Download do cats believe in god

DO CATS BELIEVE IN GOD?
Munawar Karim, Department of Physics
St. John Fisher College
While biologists are moving towards a mechanistic model of the world, physicists have, for
compelling reasons, abandoned this model. I cannot give you a name for this emerging
paradigm that physicists are leaning towards, but one of its features is the existence of a reality
beyond what we now know of as the physical reality. The advent of quantum mechanics and the
impossibility of interpreting it within a purely deterministic view has prompted physicists to look
beyond the Newtonian model of the universe. The idea of non-locality, which is essential to
interpret results of experiments in quantum mechanics, is so far removed from ordinary physical
experience that it requires a new way to look at the universe. Some of these ideas have their
counterpart in mathematics; I refer to the incompleteness theorem of Godel. The subject I will
be speaking on is an extension of the contents of books by Roger Penrose (The Emporor’s New
Mind, Shadows of the Mind) and by P.C.W. Davis, Frank Tipler among others.
The scientific method as we know it today was founded by Ibn Haytham (Al Hazen) in the 10th
century. It was the natural evolution from the ideas of the Aristotelian school, with the singular
addition of experimental verification of a conjecture. In skeletal form the scientific method
consists of a spiraling sequence of observation, hypothesis, verification, induction, followed by
further observation etc. The sequence spirals are directive, in the sense that there is a forward
movement of knowledge leading to a degree of understanding. I will argue, along with others,
that we have reached a stage in our awareness where a radical revision in the scientific method is
necessary if we are to accommodate and interpret phenomena which have been uncovered
recently. I will argue against “artificial intelligence” (AI) as an adequate model, and in favor of
expanding the scientific method to include new forms of knowledge (consciousness) which are
inaccessible to the scientific method as we know it today. Specifically I will illustrate that the
mind is capable of proving results (theorems) that are impossible using computer programs
(which are mechanisms).
I will start with examples in mathematics because it allows us to reduce complex concepts down
to essentials. The central debate is whether the mind (as distinct from a brain) is a mechanism.
If we would like to answer this question we could ask whether the mind is able to perform tasks
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that can be mimicked by a computer program (I will identify a program as a mechanism). An
external observer would not be able to distinguish the actions of either the mind or the computer.
Let us take an example from numbers
The mind has the capacity to understand. For example we take the product of two natural
numbers:
axb=bxa
This equation means that the left hand side (a groups of b objects) for a = 3 and b = 5 is:
(*****)(*****)(*****)
and the right hand side means:
(***)(***)(***)(***)(***)
Both arrangements have the same number of *’s because a x b = b x a. Let us rearrange the *’s
as
*****
*****
*****
We can read this as three rows of five *’s, or five columns of three *’s. In either case it is
obvious that we are looking at the same rectangle and so 3 x 5 = 5 x 3. This ‘obviousness’
comes from an understanding from which we can generalize the result to an array of any size. A
computer, using a program is incapable of proving this result. If for instance I were to choose
a = 7563982761059 and b = 49829804219 the mind could readily assert that:
7563982761059 x 49829804219 = 49829804219 x 7563982761059
without attempting to picture the array. A computer, on the other hand, could not enumerate the
array (it is just too big).
Let us try another example:
Find an odd number that is the sum of two even numbers. It is obvious to us that there is no such
number! A computer program written to find this number will never terminate, and therefore
does not arrive at a conclusion.
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Yet another example:
A sequence of hexagonal numbers i.e., those that can be arranged in hexagonal arrays (excluding
zero). The numbers are:
1, 7, 19, 37, 61, 91, 127 ,…
****
***
*****
****
******
* *** *****
*******
**
**
****
******
***
*****
****
The sequence of hexagonal numbers are obtained by adding to 1 successive multiples of 6 ie
1, 1 + 6 = 7, 7 + 12 = 19, 19 + 18 = 37, 37 + 24 = 61, 61 + 30 = 91, 91 + 36 = 127
This can also be diagrammed as adding an outer ring to an existing hexagonal array:
***
****
*****
****
***
The number of spots in this ring is 18 (always a multiple of 6). Adding pairs of successive
hexagonal numbers gives:
1 = 1, 1 + 7 = 8, 1 + 7 + 19 = 27, 1 + 7 + 19 + 37 = 64, 1 + 7 + 19 + 37 + 61 = 125
But these numbers (1, 8, 27, 64, 125) are cubes.
Prove that this is a general property of hexagonal numbers.
We can prove this statement by examining the diagonal plane of a cube (see diagram). But a
program will continue forever without terminating, and therefore without any conclusion.
These examples of the principle of mathematical induction where results valid for a class of
natural numbers can be ascertained from the basis of a single computation.
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Another example:
Proposition 1: Is it possible to express all integers as the sum of the squares of 2 integers?
A computer program would test the proposition starting from the smallest number and progress
upwards one integer at a time.
0 = 02 + 02
OK
1 = 12 + 02
OK
2 = 12 + 12
OK
3 = 12 +12
NO
So this proposition is false.
Let us try another example.
Proposition 2: is it possible to express all integers as the sum of the squares of 3 integers?
0 = 02 + 02 + 02
OK
1 = 02 + 02 + 12
OK
2 = 02 + 12 + 12
OK
3 = 12 + 12 + 12
OK
4 = 02 + 02 + 22
OK
5 = 02 + 12 + 22
OK
6 = 12 + 12 + 22
OK
7 = 02 + 22 + 22
NO
So this proposition is false too. We have proved by computation that these two propositions are
false.
Let us go one step further.
Proposition 3: Is it possible to express all integers as the sum of the squares of 4 integers? Let
me just tell you that if we attempt to prove the truth of this proposition our computer program
will never stop. We can conclude that no finite sequence of steps can prove or disprove this
proposition. However, Lagrange (1770) proved that indeed this proposition is true ie, that every
number can be expressed as the sum of the squares of 4 integers.
How then is the mind capable of arriving at a result which is not possible by mechanisms?
This is the first hint that there is something special or different about the mind.
We can go further; let us say that there must be a way to program a computer to use the method
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Lagrange used to prove the proposition, or even all such methods. Then indeed our computer
will be just as able to prove or disprove any theorem. If you try this something truly bizarre
occurs! For in order to generate all the methods a mind might use to prove or disprove theorems
one has to reduce the methods to the outcome of a formula, which the computer then uses to try
each of the methods. Well one of those calculations turns out to be the very theorem it is trying
to prove! What this shows is that the mind is able to work in ways which cannot be reduced to a
‘method” or an “algorithm” and therefore cannot be replicated in a program. There is no escape
from ‘self-referential limits’ of this type. The mind is not a mechanism!
We have here an example of a fact that exists in mathematics but is unprovable using
computational methods. This is what is meant by Godel’s theorem which states that any
axiomatic system is either inconsistent or incomplete.
Let me move to examples from physics. In order to understand the dilemma confronting
physicists one has to appreciate what is meant by measurement. Due to the presence of
fluctuations of the quantum field there cannot be a perfect detector. The result of any
measurement is always accompanied by an irreducible uncertainty. When a detector is applied
to the system being measured it injects its imperfectness onto the system. The resulting
measurement is only accurate to within the errors associated with this imperfectness. Does a
perfect system or a detector exist? If you accept the Copenhagen interpretation of quantum
mechanics, the answer is no. I will come back to this question later.
The Newtonian view of the universe has had such a profound effect on our thinking that we are
unaware how pervasive it has become, from its influence on Darwin’s explanation of evolution
as more or less a mechanistic adaptation of organisms to environmental influences to Marx’s
belief that man’s needs are merely food, clothing and shelter. Needless to say this irrational
belief in ‘rationalism’ has had consequences which in their horror has no parallel in history.
One view of physics is the separate existence of physical laws from the entities that the laws act
upon. For example electromagnetic fields behave according to the Dirac equation or that matter,
energy and space-time are subject to Einstein’s equations etc. Although this is the most direct
interpretation I believe there is an interpretation that is more complete. Some observations may
help to illustrate my thesis. By the end of the last century most physicists were convinced that
all that was to be known had been known except for a few loose ends, which would be explained
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in due time. The most troublesome of these was the anomalous orbit of the planet Mercury.
Despite all efforts the Newtonian view was unable to come up with an explanation. It took a
completely new interpretation by Einstein to finally resolve the discrepancy. Similar examples
from our current understanding would be the mechanism that generates the sun’s rays and the
stability of the electron. Once again despite prodigious efforts a complete explanation of these
two phenomena (among many others) is not possible. The question to ask is: how does the sun
manage to produce the energy it does? Or how does the electron manage not to explode with
infinite energy when we are unaware of the laws which govern these phenomena? Or how was it
possible for Mercury to orbit as it did when the laws of motion we were aware of could not
provide an explanation for its motion?
A model which explains these anomalies is that physical phenomena are governed by the very
properties that are associated with physical objects. Laws don’t exist separate from objects that
they act upon, instead the nature of objects themselves determine their behavior. Any discrepant
behavior is a reflection of our incomplete understanding of the nature of physical entities.
Nature never errs. It is perfection itself. Our incessant attempts to understand nature, to
understand perfection, when we ourselves are part of the system we are studying has led us to the
same self-referential limit (the quantum uncertainty mentioned earlier) as expected from Godel’s
theorem. I maintain that this is a hint that there is a reality beyond physical reality, again of
which we are aware but are unable to comprehend using the scientific method.
We are somewhat in the position of the computer that is unable to prove results which our mind
is able to prove through our understanding of mathematics. And just as the computer is limited
to algorithms to be able to prove or disprove theorems, man too is limited by the scientific
method and is similarly unable to find complete explanations of phenomena which he knows
exists. It is the unknowability of the universe that we find disturbing since it appears to be
beyond our comprehensive powers. Belief in God, whether by man or cats, is then a
manifestation of the awareness of the unknowable.
If we want to make an attempt at understanding the reality beyond the physical reality we may
have to expand the scientific method into areas beyond what we are familiar with. If nature is
indeed a perfect entity, self-governing and eternal, then this conscious being called man, who can
decide through free will to attempt or not to attempt to comprehend the incomprehensible, is the
supreme mystery. We are entering the realm of mysticism which has its origins in religion.
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