Section 1.1: Platonism and the Nature of Numbers

Platonism is rooted in the teachings of the ancient Greek philosopher Plato. Adherents of Platonism argue that numbers are indeed real entities. They contend that the concept of numbers is as tangible as any physical object, such as a book or a ball, but they categorize numbers as abstract entities.

According to Platonists, numbers exist outside the confines of time and space, which means they do not exert direct causal influence on our physical reality. However, they assert that a mathematically valid concept cannot be true unless it exists in some form—albeit abstractly.

For instance, if I say you possess three glasses, the term ‘three’ cannot hold meaning if the concept of that number does not exist in some capacity; thus, it exists in an abstract realm.

Ultimately, Platonists maintain that numbers are real, yet they inhabit a dimension beyond time and space. While this reasoning is beneficial in many respects, it also presents certain challenges.

Subsection 1.1.1: Challenges with Mathematical Platonism

What does it mean for something to exist outside of time and space? Depending on interpretation, this can lead to various understandings. If numbers are indeed abstract entities beyond time and space, how do mathematicians access these entities with precision and consistency? What is the bridge that connects our tangible world to this abstract numerical realm?

Platonists often struggle to provide satisfactory answers to these questions. This challenge introduces us to a contrasting school of thought.

Section 1.2: Nominalism — A Real-World Connection

Nominalists, like Platonists, assert that numbers exist, but they diverge in their definition of what constitutes a mathematical number. Nominalists view numbers as assertions about objects within our tangible reality. For example, the phrase “You have three glasses” uses the number ‘three’ to refer explicitly to those glasses. Without a real-world connection, the meaning of ‘three’ diminishes, according to Nominalist philosophy.

This reasoning aligns with how most people, including children, first learn about numbers. Personally, I was drawn to using numbers to count candies I could purchase. As a child, I had no intuitive sense of numbers as abstract concepts.

The main advantage of Nominalism lies in its ability to create an intuitive link between numbers and real-world objects—an aspect that Platonism appears to overlook. However, Nominalism is not without its complications.

Section 1.3: Fictionalism — The Denial of Existence

Fictionalists present a distinct perspective by asserting that numbers do not exist at all. They argue that the entire framework of mathematical discourse is an illusion and fundamentally false. However, mathematicians and scientists regularly utilize mathematical language to create real-world applications, such as computers and buildings. How can the discourse be deemed false if it yields functional outcomes?

Fictionalists contend that the practical applications merely demonstrate the utility of mathematical language rather than its truth. Among the three philosophical perspectives, this view is the most radical. Fictionalists would classify the statement “7 is prime” as false, similar to declaring “the tooth fairy is prime” as untrue. They maintain that neither ‘7’ nor the tooth fairy exist, and in essence, they deny the existence of abstract entities.

This extreme viewpoint inevitably leads to considerable debate, but I will refrain from delving into that discussion here. Instead, let us summarize the key points we have covered.