1 Causalism, Neuroscience and Persistence
– -1.1 Park and Callard
– -1.2 Neuroscience, Callard and Empirical Challenges
– -1.3 Katz and Temporal Ontology
2 Epistemology and Causalism
– -2.1 Inconceivability and Conceivability
3 Views on Abstract Objects
– -3.1 Plebani’s Combinations
– -3.2 Plenitudinous Platonism
– -3.3 Set-Theoretic Realism
– -3.4 Thomism and Mathematical Objects
– – – -3.4.1 A Thomist Consideration: Plenitudinous Fundamentalism
– – – -3.4.2 Maurer’s Unique Exposition
– -3.5 Neopythagoreanism and Mathematical Objects

4 God, Time and Abstract Objects
– -4.1 Feser’s Augustinian Proof
– -4.2 Pure Atemporalism
5 Other Views on Abstract Objects
– -5.1 The Problems of Theistic Platonism and Others
– -5.2 For DEP
– -5.3 Divine Conceptualism, Augustinianism and Classical Theism
– -5.4 Defense of ABPW
Bibliography

Notes:

First, I do not use footnotes in this post.

Second, Find sections of interest using the command ‘CTRL+F’ typing in the preferred section, then pressing enter.

Third, If you’re going to use this post in any sort of way of distribution, make sure to cite it. All fair use rules apply.

🙂.

1 Causalism, Neuroscience and Persistence

1.1 Park and Callard

In his Can Mathematical Objects be Causally Efficacious?, Seungbae Park introduces Benjamin Callard’s position where he thinks mathematical objects can causally affect the human brain and calls it mathematical causalism. Mathematical causalism, to describe, affirms the ontology of mathematics and also rejects the objection that explicitly states that humans cannot obtain any knowledge of mathematical objects. The objection, called the epistemological objection of platonism, says that because platonic objects, or rather, in this case, mathematical objects are aspatial, atemporal, and causally inert, there is in no way humans, who are what Park calls, “cognitive agents”, can obtain the knowledge of such objects because they exist, (supposedly) in a realm that has no space or time. As long as we affirm the existence of spacetime, we exist in it. Most people would say the existence of spacetime fits perfectly with our intuition and objectively exists, this view is more understandable in Aleksandar Mikovic, in his paper Gödel’s incompleteness theorems and Platonic metaphysics. From his paper a very interesting argument can be made, it goes like this: 

(1) Spacetime objectively exists, 

(2) if we affirm the existence of spacetime, we acknowledge the existence of its ontology. (3) The ontology of spacetime is geometric, 

(4) Geometric objects are mathematical objects. 

(5) Mathematical objects are Platonic objects, and (6) Math objectively exists. 

But going back to the point of this, with the existence of spacetime, anything within that realm, according to Benacerraf and his advocates of the epistemological objection, nothing within the realm can obtain knowledge of anything outside of it. Callard, the mathematical causalist, would identify this as false. Callard thinks that rather than the cognitive agent trying to obtain knowledge of the mathematical object, the mathematical object causally purports the knowledge to the cognitive agents within the spacetime realm. In the section “Metaphysical vs. Actual” Park said, “There is no contradiction, or any other conceptual or metaphysical difficulty, in accepting the claim that abstract objects impart energy to us, and thereby change us, without themselves receiving any energy or suffering any change’ (2007, 351).” (Park 2018) He went on and tried to make a refutation to this by saying “Newton’s third law of motion, however, prevents an object from imparting energy to another object without receiving energy from it. So, the law of action-reaction clashes with the causalist¬ suggestion that a mathematical object can impart energy to the brain without receiving any energy from it.” (Park 2018).

As almost anyone would do, Callard said the third law of motion is “Just an empirical truth” (Callard 2007). The empirical part of the third law of motion is that it asserts the spatial interactions between two bodies of space. It never goes on about spatial interactions occurring between abstract entities and concrete entities, therefore arguing the third law of motion against mathematical causalism would be absurd. And even more, the reason why we may also call this an empirical truth is because spatial bodies are in, what Park calls, the concretum realm, meaning they exist where all other living or non-living concreter exist, therefore the third law of motion explicates a law between empirical objects, namely any object in the concretum realm.

Park goes on, saying: “Callard’s foregoing defense indicates that he believes a mathematical object can impart energy to the brain, where ‘can’ means a metaphysical possibility, not an empirical or nomological possibility.” (Park 2018) He then goes on to further argue that an explanans cannot be a metaphysical possibility, or possibilia, whereas the explanandum is a metaphysical actuality, or an actualia. The explanandum, the statement or the fact, is that ‘all cognitive agents have mathematical knowledge’.

As we know, the epistemological challenge proposed by Paul Benacerraf in Mathematical Truth, articulated in its best form, makes it clear that we are actually having mathematical knowledge, but the problem that is insisted is on how we gain that knowledge. This epistemic problem was originally put face to face with Platonism. Now that we come back to the fact about Callard having instead put forward a metaphysically possible explanans for an explanandum, rather than a metaphysically actual explanans for the explanandum. Doing this gets us nowhere, and instead of purporting a metaphysically possible framework of interactions between concrete entities and abstract entities, we must purport a metaphysically actual framework of the interactions between concrete entities and abstract entities. But even in this there are a few holes. Metaphysics, as we know it, is almost entirely abstract in thought, and is thus the conception of abstract theories, ones not inherently founded in the empirical sciences, or not entirely theoretically accessible through empirical objects. Similar Carnap, since metaphysics isn’t wholly attainable through empirical objects or existent in a domain of empirical sciences, for better sustainability and more attainability, we shall eliminate metaphysics, and all metaphysical statements.

In “Logical Empiricism at its peak”, Carnap introduced a new concept to eliminate metaphysics called logical analysis. Logical analysis yields that the cognitive contents of metaphysical statements have terms that are erroneously believed to have meaning, or that a metaphysical statement contains meaningful words, but are arranged in a counter-syntactical way so that they do not have any meaning. “Logical analysis rests on one thesis, that all metaphysical statements are pseudo-statements thereby eliminating metaphysics and it attempts to yield to the underlying nature of reality” (Carnap 1996). But perhaps we shall be anti-realist of eliminativism of metaphysical statements in order to continue discourse.

Herein the anti-realist of mathematical causalism and realist of eliminativism of metaphysical statements about metaphysical objects and abstract mathematical entities and realist of mathematical knowledge (perhaps I shall call this anti-mathematical causalism) runs into a problem. In this case, to deny the advancements that are made by the mathematical causalist wherein there are metaphysical possibilities about the framework of the interactions between concrete entities and abstract entities, and consequently affirming the actualism of mathematical knowledge and subsequently not giving an account of the acquisition of mathematical knowledge is absurd. Some may try to object this and call the position of the anti-mathematical causalist an agnostic stance, but affirming ‘we have mathematical knowledge’ subsequently leads to the necessary prerequisite of how we obtain mathematical knowledge. And thus, any inquiry in response to “Why do we have mathematical knowledge?” as the anti-realist, requires a response, and cannot be let go. So, by epistemic necessity, the anti-realist must give some sort of account of our acquisition of mathematical knowledge instead of blatantly denying other frameworks that yield metaphysical explanations of the acquisition of mathematical knowledge. Some may be curious and ask “Why an epistemic necessity?” Say there is interlocutor 1 and interlocutor 2, or [X] and [Y]. [X] and [Y] have both entered into discourse about the philosophy of mathematics, particularly about mathematical realism and mathematical anti-realism. [X] is a mathematical platonist and is therefore disposed to the thesis that “Mathematical objects are spaceless, timeless, unchanging and causally inert”. [Y] is an anti-realist, and believes we have mathematical knowledge. [Y] contends [X]’s position of mathematical platonism with the epistemological objection to Platonism, (Benacerraf’s objection). [Y] says:

“You acknowledge that we humans exist in space time, yet adhere to the sudden and supposed fact that we have knowledge of these abstract entities existent in a realm where there is no parameter of space, and no time flows. I now move over to examine this phenomenon with an inquiry. I admit we have mathematical knowledge, but I am in condemnation with the solution you have come to explain the acquisition of our mathematical knowledge. So, I end with asking for more information. How can us, agents within a realm where space is a parameter and time flows, have knowledge of entities that are abstract, and exist in a realm where space is not a parameter, and no time flows?”

[X] responds with:

“I am now aware of the sudden conundrum. You question and condemn my solution for the acquisition of mathematical objects. But now I am becoming curious, whereas I have come to the point of wanting to expose your position. Say, you adhere to the sudden fact that us agents have mathematical knowledge, but, correct me if I am wrong, it seems to be that you have no current or actual solution in response to the question of how we obtain mathematical knowledge. So, now I ask you, why is it that you go to try and falsify expositions made on this question, rather than formulating or coming up with a solution adherent to you?”.

[X] goes on to say:

“It also is that you have this unfulfilled account that needs to be filled with a solution, as it is almost as if you affirm the ontology of mathematics, yet you cannot prove it”.

The reason this is an epistemic necessity is because there is an account of knowledge that needs to be filled by [Y] necessarily, otherwise, there will be an epistemic quandary that will get us nowhere until the anti-realist does something about it. What will be examined in this section as well is pertinently the location of mathematical objects, and particularly what Park said “Suppose that a brain state correlated with mathematical knowledge occurs, and that neuroscientists are trying to investigate the cause of the brain state. Where should they look? Should they look inside the brain or outside the brain?” (Park 2018). When we talk about brain states and neuroscience, more specifically computational neuroscience, the diagnosing of them in reference of mathematical knowledge, especially when we try to operate under the framework of mathematical causalism, it’s very hard, specifically because of the elimination of the spatial and temporal subsisting properties.

The defining of brain states is still “missing and the underlying dynamical complexity remains unknown” (Kringelbach and Gustavo 2020). But we can still give a practical definition. “The contention here is that computational neuroscience offers a mechanistic framework for characterizing brain states in terms of the underlying causal mechanisms and dynamical complexity” (Kringelbach and Gustavo 2020).

In order for us to examine brain states and how they could possibly have an interaction with abstract entities, we must operate under a mechanistic framework that includes causal mechanisms. These causal mechanisms and their dynamical complexities perform under time and space. Meaning there cannot be any causal interactions, or any interactions that include perceivability with entities outside of time and space. Callard says “It may be strongly necessary that any object receiving energy must change. But we do not have any reason to think that it is strongly necessary that if abstract objects impart energy to us (as part of the process of acquiring knowledge about them) they must themselves receive energy” (Callard 2007). This means that if “efficient causation requires energy” (Callard 2007), and if brain states can receive energy from mathematical objects, then there is a causal interaction between brain states and mathematical knowledge.

Callard goes on to say “There is no contradiction, or any other conceptual or metaphysical difficulty, in accepting the claim that abstract objects impart energy to us, and thereby change us, without themselves receiving any energy or suffering any change” (Callard 2007).

With this, if it were the case, then mathematical objects can impart energy to brain states, thereby giving them mathematical knowledge. So, where we would look is in the neuroimaging data. And just to be clearer, this neuroimaging data gives rise to the dynamical activity in the brain states and the causal mechanisms that it has started or interacted with. Most have said that the dynamic activity in the networks of the brain state can only happen within space and time, but this is only included in the study of spatiotemporal dynamics of neuroimaging data, meaning that the information given to the brain is within a spatiotemporal realm, and because this information is acquired through sense perception, it is empirical, therefore making spatiotemporal dynamical complexities in brain states empirical, and any theoretically correct claim about them is therefore empirically true. We can officially circle back to the question of where the neuroscientist should look.

1.2 Neuroscience, Callard and Empirical Challenges

I had shown a new form of anti-realism unknowingly subscribed to by the anti-realists of the ontology of mathematics. I will finalize the examination of Park’s question of where neuroscientists should look for the mathematical knowledge in the brain state. I will also present a new argument against the anti-realism of mathematical causalism. This argument will diagnose the theses that this anti-realism rests on. I said that we could officially circle back to the question of where neuroscientists could look. Since it is established that spatiotemporal dynamical complexities are empirical, any activity that is a spatiotemporal complexity is empirical.

We know that the mathematical causalist asserts abstract entities interacting in causal mechanisms with the brain, therefore, making them a part of the brain activity which shows in neuroimaging data the energy that was imparted by the mathematical object, neuroscientists should disregard spatiotemporal dynamical complexities because mathematical objects are aspatial and atemporal and therefore cannot put themselves in a dynamical complexity that is included within space and time. Meaning the inquiry has changed and is no longer where the neuroscientist should look, but rather how can the neuroscientists hypothesize the best way to account for and explain these energies that are not included within spatiotemporal dynamical complexities. This argument would simply be targeted towards the anti-mathematical causalist, for I will explain the categorical propositions the position rests on. The propositions the anti-mathematical realist rests on are (i) mathematical knowledge exists, (ii) aspatial, atemporal, unchanging, causally powered mathematical objects do not exist, (iii) metaphysical statements can be eliminated, (iv) mathematical knowledge exists in a domain of spatiotemporal dynamical complexity.

We shall further diagnose these four propositions and see if the anti-mathematical causalist is in better consideration. The anti-mathematical causalist isn’t seen to have any actual present providing of mathematical knowledge. The anti-mathematical causalist rather just says that mathematical knowledge exists and when the neuroimaging data is looked at, the mathematical knowledge is founded in a spatiotemporal dynamical complexity. Therefore, making the anti-mathematical causalist argue from ignorance. Perhaps rather than having mathematical knowledge solely empirically, dialectical partners of anti-mathematical causalism can look at apriority in mathematics in a way that can still accommodate thesis (ii). 

An argument for anti-mathematical causalism is the empirical challenge to platonism. This challenge has very similar proponents to the above arguments against anti-mathematical causalism. The empirical challenge to platonism explicates that what mathematical causalism explicates is not aligned with our empirical laws. Some of these empirical laws could be the conservation laws of energy, the third law of motion, etc. The conservation laws of energy can be argued as (i) mathematical causalism asserts that mathematical objects that are aspatial and atemporal can impart energy to concrete or empirical objects. (ii) the conservation laws of energy say that energy is only conserved throughout the concretum realm, the physical world. (iii) The mathematical causalist asserts the existence of mathematical knowledge, (iv) mathematical knowledge rests on brain states (Callard 2007). (v) So, mathematical objects must impart energy to the brain, (vi) the conservation laws of energy are true, (vii) mathematical objects cannot access energy (ii, vi), (viii) mathematical causalism is false (ii, iv, v, vi, vii).

The other empirical challenge argument that can be used against mathematical causalism goes like this: (i) Mathematical causalism asserts causal relata between concrete objects and abstract objects (aspatial, atemporal, unchanging, causally capable mathematical objects), (ii) causation between an empirical object and a mathematical object aren’t within the domain of the empirical sciences, (iii) or, causation between mathematical objects and empirical objects isn’t empirically testable and is unreliable, (iv) therefore mathematical causalism is false (ii, iii). Both of these arguments are that of the likes of an empirical dogma.

For the second argument, Callard says that “We might think that the grounds of the challenge are to be found in the idea that efficient causation must proceed by contact (or impact)—that contact necessarily involves touch, and that non-spatial objects cannot touch anything. But why think that efficient causation must proceed by contact in this (touch-entailing) sense? Efficient causal relations unsupported by contiguity relations are perfectly intelligible and (therefore) possible, even if they are not actual” (Callard 2007). Contiguity means the “touch-entailing sense” that Callard so once stressed the definition. He even quotes Dretske about causal theories of truth in epistemology, and quotes about the mathematical causalism, in which the quote says “As a child, I never found the visual exploits of Superman (seeing through buildings) incoherent or logically paradoxical. I still don’t. This was just a fanciful (“fanciful” because, as things stand, no one can see things in this way) account of an extraordinary individual who could see things in ways that no one else could. Historians tell me that the ancients had even more bizarre conceptions of how we see things” (Dretske 2000).

I would also like to take note that Callard’s arguments for it not being necessary that mathematical objects imparting energy do not change are ad hoc. He says “For (as we noted at the beginning of our discussion) it is a principle of classical Platonism that the objects of mathematics are unchanging, and it may be strongly necessary that any object receiving energy must change” (Callard 2007). This shows Callard accepting an empirical truth while disregarding other empirical truths. Why is this empirical truth applicable and others are not applicable? He dismisses other empirical truths like the laws of conservational energy and Newton’s third law of motion. I find that when there is difficulty in the construction of an argument rhetorical strategies are used, just like the ad hoc. Perhaps my view of this is wrong.

I insight into the causal relata between mental substances, which we can replace with mathematical objects, and physical substances, or physical objects.

“The problem of interactionism is a problem of immaterial substances causally interacting with material substances. Descartes himself struggled to come up with an answer to this problem. He thought that animal spirits interacted with the body through the pineal gland, a small gland between the two hemispheres in the center of the brain. In metaphysics, there are things called metaphysical impossibilities, metaphysical possibilities, and metaphysical necessities. But the main things we will be discussing are metaphysical impossibilities and metaphysical necessities. So, we are looking at the problem of the causal interactions between a mental substance and a physical substance. Since we are looking at causality, we can put this to a nomological necessity or a nomological impossibility. Since causality is a law, if it is true that a causal interaction between a mental substance and a physical substance is impossible then it will be a nomological impossibility and vice versa. We must first prove if the mind even has causal powers. To exist is to have causal powers (Armstrong 1999), (Oddie 1982) so, to prove the existence of the mind we can use Descartes argument from doubt where it is as follows; I can doubt my body exists, I cannot doubt that I exist, therefore, I am not identical with my body. This entails that the mind is an ontologically distinct substance and is distinct from the physical. Now that we have proved that the mind exists, we must warrant how it is possible for the mind to causally interact with something material. We can argue that physical actions cannot have a physical explanation so our physical actions must be reducible to the immaterial warranting that activities such as reason, subjective experience, intuition, etc., have a nonphysical/immaterial explanation. So, it makes it so that it is a nomological possibility with the ought of a duality property that our actions have either a physical explanation or a nonphysical explanation”.

1.3 Katz and Temporal Ontology

Other anti-platonist arguments or anti-mathematical causalist arguments can reside in the likes of temporality. Jerold Katz, a platonist, had said that causation can only occur between two objects that have temporal positions. The obtaining of knowledge through the acquaintance of atemporal mathematical objects to him is senseless. He says “It is as senseless to suppose that we can be acquainted with atemporal abstract objects before our entrance into the spatiotemporal world as it is to suppose that we can be acquainted with them afterward. Acquaintance requires a point of contact, some temporal position that both we and the object occupy, but there cannot be such a point in the case of objects that have no temporal location whether during the soul’s existence in this world or before its incarnation” (Katz 1997). What Katz means by “Acquaintance” is persistently the obtaining of knowledge of these abstract mathematical objects.

He verbalizes this by saying “The strategy behind the doctrine is to extend the range of the perceivable to include abstract objects” (Katz 1997). This means that the realist of abstract objects accumulates the possible view that abstract objects can enter a realm of the perceivable. But by the natural conception of a demythologization of the myth of the cave in the platonist view, this attempt of the accumulation of the perceivability view falls short. Thereby making this type of realist false. what we get out of making considerations out of this falsification of this perceivability view simply goes back to computational neuroscience. We know that from the properties of a mathematical object that they inhibit their realm differently, and by necessity and adaptation they would use concepts differently than objects in the concrete realm.

Say, we lend the quality of imparting energy to mathematical abstracta (objects). Mathematical objects, because of the realm they inhabit, would use energy differently, and when they would impart energy to the brain when the brain state occurs and you observe the neuroimaging data, you could see something entirely different than natural results of what energy being imparted to the brain would be like. So, from this, when we observe the neuroimaging data from the energy in the dynamical activity of the brain, we will have a result abnormal from previous results in neuroimaging data when the brain becomes an acquaintance to empirical information. If it were the case that abstract objects existed in the realm of perceivability then this could be combated, from this the fact of the matter would not be abnormalities in the neuroimaging data and the questions would differ and be put towards the imparting of energies from abstract objects to physical objects, again. Katz asserts another problem that involves atemporality and temporality of abstract objects and concrete objects, one that I find very interesting.

Katz says “Even supposing there are beings in another world that can make some sort of contact with abstract objects, they would have to be atemporal and then the same problem would arise when we try to imagine that we could be those beings or be continuous with them” (Katz 1997). But I take this as false. Any object in the concrete realm, on the microscopic and even submicroscopic level, is in process.

If we take a look at the theory of matter and its composition, we’ll see that any solid, liquid, etc., has its particles moving at some type of rate. We then augment this information into the fact that any object with a temporal position is potent to have its particles rearranged. At T1 F (x, y) is arranged in its initial form, but at T2 F is rearranged as F (y, x), due to external instantaneous causally sustaining factors. We know that the mathematical realist ascribes atemporality to the mathematical object. Katz said that since we acknowledge these objects, we continue our existence with them, which would be contradictory due to the original conception of mathematical objects and their properties.

But to continue your existence, your arrangement of parts must be rearranged within every temporal position you have. For the continuation of discourse, we shall take locution as a very real concept in the composition of things. For an object to be brought about in the domain of discourse, which persistently resides in describing the continuation of the existence of things, you must interact within a composition, viz. a mereological sum. This is similar to the monadism of Leibniz, (“It cannot come into existence because the simple substance is not formed by parts, it isn’t a composition (Leibniz 1720)). 

The difference is, Leibniz explains the coming-into-existence¬ rather than, continuation-of-existence. Not a mereological sum, abstract objects, because they cannot have any temporal position, (due to the fact abstract objects are aspatial and atemporal), cannot have this focal signifier denoted from them. And rather than there being one mode of existence, we shall have two, the second being only for abstract objects. This mode would be just to exist. The concrete world is the only world with objects who continue in their existence. In the abstract world, there is no space or time, therefore making it impossible to even apply continuation¬-in-existence¬ to these objects, and applying just existence, which doesn’t presuppose temporal position, is a more advantageous account than Katz’s. By and by, Classical Platonism asserts unchanging in abstracta.

This instrumentation of qualities ascribed to abstracta, distinctively shows that the applicability of temporal positions, or any additional quality that interpolates qualities of temporality, cannot be imputed to abstracta. Unchanging means that p is not under any condition of saturating with any extrinsic qualities. The intrinsic qualities of abstracta are aspatial, atemporal, unchanging, and the ineptness of causal capability, under Classical Platonism, though. Though under mathematical causalism the intrinsic qualities can vary dependent on the mathematical causalist. Habitually the mathematical causalist holds the intrinsic qualities of abstracta to be aspatial, atemporal, unchanging, and causally capable. Katz’s view also may be presupposing four-dimensionalism or perdurantism. From reasons listed former to this, it seems to be that Katz unknowingly subscribed to the temporality of abstracta while full handedly acknowledging the thesis that abstracta are atemporal.

It is necessary to note that because of the lack of spatial-ness and temporal-ness in mathematical objects, it is impossible to theorize persistence-in-time for mathematical objects. It is also important to note that because mathematical objects are unchanging, they cannot have a diachronic identity, rather they have an eternal synchronic identity, as they also have no temporal parts as they do not exist in space-time. Material objects that exist in the concrete world have temporal parts, temporal parts that can be rearranged amongst a substance. A substance is the conglomerate of the temporal parts it maintains, and while this material substance persists throughout the time it changes throughout time, and these temporal parts it maintains do not fully stop go out of existence and then the material substance simultaneously is granted new temporal parts as the material substance changes.

Rather, the material substance’s temporal parts undergo change and by the weight example, the temporal parts extend or contract into different quantities and still stay the same. Some would say that under Leibniz’s Law that the temporal parts would not be the same as they are different weights. But this is wrong. As I noted above, the temporal parts extend and contract into different quantities under the weight example. From this we can look at the temporal parts as substances, or even better, temporal parts are dispositional properties that are actualized and undergo changes. For example, a calculator’s screen at T1 is blank, but at T2 an external causal factor presses a button, say the ECF (External causal factor) presses 2 on the calculator, and the calculator’s screen begins to display 2. The temporal parts in question which are the microprocessors inside the calculator are being exercised if you will. Once this already actual ECF acts upon another temporal part that has yet to be extended, and is unchanging from point to point in time, the temporal part thus extends into a new form that is an elucidation from its earlier form rather than it being some form of a descending transformation.

The ECF is persistent through time and is also identical in qualities to other objects. In short it too has temporal parts. Does this ECF’s temporal parts extend or change when it acts upon another object in time? Metaphysically yes. When we look at it from a scientific point of view, the particle arrangements of the object moving around are still the same aggregatum of particles, just arranged differently. After acting upon another object, the particles in that composition of the ECF do not change, they are still the same. Therefore, the aggregatum of the object is in different composition, but still has the same parts that were in the aforementioned aggregatum. When we come back to the philosophical view, we see that the temporal parts of the ECF are the same temporal parts that were existent before the ECF had acted upon anything. The temporal parts are still the same, but they are in a different process, a different fashion. 

2 Epistemology and Causalism

The epistemology of mathematical realism, and more so mathematical causalism, is commonly founded in the causal theory of knowledge. The causal theory of knowledge proclaims that the knowledge of things through-and-throughout the world must be caused. Goldman, the initiator of this theory, puts it like this, “S knows that p iff S’s belief that p is causally connected appropriately with the fact that p” (Goldman 1967). This means that in mathematical causalism, all spatiotemporal subjects with the finite intellect are acquaintances of mathematical abstracta. This means that the acquaintances of mathematical knowledge rely on relata to have mathematical knowledge. But how does the causal connection work between S and p? In Callard’s view of mathematical causalism, he says that mathematical abstracta impart energy to the brain, as he says “It may, perhaps, be strongly impossible for abstract objects to receive energy” (Callard 2007). In saying this, he inherently conforms to efficient causal relations between mathematical abstracta and conscious subject requiring the imparting of energy to the brain. But then this means that for epistemic access of mathematical abstracta, a causal interaction of imparting energy to the brain is needed. More formidable explanations come into play for the shortcomings of mathematical causalism. Callard talks about the causal connections of mathematical causalism as causal relations. Mark Steiner approaches the causal theory of knowledge and Platonism with causal explanations rather than causal relations. In Platonism and the causal theory of knowledge, Steiner formulates a more plausible explication of causal theories and mathematical platonism. “(S) One cannot know that a sentence S is true unless S must be used in a causal explanation of one’s knowing (or believing) that S is true” (Cassullo 1992). This alteration of the causal occurrences between concreta and abstracta does not seem all that different. The way Callard worded it was in terms of causal relations. Though at further evaluation and analysis, a conclusion can be made and that is the coming about that mathematical abstractum under a mathematical causalist framework being causal explanations of knowledge for its acquaintances. An argument can be put like this

 (1) Mathematical causalism by it coincides with the causal theory of knowledge

 (2) Since mathematical causalism persists and, analytically coincides with the causal theory of knowledge, then it is theorized conceptually that mathematical abstracta causally affect concreta.

 (3) But the causal theorist must not start with the supposition that mathematical abstracta causally affect concreta.

 (4) Therefore, the subject p knows that Q(x) such that a is the causal explanation for p’s knowing of Q(x). 

 (5) Classical Platonism asserts that platonic objects or abstracta are not located in a spatiotemporal complex.

 (6) Rather, the abstracta are located in an aspatial, atemporal realm.

 (7) The abstracta, being in a realm where space is not a parameter and no time flows, cannot enter in causal interactions as they lack the qualities to do so. 

 (8) By mathematical causalism’s epistemological tools or assets (being the causal theory of knowledge, now useless for it to be used as a tool) now subsists in acrimony and animosity, mathematical causalism’s epistemology operations are now in vain (from (3), (4), (7)).

 (9) Now that we know and acknowledge mathematical objects not being able to be actual causal explanations, we can analyze the problem.

 (10) The absence of causal explanations to the knowledge of propositions that exemplify the actualism of the world while trying to work under a causalist framework is contradictory. Therefore, we must abandon the causalist framework if it serves as subordinacy to mathematical causalism. 

In recent times reliabilism is seen to be a better epistemological account for platonic mathematics, and the obtaining of mathematical knowledge under a platonic framework. Reliabilism is defined as “knowledge as the true belief that is produced or sustained, by a reliable process” (Mcevoy 2004). Reliabilism does not require any type of causal explanation, causal relata, or causal connection between the subject and object. So mathematical causalism in an attempt to be fused with this epistemological theory would most definitely fail. Even more, if this epistemological theory were correct, then mathematical causalism would nonetheless fail to yield any truth in the area of Platonist accounts, and would undeniably be rejected by most. Reliabilism is defined as having reliable processes sustain the concreteness of our mathematical beliefs. This means that any process of obtaining knowledge about mathematical objects is sustained and has healthy sustainability if it is a reliable process of obtaining the knowledge of mathematical objects. It should also be known that historically, reliabilism has been known as a direct descendant of the causal theory of knowledge. Though the causal theory of knowledge rather focuses just on knowledge being accessed by causal explanations, while reliabilism focuses on any reliable process that gives healthy sustainability of knowledge, this reliable process can too be a causal explanation or a causal connection between conscious subject and object. “If reliabilism is to be able to account for mathematical knowledge, it must appeal to a psychologically real process capable of yielding such knowledge” (Mcevoy 2004). This means we can grant another argument against the epistemology of mathematical causalism. If we are capable of yielding a refutation against the process of obtaining mathematical knowledge under mathematical causalism (that process being a causal process), we can find the absence of the acquiring of mathematical knowledge under mathematical causalism. It is true, that knowledge of any object in the abstract mathematical realm is non-inferential knowledge. This is obvious due to the occasion of the evidence of mathematical objects not being concrete, the evidence is rather in the mathematical object itself. The evidence-based around mathematical objects can never be found in the realm of where all concreta exist. What Mcevoy then concludes is that the only psychologically reliable process of acquiring mathematical knowledge is through the method of intuitive contact. This method of intuitive contact is generally the use of intuition. The use of intuition must not be confused with Hume’s method of intuition. Hume’s method of intuition enucleates upon the necessity of propositions as their negations are inconceivable. 

2.1 Inconceivability and Conceivability

“’Tis an established maxim in metaphysics, that whatever the mind conceives includes the idea of possible existence, or in other words, that nothing we imagine is impossible” (Hume and Selby-Bigge 2015).

In literally anything, the inconceivability of it or the conceivability of it interferes. Say, I assert the proposition “There is a blue pencil”, if I am capable of conceiving of this proposition in my mind, then it is possible. This is called the principle of inconceivability. If I assert an inconceivable proposition, then it is impossible. This is called the principle of Inconceivability. Hume’s maxim explicates these principles, and successors of Hume, such as Thomas Reid had thought that his “predecessors were mistaken in attaching great significance to the concepts of conceivability and inconceivability” (Casullo 1979). In regards to the content of the assertions mathematical causalism makes, we can happily apply this and yield at a fertile death of mathematical causalism. Mathematical causalism asserts that there is a realm where space does not exist as a parameter and time does not flow. In which there is an addition of mathematical objects that causally interact with human brains, imparting energy to them.

But to conceive of these propositions, the causalist is put at fault. When we try to conceive of a nonspatial nontemporal realm, we cannot do so such that we are existent within a space-time realm. If we conceive of a nonspatial nontemporal realm, when we observe the information of the neuroimaging data, we will find that this data is showing results of spatiotemporal dynamical complexities. Now we run to the question, “Can non-spatiality and non-temporality realm conceptions inhabit a spatial and temporal realm?”: certainly not. We are also not capable of conceiving objective propositions about such a realm, due to its non-existence. 

Reid’s first objection to Hume’s maxim asserts the understanding of the meaning of propositions rather than the conceivability of these propositions. “Reid is quickly able to dispose of the principle of conceivability simply by pointing out that one can understand a demonstrably false mathematical proposition as distinctly as one understands any true mathematical proposition” (Casullo 1979). But Reid falls into a problem of doublespeak. When he asserts one can understand a demonstrably false mathematical proposition, does he mean one can conceive of it? Or does he use the term “understanding” in its literal sense, viz., the outcome of the false mathematical proposition can never happen, and thus one cannot conceive of it. Just because one can understand the state of affairs of something impossible, doesn’t mean they have to conceive of it, and many contemporary epistemologists agree. 

“One must be able to understand sentences which describe an inconceivable state of affairs at least to the extent that one knows what he has to try to conceive” (Casullo 1979). If we apply this rule to the conceivability of mathematical causalism, we can get a coherent approach to this problem. If we make the proposition “There is a realm that is spaceless and timeless” we can most definitely understand it, but might not be able to conceive of an objective state of affairs this proposition is expressive and declarative of. Because of some reasons mentioned above, the conception of this proposition is impossible, therefore making this proposition an inconceivable state of affairs. Hume’s principles invoke the method of giving demonstrable(s) or demonstrating the possibility or impossibility of the state affairs elucidated and then rationalized by propositions, “Consequently, a demonstration is necessary to establish this relationship.

It seems both reasonable and consistent with Hume’s thinking to suppose that if the demonstration is necessary to determine the truth or falsehood of certain propositions, it would also be necessary to determine their possibility or impossibility” (Casullo 1979). But first, we shall identify what these demonstrable(s) are, and what their properties are. To define what a demonstrable is, or when something is demonstrable, we shall say it is when something “lacks logical paradox, and can then further be logically proved”. Hume’s principles also invoke the concept of intuition. The concept of intuition is used in Humean philosophy by mitigating and accounting for necessary truths. “A proposition which is knowable by intuition is necessarily true if and only if the state of affairs described by its denial is inconceivable” (Casullo 1979). This distinction between demonstration and intuition is also designated as “Relations of ideas” and “Matters of facts”. Relations of ideas are ideas that are necessary truths. For example, a square has 4 sides. Relations of ideas are essential things that the negation of it is inconceivable or self-contradictory.

The capability of imagining a 5-sided square is inconceivable, try and conceive of it, you will have trouble. So, this makes it so a square that has 4 sides is necessarily true. No matter how the world is, this relation of ideas will always be true, it can never be contingently true, they are not susceptible to anything else. Propositions like “A square has 4 sides” are merely by what Hume called the operation of thought; it takes no empirical knowledge to have logically deduced this conclusion. Matters of fact are claims where the denials of their claims aren’t self-contradictory or aren’t inconceivable. For example, I could claim that my piano is black, but since this is contingently true, denial of this claim isn’t self-contradictory or inconceivable. Say, now, we have drawn ourselves the conclusion that the proposition “The realm of which space is not a parameter and time does not flow, does not exist”. The negation to this proposition would simply be “The realm of which space is not a parameter and time does not flow, exists”. If the negation to the former proposition is true, then methodically, the latter proposition is inconceivable. 

Inconceivability, as it is, means the inability of conception. In other words, when we say something is inconceivable, we mean we cannot “have a mental representation” or a “mental image” of it in our minds, and we, therefore, cannot have any theoretical conception of this proposition. Boltzmann phrases mental imagery in the philosophy of science as is, “Boltzmann formulates the goal of physical theory as ‘constructing a picture of the external world that exists purely internally’” (Fraassen 2010). So, the question, reformed, is that “Just because it is not possible to have a mental image of the proposition, why does it make it impossible?” Yablo says “As everyone knows, ‘Alexander’s teacher’ is not a rigid designator. How though does everyone know this? Well, we imagine a counterfactual situation in which Aristotle refuses Phillip’s call, or dies of dysentery on the way to Macedonia. Such imaginings would be irrelevant to the rigidity of ‘Alexander’s teacher’ if conceivability was not evidence of possibility” (Yablo 1993). 

3 Views on Abstract Objects

3.1 Plebani’s Combinations

Matteo Plebani is a scholarly level writer from the University of Turin. I seek to refute him on his heavy advocation of Stewart Shapiro, and assertions he very much vocalized in his paper “Mathematical platonism meets Ontological pluralism?”. I will argue against his argument of plural platonism answering the epistemological question of platonism. Although in the previous section he had made solid and concrete assertions and carefully and perfectly illustrated the ideas he wanted to argue for, I am bothered by the later responses to problems that were risen in platonism. He answered the epistemological question with plenitudinous platonism. Plenitudinous platonism is the Platonist account that is quite literally and can only be presented in the form of ontological and alethic pluralism. This Platonist account has an epistemological thesis, a metaphysical thesis, and an ontological thesis that derives from its property of ontological pluralism. These are what the theses say: the epistemological thesis is that consistency is a form of validity for different mathematical sentences in the mathematical domain of discourses. Consistency is a way of finding truth in mathematics. The ontological thesis: abstracta and concreta are different modes of existence. The metaphysical thesis: all possible abstracta exist, and not all possible concreta exist (logical pluralism may be combined with this). I will also argue against his advocation for Shapiro’s acceptance of only intuitionistic logic and having it as the only subsidiary of logical pluralism.

Plebani says that Balaguer presents his form of plentinduous platonism “that every consistent mathematical theory is true of some abstract mathematical objects”. This entails and is backed up by one of the theses we had in the introduction. The metaphysical thesis had said that “All possible abstracta exist”. If there are consistent mathematical theories, then there are abstract mathematical objects, therefore if mathematical theories exist there exist abstracta by each mathematical theory.

He then forms the proposition “A theory might be inconsistent if a certain notion of logical consequence is adopted, but consistent if another notion of logical consequence is adopted.” But, with this, he accepted and only proposed intuitionistic logic as a valid way of deducting logical antecedents and consequences. You cannot have multiple ways of having criterion for the truth of mathematical theories, whereas one of the criteria make a logical consequence of propositions invalid. You can only have concrete criteria for the validity of logical consequences and mathematical theories because if there were criteria that were used solely invalidity in logical consequences and mathematical theories, that would be absurd . There is another type of Platonist account that Plebani offers, it is plural platonism.

Plural platonism is combined with ontological pluralism, alethic pluralism, and plenitudinous platonism. Ontological pluralism is that there are multiple ontological structures of the world, different modes of existence. Typically, being the modes abstracta and concreta. Alethic pluralism is “the view that sentences belonging to the mathematical domain of discourse are true in virtue of one property, say coherence or consistency, whereas sentences belonging to the empirical domain of discourse are true in virtue of another property, say correspondence” (Plebani 2017), and plenitudinous platonism, the view that all possible abstracta exist, so by all possible mathematical theory. We arrive at a problem for this position, Plebani sees this too. His definition of alethic pluralism adheres to the fact that some mathematical theories and sentences exist in an empirical domain of discourse. Plebani says that we should assume that an empirical domain of discourse is referring to empirical mathematical objects, rather than abstracta.

But we have another position that is fused with alethic pluralism and ontological pluralism, plenitudinous platonism. Which is the thesis that ascribes all possible abstracta to mathematical sentences. Although there is not a complete definition for concreta, we shall assume (just as Plebani did) that concreta are objects that can be perceived by phenomena, particularly the opposite of abstracta. Plebani named the same three theses I did in the memoir. The metaphysical thesis said that all possible mathematical theories had according to all possible abstracta in existence, but not all possible concreta.

Since we identified concrete being by what empirical domains of discourse are, the contradiction is clear. In section 4 of “Mathematical platonism meets ontological pluralism?”, Plebani proposes a plenitudinous Platonist response to the famous epistemological objection to platonism. “Plenitudinous Platonists reply to this objection appealing to the principle that consistency is the criterion for truth and existence” is what Plebani says for the first step of the objection.

Although this may be elementary to some readers, for the ones reading this who do not know of the epistemological objection to platonism it goes like this: (i) platonism states that there is a domain of mathematical abstracta, (ii) these mathematical abstracta are causally isolated mind-independent entities and are transcendent of the faculty of phenomenal intention. (iii) from this, to know about mathematical abstracta, we must be able to form reliable mathematical beliefs on them. (iv) but like stated before, these mathematical abstracta are mind-independent, meaning they exist without the faculty of phenomenal intention or essence and are transcendent of the faculty of phenomenal intention. (v) So how is it possible to form reliable beliefs about mathematical abstracta given the characteristics they hold?

Plebani’s response with plenitudinous platonism is that of a variant of Balaguer’s. (E1) we can form reliable beliefs about the consistency of some mathematical theories. (From the principle of consistency) and (E2) If plenitudinous platonism is true, then if a mathematical theory is consistent, the objects it quantifies over or refers to exist. This argument, however, does not give a valid response to the epistemological objection. If we epistemic accounts that are based on the faculty of phenomenal intention, then to try and make beliefs about the consistency of mathematical theories, that are aligned with mathematical abstracta, we are still not able to make reliable beliefs about them. I feel as though Plebani knew that something bad would happen when coming across this objection of platonism. He stated earlier in his sections that plenitudinous platonism need not be presented as a combination of alethic pluralism and ontological pluralism.

Earlier, I had already pointed out the errors and faultiness with that statement. So, it is the case that plenitudinous platonism needs to be presented as a combination of ontological pluralism and alethic pluralism. As we have stated earlier, alethic pluralism is the thesis that mathematical sentences can be looked over at in empirical domains of discourse and mathematical domains of discourse (discourses of abstracta). Empirical domains of discourse are with empirical objects. If we are not already aware of the sudden problem, we will go into I will make you aware. Mathematical platonism adheres to a platonic account of mathematical abstracta as transcendent objects. We have alethic pluralism as one of the supplementary properties of plenitudinous platonism. Alethic pluralism is the thesis that there can be both empirical domains of discourse and mathematical domains of discourse. If we have both empirical mathematical objects and mathematical abstracta, we have coincidentally contradicted one of our theses in our position. The metaphysical thesis in plenitudinous platonism states the following:

all possible abstracta exist, and not all possible concreta exist

Things purported by the faculty of phenomenal intention are of concreta. So as the plenitudinous Platonist having both concreta and abstracta is contradictory. It is even more contradictory than was before because we now have all possible concreta existing inside of the empirical domains of discourse being able to align with the mathematical abstracta. Wherefore the plenitudinous Platonist is supposed to hold that all mathematical theories exist in mathematical domains of discourse and these mathematical theories (mainly mathematical sentences) inside of the mathematical domains of discourse are the only things to align with mathematical abstracta.

3.2 Plenitudinous Platonism

Mark Balaguer is a professor of philosophy at Cal State L.A. Balaguer has written and published three books, and multiple journal articles. We will be focusing on one of his articles, “Mathematical pluralism and platonism”. In this, I sought to find problems in this paper, and surprisingly give a result of some type of refutation to the arguments he gives. I recommend the reader read two other papers before this. These two papers, give the reader a better understanding of where my view is coming from and the problems, I see in the theses that Mark Balaguer gives. Mathematical Pluralism; is the view that multiple mathematical theories are true. “Non-Euclidean and Euclidean geometrical models are both true”. Mathematical relativism; is the view that different cultures endorse mathematical theories that are true to them. “Martians can endorse the Continuum Hypothesis to be true (CH), and some other people as a collective that form a culture might endorse it to be false. To say CH for Martians is true, and CH for the collective is false, and they would both be right”. But in this case, it is only true for the collective, and only true for the Martians, hence the term relativism. Plenitudinous Platonism; is the view that all possible mathematical abstracta exist, and there are mathematical theories that explain these mathematical abstracta. For instance, the mathematical sentence “60 is a composite number” has the mathematical abstracta 60, and the mathematical sentence, otherwise known as a mathematical theory explains these abstracta. In the platonic realm, rather than there being a definitive finite amount of abstracta, all possible abstracta that can be proven by a mathematical theory exists. Sparse Platonism; is the view that there is a finite amount of mathematical abstracta already instantiated. And from this view, it could be argued that it is logically impossible for there to be a possible amount of mathematical abstracta.

On page 2 of his paper, Balaguer gives an argument for Plenitudinous Platonism (FBP) that proposes it gives the epistemic account for the knowledge of mathematical abstracta. Generally, since mathematical abstracta are wholly nonphysical, causally inert, and exist in an extra mental reality, we wouldn’t be able to obtain any concrete statements about the abstracta. The argument goes like this: “Since FBP says that there are abstract mathematical objects of all possible kinds, it follows that if FBP is true, then every purely mathematical theory that could be true—i.e., that is internally consistent—accurately describes some collection of actually existing abstract objects. Thus, it follows from FBP that to acquire knowledge of abstract objects, all we must do is produce an internally consistent purely mathematical theory (and know that it is internally consistent). This is because, again, according to FBP, every consistent purely mathematical theory accurately describes a collection of existing abstract objects. But if all we need to do to acquire knowledge of abstract objects is produce a consistent mathematical theory (and know that it is consistent), then it seems that we can acquire such knowledge. For it seems clear that (a) we are capable of formulating internally consistent mathematical theories (and of knowing that they are internally consistent), and (b) being able to do this does not require us to have any sort of information-gathering contact with the abstract objects that the theories in question are about.

Thus, if all of this is right, then FBP gives Platonists a way of explaining how naturalistic creatures like us could acquire knowledge of abstract objects, even though they do not have any information gathering contact with such objects. Another way to put this is to say that FBP gives us a sort of recipe for acquiring”.

A principle of consistency was formed here to determine truth in mathematical theories. Many philosophers like David Hilbert. Hilbert says that the consistency founded in mathematical theories is the criterion for their truth-value; whether they are true or not. But Hilbert’s position isn’t that consistency isn’t the case for ordinary sentences in domains of discourse… rather that consistency is the only criterion for truth in mathematical domains of discourse (Plebani 2020). I find this to not be the case though. Although consistency can serve as truth at points of time, it will not serve for the concreteness of a mathematical theory. Concluding the fact that it is true because of past instances is a false use of the principle of instantiation. If we were to make mathematical statements in a domain of discourse that isn’t empirical, then whether that mathematical statement or mathematical theory is true or not relies solely on the fact of whether it is true or not. But as I have said before, the principle of consistency is posed with the possibility of a mathematical theory being false.

The only time the principle of consistency is met mathematically is when we have mathematical statements in empirical domains of discourse. But mathematical statements grounded in an empirical domain of discourse cannot explain mathematical abstracta. Concreta cannot explain abstracta, concreta are simply decrees of abstracta, platonically. But there is another chance of Platonism. The inference to the best explanation grants the concreteness of maximal mathematical abstracta. From the concreta, we see in the real world, and the mathematical concreta we produce, we can use the inference to best the explanation to produce a maximal. but even this cannot help Platonism. When we look at the mathematical concreta we have produced, we look at them rather nominalistically, rather than a collective, intuitionistically. Therefore, the truth of mathematical theories cannot be verified by the principle of consistency. So, for Balaguer to use these points out a faultiness in his position for plenitudinous Platonism (FBP). This ties in with the problem of induction and is rather strung out from the problem of induction.

The problem of induction states that from past observations, and even this we can use the inference to the best explanation, that principle can give better chances for the problem of induction. In the philosophy of science, though. I would also like to reflect upon the fact that plenitudinous platonism doesn’t indispensably give an actual alternative account of epistemic access. It might be even more difficult to grasp the intelligibility of how the conscious subject can have epistemic access to the realm under this view more than other platonic accounts. Plenitudinous platonism asserts that instead of there being a sparse and finite realm of abstract objects, all possible abstract objects exist. However, it is never really amplified on what this means, so, perhaps, we shall examine this proposition and give our interpretation.

The sentence “all possible abstract objects exist” can very well mean that all the possible conceived abstract objects exist. This means that the conscious subject can perceive as many possible mathematical objects, and they will nonetheless be actual abstract mathematical objects. Though, this can be confusing. In plenitudinous platonism, a mathematical theory or mathematical sentences can be grounded in two different domains of discourse; an empirical domain of discourse and a mathematical domain of discourse. Of course, this is only the case when you are using the plenitudinous platonism of Matteo Plebani. Which is a combination of ontological pluralism and alethic pluralism. Plenitudinous platonism is met with three philosophical theses, which I had taken from Matteo Plebani, author of “Mathematical Platonism meets Ontological pluralism?”. Although I had made a refutation to that paper, I feel like these three philosophical theses hold great importance to the point I am trying to make. I feel like these theses are very coherent and adhere to what plenitudinous platonism (FBP) is defined as.

I had introduced this part in the last part where I had talked about the inference to the best explanation. “. The inference to the best explanation grants the concreteness of maximal mathematical abstracta. From the concreta, we see in the real world, and the mathematical concreta we produce, we can use the inference to best explanation to produce a maximal”. I do believe that this argument against the epistemological objection of platonism can give pathways to the truth of platonism, but in the end, I believe it fails. (In this next passage, I will introduce the three philosophical theses that carefully support the definition of plenitduinous platonism.

The Epistemological Thesis (ET): consistency is a form of validity for different mathematical sentences in the mathematical domain of discourses; consistency is a way of finding truth in mathematics.

The Ontological Thesis (OT): abstracta and concreta are different modes of existence.

The Metaphysical Thesis (MT): all possible abstracta exist, and not all possible concreta exist.

The epistemological thesis, I believe I had heavily refuted in the last excerpt. But I believe that refutation is open to the possibility of being refuted correctly as well. The ontological thesis affirms the thesis that is by ontological pluralism, which says that there are different ontological structures of being (in this case abstracta and concreta). I heavily advise the reader to not get modes of existence mixed up with the empirical domains of discourse and the mathematical domains of discourse.

The domains of discourse try to tell us about mathematical abstracta, but the modes of existence do not. The metaphysical thesis is what I am trying to use in this long excerpt, so without further ado, we shall get into it. The metaphysical thesis says that all possible abstracta exist, while all possible concreta do not. This is by the plenitudinous platonism of Balaguer. Where he says that there is a plentitude of mathematical abstracta existing in the platonic realm. When we utilize the mathematical domain of discourse to discover mathematical abstracta it doesn’t exactly lead us to mathematical abstracta if we are trying to make abstractions of things, we cannot directly know through immediate sense experience or through immediate abstract cognizing of that thing.

Another doctrine is introduced, one that is rather completely indifferent to this field. This doctrine is absolutism, specifically the absolutism of American neo-Hegelianism; that of William James. This type of absolutism is met with abstracta and concreta:

Jamesian-Hegelian Thesis (JHT): to have true and absolute knowledge of the plentitude of mathematical abstracta, we must transcend our minds, and layover ahold of the mathematical abstracta, to the point where we are on a level of owning all objects, and all relations in existence. These mathematical theories are on a level of subject cognization. They cannot lay hold of transcendent, acausal, non-spatiotemporal, abstract entities.

Another problem, saying that we will have mathematical abstracta just because we can supposedly make correct mathematical theories? This is arguing from ignorance, an informal fallacy. Balaguer saw one “consistency” in the truth of mathematical theories, and now is he using that as leverage for plenitudinous platonism. To be using the consistency of something, say, over and over, then finally reaching a conclusion based on previously perceived notions prima facie seems absurd. These conclusions are things pushed back in the past and are left only for the past. These conclusions shall be always actual, and never things that become virtual when moving on to another evaluation or analysis of a mathematical theory. However, this is not possible as to say a conclusion based on consistency is once potent then purely actual is redundant. By the term “consistency” we mean that of which is of a previously perceived instance or scenario. Thus, it shall be acknowledged that when using the term consistency to describe something, that something is an occurrence that appears to happen again and again, of course with the same result.

3.3 Set-Theoretic Realism

Set-theoretic realism is a view and variation of mathematical realism espoused by Penelope Maddy. Maddy rather focuses on the realism of sets, rather than the realism of natural numbers, though under STR (Set-theoretic realism), natural numbers too, exist. This view is explained as “That at least some sets, for example, small sets of physical objects located close to one another can plausibly be held to be perceived” (Kremer 1991). This view is closely related to the American Hegelianism of William James and Josiah Royce. When we try to acknowledge some ‘things’, we will implicitly assert them as a set. Say we see oranges when I say ‘oranges’ that implicitly asserts the set of oranges. For in the set of Oranges there is orange, a plain substantive element of Oranges. This is caused by the indispensable connotations of linguistic structures. Maddy bases her main notion of Set-theoretic realism on the fact that we develop neurophysiological processes that enable us to identify the properties of ordinary objects. From this, she says that these neurophysiological processes make it possible for us to have the ability to perceive sets of ordinary, physical objects. But why argue from possibility? Arguing for the actuality in having neuro-physiological processes yields fewer circumstances that heed in the falsity of STR. That is what it seems like for most realist accounts and even platonist accounts: Arguing for the actuality of realism because of the epistemic and metaphysical challenges it faces. We can also propose an epistemic challenge towards STR. Say we do have neuro-physiological processes of the perception of recognizing sets among ordinary physical objects.

The exposition to be made is that since there are different types of ordinary objects, do we make sets on objects that have the same properties and are thus common to each other? I say this because the elements of a set are all under the prediction of the same thing, say P predicates X, Y, and Z. All the elements in A predicated by P are all the same because they share one common trait given to them by P. e can word this into the example that was used before. The oranges in the cupboard are the set of oranges in Orange. At this glance, STR seems to be useless. This is because the process that possibly takes place, as described by Maddy, doesn’t necessarily explain anything in the physical world. Why would one be in denial of being able to perceive things as assets? For the set of common ordinary objects such as the set of oranges is surely perceivable, as oranges are the only physical objects that can clarify the percipients perceivability of the current state it is in. The technicalities that STR uses can be vaguely founded in Absolutism. JHT (Jamesian-Hegelian Thesis) is of use when addressing intrinsic applicability:

William James describes rationalism as affinity and affirmation to look at the world with wholes. On the contrary, there is empiricism, it looks at the world with parts. Thus, rationalism inclines to monism, and empiricism to pluralistic views. Abstract oneness has parts that do not change, abstract oneness does not change either. For abstract naming of the concrete things makes their nature impossible, it would include abstract oneness in their nature, which is not possible due to the fact of their concreteness. So, this functioning of naming is invalid. To grant the abstract name without the abstract consequences, we have to ascribe adjectives to the abstract name so it can be in conformity with the concreteness and so it explains the nature of the concrete thing. There cannot be abstract independence in the name as we have said before the abstract name must have adjectives for the nature of the concrete thing to be possible.

The abstract name must be Secundum quid as Lotze said. In the cat, king and queen passage, William James talks about Professor Royce’s proverb ‘the cat and the king’. Take for instance a cat(subject) who cognizes a king, and we adopt the realistic view that the king is independent of the cat’s cognition of the king. ‘This assumption, which amounts to saying that it need make no essential difference to the royal object whether the feline subject recognizes him or not, that the cat may look away from him or may even be annihilated, and the king remains unchanged’. This assumption was considered by professor Royce. James says that this assumption leads to an absurd entailment, which is that the cat and the king cannot have any linkages or ‘connexions’. The connexion between these two beings would be added to both these beings. The connexion would simply be a third being added to these other two beings. But for this additional being to be linked to these other beings there would have to be additional links connected to the third being to connect to the third being and additional links added to the aforementioned innate additional links connected to the third being, and so on ad infinitum. It would be self-evident that the additional third being could not link to the other two beings due to the potential infinite of additional links. This argument is the same as Lotze’s; where a does its influencing when it is influencing b. In Royce’s own words the cat and the queen cannot have any relations, they cannot have any ties in the same community of nature, they are not in the same spiritual or natural order, and subsequently, they form two unrelated universes.

To know the king, that cat must INTEND that king, the cat must somehow pass over and lay hold of him individually and specifically. In short, the cat’s idea of the king must transcend the cat’s separate mind and somehow include the king, for if the cat did not have an idea transcendent of its separate mind, then it would be impossible for a said cat to have known the king given that the king is independent of the cat.

As William James describes it, he says Royce interprets continuity as a union, and idealistically. Where there is a higher power that owns both objects, (take for example the cat and the king). James says that the higher power owning them as objects necessarily means there is a relation arisen between the two objects, such as witnessing each other. But if it were the case that both objects can have relations with one another, they own those relations. In Leibniz, any relations created between two different objects (substances in Leibniz) are extrinsic qualities and can be annihilated and the two objects can persist as they did before obtaining such extrinsic qualities. So, if this were the case, James continues and says that two objects cannot own the same relation, taken pluralistically both objects are “shut up” to themselves.

The relation between both objects would considerably be “betweenness”, so he is saying that both objects cannot own any part of betweenness` because they are shut up to themselves. Since, indeed, the terms used to describe the cat, king, and queen make it so that they are all distinct, in which the terms that are used are ‘independent’ and ‘indifferent’, it cannot be possible that with the abstract terms ascribed to these entities in so far as they are used to describe these entities it makes no use to describe these entities in a sense that they have relations, as the terms independent and indifferent when ascribed to entities make it so that relations cannot be ascribed to such entities. Since it is the case that they cannot possibly have any relations, it is true that the abstract terms ‘what knows the queen’ and ‘what the cat knows’ are logically distinct. William James then says that “the king breaks up into two with nothing to connect to them until a higher knower is introduced to recognize them as the self-same king concerned in any previous acts of knowledge which he may have brought about”. William James further adds to the properties and attributes of this all-knower. Saying the higher knower has all abstract terms as objects in his mind.

The intellectualistic logic that the rationalists use is that the absolute whole is there or there is nothing. The absolute whole is a logical necessity that necessitates objectivity in everything. Reality is only intelligible if the absolute whole is in existence, which according to the objectivity it exists by necessity, it is by logical necessity to exist for true intelligibility. The absolute whole, in respect of the rationalist’s view, is the minimum that can exist.

Logical proof that is used to signify a contradiction to suppose otherwise, as William James describes it, “is that you can deny the whole only in words that implicitly assert it”. Meaning, for example, if you were to deny the whole in any way to go and make it so the reality is in parts, and to describe the parts as “many” that term “many”, unifies them, to make a whole, an absolute whole, and if you were to describe those parts in any respect, the “respect” you refer to them also still unifies said parts. “In short, you fall into a hopeless contradiction”2. The world must be wholly rational or wholly irrational. “Reality being described as partly rational and partly irrational is not an admissible description of the world around us”. Only one can pervade throughout the world, if one is in it, one must be the only explanation for reality. Relations that are co-implicated and, “submerge” with the absolute individual, is rational supposition against the dilemma proposed above.

Relations that are through-and-through and consubstantial as wholly one comes to the absolute, and this is the only rational supposition to make against the dilemma above, although to say that these through-and-through relations of rational-irrational or irrational-irrational and rational-rational, it is to say both can pervade throughout reality. As we are taking the structure of reality as relations making up an absolute. But, at the same time, the dilemma presupposes a structure of reality not in relations, but merely nominalization of substantive parts. These parts all either have to be rational, or irrational, and these parts, however they are described, will unify to one, and follow the intellectualistic logic of a monistic structure. F.H. Bradley follows the same logic as Greek sophists. Where prepositions cannot just connect such disparate terms. And Bradley thinks that you can not use a logically distinct adjective to connect to an object to describe that object. For example, the proposition “man is good” is not logical because man is a man and good is good, and the proposition “is” cannot unite just disparate meanings. And within Bradley, you cannot unite a logically distinct adjective with a substantive, and even if the adjective is not logically distinct then it is the case that there is nothing to unite with the substantive.

So, in this next passage William James discovers that in Mr. Bradley’s “Appearance and Reality” Bradley experiences the same problems that Royce and Lotze had. Lotze had said that a does its influence when it influences b, but in response, William James said that “how shall an influence?” and within Royce and his relations, William James said to that “how shall a relation relate?” The self-contradiction that they face here is that any relationship that involves conjunction, or is described conjunctively say two phenomenal experiences, ones by a and b must have a third entity in between the two. And as William James says “instead of bridging the one original chasm, it can only create two smaller chasms, each to be freshly bridged. Instead of hooking a to b, it needs itself to be hooked to a fresh relation, or a and another r’’ to b”. These two new “entities” need to be freshly hooked to another relation as to go on an infinite regress. The entities in the relation, even off of face value are phenomenally distinct, must merge their terms, and thus must merge their being. Because no external-go between can logically connect due to infinite regression. To conceive of this without the absolute, is of course hard, pretty much inconceivable. But it seems as though the absolute connection between these relations is logically necessary. We cannot end at the notion of the being between because of the infinite regression. 

3.4 Thomism and Mathematical Objects

A relation between Thomism and mathematical objects may be very difficult to grasp prima facie. However, when we conceptually analyze these two concepts, we can generatively find a relation. Thomism has its roots in Aristotle and has an Aristotelian foundation for its own explicable and prominent theses. And within Aristotle, he developed a semi-platonic realism. It is called Aristotelian realism. Aristotelian realism is most often recognized as a position joined with empiricism, as it states that universals exist in the natural world, equally known as the physical world, within particulars, concrete substances that are nonetheless plain. The Thomistic foundations given to mathematical objects are nonetheless an empirical ontology, they are given an empirical reality. Thomism recognizes quantity as an accidental quality in concrete substances. Quantity is typically used for numeric measurement, meaning it is not necessarily a mathematical object. More readily, the measured entities that are the outcome of using quantity as a conceptual tool are not mathematical objects. Measured entities are the outcome of a tool used empirically, at least in the Thomistic view. Furthermore, the original conception of mathematical objects that exist independently of anything. Therefore, the outcome of measured entities from the tool of generating qualitative data from numeric measurement is the foundation for measured entities, and these outcomes which we know like numbers, by locution, are accordingly just qualitative entities from some previous datum. This is rather a mere scrutinized examination of this empirical view. We can go on further to identify more Thomistic foundations that make an empirical reality indispensable to mathematical objects. A view about the Thomistic foundations for mathematical objects was formed by Ryan Miller in ‘Thomistic Foundations for Moderate Realism about Mathematical Objects’. 

“The Thomist says that there are three persons in God, ten categories which divide being, and a multitude of the heavenly host” (Miller 2022). Miller then goes on to say that division of using the mathematical operation ‘division’ and thus uses the tool of numerical measurement. Yet again, we see in the Thomistic tradition the use of quantity as a tool to ground the existence of mathematical objects. As said before quantity cannot be used as a tool to ground mathematical objects, whether it be for the mathematical objects to be abstract, or for them to be perceivable in concrete substances. Richard E. Hennessey said, “Not only do these values seem suspiciously like numerical quantities, but logical non-identity, which they surely presuppose, can be used to give definitions for the ordinal numbers which are found in mathematics”. The so-called ten categories that divide being are called to be numerical quantities. Thomist seeks leverage for the numerical quantities to be real objective mathematical objects. 

Because Aquinas had said “is said to be divisible into infinity” by the mathematical operation of division”, Thomists try to seek further leverage because they too advocate for the indispensability of actual infinities and potential infinities needed for mathematical analysis, and thus must be mathematical objects. However, this is where the Thomists must stop. The Thomist is becoming conventionalist, as the Thomistic tradition subscribes to the assertion of mathematical objects’ unattainability to qualify for the instantiation in concrete substances. Miller himself had said this, and even quoted Frege, “After all, it seems that no concrete object could provide a referent for many important mathematical symbols: 

“No Euclidean straight line, regular polygon, etc. can be instantiated in concrete objects or can be used as a description of concrete objects. In fact, in the age of the theory of relativity and of quantum mechanics, when it becomes justifiable to represent the universe as having a finite diameter and containing a finite number of elementary particles and to conceive of space-time as having a curvature different from 0, which concrete object could be considered as a straight line without breadth and infinite in length having 0 curvature, or as an instantiation of a transfinite cardinal number greater than 2ℵ0?”.

 “If mathematical symbols are proper names for objects, then, they must be the proper names of abstract objects” (Miller 2022). Actual infinities, potential infinities, numbers, sets, etc., under the pure actual Thomistic assertions, cannot be instantiated. Though, you may ask “Why is this important”, if you truly had any intentness on reading this, you would know the Thomist follows the Aristotelian tradition, the tradition of the materiality of universals, multiplied in particulars. The lay clear now. The Thomist, an acknowledger and follower of moderate realism, a semi-platonic realism, a Thomistic Aristotelian realism, asserted the reality of mathematical objects through an empirical reality. 

3.4.1 A Thomist Consideration: Plenitudinous Fundamentalism

The Thomist, though, asserts that mathematical objects cannot be instantiated while holding that mathematical objects are multiplied and particularized in concrete substances. This is the inconsistency found within the Thomistic applications of the philosophy of mathematics. Perhaps Thomism can also never be compatible with the existence of mathematical objects, though there can be an alternative. If Thomism chooses to reject this impossibility of instantiation of mathematical objects, and reject the Aristotelian tradition of the empirical reality of mathematical objects in concrete substances, there would be no problems. Nevertheless, the use of quantity as a conceptual tool still harms and hurts the Thomistic tradition of mathematical objects. 

This conceptual tool should be abandoned if the Thomists wish for no problems and more practicality if they ever want to intervene within the domain of grounding, the existence, and the grounding of the existence of mathematical objects. Plenitudinous Fundamentalism offers a conceptual toolkit, one that can help and make more sense of the Thomistic tradition within the topic of mathematical objects.

David Lewis who was a leader in the modal revolution, (the revolution of using modality as a tool in metaphysics), made his conceptual toolkit. This conceptual toolkit included the concept of “natural properties and relations”. The type of natural property and relation that discovers objective similarities amongst things, humans, subjects and objects, is something that occurs in the fundamental laws and as ascribed with quality of fundamentality. This notion from David Lewis was also argued by Sider. It is called fundamental concepts. These fundamental concepts are used as tools to identify the structure of numerous objects. My conception of this is that these tools identify their objective similarities amongst objects, and objective relations and properties that are founded by using these fundamental concepts, and thus the objects that they find that have objective similarities with them, and have no accidental properties are nonetheless then fundamental concepts themselves. So, the realm of fundamental concepts includes qualities of being platitudinous. And then is included with all possible fundamental concepts, and thus under the view of, what I will call Plenitudinous Fundamentalism, Or PF for short. PF rests on a couple of assertions.

Plenitudinous Fundamentalism Thesis (PFT): (I) Fundamental concepts are objective, abstract, spatial, temporal, causally capable objects that interact in a realm with objects where all objects and subjects exist. (II) These fundamental concepts linger on them that we may call, subordinate prehensions. (This concept arises from Whitehead’s philosophy of processes). These subordinate prehensions are what the fundamental concepts “take aboard” with them and then casually make their lower levels of fundamental concepts. The qualities that these types of fundamental concepts hold are almost the same as the prior type of fundamental concepts I mentioned. They rather are concreta than abstracta. This means these fundamental concepts can be seen by subjects. The fundamental concepts that are concreta serve as the subordinate prehensions that the abstract fundamental concepts that prehend these subordinate prehensions. (In Whitehead it is known that there are 3 elemental prehensions, 2 eliminative and one positive. The positive can be [X], and the eliminative can be by [Y] and [Z]. The subject actively prehends the only positive prehensions which are [X], and deliberately ignores the eliminative prehensions [X] and [Y].) The subordinate prehensions work as knowledge givers to subjects, as this view can be objected to by the assertion of an impossibility of obtaining knowledge of abstracta, even if it is spatial and temporal.

The Thomistic Tradition with Aristotelian realism, the abandonment of the conceptual tool quality, can finally reach its full potential. PF gives mathematical objects subordinate objects that it owns which are the concrete substances that are appropriate to the mathematical object. We can instantiate a mathematical object, say a regular polygon, just not in its greatest form . There is a finite and definite amount of numerically measured entitic concrete substances that are in the empirical reality, the domain of empirical objects. Therefore, there must be a finite number of times we can instantiate a mathematical object, and there be a finite number of mathematical objects. Perhaps we are now by sparse platonism, the view that asserts the finitism of platonic objects in the platonic realm. So, the existence of mathematical objects is not reliant on human minds, and it is plausible that, if we take an Augustinian approach, there is a necessary interlocking system of mathematical objects that have their subordinate number of concrete substances that are all positive prehensions of its according to mathematical object. 

3.4.2 Maurer’s Unique Exposition

Armand Maurer in his paper “Thomists and Thomas Aquinas on the Foundation of Mathematics” reveals an additional insight to the Thomistic conception of mathematical objects. It is further on noticed that mathematical entities are not entia rationis but are entia realia. This means that mathematical entities are not entities existent through reason, rather mathematical entities are real actual beings. The essence in the mathematical entities is sought out to be discrete quantity in arithmetical objects or entities. A “discrete quantity” is the term used for arithmetical objects that have definitive ontology, wherefore on the contrary there is continuous quantity. The type of quantity which is described as not having a definitive ontology when ascribed to arithmetical objects. Arithmetical objects or objects in general that are seen as potentially infinite, are therefore continuous objects, as these types of objects do not have a definitive ontology. Why does the Thomist make this distinction though? Well, what is described as a continuous quantity is also described as having a fictitious quantity Maurer says. These entities with fictitious quantities are entities that are conceptual, and are entia rationis. Maurer gives an example of these fictitious quantities saying that “Recent mathematicians, Gredt continues, extend their speculation to fictitious quantity, which has conceptual but not real being; for example, the fourth dimension, which by its essence positively excludes a relation to real existence.

According to Gredt this is a special, transcendental mathematics essentially distinct from “real mathematics,” and belonging to it only by reduction”. Perhaps we once again yield at the empirical reality of mathematical entities. We have seen this before in Ryan Miller’s exposition on the Thomistic foundations of mathematical objects. Though, the essence of this literature is worded differently by using fictitious quantity only in conceptual definitive reasoning. The contrary of these fictitious quantities are nonetheless ones that are concrete and empirical objects.

3.5 Neopythagoreanism and Mathematical Objects

The Pythagoreans and mathematical objects are an interesting approach to the ontology of mathematics. Pythagoreans are commonly known as philosophers and mathematicians because they associate mathematics with nature and go on to create multiple theories that are derivative from mathematics to explain nature. So, when we speak about Pythagoreanism and mathematical objects, the mathematical concept involved is number. Aristotle in The Metaphysics describes the ontological theories of Pythagoreans, saying that they think everything can be contracted to a mathematical point. The primary questions of ancient Greek philosophy were cosmogonical and cosmological, meaning they resided in the questioning of the origin of the structural features of the arche. The Pythagoreans answered these inquiries with mathematics, particularly numbers. In Pythagoreanism, the primary generative and originative substance, the apeira and perainonta elements, were number.

This brings us back to the first Pythagorean exposition about all physical objects when contracted, are mathematical points, but not in the way we want them to. Mathematical ‘points’ aren’t necessarily numbers and are better understood in terms of vector space, viz., following the substantivalist framework of the physical world, vectorial analysis of 3-dimensional Euclidean space cannot be understood in the terms of numbers being the primary originative substance of the universe. The correlation is just not there. This Pythagorean view can be seen as one that contains the concept of multiplicity, as the number is the element of unifying reality is “The notion of limited, ordered, and defined multiplicity”. The Neopythagorean layers of Plotinus’ concept of number apply the concept of multiplicity to the primary originative substance and the concept of number as a limited and is described as ‘oneness’. Neopythagorean elements can be found in the Plotinian conception of multiplicity, as Neopythagoreanism was a huge part of philosophizing about numbers in the first and second centuries.

But the problem here is that there is a lack of writings for the Neopythagorean doctrine. The prominent writers of Neopythagoreanism, Numenius and Ammonius, are thought to be the ones who introduced Plotinus to Neopythagoreanism. Since there is a lack of writings about Neopythagoreanism in the ancient times, we can surely affirm that there is a lack of writings from the ones who introduced Plotinus to Neopythagoreanism, Numenius and Ammonius. Slaveva-Griffin enquires that we can “Perform a kind of conceptual archaeology to search and uncover the Neopythagorean remains scattered throughout the Enneads”. Numenius exposits a theory of First God as the stability and origin of motion. Numenius’ conception of number follows from stasis and is characterized from the internal elements of stasis. Numenius facilitates that number is a natural-primordial non-interlocking system of monads. It is also the progression of multiplicity, meaning there are monads after monads and so on. At the end of this sequence of monad there is monad, “This reasoning makes the first monad both the beginning of the ascending sequence of numbers and the end of the descending sequence of numbers”. Monads in the contemporary sense are reocgnized as immaterial creatures that mirror the universe.

But in this Neopythagorean sense, these creatures are not necessarily monadic in the ‘Leibnizian’ way and are rather just inert objects that are (In the Neopythagorean view), the first principles of the physical world. Another important point which I cannot put into my own words “The monad is the actual limit of quantity because there is no number smaller than it. When multiplicity is decreased by subtraction of all numbers, the naked monad (sterêtheisa), Moderatus concludes, receives onlyness (monê) and stability (stasis). Since the monad is both the starting point and the finishing end for numbers, the monad lacks motion and thus represents stability. If all numbers start with and return to the monad, the monad must be unmoved, fixed, and stable. This stability makes the monad the limit of quantity.” So, if we turn back to mathematical objects and what they would be under this Neopythagorean sense, what should we say? Under this view the monad is the representation of the beginning of number and the end of number. The monad also is the ultimate representation for number, and it can be interpreted that the monad is a natural-primordial substance for number. What is intervening between these two natural-primordial substances is number itself. The intervening substances should nonetheless be characterized as mathematical objects, arithmetical objects used with simple basic mathematical operations. Furthermore, it is said by Slaveva-Griffin that the first principle monad acquires ontological significance since it constitutes arithmetical objects as systems of monads. Let us go back to the exposition made by Numenius about the First God and stability.

Stasis, as it is, shows that there is a substance that constitutes lesser substances that are systems of it. These lesser substances, (We should signalize these lesser substances as arithmetical objects because of the context), originate from this primordial substance. This primordial substance is a signification of all order, and is the beginning of multiplicity and every lesser thing. But then, we see again in the descending order the primordial substance. We can characterize this framework as [X] owning [S], [M], and [A], and [Z] being a ‘relative’ of [X]. [X] is this primordial substance that constitutes [S], [M], and [A] and thus makes them interact in this non-interlocking but orderly system. [Z] is what is at the end of this system and therefore closes this system as it is a ‘relative’ of [X]. It is said that the arithmetical objects are generated from the monads, and that when ascension starts from the first monad and then starts to descend, the monad is there again. So, again, monad is the beginning and end of number. But another inquiry arises. If monad is the end and beginning of number, and when multiplicity is substracted from this system, it is said that monad is stable and cannot enter in events that require motion. Meaning that the monad cannot in a way causally interact with anything and is just a symbol of stability (stasis). So then how is the principle of enumerated things, arithmetical objects, generated? Or in other words, what is the explanation of the existence of arithmetical objects? Slaveva-Griffin never once considers this nor do any of the prominent figures of Neopythagoreanism do. This should not lay as an objection but rather a mere consideration of the Neopythagorean and Plotinian or otherwise Neoplatonic conceptions of arithmetical objects and the monads. 

4 God, Time and Abstract Objects

The topic of God and abstract objects has been relatively discussed within the philosophy of religion, and has been discussed at the intersection of contemporary metaphysics. Why people discuss this sort of topic rests on the sole question if “Does God depends on abstract objects or did God create abstract objects?” People like Christopher H. Menzel and Thomas V. Morris have discussed this topic and formulated that God is not dependent on such abstract objects, and rather creates abstract objects through “intellective activities”. In other cases, other people have come across the consideration that God is just another abstract object. This consideration comes around because of the fact that in Classical Platonism it is a focal point to designate that platonic objects are abstract objects which are eternal items. Whereas God is met with the quality of being eternal, and therefore God could have not created abstract objects, and is relative to abstract objects. In Classical Theism, this view would instinctively be rejected, as Classical Theism holds God being a metaphysically necessary and a metaphysically ultimate being that had created everything. 

Scott A. Davison in his paper “Could Abstract Objects Depend Upon God?” reasons of two theses that the theistic platonists could refer to. DEP is explained as the position that abstract objects depend on God ontologically. Whereas IND is explained as the thesis that, on the contrary, says abstract objects do not depend upon God. To argue and reason from DEP the theistic platonist must inherently reject Classical Platonism, as Classical Platonism asserts that all abstract objects are eternal, meaning God could have not created abstract objects such that they have existed forever, or, in other words, the framework of the persistence of abstract objects has no beginning, middle, or end.   

IND is the view that abstract objects exist independently of God’s creative act. DEP is the view that abstract objects depend upon the creative act of God. If the theistic platonist sides with IND then they are in favor of the rejection of the aseity of God. The rejection of this is the rejection of the traditional theistic framework of God, more specifically the framework of Classical Theism’s God. What Davison calls the M&M view puts forward the existence of abstract objects as the cause of the creative intellect or the intellective activity of God. They assert that God creates the framework of reality, with this, they then explain that abstract objects, and objects within the natural world are included within the framework of reality. So, whether or not abstract objects or eternal, necessary entities exist within the framework of reality. These abstract objects also coincide with the existence of objects in the natural world, such as plants, animals, trees, etc. through the intellective activity of God, he creates the framework of reality, thus creating all objects, inertial or not, in both the natural world and the realm or world of abstract objects.

From this line of reasoning, we can comfortably say that abstract objects depend upon God. The M&M view also outlines the properties of God. the properties of God are from the divine conception. Meaning that the efficacy of divine conceiving is the result of divine properties. This can also be shown within Spinoza, where his panentheism is shown to have asserted that the essence and or properties of God are conceived by God, so as is his nature. Even more, the properties brought about by human conception are that of ontologically independent, as when humans conceive of, say, redness, humans conceive of the literal property ‘redness’, but this property of redness is instantiated in an object, say a mug, and is thus a red mug. But the focal point is that humans conceive of the redness property as ontologically independent as there is no interest in the conception of a mug. And, to note, the properties of God are ontologically dependent on the divine conception. 

M&M suggest that abstract objects are constructed out of properties, and then go on to formulate that these properties are the product of divine conception. Meaning, that abstract objects are wholly dependent on the divine conception of God, or, in other words, the existence of abstract objects relies on their properties which come from the intellective activity of God. Since the natural conception of abstract objects includes necessarily existing propositions, the M&M view also constructs the framework of necessarily existing propositions through the act of God’s thoughts. They say that propositions can be seen as an extended characterization of God’s concepts being properties of abstract objects. They say that necessarily existing propositions are constructed from the thoughts of God. We can start to see more of the influence of Spinoza in the literature of the M&M view. All that follows from God’s thoughts are thought to be necessarily existent thoughts. They hone this view and say that abstract objects that are propositions follow from the intellective activity of God, are the thoughts of God and are thus necessarily existing propositions.

M&M suggests another argument that transfers over to the history of traditional theistic considerations regarding creation. This argument is called the doctrine of creation, and proposes the creation of abstract objects by God. Though, this argument is arguing from rational oughts, saying the theist rationally oughts to prefer that God created abstract objects. This is backed by supposing a consensus has been reached due to the original conceptions of God’s being, suggesting that God has full sovereignty and has the divine aseity property. Davison disagrees with this soley because of the fact that there actually has not been a consensus reached by traditional theists, and from that there is not a rational ought to make by the traditional theist to prefer God created abstarct objects. The original summarization of M&M’s argument by Davison 

“So, since there has been nothing like a consensus among theistic philosophers concerning God’s relationship to abstract objects, tradition alone does not indicate that theists ought to construe creation as absolutely universal in scope, as M & M do. So they have clearly not shown that either (i) our practice of extending the doctrine of creation to cover the contingent entities postulated by modern science or (ii) the voice of theistic tradition suggests that theists rationally ought to view creation as extending to necessary, abstract objects.”

Obviously, due to things such as historical critical method there can be such difficulties in concluding that there is a consensus among traditional theists on whether or not God really did create such abstract objects. Perhaps we should abandon this type of argument due to such difficulties which give it it’s most weaknesses. 

4.1 Feser’s Augustinian Proof

Augustinianism is a school of thought that itself derives from the literature of Augustine of Hippo or more simply put St. Augustine. The literature that includes Augustinianism and abstract objects is dialectically indisposed as there is not much to go around. The only things we can do about Augustinianism and abstract objects is create arguments from the literature of St. Augustine as there are no direct and explicit arguments made by St. Augustine. The arguments that include abstract objects from St. Augustine are arguments for the existence of God. They argue for that if there are abstract objects there must be this being that which we call God. In terms of the nature of time and temporality, St. Augustine does purport different views, specifically nine of them, most of which are popular in the secondary literature of St. Augustine. Augustine’s elucidation on the nine ways of time have been preoccupied with Neoplatonic discussions about physical divisilbity. These mathematical discussions and Neoplatonic discussions on the divisibility objects in the physical world are credited within the Augustinian nine ways of time.

Jason W. Carter exposits that “Augustine makes it clear that he believed that geometrical lengths, as well as corporeal bodies, are infinitely divisible, and that no final ‘point’ or minimum could ever be reached by a process of division”. This view can also be seen in the Eleatic paradoxes, where Zeno constructs that “All physical beings have size, and that which yields size is infinitely divisible”. Perhaps St. Augustine developed his view about corporeal spatial bodies and geometrical lengths from the view embedded in Eleaticism. It’s also shown that St. Augustine could have had a relationalist view on spatial bodies, saying that “These [living things], on the contrary, which have an infinity of divisions, are not small in themselves, but in respect to other things, and most of all by comparison with the universe itself”. In short, their size is acknowledged when in respect to other spatial bodies, whereas the relationalist view asserts the nonexistence of space and says that all spatial bodies exist and are all in relation together, Though the relationalist still believes that one could make statements about space as an adjective. 

Feser details an argument that is described as the “Augustinian proof”. The Augustinian proof asserts foremost that abstract objects are all logically related together “In such a way that they form an interlocking system of ideas.”. Feser then goes on to argue that “The reasons for concluding that at least some abstract objects exist in a necessarily existing intellect also entail that this interlocking system of ideas must exist in a necessarily existing intellect.”, which is based off of the supposition that Scholastic realism supersedes all other positions that yield either the existence of abstract objects  or the nonexistence of abstract objects. To be conspicuous, Scholastic realism asserts the existence of abstract objects in atleast one necessarily existing intellect, which is why premise 17 of his argument is based off of the supposition that Scholastic realism supersedes all other positions. But there is a problem in thinking that these abstract objects can be logically related in a way. Abstract objects in atleast the Classical Platonist conception are causally inert, aspatial and atemporal. Logical relations cannot be acknowledged amongst abstract objects because of the sole fact that all abstract objects engage in different events. It is that they cannot be logically related because of their own purpose. When these abstract objects, whether they be propositions, universals, or mathematical entities, are instantiated, their final cause is prima facie unique. Meaning, what the abstract objects do and what they are used for by the human intellect is indisputably seen as distinctive acts of the human intellect. If all abstract objects are distinctive through what they are used for, then they cannot be logically related through the indication of their purpose.

Abstract objects’ purpose is what differentiates them and makes them entirely unique from other abstract objects. So, because abstract objects cannot be logically related to one another, they cannot exist in a necessarily existing intellect, in so far as they cannot create an interlocking system of abstract objects that exist in a necessarily existing intellect. Premise 24 of his argument explicates that “If this one necessarily existing intellect were not also omniscient in the stronger sense that it knows all contingent truths, then it would have unrealized potential and thus not be purely actual.”. This premise is simply unjustified in the sense that it is an unreasonable explanation for a problem that is put towards this premise. Suppose a necessarily existing intellect exists in which that it has an interlcking system of all abstract objects that which are all logically related. This means that it only has knowledge of abstract objects. Feser here is presupposing that having the knowledge of abstract objects brings the quality of knowing everything. But this is simply untrue.

The natural conception of the nature of the realm that abstract objects inhabit is typically correlated to the infinitism of abstract objects. So, if there were an infinite number of abstract objects, and these abstract objects are all logically related and form an interlocking system of logical relations amongst them, would this necessarily existing intellect be infinite? Feser never employs any exposition on the finitism or infinitism of abstract objects. Clarifying the adherence to either the infinitism or finitism of abstract objects is really important, especially if you are employing that there is a necessarily existing intellect that which they exist in. Feser in his argument concludes that this necessarily existing intellect is really just God because of the qualities it facilitates. Feser says that it is an interlocking system of abstract objects, and that this system is contained within a necessarily existing intellect.

So, is the affirmation of containment hint towards finitism? Perhaps it does, as it is metaphysically conceivable and most likely that Feser agrees upon that Gods’ intellect is infinite, and that this containment of the interlocking of abstract objects in his intellect is rather just a point in his infinite intellect. However, this too would be incoherent. Feser, in premise 24, asserts that “If this one necessarily existing intellect were not also omniscient in the stronger sense that it knows all contingent truths, then it would have unrealized potential and thus not be purely actual.”. Meaning that he defines omniscience as the knowing of all contingent truths. What are these contingent truths? Is it that these abstract objects are contingent upon this necessarily existing intellect (Which I will now call God), thus making them objective contingent truths? Or is it that God is aware of every single truth that could have been false such as empirical or synthetical truths? Feser must comply to the first consideration, such that he asserts only the interlocking system of abstract objects are in God’s intellect, and thus we cannot infer that there is such thing as empirical or synthetical truths, rather contingent truths existent in God’s intellect. So, if God is omniscient then he has all abstract objects in his intellect. But is it really that having all abstract objects in the intellect entails omniscience?

Abstract objects are referred to as mathematical entities, universals, and propositions. Omniscience, defined, is the state of knowing everything. With this, having all abstract objects inhabited in the intellect is thus acknowledged as the knowing of everything. Knowing everything would include to be aware of the existence of all objects in the concretum, otherwise known as the physical world. But God is only aware of the existence of abstract objects such that they are in his intellect. The only way that God can essentially know everything is that if the abstract objects give epistemic accounts or epistemic access of the objects within the physical world, and the physical world itself. In spite of that, it does not seem possible.

Can abstract objects really give lead to the knowledge of every single existent? As we said before abstract objects are commonly referred to as universals, propositions, and mathematical entities. Can mathematical entities lead us to an explication of the physical world? It also seems that having multiple things that give explications to the physical world and its objects is entirely absurd. Is there any hope for Feser and his Augustinian proof? I think not, as there are many presuppositions and dialectical-illnesses¬ that are shown in this Augustinian proof. The dialectical-illnesses refer to the many claims that were met with problems, problems that analyses the sudden irresponsibility that Feser had when asserting this argument.

These irresponsbilities consisted within the fact that there was a lack of clarification in the intellect of God, that which led to the incoherency of the purpose of the abstract objects and the triviality of their existence inside of God’s intellect. These abstract objects were solely meant to show as omniscience for God, but rather were met with triviality. The Augustinian proof was originally meant for the existence of God, but I took it and dissected the inconsistencies in the description that Feser had ascribed to abstract objects, as well with the inconsistencies in the description that he had ascribed to God’s intellect. To be more specific, this Augustinian proof is more of a proof for Classical Theism, the doctrine that asserts God is a metaphysically ultimate being that is impenetrable, necessary, fully good, etc. Other proofs for Classical Theism are the Rationalist proof which coincides with the Principle of Sufficient Reason, otherwise known as the PSR. The Neo-Platonic proof which attempts to prove God with causally sustaining factor(s) for mereological sums, which is God. 

4.2 Pure Atemporalism

Pure Atemporalism is a view developed by Eleonore Stumpz and Norman Kretzmann. This view is put forth the nature of time in reference to the temporal frameworks of God. Pure Atemporalism is explicated on four terms shown by Natalja Deng in “God and Time”:

(1) A timeless being has life. This assumption distinguishes a timeless being from abstract entities, like numbers or sets. 

(2) The life of a timeless being is without limit and cannot be limited, so, for example, it cannot begin or end. 

(3) The life of a timeless being therefore involves a special sort of atemporal duration. 

(4) A timeless being possesses its life all at once, completely.

Deng in response to (4) says that “The challenge for Stump and Kretzmann is now to find a way to think of the timeless being as nonetheless “presently alive” in some sense, and of the being’s life events as still “simultaneous” in some sense, both with each other and (even more importantly) with events and beings in time.”. But this is just confused. In the Pure Atemporalism view, the temporal items being God’s life are possesed all at once. Which accommodates for the eternality that is ascribed to God’s attributes or qualities. So to say that they are challenged with having to think of God or this timeless being as “presently alive” is simply false. No present-ness can be introduced into the likes of this timeless being such that it is only served with simultaneous temporal items, meaning there can be no “present items” in the habitat of this timeless being. Deng presents both ET-simultaneity. ET-simultaneity is essentially the occurrence or existence of both what she calls “eternal and temporal items” in a timeless present, or an eternal present. An eternal present is defined as a duration that has no past, no future, and is infinitely extended. So, ET-simultaneity asserts that eternal items and temporal items both occur in a eternal present. She then distinguishes the two creating T-simultaneity and E-simultaneity. T-simultaneity is is just existence at the same time, while E-simultaneity is existence in the infinitely extended, past-less, futureless duration otherwise called the eternal present.

To make it clear, when we combine both T-simultaneity and E-simultaneity we get ET-simultaneity, which is described as the occurrence of temporal items and eternal items being simultaneous, occurring at the same time. The occurring at the same time is described at T-simultaneity, and the E-simultaneity is described as existence at an eternal present. So, both the temporal items and eternal items concur in this infinitely extended, past-less, futureless duration. But it seems like right now that ET-simultaneity is in no ways coherent. ET-simultaneity supposes that temporal items can be simultaneous with items that exist as eternal items, items in the timeless present, or the eternal present. Temporal items are items that exist in the most subliminal accommodated definition to time; tensed time, the past, present, and future. But say we suppose the moving spotlight theory, the theory that suggests “The NOW moves along the series of times from earlier times to later times.”

So, within any conscious spatiotemporal object that perceives the absolute time with a special metaphysical status, then they endure in time. The eternal present is described as a past-less future-less and infinitely extended duration, meaning it has no beginning middle or end; it is infinite. If this duration is infinite, it could not possibly be parallel to which the time the conscious spatiotemporal object perceives. Therefore, the ET-simultaneity thesis is false, and cannot give a hopeful account of simultaneity between temporal items and eternal items. A possible objection to this simple rebuttal could be in regards to the time that which the conscious spatiotemporal entity perceives. Multiple accounts of A-theory can account for the Et-simultaneity thesis, though arguing for these accounts of time can be very difficult as they can yield for bizarre instances. 

5 Other views on God and Abstract Objects

In the last , I had discussed Feser’s Augustinian proof for the existence of God, and more specifically Classical Theism. In this , I will examine more arguments that include the analyzation of the essence of abstract objects to lead to the existence of God. That of which, these arguments are Classical Theistic arguments, they all argue for the existence of a metaphysically ultimate being, God. Feser’s Augustinian proof argued that Abstract objects existed in a necessary interlocking system, this necessary interlocking system being the mind of God. For he argued that for the omniscience for God there must be actuality in the existence of these abstract objects being existent in the mind of God. This particular view of abstract objects being existent in the mind of God, can be conspicuously be characterized as “Divine psychologism”. Divine psychologism would maintain that abstract objects are mental constructions of God. Furthermore, Mark Balaguer had introduced two views in Realism and Anti-Realism in Mathematics, being actualist psychologism, and possibilist psychologism. The former view explicates that mathematical entities, otherwise abstract entities are actual mental constructions of concrete entities, namely humans. While the latter view holds the complete opposite, and asserts that abstract entities are possible mental constructions of cognitive agents. But when we characterize this view in the terms of a divine entity, and put it in relation with Feser’s Augustinian Proof, we can see that it can be viewed as actualist divine psychologism. While the contrary view would be possibilist divine psychologism. Now, possiblist divine psychologism may prima facie seem false due to the reasons Feser had listed about omniscience in his Augustinian proof, which would directly lead to the inheritance of actualist divine psychologism. So, in this I will therefore be setting forth arguments for both positions, and be giving a very vigorous and specific distinction between both of these positions. Furthermore, I will also explore other Classical Theistic proofs for the existence of God that utilize the existence of abstract objects, and also examine the essence of abstract objects. In this I’ll also outline different views regarding the nonexistence and existence of abstract objects, all of which will regard traditional theistic conceptions combined with theistic platonism, all of which can be predominantly found in the book “Beyond The Control of God?”. But first, I will start off with the problem of theistic platonism, namely its incoherency.

5.1 The Problems of Theistic Platonism and Others

The dependency problem says that necessary being Y cannot asymmetrically depend on X. meaning abstract entities which are necessarily existing cannot asymmetrically depend upon God, and rather the emergence is of a mutual logical dependence. Meaning the existence of X entails the existence of Y, not in the sense of an asymmetrical dependence, but rather a mutual dependence. And so rather, there is a mutual logical dependence between God and abstract entities. But this would not sit well with either the theistic platonism who takes to traditional theism, or the classical theist. So how can they reconcile? The theistic platonism or classical theist can take to the likes of a causal dependency between abstract entities and God. But this brings up even more problems. If the classical theist or the theistic platonism takes to the fact that there is a causal relation between everlasting abstract entities and the everlasting God, they will have turned to co-eternal existence and eternal causation. This leads to even more problems due to the skeptical adherences towards eternality and co-existence, and eternality and causal relations. The problems are explained even more by Paul M. Gould: “Is it metaphysically possible for God, or anything else, to create abstract objects? Assuming that abstract objects are everlasting, is the notion of eternal causation coherent? Does co-eternality render God somehow less ultimate? What sense can be given to the notion of one necessary being (God) creating another necessary being? What analysis of causation is required to give sense to the notion of God creating abstract objects?”. Really, at this point, theistic platonism should really just be abandoned. Moreover, the theistic platonism faces the bootstrapping worry, which was proposed by Alvin Plantinga: “God has properties. If God is the creator of all things, then God is the creator of His properties. But God cannot create properties unless He already has the property of being able to create a property. Thus, we are off to the races, ensnared in a vicious explanatory circle”. 

The ultimacy problem Gould calls it, seeks to find the problem with God being an ultimate being. It shows that because there are some abstract objects that are properties, and God has properties essentially, then these abstract objects explain the nature of God, and thus God’s sovereignty is not so unique, and we find an explanation to God’s sovereignty that isn’t one through God. The theistic platonism faces the independence of abstract objects such as possible worlds. If the theistic platonism or classical theist follows the traditional theistic conceptions of God, they will acknowledge that God is necessarily existing, and in terms of possible-world semantics, through possible worlds God exists necessarily, and therefore God needs possible worlds for his necessary existence. This means, again, that there is an independent abstract object that God needs for the completion of his divine aseity. But if we look back to the conceptions that the theistic platonism would take if he took to the traditional theistic conceptions, we would see that he solely affirms the divine aseity of God. So now the problem can be characterized like this: (i) God is a necessarily existent being through the traditional theistic conception. (ii) with reference to possible-world semantics, to be an actual necessary being you need to exist within all possible worlds, and thus consequently exist in the actual world. (iii) God needs the abstract object possible worlds to exist, and (iv) therefore God is dependent on the existence of an independent item, and causes the theistic platonism with respect to the conceptions of the traditional theist to reconcile about the true nature of God.

There are different views that which are variations of default views and then the default views. These views, however, are interesting in which they are interesting in the way they respond to the Inconsistent Triad. The inconsistent Triad is a triad of propositions that are expressive of different positions, that which the following two propositions follow from the first proposition. And therefore, the following two propositions vary differently. The Inconsistent Triad goes as follows:

(1) Abstract objects exist. [platonism] 

(2) If abstract objects exist, then they are dependent on God. [from AD] 

(3) If abstract objects exist, then they are independent of God. [platonist assumption]

The second proposition is the affirmation of what Davison had called DEP; abstract objects are dependent on God. Whereas the third proposition is the IND view; abstract objects are independent of God. So, when we speak of these propositions in this context, some of the arguments that we had discussed before when examining Davsison’s paper could possibly appear here. So, we will start with examining the arguments for the second proposition, that the existence of abstract objects is dependent upon God.

5.2 For DEP

DEP is the position that accepts (2) of the Inconsistent Triad. Positions that accept DEP can be known as theistic activism, modified theistic activism, divine conceptualism, and theistic conceptual realism. What all of these views have in common is that they all accept (2) of the Inconsistent Triad. I shall explain these views and illustrate the assertions of these views along with what the defenders of these views say:

(1) Theistic activism: theistic activism argues that the platonic tradition can accommodate abstract objects being necessarily created by God, and thus dependent on God, through divine thinking. Theistic activism when asserting this faces the problems that were explained above, being the ultimacy problem and the dependency problem. 

    (1.1) Modified theistic activism [MTA]: Rejection of (3) in the Inconsistent Triad.

(2) Divine conceptualism: All abstract entities are uncreated constructive mental constituents of a divine entity, in other words abstract entities are objects that which exist in the mind of a divine entity. 

    (2.1) Theistic conceptual realism: Rejects (3) in the Inconsistent Triad.

Theistic activism leans toward that the platonic horde is located within the mind of God. Meaning concepts, propositions, relations, etc., are all existent within the mind of God. In more explicability, propositions are the divine thoughts of God, and things like possible worlds, numbers, sets, are all divine concepts within the divine mind of God. As we did before analysing Davison’s paper or Moriss and Menzel’s paper on absolute creationism, we will again find more arguments for the position of M&M. think of this like an extension of the section. The M&M view can be looked at as theistic activism, in fact the M&M view is more notably recognized as theistic activism. If the reader remembers, M&M look at abstarct objects of the divine intellection of God. If we remember even more, we would remember that the M&M view explicates that because God exemplifies a nature, then he created that nature.

Most look at theistic activism as a position that, again, is susceptible to some objections. The first objection explained here is the one that was used against the affirmations of theistic platonism. This objection includes the presupposition of the truth in eternal causation. If we as the theistic activist accepts the fact that abstrct objects or abstract entities are necessary being, then to say that these necessary beings are created or dependent upon another being, particularly another necessary existent being God, can obstruct our position. To combat this objection of asymmetrical dependence between two necessary beings, we must affirm eternal causation, and say that God eternally enters into causal relations with abstract objects or objects within the platonic horde. We should also look at what creation is, as Plantinga does. Plantinga has this core intuition about creation involving temporality. It is how the creation of something involves that something not existing at one temporal interval, but then later that something begins to exist at a temporal interval, thus temporal becoming intervenes. Plantinga says “a thing is created only if there is a time before which it does not exist”. Gould says that the creation of necessary beings should be understood in terms of the causal or explanatory. He says “For example, God is the eternal generating cause of abstract objects”. And then for the activist he says that it would be the divine intellect that would be the causal explanations of the abstract objects or the platonic horde. 

What about the divine will or divine freedom. Obviously, divine freedom and human freedom are completely different. And the will of divine beings and the will of humans are completely different. But what about the divine freedom and the creation of necessary beings? Of God’s creation of necessary beings is accepted, then God would have had to created necessary beings necessarily due to their being necessary. So does God really have freedom? Defenders of the MTA (modified theistic activism) say that God thinks up all possible creatures and all possible states of affairs. In this creative act, God delimits all modal facts—all possible individuals and possible worlds are set—in virtue of God’s intellectual activity.

Gould says:

In addition to God’s spontaneous creation of all possible creatures via His producing divine concepts and thoughts, God creates, of necessity, a platonic horde of properties and relations that will play the role of structure making in any actual concrete universe God creates. This creating of the platonic horde is logically posterior to the Biggest Bang, and sets the stage for the Big Bang (that is, the creation of the actual contingent universe). If so, divine freedom is preserved (or so it seems) since in the first logical moment of the Biggest Bang, God spontaneously creates all possibilities even if He creates the corresponding properties and relations of necessity in virtue of the divine will.

But does this really even solve anything? God still creates the platonic horde necessarily, meaning his divine freedom is hitherto lost. But Gould has said that the theistic activist doesn’t have to care about the loss of divine freedom? Really? I think not, such that without divine freedom God has lost all of his sovereignty, as this one action of necessity has made God seem defeated, like an underachiever. The traditional theist for so long had conceived of God as metaphysically ultimate being, meaning His doings are ultimate. But if doing an action of necessity, without having the choice to not do it, are you still really a metaphysically ultimate being? Though, this problem does not really seem substantive, it’s crashing. Maybe Gould didn’t have any attractive responses, and instead went to the conclusion that the traditional theist (in addition to the theistic activist) does not have to worry about the loss of God’s freedom, otherwise known as divine freedom. Let’s talk about the next view, divine conceptualism. 

5.3 Divine Conceptualism, Augustinianism and Classical Theism

Divine conceptualism is the view that abstract objects exist uncreated inside the divine mind of God. At face value this view may be difficult to understand when there is no innate explication of the uncreation of abstract objects though existing in the divine mind. Divine conceptualism can also be described as abstract objects being identical with other parts within the divine mind. From this, it can be acknowledged that the parts of constituent entities existent within the divine mind of God, are the constituent entities that which form the final cause of God. But wouldn’t that mean God causally depends upon these constituent entities (i.e. omniscience, omnipotence, etc.) and these abstract objects for His existence? If this were true, then, again, God would lose his ultimate being, divine aseity, and sovereignty. But, this can be avoided. Divine conceptualism is the notion that abstract objects exist within the mind of a divine entity, God. So if this is the case, then these abstract objects and constituent entities are merely just entities that are uncarried out volitions, which is why they are uncreated. Instead of these constituent entities and abstract objects being the cause for the finality of God, these abstract objects and constituent entities are what make up God. The entities (that which exist within the divine mind of God) are identified as the properties of God, and are volitions that have not been carried out. As another option to understand this, we could simply say that these abstract objects and other constituent entities are rather just mereologically dependent on God for their being; for their ontology. 

If we take a look back at the first supposition that abstract objects and constituent entities are what produce the final cause of God, we can see that this supposition is an affirmation of the Aristotelian conception of what substance is. The final cause is of the last cause of the four causes that Aristotle identifies. This final cause is defined as the function of something. So, if we head back to the supposition of abstract objects and constituent entities being the parts the produce the final cause of God, perhaps we can elucidate that the abstract objects and constituent entities are what God ought to do, and are ascribing the functional properties of Himself. Say the constituent entity omniscience, if the above exposition be true, then God ought to know everything. The same goes for omnipotence, if the above exposition is true, then God ought to have power over everything. And to note, under the divine conceptualist framework, the constituent entities within the mind of God would be the aggregatum of the essential properties of God. Meaning the divine concepts that here are defined as the properties of God, are ones existent within the mind of God. These divine concepts or constituent entities would have asymmetrical dependence upon God, as they are existent within the divine mind, and are notable for being the contents of the divine intellective activity. So, if we take a look at the necessarily existing abstract objects, by virtue of divine conceptualism, necessarily existing objects would have asymmetrical dependence upon God. In these terms, it is as if the theistic activist is saved from the dependency problem, but still faces the bootstrap objection. 

The problems faced by the classical theist proposed by the rejectors of (3) in the Inconsistent Triad are that in the likes of the dependency problem. If the classical theist takes to the fact that all abstract objects are causally incapable, and are necessarily existent entities, it can be shown that there cannot be any asymmetrical dependence between abstract objects and God under classical theism. As if we look at possible worlds and necessarily existing entities, we can see that if abstract objects are necessarily existing, then they exist in all possible worlds — the same goes for God. This is crucial to the classical theistic framework of God and abstract objects, as classical theism holds that God is a metaphysically ultimate being, has divine aseity, and has complete sovereignty over reality. So how should the classical theist attend to this problem existent within platonic theism? What the classical theist needs is a metaphysically ultimate God, one which stands over all, is beyond everything, and everything else is powerless to it. What the classical theist must understand is that the only problem that is faced is the relativity between abstract objects and God. What the classical theist must do is delimit and make some of the functionality of abstract objects restrictive. We can find some the delimiting of functionality within theistic activism or divine conceptualism. In both of these notions it is an accepted fact that abstract objects are existent within the mind of God, and are thus asymmetrically dependent upon God. So is this solution for classical theism coherent? Does it solve all the worries for classical theism? Yes and no. What it does do is delimit the modal facts of abstract objects with asymmetrical dependence, but what it doesn’t do is refute the bootstrap objection. 

Perhaps it is now appropriate to talk about classical theism and divine freedom. As we had examined before, if God created necessarily existing abstract objects, then it follows that God created necessarily existing abstract objects necessarily, and he had no choice or freedom to not do so. The classical theistic framework of divine freedom is one that insists God’s will is persistent in God’s own affirmation of His good. So, if God necessarily created necessarily existent abstract objects is it an affirmation of his own good? No, for God’s affirmation of good is through his own divine freedom, and this one aspect of the creation of necessarily existing abstract objects necessarily doesn’t take down the whole entire aspect of creating contingent realities with free choice. If we adhere to the fact that God created these necessarily existing abstract objects, and that their own function is to create the ontological pegboards of reality, then it follows that through God’s own omniscience, God created the universe without a choice, and knew of the creation of the universe through the abstract objects, and had to of done so necessarily. But one may ask “If God has omniscience, and foresaw the creation of the universe, does it not follow that God could have foreseen the non-creation of the universe?”. No. Yet again, because God had to have created necessarily existing abstract objects, he would have had to create these necessarily existing abstract objects necessarily, and thus had no choice to have not created the abstract objects. It is almost like this can be characterized in the likes of God foreseeing the non-creation of the universe, but couldn’t have done anything about it. Alternatively, in the process of the creation of the universe through abstract objects, God was powerless when creating such abstract objects, and couldn’t have done otherwise. 

Is divine conceptualism a sort of causalism? Above it was noted that abstract objects within the mind are dependent upon God, and this relation between God and these abstract objects is an asymmetrical dependency; an asymmetrical dependency whether it be causal or explanatory. Take it to be that this asymmetrical dependency is a causal dependency. This means then, that abstract objects are causally dependent on the divine mind of God, for if it is acknowledged that the function of abstract objects is potential, then they need to be actualized into their final cause, that which God’s divine intellective activity is the process of actualizing abstract objects (which in this case all abstract objects, not just propositions, are the products of the divine intellective activity within God’s divine mind) into their final cause. So it seems here that when the classical theist takes traditional conceptions into account, if they adhere to this view, then they must reject the causal inertness of abstract objects. All of this prima facie seems very ad hoc, as removing only the quality of being causal inert from abstract objects puts the classical theist, and in general any theistic platonist, in a good position. Would this objection of claiming ad hoc reduction reduce this account of classical theism to absurdity? It depends as if the classical theist does not present any of these problems, and merely hypothesizes about this causal theory of dependency regarding the creation of abstract objects, then it surely cannot be ad hoc. However, in this case, we can see that because we have removed this causal inertness quality away from abstract objects, it is applicable that the affirmation of abstract objects being able to enter in causal relations and being able to be a sole effect of God’s creation is ad hoc. 

As I had said before, the solutions to the problems that were introduced to classical theism indeed work, but classical theism’s solutions could still not solve the bootstrapping objection. The bootstrapping objection is an objection that revolves around the sole essence of the properties of God. It goes like this: 

God has properties. If God is the creator of all things, then God is the creator of His properties. But God can’t create properties unless He already has the property of being able to create a property. Thus, we are ensnared in a vicious explanatory circle. God causes His nature to exist—a nature He must already possess to do the causing.

The bootstrapping worry is a huge burden placed upon the traditional theist, and replies to this objection have not been very prominent or successful. The most successful reply to this objection, in my opinion, is the se reply. The a se reply can be explained as follows:

(1) God’s aggregatum of essential properties exist apart or a se, and in short are uncreated entities.

(2) Substances are Aristotelian

Claim (1) accommodates the reality that God’s essential properties are ones that are uncreated, and it thus avoids the “God is the creator of his own properties”, and still endorses that God’s aggregatum of essential properties are still existent as metaphysical parts within God. Though this is merely confusing. If God’s aggregatum of essential properties remain uncreated entities, and exist a se from God, meaning they exist apart from God, then these essential properties are one that exist relatively to God, on the same level. So, to say that these essential properties at one point exist apart from God, yet exist as metaphysical parts to God does not make any sense. If we endorse claim (2), then we are endorsing the fact that these essential properties are what God is emergent of, and is thus dependent upon. When we say these essential properties exist at some point of time, this point of time is a time where God does not exist, then if we say at another point later in time God exists it means that these essential properties have formed an aggregatum creating God. If we also say that God does not exist at one point of time we have contradicted the most important part of the theistic tradition. To say God does not exist at a time before his creation is to say God is not everlasting, and is thus not an eternal or necessarily existing entity. If we trace ourselves back to claim (2) which affirms that substances are Aristotelian, we can put together an analysis of the origins of God:

(1) The material cause of God are God’s essential properties or (i.e., Omniscience, Impassibility, Omnipotence) 

(2) The formal cause of God would be described through his essential properties (i.e., God is the most powerful or God knows all. These clauses describe the essential properties of God, that which make up God, and we thus describe God). 

(3) The efficient cause of God is better understood in the likes of mereology. The essential properties of God are the only existence in the realm they exist in, and through examination it is logically reasoned that further it is God that has been formed.

(4) Finally, the final cause consists in what this formed object does, what its function is, namely its purpose. What would be the purpose of God? The simplest answer to this is probably the prime aspect of God’s ontology — His creative activity. This creative activity would persist in the creation of the concrete universe, and other objects. 

This examination of God through the conception of substances as Aristotelian can be faced with a prominent problem: whether or not (4) is coherent. If the essential properties of God are all existent within a realm, and are known at a further point as essential properties that form God, then it would follow that God isn’t necessarily everlasting as it is conceived that God becomes God at a point where the essential properties are identified as things that form God. Before these essential properties are identified as things that form God, they are interpreted as individual divine concepts that form nothing. So, perhaps the claiming that substances are Aristotelian in support of the explanation of the divine substance is incoherent.

Another option can be made in light of the incoherencies discovered. We can still claim that substances are Aristotelian, but we would have to abandon that these essential properties are the metaphysical constituent parts of God, and thus turn ourselves in to the notion of divine simplicity. Divine simplicity is the theological notion that God has no metaphysical constituent parts, and is thus a mereological simple. The main important assertion that divine simplicity makes is that “though nothing is constituent of God, everything distinct by God is created by God and sustained by God”. James Dolezal in his book “God Without Parts: Divine Simplicity and the Metaphysics of God’s Absoluteness” says that

ORTHODOX CHRISTIANS ARE UNIVERSALLY committed to the confession that God is absolute but they are not always agreed on how to characterize this absoluteness. Historically the doctrine of divine simplicity (DDS) has been regarded as indispensable for establishing the sufficient ontological condition for divine absoluteness. 

This is true for most platonic theists when accommodating the traditional theistic conceptions of God, when they are met with problems of God’s sovereignty and divine aseity platonic theists cannot always come up with a response or a different conception of God’s absoluteness in response to the problems. In other situations, the different type of theists who acknowledge the existence of abstract objects always have a complete consensus on how to characterize the absoluteness of God. This is in terms of what the type of theist’s position would assert about the absoluteness of God, typically in reference to His divine properties. 

Perhaps for a model of God’s absoluteness we can perform a criterion, more so a theological criterion or rationale for the model of God’s absoluteness. This model of God’s absoluteness would be characterized in terms of God’s divine properties (i.e. Omnipotence, Omniscience, Immutability, Immanence).  Most theological rationales would be of biblical data, but perhaps making a theological rationale about the essence and/or absoluteness of God isn’t the most desired way to apprehend such a model. My reasoning for this would be that biblical data in short isn’t so much included within a rationalization of things like God, as there are many different conceptions within models of God, and therefore we must attend one that of no external sources that are contingent to interpretation. Nonetheless, biblical data in reference to a model of God’s absoluteness or just God in general do give interesting ideas and should be researched extensively, getting down to a fully comprehensive model. However, we will be focusing ourselves upon a rationalized model of God that does not include any external sources that are contingent to interpretation. 

Does a model of God’s absoluteness require the DDS to be maintained by theists? Surely, for it gives the most yielding account for the creator-creature distinction. Alternatively, DDS supplies a rationally constructed theological rationale for the absoluteness of God. The doctrine of divine aseity (DDA) supplies the lead motive for maintaining a position like DDS. DDA holds that God is self-sufficient, is the fullness of being, and is a se from which when get the properties of a se and self-sufficiency we can clearly see that God is not sustained by anything or anyone. DDA is a key factor of classical theism as it holds the main conception of God in traditional theistic views. 

Let’s take a closer look back at Feser’s Augustinian proof of abstract objects proving the existence of a classical theistic God. But this time in reference to the dependency and actuality of God’s knowing of everything. 

Feser’s argument in premise 22 and 23 talks about the actuality and omniscience of God. Premise 22 says “An intellect in which the interlocking system of ideas in question existed would be conceptually omniscient”. Premise 22 surely says that God’s conceptual omniscience is dependent upon the fact that there is a interlocking system of ideas existent within this necessarily existent intellect which is God. Alternatively, it can be formulated that because Feser had put the assertion of a necessarily interlocking system of ideas was existent within this necessarily existent intellect, it could be affirmed and argued that this necessarily existing intellect’s conceptual omniscience is reliant and dependent upon this interlocking system of ideas. It should also be noted that this Augustinian proof for God through abstract objects is in fact a classical theistic one. In premise 20, Feser explicitly adheres to the fact that there is one metaphysically ultimate being, in so far as it is the only purely actual thing. He said “There cannot be more than one thing that is purely actual”. This means that this purely actual thing put this interlocking system of ideas into actuality. This is where the vicious circularity comes into play. If God is the only purely actual being in existence such that He actualizes all subordinate beings, then he actualizes the abstract objects that form logical relations and thus form an interlocking system of ideas. Two objections can be made regarding (i) the creation and essence of the necessarily existing intellect, and (ii) the dependency of the vicious circularity that occurs when arguing for the Augustinian proof. We shall explicate them and subsequently do a critical examination on whether these objections fail or succeed:

(i) If God is the only purely actual being in existence hitherto, He actualizes all subordinate objects and beings into existence, then He actualized the existence of abstract objects. From this, we can affirm that abstract objects would depend upon God under this view. If it is true that abstract objects depend upon God, then God’s necessarily existing intellect is purely absurd, for if the abstract objects are what actualize God’s omniscience into existence, but God had actualized abstract objects, how would God have had created these abstract objects with an empty intellect? For the necessarily existing intellect of God is what contains God’s creative activity, for a being of such type must have an intellect to have had any creative activity.

(ii) If God created abstract objects, and abstract objects are what actualize the omniscience of God in his necessarily existing intellect, then God created something that He Himself depends upon, but yet created, and thus when in the action of creating these abstract objects that give Him omniscience, he did not have these abstract objects that gave him the actual creative power to have created these abstract objects, and thus vicious circularity is reached.

Perhaps (ii) wasn’t explained thoroughly, but (i) definitely examines the essence of the necessarily existing intellect and the creative activity of God. The classical theist, if unable to reject and subsequently solve these objections, is then in trouble. Along with the other problems such as the ultimacy problem, and the dependency problem, the bootstrapping worry or the bootstrapping objection intertwines with (ii) as it sets out that God created properties (in this case abstract objects are also characterized as God’s divine properties) that He Himself depends upon, and thus created something that He depends upon for His creative act that sustained the creation of that something. With all of these problems, how can the classical theist reconcile? The biggest struggle for the classical theist regarding (i) and (ii) is most likely (ii). (ii) explicably argues against all traditional theistic conceptions of God, and therefore it would be better to reinforce against the objection of (ii) rather than (i). (i) sets out a case that is more focused on the Augustinian proof and solely on the Augustinian proof, whereas (ii), although characterized in the likes of the Augustinian proof, is still compatible with the bootstrapping worry. So, when we again ask the question on how the classical theist should reconcile, we should regard an objection that faces itself amongst the various kinds of the bootstrapping worry and the bootstrapping worry itself. If we want to defend an Augustinian proof version of the bootstrapping worry, (that from now on we should call the APBW), we must find ourselves directly defending the affirmations of the Augustinian proof (directly premise 19-22 of the argument). Furthermore, we can also find ourselves defending some of the Aristotelian conceptions that intervene within some of the actuality and potentiality distinctions. But first, we’ll defend the APBW.

5.4 Defense of APBW

Following all of these objections, the classical theist would sought to have had solutions regarding all of these objections. Here, I will go over a defense of APBW, which is the arguing against the Augustinian bootstrapping worry, that which forms itself as an objection to the nature of the abstract objects in an interlocking system that exists within a necessarily existing intellect. Why the object this nature pertains to the bootstrapping worry. It can be viewed as God creating the properties that he needs to be God that He did not have himself while in the creation of creating such properties. And from that, how could God create such properties, (properties that give God creative activity) without having those properties in the creation of these properties, otherwise viewed as abstract objects? The bootstrapping worry brings about a burden to the classical theistic view of God and more generally, it brings a burden to all traditional theistic views of God. If these objections are unanswered when given to the classical theist or the theist that adheres to traditional theistic conceptions, it would be clear that one would be in some trouble.

The defense given here is one that is argued by Menzel in “Problems with the Bootstrapping Objection to Theistic Activism”, but is understood in the likes of Feser’s Augustinian proof. First, we should start off with the essence of the interlocking system of ideas. This interlocking system of ideas is explained as a bundle of logically related abstract objects within the Augustinian view. This interlocking system is what gives rise to the omniscience of God, and serves as the plethora of the necessarily existing intellect within God. We can now produce what the relation between this interlocking system would be in relation to God. As we know already, this interlocking system is what gives rise to God’s omniscience. Can this sort of relation be understood in the terms of a logical relation? If there is a logical relation between this interlocking system and God, then it will be clear that this interlocking system should asymmetrically depend upon God, for it was created by God (it is also important to note that this interlocking system contains too the divine properties of God).

But inside of this interlocking system of ideas there are abstract objects that contain the divine properties of God. It is agreed by everyone that these divine properties are what make God what He is. The interlocking system includes the divine nature of God, and thus it would logically follow that God created his own nature. Since it is the case that if this interlocking system does exist, and is the concatenation of the divine nature or the purely divine properties of God along with other abstract objects, we can acknowledge that because this interlocking system of ideas is contingent upon God, it was created by God. And thus again, God created His own nature. Would it not follow that for God to create anything He would need his divine nature? For, if, God’s divine nature includes his property of being able to create everything, and more of hindrance — His creative activity — then it would fall under all of this that God created his own nature while not having the property of being able to create everything. 

The above is a careful outline of what the classical theist is very much subject to if he does not adhere to some sort of solution. Like I said before, I would be giving Menzel’s argument in the form of the Augustinian proof. So let us begin with it.

Christopher Menzel explored the bootstrapping objection in one of his most recent papers, “Problems with the Bootstrapping Objection to Theistic Activism”. Menzel says that the bootstrapping objection put forward by both William Lane Craig and Alvin Plantinga is dependent on an “unwarranted conflation of existential and causal dependence”. What is existential and causal dependence? To put it shortly, existential dependence purely means the dependence that occurs when y’s existence depends upon x, whereas causal dependence means that y causally depends upon x. Menzel continues saying “Now, let 𝐷 be God’s nature, divinity. According to activism, since God creates all the properties and 𝐷 is a property, God creates 𝐷. Hence, 𝐷 is causally dependent upon God. At the same time, as 𝐷 is a necessary being, it is necessarily the case that God exists only if 𝐷 does”. For a better understanding of what we want, we shall characterize this in the Augustinian proof tense. D would be the interlocking system of divine properties that God yields, and so God creates this interlocking system of divine properties, and thus this interlocking system of divine properties is causally dependent upon God. But this interlocking system of divine properties is necessary, and thus God creates this interlocking system of ideas necessarily for it is necessary that it exists. Hereinafter, since D is necessary and is what makes God Himself, then if D exists it is necessary that God exists, therefore God being existentially dependent upon His own nature D. Menzel further develops that this exposition epitomizes the principle of transitivity, saying:

But the conclusion that God is causally dependent on God follows only if one adopts a “bridge” principle asserting that transitivity holds across these two very different types of dependency — that is, a principle asserting that, from the fact that 𝑎 is causally dependent on 𝑏 and 𝑏 is logically dependent on 𝑐, we can infer that 𝑎 is causally dependent on 𝑐. But there is simply no reason whatsoever to think that such a principle is true.

Here, I would agree with Menzel — only for the sake of argument. For if we really want a compelling argument for APBW, then we should accommodate the arguments that are in favor of it. 

To be clearer, the actual objection to ABPW is the B&B bootstrapping objection or the BBBO. The BBBO argues in favor of the relations of logical priority and logical posteriority. Logical priority as it is, is defined as the coming before in the natural order of things, where it’s converse logical posteriority is defined as the coming after in the natural order of things. B&B in contrast to M&M argue that M&M mischaracterized the or were not able to point out the fundamental relations that which exist beyond the existential dependencies and causal dependencies. This dependency is actually logical posteriority. Meaning their version of the bootstrapping objection sketches out the varieties of dependencies now known as logical priority and logical posteriority. The traditional classical theistic piece is that God’s creative activity includes all abstract objects and more specifically all properties. This is where logical posteriority would come in. Since properties are just a small sum of the outcome of God’s creative activity, then they are logically posterior to God, insofar that they came after God. But for this to be true there must be a property in which God has to allows him to create all properties, say F. The property F would at first seem to fall under logical priority due to its innate nature. However, since it is said by the classical theist that God creates all properties, and F is a property, then F is too a property that is logically posterior to God. So, we come at the conclusion that F is both logically prior and logically posterior to God. 

So far, we have outlined the BBBO, at face value, as a convincing argument that to most is capable of taking down the classical theist view or really any theistic account of Platonism. To characterize the BBBO in terms of the Augustinian proof, we must first set out a framework in which the BBBO can be characterized. First, the right thing to do would be to identify what the objection would put into play, such as the interlocking system of logically related abstract objects, and the necessarily existing intellect contained within God. This interlocking system of abstract objects is existent within the divine mind, meaning it would incline towards theistic activism or divine conceptualism. So as to say, would it really be the case that God creates this interlocking system of abstract objects? My answer would be yes. This interlocking system of abstract objects would be of something created within the mind of God, and is therefore something logically posterior within the divine mind of God. But if the properties within this interlocking system of abstract objects is within the divine mind of God, would it too be the case that they would be identified as this existent as constituent parts of God? Clearly, no. Under divine conceptualism or theistic activism, the abstract objects within the divine mind act as divine volition, and for some all-abstract objects within the divine mind are divine thinkings or divine thoughts.

A constituent ontology cannot be applied to God at least under the divine conceptualism or theistic activism framework. Although this divine mind of God contains all the abstract objects or just the interlocking system of abstract objects and properties creates, it is not that this interlocking system or all of these properties and abstract objects are actual constituent parts of God, or make a metaphysical complexity that sums up to be God. Rather, these abstract objects and properties are more so the constituent entities existent within the divine mind of God, and what make the divine mind of God. So, it is hopefully now clear that divine conceptualism or theistic activism is in some weak way compatible with classical theism. 

What Menzel does to argue against the BBBO is give a logical framework for BBBO and then to dismantle that logical framework. Alternatively, he proposes a logical framework that makes BBBO easier to understand, further proposes the incomprehension of BBBO, and lastly says why this hindrance of comprehension makes BBBO informal. Menzel argues for logical priority not being a cogent relation to BBBO. In other words, it should be abandoned by advocates of BBBO. And he further argues on for even if you take the notion of the logical priority, there are still reasons to why it is not asymmetrical.  Menzel identifies that in B&B the syntactic entities that should be the only things acknowledged as the relata of the asymmetrical relations are sentential gerunds that he calls S-gerunds and verb phrase gerunds that he calls VP-gerunds. S-gerunds and VP-gerunds are both morphologically used in order to create sentences that describe the potentiality of, say, the creation God. So, for example, when we use S-gerunds and VP-gerunds, they can be logically formatted as “F’s exemplifying being able to create”. Where [F’s exemplifying] is the S-gerund, and [being able to create] is the VP-gerund..???/4

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