The Problem of Universals: Realist and Nominalist Solutions

Hello! Welcome to the first lesson in our new module on Contemporary Analytic Metaphysics and Epistemology.

In our last module, we concluded with Wittgenstein's radical critique of mathematical truth, where he questioned whether mathematical objects or properties exist independently of our practices. This week, we step back to look at the broader, more traditional version of this problem, one of the oldest and most fundamental in all of philosophy.

Your learning outcome for this lesson is to: Explain the problem of universals and distinguish between realist and nominalist solutions.

We will investigate the "one over many" problem: How can many different things all share a single property, like 'redness' or 'justice'? What is the nature of these properties, or universals? We will explore the two main opposing answers. Realists argue that universals are real, mind-independent features of the world. Nominalists deny this, insisting that only individual, particular things exist. This debate sets the stage for many of the core issues in metaphysics.

1. Particulars, Universals, and the "One Over Many" Problem

Let's begin by clarifying our terms. The world seems to contain two different kinds of things:

• Particulars: These are individual, specific objects that exist in a particular place at a particular time. This apple on my desk, the Eiffel Tower, the planet Jupiter—these are all particulars. They are unique and cannot be in two places at once.
• Universals: These are properties, qualities, or relations that can be shared by many different particulars at the same time. Redness can be in this apple, that fire hydrant, and that sunset. Triangularity can be a property of a yield sign, a slice of pizza, and a geometric drawing.

The philosophical puzzle arises when we try to understand the relationship between them. This is often called the "one over many" problem. How can one thing (the universal, 'redness') exist in many different places at once (in many particular red objects)? What is the ontological status of this "one" that is shared by the "many"?

To get a clear grasp of this fundamental distinction and the problem it generates, please watch the beginning of the following video.

As the video explains, the core question is whether universals like 'redness' or 'humanity' are real entities in their own right, or just convenient labels. Your answer to this question places you into one of two major camps: Realism or Nominalism.

2. The Realist Solution: Universals are Real

Realists solve the "one over many" problem by asserting that universals are real, mind-independent entities. The reason multiple apples can all be red is because they all share in, or instantiate, the very same universal: Redness. Realism comes in two main varieties, which should be familiar from our discussions of Plato and Aristotle.

To summarize the key distinction:

• Platonic Realism: Universals are transcendent. They exist outside of space and time. A particular red apple participates in the Form of Red. This is the view you encountered in our first module and mirrors the Mathematical Platonism we've discussed, where numbers and sets exist in an abstract "Platonic heaven."
• Aristotelian Realism: Universals are immanent. They exist in the objects themselves. The universal Redness is wholly present in each and every red particular. It does not have a separate existence; if there were no red things, the universal Redness would not exist.

For the realist, similarity is explained by identity: different objects are similar because they literally share the identical universal.

3. The Nominalist Solution: Only Particulars Exist

Nominalists find the idea of universals metaphysically extravagant. They adhere to a principle of ontological parsimony (often associated with Ockham's Razor): we should not multiply entities beyond necessity. Since we only ever encounter particulars, nominalists argue that reality consists only of particulars. The challenge for them, then, is to explain the appearance of shared properties without admitting universals exist.

There are several major nominalist strategies.

Let's briefly review these strategies:

1. Predicate Nominalism: This view holds that for an apple to be red simply means that the predicate "is red" correctly applies to it. There is no property of redness in the world; there are just particulars and the words we use to describe them. Critics often label this "Ostrich Nominalism," arguing that it simply ignores the question: why does the predicate "is red" apply to these things and not others?

2. Resemblance Nominalism: This is a more sophisticated approach. 'Redness' is not a universal, but the collection (or set) of all red particulars. An object is red by virtue of its membership in this set, and it belongs in the set because it sufficiently resembles the other members. This avoids universals, but faces problems like the "problem of companionship" (mentioned in the Philosophy Vibe video) and circularity (defining the set of red things seems to require a prior notion of redness).

3. Trope Nominalism: A powerful modern version of nominalism. It proposes that properties exist, but as particulars themselves. The redness of this apple is a trope—a specific, individual instance of red that cannot be shared. The redness of another apple is a different trope. These two tropes are numerically distinct but exactly resemble each other. On this view, an ordinary object is a bundle of tropes (a redness-trope, a roundness-trope, a sweetness-trope, etc.). 'Redness' in general is simply the resemblance class of all individual red-tropes.

For the nominalist, similarity is not explained by identity, but by brute resemblance or linguistic convention.

4. The Debate in Action: Abelard vs. William of Champeaux

To see how these abstract positions play out in a real philosophical argument, it is useful to look at the fierce debates of the medieval period, where this problem was a central focus. The clash between Peter Abelard (a nominalist) and his teacher William of Champeaux (a realist) is a classic example of philosophical progress through rigorous argumentation.

Your background in mathematics makes you well-equipped to appreciate the logical precision of Abelard's objections.

Abelard's critique is a perfect illustration of the power of philosophical analysis. By showing that William's realism led to the conclusion that distinct individuals are identical, he forced a refinement of the realist position and powerfully motivated the search for a viable nominalist alternative.

Conclusion

Today we have mapped out one of the most enduring problems in metaphysics. The "problem of universals" forces us to make fundamental choices about what kind of things make up reality.

Key Takeaways:

• The Problem of Universals (or the "one over many" problem) asks how numerically distinct particulars can share the same properties.
• Realism holds that universals are mind-independent entities that are literally shared by particulars.
  • Platonic Realism: Universals are transcendent Forms.
  • Aristotelian Realism: Universals are immanent in particulars.
• Nominalism holds that only particulars exist. It seeks to explain apparent shared properties without positing universals, using strategies like:
  • Predicate Nominalism: Reducing properties to language.
  • Resemblance Nominalism: Reducing properties to sets of resembling particulars.
  • Trope Nominalism: Positing properties as particular instances, or tropes.

This debate is not merely a historical curiosity. It underpins contemporary discussions in the philosophy of language (what do general terms refer to?), philosophy of science (what are laws of nature?), and, as we've seen, the philosophy of mathematics (what are mathematical properties?).

Preview of the Next Lesson:
In our resources today, you may have noticed a brief mention of the "Third Man Argument." This is a powerful objection to Platonic Realism, first formulated by Plato himself. In our next lesson, we will perform a deep dive into this argument, analyzing its logical structure and its impact on the realist project.