The relationship between mathematics and philosophy is an old one, dating back to the dawn of mathematics in ancient civilizations such as Egypt and Mesopotamia. Many mathematicians such as Archimedes were also philosophers or vice versa, such as Thomas Hobbes who made great advances in mathematics but never received a degree in mathematics [1]. For example, Newton and Leibniz, who made the fundamental breakthroughs in calculus which gave mathematics a more integral place in physics and technology.
To this day mathematics and philosophy remain inseparable. For example, one of the great unanswered questions in mathematics is whether mathematics is created or discovered [2]. On one hand, mathematicians such as G. H. Hardy claimed that mathematics was purely deductive [3], while other philosophers such as Imre Lakatos claim mathematics is creative [4]. The relationship between mathematics and philosophy can be seen outside academic circles too.
The public still hold mathematics up as an almost sacred topic due to its perceived infallibility, for instance Bill Gate’s quote: “Be suspicious of all claims for ‘the theory of everything. ‘” [5]. On the other hand mathematics is also heavily criticized by philosophers of mathematics such as Michael Dummett, John Burgess and Paul Benacerraf. The main criticism being that mathematics are just empty meaningless symbols [6]. However mathematics can also be defended from philosophy in cases.
For example when Giaquinto argues that mathematics are useful to physics [7], or when mathematical structures are argued to have an independent existence outside human knowledge in Platonism [8]. To conclude, mathematics is a fundamental topic in philosophy due to the relationship between mathematics and philosophy’s fundamental questions, but it is arguable whether mathematics is created or discovered. It seems however that mathematics will continue to remain inseparable from philosophy even further into the future.
Mathematics and philosophy are often described as separate fields of study that neither influence one another nor interact. However, mathematics has influenced the methodologies used by philosophers throughout history, while mathematics itself is rooted in philosophical ideas. The relationship of mathematics and philosophy can be seen through mathematics’s philosophical influences, mathematics’s applications to philosophy, and mathematics’s own concepts of proof. Math has been inspired by philosophy for centuries.
Many mathematicians have had a grounding in philosophy that helped drive their mathematical research forward. For example, Gottfried Leibniz was a mathematician who was well-versed in many branches of philosophy including logic—a branch which he created through his work regarding infinitesimal calculus. In fact, Leibniz stated that mathematics is the language of philosophy. In mathematics, a proof is a convincing demonstration that some mathematical statement is true.
In mathematics, proofs are required to be extremely specific and very clear, whereas in philosophy arguments are often made through inference and rhetoric rather than mathematics. Proofs have been used as similar forms of philosophical argument throughout history. For example, Socrates would have his students provide definitions for words before proving them through a logical examination of their statements—a form of the Socratic method still used by philosophers today.
Plato’s theory that mathematics could lead to eternal truths has been compared to modern-day Platonism in mathematics—the belief that abstract mathematics concepts exist outside human minds as permanent symbols waiting to be discovered. Aristotle divided knowledge into mathematics and philosophy, mathematics being knowledge of mathematics itself, philosophy being knowledge of all other things. This is known as the “two spheres” theory; mathematics would only encompass geometry whereas everything else would be part of philosophy.
However, Aristotle’s student Alexander considered mathematics to be a subtype of philosophy since mathematics was used in the sciences (such as physics), mathematics was based off philosophical principles (like logic), and mathematics had applications in virtually all fields—including politics. Early mathematicians required philosophy because mathematics did not yet exist on its own. Mathematics developed into an independent subject when it became possible to think through mathematics without first having to find philosophical justifications for it.
Philosophers now use mathematics throughout their arguments they imply that some idea is correct, but they don’t actually work through the mathematics unless they need to. Mathematics has influenced philosophy throughout history, and mathematics is currently a wide-reaching field that students study after taking a number of prerequisite philosophy courses. Mathematics and philosophy are intimately connected through mathematics’s philosophical influences, mathematics’s applications to philosophy, and mathematics’s own concepts of proof.
Now imagine philosophy. After the first interpretation of mathematics, what would your first interpretation of philosophy be? Maybe philosophy becomes something scribbled on similar paper to mathematics or maybe they are inseparable. Once again, these two entities are seemingly separate but still remain connected in some way that we may never fathom. However, there seems to be more than just that connection between mathematics and philosophy. According to Dr.
Zbigniew Pelczar’s article “Philosophy versus Mathematics” (1), he states that mathematics itself is based off of philosophy. Since mathematics is a branch of philosophy, the relationship between mathematics and philosophy can also be defined in two ways: mathematics as a part of philosophy or mathematics having its own philosophical implications. Imagine mathematics in the same way that Pelczar describes it; mathematics is based off philosophy with each other being interrelated in one way or another.
For example, Plato believed that universal forms enable us to recognize objects (2). These thoughts are reflected when he states that mathematics then becomes the only discipline besides dialectic that has access to true reality (1). This means that mathematics acts as an entity of knowledge unlike any other because nothing else possesses these capabilities except mathematics. On the other hand, there are times where what we find mathematics to be today may even be different from what mathematics was or will become in the future.
Take for instance, mathematics before the discovery of irrational numbers, mathematics after the discovery of irrational numbers, and mathematics after their acceptance into the mathematical community. Before they were accepted mathematics had an “axiom” that states that all equations must have solvable numerical values (1). However, when irrational numbers were discovered this axiom was broken forcing mathematics to branch off in a new direction without this rule. Once these newly found ideas are accepted into mathematics it leads mathematics down a path that is completely different than what mathematics used to be like in Plato’s time.