Platonism vs Formalism in Mathematics

Platonism in mathematics is the belief that numbers like planets are discovered and not invented. “Just as electrons and planets exist independently of us, so do numbers and sets.” (Linnebo, 2018). A more formal definition of Platonism given by the Internet Encyclopedia of Philosophy is “Mathematical Platonism is any metaphysical account of mathematics that implies mathematical entities exist, that they are abstract, and that they are independent of all our rational activities.” (Cole). I believe that Platonism is not a plausible view to take because I believe numbers are invented and not discovered therefore, making me a formalist.

Platonism was inspired by Plato and his belief and practices. According to Hilary Armstrong “Platonism, any philosophy that derives its ultimate inspiration from Plato. Though there was in antiquity a tradition about Plato’s “unwritten doctrines,” Platonism then and later was based primarily on a reading of the dialogues.” (Armstrong, 2017). Plato was a firm believer in his theory and believed that people did discover different components of mathematics but ultimately considered them to just be a medium to get to the knowledge that already existed. “It exists in an abstract world, a world of ideas, irrespective of the people who discover it and of the phenomena it describes. The way to discover it is by using the power of logic and logical arguments.” (Artstein, 2014).

Formalism on the other hand, originates from ancient Greek metaphysics and is according to Weir, “The guiding idea behind formalism is that mathematics is not a body of propositions representing an abstract sector of reality but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess.” (Weir, 2015). I believe that numbers are a language and we, humans created different kinds of languages. In a sense numbers are just another way for us to communicate such as; when someone is programming a computer, we use numbers as a way to communicate with that said computer. Which then in turn produces a bunch of numbers that translates into the language that we see onto our computers. Without someone to define those numbers they wouldn’t exist.

Other great philosophers such as Aristotle believed in formalism as well. “He claimed that mathematics did not exist as an independent entity. Mathematics is simply the outcome of logical operations based on syllogisms, and the search starts with axioms. Mathematics is created, not discovered.” (Artstein, 2014). However, sometimes formalism is a hard theory to grasp. Which is why Platonism is more popular theory.

There are many different views one could take such as; formalism, mathematicism, platonism and logicism. However, the most common view amongst people is Platonism. In an article written by Cole he states, “Mathematical Platonism enjoys widespread support and is frequently considered the default metaphysical position with respect to mathematics. This is unsurprising given its extremely natural interpretation of mathematical practice.” (Cole). Also, Platonism gives us an explains that everything above that most people would consider supernatural would still exist even without us here.

On the other hand, mathematics is so broad and has many different aspects, it couldn’t possibly be reduced to just one viewpoint or methodology to help us understand mathematics place in our lives. With it being such a narrow mindset to have, people often go between being a Platonists and a formalist. “Thus, so the story goes, it is quite normal for a mathematician, when asked if he or she is a Platonist or formalist, to give one answer on weekdays and the opposite answer on the weekend.” (Artstein, 2014). Another disadvantage of Platonism is that there is no evidence to support the theory. We have no way of knowing or proving that the numbers would exist without us being here. There is not a way for us to disappear and see if they would still exist or a way to go back in time to before we were here to see if they indeed still exist.

In conclusion, Platonism is the belief that numbers are discovered and I do not believe that it is a plausible view to take. To this day no one has truly determined the methodology of mathematics place in our lives. “The debate initiated by Plato and Aristotle on the essence of mathematics has continued for two thousand four hundred years and continues to this day.” (Artstein, 2014). However, while this is a topic that is heavily discussed in mathematics it does not have any direct effect on mathematics itself.

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Works Cited

A. Hilary Armstrong and Henry J. Blumenthal, “Platonism”, Encyclopedia Britannica Encyclopedia Britannica, Inc. June 15,2017 URL =

Artstein, Zvi, “Mathematics and the Real World: Role of Evolution in the Making of Mathematics”, 2014 URL =

Linnebo, Øystein, “Platonism in the Philosophy of Mathematics”, The Stanford Encyclopedia of Philosophy (Spring 2018 Edition), Edward N. Zalta (ed.), URL = .

Weir, Alan, “Formalism in the Philosophy of Mathematics”, The Stanford Encyclopedia of Philosophy (Spring 2015 Edition), Edward N. Zalta (ed.), URL = .