﻿ ﻿Nominalism, the nonexistence of mathematical objects - Views on the Meaning and Ontology of Mathematics - Mathematics, Substance and Surmise

## Mathematics, Substance and Surmise: Views on the Meaning and Ontology of Mathematics (2015)

### Nominalism, the nonexistence of mathematical objects

Jody Azzouni1

(1)

Department of Philosophy, Tufts University, Miner Hall, Medford, MA 02155, USA

Jody Azzouni

Email: jodyazzouni@mindspring.com

Abstract

Nominalism is the view that, despite appearances, there are no mathematical entities. The ways that nominalism is both compatible with there being mathematical truths and falsehoods, and compatible with mathematical truths being valuable in scientific applications are explored in this paper. Some of the purely psychological reasons for why nominalism is so hard to believe in will also be discussed.

There is an amazingly large number of different positions in the philosophy of mathematics—almost as many as there are professionals who have written on the subject area. But as far as the ontology of mathematics is concerned—as far as the question of whether or not mathematical objects exist—every position boils down to one of two.

(i)

Mathematical objects exist.

(ii)

Mathematical objects don’t exist.

For purely historical reasons, and to adhere to common nomenclature, call Platonism the position that mathematical objects exist. Call nominalism the position that mathematical objects don’t exist. Between these two choices, Platonism is the far more natural position to adopt for two reasons. I’ll discuss the first reason now, and take up the second one in a couple of paragraphs. The first reason is that the language of mathematics is pretty much the same as that of ordinary language. That is, it shares its grammar and much of its vocabulary with ordinary language. The language of Euclidean geometry, for example, is just ordinary language plus some additional specialized vocabulary that’s been refined from ordinary life, for example, point, line, triangle, and so on.

The reason that this is relevant to the issue of whether mathematical objects exist or not is that the noun phrases that occur in natural language and that occur in mathematics play exactly the same grammatical and semantic roles. Here is an example:

(iii)

There are finitely many grains of sand.

(iv)

There are infinitely many prime numbers.

It’s very natural to think of (iii) as indicating not only that there are a finite number of grains of sand, but also that these grains of sand exist. It seems weird, after all, to think that grains of sand don’t exist but there are nevertheless only finitely many of them. In the same way, it’s very natural to think of (iv) as indicating not only that there are infinitely many primes, but also that these primes exist as well. Surely (iii) and (iv) are very similar in meaning—so similar in meaning that if one indicates that the objects it is about exist then the other one surely indicates the same thing.

I mentioned that there are two reasons for Platonism being a much more natural position than nominalism. The second is this. (iii) and (iv) are both true. This is something that the ancient Greeks, Plato in particular, were very sensitive to. If something doesn’t exist, here’s a way to put the point, then how is it possible to say something true about it? For that matter, how is it possible to say something false about it? What, after all, would it be that we were saying something true (or false) about?

So that’s one big issue. It seems that mathematical statements, generally, are true (or false, for example, 2 + 2 = 5) but this can only be because they are about particular objects. They are true if they describe the mathematical objects they are about correctly (2 + 2 = 4) but otherwise they are false (2 + 2 = 5). If mathematical statements aren’t about mathematical objects then it’s hard to see on what grounds we could sort them into the ones that are true and the ones that aren’t.

There is a second point to make about this that connects truth to the empirical application of mathematics. Ancient Euclidean geometry already exemplifies this important aspect of mathematics, one that’s more dramatically illustrated in contemporary physics, and in many other sciences as well. This is that mathematics is used in scientific applications. The way that mathematical statements being true bears on this fact about the scientific application of mathematics is this. Our empirical theories—ones in physics, for example—are usually deeply intertwined with mathematics. These empirical theories, however, are ones that we draw implications from. A major scientific event (for example, the discovery of Bell’s theorem) is often the derivation of an important scientific result from a scientific theory, a derivation that involves a great deal of mathematics. But the steps of any derivation have to be truth-preserving. That is, what makes valuable the derivation of a result from a scientific theory is that if the scientific theory is true, then the result derived must be true as well. This truth-preserving nature of scientific inference, however, will be short-circuited if mathematical statements aren’t true. For anything can follow from what’s false: something false or something true. But this means that we wouldn’t know that a result derived from a scientific theory were true if that derivation involved mathematical steps.

Some philosophers have entertained the idea that mathematical statements are part of one or another mathematical game, that there can be something like “truth in a game,” and that this is the kind of truth that we have, say, when we use a system of mathematical axioms, either in pure mathematics or in empirical applications. So, instead of saying that 1 + 1 = 2 is true, we can instead say that 1 + 1 = 2 is true-in-arithmetic, although we can go on to say that it isn’t actually true. The suggestion is that this is just like what happens with fictional truth. We can say that Hamlet is the Prince of Denmark is true-in-the-play-Hamlet, although of course it isn’t actually true.

The problem with this suggestion is that it forces all the empirical results that follow from our scientific theories using applications of mathematics to yield result that are only true-in-mathematics, and not simply true, as we need all our empirical results to be. Imagine, for example, that D is some scientific theory, one that we take to be true. Let M be some arithmetic that we don’t take to be true, but only true-in-arithmetic. Suppose we can derive from M+D some important empirical result C. We aren’t licensed to assert that C is true, which is what we need, but only that C is true-in-arithmetic, a much weaker result. Imagine, for example, that we calculate the area of a rug based on its boundaries. It’s a peculiar conclusion to draw that results about the rug’s area are only true-in-arithmetic and not true outright. The size of a rug being true-in-arithmetic and not true outright sounds just like the tricky sort of thing a dishonest salesperson might say who was trying to sell us a rug that’s different in size from what it’s been advertised to be.

The upshot. Here are some of the reasons to think that mathematical objects exist and that mathematical statements are about those existing objects. First, mathematical statements are true and false the way that other statements normally are. Second, in general, statements seem to be true and false precisely because they describe accurately (or inaccurately) the properties of the objects they are about. Since mathematical statements are true and false, they must be about the objects they describe accurately (or inaccurately). Finally, mathematical statements are ones that we use in empirical science. This last point is what prevents us from replacing the apparent truth and falsity of mathematical statements with something weaker like truth-in-arithmetic and falsity-in-arithmetic, or the like.

2 Given the above, why would anyone think nominalism was a sensible position?

If mathematical language is just like ordinary language, and presuming the importance of mathematical statements being true (or false) as well as the importance of the empirical applications of mathematics, why would anyone want to be a nominalist? Why would anyone want to deny that mathematical objects exist? The primary reason is that there don’t seem to be any good candidates for mathematical objects among the things that we already take to exist. Let’s consider Euclidean geometry again, specifically Book 2 of Euclid’s Elements, which is where Euclid describes the construction of various diagrammatic proofs of results like: the sum of the interior angles of a triangle is one hundred and eighty degrees. A possibility for the ontology of Euclidean geometry is that the mathematical objects, triangles, rectangles, points, lines, and so on, that are proved to have the properties shown in that book are the actual diagrammatic items—or classes of them—that are drawn during the construction of the proofs of the properties of these things.

One problem with this suggestion is that the mathematical objects, points, lines, and so on, are traditionally described in ways that seem to rule out this possibility: points are described as having no dimension, lines are described as having one dimension and as being perfectly straight, and so on. This is obviously false of anything we draw: drawn lines—regardless of how sharp the pencil is—have various thicknesses, and are genuinely fuzzy, for example. Points have (at least) two dimensions, and arguably three, as well as having fairly irregular contours.

The real problem with this suggestion, however, isn’t merely that points, lines, triangles, rectangles, and so on, are described in ways that rule out their being drawings of one sort or another. It’s that drawings of triangles, lines, and so on, don’t have the properties that these items are proven in Euclidean geometry to have. A drawing of a triangle, for example, doesn’t have angles that sum to exactly one hundred and eighty degrees simply because the lines composing it aren’t straight. There are other similar failures. For example, since diagrammatic points have dimensions, various uniqueness results (like any two lines intersect in one and only one unique point) are simply false, unless we attempt to define, for example, a point as the maximal area that two lines overlap on. (Doing this, of course, is going to cause lots of problems elsewhere in Euclidean geometry because, on this kind of view, points will have different sizes relative to the lines they occur in.)

This is one sort of issue we have with finding suitable objects for mathematical statements to be about. The candidate objects usually fail to have the properties that are attributed to mathematical objects by a mathematical discipline. There is a second problem that’s more important, however. This is that any candidates for mathematical objects have to be ones that the mathematical statements in question are clearly about. To make the issue clear, consider the case of an empirical theory about a certain kind of micro-organism. What’s needed to force the sentences of that theory to be about a specific kind of micro-organism (and not something else entirely) is more than a lot of theoretical descriptions of that kind of organism. The descriptions provided by a theory—especially an empirical theory—are always incomplete and can apply to more than one kind of thing. This is why we can always empirically discover new facts about the kinds of things that an empirical theory is about, results that go, often, far beyond what the theory itself says about the things in question.

What makes an empirical theory about a certain kind of micro-organism a theory about that kind of micro-organism isn’t theory at all. It’s the facts about how scientists—as it were—make contact with the micro-organisms in question. In other words, it’s the instrumental and perceptual facts about how scientists empirically study, say, specific micro-organisms. The micro-organisms in question, for example, may be ones that live in the human stomach, and ones that have certain effects on humans that we can measure and study. Or it can be that we take samples (of various types) and we can actually see, using certain instruments, the specific micro-organisms that we’re theorizing about. In general, it’s this making-contact with particular micro-organisms and not other micro-organisms (or something else entirely) that forces our sentences to be about those micro-organisms. Furthermore, it’s this making-contact with these specific micro-organisms that allows our theories about these specific micro-organisms to be wrong. It could be, for example, that the theory in question places them in one microbial family but we discover by instrumentally interacting with them that in fact they belong to a different microbial family. The terms naming the microbes in the theory about them succeed in referring to those microbes despite the theory being false only because of the instrumental and perceptual connections we have to those microbes (and not to other microbes).

The problem of applying to mathematics this making-contact explanation, for what kinds of entities an empirical theory is about, however, is that there isn’t anything like instrumental or perceptual connections to objects that cements what mathematical terms refer to. Mathematics is, as it were, pure theory all the way down. Here’s a somewhat idealized picture of how we practice mathematics: we set down a set of axioms and then derive some consequences of those axioms. We could imagine an empirical subject matter being axiomatized in a similar way, an axiomatization of a theory of a kind of cellular structure, for example. In that case, we could write down a set of characterizations of these structures, and then derive results about the structures from those characterizations. But the characterizations have to be localized to particular cellular structures to begin with and not to some other ones precisely because we instrumentally or perceptually fix on those particular structures to initially test our characterizations on. It’s this first step that’s simply missing from mathematical practice.

So the deep point supporting nominalism is this. There doesn’t seem to be a role in mathematical practice for mathematical objects. This absence of a role is in the sense that nothing in how we do mathematics makes contact with mathematical objects in a way that we can understand as forcing mathematical terms to refer to those (specific) objects, and not to some other set of objects.

3 How does the nominalist handle the problem of the truth of mathematical statements?

My job in this paper is to defend nominalism. So I won’t consider how the Platonist might try to respond to the challenge to the position that I raised in section 2. I’m going to instead take that challenge as motivating nominalism, and take up the first set of challenges to nominalism that I described in section 1. I’ll defend nominalism by describing some of the responses available to nominalists.

Let’s start with the first problem I raised for the nominalist. This is the problem that mathematical statements seem to be true, and false, and not merely “true-in-mathematics” and “false-in-mathematics,” or something like that. 2 + 2 = 4, for example, is simply true, and 2 + 2 = 5 is simply false. How is this possible if numbers don’t exist?

Thinking about other subject areas quickly dispels this concern because a little thought shows that there are lots of statements that are true (or false) but are nevertheless about things that we take not to exist. Here is a handful of examples (it will be easy for the reader to think of lots more).

· Hamlet is depicted as the Prince of Denmark in Shakespeare’s play Hamlet.

· Hamlet is depicted as the Prince of Latvia in Shakespeare’s play Hamlet.

· James Bond is depicted as a spy for the United Kingdom in Ian Fleming’s novels.

· James Bond is depicted as a spy for the Soviet Union in Ian Fleming’s novels.

Notice that despite the nonexistence of Hamlet, the first statement is true of Hamlet, although the second isn’t. Similarly, despite the nonexistence of James Bond, the first statement about James Bond is true, and the second isn’t. Indeed, it’s clear that we often speak about nonexistent beings, and say all sorts of true (and false) things about them. Here are a few more examples.

· There are as many prominent Greek gods as there are Greek goddesses.

· Most of the people I dream of at night are far more interesting than real people.

· Scientific history would have been quite different if phlogiston had existed.

In section 1, I mentioned the possibility of truth-in-fiction, that a statement can be true in a fiction although it isn’t genuinely true. Notice that none of these examples are ones of truth-in-fiction, or anything similar. To say that Hamlet is the Prince of Denmark is to say something that—strictly speaking—is false, although it can be taken to be true-in-the-play-Hamlet. But this isn’t the case with my first example above, “Hamlet is depicted as the Prince of Denmark in Shakespeare’s play Hamlet.” This sentence is simply true: it’s making a claim about a fictional character, Hamlet, and the role of that character in one of Shakespeare’s plays. It’s not instead a fictional statement that’s part of the play itself, as “Hamlet is the Prince of Denmark,” must be taken. It’s easy to see that all of my other examples are the same, although they are also about objects that don’t exist. For these sentences, too, are simply true (or false), and not true-in-a-fiction (or false-in-a-fiction), or true-of-a-mythology, or true-of-a-dream, or anything like that.

That so many statements, that are about things that don’t exist, are true (and false) raises interesting and important questions in philosophy of language—specifically in the science of semantics. One simple result that follows is this: our ubiquitous practice of talking about the nonexistent shows that the little story I gave in section 1 about how truth and falsity work is wrong. That a statement is true because it describes accurately what it’s about, and otherwise a statement is false, are facts that only hold of those statements that are about things that exist. Truth and falsity work in some other way entirely when it comes to things that don’t exist. And mathematical statements are true and false in the same way that other statements about things that don’t exist are true and false. This is what the nominalist will claim.

I mentioned in section 1 that mathematical statements being true is important because so much scientific work involves the inference of results from scientific theories, and inference has to be truth-preserving. If the mathematical statements, that are used in empirical science, are true—as the nominalist claims—then this important fact about scientific inference doesn’t put the Platonist in a better position than the nominalist, as it appeared to in section 1. The Platonist may nevertheless claim that the nominalist still has a problem with the application of mathematics. This is because it seems odd that statements that are about things that don’t exist have any useful empirical applications at all. Here is a way of posing the problem. If a statement is about something then its usefulness in applications makes sense to begin with. It’s about something, and it describes that something accurately. That’s what makes its application to that something valuable. But if a statement is only about what doesn’t exist, it’s hard to see how it could be valuably applied to anything. I take up this challenge to nominalism in the next two sections.

4 An example of a Platonist explanation of the value of applied mathematics

What’s being asked for is a kind of explanation. We have a subject area, a particular branch of mathematics, say, and the question is why that branch of mathematics is valuable for certain empirical applications. By using the phrase “for certain empirical applications,” I’m signaling that I don’t intend to give a global explanation for the value of all applied mathematics, one that explains all at once why mathematics is empirically valuable. I don’t intend to do this because I don’t think a global explanation for all of successful applied mathematics is possible—or even reasonable to expect. Explanations for why particular branches of mathematics are valuable for specific applications turn on details about the mathematics in question and on details about the specific applications. But the way the challenge to nominalism was posed at the end of the last section makes it hard to see how any branch of mathematics can be valuable for any specific application if there are no mathematical objects. So presenting how this works for a specific case will be valuable. Having said this, I should add that I do intend to offer a kind of explanation that is generalizable to more cases than the one I apply it to.

My strategy is to look again at the case of applied Euclidean geometry, but I’ll first consider how a Platonist explanation for its value goes. Then, in the following section, I’ll investigate whether there is anything in that explanation that requires mathematical objects to actually exist, as the Platonist presumes. To anticipate, I’m going to show that in fact there is nothing about Euclidean objects existing that the Platonist explanation for the value of applied Euclidean geometry turns on. So the nominalist can comfortably borrow the Platonist’s explanation for the value of applied Euclidean geometry.

So let’s consider Euclidean geometry. The Platonist, recall, will claim that points, lines, and triangles exist, and that this is true of all the items referred to by terms in the language of Euclidean geometry. These things aren’t, however, the diagrammatic items that we draw to prove various properties of points, lines, triangles, and so on. Instead, the real points, lines, triangles, and so on, exist somewhere else. Outside of space and time, say. That’s one view, originating in Plato perhaps, but still popular today. There is a second option, however, one that’s common among philosophers and certain philosophizing physicists. This is to say that Euclidean objects (or, rather, their relativistic/quantum-mechanical/string-theoretical/etc. generalizations) are located in one or another spacetime-like structure. Spacetimes, for example, are taken by many philosophers and physicists to be composed of points. And everything else that can be composed out of points—triangles, of one or another sort, for example—therefore exists in the various spacetimes as well.

Even if a spacetime itself is taken to be composed of some sort of physical item—quantum foam for example—it still turns out that mathematical objects end up appearing as the physically basic units on views like this. For the strings of string theory, for example, live in a (many-dimensional) space, and are themselves composed of coordinate items that are points, at least as far as mathematical purposes are concerned.

Let’s return to the simpler case of Euclidean geometry. On any Platonist view of the location of the mathematical objects of Euclidean geometry, it’s still true that the actual objects that Euclidean geometry is applied to, rugs for example, don’t have the properties that mathematical proof shows Euclidean objects to have. How then is Euclidean geometry nevertheless empirically valuable?

The answer, of course, is that although the actual shapes that Euclidean geometry is applied to aren’t themselves Euclidean shapes—they aren’t (for example) bounded by curves of one dimension—they nevertheless approximate Euclidean objects in a way that enables the application of Euclidean geometry to them to be successful. A square rug, for example, isn’t exactly square. At best it’s squarish because its borders are ragged and fuzzy in various ways. Despite this, the deviations from the exactly square can be (roughly) measured, and correspondingly, the deviations of the actual area of the rug from the idealized result Euclidean geometry predicts can be measured as well.

It’s worth noting that this approximation story has wide—although probably not universal—application. The reason that Newtonian physics is so valuable, despite its strict falsity, turns on the fact that the slower objects move the closer their physical properties approximate to the predictions of Newtonian physics. So this approximation story goes well beyond providing explanations for applied mathematics.

5 How the nominalist can borrow the Platonist’s story about the usefulness of applied mathematics

Here is what the nominalist is required to do to borrow the Platonist’s story about the usefulness of Euclidean geometry. The value of approximation has to be separated from the notion that what’s going on is that the properties of empirical objects (for example, the properties of a rug) approximate the idealized properties of mathematical objects (for example, a square on a plane).

How is this to be managed? Consider a triangle that you’ve drawn on a piece of paper. What it has to mean (according to the nominalist) that the triangle that you’ve drawn approximates a Euclidean triangle can’t be that it resembles a Euclidean triangle. This is because something that’s real can’t resemble something that doesn’t exist. What could that even mean? Things that don’t exist don’t resemble anything because they aren’t anything.

What it must mean can be illustrated instead this way: The narrower and straighter you draw actual lines, and the more carefully you measure the interior angles of the actual lines you draw, the closer the result will be to one hundred and eighty degrees. This is true about everything you can prove in Euclidean geometry about Euclidean figures: there are ways of drawing real figures so that the results of doing so will more and more closely approximate results from Euclidean geometry. That is, we explicitly shift the explanation of the value of Euclidean geometry, that real objects approximate in their properties Euclidean objects, to one in terms of the theorems of Euclidean geometry. That is, what we can empirically demonstrate as true of empirical objects can be made to approximate the theorems of Euclidean geometry. We’ve shifted the grounds of the explanation from ontology to theoremhood. Notice that crucial to this explanation is that the theorems of Euclidean geometry are true. But it’s already been demonstrated that the nominalist has access to the true theorems of Euclidean geometry without being forced to take the terms in those theorems to refer to actual objects.

So, as just noted, this explanation for why Euclidean geometry is empirically valuable avoids treating Euclidean objects as ones that exist (in one or another weird way)—in particular, this explanation avoids talking about nonexistent entities resembling real things. Plato, by contrast, hypothesized that circles on Earth resembled “perfect” circles—the ones studied in Euclidean geometry. This would also explain why Euclidean geometry is empirically valuable, as I indicated in the last section. If things in domain B resemble things in domain A, then studying the things in domain A will shed light about the things in domain B. But an explanation like that requires the objects in both domains to exist. I’ve given a different explanation: it doesn’t focus on the objects of Euclidean geometry at all—which is a good thing, the nominalist thinks, since those objects don’t exist. It focuses instead on the truths of Euclidean geometry. It focuses on the sentences of Euclidean geometry and not on what the sentences are supposedly about. And it explains why the truths about the objects we draw on (relatively flat) pieces of paper or in sand, or whatever, are going to approximate the truths of Euclidean geometry under certain circumstances.

Notice what has been done here. We’re starting, say, with a drawing of a triangle on a piece of paper that it’s empirically profitable to analyze the properties of in terms of Euclidean geometry. This triangle, however, isn’t a Euclidean triangle precisely because it’s a drawing on a piece of paper, and so it doesn’t have the standard properties that Euclidean triangles are taken to have. The question is: why is it valuable to treat the drawing, when empirically applying Euclidean geometry to the triangle, as if it is a Euclidean triangle? The Platonic answer isn’t that the drawing of a triangle is a Euclidean triangle because everyone can agree that it isn’t. The Platonic answer is that Euclidean triangles exist, and the drawing of a triangle sufficiently resembles Euclidean triangles so that treating the drawing as if it’s a Euclidean triangle is valuable. My alternative answer is this: Consider a truth of Euclidean geometry: The sum of the interior angles of a triangle sums to one hundred and eighty degrees. We can apply this theorem to a drawing, by treating its angles as if they are Euclidean angles (bounded by straight lines), and the result will more closely approximate the answer: one hundred and eighty degrees, the sharper and straighter we draw the boundaries of the triangle.

So far, in this paper, I’ve focused on what might be called the real obstacles to adopting nominalism. These are issues about the nature of applied mathematics, and the nature of mathematical truth. If the nominalist can’t make sense of mathematical truth or of the success of applied mathematics, on a view of mathematical objects not existing, then that’s a genuine deficit for nominalism—perhaps a fatality for the position. I’m going to turn now to what might be described as a softer consideration against nominalism. This is that the way that we think about mathematics seems to require that we think about mathematical objects of some sort—numbers, functions, spaces, and so on. The question, therefore, is whether the psychological need to think about mathematics in terms of kinds of objects is any reason to think that nominalism must be wrong. The question is whether the fact, if it is a fact, that we have to think through our mathematics in terms of mathematical objects requires that mathematical objects exist.

6 How we think about nonexistent objects

As before, it pays off to initially discuss a nonmathematical area of discourse where we think about objects that we antecedently recognize not to exist. When we are fantasizing, making up stories, dreaming, or otherwise thinking up or thinking about imaginary beings, for example, our psychological methods of thinking about real objects is the only mental tool we have to manage this.

The mental tool we use is an interesting capacity for “object-directed thinking,” one that we can detach the operations of from the real things that we normally think about, and that we can continue to use even though we recognize that there are no objects involved any longer. And, along with this, we can also borrow our ordinary ways of talking “about” objects when fantasizing (for example). That’s why we can successfully speak of “thinking about” hobbits or elves or dragons, even though we know—or most of us do, anyway—that there are no hobbits or elves or dragons.

Nevertheless, we experience our talking about (or thinking about) hobbits as different from our thinking about dragons—even though there is nothing we are thinking about (or talking about) in both cases. This is reasonably described as a cognitive illusion, in particular, as an aboutness illusion. When thinking about something we recognize not to exist, such as a fantasy character, despite our recognition that it doesn’t exist, we nevertheless experience that thinking as thinking about a particular thing.

What’s really going on when we are thinking about dragons and hobbits? Well, there are different words, and different concepts that we use. Strictly speaking (since there are no hobbits or dragons), there are only thought vehicles and language vehicles involved, and nothing else. I just spoke of aboutness illusions. Aboutness illusions misfocus our minds. Consider the two sentences.

(v)

Pegasus was believed to be a flying horse.

(vi)

Hercules was believed to be a flying horse.

Instead of our feeling the difference between talking about Pegasus, as we do when we utter (v), and talking about Hercules, as we do when we utter (vi), as a matter of differences in the truth values of sentences in which the words “Pegasus” and “Hercules” appear, we instead feel the difference as due to there being different objects we are talking about that have different properties. This feeling occurs simultaneously with the awareness that there are no objects such as Pegasus and Hercules—that they are complete fictions.

What makes (v) and (vi) differ in truth value, of course, isn’t that they are about different objects with different properties. What makes them differ in truth value is that they allude to a mythological tradition in which certain beliefs (about Hercules and Pegasus, in particular) circulated. Objects don’t come into it at all; thinking they do—even for a moment—is to fall prey to aboutness illusions.

Unfortunately, telling ourselves (or others) all this stuff about aboutness illusions won’t eliminate our experience of these illusions that arises whenever we “transact” with the nonexistent. These illusions are like optical illusions: No matter how much we stare and stare at an optical illusion, we can’t make it go away just by saying to ourselves (for example): “I know those lines are the same length even though they appear not to be.” Here too, we’ll always have the overwhelming cognitive impulse to experience our thinking “about” Pegasus and our thinking “about” Hercules as kinds of thinking about objects—and objects that are different. Even when we know it isn’t true. There is no way to escape these aboutness illusions. Not for us. Not for us humans.

It seems that the above yields a promissory note. This is that I’ve described us as thinking about what doesn’t exist by psychologically borrowing the object-directed mental tool that we use when we think about real objects. But is there any other way to manage this? I’ve already suggested a way that it could be done, although not by us. Consider the following two statements.

· Hobbits are depicted in the Lord of the Rings as being very short.

· Wizards are depicted in the Lord of the Rings as being very short.

These statements have different truth values, the first is true and the second is false. We humans naturally think about these two sorts of sentences as respectively about two kinds of objects, hobbits and wizards. The truth of the first sentence is due to the fact that hobbits are indeed depicted as quite short in the Lord of the Rings. The falsity of the second sentence is due to the fact that wizards, instead, are depicted as quite tall in the Lord of the Rings. So we naturally think of one sort of object—hobbits—being depicted as short, whereas wizards—another kind of object—are depicted as being tall. But we could have been a different kind of creature, one that never thinks about objects when it recognizes that terms, like “hobbit” and “wizard,” don’t refer to anything. Instead, this creature would instead automatically think about the sentences of a novel, and that certain sentences appear in that novel and not other sentences. If such a creature understood “depiction” in a novel, for example, simply as the appearance of certain sentences in that novel, then that creature might think that “Hobbits are depicted in the Lord of the Rings as being very short,” means only that certain sentences appear in the novel, such as “The average hobbit is about four feet.” That creature might find it peculiar to think of hobbits as any sort of object at all, as opposed to there just being a term “hobbit” that appears in sentences with specific truth values.

I should make this point as well. Novelists, when writing novels, and the rest of us, when reading novels, almost always read through the sentences of the novel to what the sentences depict. That is, we always read novels by employing the object-directed thinking that arises because the sentences of the novel make us think of characters and their properties—and often in great detail. But this should not suggest to anyone that fictional characters therefore exist. Rather, it should be seen the way I’ve presented it in this section. We engage in a complex psychological process of thinking of objects that we simultaneously recognize don’t exist. It isn’t that because we must think of the characters of a novel as having specific properties (and not other properties) that we should therefore think that these characters must have some sort of existence. This would exactly be the sort of peculiar mistake that, usually, only philosophers make.

I conclude this paper, in the next section, by drawing on behalf of the nominalist the same conclusion about the mathematical objects that mathematicians (and others) think of when doing mathematics.

7 Object-directed thinking in mathematics

Broadly speaking (and with a bit of inaccuracy), mathematical thinking comes in two forms, what we might describe as geometric thinking and as algebraic thinking. By algebraic thinking, I mean a kind of “object-blind” manipulation that we often engage in when maneuvering formulas into a suitable form to extract certain results from them. Although we do think of numerals as names for numbers, it’s often the case that complex numerical manipulations are similarly “object-blind,” especially when “short-cut” algorithms, for example, for multiplication and division, are employed. Manipulating code according to rules is yet a third similar sort of experience. In all these cases, we aren’t thinking of mathematical objects that we conceive of in a certain way but instead, and more directly, we are thinking of the various kinds of formulas we are writing down on a piece of paper (or manipulating on a computer screen).

The second kind of mathematical thinking—broadly speaking—is geometric. In this case, we are thinking of the mathematical language we are manipulating as diagrammatically depicting the mathematical objects we are attempting to prove results about. This is the case, of course, with Euclidean geometry; but this kind of thinking shows up with conceptual proofs about, say, manifolds, or the diagram-chasing that takes place in topology or knot theory.

There are mixed cases of course. And, it might even be argued that attention to the “form” of a discourse, as we do with the mechanical manipulation of mathematical formulas (according to rules) is reflected somewhat in fiction writing, especially with those authors who are concerned with the formal properties of their sentences—such as rhythm, for example.

Informal-rigorous mathematical proof almost always has conceptual elements that induce object-directed thinking. I mean by this that there is always an aspect to it where it is most natural for us to be thinking of a kind of object that we are trying to prove has certain properties or other. The most familiar case of this, of course, is number theory. But it’s important that informal-rigorous mathematical proofs can in principle be replaced by formal derivations, where each step follows mechanically from previous steps, and all the assumptions—implicit and explicit—employed in such a proof can be axiomatized. This shows, as I indicated in section 2, that objects play no essential role in mathematical practice. The fact that we think about mathematical objects the way we do (and when we do) is an interesting psychological fact that’s very important. It’s just not very important for ontology.

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