LOGIC, GOEDEL’S THEOREM, RATIONALITY, AND
SPIRITUALITY
Jagdish N Srivastava
CNS
Research Professor,
Colorado
State University,
Ft.
Collins, Colorado, 80523, USA.
[ This article has appeared in “ T.D. Singh et al (Eds.) Thoughts on
Synthesis of Science and Religion, pages 157-174. Bhaktivedanta Institute.
Kolkata. Singapore. Rome.” The Bhaktivedanta Institute has copyright on this
volume. As the author, I am sending this to interested friends for their
non-commercial use.]
SUMMARY
In this article, we examine the role of
‘Rationality’ both in the materialistic and the spiritual contexts. Since
‘rational’ refers to that which “manifests, or is based upon, reason or logic”,
it is necessary to consider the latter as well. Goedel’s theorem is perhaps the
most famous result in logic, with profound implications in various directions.
It is argued that what is called ‘rationality’ in everyday worldly life
corresponds to optimization over a subset, while real spirituality basically
means optimization over the Whole set. Since the Whole set contains the subset,
the optimum over the Whole set cannot be inferior to the optimum over a subset.
Thus, spirituality is seen to be supra-rational.
1.
INTRODUCTION
It would be instructive to briefly relate my own
story as to how I landed in this field. Even though I was born in a family that
was very religious, at the age of sixteen I developed an atheistic outlook
because of social influences. I argued with my friends insisting that there is
no God, asking them to prove that there is one. This continued for many years until one day I began to feel that
my arguments could be reversed. If some one asked me to prove that there is no
God, how could I proceed? I could not
find an answer to this, and my social patrons could not help. I became an
agnostic.
My Ph.D., in 1961, involved
Statistics and Discrete Mathematics, and in 1964, I found articles connecting
the latter to Physics, attempting to find a discrete base for the same. I
studied Quantum Mechanics. It led me to Logic and Goedel’s Theorem (GT). All
this time, as an agnostic, I considered Science to be my deity. However, my
faith in Science was shaken when I understood the implications of GT. Briefly speaking, one of the results that
follows from GT, in layman’s terms, is as follows. If M is any mathematical
system which involves the natural numbers 0, 1, 2, …, then there are questions
inside M which can not be answered using the axioms of M. To answer a given
question of this kind, one could expand the set of axioms by adding an
appropriately chosen new one. But now, the new system will again be subject to
GT, and there will be other questions that cannot be answered. Science is now a
system such as M, and shall ever remain so, even after billions of years of
research. Thus, GT ensures that there will always remain unanswered questions.
In other words, Science will never be able to fathom the depths of Reality.
While the agnostic in me struggled
in confusion, I wondered if there is
any
other way. I recalled vaguely that my
father used to say that according
to
the sages, true and deeper knowledge
often comes by ‘direct perception’ (DP). Besides receiving knowledge
from their teachers, the sages had received some by DP as well. I realized that
DP is often called ‘intuition’. Soon, I felt that the very basic ideas in much
of my own technical research had come by DP. I independently checked this with
my teachers and others, most of whom happened to be fathers of many fields and
were among the greatest scientists of the century. They all strongly
confirmed that, yes, DP had played a great role in their own work. Since then,
over the decades, I have found that among prominent scientists who touch on
this subject in their works, the importance of DP (and the need to study it (and,
consciousness in general,)) is echoed quite often. See, for example, articles
in Singh et al (1986), and Singh and Gomatam (1988). To illustrate, in one of
the articles in the former book (p.77), the distinguished computer scientist J.
Weizenbaum says “…So, the idea that science is the only source of knowledge in
the world is ridiculous….Indeed, all the important things that we know about
the world are the result of transcendent thinking, not of , let me say,
instrumental thinking…” Similarly, in the latter book (p.31), medicine Nobel
Laureate M H Wilkins says “The logical and the rational is a very important
element in science. But, emphasis on that tends to make people not notice the
essential role played by intuition.”
We
now proceed to consider the above topics in more detail. Our intention is,
obviously, not to go into the technicalities of these large and complex fields.
We only wish to bring out some features that are of general interest from the
point of the commonality between Science and Spirituality. Goedel’s theorem is
a result in the general subject of logic, and rationality is also, in a sense,
logical behavior only. So, we start with logic.
2. LOGIC
Logic is the science of
valid reasoning. We have deductive
logic when, given a set of statements, we wish to draw valid conclusions from
them. For example, suppose whenever C
sings then D dances, and whenever D dances then C sings. Then, we deduce that the singing by C and
dancing by D always occur at the same time. Similarly, we have inductive logic,
when we make inference about a set from the information on only a subset. An
example would be when we wish to get an idea of the average score of the
students in a class of size 50 from a sample of only 5 students from the class. Logic is also the study of the principles of
correct reasoning. It deals with the
appropriate use of the words ‘and’, ‘or’, ‘not’, etc. In language settings,
this can get very complex. Logic deals with the ‘structure’ of statements
rather than their ‘content’. Thus, the above statement about C and D has the
same structure as “whenever C plays then D sleeps, and whenever D sleeps then C
plays”. The structure in both is “whenever C does c then D does d, and whenever
D does d then C does c”. The logical conclusion is “C does c and D does d
always at the same time”, and involves only the ‘structure’. The ‘content’
deals with the specifics of what c and d are, and is clearly immaterial so far
as logical analysis is concerned. The structure can be deceptively complex even
in simple looking situations and even when the content is reduced to a very cut
and dry form. For example, for fun, answer the question: “ Given that whenever
C sings and D dances then E eats, and whenever D dances and E eats then C
sings, is it true that whenever E eats and C sings then D dances? ”
We
now explain the concept of a logical-mathematical universe, starting with an
example. Continuing with song and
dance, assume now that there are some singers and dancers, such that the
following conditions (or ‘axioms’) are satisfied: (1) every pair of singers has
exactly one common dancer (with whom they work), (2) every pair of dancers has
exactly one common singer, and (3) there is a group of four dancers, no three
of whom work with the same singer. Let U be a collection of systems, such that
each system in this collection satisfies the above axioms. For this, and other
reasons (not given here), we shall call U a universe. Can there be a system of
singers and dancers like the above? The answer is positive, an example being A
(123), B (145), C (167), D (246), E (257), F (347), and G (356), where the
singers are named A, B, C, D, E, F, G, and the dancers are named 1,2,3,4,5,6,7,
and where for each singer the dancers are given in the parenthesis. Notice that
the singers E and F have the dancer 7 in common, and dancers 3 and 4 have the
singer F in common, etc. The third axiom is also found to be satisfied, if we
take the four dancers to be 4,5,6, and 7.
The above system has 7
singers and dancers, and is an example of a ‘non-Euclidean geometry’. To see
this, we change the ‘content’ of the logical aspect of U (while keeping the
‘structure’) by calling each ‘singer’ a ‘line’, and each ‘dancer’ a ‘point’,
and where a point ‘lies’ on a line if the corresponding dancer and singer work
with each other. Notice that while in our ‘ordinary’ geometry, there is a
concept of ‘parallel’ lines, there is no such concept here, because here any
two lines must intersect in one point! Thus, the new system, while contrary to
our ordinary geometrical intuition, is still an existent system, and it affords
us an example of a system where the axioms are actually satisfied. Thus, the
axioms do not mutually contradict, because if they did, then no system could
exist which satisfies them. A set of axioms is said to be ‘consistent’, if the
axioms do not mutually contradict each other.
Thus the above universe U is ‘endowed’ with a set of consistent axioms.
But, could there be any use of such a weird thing? Yes, it is used for
computers, for communication, for increasing the yield of wheat, and so on!
Inside
the above universe U, we can ask questions. Can we have a system in U, which
has 14 points instead of 7? The answer
is no. Indeed, it can be proven that in every system, the number of points must
equal the quantity [pxp+p+1], for some number p, where p is an integer. There
is no integer value of p, for which [pxp+p+1] equals 14. Hence, there is no
system, in which the number of points (or lines) is 14. It can also be shown that a system does exist
with [pxp+p+1] points if, for example, p is a prime number. Thus, our universe
under consideration has a lot of systems, indeed, an infinite number of them.
The above example with 7 points corresponds to p=2. For p=3, we will have a
system with 13 points, for p=7, we will have 57 points, and so on. All such
systems are within the universe.
Now consider the
question: Does there exist a system
corresponding to the case p=10, which would give 10x10+10+1=111 points? No one
has made such a system. Also, no one has proven that such a system cannot be
made. Thus, the answer is yet unknown, even though a lot of effort has been put
into it, particularly in the last 40 years. Here is an example of a situation
where a universe U got formulated with relative ease by putting forth three
simple axioms, and yet there are simple questions within U whose answers are so
hard to find even with the usage of the immense power of the modern computers.
( I may add that even simple discrete mathematical problems go far outside the
range of any conceivable computers. For example, a search for the solution to
the above problem for p=10 done in a ‘brute-force’ manner, without using any
mathematical thought, would require looking at about (10^900) cases, which is
one billion billion …(repeated 100 times). Thus, any thought that powerful
computers of the future will solve all problems is scientifically unsupported.
We will have to fall back upon intuition.)
With these observations, we
now enter into even deeper waters.
3.GOEDEL’S THEOREM
We
continue with our (necessarily imprecise) discussion of logical-mathematical
universes. Let W be such a universe. In the last section, we had the example
where W=U, and where U is endowed with a set of 3 consistent axioms. We now
discuss the following problem. Consider W, and assume that W is endowed with a
finite number of consistent axioms. Let s be a statement pertaining to W. Then,
is it always possible to decide whether s is true or false by using the axioms
of the system? Goedel’s theorem answers
this problem. The answer is that whether a statement s is decidable by the use
of the axioms will depend on s. But, very importantly, if W is ‘rich enough’
(in a certain mathematical sense), then there will exist a statement (say, t)
in W which is true, but whose truth cannot be decided by using the axioms. In
order to show that t is true, it may be possible to add a new axiom to the
universe (consistent with the old ones), such that this enlarged set of axioms
implies that t is true. However, now the Goedel’s theorem will be applicable to
the universe with the enlarged set of axioms, and there will be some new
statement (say, u) such that u is true, but its truth cannot be established by
using even the enlarged set of axioms. And, so on.
I may add here that a full
explanation of the qualification ‘rich enough’ requires technical details
beyond the scope of this article. However, this condition is not very
restrictive, and is satisfied in large numbers of interesting situations.
In our example of the
universe U, let s2 be the statement “Inside U, a system with ‘p=2’ exists.” In
this case, it can be checked that using the axioms one can construct the system
in question. Thus, the statement s2 happens to be decidable by the use of the
three axioms, and the axioms imply that s2 is true. However, let s10 be the
statement “Inside U, a system with ‘p=10’ exists.” In this case, under our present stage of knowledge, we cannot say
whether s10 can be proven to be true or false by using the three axioms of U.
In other words, we do not know whether s10 is a statement like t mentioned
above. If some one could show that s10 is a statement like t, then further
attempts to decide on it using the axioms are obviously futile. If it can be
shown that s10 is not like t, i.e., it is decidable using the axioms, then we
should continue to try to settle it using the axioms. At present, we do not
know whether s10 is like t or not.
It is thus clear that,
basically, Goedel’s theorem says the following: Under certain fairly general
conditions, a (finite) consistent set (say, X) of axioms for a universe W is
‘incomplete’, in the sense that there exist true statements t in W whose truth
cannot be established using X.
How did Goedel come about
this result? The proof, of course, is very complex, but there is an idea in it
that is of general interest, and is instructive. The idea is a logical
situation known as ‘self-reference’, or
‘tangled hierarchy’. Here, the predicate refers back to the subject.
Normally, the subject in a sense acts on the predicate. But, here. the
predicate acts back on the subject. Consider the statement “I am a liar”. Here,
‘I’ is the subject and ‘liar’ is the predicate, which is referring back to the
subject, so that the hierarchy (i.e., which one is acted upon by the other), is
tangled. If the statement “I am a liar” is true, then it implies ‘I am speaking
a lie’, and therefore the statement “I am a liar” made by me is not true! On
the other hand if the statement is false, then it implies that ‘a lie is being
spoken’, so that the statement ‘I am a liar ‘ is true! Thus, if you assume the
statement is true, then you must conclude that it is false, and if you assume
the statement to be false, then you must conclude that it is true! Another example is the Russell’s paradox. Let Z be a set with the property that it is
a collection of all sets that do not contain themselves. The question is: Does
Z contain itself or not? If we assume that ‘Z contains itself’, then we
conclude that ‘Z does not contain itself’, because Z is the collection of only
those sets that do not contain themselves. On the other hand, if we assume that
‘Z does not contain itself’, then it follows that ‘Z contains itself’, because
Z is the collection of all sets that do not contain themselves. In the first
example, the predicate ‘liar’ refers to the subject ‘I’, and redefines it. The
redefined subject then redefines the predicate, which in turn redefines the
subject, and so on. In the second example, the definition of Z involves Z. In
both cases, there is a reference to the ‘self’.
One may wonder what is the
use of looking at such artificial and crazy problems. The answer is that it is
really the crux of the matter in many subtle and important conceptual issues.
For example, consider U again. When we ask a question about U (such as whether
it has a system in it, for ‘p=10’), we are asking a question about its ‘self’.
Now, if the axiom set is not complete, we are asking a question about something
which is, in a sense, not ‘sufficiently’ well defined. So, it is somewhat like
the situation in Z. This ‘self-reference’ is a conceptual entity, sometimes
called ‘Goedel-knot’, since Goedel used it in his proof. I feel that this
‘self-reference’ needs much more study. Goedel has probably only touched the
tip of the iceberg.
Indeed, I feel it is no
coincidence that in Indian spiritual literature, God is often referred to as
the ‘Self’. So, it is also in Soofee-ism, where the word ‘Khudaa’ is often used
for God, because this word is derived from ‘khud’, which means ‘self’. We shall
come back to this after the next section, where we discuss what is considered
to be perhaps the most desirable aspect of our day-to-day life.
4. RATIONALITY
As
stated earlier, ‘to act in the rational way’ is ‘to be logical’ and ‘reasonable’.
Consider the statements ‘It is raining outside’, ‘I must walk outside’, ‘If I
do not have an umbrella, I will become wet’, and ‘I do not want to become wet’.
The logical consequence of this is ‘I should carry an umbrella’. Thus, to be
rational, I should accept the logical argument and carry an umbrella. What
happens if I don’t? Well, in that case, I will incur a ‘loss’, namely, that I
will become wet. So, ‘rationality’
demands that I avoid losses, which is why it entails that I pay heed to
logic. Clearly, ‘rationality’ is a
utilitarian concept, and is necessary for a successful life in the world.
As
would be expected, it is quite important in economic contexts. Indeed, the
great mathematician von Neumann helped develop the field of ‘game theory’, which
does influence the actions of business corporations today. Imagine there are two players (corporations)
G and H. In any given situation, G can
take (one of the) actions a1, a2, a3, … , and H can take actions b1, b2, b3, …
. If G takes action a1, and H takes b1, then G pays the amount u(a1,b1) to H,
where the symbol u(a1, b1) denotes a monetary amount. If u(a1,b1) is positive,
then G has a loss, if u(a1,b1) is negative, then G has a gain. Thus, a loss is
a negative gain, and vice versa. The loss (or gain) of G is equal and opposite
to the gain (or loss) of H. This is called a ‘zero-sum’ game, since the [(loss
of G)+(loss of H)] equals zero.
Now if, for example, G and H
respectively take actions a4 and b3, then the payment by G is u(a4,b3). Thus, given any action a of G and any action
b of H, the payment by G equals u(a,b), where the symbol u(a,b) depends only on
the actions a nnd b. The symbol u is called a ‘loss function’, and indicates
how the loss depends upon the pair of actions chosen by the players. Looking
from G’s perspective, ‘Rationality’ requires that G should play so as to reduce
his loss. Game Theory deals with the question of how G should play so that the
loss to G is minimized in some sense. Thus, game theory provides a model for
rational economic behavior.
How are the quantities
u(a,b) decided for different a and b? If the players are gambling, they make a
mutual agreement. In business, the amounts are estimated by considering how
much the corporations would actually lose or gain by their economic behavior. A
study of the same fixes the function u. Of course, the situation presented here
is very simplistic. But, it does bring us to the fact that, at least from the
worldly point of view, a loss function needs to be considered if we wish to be
rational. In applications, often, the
above becomes a model for ‘cut-throat competition’. Actually, the mathematical
theory is not at fault. Indeed, in real life, there are many varieties of
losses, not just monetary losses. In other words, there can be many loss
functions, u(a,b), v(a,b),…, etc. So long as each type of loss can be expressed
numerically, the theory can be used. Conceivably, ethical parameters could be
brought to bear on the functions u, v, …, etc.
If this is done, then we would have a rational behavior, which though
economic in nature, is ethical in its core.
Unfortunately, the world of
today does not come arranged that way. There is a many-person game going on,
which is not zero-sum, but in which the sum of the losses of all parties taken
together, is very large and positive.
To give a simplistic example, consider the above game again, but with
new players G` and H`. Assume that, besides paying to H`, the player G` also
puts an amount t(a) into the ‘trash can’, when he takes action a. Similarly, H`
puts t(b) into the trash when he takes action b. Here, t(a) and t(b) are
positive quantities for all actions a and b. In this case, (loss of G`)+(loss
of H`)= t(a)+t(b), which is a positive quantity. Thus, this is not a
zero-sum game. Here, on the whole, both players lose. The situation under
‘cut-throat competition’ is similar (irrespective of how rational such
competition seems while one is engaged in it). A big part of the energy of
every player in the game goes towards ‘destroying’ the competitor(s). This
often corresponds to putting valuable resources into the ‘trash can’, i.e.,
into wastage.
Most readers would consider
the players G, H to be wiser than the pair G`, H`. Notice that both pairs are
‘rational’ in their own way. G and H do not mind losing to each other as much
as they mind dumping into the trash. On the other hand, G` and H`, in the fury
and frenzy of competition, may wish to play the different game, because each
thinks that he may force the other out of business by making him pay large
amounts to the trash. Clearly, the ‘axioms’ of the ‘logical universe’ of the
pair G, H are different from those of the pair G`, H`. It is thus seen that
‘rationality’ is not the same procedure under all circumstances, but depends on
the ‘logic’ and ‘axioms’ of the world in which utility is defined. Depending
upon the nature of the logical universe and its axioms, different procedures
will turn out to be rational. An important example of this fact is the case of
‘nested’ universes.
To illustrate, consider the
sequence of universes U1 (home), U2 (housing complex), U3 (town), U4 (county),
U5 (state), U6 (region), … , and so on. Notice that these universes are
sequentially nested, in the sense that U1 is contained in U2, U2 in U3, and so
on. Let m be an action, such as buying a motorhome. From the viewpoint of the
people in the home (U1), i.e., the family, the action m may be very exciting,
so m may be perfectly rational. But then, the housing complex (U2) may not
allow it to be parked at the home. So, may be, m is not such a good idea.
Further, the town (U3) may levy a huge tax, so perhaps m should be discarded.
But then, the motorhome may be very useful in serving the county (U4), which
may give a large reward. Thus, at the level of U4, considering everything,
doing m may be quite rational. But, at the level U5, m may cause major trouble,
because drivers in U5 may not generally like motorhomes and may give a hard
time. However, the state border may not
be far from home, and the region (U6) may be scenic, and motorhomes may be
quite popular in U6 (except for U5). So, at the level U6, m may be very
rational. And, so on. It is thus clear
that what is rational depends upon how far you look.
However, at any time, a
person has only a relatively small amount of knowledge, this being enveloped by
ignorance. In the above case, one person may take decision by looking at only
U2, while another may go up to only U4, and another only up to U5. Thus,
different people may take different decisions, each considering himself to be
rational. The question arises: What is
truly rational in this case? To answer this, one should perhaps look at some U7
(which includes U6), but then there may be a U8 containing U7, and so on. Where
does one stop? What is the ‘rational’ answer to the stopping problem?
Obviously, one should stop only at TR, the Total Reality, because it contains
all possible universes. It may be added that TR is only another aspect for the
Divine.
How can one find out what is
right at the TR level? The fact is that it is awfully simple (though it may be
quite hard for many people). I will describe the method (called here,
‘procedure’ P*), which I followed starting around 1980, and which I now use
very often. Suppose you have a question q, and you wish to have Rational
guidance from the Divine, i.e., guidance at the TR level. Go sincerely into
your heart, and ‘ask’ the Divine for guidance. The ‘asking’ can be achieved by
setting up some simple experiment (say, E) whose result is random and depends
upon chance. Different possible answers (to q) are put into correspondence
(before E is performed) with the different possible random results. You first
pray to the Divine without asking for anything. Next, pray to the Divine to
produce that result which corresponds to the correct answer. Believe that the
Divine has agreed to your request. Then, you do the experiment E, observe the
result that you got, and interpret the result according to the correspondence
you have set up.
Know that the Divine is your greatest well-wisher, even more than
you yourself are. Pledge that you will always do whatever the Divine tells. and
will never try to edit or modify it. Pledge that you will practice
‘contentment’, in the sense that you will totally accept (and, not ‘evaluate’
or dislike) any consequences that flow from your obeying the guidance of the
Divine. Remember that what you believe to be the best thing that should happen
to you, may not be the same as what is actually best for you (considering
Everything). Know that the Divine will give you only that thing which is
actually the best for you in the overall perspective. Also, unless your faith
becomes very strong and your practice mature, do not ask questions whose
results are verifiable. Even when you feel that you got some answers that seem
to be wrong, ignore this and think that whatever answers you were given were
really for your own good. When you are adequately mentally prepared this way,
you are ready to be told what action is Rational for you, and (through your
experiment E) you will be provided with the same.
Even when you are far below
this level of mental preparation, you can start the above practice. If you are
keenly observant, you may begin to see a subtle but distinct positive-ness in
your results that, occasionally, might be independently and voluntarily
confirmed by other people. (However, as stated earlier, do not try to evaluate
or verify the ‘correctness’ of the answers you get from your experiment E.) All
this will tend to increase your faith, which in turn would improve your mental
outlook, which would give even more faith. Thus, slowly, you will ‘ascend’.
Thus, a key to this whole procedure P* is faith. As your faith increases, the
practice will begin to bear positive fruit that will be easily visible. Again,
I would like to stress that ‘attitude’ is very important here. In Science, one tests a given scientific
hypothesis by doing repeatable experiments. Remember that this is not what we
are doing here. Nothing is being ‘tested’. Never try to test the Divine.
If one has the attitude “Let me try the experiment which is to be done under
P*, and see if it works and gives me answers which are reasonably good”, then
one is not really doing P*, and will get random answers only. Getting
‘correct’ answers by using P* is a result of faith and the pledges made.
Why? These and other similar questions
will be considered in the next section.
5. THE SPIRITUAL PERSPECTIVE: FOUNDATIONS OF REALITY
So, continuing the above
discussion, the question is: Why is it that faith and pledges come into the
picture in the procedure P*, but not in the ordinary scientific
experiments? Of course, the spiritual
leaders have always been stressing the importance of faith, etc. So, their
authority is behind it. However, apart from authority, it is also
(heuristically) implied by the author’s (skeletal) mathematical theory of the
Foundations of Reality (FR).
In the author’s theory
(denoted by atfr) of FR, deep down, we have only mathematical structures. These
constitute ‘Nature’ (denoted by ‘N’). ‘Nature’ is the ‘nature’ of the Divine
(denoted, henceforth, by ‘V’). A person’s (in general, a living entity’s)
‘consciousness’ converts (or, interprets) a particular structure (‘near’ the
entity) as his/her/its immediate ‘reality’. (Thus, for example, in atfr, what
is called ‘matter’ in ordinary physics and chemistry is only a mathematical
structure. Our consciousness converts this structure into the perception of
matter.) Consciousness is an attribute of V alone; V is conscious of all of N.
An entity that is ‘alive’ has the form (V, W), where W is like a ‘web’ formed
by (some set of) ‘links’ of V to N. For different living entities, W is
different, but V is the same. The consciousness of (V, W) is far less than that
of V, because W muffles it. [Here, ‘muffle’ is used only in a ‘logical’ sense.
Ordinarily, to ‘muffle’ something means to wrap it up in a cover. Here,
however, it refers more to logical restrictions on V. For example, it may
corresponds to ‘desires’ (only directions d1, d2, .., should be taken), or
aversions (directions a1, a2, .., should not be taken), or ‘attachments’ (stay
inside (the sub-universes) s1, s2,..), etc.].
Living entities may or may
not be ‘embodied’; an embodied living entity has the mathematical form (V, W,
X), where X is a structure inside N. The part (V, W) may be called the ‘soul’
of the entity (V, W, X). As time
passes, X may change, decay, and die, W may change to W*, and the entity (V, W,
X) may change to (V, W*). As W goes to ‘zero’ (in the universe N# (a part of N)
where X is defined), the soul (V, W) tends to become close to the ‘super-soul’
V. However, it should not be concluded that (V, W) ‘becomes’ V. Indeed, in a
sense, V is defined in all possible universes. W is also defined in many
universes. Thus, although, W may approach zero inside N#, there may be
universes where W is defined which have nothing to do with this. For example,
one such universe is N~, which consists of all entities which possess the
awareness that they have had a sequence of bodies. A particular W may contain a
‘link’ of the form (V: N~), which in some manner connects V to N~. This is a
somewhat logically tangled situation.
No structure inside N has V
in it, although N has evolved out of V. Thus, ‘life’ arises only from
‘life’. Structures X may ‘evolve’
inside N# (which is inside N), but V is fixed and does not change or evolve.
For example, on our earth (which is a part of N#), Darwin’s theory must really
concern only with the evolution of ‘bodies’ X, not of ‘life’ V. Thus, according
to atfr, instead of saying that ‘life evolved’ on earth, one would say that
‘bodies evolved’.
Under atfr, N consists of
innumerable (mathematical) structures (indeed, all possible ones). N# is one
such mathematical structure inside N. Some of these structures are rich enough
to be called a universe (denoted by U), and some of them (like N#) may contain
self-reproducing systems X, called ‘bodies’, through which a pair (V, W) may
operate. Thus, the physical aspect of a scientific experiment is contained in a
universe like U. The procedure P* of the last section involves invoking V,
which is not a part of N. Thus, P* happens outside N, in another domain (say,
N*). The domains N and N* are like two branches in the flow chart of a computer
program. (I may add that, under atfr, our physical universe has the structure
of a logic-tree, i.e., it is like a computer program.) In a computer program,
at a particular juncture, we may have a variable v, and from this juncture two
branches may go forth, one branch corresponding to (say) v=0, and the other
corresponding to (say) v=1.
The situation here is very
similar. The case v=0 corresponds to N, where V is ignored, and v=1 corresponds
to N* where V is invoked. Let P denote the procedure P* when the faith and the
pledges are lacking, i.e., V is ignored. Then, P occurs in N, while P* occurs
in N*. Since P occurs in N, it obeys the ordinary rules of probability theory,
and thus P will lead only to random answers. Now, in a sense, N* is the version
of N where V is invoked, i.e., where V plays ‘active’ role. (As I stated earlier, N evolves out of V
only, but in N the role of V is ‘passive’.) In N*, V acts on the experiment E.
How? Notice that you (the person doing the experiment E) are of the form (V, W,
X), your operative part being (V, W). Your faith in the Divine and the
(sincere) prayer and pledges temporarily ‘remove’ W from inside N# (i.e., from
inside N), so that (V, W) becomes like V, so far as N# or N are concerned. At
this point, your will (to obtain the Rational guidance) is close to the will of
V, which in turn acts on E, and produces the correct result (under the Overall
Perspective, i.e., considering Everything). Thus, the procedure P*, indeed,
does work.
If your attitude is one of
doubt, then the W in (V, W) tends to (logically) obstruct the role of V, which
then does not act to modify the result of E to make it correspond to the
Rational answer to your question. Notice also that if you have too much of
selfish motive and worldly desire behind asking the question q, then your
pledges (such as ‘contentment’) cannot be sincere, and P* will tend to reduce
to P, and the Divine guidance will not come.
To summarize the
mathematical aspect of the above discussion, the situation is this. Looking at
the flow chart of the logic-tree which corresponds to our Reality, we have the
choice to execute one of two procedures P* and P, which are definable only in
two disjoint worlds N* and N respectively. Doing P is simple, because we are in
N, and doing P will give us results according to the rules of N (i.e.,
according to probability theory). However, doing P* requires us to be in N*. In
order to go to N* and work in N*, one must first acknowledge the existence of
N*. This simple but important mathematical fact corresponds to what is called
‘faith’ in human experience. This faith can give us entrance into N*. Next, in
N*, N gets transcended. This happens because the person (V, W), (‘subduing’ W
through faith, prayer and pledges), temporarily comes close to V. As W
decreases inside N#, the ‘will’ of (V, W) and the will of V become similar. The
will of V thus changes the probability distribution of the result of E, giving
probability 1 to that result of E which would correspond to the Rational answer
to the question q. Thus, the key is to get rid of W.
How does V influence E in
N*? Since N and N* are logical
structures they are ideas, and so is E and the set of its results, and the
interpretation of these results. In view of these interpretations, the Rational
answer to q is one particular result (say, r) of E. Thus, the will of V
corresponds to getting r from E. Since all of N and N*arise out of V only, the
result r receives probability 1.
It should be remarked that
the above discussion is rather close to the point where it can be the guiding principle
of a mathematical formalism for revealing the (already existent) bridge between
Science and Spirituality. The reader is invited to future technical papers of
the author in this field, where such formalism is attempted.
We see from the above discussion
that to be truly rational, we have to go beyond the ‘rationality’ arrived at by
looking at narrow universes, to the level of TR, which is done by cutting down
on W. But, that is nothing but the
practice of Spirituality. The same conclusion is reached by Goedel’s theorem in
two ways. The first is the realization that every universe (of reasonable
richness) has an incomplete set of (consistent) axioms. Doing Science is
discovering such axioms. But, they will always remain an incomplete set. Thus,
the knowledge obtainable by the ‘scientific process’ will always remain
relatively insignificant. Yet, every universe is open to ‘direct perception’,
which occurs more and more as W decreases, so that we are again led into the
lap of Spirituality.
The second implication of
Goedel’s theorem is that it leads us to the consideration of the ‘Self’.
Arguments can be given pointing towards V as being the ‘Self’. This is a large topic, and its elucidation
will be done elsewhere.
6. BEYOND RATIONALITY?
What
is the upshot of this article? It is
that rationality should be practiced in the true sense, i.e., we should go
beyond ordinary rationality to Rationality. Also, Goedel’s theorem tells us
that whereas each universe is a spiritual realm reachable by direct perception,
Science can take us only to negligibly small portions of the same. Both of
these conclusions send us into the lap of Spirituality, which is seen to be
relevant in all cases. We may therefore sing:
Spirituality is the answer.
It’s blowing with the wind.
The answer, my friends,
Is blowing,
With the wind.
7. REFERENCES
Singh, T. D. et al (1986) Interviews with Nobel
Laureates and Other Eminent
Scholars, The Bhaktivedanta Institute, San Fransisco, Bombay
Singh, T.D., and Gomatam, R. (1988) Synthesis of Science
and Religion, The Bhaktivedanta
Institute, San Fransisco, Bombay
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THE END
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