On James Bradley's "The Speculative Generalisation of the Function: A Key to Whitehead"

Wassily Kandinsky, Succession, 1935

I do not remember hearing about James Bradley from any Whitehead scholars before around 2020 or so. In those more recent years, he seems to have gotten more attention among those interested in Whitehead, Peirce, the British Idealists, Schelling, and the various associated movements in contemporary philosophy. Even his biography was difficult for me to piece together: as best I can tell, Bradley studied English literature at Cambridge in the late ‘60s, later obtaining his MA in theology and his PhD in the Faculty of Divinity at Cambridge with a thesis on the philosophy of F.H. Bradley. He would later, around 1986, become a professor at Memorial University of Newfoundland, working there until his death from cancer on May 17, 2012. A collection of his essays was published in 2021, contributing no doubt to his recent increase in readership. I may have first heard of Bradley through Matthew David Segall; a couple years ago, he and Timothy Jackson engaged in a dialogue posted on here discussing his essay, “The Speculative Generalisation of the Function: A Key to Whitehead”:

The Generalization of Function Across Scales

Upon first discovering Bradley, I found him intriguing but too difficult to really be able to comment on. Upon returning to this essay in particular, I can now hazard some more substantial reflections on his reading of Whitehead.

Bradley begins this essay by opposing speculative philosophy to analytic philosophy, stating that the former “is centrally concerned with the concept of activity, understood as the activity of actualization which makes things what they are” (253). He identifies Whitehead as part of this tradition of speculative philosophy, but with the distinction that “he fuses together a speculative philosophy of activity and logical analysis” (253-254). Bradley is thus, as is common with Whitehead scholars, approaching Whitehead as part of a broader metaphysical tradition in some ways opposed to more recent developments in 20^th century analytic philosophy—but with the distinction that he is nevertheless attempting to understand Whitehead’s metaphysics in terms of his earlier mathematical logic.

Bradley identifies Whitehead as part of the Fregean tradition that placed functions at the foundations of mathematics, such that all other mathematical entities were definable in terms of functions. The main thesis of his essay, in turn, is that insofar as Whitehead is—qua speculative philosophy—providing a concept of activity (or the “activity of actualization”), this concept is in turn a generalization of a mathematical function.

Bradley focuses on Whitehead’s concept of creativity and what he termed “the category of the ultimate,” this being “the general principle pre­supposed in the three more special categories” of “creativity,” “many,” “one.” Creativity, in Whitehead’s special sense of the word, is the “ultimate principle by which the many, which are the universe disjunctively, become the one actual occasion, which is the uni­verse conjunctively. It lies in the nature of things that the many enter into complex unity” (PR 21). Bradley takes this process, whereby the many entities in a universe “become” the one actual occasion, as referring to the definition of a function as a many-to-one relation. Indeed, Bradley takes Whitehead to here be “defining the concept of the ultimate as the concept of the function in general” (103).

In a mathematical function, every element of the domain is assigned exactly one element from the codomain. This is the sense in which a function is “many to one”—though this does involve cases of injective functions that are one-to-one (that is, at most one element from the domain maps to each element in the codomain).

Whitehead’s concept of creativity, on the other hand, involves what he would term the “concrescence” of an actual entity. This concrescence is the process by which an actual entity comes into being through its feelings of the entities making up its world. The actual entity, in this sense, is a process by which a complete perspective on the world is formed. In a concrescence, the “many” in question are the many entities in the given world that will be felt, as data, by that actual entity. The “one” is the actual entity who will feel all said data. Given this, if a concrescence is considered a function, it is a peculiar kind: it would be a constant function where no matter the input it will always return the same value. That is, as an activity resulting in the creation of the one actual occasion, the concrescence takes each entity in its universe and returns that one actual occasion. It is left unexplained, then, why Bradley takes Whitehead’s “principle of the ultimate” to in some way be identical to the concept of a function in general, or even a generalization of the concept of the function. It appears, to the contrary, to be a highly specific type of function—if it is functional at all.

The essay, from there, explores a notion of function as “mapping activity” in a kind of universal sense—not presupposing any particular functions either extensionally in terms of their elements or intensionally in terms of their mapping rules. This universal mapping activity, in this way, is a transcendental condition (in the Kantian sense) presupposed in all other determinations: in some way, before an element can be differentiated as a singular element, it must be constructed by a function. The sense in which this kind of transcendental concept of a function is applicable to Whitehead’s philosophy, however, remains obscure insofar as the fundamental identification of Whitehead’s “category of the ultimate” with the concept of a function is obscure.

We may be able to clarify the more obscure problems of Bradley’s essay, however, by focusing on one issue where Bradley is more clearly in error. This is Bradley’s misreading of Whitehead as part of the Fregean tradition that takes functions as basic. In both his work with Russell in the Principia Mathematica and his own later work on the foundations of mathematics in his 1934 article “Indication, Classes, Number, Validation” (which reproduces much of Principia’s logical apparatus, minus its type theory), Whitehead does not take the concept of function as basic or ultimate, but rather that of a propositional function or, to put it in more ontological than linguistic terms, a relation.

As Bernard Linsky, in his essay “Logical analysis and logical construction” published in the 2007 book The Analytic Turn, has noted: “The notion of function, although primitive and indeed basic in Frege’s thinking, was viewed as mathematical by Russell [and, thus, not acceptable to be taken as basic in a logicist program], and so the theory of definite descriptions allows the reduction of talk of functions to that of relations or propositional functions with two arguments” (110). In the Principia, functions are definable in terms of functional relations in the following manner. A two-place relation R can be considered functional when for all x, y, and z, (Rxy & Rxz) → y = z. Given an interpretation of a function f in terms of a functional relation R, the term f(x) can then be defined in terms of definite description, as ιy(Rxy). This is thus close to the typical set-theoretic definition of a function, but in Principia’s system the relations are taken as basic, not capable of further analysis (rather than being interpreted in terms of sets of ordered pairs).

We can thus distinguish between the Principia’s logic, consisting of a relational theory of types, and later forms of type theory based on Alonzo Church’s lambda calculus. It is this functional theory that, in turn, has become influential in approaches in computer science. There are many live debates in the foundations of mathematics and computer science involving these concepts. Still, relations and functions are—at least in many logical frameworks—interdefinable, and thus the importance of taking one rather than another as basic is, from the standpoint of logic, arguably only a matter of convention (although, for an argument to the contrary, see Paul Oppenheimer and Edward Zalta’s “Relations Versus Functions at the Foundations of Logic”).

From an ontological standpoint, however, there seems to be a clear consequence to taking relations as ultimate. An ontological correlate to the category of relation is the category of fact. A statement about a relation between individuals is made true by those individuals exemplifying that relation in fact. A relational theory thus corresponds to a factualist ontology. This, at least, was what Russell advocated in the introduction to the Principia. There, the universe is asserted to consist of complex objects, or facts, such as “a-in-the-relation-R-to-b.” Thus the statement “Rab” is made true by there being a corresponding fact “a-in-the-relation-R-to-b”; we, in turn, are capable of judging the truth of statements by perceiving such facts that are their truthmakers. This is the basic starting place of a factualist ontology; of course, it does not remain so simple in Russell’s later work or in the work of other logical atomists.

Just as Whitehead’s later logical work continues to involve a relational theory, his ontology also still bases itself on the category of fact considered as an exemplification of a relation (or, in the limit case of a one-place relation, the exemplification of a quality/property). This is what his category of “prehension” is: prehensions are, as he put it in his table of categories at the start of Process and Reality, “Concrete Facts of Relatedness” (22). I have explored this factualist aspect of Whitehead’s theory of prehensions in some prior writing on my substack:

Whitehead’s theory of creativity consists of an analysis of how prehensions come into being: it is a theory of dynamic facts, or how actuality moves from the mere possibility of something being the case to the actuality of it being the case. An actual occasion’s “concrescence,” or coming into being, consists of the process by which possibilities for how it could feel the world develop into one complex fact of the subject feeling everything in its world. The “many become one” in the sense that the complex fact that is the final product of this process has as its parts various facts of the subject feeling its many data.

This raises the question whether the relations of feeling making up this complex are functional relations. If they are not functional in this way, then Bradley’s main premise—that Whitehead’s “category of the ultimate” in some way is identical to the concept of the mathematical function—is made all the more questionable. In one sense, it is clear that the relations exemplified by prehensions are not functional in this way. The same datum can be felt by many distinct subjects. Indeed, the ability for distinct subjects to feel the same shared entities is fundamental to the manner in which Whitehead is a realist: it is how we can be said to exist in a shared, public world. There is thus, for any datum x, no unique actual entity that qualifies as the y such that x is prehended by y. (To get an intuitive sense of how this differs from a functional relation, consider the successor relation by which for any number n there is the successor that is n+1.) For any datum x, there are many actual entities that could be described as prehending x. If there is some way to rescue Bradley’s thesis and provide an analysis of concrescence in terms of functional relations, it is not obvious to me how to do so; I would be interested to hear any interpretations along these lines, though.

Bradley’s essay is an interesting attempt to connect Whitehead’s logical work to his metaphysics (and to transcendental philosophy), but it does not come across to me as helpful in reading Whitehead himself. Insofar as we wish to read Whitehead’s metaphysics in terms of his earlier work in mathematical logic, we need a closer understanding of this work itself. My impression is that the Principia Mathematica is little read today even among logicians expert enough to understand it; Whitehead’s later revision of the Principia logic in his 1934 essay is read even less. This perhaps explains why Bradley conflated this logic with the kind of functional type theory that has since become more widely discussed. The fact that varieties of relational, rather than functional, theories are still being explored in logic and computer science might indicate the relevance these works could still have, though. At the least, I do believe that so long as these aspects of Whitehead’s thought are poorly understood it is difficult to grasp the full meaning of his work.

Works Cited

Bradley, James. “The Speculative Generalisation of the Function: A Key to Whitehead.” Collected Essays in Speculative Philosophy. Edited by Sean J. McGrath, Edinburgh University Press, 2021.

Linsky, Bernard. “Logical analysis and logical construction.” The Analytic Turn. Edited by Michael Beaney, Routledge, 2007.

Whitehead, Alfred North. Process and Reality. 1929. Edited by David Ray Griffin and Donald W. Sherburne, Free Press, 1978.