Yes, Non-Existent Entities Exist (Part 1: The Nature of Abstract Objects)

why and how abstract objects exist

Do non-existent entities exist? This question seems to answer itself, but I mean "exist" in two different senses. By "non-existent," I mean non-physical, non-causal entities, like abstract objects. And by "exist," I mean existing independent of mind, in other words, are objective rather than subjective. Rephrased, the question can be, "Can something that exists be non-physical, non-causal, and non-mental?" If not, how can we explain mathematics, the quintessential domain of "abstract entities" we call numbers? And if so, then how do they exist?

Philosophers have been debating the existence or non-existence of abstractions since at least Plato’s Theory of Forms, creating a whole slew of different views along the way, including Platonism, Universalism, Nominalism, Fictionalism, Pragmatism, and Meinongianism. Only 37% of philosopher respondents in the 2020 PhilPapers Survey even "lean towards" Platonism, the view which states abstract objects exist. Notwithstanding the weight we give to its namesake, Platonism is a minority view. Occam's razor places the burden on the Platonist to explain how abstract objects exist, who have provided diverse yet unsatisfying answers.

And the lay feels similarly about the lesser existence of abstractions. We might even call something "abstract" as an epithet for fake (example: “The politician was just making abstract statements, rather than promising concrete results.”), as opposed to something “real” and “physical.” Abstract entities, even if they were to exist, still tend to be placed on a lesser plane of existence than the physical. Physical objects are those self-evident things that we can know with our five senses. Whereas if something is "abstract," we have difficulty grounding it to reality. It's stuck in our heads—if it exists, it feels subjective rather than objective.

Why Thoughts Exist: An Argument for the Existence of Thoughts

As I've argued, thoughts, not matter, are the true subjects of the laws of logic (which are more appropriately dubbed the laws of thought) and are the proper unit of analysis for philosophy. Philosophers don't run experiments in the world but instead run "thought experiments," on thoughts.

However, many have falsely equated physical with the real, leaving meta-physicians ill-equipped to explain the existence of the non-physical. Without an explanation, then how can we even say that "holes" exist?

I will defend the view that, similar to Platonic forms and Fregeian senses, "thoughts" are objective, above and beyond the physical, to which our minds provide access.

First, let's establish one important premise: "I think therefore, I am," popularly known in its original Latin as "Cogito, ergo sum," an affirmation made famous by Descartes (this statement can be called the "Cogito" for short). The Cogito hasn’t been free from critics, particularly those who reject its dualist implications. However, anyone who rejects the Cogito would still have the burden of proving existence without thought. And it is impossible to deny the subjective experience of thought. 

The Cogito is convincing. If we were to doubt all of our experiences, we could at least still say with 100% certainty, that we are nonetheless thinking; our act as thinking counts as certainty for “something,” so “something” must exist. Therefore, I exist as this something thinking thing. The act of thinking is proof of my existence. And for thinking to be proof of existence, thinking itself must exist.

Thinking cannot be proof of my existence if thinking itself doesn't exist. And if thinking exists, thoughts must also exist as the product of my thinking.

You might respond that "thoughts" and "thinking" are in different categories altogether, as the former refers to a noun and the latter a verb. But this view fails to recognize their essential identity. There cannot be thinking without a thought that becomes generated, and there cannot be thought without thinking that has occurred. Try "thinking" without having a "thought." "Thinking" and "thoughts," are necessary and sufficient conditions of one another, one cannot exist without the other.

Once thinking gives you certainty over subjective experience, anything within that experience must exist with certainty, including thoughts, because something that doesn’t exist can’t then be used as evidence for existence. Take the premises below as a summary.

1. If I can think, I must have thoughts.

2. For my ability to think to be proof of my existence, thinking must also exist.

3. If thinking exists, thoughts must exist.

4. I know that I am because I can think (Cogito).

5. Because my thinking is proof of my existence (Cogito), proof of existence must also exist (P2), and thinking necessitates thoughts (P3), thoughts must therefore exist.

Although the above argument doesn't reveal how thoughts exist, only that thoughts exist. Once we accept that "thoughts" exist, we can see how entities that are outside of time, space, and mind (i.e., abstract objects) exist as these thoughts.

How Thoughts Exist: A Positive Account of Abstract Objects

Arguments for “abstract objects” have been met with skepticism, a major point of criticism being their purely negative definition. If abstract objects are non-spatial, non-temporal, non-causal, non-mental, then they must be non-anything. But of course, we mean something when we describe our abstractions—as these abstractions exist in the form of thoughts.

In reference theory, if a word fails to refer to a thing in reality, then the word is false/meaningless for a failure to refer. So if the meaning is grounded in referent, and statements about abstract objects have no referent, then statements about abstractions lack meaning.

But we can clearly have thoughts, and convey those thoughts, without the need for a referent to exist. When I say that I "thought" of a Batman, a person who doesn’t exist in the world, you still know what I mean; Batman exists as a "thought” and we can take about our shared objective “thought” of Batman. Nerds don’t engage in pure “gibberish when they discuss events in Batman comics, they can both talk on the same objective playing field.

So if abstract entities exist as "thoughts," then what are "thoughts"?

I will define a “thought” as something that a mind can create and understand. I'll justify the "created" and "understand" requirements below.

Thoughts are Created by Mind

Thoughts are a product of the mind. A mind is only a mind so far as it can produce thoughts. This requirement is obvious enough.

But we shouldn't confuse thoughts as a product of the mind with being mind-dependent, at least not in the most ultimate sense. The number 84313158935150 may have never been thought of in the history of humanity, but assuming it hadn't, I didn't just "create" that number. It would be more appropriate to understand that I had “picked” it out to use as an example, which presumes its mind-independent existence. 84313158935150 was a thought that I had, but it wasn't a thought that I independently created.

84313158935150 had existed before humanity had thought of it, and it will continue to exist well after humanity, for anyone else to pluck out.

To use an analogy, our minds might be the retailers of thoughts, but even after our retail store goes out of business, thoughts will still exist with the manufacturer. They may not be "consumed" by anyone, but they will still be in storage.

Yet thoughts can't be just any mental product. For thoughts to exist objectively, they must be comprehensible to others. They must be able to exist in other minds to qualify as truly objective.

Thoughts are Understood by Mind

Most can agree that a thought needs to come from a mind. But why must a thought also need to be comprehended?

Thoughts aren't just anything we can describe. "Colorless green ideas sleep furiously," or "This sentence is false," may be products of the mind, but other minds can’t then understand them. They are meaningless phrases that fail to convey a thought.

Objective thoughts are like food. Like thoughts, food can't just contain physical substances, but to qualify as food, it must be digestible. If other minds can't digest thoughts, they would be purely private mental experiences that could never go beyond our own skulls. Someone could have an "idea" of what the statement, "This sentence is false means", but unless they can explain that meaning to others (i.e., have only a "private language"), then that ""idea" couldn't be a "thought."¹ It would be subjective rather than objective.

And for a thought to be understood, it cannot be a contradiction. It needs to be coherent and sensible. Anyone else should be able to grasp that thought for themselves, at least in principle. 1=2 is not a thought, for what would that even mean? In fact, everyone could have their own interpretation of what 1=2 could mean in our their private language. However, we use only one standard interpretation for 1=1, making it an objective thought, and we have dubbed it the "law of identity".

Thoughts can take the form of a proposition, command, belief, memory image, concept, etc. And they can be conveyed through methods like language, art, gestures, interpretive dance etc. If you can think of something and can (at least in theory) convey that idea to others, you have an objective thought.

Objective and Subjective Thoughts

Thoughts aren't purely subjective. For thoughts to be truly objective, they must be capable of being independently understood by others. While methods of communicating our thoughts are clearly imperfect (just see errors in language, physical gestures, social media, etc.), in principle, a thought must be intelligible to others. If you think of something that no one else could possibly understand, then it cannot be a true thought that can be shared with others. It would be like a private language where you are free to change the rules of that language at will. How can anyone know the meaning of a purely subjective system? That kind of idea would be purely subjective.

Meanwhile, a thought can be said to be objective and can be grasped through any act of thinking. If someone else, at the very least, could understand an idea, that idea is an objective thought and can be shared as one. 1=1 is a universal truth whose meaning anyone could grasp, so it is a thought. Although "1=1" couldn’t refer to anything in the world (for numbers exist only as thoughts), we know what it means, and we know that its true.

Now that we have positively defined thoughts, showing both (1) why they exist and (2) how they exist, we will apply this ontology to counterfactuals, fictions, and contradictions later in this series.

Conclusion

Abstract entities exist with 100% certainty. Because my existence is evidenced by thought, thoughts must therefore exist. Otherwise, thoughts (or thinking) cannot be evidence of existence.

A “thought” is created and comprehended by minds. If the mind does not produce an entity, then it's not an "idea". And if the "idea" cannot be understood by others, then it's not a "thought." Abstract entities exist as thoughts, and thoughts are produced and conveyable by the mind.

Next, we’ll put this general theory into practice.

1. It would be what Gottlieb Frege called an "idea", an ontologically subjective entity, rather than a "sense," which is ontologically objective.