Evaluating Anti-Platonist Approaches to Mathematics

In this paper, I will first explain Platonism and its relation to mathematics and reconstruct arguments against it to show how mathematical objects ultimately cannot exist. I will then explore logicism and formalism in order to critically evaluate how they create truth for mathematical propositions and the problems that they have that could or should prevent mainstream philosophical adoption. Finally, I will explain Benacerraf’s structuralism and why I think it is the best Anti-Platonist explanation for the philosophical foundation of mathematics.

Platonism in mathematics is simply the idea that the mathematical entities that we refer to actually exist, and in virtue of them existing, they have actual properties that we can describe, and that is where mathematical propositions derive their truth value. This comes about from Plato’s ontology where he posited that there were two realms: the Changing and the Unchanging. The changing and impermanent world of material is the one that people live in, while the eternal, unchanging world of the Forms is not accessible to us directly. Plato thought that people learn or gain knowledge through the recollections of the Forms from a time when our souls were united with them. In this way, we can come to know truths about Forms by inquiring about the object without having to assume anything.

While the idea of mathematical objects existing in a separate realm certainly sounds nice and is very useful in particular for fixing a referent for mathematical entities, there are a number of issues with Plato’s Theory of Forms. It is none other than Aristotle who I believe offers the best criticism of the Forms. “Further, of the ways in which we prove that the Forms exist, none is convincing; for from some no inference necessarily follows, and from some it follows that there are Forms of things of which we think there are no Forms. […] Further, of the more accurate arguments, some lead to Ideas of relations, of which we say there is no independent class, and others involve the difficulty of the ‘third man’.” (Aristotle 78). The third man argument has to do with the interconnection of Forms and how there would be an infinite number of Forms because a man is a man because he partakes in the Form of Man. However, then a third form would be necessary to explain how man and the form of man are both man, and so on, to an infinite regress.

While Aristotle provides good arguments against the theory of forms, there are different approaches to mathematical Platonism that distance themselves from the Theory of Forms, but still hold that mathematical objects are real. I think Benacerraf in his paper, What Numbers Could Not Be, logically shows how numbers, the foundation of mathematics, could not be objects at all. “Ernie pointed to his theorem that for any two numbers, x and y, x is less than y if and only if x belongs to y and x is a proper subset of y. Since by common admission 3 is less than 17 it followed that 3 belonged to 17. Johnny, on the other hand, countered that Ernie’s “theorem” was mistaken, for given two numbers, x and y, x belongs to y if and only if y is the successor of x.” (Benacerraf 54). This debate results in the following set to be constructed by Ernie to represent 3: {Ø, [Ø], [Ø, [Ø]}, where Johnny’s 3 is represented by [[[Ø]]]. These theorems posited by Ernie and Johnny are incompatible because they both represent the same number, but with different sets.

Frege had a slightly different account of what set a number was, which had to do with an equivalence class, which is the class of all classes equivalent with a given class. Under this view, the number 3 would be the set of all sets with three members. This view initially has merit to it because the idea follows from the idea that number words are class predicates. Under any consistent set theory, i.e one that does not fall prey to Russell’s paradox, it does not seem like we can have a set of sets. Therefore, number words need not be class predicates, so then it is not necessarily true that 3 is the set of all triplets. After this, Benacerraf explores three different problems to support his claim that numbers could not be sets: identity, explication, and reduction. I think the most persuasive of the three problems that Benacerraf explores is about identity. The problem of identity results from trying to assert n = s, where n is a natural number and s is a set representation of n. He says that when making statements of identity, they should be of the same category C or otherwise the statement is unsemantical, because to compare the identity of two random entities makes no sense. In considering this, it would be unsemantical to compare anything with a number except another a number, unless we knew that they shared some category C,and there are conditions which can individuate objects of the same C. After analyzing the problems of identity, reduction, and explication, Benacerraf ultimately concludes that we cannot identify numbers with any objects at all, and lands on his view, which we can call structuralism, which will be explored later in this paper.

Logicism as a school of thought in the philosophy of mathematics starts with Frege, where he is primarily responding to a long tradition of thought stemming from philosophers like Kant where mathematics and mathematical truths were not analytic truths, nor were they truths about the actual world. Under Kant’s picture our geometric and algebraic concepts are things that get organized by our minds and structure our experience, and are not things that are fundamental to the structure of the world. “A typical crudity confronts me, when I find calculation described as “aggregative mechanical thought.” […] The present work will make it clear that even an inference like that from n to n + 1, which on the face of it is peculiar to mathematics, is based on the general laws of logic, and that there is no need of special laws for aggregative thought.” (Frege 316). Frege thinks that mathematics based on thought or sensation are characteristically indefinite, which stands opposite to the definiteness of mathematical thought. To Frege, math was one particular way, a definite way that was independent of human thought, which could be found via logical reasoning.

Mathematical truth for the Logicist comes from the idea that mathematics is ultimately reducible to logic, which is known a priori. “The present work will make it clear that even an inference like that from n to n + 1, which on the face of it is peculiar to mathematics, is based on the general laws of logic, and that there is no need of special laws for aggregative thought.” (Frege 316). Frege believed that there were two kinds of truth, one of which could be purely logically supported, which was the firmer proof of the two. Since math is based on logic, then it too could be known _a prior_i. However, the formalist, Hilbert in particular, did not believe that logic should or could be the sole foundation of mathematics. However, Hilbert thinks that the idea that all things are reducible to logic, otherwise known as the axiom of reducibility, is an assumption in order to demonstrate consistency. Which then shifts the question to if we can really assume reducibility. In order to have and demonstrate the consistency of axiomatic systems, we can’t reduce the proof of consistency to logic, but instead something else. I think that this is a very important consideration because if mathematics is to be based solely on logic, it must be quite certain that the underlying axioms are correct.

Formalism, as touched on a little bit previously, is primarily pushed forward by Hilbert, who believes that instead of the foundation of mathematics being logic, it was the symbols and the rules used to manipulate said symbols. “As a further precondition for using logical deduction and carrying out logical operations, something must be given in conception, viz., certain extralogical concrete objects which are intuited as directly experienced prior to all thinking. For logical deduction to be certain, we must be able to see every aspect of these objects, and their properties, differences, sequences, and contiguities must be given, together with the objects themselves, as something which cannot be reduced to something else and which requires no reduction.” (Hilbert 370). These intuitive concrete objects that don’t need logic are the symbols, sometimes referred to as signs. They can be thought of in terms of physical objects, if necessary, but they are not nor need not be actual physical objects. Additionally, these symbols don’t have any kind of meaning in themselves, but they can be operated on by a series of rules and be compared to each other. In this way, Hilbert believes that he can solve the problem of consistency where he thinks the Logicists could not.

The only logical condition for the Formalist that is of importance, of utmost importance in fact, is the consistency of the logical formal system. In an inconsistent system every logical syllogism that is syntactically valid can be proven, which ultimately makes the system useless. This, in turn, is why it is so important that our foundations are consistent, so that they don’t produce any contradictions. Unfortunately, Gödel showed with his Incompleteness Theorems that no formal system could prove consistency within the framework of its own system, which was a big hit to the Formalist project. This does not mean that Formalism is completely over with, the idea that mathematics is symbols and rules is still completely valid, it just isn’t provable. However I think that Formalism, particularly Game Formalism or Deductivism lean too heavily into the fact that mathematics is just a “game” and the rules of such a game are not necessarily true, but just serve a means to an end. This runs against normal intuitions about mathematics being defined: a priori and analytic, as the Logicists like Frege were trying to work out.

With Logicism and Formalism ultimately failing to provide a strong foundation for mathematics, we turn to Benacerraf. Benacerraf’s structuralism can be thought of as taking the natural numbers to be defined by a recursive progression with a less-than relation, for “To be the number 3 is no more and no less than to be preceded by 2, 1, and possibly 0, and to be followed by 4, 5, and so forth.” (Benacerraf 70). Any investigation into the properties of numbers are actually just the characterization of an abstract structure separate from the numbers themselves. Numbers, or the individual elements of the structure have no properties other than the relationship to other elements within the same structure. This then makes Number Theory about the properties of all structures that follow the order type of the numbers, according to Benacerraf. This is a good thing because it both shows how we can learn about numbers and other mathematical truths, but also shows how mathematics is analytic because the properties of this structure are unchanging and necessary. This maintains a strong Anti-Platonist stance while being able to explain the properties of numbers via abstract structures. A plus side of this view is that it does not require such strong logical foundations such as what the Logicists and Formalists were working with. Additionally, it doesn’t require any rewriting of old mathematical theory, like Intuitionism or Nominalism would have.

Logicism seems promising initially, since mathematics and logic seem so connected that it would make sense that mathematical truth is but a subset of logical truth. However, it becomes quickly clear that it presupposes reducibility of mathematics to logic, which seems like something that they would have to demonstrate first. Additionally, their project would be stumped by Gödel’s Incompleteness Theorems, just as the Formalists did. The Formalists seemed promising as well, since it did not have a purely logical background. Rather one that allowed for (almost) abstraction of mathematics from experience, which provides good insight into how people come to learn mathematical truth. However, just like the Logicists, they still do lean on logic, and it was shown by Gödel that consistency of a formal system could not be proved from within that system. Additionally, the ideas of some strains of Formalism view mathematics as ends-based rules to solve the game of mathematics, which feels contrary to most intuitions about mathematics being analytic. Which finally leads to Benacerraf’s structuralism which is a theory that doesn’t lean too much onto logic, and instead characterizes the foundations of mathematics as working with the properties of an abstract structure. It explains enough without having to rewrite any previous mathematical work, and allows for sets and other abstractions to be used as normal, just with the understanding that they are just representations and not actually objects.