The abstract realm

Lastly we come to the third realm, the domain of abstract entities. Do we really need those? The term itself may evoke some ethereal space of eternal Forms. But the abstract is not (at least, not only) the product of a magical feat of mind, the denizen of some remote immaterial eternity, imprecise, the rational, the incorporeal, the ideally perfect, the non-temporal, the primordial or ultimate, the deficient, the potential, the unreal, the symbolic, figurative or merely representative, nor is it the vague, confused and indefinite.

Rather, most elements from our ordinary conversations are prima facie abstract. Imagine you’re talking to a friend about a film: "Did you see All the Vermeers in New York? Emmanuelle Chaulet plays Anna, a French student who loves Proust’s books, and meets a financial broker... but romance does not ensue!". Nothing here seems to refer to our first two realms. "Romance" is a concept. "Anna" is a fictional character. "Financial broker" is a property defining a class of humans. The film’s title itself refers to an artwork, not to a physical token.

Characterization

In what precedes, we’ve tried to characterize the physical and the mental. Can we attempt the same thing here? Three main avenues have been explored:

Negation

: traditionally, abstracta have been defined by what they’re not. They’ve been characterized by aspatiality, atemporality, acausality, or any combination thereof. All options seem very problematic from my perspective. Take aspatiality: if I imagine a unicorn, my mental image is most definitely aspatial, yet it belongs to the mental (the second realm), at least as we’ve described it. Atemporality is frequently invoked but is quite an unclear concept. Take the reference of the phrase "the battle of Waterloo": this set of military maneuvers and various human actions seems to belong to the abstract. Is it situated in time? The battle itself occured on June 18th, 1815, but since I’m writing about it today, the concept itself cannot be temporally confined to that date. I’ve got two options then: either the concept "was born" in 1815 and has persisted since, in which case it is not atemporal, or it truly doesn’t exist in time and so in principle could have been grasped billions of years ago. The latter option is quite counter-intuitive but logically defensible; in that case it would be both necessary and sufficient for a thing to be out of time, in order to be abstract. But since it captures some concepts that are not intuitively atemporal, like famous battles, it’s a poor characterization. Acausality is perhaps even more confounding. For instance, it is a tired cliché that "ideas change the world". Undeniably, concepts such as freedom or nation have influenced the train of thought of countless humans, which in turn has led to myriad of physical acts. Whether that constitutes causality is entirely up to debate and depends on one’s tenets about it. On one end of the spectrum, there are such thing as mental causation and conceptual causation — whereby the existence of concepts can trigger chain of events that leads to certain mental representations; in that case, abstracta are not acausal. On the other end, one may insist that causation is a strictly physical process (a rock hitting another rock), in which case it is self-evident that abstract entities are acausal, since they are not physical... Finally, on a general note, modern views of space, time and causality under the prevalent physical theories of our time make this sort of characterization at least shaky, if not totally inoperative.

Abstraction

: sometimes, abstract objects have been defined constructively, as the output of a well-defined process of abstraction. Most often, abstraction relies on the concept of equivalence classes. If \(R\) is an equivalence relation (that is, reflexive, symmetrical, transitive), then every \(x\) such that \(xRa\) belongs to the equivalence class \(\widetilde{a}\). This seems promising for a wide range of abstracta. "Citizen Kane" as an artpiece is an abstract entity; but the copy shown at the premiere of that movie is a physical token. If we can define a relation \(R\) as "being the same film as", such that a subsequent copy in film form, a disc, a tape, are all "the same film as" that initial token, the corresponding equivalence class seems to be what we mean by "Citizen Kane". Of course this characterization does not go much deeper than an existence statement. The abstracted artpiece does not carry over properties of the physical tokens (weight, materials...) and has properties of its own (duration, title...). Unfortunately, this simple process seems hard to generalise. Thinking of abstract objects like justice, Baudelaire’s style, or Raskolnikov, any attempt to build them from an equivalence relation would have to be extremely contrived or circular. Systematic abstraction thus seems insufficient to characterize the wide universe of abstract entities.

Complement

: the last of the main strategies simply posits that the abstract world encompasses all things that are both non-physical and non-mental. We have already proposed characterizations of the physical and the mental. For our definition to be useful, we would need to have an idea of what "all things" means. Simply put, we’ll define this as: everything one can think of, or talk about. More technically, and this ties up with our section about the mental: everything that refers or that can be an intentional object. Each such object will then have to belong to one of the three realms. I can talk about Emmanuel Macron — most theories of meaning will accept that the intentional object associated with this proper name is the physical being it designates. I can talk about my feelings — those are mental states. I can also talk about the French party system — it does not seem to fulfil the criteria we’ve elaborated for the physical and the mental, therefore it is part of the third realm, an abstract entity.

A note...

: In line with similar comments in previous sections about the physical and the mental, I’m not taking a position on the existence or nature of the entities making up this realm (at least not yet). As we shall soon see, storied philosophical traditions defend the existence of all abstract entities hereby defined, while others deny it. For now though, we’re merely aiming to describe the relevant entities as they appear, regardless of their ultimate nature. At the moment, let’s be abstracta taxonomists, rather than ontologists.

Common universals & particulars

Properties

(or attributes, features, qualities...) are the most common abstract entities in philosophical discourse. They’re borne out of the simple realization that our world is not an undifferentiated bog. Things are a certain way, and importantly, we can notice repetition and regularity in those ways. Properties are these ways things are, predicates attributed to objects. Conversely, we can say that things exemplify, or instantiate, properties. By saying "this leaf is green", I’m attributing the property "greenness" to the leaf, and the leaf is an instance of greenness. Properties are often thought of as abstract universals, as they can be instantiated by distinct objects that are remote in space, time or kind. The plant in front of me, the giant Bahia emerald, and the mythical oak of Dodona are all green /textemdash  they instantiate the same property of greenness. Properties roughly match what we expect from Platonic forms: "Beauty" and "Justice" can be simply identified with the properties "being beautiful" and "being just".

Tropes

are the particular version of universal properties. The exact shade of green of a particular leaf is a trope, as are its shape and weight. It may be that a different leaf has the same shade of green, as far as my eyes can see, but that green-trope is in any case a distinct entity from the first one. Sometimes, tropes are used by theorists who are wary of accepting universals, but don’t wish to deny all abstract entities. Tropes can also co-exist with properties, most famously in Aristotle’s view. More recently, E.J. Lowe adopted that approach, whereby tropes exemplify properties in the same way as individual objects exemplify kinds.

Kinds

represent categories in which we sort objects. Most common nouns in any language designate kinds: woman, zebra, apple, chair... Philosophical attention has long been focused on natural kinds, which typically feature in scientific theories and are supposed to reflect some relevant aspects of the physical universe. They contrast with artifact kinds (categories of man-made objects or concepts, e.g. "breakfast food") and with random, accidental groupings (e.g. "item for sales!"). In one view, once prominent, kinds are universals. Indeed, it seems that kinds are predicable: "Toby is a dog" does not seem that far away from "Toby is white". A simplistic account would say that the kind "dog" is just the extension of the property "being a dog", which itself is a bundle of simpler properties, like "being an animal, who walks on four legs and barks". That is a decent account for elementary physics and chemistry (whose bricks are defined through a finite number of clear, intrinsic properties), but for biology or other sciences, a "cluster" or "family resemblance" approach is deemed more appropriate. Lastly, even if one accepts that kinds may be universals, it is not automatic that they are equivalent to an assemblage of properties. Aristotle, for one, clearly distinguished between the two, describing kinds as substantial universals while properties are accidental. Simply put: there may not be green or red swans in nature, but if one showed up, it would still be a swan. Similary, if my white dog Toby suddenly turned brown, or if he unfortunately lost a leg, it would still be my dog Toby. If he became a cat, it would be hard to argue he’s the same being.

Types

are another form of category. It has been defined in contrast with tokens and occurrences. A typical example is as follows: how many letters are there in "taramasalata"? I might argue that the word referring to that delicious dip has 12 letters. I could also say it has 6: "t, a, r, m, s, and l". To dispel the ambiguity, we should say that the word is made of 6 letter-types, but has a total 12 letter-occurrences. When I typed the word, 12 physical letter-tokens appeared on the screen. Another oft-used example is: "Great Expectations" as the book-type; my personal copy of it as a book-token. Conceptually, there may not be much difference with kinds. In practice, distinctions are often made, mostly on metaphysical grounds, either when the type of case outlined above is not a good fit for the prevalent theory of universals, or because universals’ existence are simply denied.

Sets

may belong to the broader category of mathematical objects, but are worth mentioning in the same breath as kinds and types as set theory is often involved in discussions thereof. Kinds and types are usually distinguished from random, amorphous collections, but those can always be associated with sets.

Relations

are ways in which things stand to each other. For instance: "being taller than", "being the father of", "being the same color as" are relations. More generally we could call those dyadic relations, since they involve two things, and expand our definition to n-adic relations between an arbitrary number of things. Taking a page from mathematics, it is arguable that properties are just monadic relations. Different positions have been developed. For some, all relations are ultimately properties, since a dyadic relation can be seen as the property of a pair: we can say indifferently "Bill is the father of Chelsea" or "{Bill, Chelsea} are a father/child pair". Conversely, some have argued that properties are relations. For instance, "Toby is white" could be construed as a relation between Toby and whiteness.

Mathematical objects

, as a class, encompass a wide array of particulars that are commonly considered as abstract entities. We won’t delve too much into the nature of these objects, which is the focus of mathematical philosophy. Suffice it to say they’ve been considered from very early on as archetypes of what abstraction could mean. Numbers and geometrical figures, in particular, have been at the forefront, and fittingly played an important role in, respectively, Pythagoreanism and Platonism.