One of the prominent positions in philosophy of mathematics is mathematical fictionalism. As the name suggests, it uses our view of fictions to explain how mathematics helps us to understand and describe the world as it is seen through the lens of science. For the uninitiated reader, this may come as a surprise. After all, we relegate fictions to the subjective realm of the imagination. As we typically see it, games of make-believe do not come into play in the supposedly objective realm of science. Yet, a closer look at our most successful scientific theories shows otherwise.
To illustrate the point above, consider, for example, the presence of idealizations and theoretical entities in our best scientific theories. Idealizations can be described as intentional simplifications of real phenomena so that they are easier to analyse or understand. A common example of an idealization in physics is treating fluids as if they flow smoothly without gaps. Even though we know real fluids are made of separate molecules, we use this simplified idea because it helps us to understand how fluids move in the real world. In doing so, we can say that we engage in a game of make-believe when we adopt the continuum model in physics. The same act of engaging in a game of pretence can also be seen when we presume the existence of theoretical entities in particle physics (e.g. axions, tachyons). We pretend that these entities exist even if we do not have evidence of their actual existence. In the case of idealizations and theoretical entities then, we can see how our scientific theories require us to engage in games of make-believe so that we can understand our existing explanations of events in the world.
That we engage in a game of pretence when we use our scientific theories can also be seen in how we treat the mathematical objects in our best scientific theories. Consider, for example, how we treat the mathematical objects in Newton’s second law of motion. This law tells us that the force acting on an object is equivalent to the mass of an object multiplied by the acceleration of an object. This is represented by the equation F = ma. With its use of real numbers to represent the force, mass and acceleration of an object, this law is mathematical in its entirety. In this case then, when we use Newton’s second law of motion, we also engage in a game of pretence where we assume that mathematical objects exist since they are a part of a successful scientific theory in the form of Newtonian mechanics.
As we can see, the use of fictions in the form of idealizations, theoretical entities and mathematical objects is standard practice in our best scientific theories. Regardless of their fictional character, they are included in our best scientific theories due to their usefulness in explaining various facts about the world.
With these in mind, the goal of our discussion is to demonstrate how mathematical fictionalism shows that we engage in games of make-believe when we look at the role of mathematics in our best scientific theories. More specifically, we aim to show how Mary Leng’s version of mathematical fictionalism considers mathematics as fictions in our scientific theories.
Leng argues that science should be our guide when we determine what we should believe. She also claims that if we look at science, we do not have good reasons to believe in the existence of mathematical objects. For Leng, all that science tells us is that mathematical objects play an instrumental role in our best scientific theories. To understand this, it would be helpful to show how Leng makes use of Kendall Walton’s theory of make-believe.
‘... if we want to engage or play in a game of make-believe, we have to agree to imagine certain things as if they were true.’
Walton maintains that if we want to engage or play in a game of make-believe, we have to agree to imagine certain things as if they were true. What we imagine is guided by rules called principles of generation that help us treat real-world ideas as if they were true in a fictional world. To illustrate how a fictional world works, let us consider one of the examples in Walton’s text. In his example, Eric and Gregory engage in a game of make-believe where tree stumps are bears. Thus, in this case, if Eric and Gregory pass by a big tree stump then they can state that they came across a huge bear. In this scenario, the tree stumps act as props which generate a fictional truth that tree stumps are bears. In this case, props, the objects in the external world, are important because they give structure and constraint to Eric and Gregory’s game of make-believe.
Now that we have shown how props and rules of generation work in a game of make-believe, we can show how Leng uses them in her characterization of scientific theories. In her view, we need a way to explain how mathematics connects to the world so that we can treat scientific laws written in mathematics as part of a fictional story. One way to do this is through set theory that includes real-world objects (called urelements). In this story, facts about real objects help create what we imagine to be true. In other words, the way things are with real objects (i.e. the props) makes certain mathematical statements fictional. And by saying a mathematical statement is fictional in this make-believe story, we can suggest something about how the real-world props actually are.
To understand the role of props in our theories, it is helpful to differentiate between content- and prop-oriented make-believe. What differentiates the two is in their focus in their respective games. Whereas the former focuses on the internal fictional narrative, the latter focuses on the real-world objects used in the game. For instance, in reading a book, we may be engrossed in finding out what happened to the protagonist at the end of his duel with the windmill. In this case, our focus is on the content of the fictional narrative. On the other hand, we may claim that there is a dragon whenever we see a windmill. In this case, our focus is on the prop of the game.
With this being said, adopting Walton’s view about prop-oriented make-believe allows Leng to differentiate between the fictional as opposed to the literal content of a scientific theory. It also enables her to distinguish between the instrumental as opposed to the descriptive content of a scientific theory. Finally, it lets her argue that our best scientific theories work just fine without needing to assume that numbers actually exist.
Portraying mathematics in our scientific theories in this way allows a more accessible way of viewing mathematical objects in our theories. For instance, it does not require us to believe in the existence of mathematical objects given that they are non-spatial, non-temporal, causally impassive and inactive objects. Engaging with scientific theories, in this case, becomes similar to engaging with the games that we play which involve the use of fictions in order to help us explore, represent and reason about the real world without assuming that the fictions themselves are literally true.
