Eriugena and Platonized Naturalism
A Modern Interpretation of a Medieval Irish Theology
Presentation paper from CSANA Conference 2023
One of the key topics present in metaphysical discourse is the real existence of abstract objects, their interactions within the physical world, and how we obtain knowledge of their existence. The existence of these objects or abstracta, have been debated among philosophers and theologians over the past millennia. The existence of abstracta play a significant role in Platonic philosophy, where their existence as real objects is asserted as true. This dialogue is also present in Christian Platonism, particularly in the works of John Scottus Eriugena. An Irish theologian from the 9th century, Eriugena discussed the existence of abstracta in his seminal work Periphyseon. His ontology is illustrated in the four divisions of nature, where all real objects are differentiated into four species. This dialectic provides a novel insight into the existence of abstracta but also their relationship to the physical world.
The existence of abstracta still poses issues and are debated among contemporary analytical philosophers. Bernard Linsky and Edward N. Zalta addressed the issue of abstracta in their paper Naturalized Platonism vs. Platonized Naturalism, offering a traditional approach to Platonism that is consistent with naturalism. Their approach, entitled “Platonized Naturalism” posits three parameters to establish their argument: a Principled Platonism, a philosophy of mathematics, and their argument that asserts Principled Platonism is consistent with naturalism. Through the application of these parameters, Eriugena’s ontology and epistemology are given a significant amount of clarity. The purpose here is to place Linsky and Zalta’s Naturalized Platonism in dialogue with Eriugena’s philosophy, providing additional insight into the theologian’s views on the relationship between physical objects and abstracta. This offers a new perspective on the continuity of thought between early medieval Christian Platonists and contemporary analytical philosophers, offering a new perspective on Celtic Christian theology.
To begin, let’s first establish Eriugena’s ontology while providing clarification using Linsky and Zalta’s philosophy of mathematics. In the Periphyseon, Eriugena begins by stating that nature is a “general name for all things, whether or not they have being.” This provides a basic definition from which nature is understood as a genus consisting of two distinct categories: things that possess being and things that do not possess being. The distinction between being and non-being will be important in understanding Eriugena’s epistemology. Having established a definition of nature, Eriugena posits his argument: “The ﬁrst is the division into what creates and is not created; the second into what is created and creates; the third, into what is created and does not create; the fourth, into what neither creates nor is created.” This is not a hierarchical structure, reminiscent of the Platonic schema in late antiquity, but rather what Dermot Moran calls a “set of mental acts of intellectual contemplation.” It is a philosophical model where nature is meant to be understood as a confluence of being and non-being. The first and fourth divisions represent God as the Beginning and End of all things, considered to be part of nature and yet transcending it, while the second and third divisions are the unity of intelligible and sensible objects related to one another through causality.
The second and third divisions are particularly relevant, as they pertain to the existence of abstracta and material objects. Within Eriugena’s ontology and the philosophy of mathematics posited by Linsky and Zalta, mathematical objects are real and essentially distinct from material objects, as they are not subject to physical conditions. They are incapable of possessing physical properties, such as texture, but can still be considered complete objects. This includes the view that numbers are real as individual objects, while individual mathematical theories are understood as abstract objects. This is summed up by Linsky and Zalta, stating that “If the mathematical theories are different, the mathematical objects are different.”
Now that a basic philosophy of mathematics has been established, let’s clarify the ontological properties of abstracta. While he establishes their existence as real, Eriugena does not make it clear on how they exist. Through their Principled Platonism, Linsky and Zalta offer insight through the encoding and exemplification modes of predication. They define encoding under three principles: 1) “for every condition on properties, there is an abstract individual that encodes exactly the properties satisfying the condition,” 2) “if x encodes a property F, it does so necessarily,” and 3) “If x and y are abstract individuals, then then they are identical iff they encode the same properties.” Linsky and Zalta also recognize the implications of the first and third principles, that “there couldn’t be two distinct abstract objects encoding exactly the properties satisfying a given condition if distinct abstract objects have to differ by at least one encoded property.” This provides a concise paradigm for establishing how abstracta can be real through an axiomatic paradigm.
However, these principles do not apply to all abstracta universally as some will “necessarily fail” due to the necessity for an abstract object to encode a set of properties exactly. For this reason, mathematical objects become the focus of Principled Platonism, as mathematics is the discipline of “properties encoded by abstract mathematical objects.” As stated by Linsky and Zalta, “Mathematical objects certainly exemplify properties that are characteristic of their abstract nature, but the fact that they exemplify such properties is extra-mathematical.” Mathematical objects are more robust than most other abstracta, as they encode all and only their structural properties which is more essential to them than the properties they exemplify.
While this addresses the nature of abstracta and how they exist as real objects, we need to address how they interact with the physical world. Eriugena does not offer much regarding how abstracta and physical objects interact in particular, though Linsky and Zalta’s position may offer insight on this issue. Due to the distinction between abstracta and physical objects, we cannot model physical and abstract relations as equivalent but rather treat them as distinct. This is due to the lack of an appearance/reality distinction for mathematical relations and the incomplete nature of mathematical theories. Linsky and Zalta suggest “they are just the way we specify them to be— they are creatures of theory just as much as mathematical objects, and as such, are indeterminate.”
Now that we have established the existence of physical and abstract relations, and their innate distinctions, we need to address how physical and abstract objects interact. While Moran states the physical/abstract object relation is an “isomorphism between intellectual structures and the structures of the real,” it can be observed in Eriugena’s model that he intended to assert that the entirety of nature interacts each to their own capacity. Linsky and Zalta concur by affirming that physical science is successful in the application of nonspatiotemporal things to spatiotemporal objects. They state this is possible due to the “structural relationships between different mathematical objects and features of the world.” As the structures within mathematical objects and physical objects have a high degree of affinity, interactions between both are possible as they are capable of exemplifying relations between one another.
Now that we have established how abstracta are real and how they interact, let’s explore the epistemology of how this can be proven. In the Periphyseon, Eriugena delineates his epistemology as “five means of interpretation” or on things can be said to have being or not have being. The first mode pertains to things “accessible to the senses and the intellect,” or things from which knowledge can be obtained a priori and a posteriori. The second mode pertains to the “orders and differences of created beings” where “an affirmation concerning the lower order is a negation concerning the higher, and so too a negation concerning the lower order is an affirmation concerning the higher.” The third mode pertains to things that have come into being and things that do not yet have being, resting in the “most secret folds of nature.” The fourth mode pertains to things that are “contemplated by intellect alone” and things subject to “generation and corruption.” The fifth mode is more theological, pertaining to those sanctified by grace are said to be and those who reject divinity are said not to be. In summation, Eriugena posits five epistemological methods: a means of obtaining knowledge a priori and a posteriori, a hierarchical categorical method, a method of differentiating things as the effects of causes and things that are virtual, a method for differentiating abstracta and physical objects, and the differentiation between the sanctified and those who renounce.
For our purposes, we will focus on the first and fourth modes as they are relevant to determining how we can know abstracta exist and their identification. Linsky and Zalta develop a similar epistemological model but are careful to address that identification and reference between abstracta and physical objects are not the same. While physical objects are capable of identification and reference through a combination of causal processes, referential intentions, and descriptive properties, abstracta are identified only by their descriptions. This is evident of the comprehension and identity principles in their Principled Platonism, that for “every condition on properties there is a unique abstract object that encodes just the properties satisfying the condition.”
To obtain knowledge of a particular abstract object does not require a causal connection to something else, but rather knowing them one-to-one is accomplished through description. If we understand the comprehension principle well enough, then we can know an abstract object completely. As such, it fits within Eriugena’s first epistemic mode as an a priori and synthetic means of obtaining knowledge that asserts the existence of abstracta encoding certain properties. As they are not identified through the observation of the physical world – they are not subject to confirmation or refutation through empirical evidence – the fourth epistemic mode asserts they are differentiated from physical objects as things “contemplated by intellect alone.” This does not change how we view physical world as Linsky and Zalta state “the objects of the natural world are still mind-independent, objective, and sparse, and the truths about them are discovered a posteriori. Eriugena’s first and fourth modes are concurrent with this position, affirming that both a priori and a posteriori are valid means of obtaining knowledge.
To conclude, we can establish a foundational philosophy of mathematics that posits abstracta are real through Eriugena’s ontological dialectic and epistemology. While his treatment of abstracta is unique within the history of Christian Platonism and Irish Christianity, he lacks a sufficient argument or a rigorous treatment on how exactly they interact and how their existence as real objects can be proven. It is through the application and dialogue of Linsky and Zalta’s Platonized Naturalism that Eriugena’s philosophy is clarified. This provides a robust understanding on the existence of abstracta, and their interactions within the physical world, providing a philosophical position within Celtic Christianity that is concurrent with contemporary analytical philosophy.