The Origins of Mathematical Concepts


By Billy Pierre, Staff Writer

Mathematics is often considered the mother of all sciences, a belief shared by more than one. It is the basis of all scientific disciplines, and everything tends to rationalize using mathematical concepts. Galileo himself said that the “[universe] is written in a mathematical language,” a way to extol the virtues of this science. While everyone seems to recognize the merits of this discipline and how it is the backbone of every topic, the origin of its concept remains a controversial topic. Philosophers are unable to agree on if it derived from the experience of our senses or instead purely of our reason. This introduces us to two schools of thought: empiricism and rationalism.

Empiricism is a philosophical theory that “the origin of all knowledge is sense experience.” The origin of the word itself is “empeiria,” an ancient Greek word that translates to “experience.” This theory was developed in the 17th and 18th centuries, expounded by John Locke, George Berkeley, and David Hume. Empiricism was established as a critique of the nativist theory that believes that knowledge is not acquired, but rather innate. Thus, they argue that all our knowledge is derived from our sensory experiences.

They also agree that mathematical concepts are inspired by our perception. First, they prove that geometric shapes are objects of nature. For example, the idea of a circle is probably inspired by the shape of the Sun or the moon. They continue to say that the concept of motion is given by the physical experience of displacement. Furthermore, they argue the plurality of real-world objects should already evoke the idea of numbers, the same way that recognizing that these objects might be big or small should directly invoke the notion of space.

It is true that the arguments of the empiricists are compelling. However, we cannot overlook the fact that each experience is unique and relative to everyone. In other words, it must be admitted that the experience is subjective. Also, we know that sensations change over time. Can one build knowledge on such futile grounds? Considering Plato’s “Allegory of Cave,” we realize the sensations are often a source of illusion.

Moreover, in their thesis, the empiricists have neglected the role of the human mind in the design of mathematical concepts. That is in fact what allowed the emergence of a new group of philosophers with an argument opposing that of the empiricists. These are the rationalists.

For rationalists, like Goblot and Descartes, our knowledge is not derived from the senses, but from our reasoning. They say that mathematics does not need their objects to be real. The mathematician only needs his thoughts to build a science and whose objects have reality just in the mind. According to St-Augustin, numbers that the empiricists seized from the plurality of material objects are already inspired by the divine revelation. Numbers, as well as geometric figures, exist in the form of eternal essences in the divine mind since their correlation remains eternally the same. With respect to the idea of space, Immanuel Kant will tell us that it is a pure necessity of the mind, in other words, a priori form of sensitivity.

In their thesis, that shows a lot of similarities with the nativist (who support the theory that concepts, mental capacities, and mental structures are innate rather than acquired by learning) the rationalists presented the mathematical sciences as an abstract discipline and whose concepts have no connection with the reality outside. Mathematical thoughts can also be inspired by the elements of nature. Thus, the thesis of the rationalists sees itself as insufficient to explain the genesis of mathematical concepts.

In mathematics, as in any scientific discipline, it is impossible to rely only on our sensitive experiments without the judgments of reason, as it seems impractical to settle for pure imagination of our understanding in our quest for universal truth. Thus, we cannot overlook one of the theses to promote the other. We must instead recognize that mathematical concepts are derived from a synthesis of our sensory experiences and our ability to reason with these facts to become general laws.

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