by Dr. Hector Rosario
Dr. Hector Rosario holds a Ph.D. in mathematics education from Columbia University and was a tenured professor of mathematics at the University of Puerto Rico at Mayagüez (2003-2015). He is currently the Director of Outreach and Festivals for the Julia Robinson Mathematics Festival, an organization whose mission is “to inspire joy in mathematics through exploration and collaboration.” Dr. Rosario has published three books: Mathematical Outreach: Explorations in Social Justice Around the Globe (2019), Math Makes Sense! A Constructivist Approach to the Teaching and Learning of Mathematics (2016), and Mathematics and Its Teaching in the Southern Americas (2014).
(iAbstract: I (de)construct an ontological argument based on Gödel’s proof and Oppy’s parody. This approach addresses (1) omnipotence as a defining characteristic of God; (2) the existence of the soul; and (3) God’s existence and individuality.
Introduction
It is generally accepted that God’s existence can neither be proved nor disproved scientifically. However, Tarski argued that every mathematical science is deductive, and every deductive science is mathematical. Therefore, I propose that the existence of God can be proved scientifically – using mathematical logic or some other deductive method – provided we agree on a definition of God, a set of axioms, and some rules of inference.[1]
The first recorded ontological argument is attributed to St. Anselm, which was later refined by Descartes and Leibniz. This argument reached the pinnacle of abstraction in the realm of symbolic logic in the twentieth century thanks to Gödel. Nevertheless, the arguments have fierce critics. In this article I will briefly review the history of ontological arguments and discuss a major hurdle in the traditional approach to the subject by philosophers of an Abrahamic background. This obstacle is related to omnipotence and its consequences – like the origin of evil – in the definitions of God provided by adherents of ontological arguments.
Toward the end, I (de)construct an ontological argument for the existence of the soul based on Gödel’s ideas and Oppy’s parody of the argument. The goal is to adjust Oppy’s contrivance to reflect an aspect of Vaishnava theology based on Bhagavad Gita, Srimad Bhagavatam, and Vedanta Sutra. To accomplish this, I present the philosophical foundation of my approach and contrast it with some key Abrahamic theological needs. Subsequently, the existence of God and the concept of logical truth are addressed.
A Very Brief History of Ontological Arguments
Ontological arguments are proofs for the existence of God based on a definition of God, a set of axioms, and accepted rules of inference. Gödel’s ontological argument, like most ontological arguments, is based on St. Anselm’s eleventh century work Proslogion. Anselm defines an omnipotent God as “a being than which nothing greater can be conceived.” A crucial aspect of his argument is that contingent existence is greater than possible existence, and that necessary existence is greater than contingent existence. (Exactly in what way one is greater than the other may reflect a Platonic inclination.) He asserts that even an atheist would agree that God’s existence is possible, but that such existence is simply a contingent falsehood. Just as Leonardo da Vinci must have envisioned L’Ultima Cena before creating it, an 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, Descartes followed in Anselm’s footsteps by furthering an ontological argument in the Fifth Meditation. In the eighteenth century, Leibniz 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. In the twentieth century, Hartshorne and Gödel, among others, furthered Leibniz’s ideas, setting the dispute of demonstrating God’s existence in the realm of symbolic logic.
Modal Logic and Ontological Arguments
Symbolic logic is a mathematical model of deductive thought, just as modern probability theory is a model for situations involving randomness and uncertainty.[2] In general, the purpose of logic is to characterize the difference between valid and invalid arguments. In particular, modal 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.
Some detractors of ontological arguments adhere to Kant’s objection, who stated in the eighteenth century 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. Perhaps the argument holds in predicate logic – the underlying logic of mathematics – but the argument certainly fails in modal logic. In fact, Gödel’s ontological argument does not explicitly assume that existence is a predicate, but that the modal status of an individual can be regarded as a property. For instance, a number greater than 1 exists necessarily (at least Platonically), whereas the Statue of Liberty – American sentiment notwithstanding – exists contingently. The distinction between them, that is, whether an object exists necessarily or contingently, can be regarded as a property of each object. Incidentally, Anselm argued that a being that exists necessarily is greater than a being that exists accidentally. Greater, I think, from the viewpoint of causality.
The resistance and attacks against ontological arguments have motivated the development of finer, perhaps even simpler, ideas. I attempt to address a sense of inadequacy springing from the definitions of God given by advocates of the arguments, as omnipotence is tacitly denied or compromised by them.
Philosophical Framework
Omnipotence is a core tenet of Vaishnava philosophy, as much as it is for Abrahamic traditions. Vaishnavism also satisfies another Abrahamic theological need – monotheism –although as a result of omnipotence, God manifests polymorphously. Moreover, as an omnipotent entity, God must also have the ability to experience and exchange sentiments.
Srila Prabhupada comments on Srimad Bhagavatam 2.7.24:
The Personality of Godhead has every sentiment of a sentient being, like all other living beings, because He is the chief and original living entity, the supreme source of all other living beings. He is the nitya, or the chief eternal amongst all other eternals. He is the chief one, and all others are the dependent many. The many eternals are supported by the one eternal, and thus both the eternals are qualitatively one. Due to such oneness, both the eternals constitutionally have a complete range of sentiments, but the difference is that the sentiments of the chief eternal are different in quantity from the sentiments of the dependent eternals. […] As it is said in the beginning of the Srimad-Bhagavatam, the Absolute Truth is the source of everything, so the Absolute Person cannot be devoid of the sentiments that are reflected in the temporary mundane world. Rather, the different sentiments found in the Absolute, either in anger or in mercy, have the same qualitative influence, or, in other words, there is no mundane difference of value because these sentiments are all on the absolute plane. Such sentiments are definitely not absent in the Absolute.[3]
[God] is the origin of everything, animate and inanimate, and is the cause of all causes – sarva-karana-karanam. He is the Supreme Controller and the prime mover of all cosmic manifestation. He has Universal Consciousness and He is the well-wisher of every living being. He is beyond the perception of the material senses; however, His symptoms are visible in the effects (products) of His creation. […] He is Supreme Eternal among all eternals and Supreme Consciousness among all consciousness – nityo nityanam cetanas cetananam (Katha Upanisad 2.2.13). […] Every living being has a jiva, or soul, in it. In other words, all microorganisms, insects, aquatic living beings, plants, reptiles, birds and so on have souls. […] In the Bhagavad-gita (15.7), we find the following: mamaivamso jivaloke jivabhutah sanatanah, which means that all living beings are eternal and conscious particles of the Supreme Lord. […] However, the difference between Isvara, God, and jiva, the living entity, is that the consciousness of the jiva is localized. In the words of Srila Bhaktisiddhanta Sarasvati Thakura, Isvara is Absolute Infinity, and jiva is absolute infinitesimal. In other words, the living being has the same spiritual quality as that of the Supreme Lord, but the capacity of the living being is limited whereas the capacity of the Supreme Being is unlimited.[4]
Omnipotence, Personality of God, and Gödel
A maxim found in Srimad Bhagavatam 3.26.49 states “that the cause exists in its effect. Following this argument, since we all experience very clearly that all developed living beings have individual personalities, it is certainly conceivable that the Supreme source of all must also possess [a] personality”.[5] Gödel concurs that God must be personal in the sense we are discussing, for if God lacks the ability to play the role of a person, then his omnipotence is compromised. 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”.[6] Gödel in turn thought “Einstein’s religion [was] more abstract, like Spinoza and Indian philosophy. Spinoza’s god is less than a person; mine is more than a person; because God can play the role of a person”.[7] Therefore if God existed, reasoned Gödel, then he must be able to play the role of a person. An opponent may counter that this seems like a rather peculiar argument from potentiality for God’s personalism, as the same argument could be advanced for God’s playing the role of a stone, a girl, or a lion. In fact, omnipotence, as understood in Vaishnavism, allows God to take these shapes, or at least, will not constrain God’s potential “to play the role” of one creature or another, as God so wishes.
As the supreme independent entity, God is at will to manifest certain attributes, while keeping others hidden. God may even show an attribute and its negation simultaneously. This might be unacceptable for Gödel since he defines a God-like entity to be the possessor of all positive attributes, but not of their negations. For instance, Anselm assumed that God cannot lie, for he considers that a weakness and not a potency. However, God’s omnipotence is then compromised by Anselm’s imposition of his value system on God. This observation is crucial in the development of a Vaishnava ontological argument. For a Vaishnava, omnipotence means the potency to do everything, including the null action.
Vaishnavism on Free Will and the Problem of Evil
It appears that many intellectuals, who admit to believe in God, either believe in some kind of impersonal god or some type of an evolving deity. The problem of evil is one of the issues that most troubles the adherents of these doctrines. They see the omnipotence of God as inconsistent with God’s absolute benevolence. Unable to reconcile the existence of evil with the omnipotence and absolute goodness ascribed to God, they opt to drop omnipotence from the list of God’s attributes. They feel more comfortable with that approach rather than facing the inferential rejection of God’s absolute benevolence. However, I argue that this confusion springs from a misunderstanding of the interplay between omnipotence, evil, and free will.
Regarding an evolving deity, to say that God evolves places God at the whim of an objective nature – apparently more powerful than God – that determines God. Under this prism, God is reduced to a mere sorcerer who may know much, have some power, and might even be eternal (or able to grant you eternity), but who is nonetheless subject to higher forces.
Some people see God as a supplier of pleasure if one pays the market price (in the form of penance, prayer, mercy, among other practices). In contrast, Vaishnavas see God as someone we can choose to serve without any other motive than to love him. Furthermore, since God is the Creator of all that exists and the origin of all that be, then he must be the origin of evil itself. This is where many theologies either collapse or look for explanations to resolve the problem of evil.
Vaishnavism maintains that God is the origin of everything, good and bad. In fact, he is the origin of evil itself – which manifests via free will. Free will is the (limited) ability to choose. God is the possessor of absolute free will (complete independence), whereas souls, as parts and parcels of God, have only a limited amount of said free will (incomplete independence). Hence, because we possess free will we can either choose to walk towards God (and love him) or walk away from God (and rebel against him). Free will allows for the possibility of a loving relationship with God, since clearly love never exists in slavery. However, if we misuse our independence, all kinds of havoc occur.
The concepts of karma (action) and reincarnation may be introduced to fully address the question of evil. However, as it stands, the question of evil is no longer a problem, but a mere consequence of free will, which in turn exists as a primary constituent of God’s infinite potencies. Thanks to free will we can relish a relationship with God.
B.R. Sridhar Goswami states:
Once, an Indian political leader, Syamasundara Chakravarti, asked our spiritual master, Srila Bhaktisiddhanta Saraswati Thakura Prabhupada, “Why has the Lord granted such freedom to the jiva (soul)?” Prabhupada told him, “You are fighting for freedom. Don’t you know the value of freedom? Devoid of freedom, the soul is only matter.” Freedom offers us the alternative to do right or wrong. Once, Gandhi told the British authorities, “We want freedom.” They replied, “You are not fit to have self-government. When you are fit, we shall give it to you.” But finally, he told them, “We want the freedom to do wrong.” So, freedom does not guarantee only acting in the right way; freedom has its value independent of right and wrong.”[8]
The concept of free will is not addressed by ontological arguments. However, this is needed for a Vaishnava ontological argument to complement our definition of God as the possessor of all qualities, irrespective of their categorization.
Gödel’s Ontological Argument
Much has been written about Gödel’s ontological argument. Gödel showed his proof to Dana Scott, who in turn presented it at Princeton University during the fall of 1970. However, according to Gödel’s biographer, John Dawson, Gödel never published his ontological argument for fear of ridicule by his peers. It was finally published posthumously by Sobel in 1987 and was later included in Volume 3 of Gödel’s Collected Works. Many recent discussions of Gödel’s ideas have centered on Sobel’s (mis?)understanding. He claimed that Gödel’s axioms would lead to a modal collapse; that is, that all truths are necessary truths. This is obviously unacceptable for a philosopher. In 1990 Anderson provided an emendation in an attempt to salvage Gödel’s argument from Sobel’s conclusions. Although Sobel’s objection has been questioned and Gödel’s original argument revisited by Fitting, Small and others, I will base this discussion on Anderson’s emendation, and will adapt what Oppy intended to be a parody of Gödel’s work. Here is Anderson’s emendation as presented by Oppy.
Definition 1: x is God-like if x has as essential properties those and only those properties which are positive.
Definition 2: A is an essence of x iff (if and only if) for every property B, x has B necessarily iff (if and only if) A entails B.
Definition 3: x necessarily exists iff every essence of x is necessarily exemplified.
Axiom 1: If a property is positive, then its negation is not positive.
Axiom 2: Any property entailed by a positive property is positive.
Axiom 3: The property of being God-like is positive.
Axiom 4: If a property is positive, then it is necessarily positive.
Axiom 5: Necessary existence is positive.
Theorem 1: If a property is positive, then it is consistent.
Corollary 1: The property of being God-like is consistent.
Theorem 2: If something is God-like, then the property of being God-like is an essence of that thing.
Theorem 3: Necessarily, the property of being God-like is exemplified.
It is established that the theorems listed do follow from the axioms. Nevertheless, some have questioned the appropriateness of axioms 2, 4, and 5. This is related to the belief that axioms must be inherently true, as opposed to being true simply by convention. I will address this when we consider the concept of logical truth.
Sketch of a Vaishnava Ontological Argument
I use Anderson’s formulation of the Gödelian argument, as presented by Oppy. Then I make some emendations to Oppy’s parody, which he uses to claim that “[Gödel’s] proof is demonstrably no good even if the conception of properties and/or the modal logic are accepted”.[9] He creates the following template for constructing parallel arguments that lead to “absurd results.”
Definition 1*: x is God*-like iff x has as essential properties those and only those properties which are positive, except for P1,…, Pn.
Definition 2: A is an essence of x iff for every property B, x has B necessarily iff A entails B.
Definition 3: x necessarily exists iff every essence of x is necessarily exemplified.
Axiom 1: If a property is positive, then its negation is not positive.
Axiom 2: Any property entailed by a positive property is positive.
Axiom 3*: The property of being God*-like is positive.
Axiom 4: If a property is positive, then it is necessarily positive.
Axiom 5*: Necessary existence is positive, and distinct from each of P1,…, Pn.
Theorem 1: If a property is positive, then it is consistent. Corollary 1: The property of being God*-like is consistent.
Theorem 2: If something is God*-like, then the property of being God*-like is an essence of that thing.
Theorem 3: Necessarily, the property of being God*-like is exemplified.
Oppy’s main objection is that “for each collection of positive properties which includes necessary existence, there is a being which has exactly those positive properties, and which has no other positive properties. Even if there are only (n+1) positive properties, this gives us 2n distinct necessarily existent beings. Not good. (Perhaps n isn’t finite. If so, things are even worse.)”[10] In summary, he claims that with this version one can conclude the existence of multiple (perhaps an infinity of) gods. Oppy further claims that he “is inclined to draw the conclusion that the defender of the original argument is playing a losing hand. Perhaps I am wrong about this; but I do not think that I will be alone in thinking that there are no prospects for the production of a Gödelian ontological argument which does not admit of these kinds of damaging parallels.”[11]
However, Oppy’s invention can be slightly amended to produce an ontological argument consistent with Vaishnavism where such “damaging parallels” are reinterpreted. However, let us note first that Oppy’s exclusion of a finite number of positive properties from the definition of God must be rejected by adherents of omnipotence – even Anselm’s conception – since such a definition would present a serious limitation on God. Nevertheless, let us take Oppy’s template and simply replace ‘God-like’ by ‘soul-like’, since I see no conflict if this limitation is applied to individual, non-omnipotent souls.
Thus, the existence of soul-like entities is established. These Platonic entities cannot be inanimate due to the nature of the attributes. Moreover, these entities are eternal, with positive essential attributes or their negations. Besides, according to Vaishnavism, souls are not created but are eternal individual portions of the Lord. Hence, it is valid and sound to claim that the souls’ existence is not contingent, but necessary. I concede this is a significant difference from Abrahamic traditions.
Furthermore, the question of the multiplicity and perhaps infinity of soul-like entities is no longer a concern but a desired consequence. Let God be the class of all soul-like entities with all positive properties and their negations. (A bolder step may be to define God as the Set of all sets.) Then God possesses every attribute and its negation, including the property of being a person or not. This should be acceptable since, as the origin of all that exists, God is also the seed of all contradictions. Therefore, one may conclude not only the existence and uniqueness of God (by construction), but also of the Platonic reality of his personality.
An opponent may argue that to define God as the Set of all sets one does not need an ontological argument at all. Besides, the convention rules against the existence of such an entity. Am I playing a losing hand? I think not, because, echoing Anselm, “God is a being than which nothing greater can be conceived.” What can be greater than the Set of all sets? (Interestingly enough, Errett Bishop dismissed set theory as “God’s mathematics, which we should leave for God to do.”)
However, another opponent may counter that God can do anything other than that which is logically impossible. Nevertheless, what is logically possible – like the soul-like entities of our discussion – is a consequence of agreed upon definitions, axioms, and rules of inference. As an individual, I can submit to that limitation, but I see no justification to apply it to an omnipotent entity. Perhaps we should revisit our faith in logical truth.
The Pythagorean Religion
The previous point reflects a profound philosophical bias in favor of logical or mathematical truth. Edward Nelson says it beautifully.
Numbers were invented (or revealed, as believers would maintain) by Pythagoras. Numbers are divine, the only true divinity, the source of all that is in the world, holy, to be worshiped and glorified. Such is the Pythagorean religion, and such is the origin of mathematics. This is the religion from which I am apostate. […] From the perspective of monotheistic faith, we reject the religious idea of numbers as divine and uncreated. What then are they? Are they created? Everything in creation is contingent; every created thing is dependent on the will of the Creator for its being. If numbers are uncreated, they are divine – this we reject. If numbers are created, they are contingent – this we find absurd. What other possibility is there? Simply that numbers do not exist – not until human beings make them. Despite the assertion of William Butler Yeats that “Things out of perfection sail,” very few would maintain that the poem of which that is the first line existed before Yeats made it. Why do we mathematicians, makers like poets and musicians, describe what we do as discovery rather than invention? This is Pythagorean religion.[12]
Conclusion
I have purposely used the term existence rather loosely trusting that the reader may have a better sense than I about what that is. With that concession, I have given a sketch for a scientific proof of God’s (Platonic) existence. That is, for a God that is consistent with some essential aspects of Vaishnavism, like omnipotence. What does that mean? Absolutely nothing, for ascribing reality to such a god would imply ascribing reality to all logical truths – to this I object. (In this age of religious intolerance, I should guard against Pythagorean fundamentalists.) But why would anyone create an ontological argument and then whack it like a 5 de Mayo piñata? Well, deductive thought is seductive. Yet, the mystery of spiritual life often escapes our purview.
References
[1] A reader may invoke Gödel’s Incompleteness Theorems as a counterargument. For such a potential misunderstanding, I recommend T. Franzén’s Gödel’s Theorem: An Incomplete Guide to Its Use and Abuse, 2005.
[2] H. Enderton; Mathematical Introduction to Logic, Harcourt Academic Press, San Diego, 2001, p. xi.
[3] A.C. Bhaktivedanta Swami Prabhupada, Srimad Bhagavatam, Bhaktivedanta Book Trust, Los Angeles, 1972, pp. 44-45.
[4] T.D. Singh; “Vedanta and Science I – Human Life and Evolution of Consciousness,” Savijñanam, 2002, Vol.1, pp. 54-55.
[5] T.D. Singh, God is a Person, The Bhaktivedanta Institute, Kolkata, 2006, p. 67.
[6] A. Einstein; Telegram to New York’s Rabbi Herbert S. Goldstein, April 24, 1929.
[7] H. Wang; A Logical Journey: From Godel to Philosophy, MIT Press, Cambridge, MA, 1996, p.152
[8] B.R. Sridhardev Goswami, The Search for Krishna, Guardian of Devotion Press, West Bengal, 1989, p. 69.
[9] G. Oppy, “Gödelian Ontological Argument,” Analysis, 1996, Vol. 56 (4).
[10] G. Oppy, “Gödelian Ontological Argument,” Analysis, 1996, Vol. 56 (4).
[11] G. Oppy, “Gödelian Ontological Argument,” Analysis, 1996, Vol. 56 (4).
[12] E. Nelson, “Confessions from an Apostate Mathematician,” http://www.math.princeton.edu/~nelson/, 2000.