Is Category Theory Important for Philosophy?

The following snippet is about Category Theory. If there is anything likely to scare the troops, it’s Category Theory. A long while back I met a well-known Australian philosopher of the analytic tradition. He was into mereology and made his long career in this kind of area. Not many people have heard of mereology. You might have to try Google or look at my last post here. I don’t want to belittle mereology because after all Whitehead was quite involved with it, and he was no lightweight. Pity he spent so much time with Russell. No one is perfect. I asked the professor what he thought about Category Theory. He replied that he didn’t know anything about it except that it was hard. I was flabbergasted. I still am.

In the snippet, which follows, I describe a very first glimpse of Category Theory and its philosophical implications. Category Theory, whether you like it or not, sits at the top of formal axiomatic mathematics. In other words, it sits at the top of abstract science, as we customarily know it. It articulates the shape of abstract (Platonic) ideas in the form of (so-called) universal constructions (limits, colimits, pushouts, pullbacks, and equalisers for example). Who cares? Good question. These universal constructions are not actually universal but more in the line of being very general. There is an important difference between the universal and the general. The general applies to everything but not everywhere. The universal applies everywhere but not to everything. People confuse the two. Traditional sciences home in on the general. These sciences, including axiomatic mathematics, are based on generalisation, on abstraction. I call them left side sciences. The true universal is not abstract but generic. I call this right side science. Developing right side, truly universal science, is my project. Lucky for me, I am not the first. The Stoics, and in particular Chrysippus, appear to be the first, at least in the Western World.

It is a long story, much simpler than you think. If you have never really understood Category Theory, this thing that is important but “hard,” the following snippet might get us into the foothills. It might be worth the effort. Pleas keep in mind that our approach is not at all orthodox and might upset mathematicians.

In the final thrust of things, the difference between the general (Platonic) constructs of traditional left side science and the true (generic) universals of right side science is that the latter exist, and the former do not. Abstractions are useful but they don’t exist, as such.

The idea is to look at the “arrow theoretic” approach of Category Theory as a way of explaining things without words. You use arrows and diagrams instead. This works very well for deep abstract mathematical kinds of concepts. That is what Category Theory is about. For the abstract mathematician, this is as good as it gets. But is there another non-abstract way of explaining things with arrows? A generic way, for example.


That is enough of the hyperbole; let us look now at the left side version of the arrow theoretic method. We start with a practical example from Category Theory. Figure B 4 shows the categorical representation of the categorical sum and the categorical product of objects in a Category. We won’t attempt to understand the diagrams for the moment. We simply note that the diagrams look quite neat and clean, quite beautiful in fact.

The notion of the product and sum is fundamental to mathematics and crops up in many guises. Already, the Categorical diagrams reveal something that can be difficult to describe otherwise: In general, product operations are a kind of dual of sum operations. This can be seen simply by reversing the direction of the arrows for the sum diagram to produce the dual diagram. This corresponds to the categorical product.

Diagram of categorical sum
Figure B 4 The arrow diagrams for the categorical sum and the categorical product of two objects A and B in a category.

In order to get an understanding of the arrow theoretic methodology involved we will assume that the objects of the category are sets and that the sum of two sets will be set theoretic union and the product will be set intersection. Now consider the set theory problem illustrated in Figure B 5(a). The problem is to work out the set theoretic union of two sets without using labels. The answer is that it can’t be done. However, in Figure B 5(b), we have invoked the Axiom of Choice, which says that any set can be labelled. The elements of each of the two sets have thus been dutifully labelled and the corresponding result of set theoretic union is shown on the right, problem solved. Of course, if we had chosen a different labelling, we would get a different result. The conclusion here is that the ability to label is very important in Set Theory. That is why Set Theory cannot do without the Axiom of Choice.

Figure B 5 (a) Set theory sum of two sets without labels. (b) With labels.

Let us now start moving towards a categorical approach to the problem and see how Category Theory can get by without any explicit labelling technology whatsoever. The labelling used in Figure B 5(b) was quite ad hoc. We now interpret this ad hoc labelling as a very nuts and bolts model of the Category Theory representation of categorical sum. Using this more elaborate labelling technology, the ad hoc labelling is replaced by a system of labelling as shown in Figure B 6. In this diagram, the set C consists of a large enough set of labels to do the job. The sets A, B and their sum are each labelled by bunches of connections between each of the sets and the labels in C. The three bunches of lightly drawn arrows are instances of the morphism arrows f, g, and h of the highly abstract categorical representation of sum shown in Figure B 4. The Category Theory represents all the possible indexing instances possible for determining the abstract mathematical notion of sum and does so at the most abstract level.

We will now look at the specificities of this kind of this left side arrow theoretic methodology. Our discussion is motivated by the quest for a sibling right side arrow theoretic methodology. Our aim is to resolve the Kantian problem and so the discussion has a philosophical and even a psychological tone. One of our objectives is to cure mathematics of the hemi-neglect syndrome mentioned in Part 1. Against blatant neglect, we must show that there is another way, another side to it all.

Categorical sum diagrams
Figure B 6

Figure B6 The Category Theory diagram can be thought of as an abstraction of a labelling system. In Category Theory all this detail is superfluous and no explicit labelling is necessary..

Traditional Arrow Theoretic Methodology

Our immediate task is to characterise the left side arrow theoretic methodology of Category Theory. We will look at how Category Theory respects First Classness (FC), how and where it violates FC, and its overall architectural characteristics.

The first startling realisation is that Category Theory, right from the start, formally articulates the core requirement of FC notably that no entity be anterior to any other. What this means is that in any Category Theory diagram, no arrow can be said to be before or after any other arrow. There can be no concatenation of arrows. In other words, Category Theory starts off without any appeal to the composition of arrows. However, there are two caveats. Firstly, we must add the caveat that this FC only applies to the non-dotted arrows. The dotted arrow is added ‘to make the diagram commutes.’ The orientation and location of the dotted arrow is such that it always violates FC as it appeals to the fundamental structural mechanism of Category Theory, associativity and the composition of arrows.

There is a second caveat that we add in small print. This is to do with identity arrows, which clearly violated FC. In practice, this is the case anyway as identity arrows are usually only explicitly incorporated in the diagram when pertinent and that is usually after adding the dotted arrow, not before. We will sweep this aspect under the carpet by saying that, except for the identity arrows, the rest of the arrows in an arrows diagram, excluding the dotted arrow, all respect FC.

At this point, we will introduce some terminology for describing the two kinds of arrow in an arrow diagram. We will refer to the dotted arrow as being real and all the other arrows as being imaginary. Thus, every arrow diagram has an imaginary part and a real part. The real part consists of one single arrow, the specificity of which is represented by the geometric configuration of the arrows making up the imaginary part of the diagram. The imaginary arrows, relative to each other, respect FC. They all determine the specificity of the real arrow.

Arrow diagram of Category Theory is a technology for representing the specificity of a “real” entity, the entity represented by the dotted arrow added to the diagram in order to make it commute. All of the specificity is encoded in the topological configuration of the imaginary arrows relative to each other and to the real. There is no need for labels. In practice, mathematicians label the various arrows making up the imaginary part and real parts of the diagram, but this is only to make it easier to talk about the structure. The labels themselves impart no additional structural specificity to the diagram.

This is the way of pure Category Theory. Workers in Computer Science muddy the waters by considering the imaginary arrows as signifying real things such as datatypes. Thus they treat the imaginary arrows as types, giving arrows labels with meaning, a datatype meaning. Even worse, they use the same label for different arrows in the same diagram, declaring that this construct represents a “polymorphic” data type. Such a practice violates FC as it destroys the monism of the real entity, sprinkling real typing of arrows willy nilly. Instead of the imaginary structure based on FC articulating the abstract essence of one single real entity, the imaginary structure, that pure attribute of the real, gets polluted by real world side effects. This doesn’t mean that such an approach is without value. It merely means that the approach properly belongs to a Computational Category Theory, but not to Category Theory pure. We are concerned only with the latter. Curiously though, the polymorphic types that the Computer Scientists want to graft onto Category Theory, start to take on a different allure when we go over the right side alternative to classical Category Theory. However, the types don’t arise from the everyday concerns of Computer Science, but from the very demands of FC itself. In other words, the polymorphic types must be based on ontological gender.

Representation of Generic Structure

I categorise the traditional sciences, including mathematics, as left side science. The science without any a priori conditions, I call right side science. Lefts side science deal with knowledge of objects. Right side science deal with the science where the subject is always present and is an integral part of the action. To resolve the Kantian problem, we need a right side science. This is my project. Instead of mathematics, we need anti-mathematics.
The technical core of the paper will not be presented in the blog.

Formalising Choice
The process of formalising knowledge in the left side science is a relatively straightforward affair. The basic technology is already in place: It is called mathematics. Mathematics provides the tool for formalising the traditional left side science knowledge. When it comes to formalising right side science, one immediately comes up against a brick wall. None of the mathematics works. The obstacle is the draconian constraint of FC. FC must not be violated. The problem is that traditional axiomatic mathematics violates FC right down to its very core.

However, all is not lost. Because axiomatic mathematics is a formal system, it can be exploited to formalise the obstacle to formalising an FC compliant system. Mathematic formalises the way the problem must not be tackled. Axiomatic mathematics formalises the wrong way to go, that is to say, the wrong way to tackle the Kantian problem. Having a formal statement of the obstacle to progress, all we have to do is to find the way around the obstacle. If we cannot do it with mathematics as it stands, we will need something else.

Looking down at the very foundation of mathematics, we come to Set Theory, the elementary mathematic of collections. Without a formal notion of collections of things, there can be no formal mathematics. There are many axiomatic systems that claim to formalise Set Theory. Each system has a different set of axioms, but all systems contain one pivotal axiom, the Axiom of Choice. Faced with a Set of elements, which may even be infinitely denumerable, how can you distinguish one element from the other? How do you choose? The Axiom of Choice imposes sufficient structure on the system to solve the problem. Equivalent to the Axiom of Choice is Zorn’s Lemma, which is easier to understand. The lemma effectively states that the elements of any set can be uniquely labelled with real numbers. Thus, using real numbers as labels, there always exists a unique labelling of elements such that one element can be distinguished from the other.

The very reliance on an axiom, any axiom, violates FC, as no such a priori constructs are permissible in a First Class system. What is of interest with the Axiom of Choice is that it situates the way that mathematics resolves the distinguishing problem. Firstly, it has to resort to a construct at the axiom level. Secondly is equivalent to using an ad hoc labelling technology, a characteristic of all left side sciences. The Axiom of Choice, and its fundamental lemma, thus articulates quite clearly, the way not to proceed: Don’t use labels.


Structure is in the mind of the beholder. For the left side sciences the beholder is the impersonal subject providing the much sought after ‘mind independent’ point of view. This primary opposition between the impersonal subject and its object is ignored by left side science and replaced with an opposition of its own making, that of the rigid dichotomy between abstract theory and its object. In left side mathematics, the primary dichotomy becomes that between a set of axioms and a world of deductively explorable mathematical objects so predetermined, either explicitly or implicitly.

For right side science, the mind of the beholder is of primordial importance and is always present. Not only is the impersonal subject present, but also the personal. There are many ways of interpreting these two kinds of subject. As mentioned previously, the subject as placeholder and the subject as value is one possibility. A more mathematical flavour might be to call them the “covariant” and “contravariant” subjects, but one must be on guard not to slip into abstraction ways of thought. Both these two kinds of subject are simultaneously present in any whole considered by right side science, The science of wholes is the speciality of the right side of the epistemological brain. What matters is the generic subject formed by a highly primitive, primordial Clifford-Grassmann style “geometric product” of these two subjects (together with their respective worlds). The end result is a the generic subject in the form of a “quaternion” kind of Three-Plus-One structure, a semiotic square which can be more formally understood in terms of the ontological gender typing construct. In the right side science paradigm, this artifice occupies centre stage at all times. One could even say that it is centre stage.

One way of understanding the generic subject is to realise that it suffers from an incurable disease. The disease is called monism. Patients suffering from monism exhibit the pathological symptoms of being totally incapable of distinguishing the difference between the real world and their conception of it. Both appear to be the one and the same thing. Curiously, most human subjects, at least when not on hallucinogenic drugs or suffering from a deep schizophrenic episode, also seem to exhibit these symptoms .

Right side science not only must articulate the basic architecture of the generic subject but also of the generic objects. There are four types of generic object, four bases distinguished one from the other by binary gender typing. The typing of bases is determined relative to each other and ultimately compatible with the polarity conventions established by the subject, the ultimate arbitrator of type. These four bases can be represented by four binary gendered typed arrows. The problem now is to establish how these arrows can be combined to form elementary structures, without violating FC.

From a left side science perspective, if a right side science were at all possible it would present as some kind of meta science, metaphysics or meta mathematics equipped with its own metalanguage. Such a science is not possible under the ambit of left side paradigm dominated, as it must be, by its atomistic and dualistic worldview. However, even though fundamentally incompatible with FC, some accommodations can be made to achieve a kind of Partial First Classness (PFC). The resulting science will not be a true metaphysics but at least pass as a poor man’s cousin.

The Sad Story of Mereology

One such accommodation is the rather obscure quasi-mathematical discipline called mereology, a left side attempt at a science of wholes and parts. Mereology is an exercise in mathematical logic. It achieves PFC by removing the rigid set theoretic dichotomy between sets and the elements that they contain. This is achieved by ignoring any explicit reference to the elements of a set and only considering containment relations between sets. Sets do not contain elements, they contain other sets. Contained sets are parts of the containing set. Different axiomatic schemes are set up to formalise this kind of structure where wholes contain parts and PFC is achieved by both parts and wholes being sets.

Mereology is of interest because it is essentially an attempt to formalise has-a relations between entities. Such structure finds echoes in the class inheritance structure of Object Oriented programming systems, for example. There are also echoes with our initial development of right side science where the has-a relation is paramount. However, right side science grants comparable prominence to is-a relations. In fact, the basic building block involved the gender construct where the feminine ontological gender corresponds to the has-a relation and the corresponding masculine gender to the is-a relation. The core of right side science, with its ontological vocation, consists of the dialectic of the has-a and is-a relationship. In mereology the has-a relation is axiomatised in terms of some kind of partially ordered structure such as set inclusion. As for any ontological is-a structure, that is hard wired into the axioms. Being a left side science mereology does not entertain any kind of is-a has-a dialectic.

A. N. Whitehead, in his philosophical quest for a holistic rationalist science, extended mereology concepts to geometry and achieved a geometric PFC (Whitehead, 1919). In this case, the rigid dichotomy between geometric objects with extension and geometric objects with no extension (points) was avoided to produce a pointless geometry. A pointless geometry is a right side kind of geometry. However, the geometry was caste in a left side, abstract, dualistic, atomist framework. In the final count, the system inevitably violates FC on practically every other front. Nevertheless, mereology is worth mentioning here as it expresses many of the aspirations of right side science even though it fundamentally lacks the necessary equipment to deliver the goods. In this respect, the mereology-based paper “Steps Toward a Constructive Nominalism” (Goodman, et al., 1947) is notable. In espousing constructionism and nominalism, the paper articulates important hallmarks of right side science. In addition, the authors start the paper with the doctrinal declaration: “We do not believe in abstract entities. No one supposes that abstract entities—classes, relations, properties, etc.— exist in space-time; but we mean more than this. We renounce them altogether.” This rejection of abstraction is yet another fundamental tenant of right side science. However, declared within the confines of left side abstract axiomatic technology this anti-abstract belief becomes a bit of an oxymoron. It is like the Christmas turkey that struts into the kitchen valiantly declaring that it does not believe that turkeys are food.

From our perspective, mereology is interesting more for its aspirations than its achievements. What we want is a left side discipline that can properly formalise the very essence of mathematics and that comes from within mathematics itself. What we need is an abstract theory of abstract mathematics, and that naturally leads to Category Theory, the meta-mathematics of mathematics. It is with Category Theory that we can find a formal specification of the kind of structure that is anathema to our right side science. We will use Category Theory as a formal negative indication of what we are up against in trying to resolve the Kantian problem.

Category Theory Structure Violates FC

Category Theory provides abstract representations of mathematical structure in terms of a collection of objects and a collection arrows or morphisms between the objects. The specificity of mathematical structure is represented by the arrows and in no way by any explicit internal structure of the objects. The approach is thus structuralist in nature. Representation of the most elementary mathematical structure starts with placing two arrows end to end. This represents the composition of two arrows. Composition of arrows must satisfy two axioms, identity and associativity relying on the structures illustrated in Figure B 3 . Both of these structures violate FC.


Figure B 3 Even the most elementary structure necessary for a mathematical category violates FC..

Figure B 3(b) represents the composition of two arrows f and g to determine a third arrow h thus satisfying associativity. This violate FC because the arrow g is in an absolute ordering relationship with the arrow f. In a system satisfying FC, no entity can be absolutely before or after any other. Thus, even the two arrows shown in Figure B 3(c) is an FC violation. Thus not only is associativity prohibited but any kind of composition. We could call this disallowing of any absolute ordering relationships, the Parmenidean condition. For FC, the only thing that is must be immediate, not anterior, nor posterior.

A mathematical category requires the notion of composition identity defined for each object. This requires arrows that close back on themselves to form a loop as shown in Figure B 3(a). This structure also violates FC as it infers than the same entity can be different to itself. We will call it the Heraclitus principle expressed by the saying that “You can’t put your foot into the same river twice”. It is a special case of the Parmenidean condition. This prohibition is a subtle one but suffice to say that it can be represented by a prohibition on circular arrows.

Without getting into messy details, it suffices to say that the formal axiomatic mathematical Category abstractly states the minimal structural characteristics that a system must possess in order to qualify as mathematics. What interests us is not mathematics, but its opposite, anti-mathematics. We informally define anti-mathematics, as being everything that mathematics is not. At the abstract pinnacle of mathematics, we find Category Theory. The anti-mathematical counterpart will be the Anti-Category. The only thing in common between the Category and the Anti-Category will be that they both exploit an arrow theoretic one way or another.

The Anti-Category and the Kantian Conditions

The conditions on the Anti-Category can be summed up as:

  • Unlike the Category, the Anti-Category cannot be abstract. This can be achieved by prohibiting dualistic structures, the essence of abstraction.
  • Thus, unlike the Category, the Anti-Category cannot tolerate a duality between a collection of objects and a collection of arrows. For the Anti-Category not to violate FC, the mantra is that all entities are arrows. In this way, any entity will possess extent. From an ontological point of view, we reiterate the Stoic mantra that only bodies exists. Point like entities do not exist.
  • Unlike the Category, there can be no identity, no associativity and not even composition of arrows.
  • There can be no axioms as any such predetermining structure violates FC.

We will call these conditions, the Kantian conditions for determining a formal structure that is totally devoid of any predetermining considerations. Realise an apparatus that satisfies the Kantian conditions and one has resolved the Kantian problem. In other words, one would have provided a formal basis for right sides science, the monistic counterpart of the dualistic left side sciences. Bot easy, but it can be done.
The axiomatic formalisation of mathematical categories is quite precise. Taking these conditions in the negative provides draconian requirements on the right side counterpart to the Category, the Anti-Category. Briefly, arrows determining anti-categories cannot form loops or be concatenated end to end. This leaves plenty of slack for finding a solution to the riddle. At least the Kantian problem is starting to look tractable.

Arrow Theoretic Methodology

Category Theory is based on an arrow theoretic methodology. It expresses its fundamentals in terms of arrow diagrams. Our task is to develop the right side counterpart of the Category in terms of the Anti-Category. If we can achieve this objective then we will have made a breakthrough is resolving the Kantian problem, the fundamental thrust of this paper. Thus, we claim that, in addition to the known left side arrow theoretic methodology of Category Theory, there must be a complementary right side arrow theoretic methodology. Our task is to bring this right side version of arrow theoretic methodology into the light of day. In the process, we will see that the traditional left side version specialises uniquely in the syntaxical aspects of structure and is virtually devoid of fundamental semantic considerations. On the right side of the equation, we will demonstrate that right side arrow theoretic structures are virtually syntax free, concentrating uniquely on semantics. One could say that traditional left side abstract approach to semantics leads to a syntax only account: Abstract semantics distils down to syntaxical expression. On the other hand, rhea right side paradigm approach to semantics leads to generic, non-abstract semantics. This kind of semantics is ultimately expressed in the gender calculus in the form of a syntax free generic code, a code capable of coding any semantics whatsoever that is compatible with FC.