The Problem of God and Abstract Objects: A Reassessment

1 Introduction

According to Gould (2011, 2014), there is an incompatibility between ‘Platonism’ – the view that there exists a realm of ‘abstract objects’, conceived of as entities that are non-spatiotemporal, causally effete and necessarily existing, and which are picked out by terms and predicates such as ‘property’, ‘proposition’, ‘relations’, ‘set’, ‘possible world, ‘number’ – and traditional ‘Theism’ – the belief in the existence of a personal being: ‘God’, identified as the perfect, independent, self-sufficient and ultimate source of all reality distinct from himself. That is, as noted by Gould (2011, 257; 2014, 2), one can consider the following three jointly inconsistent statements that are, however, required to be affirmed by an individual that ascribes to the combined thesis of ‘Platonic Theism’:

The three claims that are at the heart of (1) seem to be founded on good grounds, with the rejection of these claims leading to further issues. That is, if (i) is rejected, then, firstly, one loses out on the great benefits provided by Platonism to the nominalist/realist debate concerning the existence of universals (Gould 2014). Secondly, and more importantly, one is also presented with the task of having to explain away the indispensability of mathematical entities (such as numbers and sets) for contemporary science. That is, according to Quine (1960) and Putnam (1979), one should be ontologically committed to mathematical entities due to the fact that, as Colyvan (1998, 40) notes in further elucidating this ‘Quine-Putnam Indispensability argument’, the existence of mathematical entities is indispensable to our best scientific theories and thus – based on the widely held thesis of ‘naturalism’ – as we (ought to) have ontological commitment to all and only those entities that are indispensable to our best scientific theories, then one can conclude from this that we (ought to) have ontological commitment to mathematical entities – that is, one ought to be committed to a ‘platonic ontology’ that incudes within it a realm of abstract (mathematical) objects. So, given the use that Platonic entities perform in contemporary philosophy (and theoretical science), the rejection of (i) appears to be problematic. A rejection of (ii) leads to the problem of denying God’s ultimacy. More fully, for God to be ultimate, as Plantinga (1980, 78) notes, is for him to have ‘created everything distinct from himself…everything distinct from him depends upon him, and…everything is within his control’. However, as Plantinga (1980, 80) further writes, it is ‘is easy to see why the Platonic menagerie should be objectionable. If these abstract objects exist necessarily and have some of their properties essentially, then that they exist and are constituted as they are is not up to God’. That is, a rejection of (ii) is a denial of God being the creative source of abstract objects, thus him being the ultimate explanation of everything that exists – let’s term this the Ultimacy Problem. Hence, in warding off the Ultimacy Problem, and thus upholding the ultimacy of God and the existence of abstract objects, one must affirm the fact that all abstract objects are dependent on God for their existence, which is therefore a clear denial of claim (iii). So, if one wants to maintain claims (i) and (ii), then one must reject (iii). Yet, if one is, in fact, to reject claim (iii), then one must provide a solution to the Dependence Problem and the Bootstrapping Problem. For the former problem: the Dependence Problem, if one takes platonic entities to depend on God for their existence and nature, then one must detail the nature of the relation that can connect to types of necessary beings. Yet, it cannot be a relation of logical dependence (i.e. if x then y), as given that a necessary being must exist, then necessarily, if one of the necessary beings exists, then there is a mutual entailment of the other necessary being, and thus there is not an asymmetrical dependence relation between them (Gould 2014). Maybe the relation that is sought is not that of a logical dependence relation but that of a relation of causal dependence – such that abstract objects are causally dependent on God. This causal dependency between God and abstract objects, as Gould (2014, 4) notes, ‘seems to be just what we are looking for – an ontologically significant, asymmetrical or one-way relation of dependence running from each nondivine object to God’. Thus, the platonic theist can maintain the position that God, as the creator of all of reality that is distinct from him, eternally and necessarily creates (i.e. causes) the existence of abstract objects. However, in maintaining this position, the platonic theist is faced with the issue of explaining how it is metaphysically possible for any entity (including God) to create abstract objects? One can indeed also ask the questions of if the co-eternality of abstract objects with God renders him as less ultimate? And what notion of causation is required for one to affirm the fact of God eternally causing these entities? Finding answers to these questions, in addition to the problems raised concerning there being a logical dependence relation between abstract objects and God, has led some contemporary philosophers such as van Inwagen (2009, 17) to write, ‘no one, not even God, could be the creator of an abstract object’. This is based on the fact that, as noted by Gould (2014, 3), it is ‘logically impossible for any necessary being to asymmetrically depend on another’. Thus, given these issues, things seem to be bad for the platonic theist. However, the issues faced by the platonic theist can be further deepened by turning our attention to the latter problem: the Bootstrapping Problem.

The issue raised by the Bootstrapping Problem can be understood best when one considers properties, as, specifically with this class of abstract objects, it seems to be the case that one is faced with an incoherent state of affairs in affirming the veracity of Platonic Theism. As if, on the one hand, one takes God to be an entity that exemplifies properties, and if, on the other hand, one takes God to be the creator of all of the things distinct from himself, then he must also be the creator of his properties – given that his properties (i.e. abstract objects) are entities that are not identical to him. Yet, God cannot create any properties unless he already exemplifies the property of being able to create a property. Or for another example, as noted by Craig (2016, 60), ‘in order to create the property being powerful, God would already have to be powerful. An impotent God obviously could not create anything. Thus, God would have to already possess a property in order to create it, which is viciously circular’. That is, as Craig (2017, 144) further writes concerning this issue, ‘there seem to be good candidates for genuine properties which God must possess in order to be able to create properties. For example, being powerful is surely a genuine property which God shares with creatures and which is obviously a prerequisite for being able to create anything. But then we are stuck in a vicious circle again’. In short, one is thus entrenched in a vicious explanatory circle in which God must be the creator of all of his own properties that are logically necessary for his creative activity – which clearly seems to be incoherent. These issues, as Gould (2014, 4) notes, seem to show, firstly, the apparent intractability of the Dependence Problem in particular, and the problem of God and abstract objects in general, and, secondly, the multilayeredness of the problem that is presented to a platonic theist – as reasons provided by philosophy seem to lead one to affirm the existence of abstract objects, whereas reasons provided by theology seem to lead one to affirm the ultimate sovereignty of God. However, by one being led in both directions, and thus endorsing the thesis of Platonic Theism, one is presented with the dilemma of the Inconsistent Triad. Yet, a resolution to the dilemma by a rejection of one of the tenets of the triad leads to additional problems, which, in following Gould (2014, 4), one can state more succinctly as such:

The central question that is now presented to a platonic theist, who does not want to reject realism about abstract objects (i.e. Platonism), or adopt an ‘unorthodox’ construal of Theism, is: is there a means that one can provide to deal with the problem of God and abstract objects by finding a way out of the dilemma presented to by (1)? I believe that there is, by proceeding to reject claim (iii) of (1): the Platonist Assumption, however, in a manner that does not result in one being saddled with the Dependence and Boostrapping Problems. How this end can be achieved is by utilising some important notions from contemporary metaphysics to formulate a concept of God, and the relation between him and abstract objects, that is free from the problems raised. More specifically, in rejecting claim (iii) of (1), the Bootstrapping Problem will be warded off by adopting the thesis of Theistic Aspectivalism, which is a concept of Theism that employs the notion of an ‘aspect’ to reconceptualise the nature of God as that of an omnipotent being that has aspects,^[1] rather than properties. Moreover, the Dependence Problem will be warded off by one adopting the thesis of Theistic Essentialism and thus re-conceptualising the relation between God and abstract objects as that of an essential dependence relation, which will be taken to be the needed asymmetrical relation that can connect God to these entities – without, however, facing the issues of vagueness that was raised against the concept of the relation as a logical dependence or causal dependence relation. And thus, at the end of our analysis, an account of God’s relationship to abstract objects will be provided that is free of any dilemma presented by (1) and the implications of rejecting the tenets of it found in (2).

Thus, the plan is as follows: in section two (‘The nature of essentialism), I provide an explication of the notion of essential dependence provided by Kit Fine and Katherin Koslicki. And, in section three (‘The nature of aspectivalism’), I provide an explication of the notion of an aspect, provided by Donald L.M. Baxter. Then, in section four (‘The nature of theistic essentialism and aspectivalism’), I apply these theses within a theological context to provide a re-conceptualisation of the nature of God, and relation between him and abstract objects. Subsequent to this, in section five (‘Response to the problems’), I show how this re-conceptualisation provides an account that is free from the Dependence and Bootstrapping Problems, which will thus allow us to affirm the consistency of the thesis of Platonic Theism. After this section, there will be a final section (‘Conclusion’) summarising the above results and concluding the article.

2 The Nature of Essentialism

The thesis of Essentialism centres on the two philosophical notions of an ‘essence’ and ‘essential dependence’. The ‘essence’ of an entity expresses what it is to be a given entity, with one entity ‘essentially depending’ on another entity if the latter entity plays an important role in determining the identity of the former entity. More specifically, we can construe both of these concepts more succintly as follows:

Focusing our attention first on (3): the notion of an essence, as construed here, expresses what it is to be a certain thing. More precisely, Fine – and Lowe (2008, 2012a, 2012b, 2013), who follows in Fine’s footsteps on this matter – believes that one must proceed to detail the nature of essence through a non-modal approach to essence that is termed ‘serious essentialism’, and is focused on the further notion of a real definition. Serious essentialism, as noted by Lowe, seeks to follow Aristotle and, to a greater extent, John Locke, in construing an essence as ‘the very being of anything, whereby it is, what it is’ (Locke 1975: III, III, 15, quoted in: (Lowe 2008, 34)). Importantly, however, this specific approach to essence provided by serious essentialism does not make the further move of reifying essences and thus taking an essence to be a further entity in addition to the entity that possesses it. Rather, all entities have essences, but essences are not entities (i.e., an objectively real thing). Thus, as Lowe notes, an entity’s essence ‘does not literally contain any entities as parts or constituents, since only entities can have other entities as parts’ (Lowe 2013, 195). The ’parts’ that feature in an essence are parts of the real definition which express these essences. Hence, the central element of the non-modal construal of essence is thus that of essence acting (in some manner) as the definition of the entity in question. As Fine notes, ‘my overall position is the reverse of the usual one. It sees real definition rather than de re modality as central to our understanding of the concept’ (Fine 1994b, 3). And, as he further writes, that ‘just as we may define a word, or say what it means, so we may define an object, or say what it is’ (Fine 1994a, 2). In other words, what the essence of some entity x is, is what x is, or what it is to be x (Lowe 2008). An essence is thus the whatness of an entity and therefore constitutes its identity.^[2] Hence, serious essentialism seeks to provide a characterisation of the notion of essence that provides a means for one to identify, in a perspicuous manner, what an entity is. Essence is thus taken by Fine and Lowe to be analogous to a linguistic definition, in that, in a similar manner to the latter, which states what certain terms mean, essences, acting as real definitions of entities, state what those objects are. For Fine and Lowe, a statement of essence is a real definition, through it specifying what it is to be that certain entity. That is, it reveals and explains the essence of an entity (without being a distinct entity from that essence), as Lowe (2012a, 935) further writes:

a real definition of an Entity, E, is to be understood as a proposition which tells us, in the most perspicuous fashion, what E is, or, more broadly, since we do not want to restrict ourselves solely to the essences of actually existing things, what E is or would be.

Real definitions thus serve as explanatory principles and are (usually) formulated through a ‘<To be___>’ construction, such as ‘<To be X is to be Y>’ that, if successful, express the identity of the specific entity and provide definitive answers to questions concerning what that entity is. These types of questions, as Sam Cowling (2013, 4) notes, can be termed what-questions, which ‘ask for the metaphysically significant features of an individual and are answered only if they explain that some individual really is’. The essence of an entity and, more importantly, the real definition that is associated with it, thus provide proper answers to what-questions – in other words, a statement concerning the essence of an individual should provide a correct, if partial answer, to questions concerning the identity of a given entity. And these ‘answers’ can be provided at a more ‘general’ level or more ‘specific’ (‘individual’) level, as the essence of an entity comes in two different types: a general essence and an individual essence. Now, what is central to this distinction is the fact that a given entity x must be a thing (i.e. an instance) of some general kind – at the very least, it must belong to an ontological category. And thus, as Lowe writes, if ‘x is something of kind K, then we may say that x’s general essence is what it is to be a K, while x’s individual essence is what it is to be the individual of K that x is, as opposed to any other individual of that kind’ (Lowe 2008, 35). For example, we can take Socrates to be an instance of the general kind Human (i.e. he is a human-instance), which results in Socrates’ general essence being what it is to be human, and his individual essence being what it is to be Socrates, as opposed to any other human. This definitional characterisation of essence – that is now specifiable as a general or individual essence – is thus the one that is needed for the task of accurately elucidating the nature of an entity, as the essence of an entity is those propositions that are part of the entity’s ‘definition’ and thus this approach, unlike the modal approach, enables only relevant propositions to be included within the essence of an individual (Fine 1995). In summary, the notion of an essence within the non-modal essentialist framework under question is thus the collection of the propositions that are true in virtue of the nature of that entity and thus express, through a real definition, what it is to be a given entity in the most perspicuous way possible With this conception of an essence in hand one can thus further understand the nature of (4): the notion of essential dependence.

At a general level, the notion of ‘essential dependence’ finds its home in the wider category of ‘ontological dependence’ relations. Ontological dependence, as Fabrice Correia (2008, 1,013) writes, is a term that stands for ‘a non-well delineated, rich family of properties and relations which are usually taken to be among the most fundamental ontological properties and relations – along with part-whole, exemplification, or again existence’. This notion of ontological dependence – which includes within it such relations as rigid existential dependence (i.e., x rigidly depends upon y _=df Necessarily, x exists only if y exists) and non-rigid existential dependence (i.e., x non-rigidly depends upon F _=df Necessarily, x exists only if the F exists) etc. – is distinct from the many types of dependence relations postulated by philosophers. As noted previously, one type of dependence relation is that of causal dependence, which relates cause and effect, in accordance with the notion of causation, such as the way in which Julius Caesar’s death causally depends on Brutus’ stabbing (Koslicki 2012). Another type of dependence relation, as also noted previously, is that of logical dependence, which obtains between propositions, in accordance with the laws of logic, such as the way in which the truth of the conclusion of a valid argument logically depends on the truth of the premises (Koslicki 2012). Moreover, a further type of dependence relation is that of probabilistic dependence, which obtains between different entities, in accord with some probabilistic theory, such as the way in which the decay of a particle probabilistically depends on the half-life of the particle or the way in which an individual’s recovery from an illness depends on them having taken some medicine (Koslicki 2012). These three types of dependence, even though they play a prevalent role in various areas of philosophical discourse, do not express a relation in the deep, ontological sense. The specific variety of ontological dependence that does fulfil this role, according to Koslicki (2012, 2013) is that of essential dependence. The relation of essential dependence is best understood as that of an entity x, being ontologically dependent on another entity y, if y is a constituent of a proposition that it is true in virtue of x’s nature (or x is a constituent of an essential property of y) (Koslicki 2012). Given this sort of definitional relationship, the entities which feature within the real definition of another entity are said to be in a constitutive relationship. In the words of Koslicki (2013),^[3] we can construe this constitutive relationship as one of essential constituency, which can be defined as follows:

The entities that fulfil the role of being the essential constituents of another entity pertain to (or feature within) the real definition of that specific entity and therefore contribute to defining the entity as it is. This constitutive relationship can hold between an entity and their essential constituent, in regards to its general essence – where the essential constituent of the entity has its general essence (i.e., kind identity) determined by this entity through it being a constituent of the part of its real definition that expresses what it is to be of its kind. Moreover, this constitutive relationship can hold between an entity and their essential constituent, in regards to its individual essence – where the essential constituent of the entity has its individuation determined by this entity through it being a constituent of the part of its real definition that expresses what it is to be that specific individual. Taking this all into account, we can thus further illustrate these notions through the Table 1.

At a general level, a given entity depends on another entity essentially if the latter entity fulfils the role of being an essential constituent for that entity – that is, it is a constituent of a proposition that is part of the real definition of that entity. Hence, if an entity depends on another entity in this way, then the former entity cannot exist without the latter entity, as Fine (1995, 279) writes, ‘What is maintained is that, given that one object is ineliminably involved in the nature of another, then it is not compatible with the nature of the second that it should exist without the first’. An entity cannot exist unless everything upon which that entity’s identity depends on also exists as well. Hence, a relation of essential dependence holds when the essence of a specific entity involves that of another entity – such that one entity is part of what it is to be the other entity by it serving as a constituent of the real definition of that entity. The existence and identity of the dependent entity are determined by the entity that it is essentially dependent on. There is thus an asymmetry in this type of relationship, in that the dependent entity has the entity that it is dependent on as a constituent of its real definition, but the latter entity does not have the former entity as a constituent of its own real definition. Thus, in summary, in an essential dependence relationship, the essence (i.e., identity), and thus existence, of an entity is asymmetrically determined by another entity – on the basis of the latter entity being an essential constituent of the former. Essential dependence is thus a fine-grained, asymmetrical dependence relation of great ontological significance that can be put to great use. Before we do that, it will be important to now turn our attention to the second philosophical framework under analysis: Aspectivalism.

3 The Nature of Aspectivalism

The thesis of Aspectivalism centres on the philosophical concept of an ‘aspect’.^[4] The concept of an ‘aspect’ plays a helpful role in providing a coherent conceptual foundation for the notion of ‘qualitative self-differing’. More specifically, we can construe this concept succinctly as follows:

For the notion of an aspect, as posited by Baxter (1999, 2016, 2018a, 2018b, 2018c), qualitative self-differing (hereafter, self-differing) is the qualitative differing of some entity in one way (or respect) from itself in another.^[5] Self-differing is thus the qualitative differing of numerically identical aspects possessed by an individual (Baxter 1999). That is, the same individual can possess qualitatively differing aspects that are nevertheless numerically identical with the individual, and thus, given the transitivity of identity, with each other. To help motivate the existence of aspects within this context, we can consider a case in which an individual is torn about what to do (or how to feel) in a certain situation:^[6]

David is an ardent philosophy professor and is also a loving and faithful father of two children, Jacob and Melissa. Now suppose that, firstly, David has an upcoming philosophy conference in which he is the keynote speaker and, due to other work commitments, has not prepared his speech yet. Secondly, suppose that David had previously promised that he would reward his children with a camping trip this upcoming weekend if they achieved A* grades in their A-Level results. And, thirdly, suppose that Jacob and Mellissa have both, in fact, recently achieved A* grades in their A-Level results.

In this specific scenario, David is in a situation of self-differing as he knows that he has an important keynote speech that he needs to prepare. David being an ardent philosophy professor results in him wanting to fulfil this commitment and thus complete his speech. So, the following proposition would be true: David ‘does not want to take his children on a camping trip this upcoming weekend’. However, having promised his children that he would reward them for their academic achievement, and being a loving and faithful father, he wants to fulfil his promise to them. So, the following conflicting proposition would also be true: David ‘wants to take his children on a camping trip this weekend’. David is torn. He is in conflict with himself. He thus differs from himself. David’s struggle is between two aspects of him: David insofar as he is a philosopher versus David insofar as he is a father. This, and other cases of internal conflict, are cases of self-differing, where the subjects of what differs are the aspects of the individual that self-differs. Thus, for the case to be one of differing, one aspect must possess a quality that another aspect lacks. And for it to be a case of self-differing, the aspects must be numerically identical to the individual that bears them (Baxter 2018a, 907). Off of this introduction to the notion of an aspect, we can further elucidate this notion at two levels: the semantic level and the ontological level.

At the ontological level, according to Baxter, aspects are difficult to distinguish from other entities.^[7] However, we can begin to acquire an understanding of their nature by describing their functional role and the relationship to the individuals that bear them. Primarily, the aspects of an individual function as the particular ways of being of that individual. A way of being is a conceptually primitive notion that, as noted previously, can be glossed in part by taking it to be the way or manner in which an entity exists. Thus, aspects function as the particular ways in which individuals are. However, as ways of being of an individual, aspects are not qualities (or properties) as they can, themselves, possess qualities (or properties) due to their numerical identity to the individuals that bear them.^[8] Aspects, however, do not possess all of the qualities that the particular individuals that they are aspects of have. Moreover, in a similar manner to their bearers, they are particular entities, rather than universals, through Leibniz’s Law (in an unrestricted sense) failing to hold for them. Secondly, despite the numerical identity between individuals and their aspects, aspects are not ‘complete individuals’ due to the fact that complete individuals are entities that can exist independently. Instead, according to Baxter, aspects are ‘incomplete entities’ due to them ‘having fewer properties than it takes to exist on one’s own’ (Baxter 2018a, 916). Aspects are thus incomplete in the sense of them being dependent upon the complete individuals that they are numerically identical to.^[9] The nature of a complete individual determines the aspects that they have, in that they depend entirely upon how that individual entity is – once we have the individual, we also have its ways of being (Giannotti 2021). Thirdly, aspects are not mereological parts of the individuals that they are aspects of, as, again, they are numerically identical to, rather than a ‘part’ of, these individuals (Baxter 2018a). Lastly, aspects are not mental abstractions. That is, even though a complete individual’s aspects are abstract entities (through them failing to exhaust the content that they are aspects of),^[10] that can be considered by means of abstraction (where one abstracts a way that an individual is), it is important to note, as Baxter writes, that the difference between a complete individual and their aspects is ‘a less-than-numerical distinction but more than a mere distinction of reason’ (Baxter 2016, 99). Baxter terms this distinction, an aspectival distinction, which results in the aspects of an individual only ever being two (or more) in a ‘loose’ sense – when they are counted based on qualitative distinction. However, in a ‘strict’ sense – when the aspects are counted based on a numerical distinction – they are only ever one. Thus, aspects, as Baxter notes, provide a ‘complexity to the simple, i.e., a qualitative complexity to the quantitatively simple’ (Baxter 2016, 178). Hence, in regards to the notion of self-differing, we having being best understood as the qualitative differing of numerically identical aspects possessed by an individual with aspects being able to differ in their qualities without the resultant differences indicating numerically distinct individuals (Baxter 2016, 175). Thus, what we have with the aspects of an individual is that of the negation, as Baxter (2016, 104) writes, being internal ‘that is, has short-scope relative to the nominal qualifier and so there is no contradiction’. Thus, it is the aspects of Jane that have the conflicting qualities noted above, but not Jane (unqualified). That is, one can block the secundum quid ad simpliciter inference, which, following Baxter (2018a, 913), is that of it being the case in aspectival context that ‘it doesn’t follow from the fact that an aspect of a complete individual x is F that x is F’. That is, according to Baxter (2018a, 913), an individual insofar as they are a particular way bearing a particular quality does not entail that the individual unqualified bears that same quality. Yet, despite the distinct possession of a quality by an aspect, it is important to continue to maintain the fact that each of the aspects of an individual is identical to the individual. Hence, the position that has been reached here is that of the possibility of the same individual possessing qualitatively differing aspects that are nevertheless numerically identical to the individual that bears them and also with each other. Aspects thus allow, as noted by Baxter, ‘contradictories to be predicated of the same thing in a way that Leibniz’s Law is silent about’ (Baxter 2018a, 172). Rather it is important to consider the domain of quantification for Leibniz’s Law. That is, according to Baxter, Leibniz’s Law (i.e. the Indiscernibility of Identicals) solely applies to individuals (i.e. complete/independent entities) and thus does not generalise over to aspects (i.e. incomplete/dependent entities). Thus, on this basis, we can conclude with Baxter that Leibniz’s Law does not apply to aspects, and thus it is coherent to posit the existence of qualitatively differing, yet numerically identical aspects. In summary: an object can bear aspects which are conceived of as qualitatively differing, yet numerically identical abstract particular entities that function as ways in which that object is – which ultimately enable it to bear a range of incompatible qualities, without an entailment of a contradiction. More can indeed be said here concerning the nature of the entities featured in the theses of Essentialism and Aspectivalism; however, for the task at hand, we can take the central components of the notion of an essence/essential dependence and an aspect to have been laid out. We will now turn our attention to applying these two notions to the task at hand so as to provide a conception of God that is not subject to the Bootstrapping Problem.

4 The Nature of Theistic Essentialism and Aspectivalism

In firstly applying the thesis of Essentialism within a theological context – which we can term a Theistic Essentialism – we can now provide a new conceptualisation of the relationship between God and abstract objects and thus provide grounds for dealing with the Dependence Problem. Within Theistic Essentialism, God is taken to be the source of all things distinct from himself, which includes that of abstract objects. How God can fulfil the role of being the creator of these types of objects, which are taken to be necessary, eternal and causally effete entities, is through there being a relation of essential dependence between God and each abstract object. More specifically, an essence, as previously noted, focuses on stating what it is to be a certain thing through the notion of a real definition – a collection of all the propositions that are true in virtue of the nature of an entity. As all entities have essences, all abstract objects are taken to have an essence – and thus a real definition that expresses this essence. The question is: do abstract objects have their essences (definitions) in and of themselves, or are they essentially dependent upon another entity? The Platonist Assumption of (iii), noted previously, urges us to affirm the independence of abstract objects. However, following Lowe (2012b),^[11] we can take an abstract object to be one that is dependent on what it is abstracted from, and thus it is right to take these types of entities to be mind-dependent entities. That is, an abstract entity is one that is, by its very nature, one that is abstracted from or drawn out of, or away from something else. Hence, any such entity must thus be understood as an entity that depends for its existence on that which it is ‘abstracted’ (Lowe 2012b). The primary candidates for being abstract objects: numbers, propositions, sets etc., are objects of reason – that is, they are entities that stand in rational relations (such as mathematical and logical relations) to one another. Hence, plausibly, it does not seem to make sense, as Lowe (2012b, 189) notes, ‘to think of such entities as existing and standing in such relations independent of some actual or possible mind that could contemplate and understand them’. Hence, there is thus a good candidate for the sort of entity that abstract objects can be abstracted from – namely, a mind of some kind upon which they would thus depend on for their existence. Given this, one can thus take it to be the case that an abstract object is indeed one that is a dependent entity – it is dependent on a mind – with the type of mind that is needed to contemplate and understand all of the range of abstract objects being possessed by God. Hence, abstract objects are dependent on the mind of God – from which they are abstracted. This dependence though can be construed not only existentially, but, more importantly, we can take it to be one that holds essentially. More precisely, each abstract object is taken to be essentially dependent on God in the sense of God being a constituent of a proposition that is true in virtue of that abstract object’s nature. That is, God is an essential constituent of abstract objects in the sense of him being a constituent in the real definitions of them. In other words, we take it to be the case that God is in a constitutive relationship with abstract objects by him being featured in the real definitions of them – and thus contributes to defining these objects as they are. This constitutive relationship between God and abstract objects holds specifically in regards to their general essence – as given that these entities are taken to be abstractions from the mind of God, the kind identity of each abstract object – which is that of them being abstracta – is determined by God being a constituent of the part of their real definition that expresses what it is to be of their kind. However, for these types of objects, God does not serve as an essential constituent in regard to their individual essence. That is, the individuation of each abstract object – which is that of them being, for example, a property, set or relation etc. – is determined by other entities that are constituents of the part of their real definition that expresses what is to be that specific individual. Hence, for abstract objects, there is a dual dependence in play: one stemming from God, who serves as the essential constituent of their general essence, and another stemming from other entities that serve as the essential constituents of their individual essence. We can further illustrate the position here through Table 2.

God thus stands in a relation of essential dependence with abstract objects as the general essence of these objects involves him. God is part of what it is to be an abstract object by serving as a constituent of their real definitions, and thus their general essence – with other entities s also serving as constituents of their real definition; however, this is solely in regard to their individual essence. Thus, the existence and identity of an abstract object is ultimately determined by God as the primary entity in which they are essentially dependent on – as, without God, these entities would not be instances of the kind that they are – namely, that of abstracta (by being ‘abstracted’ from the mind of God). Thus, there is an asymmetry in the relationship between God and abstract objects – as each object has God as a constituent of its real definition, but God does not have any of these objects as a constituent of his own real definition. Hence, in the relationship between God and abstract objects, the essence (i.e. identity), and thus existence, of these objects is asymmetrically determined by God. We thus have within a theistic essentialist framework the relation that we are looking for: an essential dependence relation which is, in a similar manner to a causal dependence relation, an ontologically significant, asymmetrical or one-way relation of dependence that runs from each of the abstract objects to God. Thus, instead of the dependence relation between God and abstract objects being one of a logical dependence relation or a causal dependence relation (as was previously posited in a ‘hypothetical’ sense by Gould (2014), we now take it to be one of essential dependence, which allows one to affirm the position of God being the ultimate source of all abstract objects in the sense of these objects being essentially dependent on God – that is, God is a constituent of the real definitions of these objects, and thus they ultimately depend for their existence and identity on him.

Now, in applying the thesis of Aspectivalism within a theological context which we can term Theistic Aspectivalism – we can now provide a new conceptualisation of the nature of God and thus provide grounds for dealing with the Bootstrapping Problem. Within Theistic Aspectivalism, God is identified as an omnipotent being. Given his omnipotence, God would be an entity that is unlimited in knowledge, presence, freedom and goodness. That is, these attributes – omniscience, omnipresence, perfect freedom and perfect goodness – are derivable from the supposition that God is an omnipotent being.^[12] However, in construing God as a omnipotent being, we can see that the derivable attributes of God, which are normally taken to be attributes that are related to God (and each other) by an ‘entailment relation’ are, in fact, now within the aspectival framework, related to God (and each other) by a relation of ‘numerical identity’. More specifically, God does not exemplify any numerically distinct properties; rather, any intrinsic characteristic ‘attributable’ to God must be numerically identical to him. For example, if the intrinsic property of goodness is attributed to God, then one is not properly attributing to him an ontologically distinct property that he exemplifies. Rather, God is instead taken to be identical with his goodness (and all the other properties that are attributed to him as well). Moreover, given that God is identical to each of his attributes, one must also infer that his attributes are identical to each other due to the transitivity of identity. Thus, God’s identity with his goodness and his power entails the fact of his goodness being identical to his power (and, again, for all of the other properties that are attributed to him). Now, the manner in which we can best understand this is by taking God to possess aspects rather than that of properties. That is, instead of the omnipotent being that God is possessing properties through the entailment of these properties from his omnipotence, we can now ‘convert’ these properties into aspects, which will also result in the entailment relation being converted into a relation of numerical identity.

Given these conversions, one can now further understand the nature of these ‘aspects of Omnipotence’ by focusing on their functional role and the relationship that they have to omnipotence, which allows us to say that they are not properties, complete entities, or mereological parts. Rather, they are incomplete abstract particular entities that are numerically identical to a specific complete individual and function as his ways of being. More fully, each of the aspects of Omnipotence is numerically identical to omnipotence, yet they do not possess the same characteristics as it – they are each not the ability to perform any logically possible action. Lacking this characteristic, the aspects of Omnipotence are thus incomplete entities, in that they are dependent on the omnipotence, which exists as a complete entity (i.e. an independently existing entity). These aspects of Omnipotence do not exhaust the content that they are aspects of (i.e. they each do not exhaust omnpotence), and they each function as ways that this omnipotent being exists, which we can consider through a process of abstraction. This aspectival construal of the divine properties thus allows us to re-define the traditional set of divine properties as such (Table 3):^[13]

At a specific level, these aspects of Omnipotence are focused on the different particular ways in which the omnipotent being is. That is, by being having (or, more specifically, being) the singular-character of omnipotence, it would exist in a particular manner and have certain limitless abilities that enable it to fulfil different roles. This functional role fulfilled by Omnipotence allows one to establish an aspectival distinction that takes these ways to be aspects of this specific being. Therefore, as was seen in our previous example, we have a case of self-differing here. The subjects of this differing would be the aspects of the omnipotent being, with each aspect possessing a ‘quality’ that each of the other aspects lacks, yet, despite their differing, these entities are numerically identical to the one omnipotent being and thus each other as well. That is, within an aspectival context, the same thing: the omnipotent being, which is numerically identical to God, is discerned in multiple ways without absurdity. More specifically, within this aspectival framework, there is one being, an omnipotent being, that is identical to multiple aspects, which are, in turn, identical to one another. In short, God is an omnipotent being, and this omnipotent being is the qualitatively differing aspects of Omnipotence, and the qualitatively differing aspects of Omnipotence are one another. The traditional understanding of the possession of ‘many qualities’ by God is thus, in fact, the possession of many qualitatively differing, yet numerically identical aspects. The aspects of Omnipotence provide a certain ‘complexity to the simple’ – a qualitative complexity to the quantitatively simple omnipotent being which God is. Thus, by utilising an aspectival distinction here, in a ‘loose’ sense, focused on qualitative distinctiveness, we can indeed count a multiplicity of aspects within God. Yet, in a strict sense, focused on numerical distinctiveness, there is solely one self-same entity, the omnipotent being, which is differently considered. This is thus the conception of the nature of God that is affirmed within a theistic aspectival framework. And thus taking into account the position reached here, and the philosophical frameworks of Theistic Essentialism and Aspectivalism, we can now turn our attention now onto addressing the Dependence Problem and the Bootstrapping Problem – both of which can be avoided within the philosophical frameworks that we are operating within.

5 Conclusions

In conclusion, this article focused on providing a re-assessment of the relationship between God and abstract objects. This goal was achieved by utilising the philosophical framework of Essentialism and Aspectivalism, which provided the metaphysical basis for dealing with the charge of inconsistency raised against the combinatory thesis of Platonic Theism. Within a Theistic Essentialist and Theistic Aspectivalism framework one can indeed proceed to affirm Platonic Theism without, however, being charged with incoherence concerning God’s creation of causally effete, necessary and eternal abstract objects. That is, one can thus maintain the position of God being the sovereign creator of all things, including that of the category of abstract objects that postulates the existence of entities that have come to play an important role within contemporary philosophical (and scientific) theorising.