The problem of universals
Having reviewed the different positions relative to the mind-body problem, let’s now move on to the case of realm \(\mathbf{3}\), the abstract world. Although a philosophical worldview is a holistic affair, I would still argue that the one-to-one metaphysical relations between \(\mathbf{1}\), \(\mathbf{2}\) and \(\mathbf{3}\) are fairly orthogonal. That fact is obscured by the many layers of historical terminology — yet an "objective idealist" like Hegel who believes in an absolute Idea and a "subjective idealist" like Berkeley who believes experiences to be the only ontological primitive: the former is a statement that \(\mathbf{1} \prec \mathbf{3}\), the latter that \(\mathbf{1} \prec \mathbf{2}\). I’ve chosen to first examine the \(\mathbf{1}-\mathbf{2}\) dyad in isolation, and to now consider its relation to \(\mathbf{3}\). To simplify notations and when appropriate, I will use the shorthand \(\mathbf{\aleph}\) in lieu of \(\mathbf{1}-\mathbf{2}\) (which could in itself be a monism, a dualism, etc.).
A previous section of this book has looked in details at various classes of abstract objects — it’s important to note the positions outlined here are very general, and could potentially be differentiated according to the type of objects. For instance, Quine at some point of his career believed that sets exist, but no other abstract objects. As to the title of this section, "the problem of universals", it is a reference to the medieval quarrel that set the terms of the modern debate, and from the tenth century opposed "radical realists" like Guillaume de Champeaux to nominalists like Roscelin de Compiègne. The reference should not be taken at face value though: the various positions described herein pertain to abstract objects in general, not just universals.
\(\{\mathbf{\aleph}\}\) : nominalism
Nominalism refers to a cluster of positions that deny the existence of abstract objects (this historical term is indeed synonymous with that of anti-realism about the abstract).
This may seem hard to swallow at first. "Color" is an abstract idea, yet how could one deny that color exists? — it is there, everywhere I look! Nominalists do not deny that there are colored things; they also accept abstract terms such as "color". But they deny that color itself is a thing in the world that can be wholly present within physical or mental objects.
Nominalist strategies
Two types of strategies are used. First, we can start from the medieval insight that our discourse about apparent abstract entities can be analyzed in terms of familiar concrete particulars. "Red is my favorite color" is just another way of saying "I prefer red objects". There is no Red or "redness" that is instantiated in red objects, those objects just happen to fall under the predicate "being red". This stance, called predicate nominalism, is an example of the more general attitude of relational nominalism, whose goal is to explain abstract entities through relations between particulars: "being red" would then merely reflect the relation between an object and the light that is reflected from it.
Another strategy, variously named deflationary or austere nominalism does not even seek to explain the attribution of properties to individuals; instead it implicitly accepts the intrinsic and primitive nature of the attribution: the apple is red because it is red.
These are uneasy positions to hold, for sure, but they are motivated by some powerful arguments. Ockham’s razor is such a famous image it is now an everyday figure of speech: accepting abstract entities quickly leads to an over-populated ontology, and potentially to some vicious infinite regresses (relations between properties an objects, relations between those relations, etc.). Ockham’s razor is not sufficient to deny those entities, but it compels us to at least look for alternatives, and if we find acceptable ones, to adopt them in the name of ontological sobriety.
Another argument is reminescent of the problem of mental interaction: if abstract objects exist then how do our minds know about them? Conversely: under which circumstances can our thoughts or actions "create" abstracta? This puzzle does not prove nominalism, but it does point to an important gap in realist positions.
One last argument that has often been used invokes the unintelligibility of abstract entites: in other words, the difficulty to define precise identity conditions for them (what does "being tall" mean)? The rise of more formalized metaphysics, from Leibniz to Zalta, may have alleviated those concerns in large part.
Nominalism in mathematics
Now, the traditional nominalist stances are by design geared towards universals (properties, relations...) and often extended to objects like propositions (arguably, Russellian definite descriptions help nominalize propositions as complexes of concrete particulars and properties/relations, which themselves can be supposedly nominalized in the aforementioned ways).
But a much thornier issue is that of mathematical objects, for two main reasons. First, the nominalization process whereby universals are analyzed typically make "backdoor" use of basic mathematical objects, like sets or classes. A predicate nominalist may well argue that talking about "red" is a way of talking about "all objects that happen to be red", but that implicit use of a set and a universal quantifier should be problematic to a full-blooded nominalist. Second, our "best scientific theories" are expressed in mathematical notation. Modern developments make it look more and more like mathematics are indispensable to a scientific description of the world: you could conceivably reduce "the color blue" or "softness" to some paraphrase that cleverly avoids any abstract concept, but it’s less obvious how to do that with Einstein’s field equations.
Consequent nominalists about mathematics would have to do just that though. The most famous approach developed in the twentieth century is mathematic fictionalism. That doctrine accepts the main consequence of mathematical nominalism: that all mathematical statements are false. Are we really supposed to believe that sentences like "\(1+1=2\)" or "\(3\) is prime" are false? Strictly speaking, yes. However, fictionalists introduce a modal operator, such that according to arithmetics, \(1+1=2\) and \(3\) is prime. That’s where the "fictionalist" moniker comes from: this type of modal statement is similar to something like "according to Conan Doyle’s stories, Sherlock Holmes lives on Baker st.".
From there on, fictionalists have to accomplish two things. The first one is to re-frame the aim of mathematics, which is not to find the truth about objects that don’t exist, but to be a "useful fiction" — one criteria for that usefulness is conservativeness, the fact that mathematics ought to be consistent with our theory of the physical world. The second one is precisely to provide a fully nominalized physical theory, making no use of numbers nor functions. Thus, every physical fact from mathematical physics should be provable in the nominalized theory; the "fictional" world of mathematics, thanks to the conservativeness criteria, allows to shorten the derivation of these facts, but does so without any need to ontologically commit to the objects it sets into motion. This project presents attractive feature from a philosophical perspective. However, it must be said that the nominalization project, started by Hartry Field for Newtonian physics by way of Hilbert’s geometrization, has not picked up much steam — at least not enough to suggest similar methods could be extended to the "best theories".
\(\{\mathbf{1}\mathbf{\succ}\mathbf{3}\}\) : immanent realism
Advocates of this position, unlike nominalists, don’t deny the existence of abstract objects. However, they agree that abstracta do not form an independent, fundamental realm of the world. They believe those entities to be grounded in the physical world.
This can be traced back to Aristotle. Whilst Plato saw universals as aspatial, atemporal entities that happened to be exemplified by particulars, Aristotle thought that universals existed in particulars. In terms of our modern inter-level metaphysics, we could argue that their existence is grounded in that of concrete particulars.
The idea seems quite plausible, initially. If I’m looking at a blue ball, and I think that blueness exists, then it’s perhaps natural to say that blueness is spatiotemporally co-located with that ball. The same kind of rationale has been extended to an Aristotelian view of mathematics: the number 3 could be seen as immanent within every pile of 3 objects. Bringing a pile of 3 and a pile of 2 objects together yields a pile of 5 objects, so arithmetic statements like \(3+2=5\) can be interpreted in physical terms, turning mathematics into a sort of generalized natural science.
Unfortunately, this doctrine is riddled with problems that make it a very unstable position between full-blown nominalism and Platonism. If the universe is finite, it’s hard to account for infinities. (Some, like Penelope Maddy, have dealt with that issue by highlighting that a single physical object could ground infinities in itself: the number of its atoms, the number of sets of atoms, the number of pairs of sets of atoms etc. would provide a never-ending increasing series. But it’s arguable that this is taking us away from Aristotle’s original view and sliding ever so slightly into Platonism). Even more problematic, if properties are grounded in physical objects, then non-instantiated properties don’t exist. There is no fire-breathing animal, so any proposition about a dragon is false — this would ultimately force us, to preserve the non-realist stance about properties, to delve into nominalist solutions to this issue.
\(\{\mathbf{2}\mathbf{\succ}\mathbf{3}\}\) : psychologism
Psychologism, sometimes called conceptualism or mentalism endorses the idea that abstract entities are mind-dependent, and therefore grounded in mental states. Simply put, universals, mathematical objects, propositions and other abstracta would then just be "ideas in our head".
John Locke and J.S. Mill are often considered to be precursors of this strand of thought, which became especially popular in the nineteenth century. It stems quite naturally from a simple observation. Human minds perceive the world and through abstraction, create discourse about universals; human minds also manipulate mathematical objects and prove theorems. Therefore, it seems reasonable to assert that all abstract entities are nothing but mental objects.
Unfortunately, this stance suffers from most of the same issues raised by immanent realism: the question of infinities and never-thought-out universals remain. Besides, despite its superficially "common-sense" aspect, it looks bizarre upon closer examination. Given the subjective nature of mental processes (historical proponents of psychologism never mentioned panpsychism), the process whereby abstract entities are created is questionable. Evolution is a continuous process, so when are basic abstracta like numbers supposed to have come into being? Does an original thought that is never shared with others exist in the same way as the number 6? Would it pop in and out of existence when it is conceived and then forgotten? Should we infer that numbers will cease to exist when life disappears? All of this makes mentalist abstract entities look pretty flimsy, with vague identity and existence conditions.
Having examined these problems, we find ourselves backsliding into nominalism once again. Psychologism wants us to think that the number 6 is identical to "6-ideas" in our minds, or at least that it is grounded into them. However, 6-ideas didn’t exist before minds appeared, and won’t exist once the last one is extinguished. Since the notion that the number 6 is temporally located is hard to accept, it might be easier to accept that 6-ideas undeniably exist, but refuse that the number 6 itself exists.
\(\{\mathbf{\aleph}\mathbf{\parallel}\mathbf{3}\}\) : platonism
Platonism as a term has changed meanings quite a bit these past two millenia. By today’s standards, Plato may not be a platonist... The most common understanding nowadays is that platonism advocates the existence and independence of abstract objects. To a platonist, there are entities in existence that are neither physical nor mental, and whose existence is not grounded in physical or mental facts.
Existence:
To prove the existence of abstract entities seems quite daunting. Descartes’s "cogito ergo sum", Moore’s proof of an external world ("here is one hand") are not workable here. Squeezed down to their barest expression, most arguments about the abstract would follow this template:
If a simple, irreducible sentence is true, then its singular terms exist.
There are singular terms in simple, (a) irreducible, (b) true sentences that could only denote an abstract entity.
Therefore, abstract entities exist.
Singular term is taken here in its broadest linguistic meaning: a word or phrase that denotes a single thing, like "Roger Federer", "this chair" or "the Sears tower". Simple sentences are basic clauses like "Meg is tall" or "Beth is angry at Amy". Lastly, an irreducible sentence is one that cannot be paraphrased away. For instance, "Eurythmics" is a prima facie singular term: it is just one word. But to accept that "Eurythmics is British" does not entail that this pop duo exists in the same way as individual humans. The sentence could be paraphrased as "Annie Lennox and Dave Stewart are both British", which means it was actually reducible.
Now, the second step of our argument assumes that certain terms "could only denote abstract entities". Aren’t we putting the cart before the horse and already assuming such entities exist, when the whole point is to try and prove it? Not so: a virtue of the characterization we’ve favored, the via negativa, is that "abstract" is simply taken as a synonym for "neither physical nor mental". One could easily refuse that \(6\) be a physical object (it is not composed of elementary particles, doesn’t have spacetime extension, etc.), and be confident it is not a mental state either. We coud then say that \(6\) "could only be abstract" if it existed, but avoid committing to its existence.
This argumentative strategy actually helps retrace the anti-realist options we’ve discussed above. Rejecting (1) would be the wont of austere nominalists, who accept, for instance, that "\(3\) is prime" is a true sentence, but refute the existence of \(3\). Rejecting (2a) is a form of paraphrase nominalism: it assumes that all sentences involving prima facie abstract terms are reducible. Rejecting (2b) is a fictionalist approach, which accepts that there may be simple, irreducible sentences about entities that "could only be abstract", but simply has it that they are false.
Independence:
Existence alone only proves that realism about the abstract is true. However, as we’ve seen in our discussions of immanent realism and of psychologism, one could suppose that all abstract facts are grounded in physical or mental ones. Platonists thus need to refute those positions — we’ve already outlined their main weaknesses, and highlighted that nominalism and Platonism are acting as attraction poles for these unstable stances.
Selective platonism:
Platonist arguments have been developed for most of the entities we had identified as abstract so far: numbers, universals, relations, propositions, senses, fictional characters, etc.
To be successful, one needs to either refute all the alternate positions (austere/paraphrase nominalism, fictionalism, psychologism, immanent realism), or to prove that the entity under examination can be derived from another, but one we’ve already proved is abstract (e.g. that universals are actually sets of objects, or that propositions are sets of worlds...).
It is important to realize that one can be a platonist about some objects, but not others. Mathematical platonists (about numbers) can be nominalists about properties; Maddy is a platonist about sets, but immanent realist about numbers; and so on... To be a platonist about one thing only is enough to validate the fundamental nature of the abstract realm.
The access dilemma:
One major challenge to Platonism, which we’ve already evoked as an argument for nominalism, was formulated by Paul Benacerraf as an epistemological dilemma. It is reminiscent of the mental interaction issue. If abstract objects exist independently from us, and we are nevertheless able to utter true sentences that refer to those, then how did we acquire this knowledge? There seems to be a severe epistemological gap there. It only grows wider when we consider mathematics: how can we accept that mathematicians are able to prove new theorems without any access to the entities they’re manipulating?
One way to face the challenge is to abandon Platonism altogether (and indeed, modern nominalism, fictionalism, post rem structuralism can all be said to be motivated by that initial insight). Another one is to reject the no-access premise: Gödel, Parsons and others do exactly that, alleging that our minds can access some abstracta via some kind of special perceptual means (Plato himself suggested that our souls had been denizens of the world of Forms before our birth). However this type of approach seems very hard to square with the independence tenet; we would still insist that two fundamental ontological domains need to be causally remote.
Modern Platonists have confronted the issue heads-on, seeking to demonstrate how we can know and refer to abstract entities without access. Most theses have followed this template: "what we refer to are indeed existent abstract entities, however we don’t need access to them in order to make true statements: instead we rely on "something else" (the empirical world, abstraction principles, definitions, our ability to build consistent set of axioms, etc.).
A first, famous implementation of that strategy can be found in Quine’s naturalism. Quine started out as a would-be nominalist, but ultimately saw no contradiction between our knowledge of mathematics and its abstract remoteness: indeed mathematical truths are embedded in our physical theories, which are empirically confirmed. Even though mathematical objects are by definition not physical, mathematical knowledge doesn’t differ in its acquisition from other types of knowledge. After all, the fundamental bricks from our "best physical theories" like quarks or bosons aren’t directly observed: we detect macroscopic phenomena which can be interpreted as fairly distant manifestations of their existence. Through confirmation holism, we then accept that entire theories are verified through those isolated specks of evidence. Despite the seductive simplicity of naturalism, the transposition of this method to modern mathematics is bound to be clunky: it cannot easily account for the fact that mathematicians could discover true theories before those are ever applied in an empirical context.
Other approaches can be (roughly) grouped under the banner of ante rem structuralism. This was formalised by Shapiro and Resnik, as well as by Hale and Wright under the guise of abstractionist structuralism. The basic idea is that the abstract ontology is populated, not by what we normally conceive of as mathematical objects such as \(6\) or the \(exp\) function, but by structures. Mathematical objects are then the name we give to specific roles within these structures. Ante rem structuralists are normally realists about structures — but some are not realists about the role-fulfilling mathematical objects. To avoid the epistemological dilemma, which is a key motivation of structuralism, it is necessary to rely on abstraction principles, which we need to demonstrate can be known a priori. This allows to build a neo-logicist programme whereby we can refer to mathematical structures without accessing them, because the abstraction principles act as a dictionary that helps express our statements as references to objects which we can access (this could easily veer into a way to be "paraphrasing away", hence the fine line with the nominalistically inclined post rem version of structuralism). A different flavor of ante rem structuralism is often called plenitudinous platonism and has been defended by Balaguer, Zalta and Linsky, among others. The core tenet is that all possible consistent mathematical structures exist. This supposedly solves the epistemological dilemma automatically. Balaguer gives the example of Nepalese villages. It would seem odd for me to comment on the daily life of a specific village in Nepal if I had no way to access information about it. However, under the assumption that "every possible Nepalese village exists", anything consistent that I say about a hypothetical Nepalese village automatically refers, and I didn’t need any access for that. Of course, the challenge here is in the concept of consistency.
There is not enough space for a long discussion of these positions: mathematical structuralism is an active research domain at the time of writing, with a dense taxonomy of positions and schools of thought. The only general (if uncharitable) comment I would dare is that many of those seem to be merely moving the goalposts. Sure, we may have a coherent "no-access" epistemic story, but we still need to explain what makes a "good" abstraction principle, or a "consistent" mathematical theory.
\(\{\mathbf{\aleph}\mathbf{\prec}\mathbf{3}\}\) : abstractionism
What would it mean for the abstract realm to be the ground of the physical and the mental — a position I call "abstractionism"? Remember that we chose to define the abstract in complementary fashion as "neither physical nor mental". Platonism asserted that not only some things exist that are not physical or mental, they are ontologically independent from those realms. Abstractionism tells us instead that they are not really independent: physical and mental facts are fully derived from abstract facts.
There’s something immediately seducive about this hypothesis: it has the potential to dispel the thorny "access problem" which plagues Platonism. However it imposes upon us to explain the grounding relationship that’s at its heart. Even though his view lacks the rigor and formalism that would be expected today, one may say this is what Plato himself sought to achieve with his theory of Forms: unlike modern platonism, its namesake didn’t talk of two or three worlds, but of one "real world", that of the Forms. With his cave, Plato offers an allegorical explanation of the physical world’s emergence, as it is considered a mere shadow of higher, abstract entities. Despite the naming conventions borne from history, abstractionism may then be closer to Plato’s own stance than "platonism" (itself more aligned on medieval scholastic interpretations of the Athenian’s work).
The issue, as always, lies in some key discrepancies between the grounded and the grounding entities. We haven’t sought to characterize the abstract realm, which is indeed a bestiary of various classes; however we’ve tried to characterize its complements. Mental states are subjective, private, qualitative, and bear phenomenal or intentional properties. The question of how those could be explained by facts about entities that (apparently) lack any internal or intrinsic feature seems very similar to the main challenges of physicalism.
For the physical to be grounded in the abstract raises more unusual questions. In our talk of the physical world, we’ve foregone old arguments about spatial extension or causal efficacy, giving up such characterizations for a somewhat underwhelming proposal that physical objects be the ones described in our "best theories". In other words: if the Standard Model is one of those theories, and it is expressed in terms of mathematical objects that include an "electron wavefunction", which in turn predicts the likelihood of observing an electron with certain characteristics, then electrons belong to the physical world. But, as was raised when we discussed Russellian monism earlier, physical theories describe their objects in purely relational and extrinsic terms: if we only account for the mathematical underpinnings of the theories, something is an electron in virtue of what it does, not of what it is. Metaphysically speaking, this is consistent with the view of objects as mere bundles of properties.
In contrast, one may insist that equating an object with its properties is a category mistake: what remains when one "peels away" the properties, or "unhook" them one by one is some characterless substrate that’s often called substance, after Aristotle. Substance arises in the context of ordinary objects and their identity or persistence conditions, which we’ve evoked. The question of substance in the context of modern physics is much more delicate and steeped in the impact of quantum physics on traditional metaphysical categories; we shall come back to this in the next chapter.
However, we can already note that a theory of pure substance (where one can talk of property-less bare particulars) can hardly be accommodated with abstractionism: to borrow from Mally or Zalta, abstract objects encode properties, and that’s enough to fully characterize them. The number \(2\) fulfils a specific role in a certain mathematical structure. Remove one of the definitional elements of that role, and you end up with something that’s not \(2\). Take Sherlock Holmes and encode him as a baker instead of a detective — it’s not Sherlock Holmes anymore: there’s no substance there beyond the bundle of properties we associate with him.
Conversely, bundle theories seem to be a good match for abstractionism. The difficulty then stems from how we could differentiate at all, metaphysically speaking, the physical from the abstract. If I was able to conceive and run a complete mathematical simulation of a universe, in what sense would that universe be less "physical" than ours? This line of questioning leads us very fast down a slope towards "mathematical universe" proposals, similar to Max Tegmark’s, according to which the physical world is literally a mathematical structure — the one that a Theory of Everything would represent. Since it’s hard to argue why any consistent mathematical structure should be deemed nonexistent, a corollary seems to be the embedding of our physical world into a much larger multiverse.
Can we prevent abstractionism from becoming a monism, and establish a distinction between physical worlds and the rest of the realm? It’s one thing to say that mathematically "neighbouring" universes (say, ones with different initial conditions or universal constants) exist, it’s another to say that any mathematical structure (like \(\mathbf{Z}/6\mathbf{Z}\) or the hyperbolic plane) is qualitatively undistinguished from the universe. It then becomes the philosopher’s job to establish credible criteria for "physicality" of abstract structures that allow to accept the derivability relation without losing sight of a healthy distinction between abstract and concrete.
\(\{\mathbf{3}\}\) : nihilism
This purely abstract monism implies that our concepts of "physical" and "mental" are meaningless, and that any apparent difference between the corresponding objects and stereotypical denizens of the abstract realm such as numbers, universals or sets, is sheer illusion. Although strictly speaking, this doctrine asserts that there is something, it is close to a form of pure ontological nihilism.