Look around. You’ll see a vast array of things that seem to be independent individuals—”Atoms” of the world. One of the “proofs” of their purported independence lies in their “properties.”
“This pencil is blue” means it has the property of “being blue.” Yet, one may argue that colors are not properties of things in themselves; rather, they are “added” by a viewer, a subject. This is why we have historically considered color, smell, and other sensible features of things secondary properties. They reside in the subject, so to speak, in something other than the thing. Primary properties, in contrast, should exist within the things themselves. A typical example in this context has been size. The pencil measures 20 cm. This is an objective property. But what is a measure in general? It is a formal comparison. Measuring means to count. We count how many centimeters “fill” the size of our pencil. Counting is a purely relational action. We associate two sets: in this case, centimeters and the indeterminate size of our pencil.
Centimeters are not in the pencil. The measure is not in the pencil, but in the operation of a function associating the magnitude of an object with a set of numbers. This is not to say that “distance” or “size” are not related to the object at all. They are, but not as something “internal” to the object. To avoid confusion, let’s suppose we omit all numeric references and say that the object at stake, the pencil, is simply something “extended.” We may measure the extension of an object in centimeters or inches, but this is arbitrary. Extension, on the other hand, is not. But what is extension? Even if we avoid all determinate magnitudes, we are dealing with a mathematical structure: real numbers. The “extension” of every object is a subset of the real numbers. Real numbers form a set. If we want to characterize the size or extension of things properly, we need to provide the set with some operations, like addition or multiplication. Numbers can indeed be added together and multiplied. We obtain algebraic structures by defining sets with certain operations that satisfy specific axioms. These structures are useful because they contain operators (+, *) that allow us to combine different numbers. This is how we ground arithmetic.
But this might sound very far-fetched and strange. To clarify, let’s use scientific examples. Let’s look at scientific objects, which are described using mathematical equations but refer to tangible items. These items require instruments for measurement. Science may be mathematically defined, but it still relies on data. Scientific objects are not defined by words but rather by equations. These equations can be translated into language, but their essence lies in their construction. Consider Newton’s famous equation: F = ma. We don’t provide definitions for concepts such as force, mass, or acceleration. Instead, we explore their relationships. Formulas show the variation or covariation of the terms. Variations are formal actions on the terms. If we change the value here, we obtain another value there. These are called functions. A function is, roughly speaking, a rule that takes elements from one set to generate another. This is what we call an action because it effectuates a transformation.
Let’s revisit the issue of properties. One might argue that mass is an intrinsic quality, representing something essential about an object. This perspective can be useful in certain contexts, such as classical predication. For example, the statement “This pencil weighs two grams” suggests that it possesses the property of weight and a specific value of two grams. However, in Newton’s formula, mass only has meaning in relation to force and acceleration. This is what we call weight: the product of mass and the earth’s acceleration. Weight is a relative value, but so is mass. In other words, mass is not something “interior” or “inside” physical objects, but rather something that can affect other values and operate on other objects. In short, a property is the capacity of an object to influence other objects (including itself). Properties connect objects, defining how they can interact, either formally (through morphisms) or physically (through effects).
To say that “S is P” means, classically, that there is a substance or subject, S, and an attribute or property, P. Can we rephrase this expression without resorting to a substantial approach? It is acknowledged that S and P can be treated as sets, associated through a function. For the proposition “cats are animals,” we don’t say that “cats” have the property of being animals. We select the elements of the set “animals” by applying the rule “being a cat.” Functions provide a more flexible way to describe things than predication. Rather than assuming that substances have certain properties residing within them, we can examine the relationships between sets. We can now expand the universe of set-theory language to categories through the concept of morphism. Morphism is the term used in category theory to describe relationships. Objects are not conceptualized as isolated entities with inherent characteristics, but rather as nodes of a graph constructed by a set of relationships with every other object within the same category. To state it intuitively: a thing is defined by what it can do to other things in a given universe. This shifts the focus from properties to the possible actions that things can perform on one another. Things may have an interior in the sense that they may be complex, i.e., constitute a universe in themselves. However, in a given universe, there are no private properties. They are possible interactions. Current physics acknowledges that even mass is not a “property” in the classical sense, but rather the consequence of the interaction of particles with the Higgs boson.
Objects are not conceptualized as isolated entities with inherent characteristics, but rather as nodes of a graph constructed by a set of relationships with every other object within the same category.
Are we asserting that category theory is the ultimate logos of the real? Not at all. Metaphysical claims are always a wager. Many considerations, such as explanatory power, plausibility, and coherence, play a role. The use of category theory to illuminate the classic concept of “property” has a clear goal: to demonstrate that objects are fundamentally relational, distancing themselves not only from substantive but also from atomistic approaches to the real. Objects arise from and point towards other objects. There is no interiority. This does not imply that objects are fully defined by their actual relationships. Properties create a space for possible interactions. Moreover, objects are not fixed entities; they emerge as individuals. In other words, their individuality is a result, not a starting point, as is the case with all atomist approaches.
Actions are effectuations of properties that simultaneously create and relate individuals. This requires transcending the notion of isolated points in space (as objects filling the void). Objects “unveil” or “manifest” or even “actualize” other objects. This is the reciprocal action of things on things. They are ontologically directed at each other. Now, the proposed model may function only if relations are linked to evolution and the phenomenon of emergence. In other words, we should not view morphisms as a set of preexisting relationships between established objects in a particular universe. Instead, they should capture actions, transformations, and innovation. Morphisms are often represented as arrows, highlighting their ability to symbolize change, or “becoming.”
Actions are effectuations of properties that simultaneously create and relate individuals.
Arrows connect interactions and becoming, or to phrase it more poetically, encounters and emergence. This is the moment when we need to leap into the real world to test the capabilities of our formal apparatus. Evolution is a blend of randomness (encounters) and constraints (rules). Pure chance produces the most common possible states. Scatter some dust in the air, and no significant shape or pattern will materialize. We can repeat this a million times. The forms that fill our universe do not emerge from pure chance but from chance within a specific order or system. Genes, for example, do not behave like dice, where all potential outcomes are predetermined, nor can they be characterized as a random scattering of particles. The former represents a self-contained system of combinations, while the latter is indistinguishable from sterile entropy. In reality, arrows should illustrate existence as a balance between unpredictability and limitations.
However, if we truly consider emergence, two fundamental “hard problems” remain to be explained: consciousness and freedom. Arrows should also be able to shed light on such problems.
