Philosophy and Mathematics Are Closely Related

Pythagoras, a renowned figure in ancient Greece, was a multifaceted intellectual known for his contributions to both philosophy and mathematics. As a philosopher, he founded the Pythagorean school, emphasizing spiritual and ethical teachings. In mathematics, he is best known for the Pythagorean theorem, fundamental in geometry. His dual legacy as a philosopher and mathematician has left an indelible mark on Western thought, influencing not only mathematical principles but also broader philosophical inquiries into the nature of reality and the relationships between numbers and the cosmos.

Often times in society, philosophy and mathematics are seen as two distinctly separate fields of study in which there is little relation (you know cause one is a liberal arts, the other is not type-of-argument.).

But, this is unequivocally false.

Outside the Platonic Academy (the school in which Plato lectured), the entrance read:

“May no ignorant of Geometry enter here.”

Now why would Plato, a philosopher, care so much about mathematics?

Well because mathematics is essential to the development of philosophy, and vice versa.

Let me offer a case,

1. Plato’s Theory of Forms: a dualistic reality comprising the imperfect physical world and the non-material world of Forms. In this abstract realm, universal and eternal Forms exist in their purest state, serving as the true reality behind the imperfect copies found in the physical world. Plato argued that physical objects participate in or imitate these Forms, and true knowledge lies in understanding these ideal entities. The Allegory of the Cave vividly illustrates this concept, portraying individuals as prisoners perceiving only shadows on a cave wall (representing the physical world) while remaining oblivious to the true reality outside (representing the world of Forms). Knowledge, for Plato, involves transcending the illusory physical realm to apprehend the unchanging and perfect Forms — “the non-physical, timeless, absolute, and unchangeable essences of all things, of which objects and matter in the physical world are merely imitations.”

Many individuals during their study of philosophy are aware of Plato’s idea of the imperfect and perfect world, but some are unaware of it’s connection to mathematics.

In searching for the objective truths of the world, one finds that objects (tangible entities) are often the source of truth (for example, the three-dimensional space is understood by a cube). However, don’t must of us agree that 1 + 1 = 2?

In a world that has agreed on the Peano axioms, 1 + 1 naturally equals 2. But wait… according to one of the most prevalent theories of truth, correspondence theory, numbers are considered not to exist as objects (you can’t use any of your senses to verify the existence of numbers). Nonetheless, Plato and many other followers of the correspondence theory believed in the truth of mathematics.

Specifically, Plato found numbers to be a Form, a non-physical, timeless, absolute, and unchangeable abstraction that exists in the perfect world.

Clarification: arguing the existence or non-existence of the Forms is not the point. Instead, I am attempting to show individuals the NATURAL link between philosophy and mathematics.

In contrast, Aristotle disagreed with his teacher (Plato) and said that mathematics existed in the physical world as mathematical concepts are realized by individuals: imagine you buy a bag of apples and count the apples in the bag, you are perceiving the amount of apples in the physical world; thus, mathematics is present in a non-abstract reality.

Despite the differing idea of truth, mathematics is developing the study of philosophy in both scenarios.

Not only has mathematics progressed philosophy, but philosophy has also progressed mathematics: the development of Peano axioms would not be necessary without philosophers questioning the reality of numbers.

By showcasing the interconnectedness of philosophy with other academic disciplines, I hope to inspire others to fully embrace the world of philosophy.