Earlier
Efforts
of Western Sciences
f.)
Kurt Godel
Faith and Science
Reflections
on Kurt Gödel’s Mathematical and Scientific Perspective
of the Divine: A Rational Theology
Hector
Rosario, Ph.D. Department of Mathematical Sciences, University of
Puerto Rico, Mayaguez Campus December 9, 2006
Abstract: Kurt Godel had a profound rational theology. Godel was
not only a theist, but a personalist and a believer in the afterlife.
I will explore his philosophical stance through exchanges with Albert
Einstein and others to understand how such a foremost mathematician
and physicist held such views. I will also address Godel’s
ontological argument.
Kurt Godel, the preeminent mathematical logician of the twentieth
century, is best known for his celebrated Incompleteness Theorems;
yet he also had a profound rational theology worthy of serious consideration.
“The world is rational,” (Wang, 1996: 316) asserted
Godel, evoking philosophical theism, “according to which the
order of the world reflects the order of the supreme mind governing
it” (Yourgrau, 2005: 104-105).
Godel’s “Incompleteness Theorems” are an “extraordinary
comment on the relationship between the mission of mathematics and
the manner in which it formulates its deductions” (Mazur,
2006: 3-4). They have been interpreted as a limitation on rationality,
since a possible semantics for the results is that, in any axiomatic
and consistent system capable of doing arithmetic, there are truths
that cannot be proved within the system. This has very profound
philosophical implications that shattered the hopes of many a previous
mathematician and philosopher, including thinkers of the stature
of David Hilbert, Bertrand Russell, and Ludwig Wittgenstein. Frustration
notwithstanding, “[Godel’s] works on the limits of logic
have inspired awe, respect, endless development and speculation
among mathematicians, and indeed among all theoretical scientists”
(Davis, 2002: 22).
Among the theoretical scientists influenced by Godel was his friend
Albert Einstein. Between the years 1940 and 1955 they developed
an intimate relationship as colleagues at the Institute for Advanced
Study in Princeton. According to colleague Oskar Morgenstern, the
co-founder of Game Theory, when Einstein had lost enthusiasm for
his own work, he went to his office “just to have the privilege
of walking home with Kurt Godel” (Wang, 1996: 57). Indeed,
according to Institute colleague and physicist Freeman Dyson (the
discoverer of combinatorial proofs of Ramanujan’s famous partition
identities), Godel was “the only one who walked and talked
on equal terms with Einstein” (Dyson, 1993: 161). However,
I would argue that Godel’s intellect was in many ways subtler
than Einstein’s, in philosophy and perhaps even in physics.
God and Godel
As
his correspondence with Burke D. Grandjean attests, Godel was a
self-confessed theist, going as far as developing an ontological
argument in an attempt to prove the existence of God. He chose the
framework of modal logic, a useful formal language for proof theory,
which also has important applications in computer science (Blackburn,
de Rijke & Venema, 2001). This logic is the study of the deductive
behavior of the expressions ‘it is necessary that’ and
‘it is possible that,’ which arise frequently in ordinary
(philosophical) language. However, according to his biographer John
Dawson, he never published his ontological argument for fear of
ridicule by his peers.
An important aspect of Godel’s theology – one that has
been greatly overlooked by those studying his works – is that
not only was he a theist but a personalist; not a pantheist as some
apologetic thinkers may portray him. To be precise, he rejected
the notion that God was impersonal, as God was for Einstein. Einstein
believed in “Spinoza’s God who reveals Himself in the
harmony of all that exists, not in a God who concerns Himself with
the fate and actions of men” (Einstein, 1929). Godel in turn
thought “Einstein’s religion [was] more abstract, like
Spinoza and impersonalist philosophy. Spinoza’s god is less
than a person; mine is more than a person; because God can play
the role of a person” (Wang, 1996: 152). This is significant
since a God who lacks the ability to “play the role of a person”
would obviously lack the property of omnipotence and thus violate
a defining property universally accepted as pertaining to God. Therefore
if God existed, reasoned Godel, then He must at least be able to
play the role of a person. The question for Godel was how to determine
the truth value of the antecedent in the previous statement.
A relevant issue in Godel’s discussions on the Divine with
Einstein is his mention of “Indian philosophy.” Godel
considers Spinoza’s concept of God and the Advaitavada impersonalist
concept to be in the same category, which is not a correct understanding
of these notions. Spinoza’s stance on God is impersonal, akin
to Sankaracarya’s monism (c. 788-820 CE). Unfortunately, although
familiar with such Sankara’s view, Godel was apparently unaware
of the philosophical conclusions of Ramanujacarya (1017-1137 CE)
and Madhavacarya (1238-1317 CE), who would also reject Spinoza’s
God. The rejection comes not because they deny God’s presence
in all that exists, but because such view is considered subservient
to one in which a personal relationship with the Supreme can be
established and nurtured. Taking omnipotence seriously, “playing
the role of a person” is one of God’s unlimited potencies
which these sages do not compromise in their theology.
Certainly, Godel was also unaware of the philosophy of Caitanya
Mahaprabhu (1486-1534 CE), who follows Ramanujacarya and Madhavacarya
in the essential points. However, the detailed description and practice
of divine love in service of purusottama –devotional service
to the Supreme Person – given by Caitanya Mahaprabhu and his
followers arguably make this a subtler and more revealing theology
than those presented by his predecessors. In it Godel would have
found his theological conclusions realized in completion five centuries
earlier.
Godel’s Philosophy of Physics
In
physics, Godel’s contributions are well-known. However, physics
was not a detour Godel took to amuse himself, but rather an essential
part of his philosophical fabric. In 1949 Godel expressed his ideas
in an essay that in Einstein’s own words, “constitutes
[…] an important contribution to the general theory of relativity,
especially to the analysis of the concept of time” (Schilpp,
1949: 687). Unfortunately, even with Einstein’s high estimation
of Godel’s work, modern physicists have been wont to discard
Godel’s ideas, trying (without success) to find an error in
his physics (Yourgrau, 2005: 7-8).
Godel’s unsuspected solutions to the field equations of general
relativity, solutions in which time undergoes a peculiar transformation,
made the discussion of time-travel respectable in scientific circles.
In fact, Godel concluded that time travel is indeed theoretically
possible, rendering time, as we know it, meaningless. Time, “that
mysterious and seemingly self-contradictory being,” as Godel
put it, “which, on the other hand, seems to form the basis
of the world’s and our own existence,” turned out in
the end to be the world’s greatest illusion (Yourgrau, 2005:
111). For Godel, time was a crucial philosophical question, but
I am unaware of any direct connection Godel might have made between
time and God. However, his belief in the afterlife might give some
insight into how he understood the relationship between them. Godel
expressed his belief in the hereafter in the following terms, “I
am convinced of the afterlife, independent of theology. If the world
is rationally constructed, there must be an afterlife” (Davis,
2002: 22). “His arguments were, as always, rationally based
on the principle that the world and everything in it has meaning,
or reasons. This is closely related to the causality principle that
underlies all of science: Everything has a cause, and events don’t
just ‘happen’” (Casti & DePauli, 2000: 87).
Mathematics, Science, and Faith
An
ultra-rationalist like Godel was a theist, a personalist and a believer
in the afterlife, and he appealed to reason as his witness. Atheists
and agnostics usually portray their philosophy as rational, discarding
the theist conclusion as a mere psychological refuge of the ignorant
or self-deceiving. Nevertheless, ultra-rational thinkers like Godel,
Leibniz,and Descartes have reached the theist conclusion. Is there
an apparent disconnect between rational thinkers and rational thought,
or is it that the theists’ view is the rational conclusion,
even if often embraced by fanatics in unimaginably irrational ways?
An objector may argue that science and mathematics are outside the
realm of faith, where theology may belong. However, a closer look
at the foundations of physics and mathematics, as well as to the
history of these subjects, seems to yield a different conclusion.
This closer look reveals a delicate membrane that conjoins these
experiences: Faith. This is the greatest common denominator of science,
mathematics, and theology.
Consider the nature of axioms in any formal system, including mathematical
systems. Once the axioms have been chosen, the accepted rules of
inference can potentially be entered into a computer to verify the
validity of any argument, but the axioms themselves are arbitrary.
For instance, the now-indispensable Axiom of Choice has troubled
many mathematicians since it was formulated by Ernst Zermelo in
1904. In addition, the elimination of the parallel lines postulate
in Euclid’s rendition of geometry has given rise to other
geometries. One of these geometries, hyperbolic geometry, finds
an important application in the theory of relativity.
Axioms may be useful, but there is no inherent truth in them. Changing
them alters the system and the true sentences produced by such a
system. If we consider that at present all that mathematicians have
are “axiom systems for which no one can give a convincing
demonstration of consistency,” the situation turns even more
discouraging (Nelson, 2002:5). Certainly, this is not the way most
mathematicians do mathematics, but the belief that one should at
least be able to theoretically place any mathematical
statement within the framework of a formal system is sacredly held
by the majority of mathematicians.
Furthermore, many thinkers believe that mathematics is the most
certain means of acquisition of knowledge, the consecrated pinnacle
of intellectuality. “This misperception leads to such embarrassments
as the pseudo-Euclidean form Spinoza gave to his Ethics. These writers
are too pedestrian in their view of mathematics and yet they give
us too much credit” (Nelson, 2002: 5). “Why do we mathematicians,
makers like poets and musicians, describe
what we do as discovery rather than invention? This is the “Pythagorean
religion” (Nelson, 1995: 3). According to Edward Nelson, most
mathematicians are devout followers of this religion, although they
attribute it to Plato, born over fifty years after Pythagoras’
death. Moreover, faith plays a vital role in science as well. When
considering the nature of energy and matter, the laws of physics
are taken as axiomatic. Certainly, if we believe “the world
is rational” and imbued with inherent order, as Godel did,
then taking the laws of physics as axiomatic might be acceptable;
yet, as in theology, faith remains a preliminary step to understanding.
Many scientists would argue that even though they cannot completely
(or partially) explain the origin of the universe – or the
origin of life, or the nature of consciousness, or the nature of
time – the answers would certainly not involve God. They have
placed their faith in their cognitive processes and in their colleagues.
They submit to those authorities;
but faith they have, nonetheless.
If we define faith as “belief based on the authority of the
information source,” be it Scripture, scientists, a friend,
a teacher, a digital picture, a DNA test, our own cognition and
experiences, or even politicians (for the really insane), we will
realize that faith plays an essential role in the development (or
destruction) of knowledge. Why is it acceptable in science and mathematics
to have faith, not only in the axioms or laws of nature, but also
in the peer-review process and the causality principle, while faith
in the religious realm is viewed as superstitious at best? As Godel
states, “Religious institutions are, for the most part, bad,
but religion is not” (Wang, 1996: 316).
George Berkeley had already questioned this attitude in 1734. In
“The Analyst” he wrote:
“Whether Mathematicians, who are so delicate in religious
Points, are strictly scrupulous in their own Science? Whether they
do not submit to Authority, take things upon Trust, and believe
Points inconceivable? Whether they have not their Mysteries, and
what is more, their Repugnancies and Contradictions?”
Perhaps, not being a mathematician himself, Berkeley was considered
“too pedestrian” in his view of mathematics, which accounted
for the dismissal of his ideas. To counter similar objections, rational
theists have tried to justify their beliefs by submitting to the
accepted rules of inference. However, one may argue that “[i]nferential
arguments are employed in a case where the existence of the thing
to be inferred is considered of doubtful character” (Sinha,
1999: 5). Yet, as remarked by Ludwig Wittgenstein, a philosophical
antagonist of Godel’s, those who want to provide an intellectual
basis for theism furnish arguments in favor of the existence of
God, although their actual belief is not based on the argument itself
(Davis, 2002: 22). Besides, the experience of the divine might well
be one of the limitations of rationality.
Godel’s Ontological Argument
Godel’s
ontological argument, like most ontological arguments, is based
on St. Anselm’s eleventh century work Proslogion. Anselm defines
God as “that thing which nothing greater can be thought”
(Small, 2006: 16). He asserts that even the atheist would agree
that God’s existence is possible, but that such existence
is simply a contingent falsehood.(Small, 2006: 16). Just as Michelangelo
must have envisioned his David before metamorphosing marble, the
atheist might argue that he can conceive of a world in which God
exists even if that world is not the true world. In the seventeenth
century, Rene Descartes, using an analogy with Euclidean geometry,
followed in St. Anselm’s footsteps. In the Fifth Meditation,
Descartes furthers the claim that “there is no less contradiction
in conceiving a supremely perfect being who lacks existence than
there is in conceiving a triangle whose interior angles do not sum
up to 180 degrees. Hence, […] since we do not conceive a supremely
perfect being – we do have the idea of a supremely perfect
being – we must conclude that a supremely perfect being exists”
(Oppy, 2002). (Ironically, in non-Euclidian geometries the interior
angles of triangles do not sum up to 180 degrees.)
In the eighteenth century, Gottfried Leibniz, co-creator along with
Isaac Newton of the Calculus, attempted to improve Descartes’
argument. He asserted that Descartes’ argument fails unless
one first shows that it is possible for a supremely perfect being
to exist. Leibniz argued that, since perfections cannot be analyzed
objectively, it is impossible to demonstrate that perfections are
incompatible – and he concluded that all perfections can co-exist
in a single entity, namely, God (Oppy, 2002).
This is the intellectual and historical framework Godel used to
devise his ideas. He admired Leibniz and attempted to improve on
his ontological argument. Some have questioned the validity of the
underlying modal logic, while others have objected to his set of
axioms and definitions. That is all they can do to the Godelian
argument since they cannot find fault with his flawless reasoning.
Some objectors adhere to Immanuel Kant’s position, who in
the eighteenth century argued against ontological arguments in general
stating that existence is not a predicate. That is, existence is
not a property of individuals in the same way being blue or strong
is; hence, existence cannot be proved (Small, 2006:18). Perhaps
the argument holds in propositional logic – the underlying
logic of mathematics – but the argument certainly fails in
modal logic. Godel’s argument, even if sound, does not settle
the question of a personal God, which was part of Godel’s
ethos. Neither does it address the question of uniqueness, at least
up to isomorphism. Nonetheless, even if his argument is not accepted
as a proof because of the questionability of the axioms chosen,
it still suggests a via positiva to understanding the idea of God
rationally (Small, 2006: 28).
Conclusion
“However,
as Bertrand Russell observed, it is much easier to be persuaded
that ontological arguments are no good than it is to say exactly
what is wrong with them” (Oppy, 2002). Yet, “[t]hose
who find the assumptions of the ontological argument suspicious
should ask themselves whether their suspicion is based […]
on an unwillingness to accept the conclusion of the argument”
(Small, 2003: 25). Likewise, those in favor of the argument should
ponder whether they have been lenient in their philosophical rigor.
Ultimately, however, existence is independent of belief. We may
argue for eternity whether God exists or not and it will not affect
God’s existence. However, it may affect ours. We should not
be naive and think we can convince any purportedly rational being
to accept theism. In spite of all our efforts in attempting to rationally
prove the existence of God, we must agree that we may fail to convince
even a single obstinate atheist shrouding his arguments with scientific
or philosophical jargon. What is remarkable about Godel’s
theological inclinations is that whereas “ninety percent of
philosophers these days consider it the business of philosophy to
knock religion out of people’s heads,” said Godel (Wang,
1996: 152), “he exploited the machinery of modern logic to
reconstruct Leibniz’s ontological argument” (Yourgrau,
2005: 13).
Blaise Pascal, fundamental in the development of probability theory,
might induce them to reconsider their position with his famed wager
published in 1670: God is or He is not. Let us weigh the gain and
the loss in selecting ‘God is.’ If you win, you win
all. If you lose, you lose nothing. Therefore, bet unhesitatingly
that He is. (Pensées)Hence, as an exponent of theism, Godel
is sempiternally victorious.
Towards a New Ontological Argument:
The
Existence of the Soul vis-à-vis The Existence of God
Abstract:
Ontological arguments are philosophical discourses that purport
to deductively prove the existence of God. The first recorded argument
in Western thought is due to St. Anselm, which mathematicians Descartes
and Leibniz later refined. This argument reached the pinnacle of
abstraction in the realm of symbolic logic in the twentieth century
thanks to Gödel. Nevertheless, the argument has met with fierce
critics and opponents. In this article I will briefly review the
history of the ontological arguments and discuss two major hurdles
in the traditional approach to the subject. The first obstacle is
a result of the consequences related to the definitions of God used
by the proponents of the ontological arguments. The second obstacle
lies in addressing the existence of God without first addressing
the existence of the soul. I will build on Gödel’s ontological
argument – whose theology is essentially Judeo-Christian –
and combine it with the philosophical framework of the Vaishnava
school of thought based on the Bhagavad Gita and the Vedanta Sutra.
I will present my rationale and plans for advancing an ontological
argument that first considers the existence of the soul, and once
that is established, proceeds to conclusively address the existence
of God.
Biography
Héctor
Rosario teaches mathematics at the University of Puerto Rico, Mayagüez
Campus. He was born in Puerto Rico in 1974 and earned his Ph.D.
from Columbia University in New York in 2003. His current academic
interests are the ontological arguments, the foundations of mathematics,
and the philosophy of science and mathematics. He is also an ordained
Gaudiya Vaishnava priest known as Ananta Ram das Adhikari.
Commentary by Swami B.A. Paramadvaiti:
Kurt
Godel’s contribution to OIDA therapy is very significant.
A scientist who is trying to prove the existence of a personal God
and any other features God may have. Faith is the foundation of
all thought, even the faith that there is a validity in combining
letters into words and to use them to express opinions. Godel says
“I am convinced of the after-life”. I am convinced,
I believe, I am sure of or I imagine are simply different degrees
of intensity of faith, and to say I know is just another variety
of it. OIDA Therapy steps back from this highly intellectualized
approach. OIDA-Therapy is concerned about the overall good of our
children, misguided by an educational system which claims to be
based on rational thought but is actually nothing more then the
belief that there is no God and that there is no meaning behind
the individual challenges we have to face. But psychologists such
as Jung and Maslow or great scientists like Godel, in their approach
towards the subject of faith are rather non denominational. But
in order to practice a particular faith and obtain the healing effect
of it, either an attitude of universal love with ecumenical broad-mindedness
or a process guided by a true mystic within a certain denomination
has to be applied. It does so appear that the universal love approach
does not have a concrete forum in our present day and thus it remains
a very abstract proposal, but the need for healing for many people,
if not all, is an immediate need. Therefore OIDA-Therapy encourages
us to accept the values of our faith to see how we can be more actively
involved in following our hearts progress. When the believer in
his mystical tradition follows an authorized process there should
also be a positive result regardless of the denomination. If one
searches for God without harming others he is on the right path,
and when he discovers that very highly advanced intellectuals came
to the same conclusion that faith is what is guiding all of us he
doesn’t have to feel ashamed or embarrassed anymore and can
follow his spiritual discipline with enthusiasm.
~~~
a)
Abraham Maslow
b) Erich Fromm
c) Viktor Frankl
d) C.G. Jung
e) Werner Heisenberg
f) Kurt Godel
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