
result is a combination of the two waveforms. Pythia transfers the combined waveform to Troi
who must decide whether the combined waveform is ‘good’ or ‘bad.’ If Troi ‘likes the sound of
it’ she perceives ‘resonance’ in the combined waveform she pronounces Kirk’s potential play as
‘good’ and advises Kirk to make his potential play an actual play.’ If Troi perceives
‘dissonance’ she pronounces Kirk’s potential play as ‘bad’ and advises Kirk not to make it an
actual play. The routine is iterated until all 48 cards are played by Kirk and his potential plays
have become actual plays.<5>
Finally, Kirk orders Spock to make an evaluation of the Pythia-Troi team’s ability to
prognosticate. Spock’s brain has been programmed to compute the odds against the successes
actually achieved by Pythia and Troi with those of the team of Data and Spock. Regretfully,
Data and Spock must concede that Pythia and Troi have achieved a much more surprising set of
statistics.
The Second Law and The Arrow of Time
There is only one well-established law of physics that restricts the Arrow of Time to the direction
of Present > Past. Intuitive time seems to flow in this direction, and yet the laws of physics are
invariant to the direction of time except for the one physical law that is called the Second Law of
Thermodynamics. In this paper we will shorten the name to “The Second Law.”
Stated in layman’s terms, The Second Law asserts that as time flows in the direction Present >
Past, all isolated physical systems tend from a less chaotic system towards a more chaotic
system. The mathematics behind this Law are far too subtle and complex to be discussed here,
but it may be understandable by reconsidering Data and Spock’s game of pocketless pool.
The Second Law is a law about the statistics of actua. In the Yang-Realm of countable events it
is necessary to decide which actual events are distinguishable from other actual events. Once it
has been decided that event-A is distinguishable from event-B, it is possible to assign a unique
number to event-A and another unique number to event-B. In Spock’s computer-brain there has
to be a device called a ‘RAM’ (Random Access Memory) for storing the numbered events and
assigning an address location to each event. Once done, the two events can be accessed and used
in Spock’s computation algorithm for predicting Data’s best future plays and their future
outcomes.
Before Data’s game of pocketless pool, for example, the subset of RAM defining the locations
of the fifteen balls on the billiard table required the definition of an imaginary grid on the surface
of the table marked off in the minimum distances that are discernable and usable by Data. The
engineering decision involved a mix of physical measurements and guesswork. A rough guess
would be, say, a spacing of 1-millimeter between parallels of the imaginary grid, resulting in a
requirement of about 4-million squares 1-millimeter on each side. Thus the number of
addressable locations in the RAM needed to define the meaningful positions of the balls on a
standard billiard table would be roughly 4-million. Similar subsystems of RAM are required to
store possible future events such as the momenta (speeds and directions) of the struck balls, the
‘english’ (angular momenta) on the struck balls, and so on. One can see how the required
memory space builds up pretty fast, quickly reaching the limits of what is feasible on a laptop.
But more to the point, there will always be a limit to the accuracy of the prediction of each strike
by Commander Data. Not only is there a finite limit to the precision of Spock’s computations,
there will always be factors that weren’t programmed into Spock’s brain or which are beyond
Data’s control, factors such as microscopic irregularities in the surface of the billiard table,
variable air drag, floor vibrations, errors in table leveling, and-so-on. The errors in the predicted
positions of the balls are carried forward and increased from strike to strike until the strategy for
successful plays on the billiard table becomes incomputable.
But, one might ask: “Suppose we imagine perfect knowledge of mechanical dynamics, perfect
precision in Spock’s computations, a perfect table, perfect balls, a perfect cue, perfect isolation
from outside forces .... . Could we argue under these conditions that The Second Law would not
prevail over the game?”
Yes, if such perfect conditions could exist in the Realm of Yang, perhaps Data could play a
perfect game. The problem is that in the Realm of Yang, everything is countable, even in the
limiting case of a ball’s position and momentum. Since Planck’s discovery of the countable
quanta of action, we now know that there is a tradeoff between the knowable precision of a ball’s
position and the knowable precision of its momentum.<6> If Spock could ever know the
position of a ball perfectly, he would know nothing of its momentum. Similarly, if he could ever
know its momentum perfectly, he would not know where it is. Once Data plays a ball in the
Realm of Yang, every action becomes countable and every countable action has its limits of
precision.
Time Reversal
Let us imagine that there is a motion picture camera over the billiard table that records the
progress of Data and Spock’s game of pocketless pool in ‘normal’ time. If the motion picture is
viewed in its normal forward mode, it will be evident that the positions of the balls do, indeed,
proceed from Order > Chaos with the normal directional flow of time Present > Past. But if the
motion picture is played in reversed time mode (Past > Present), the positions of the balls appear
to evolve from Chaos to Order, finally arriving at the well-ordered triangular array at one end of
the table.
The success of the Pythia-Troi team can be interpreted as a reversal of ‘The Arrow of Time.’
Contrary to the game of pocketless pool, Pythia and Troi start out with a game that is initially
chaotic, a well randomized deck of 48 cards, totally opposite in principle to the initialization of
the game of pocketless pool. Again, the challenge is to overcome the chaos of the system, but
here there are no physical laws of mechanical dynamics to aid Pythia and Troi. The motion
picture camera over the card table displays the evolution of the game proceeding from Past
Chaos to Present Order when viewed in its normal time mode Present > Past. The motion
picture must be run in reversed time mode (Past > Present) to display the ‘normal’ evolution
predicted by the Second Law from Order to Chaos. It is in this sense that Pythia and Troi are
able to reverse the Arrow of Time.
The squeamish reader may find the notion of time reversal to be beyond the boundary of belief.
In fact, there is no known a priori reason to reject the notion. The reality of time has been the
subject of profound philosophical and scientific debate for thousands of years, and is still hotly
debated. Some of the great thinkers who have tackled the problem of time include Plato,
Augustine, Newton, Kant, Maxwell, Minkowski, Einstein, Bergson, Heidegger, Gödel, ..., the list
goes on. In the fourth century Augustine argued persuasively that countable, measured time is
not the same as intuitive time.<7> Seventeenth century scientists like Isaac Newton argued
persuasively that time is absolute and countable, that the hands on a proper clock will read the
same on Earth as on Mercury. In the twentieth century Einstein proved mathematically that
Newton’s theory is blatantly false, and modern experimental evidence has proven Einstein to be
correct – that time counted by a proper clock will read differently at different locations in the
universe. Not only is time not absolute, its measured value depends upon the platform from
which it is measured; it can slow down, even crawl towards zero when certain physical
conditions are met. The latest round in the debate over the ultimate reality of time was fired by
Einstein’s close friend -- Kurt Gödel. (Kurt Gödel is the same genius who proved that every
model that can be fed into Mister Spock’s computer-brain will be incomplete.) Starting with
Einstein’s famous field equation, Gödel proved mathematically that the Arrow of Time can be
reversed in Einstein’s physical universe by a space craft with finite, albeit economically
prohibitive, thrust. Gödel’s conclusion was that countable time as measured by Einstein’s proper
clock results in a paradox and therefore the sort of time envisioned by Minkowski and Einstein
as a ‘fourth dimension’ does not exist. At a minimum, Gödel’s argument about time proves that
Augustine was right – measured, countable time is not the same as intuitive time.
APPENDIX
A Walk in the Valley of Terra Incognita
The distinction between the countable and the uncountable was explored to great depth
by the mathematician Georg Cantor in the late 19th century. By countability, Cantor meant that
the elements in a countable set could, in theory, be placed in a one-to-one correspondence with
some or all of the cardinal numbers 1, 2, 3, ..., ... . Cantor then proved that there are a host of
transfinite (a.k.a. infinite) numbers that are uncountable, one of which will be discussed more
fully in later paragraphs.
Our use of the words ‘countable’ and ‘uncountable’ refer obliquely to the same concepts
explored by Cantor, but we rely heavily on intuition to distinguish between the two concepts.
We may have to revisit the problem in future discussions, but for now, we argue that all events in
the realm of actua are countable simply because they are distinguishable. As examples:
theoretical physicists claim that there are approximately 10E+34 distinguishable increments in
one meter, and approximately 10E+43 distinguishable increments in one second; cosmologists
believe that there are about 10E+79 distinguishable particles in the observable universe and there
have been about 10E+121 distinguishable interactions between those particles since the Big
Bang.
Further, we argue that the realm of potentia is uncountable since the realization of an
event does not alter the efficacy of the potentia that will permit it to recur. We are arguing, for
example, that in the realm of potentia the ideal meter rod has an uncountable number of
increments, but when we attempt to realize the ideal meter rod within the realm of actua we find
that there are only 10E+34 countable, distinguishable increments on the rod. For whatever
reason, all other increments on the rod are indistinguishable and are, therefore, forbidden
increments.
The smallest transfinite (a.k.a. infinite) set is the open set of countable cardinal numbers
[1]
{N = {1,2,3,....
For convenience, it is said that any set of objects, events, etc that can be placed in a 1-to1 correspondence with the set {N has the cardinality k{N.
The first uncountable transfinite set discovered by Cantor is the open set of positive real
numbers {R. We will say that any set of objects, events, etc that can be placed in a 1-to-1
correspondence with the set {R has the cardinality k{R.
This concept will become clearer when we consider the number of potential outcomes in
the game of psychicards. For now we point out that the cardinal numbers are a subset of the real
numbers.
An example of the open set {R is the set of increments in the ideal meter. It has
countable increments corresponding to decimeters, centimeters, millimeters, etc. But between
each countable increment, there is an uncountable transfinite number of increments.
[H]
Hypothesis: Pythia has access to an uncountable set of potentia.
To prove the hypothesis, consider the following argument:
We first consider the game of psychicards in which the deck is 48 cards. If Pythia and
Troi were able to achieve a perfect score with a deck of 48 cards, there would still be a very
small probability that they may have achieved their success through some freak accident. The
odds against 48 successes in 48 trials has already been computed and is exhibited as the last
entry in Table One of Parapsychology? Or is it Paraphysics?. Although the probability
(1/odds) of achieving perfect success is unimaginably small (of the order of 10E-30 ), yet it is not
identically zero. In order to prove the hypothesis to be absolutely True, we have to imagine a
game in which there are a countably infinite number of cards in the deck and which can be
played a countably infinite number of times. The set of plays is a countable array of dimensions
{N x {N.
Such an array exists only in imagination, but we can prove the hypothesis by considering
a small subset of the array, say 5 plays wide and 5 shuffles deep.
Let:
S = Spades
H = Hearts
D = Diamonds
C = Clubs
The first five cards and shuffles (actually drawn from the 5 top cards of 5 shuffles) might
look like this:
[2]
Card:
{1 2 3 4 5 . . . . .
__________________________
|Shuffle (1): {S C C H S . . . . .
|Shuffle (2): {H S C C D . . . . .
|Shuffle (3): {C C H S H . . . . .
|Shuffle (4): {D D D S C . . . . .
|Shuffle (5): {D H H S C . . . . .
|Shuffle (. ): {. . . . . . . . . . . . . . . . . .
where each random shuffle is represented by a countable row of dimension {N and there are {N
countable columns. It is easy to show that the total number of elements among the {N rows in
[2] are also countable and therefore the totality of shuffles is still of cardinality k{N.
Now if Kirk plays the cards without listening to the advice of Pythia and Troi, one fourth
of the cards will, nonetheless, end up in the correct stacks. We denote the incorrectly placed
cards with parentheses ( ), and the randomly correct plays without parentheses. The new array
with one fourth of the cards correctly played by accident might look something like this:
[3]
Card:
{1 2 3 4 5 . . . . .
___________________________
|Shuffle (1): {(S) (C) C (H) (S). . . . .
|Shuffle (2): {(H) (S) C (C) (D) . . . . .
|Shuffle (3): {(C) (C) (H) S H. . . . .
|Shuffle (4): { D (D) (D) (S) C. . . . .
|Shuffle (5): {(D) H (H) (S) (C). . . . .
|Shuffle (. ): {. . . . . . . . . . . . . . . . . . . .
The next step is to remove all cards from the array that Kirk correctly plays by accident. .
We are justified in doing this because Pythia and Troi will not change those plays. The new
array may look like this (after the fifth column is completed):
[4]
Card:
{1 2 3 4 5 . . . . .
____________________________
|Shuffle (1): {(S) (C) (H) (S) (H). . . . .
|Shuffle (2): {(H) (S) (C) (D) (S). . . . .
|Shuffle (3): {(C) (C) (H) (C) (S). . . . .
|Shuffle (4): {(D) (D) (S) (C) (D) . . . .
|Shuffle (5): {(D) (H) (S) (C) (C). . . . .
|Shuffle (. ): {( . ) ( . ) ( .) ( . ) ( . ) . . . . .
The rows of array [4] represent all of the shuffles in the never-ending game of
psychicards that Pythia and Troi must identify. The totality of the elements in the rows of array
[4] are still countable of cardinality k{N.
After Pythia and Troi make their recommendations to Kirk, and provided Kirk follows
their advice, all of the parentheses disappear in the array and are different from the plays that
Kirk would have made without their advice. The resulting array will be something like this:
[5]
Card:
{1 2 3 4 5 . . . .
____________________________
|Shuffle (1): {S C H S H. . . . .
|Shuffle (2): {H S C D S. . . . .
|Shuffle (3): {C C H C S. . . . .
|Shuffle (4): {D D S C D . . . .
|Shuffle (5): {D H S C C. . . . .
|
.
{. . . . . . . . . . . . . . . .
Following in the footsteps of the great Georg Cantor, all that is required is to show that
the number of infinite sets of random numbers among the rows of array [5] is incomplete. We
have postulated that the rows and columns are unlimited in number and the game can go on
forever. But regardless of how many games are played, the elements in the rows of array [5] are
still countable. If we can find just one random sequence that is not included in the rows of [5],
then the array is incomplete.
Consider the main diagonal of the array [5]. We take the first card in Shuffle (1), the
second card in Shuffle (2), the third card in Shuffle (3), the fourth card in Shuffle (4), the fifth
card in Shuffle (5), and so on. After Pythia and Troi make their recommendations and Kirk has
the good sense to follow their advice, this particular diagonal subset becomes the open set
[6]
{S S H C C . . .
The next step is to change each of the elements in [6] to some other suit. In order to
maintain the equality of the number of cards in each of the four suits, we change the suits in the
diagonal set [6] using the process
[7]
S >> H
H >> C
C >> D
D >> S
The result of this transformation in [6] is
[8]
{H H C D D . . . .
By inspection, one sees that even in this small, five-by-five subset of [5], the diagonal
subset [8] is not a member of any row in the countable set [5]. Indeed, regardless of the number
of shuffles infinitely long, the open set [8] will differ from the countable rows in at least one
place. Therefore, the infinite set of plays that Pythia and Troi may encounter and identify is not
included in the countable set of cardinality k{N. This completes the proof of the hypothesis [H].
Now suppose that Spock attempts to ‘fix’ array [5] by placing the open set [8] at the top
of the array [5]. The newly ‘fixed’ 5-by-5 array will look like this:
[9]
Card:
{1 2 3 4 5 . . . .
____________________________
|Fix (1):
{H H C D D . . . .
|Shuffle (1): {S C H S H. . . . .
|Shuffle (2): {H S C D S. . . . .
|Shuffle (3): {C C H C S. . . . .
|Shuffle (4): {D D S C D . . . .
|
.
{. . . . . . . . . . . . . . . .
We can see that there is a new open set along the diagonal with the elements
[10]
{H C C C D . . .
Repeating the procedure [7] of changing each element of [10] we generate a new set
[11]
{C D D D S . . .
Tough luck, Spock! The newly discovered set is not among the rows of array [9] either.
The only thing Spock has achieved by trying to ‘fix’ the array is to enlarge the number of
uncountable diagonal sets. The array can be ‘fixed’ an infinite number of times and Spock will
suffer the same frustration.
Now suppose that Spock has discovered a sequence of cards that seems to imply some
‘cohesion’ between them and induces that the next card will be of the same suit. This would be
the case, say, that the first 20 cards in Shuffle 1 were all Clubs. Spock reasons:
“There have been twenty Clubs in a row. This would be nearly impossible if the events
are random. Therefore the next drawn card will also be a Club.”
It is easy to see the fallacy in Spock’s reasoning. There is simply no way for Spock to
predict the outcome of the next draw of something that is basically random. There is no
cohesion in a sequence of random events. The very next drawn card has 3 chances in 4 of being
something other than a Club. Again, tough luck, Spock!
On the other hand, Pythia has access to the realm of the uncountable. We conjecture that
the uncountable realm is Plato’s Ideal Realm of Yin in which all colors, shapes and sounds are
eternally beautiful, meter sticks are eternally straight, and triangles have angles that always add
up to 180-degrees. Pythia is able to explore this Ideal Realm of Yin and to discover eternal
realities that serve as tools for the advancement of the arts and sciences in the countable Realm
of Yang. Each new advance in the countable Realm of Yang results in the demand for new
discoveries in the Realm of Yin, and Pythia must be asked to take yet another walk in the Valley
of Terra Incognita.
Notes
<1> The Pythia (Oracle) of Delphi made a distinction between future events that could be
prophesied and those that could not be. Events that could not be prophesied, were still under the
aegis of the goddess of Fortune (Tyche). In contrast, events that could be prophesied were under
the aegis of the goddess of Fate (Ananke). It is interesting that when the event was still under the
aegis of Fortune, the Pythia would be reluctant to prophesy an outcome. If the client persisted,
she might utter an ambiguous prophecy that contained all possible outcomes. One can imagine a
game of coin tossing in which the coin is truly balanced and the toss is truly random. In that
case, the Pythia might answer the persistent client with “The outcome will be both a success and
a failure.” On the other hand, once the coin has been tossed into the air, the event is under the
control of Fate and only one outcome is possible.
<2> See Appendix.
<3> One important assumption of physics is the theory of ‘locality.’ By this theory we are
encouraged to induce physical theories and to perform experiments that are demonstrably valid
in the immediate locality of the observer, and then to extrapolate the validity of those theories to
non-local, unreachable regions of the universe. Certain experiments in quantum physics
apparently violate the principle of locality, especially the phenomenon called ‘quantumentanglement’ (a.k.a. ‘quantanglement’) that will be discussed at the appropriate place. The
impossibility of logically ‘True’ induction was pointed out by David Hume in his Treatise on
Human Nature and explored in depth by Immanuel Kant in his Critique of Pure Reason.
One can distinguish local inductions from non-local inductions depending upon the
domain for which the induction is considered ‘valid.’ For example, inductions in the domain of
physics attempt to define laws that are non-local and applicable anywhere, anytime in the
observable universe, while inductions in the domain of anthropology attempt to define patterns
of human behavior that are localized to specific cultures and stages of evolution.
<4> This follows from the ‘incompleteness theorem’ proved by the master logician Kurt Gödel
in 1931. Relatively few mathematicians are able to follow this profound proof, but no one has
been able to refute it. In effect, Gödel proved that there will always be conjectures that are
intuitively True but which can never be demonstrated to be True using any finite set of axioms.
As an example, consider Goldbach’s Conjecture, to wit: ‘every even number greater than 2 is the
sum of two different prime numbers.’ (That is, 4 = 1 + 3; 6 = 1 + 5; 8 = 3 + 5; etc.) Apparently,
this conjecture is not demonstrably True using the standard axioms of arithmetic but is accepted
as ‘intuitively True.’ According to Gödel’s Proof, any attempt to make Goldbach’s Conjecture
demonstrably True by enlarging the set of axioms will introduce further conjectures that are
intuitively True but which are not demonstrably True.
Now any model of Reality that Spock can deal with will fall into the incompleteness trap
discovered by Gödel. Spock will be able to derive many demonstrably True results of any set of
true postulates. To this extent, the model that Spock’s brain is programmed to handle will
correspond to ‘Truth.’
But in one unmapped quarter of the galaxy, Data detects an anomalous event that is
beyond Spock’s existing capabilities to handle. Spock’s attempt to make predictions based on
this anomalous event engenders Chaos on the bridge of the Enterprise. The Chaos on the bridge
is initially perceived as an inexcusable lapse of discipline by Captain Kirk. But on hindsight,
Kirk perceives the incident as an Opportunity to add computing power to Spock’s brain to make
him more effective in future encounters. The ambitious Kirk decides not to tender his
resignation right away and orders Counselors Pythia and Troi to help Spock come up with an
improved model of Reality. Pythia enters the realm of potentia to ‘record the sounds’ of various
program models when confronted with the anomalous event observed by Data. Pythia reports
the sounds of the various models to Troi who decides that certain models sound beautiful while
others are disharmonious claptrap. Troi relays those models that have Beauty on to Spock, who
decides which models, if any, are consistent with those parts of his brain that he already ‘knows
to be True.’ The process is iterated until a model is found that is both Beautiful and more
Truthful than the older model. Kirk says OK to the improved model, orders Spock to be
reprogrammed, and somehow Kirk forgets to report the inexcusable lapse of discipline to
Admiral Striker back at Starbase C3.
<5> Here may be an opportunity to investigate the exact moment of Pythia’s comparison of
waveforms and Troi’s pronouncement of good or bad. The experiments of Dr. Katherine
Benziger indicate that certain sections of the human cortex ‘light up’ on the MRI screen when
either Data, Spock, Pythia or Troi are active. It might be another worthwhile experiment to use
the services of a hypnotist to detect the mystic moment of quantanglement between Kirk and his
drawn card.
<6> Heisenberg uncertainty is not limited to the microscopic level. Radar and sonar engineers
routinely encounter this problem at the macroscopic level in the design of transmitted waveforms
and the reception of echos. The ideal transmitted waveform for the determination of a target’s
velocity is continuous wave of infinite duration. The ideal transmitted waveform for the
determination of a target’s position is a single impulse of zero duration. Any other waveforms
result in ambiguities of both speed and position of the target. You might find this fact useful the
next time you get a speeding ticket.
<7> “It is in my Psyche that I measure time... . The impressions that things make upon my
Psyche as they pass from the present moment into the past are what I measure. ... I might even
say that I do not measure time at all.” - Augustine, Confessions, 11:23
Further Reading
Physics, Transfinite Numbers, Quantanglement
Roger Penrose, The Road to Reality, Vintage Books, NY, 2007
Time
Palle Yourgrau, A World Without Time, The Forgotten Legacy of Gödel and Einstein, Basic Books, NY,
2005
J.T. Fraser, Editor, The Voices of Time, The University of Massachusetts Press, Amherst, 1981
Essays, St. Augustine, His Age, Life, and Thought, Meridian Books, The World Publishing Company,
Cleveland, New York, 1969