In my last post, I began the next major part of my blog by illustrating a tripartite view of reality, dividing what we seem to experience into three realms: the physical, the mental, and the abstract. Now I am going to start exploring the question of which of these three realms actually exist.
I was going to begin by looking at whether the physical realm exists, since it seems like the answer to that question, at least, is fairly obvious. Instead, for some reason I have decided to reverse course completely and talk about the realm of abstractions. It is a little more philosophical of a subject, but I think it can help to clarify things.
Abstract objects are somewhat hard to define, but I can give some examples to illuminate the concept:
- Propositions are abstractions of meaning. As an abstract object, a proposition is usually taken to exist independently of any sentence that has that proposition as its meaning, and two different sentences can express the same proposition.
- States of affairs are abstractions of reality, descriptions of ways that reality can be. States of affairs are usually taken to exist whether or not reality actually is the way that they describe.
- Things that exist have properties, such as being red, or being made of wood. Two different things can have the same property, and this is often said to explain how things can resemble each other.
- Similarly, things that exist can stand in relations to each other, such as the relation of one thing being taller than another.
- Mathematical objects such as numbers or sets are paradigmatic examples of abstract objects.
If they exist, abstract objects are thought to lack any kind of causal powers. Numbers, for example, can’t cause any kind of change in reality. This is commonly taken to be the defining feature of being abstract, next to being neither a physical nor a mental entity. Abstract objects are contrasted with concrete objects, which are physical or mental entities and can affect reality in some way.
Many of the above examples are also thought to exist necessarily, outside of space and time. But there are things that are reasonably construed as abstract that lack these features. For example, the equator can be thought of as an abstract object with a definite location that came into existence at a certain time (when the Earth formed into its approximately spherical shape).
So much for what abstract objects are. The interesting question now is, do they exist?
Platonism is the philosophical position, named after the ancient Greek philosopher Plato, which holds that abstract objects are real things that exist, and that they exist just as fully as concrete objects. If platonism is true, numbers, for example, exist independently of any minds, outside of space and time. Nominalism, on the other hand, is the belief that platonism is false, and that abstract objects do not exist. But if nominalism is true, we have to find some way to explain how we can talk about things like numbers.
NOTE: Since writing this post, I have come to realize that the question of abstract objects is more nuanced that I presented it here – in particular, there are more positions than just platonism and nominalism, and when I conclude in the next post that “abstract objects such as properties are just a means of speaking about the way that things exist,” what that ends up looking like when formulated more precisely is something more like platonism and less like nominalism than I originally thought. More to come.
An Argument for Platonism
One of the oldest arguments in favour of platonism is this: consider a red apple, and a little red wagon. How is it that both of these things can be red? How can they resemble each other, seeming to have the very same redness? Platonism explains this by saying that the apple and the wagon exemplify a property, the property of being red, and it is the very same property that they both exemplify. Properties are universals, things that can be exemplified in more than one place at once (so that more than one thing can be red), and that exist changelessly across time (so that redness today is the same as redness tomorrow).
Philosophers nowadays have generally recognized that the issue of resemblance in this argument is a red herring. If you can explain how the apple is red, presumably you can explain how the wagon is red in the same way. And then nothing more is needed to explain the fact that they are both red. So the real challenge is to explain how something can be red.
Nominalism responds by saying it can do this just as well as platonism can. Things are red because (i) they reflect certain wavelengths of light, and (ii) we perceive those wavelengths as red. In principle, we can continue offering explanations for facts (i) and (ii) until we arrive at a number of statements that will presumably be about the basic constituents of reality. All of these statements can be put into a form like this: Xs are Y. (For example, electrons are negatively-charged.)
The platonist explains facts like these by saying that Xs have the property of Yness. But the nominalist says that this is not an explanation – saying that Xs are Y because they have Yness is no more informative than saying that Xs are Y because they are Y. We might as well just accept that when we reach the fact that Xs are Y, we have reached an irreducible fact about reality. (This is especially plausible if all of the fundamental facts we have reached in this way are necessary facts, facts that could not possibly have been false, since necessary facts are often thought of as being explained by their own necessity.) There is no explanatory gain in accepting the existence of properties like Yness.
NOTE: In fact, I have realized that what I have said in the last paragraph may very well be false. This is in part because Platonism is not trying to explain statements like “Xs are Y” so much as it is trying to understand what such statements actually mean – what it actually means for Xs to exist in such a way as to be Y. The nominalist response that I gave, in that case, is missing the point.
So the nominalist thinks we can talk about properties as a convenient way of expressing the ways that things can exist and resemble each other, but that we don’t need to actually believe they exist, at least not for any explanatory power that they offer.
Another Argument for Platonism
Here is another argument in favour of platonism. Even if we can sometimes paraphrase statements to avoid talking about abstract objects (instead of saying that Xs have the property of Yness, just say that Xs are Y), we can’t always do this. And often when we talk about abstract objects, we don’t have any paraphrase in mind at all – we just mean what we are saying.
So if we say something like there are infinitely many prime numbers, and we think this is true, we are committed to believing that the things we are talking about exist. Prime numbers have to really exist, the platonist argues, in order for it to be true that there are infinitely many of them. And some statements like these have to be true – they are indispensable for our understanding of reality. Mathematics, for instance, is a necessary foundation of our scientific theories. Arguments like this can be made for the existence of many different kinds of abstract objects.
Can we mean what we are saying without committing ourselves to the existence of what we are referring to? I want to suggest that we can. In fact, we do it quite often. You might say something like “she has a heart of gold,” and mean it, without believing that anything medically odd is going on. Or you might say “I’m overflowing with joy,” without believing that an emotional state has somehow liquefied and started pouring from you into the surrounding environment. Or you can say “the price of apples is skyrocketing” without believing that anything is being rapidly propelled into the air.
In other words, we can speak figuratively, and we can even do this without being aware of it. There is a theory in the field of cognitive linguistics, the idea of conceptual metaphors, that metaphors are a fundamental aspect not just of our language, but of our whole way of thinking, and that we use them often without realizing it. (A great book to read on this subject is Metaphors We Live By, by George Lakoff and Mark Johnson.) Completely literal language is the exception, not the norm.
“Cognitive discourse at its most dryly literal is largely a refinement. … It is an open space in a tropical jungle, created by clearing tropes away.” – W. O. V. Quine
If we have reason to think we are speaking figuratively, and not literally, when we are talking about abstract objects, then this argument for platonism falls flat – we don’t need to believe that abstract objects exist in order to meaningfully speak as if we did.
Even more than that, abstract objects do not need to exist for our figurative statements about them to be objectively true or false. Using the example of numbers: if our concept of numbers is sufficiently determinate – if the laws of logic determine what numbers have to be like, given our conception of them – then there are objectively right or wrong answers to questions we ask about them when we are speaking as if they exist, whether or not they actually exist.
And if our concept of numbers is not sufficiently determinate, their real existence is no help. Since in that case the laws of logic leave some things about numbers undetermined, the truth of a statement like there are infinitely many twin primes could either be true or false, and it would just be a coincidence if our belief about that statement lined up with reality. (Compare this with what is usually thought to be true, which is that our concept of numbers is determinate, and that the above statement is either necessarily true or necessarily false – whichever way it is, it could not possibly be any other way.)
In my next post, I’m going to look further at the idea that our abstract object talk is figurative language, or something similar to it. I think this theory spells trouble for belief in the existence of the abstract realm.