If we consider the problem of Being as the question of whether it is fundamentally One or Many, then Plato’s dialogue Parmenides seems to answer with the undecidable. According to the theologian Cornelius Van Til, this conclusion is reached if no distinction is made between eternal or infinite categories and temporal or finite categories. This is precisely what Christian theism invites us to do, by discovering in the doctrine of the ontological Trinity the One and the Many as equally ultimate. The most radical distinction of Being should therefore be sought rather in the duality between the Infinite and the Finite, which reappears strangely in the theory of sets, the Cantorian foundation of mathematics and the dialectic of the One and the Many.

1 The ontological problem of the One and the Many

1.1 Multiplicity and Form in Plato’s Parmenides

When we speak of the ‘universe’, we mean both the unity of all things and their diversity or multiplicity. As far back as antiquity, the question of how to think beyond the universe to Being was raised, and we know that by the time of Socrates the concept was already fraught with difficulties and insurmountable paradoxes. “Parmenides” is one of Plato’s most complex and enigmatic dialogues, featuring the philosopher of the same name and Socrates. It explores in depth ideas about the nature of existence, the theory of Forms (or Ideas) and various dialectical methods.

The dialogue begins with a meeting between the young Socrates, Zeno and Parmenides. Zeno first presents his arguments against the pluralists, who believe that there is a multiplicity of real things, and then Socrates proposes his theory of the Forms, which holds that there are immutable and eternal realities (the Forms) that are more real than the sensible objects of our experience, which ‘participate’ in these Forms. Parmenides raises several objections to the theory of the Forms, and the second part of the dialogue consists of a series of dialectical exercises led by the master, which explore various hypotheses concerning the existence or non-existence of the Forms.

1.2 The problem of Being: is it One or Many?

The ‘Parmenides’ thus tackles the question of Being from the angle of the One and the Multiple, through this series of dialectical exercises. The purpose of these exercises is to examine the logical consequences of various hypotheses concerning the existence of the One as Form; Parmenides and Zeno then highlight the paradoxes that can arise when we try to conceptualise Being in terms of unity or multiplicity. These paradoxes are illustrated by a series of deductions which show that, whatever hypothesis one adopts (that the One is or is not, that Being is multiple or not), one is confronted with contradictions and logical impasses.

The dialogue does not provide a definitive answer to these problems, but it does illustrate the difficulty inherent in conceptualising Being, whether in terms of unity or multiplicity. This discussion is often seen as a critique or assessment of the limitations of Plato’s theory of Forms itself, but the conclusion we want to draw here, following Parmenides, is that the question of whether Being is One or Multiple is undecidable. The paradoxes raised by the philosopher raise questions about the capacity of human rationality to fully grasp the nature of Being, and the theologian Cornelius Van Til proposes to overcome these limits by introducing an ontological distinction between ‘eternal’ and ‘temporal’ categories (categories are the dimensions of Being).

2 Van Til’s theistic solution to the problem of Being

2.1 Interpreting reality through biblical presuppositions

Cornelius Van Til, an American Reformed theologian of Dutch origin, founded a school of apologetics (a branch of theology devoted to the defence of the Christian faith) that is now called “presuppositional” or “presuppositionalist”. His radical method is distinctive in that he sets out to establish that all non-Christian philosophies are fundamentally incoherent or irrational, and that only a worldview based on the Bible can truly give a coherent or rational account of reality.

As a good apologist, Van Til was well versed in the philosophical tradition, particularly that of Greek antiquity. In this respect, the problem of the One and the Many, as addressed in Parmenides, was for him a direct link to the knowledge of God. Without necessarily embracing all the aspects of his apologetic approach, some of which are intrinsically problematic, we can nonetheless follow his idea of looking to Christian theology for the metaphysical presuppositions that will allow us to propose, if not a resolution, at least a displacement.

2.2 Distinguishing between the eternal and created One and Multiple

According to the Presbyterian apologist, God, as an eternal and transcendent Being, is both One and Multiple. He is One in essence in that he is a single personal God, but Multiple in persons in that he is eternally a Tri-unity: the Father, the Son and the Holy Spirit. This Trinitarian nature of God allows him to be both absolutely unified and diversified at the same time, and is a ‘solution’ to the problem of the One and the Many which, according to Van Til, is unique to Christianity. Let us add that for Van Til, in God neither unity nor plurality is in itself ultimate, in the sense that one would derive from the other: thus the Parmenidean impossibility of deciding whether Being must be conceived fundamentally as One, or fundamentally as Multiple, remains. For Van Til, Parmenides errs in that he poses the question of Being ‘in general’, without distinguishing between God, who possesses Being in himself, and creation, whose Being derives from God. This duality of the One and the Many thus refers back to the Trinitarian God revealed in the Bible.

Creation, as a derived and contingent reality, reflects this duality. It is multiple in its countless manifestations, but it has an underlying unity in that it derives from the creative and providential act of a single God: here we find the composite idea of “universe”, but reinterpreted in the light of Christian theism. It is hardly surprising, then, that the formal, undifferentiated approach set out in “Parmenides” fails to solve the problem of Being. Van Til argues that it is only the biblical presuppositions that allow for rational coherence between unity and diversity, and that without these presuppositions, philosophy falls into incoherence, as this dialogue by Plato demonstrates, in his view. And yet this leads us to consider the ‘Cantorian’ handling of the One and the Many in the mathematical theory of sets, where we find an echo of the problem of Being, and its original solution proposed by Van Til.

3 Mathematical echoes of the question of the One and the Many

3.1 The mathematical ontology of set theory

If the duality between the One and the Many is constitutive of reality, then, to use Paul Tillich‘s terminology, we would rather call it a polarity: each of the poles is necessary in order to think together being in all its complexity (Van Til speaks with the idealist philosophers of ‘limiting concepts’). Now, this polarity between the One and the Many is itself a tension, which is played out in set theory, the foundation of modern mathematics laid by the (Lutheran!) mathematician Georg Cantor. Sets are by definition “pluralities” or “multiplicities” made up of objects called their “elements”, and can themselves be considered as units and elements of other sets. In other words, Cantor’s theory systematises precisely this polarity between unity and multiplicity.

Cantor’s theory essentially makes it possible to reconstitute all the classical mathematics developed since Antiquity, based on this single concept and the so-called membership relationship between objects and classes (the ‘ultimate’ version of the concept of set). In other words, numbers, figures, space, time, relations, functions… all mathematical objects can be described as multiplicities. Now, the class of all these objects, the mathematical ‘meta-universe’, is intrinsically infinite, in a rigorous sense that makes it possible to reduce all mathematical infinities to a purely mereological conception (referring to the relationship between the whole and the parts) that set theory makes it possible to define precisely. The entire natural mathematical structure, starting with the natural integers of elementary arithmetic, is thus reconstituted on the basis of an intrinsically infinite reality.

3.2 The finite-infinite duality as a radical structuring of Being

The notion of finitude, the dual concept of infinity, is also fundamental to mathematics, and the duality between finite and infinite runs through and structures the entire mathematical universe and discourse. Now, the intuitive definition of mathematical infinity proceeds from a negation of the finite, just as certain attributes of the divinity are defined in ‘apophatic’ theology, and while the intuitive definition of the finite set, i.e. that of a set that can be enumerated by the first integers 0,1,…,n up to a certain integer n, is positive, it is nonetheless extrinsic, in that it calls on objects external to the multiplicities under consideration, the natural integers. On the other hand, with the German mathematician Richard Dedekind, we arrive at an intrinsic positive definition of mathematical infinity – that which is “equipotent with a proper part” – from which the finite is defined intrinsically in a negative way.

Thus, the analytical elucidation of these two limiting concepts of the finite and the infinite through the mathematical foundation of set theory reveals the infinite as a positive and therefore supposedly original notion, and the finite as a negative and therefore derivative notion. Van Til’s ontological motif is found here in the foundation of mathematics, when it is rooted in a theory of the One and the Many. And if pure mathematics is, as Edmund Husserl put it, a ‘formal ontology’ – ontology is the ‘discourse on being’, and thus in philosophy the ‘theory of being’ – we have here a solid, lay argument to support what is perhaps the essence of Van Til’s thesis on a strictly philosophical level: the polarity between the One and the Many is not the primary structuring of Being, but it is the duality between the Infinite and the Finite that provides a radical answer to the ontological problem: the infinite understands itself in itself and therefore comes first. And this is perhaps no different from what Tillich is saying when he boldly reinterprets the God of Christian theology as the “infinite foundation of being”. Or, to put it with Wolfhart Pannenberg:

[God] is there as the indeterminate infinite, which constitutes the primordial intuition of our perception of reality in general, the horizon on which we conceive all things, by limitation. The idea of the infinite, which Descartes says is a condition for our perception of finite objects, is not an awareness of something determinate, so it is not an awareness of God. It is the indeterminate perception of a something, which, as finite objects are constituted, will become aware of that which goes beyond them, as well as the world as a whole. Present in our lives and in our world, something is there beyond all finite objects. Something is there in the world, encompassing all finite objects and at the same time transcending them, acting in our world as in our own life. He can and will be called ‘God’ in the context of a concrete revelation, a religious experience and interpretation of the world.

Wolfhart Pannenberg, Systematic Theology I