When Plato implied ‘Mathematics is good for the Soul’
“Mathematics, rightly viewed, possesses not only truth, but supreme beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.”
- BERTRAND RUSSELL, Study of Mathematics
“When will I ever use the quadratic formula in real life? Why were we taught triangles and not doing taxes during school?”
For someone not belonging to mathematical or scientific academia, these recurring thoughts must have acted as pestering thorns on petals of mind. Why is it that a set of equations trigger anxiety in the majority of the masses today? Why is it treated as “he who must not be named” in this magical world? Is studying mathematics truly difficult or is it just taboo? Is there something lacking in this evolutionary process?
The answers may differ for people depending on how deeply one tries to search for them. But a conclusion can be drawn from each narrative that serves as the undertone for humanity’s pursuits of knowledge, life, and most importantly the understanding of being human. Perhaps, evolution has been kind to us in the sense that mathematics is bestowed upon homo sapiens as a ‘sense’, just like sight or touch. A sense for recognizing, creating, and appreciating patterns, relationships, and logical connections - capable enough of becoming mathematicians, although perceived differently. What is an artist if not a mathematician at work? A musician composing a symphony, a poet harmonizing a sonnet, a painter stroking a canvas, a cinematographer directing a lens is but dictations of mind’s conscious and subconscious approach for creating patterns and exploring.
“What is in a name?”
Well, at least in this context, everything!
The word mathematics comes from the Greek (mathema) meaning “science, knowledge, or learning”; (mathematikos) meaning ‘fond of learning’. Mathematics is incorporated in and around everyone which humans have begun to appreciate again, only now. Since the dawn of civilization, mathematics has been used as a guiding tool as well as a domain of deep philosophical insights — from measuring time-based on stars, the birth of number system to count herds, weapons, and monthly cycles of women to the development of computers and technologies of today. From finding geometrical patterns that stood the test of spacetime (fractals, Fibonacci sequence, golden ratio, etc.) to discovering more and advanced applications (computers, cryptocurrency, etc.), each direction is surrounded by mathematics and it is perhaps, as crucial as the breathing air, yet it is taken for granted, just like the latter. The fact that such a language is capable of unfolding the entire cosmos in front of the eyes, is nothing less than an overwhelming realization. Reportedly, Galileo is credited with the saying:
“Mathematics is the language with which God wrote the Universe”
(The author firmly agrees with this notion for every equation, every theory humans comprehend and solve, the closer they get to God.)
To argue for or against this perspective, one ought to dive deep into various modern and contemporary schools of thought of mathematics, and comprehend perspectives of philosophers throughout the ages; a conversation for another time.
The implications of mathematics as a language, as an art, have been archived for generations. The most ancient mathematical texts available are recorded to have originated from India (1500–500 BC) (Rigveda-Sulba Sutras) and ancient Egypt in the Middle Kingdom (1300–1200 BC). Interestingly, the Pythagorean theorem is found in all these texts, apparently making it the most widespread mathematical development along with basic arithmetic and geometry. In the West, especially Greek philosophy on mathematics was strongly based on geometry. Plato studied the ontological status of mathematical objects, while Aristotle’s focus was on logic and issues related to infinity.
Plato’s take on mathematics comes into knowledge in the book The Republic in which the emphasis is laid upon the importance of mathematics and why it is crucial for making a good philosopher and a political leader. Plato’s philosophy is based on the construct of two worlds — the physical world which is the one here and now, and the abstract world (the realm of the “forms” ) which contains unchanging truth, mathematics being one of them. The physical world is accredited imperfect, mutable, and transient, and is a mere reflection of the abstract world which is perfect, immutable, and eternal. One mathematical school of thought, eventually, is based on Plato’s theory of forms. Platonism, suggests that mathematical entities exist in an abstract form beyond the general notion of space and time, and are eternal and unchanging. Such an ultimate reality could only be imperfectly approximated by beings of physical worlds. Plato’s understanding of mathematics was majorly derived from Pyhtagoreans’ beliefs that the world, was quite literally, generated by numbers. This notion had its fair share of criticism, however, it also helped in the development of other schools of thought in times that followed.
Plato, through the dialogues of Socrates, in The Republic, advocated why everyone must be proficient in basic mathematics with the underlying purpose of attaining “the goodness” and the ultimate truth, which was the highest form in the theory of forms. It also plays a crucial role in making a person more “just”, as for mortal beings it is their duty to seek the objective truth and try attaining and comprehending it through the means of mathematics which caters directly to the soul. In the book, apparently persistent, Plato suggested that to become a prominent person of philosophy and government, one must undergo several years of astounding mathematical training. He believed that no one can become a ‘moral hero or saint’ without ‘discipline in sheer hard thinking’. In Plato’s academy which he established and where he taught mathematics, read:
“Let no one unversed in geometry enter here”
at the entrance. Therefore, Plato’s ideology revolved around the fact that one must spend ample time learning mathematics if one is to become a good political leader and thrive in other areas as well.
Disregarding momentarily, the deeper philosophical dimension of mathematics which Plato intended for people to accept and follow, and focusing solely on the practical implications of the discipline in the physical, more real world, one can notice that his reasoning still holds and shall stay true in the future as well. For instance, Socrates in a dialogue mentioned that mathematics is not just limited to mind training. Arithmetic must be studied by everyone for the following reasons:
- It forces the soul to get a better understanding of what numbers are in themselves, thereby focusing the mind on a realm of unqualified truth and being.
- Learning and practicing arithmetic makes one quicker at other studies, all of which involve numbers in any way.
- It is a good test of intellectual and moral caliber as it is demanding to learn and practice.
This ideology can be seen in the practice in today’s time as well. A person aiming for a career in any scientific or non-scientific domain is required to show a certain level of logical reasoning and aptitude through clearing some exams based on mathematical thinking and aptitude. The whole purpose of this exercise is to check the cerebral capacity of a person and how capable one is of critical thinking. A common practice that almost every generation has seen in one way or another.
Not only does mathematics help curate a great mental discipline, but it also develops the capability of abstract thinking. This has led to the birth of what mathematicians today call pure mathematics (This perhaps, has led to the birth of a lot of artists as well! ). However, people who pursue pure mathematics and have studied in the past, have remarked that there aren’t any realistic applications of the subject as such. Henri Poincare, a French polymath of his time said that pure mathematics is not useful in the real world but it is studied because it is delightful and beautiful. Analogous to an artist at work falling in love with his art. Paul Erdos’ words fit perfectly in this context:
“Why are numbers beautiful? It’s like asking why is Beethoven’s ninth symphony beautiful. If you don’t see why someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.”
G.H. Hardy, another mammoth in the field of mathematics, and mentor of Srinivasa Ramanujan proclaimed in his most famous essay, “A Mathematician’s apology” that trivial mathematics, on the whole, is useful, while real (pure) mathematics, on the whole, is not. However, nature seems to refute this statement, for mathematics, with a certain sense of irony, has its way of coming back. It is all about understanding and uncovering the intertwining of mathematics with all other aspects of life.
When such an argument is posed about the utility of mathematics in real life, it perhaps stems from one’s ignorance and unwillingness to observe the skeleton that holds and connects all bodies of work. What humans lack is the basic understanding and acknowledgment of interconnectivity. In today’s time, the impact of society’s persistent attempt at compartmentalizing everything is shadowing the importance of interdisciplinarity. A specific theorem or a mathematical concept may not be used directly, but one could never know what applications it might lead to. To state a famous example, if it were not for the Theory of Relativity by Einstein, the world might not have gotten the accurately functioning GPS. Eugene Wigner in his essay, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences “ said:
“Mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomenon in these connections.”
The remarkable usefulness of mathematics in natural science, especially physics is mesmerizing and surprising at the same time:
“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.”
Mathematical concepts and geometries lie at the heart of everything. Observe a sunflower blossoming naturally or notice a pinecone’s spiral and learn the Fibonacci sequence from it. Watch crown shyness of trees and instantly be reminded of similar patterns of blood vessels in a human body, river deltas, and lightning during a thunderstorm, and viola! one just learned about fractals. Such examples are everywhere around and within our species, and perhaps true beauty lies in the essence of acknowledging it. This sense elevates all other human senses, and ultimately, the soul.
Once a person understands what it means to channel their inner mathematician, it is a matter of time to come up with ways in which one’s mathematical prowess can be uplifted. It’s never too late to rectify our perception and approach towards learning the discipline. If parents of today and tomorrow are able to build an informal teacher-student relationship with their children, without being afraid of not knowing all the answers, this relationship could have the potential of raising not only the thinking and decision-making abilities of the child and parent alike but also strengthen their interpersonal relationship.
The relationship of the teacher and the learner ought to be symbiotic for they together learn and discover something new in the process. The miseducation of mathematics is so common all over the world that it has resulted in a strong repulsion of people from the discipline akin to that of the same poles of magnets kept close. Everyone is in a rush to solve the given problem and follow the prescribed solutions (quite analogous to the way of living in today’s world) or do the math to pass a class. A person understands and learns the best of something when the subject becomes personal. The approach is no different here. Spending some time with the problem at hand should be taught everywhere, formally and informally, instead of jumping right for the solution. This helps the learner engage, question, doubt, and apply reasoning to the subject and the problem. According to the mathematician, Dan Finkel:
“Thinking happens only when we have time to struggle.”
As a communicator of subject knowledge, a child’s curiosity should be enhanced by helping them to be patient and courageous while persevering a given problem. As a teacher, one is not under any obligation to know all the answers. “Not knowing is not a failure” as Finkel remarked in his talk.
On an individual level, one ought to acknowledge ignorance and limited knowledge and look for ways to expand their horizon. Math is not only about following rules or finding answers, it is also about developing critical thinking to help one ask the right questions. This caters to viewing a problem from different perspectives and helps tackle it with insight mixed with creative imagination. The more one allows the self to experiment and play with mathematics and numbers, the more one learns, discovers, and grows in more than one way, irrespective of career or academic background. Perhaps that’s why Albert Einstein famously said:
“Play is the highest form of research.”
Therefore, indulging in mathematics doesn’t only pertain to solving equations and numbers, rather it has a universal application in shaping a person’s mind, attributing to mental sharpness which can be applied to multitudes of disciplines and aspects of life.
For Plato, mathematics was a tool serving as a preliminary framework to achieve a deeper meaning of objective truth and one ought to train by practicing mathematics rigorously to sharpen the intellect and thrive in polymathy. For anyone, mathematics could be anything, once the standard and tabooed perception of this ‘sense’ deviate from the norm. One must create a narrative of their own and witness new dimensions unfolding about discovering the cosmos and oneself!
References:
- PLATO. (1943). Plato’s The Republic. New York: Books, Inc.,
- BURNYEAT, M.F. (2000). T. Smiley (ed.), Mathematics and Necessity: Essays in the History of Philosophy. pp. 1–81.
- FRANCIS, JOHN. Philosophy of Mathematics. Global Vision Publishing House, 2008.
- BERNAYS, P. (1935). Translated from the French by C. D. Parsons from L’enseignement mathematique, Vol. 34 (1935), pp. 52–69.
- WIGNER, E. (1960). The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Communications in Pure and Applied Mathematics, 13:1–14.
- HARDY, G.H., 1877–1947. (1967). A Mathematician’s Apology. London: Cambridge U.P.,
- Talks, TedX. [TedX Talks]. (2018, July 24). Mathematics is the sense you never knew you had | Eddie Woo | TedXSydney [Video]. YouTube. https://youtu.be/PXwStduNw14
- Talks, TedX. [TedX Talks]. (2020, March 14). What makes people engage with math | Grant Sanderson | TedXBerkeley. [Video]. YouTube. https://youtu.be/s_L-fp8gDzY
- Talks, TedX. [TedX Talks]. (2016, February 17). Five principles of extraordinary math reaching. [Video]. Youtube. https://youtu.be/ytVneQUA5-c