Unseen Beauty

Obviously noted is the popular disdain for mathematics, both in principle and in practice.  One need not look far to encounter someone who, maybe even proudly, claims their inability to understand the subject or gleefully shares their ignorance about relatively basic mathematical concepts.  This seems strange.

Upon learning that someone knows little about history, politics, or even a basic news story, there is a reaction that points to the speaker being somehow lacking in an appropriate level of general knowledge.  Virtually any American would, for example, at least know something basic about Abraham Lincoln, and rightfully so.  Anyone who looks at you funny in response to this name raises some ambient level of questions about their education and understanding of the world.  And so too with countless other sorts of subjects:  Mark Twain is only a pen name, human beings are a species of African apes, and the reason that the earth has seasons is mostly due to the fact that the earth has an axial tilt.  Julius Caesar is some Roman emperor, Beethoven some guy who wrote music, and a significant portion of human communication is done nonverbally.  How can it be that among a vast landscape of knowledge on all sorts of subjects ranging from immunology to cosmology that math rarely makes the cut in the popular sphere?  Why will you hear about Tesla or Einstein, but never about Gauss or Euler (pronounced “Oil-er”)?

Assigning blame in this situation is difficult.  It is definitely true that the way in which math is taught leaves much to be desired.  Obscuration of the subject in the schools from testing standards, not enough time, and teachers who do not approach the subject appropriately are all certainly important factors.  However, these are not the focus here.

Subjects like history and politics are relatively easy in which to get involved, if not emotionally invested.  Being riveted by figures such as Rasputin or the idea of monarchical rule seems easy, even for children.  The reason for this seems to be the simple fact that these concepts are relatable and deal with people at some basic level.  Imagining how someone acts, or how a group of people behaves is substantially easier than getting a “feeling” of what a linear equation does in our lives or society.  The same can be said for all of the facets of fiction: films, novels, plays, etc.  We can all tap into the inner lives and the inner feelings of these people and their situations.  Through this direct connection concepts can be picked up much more readily.  Music has an uncanny grasp on our imaginations and so therefore also holds significant weight in the popular sphere.  Even science, which on paper seems cold, away from the individual, and merely descriptive, captures the imagination of most people—you might be hard-pressed to find someone who hasn’t heard of a black hole or supernova.  This is because science can be easily made to interact with the human element.  Walking on the moon, flying, and computers are celebrated examples of the popular notion of science.  This brings us to an inescapable observation:  mathematics is abstract.  If a concept rarely collides with ordinary life, what value might there be in reaching out to it?

Euler’s Equation is often reffered to as one of the most beautiful mathematical relationships.

At its core, mathematics is born out of attempting to describe the natural world.  Fundamental concepts that we easily grasp onto like numbers and addition actually have deep philosophical implications, assumptions, and developments that are hidden from view.  Indeed, at some point for the entire system to work there must be essential axioms, or things that we simply assume are true without a logical proof.  In what way can a child be made to appreciate what the study of math accomplishes?  The French Revolution has clear and obvious implication upon life, but what about trigonometry?  For a start, there are genuinely useful things one can do with concepts learned beyond arithmetic.  Modeling one’s own investment portfolio growth relies on differential equations, and calculating the proper angles by which a stable roof can be built relies on geometric principles.  Figuring out how populations of animals grow or migrate, or the likelihood of a given outcome based on statistics is useful.  Where does the excitement come in?  There are genuinely exciting concepts, even simple ones, which can pique people’s interest.  What happens if you treat negative numbers as existing mathematical objects, even though in no sense in the universe we actually ‘have’ a negative existent thing?  Why do the prime numbers start getting so far apart, and is there a pattern to them doing so or any decent reason?  Does taking the sum of infinitely small values have a meaningful answer?  How can the value pi be a non-terminating decimal that describes exactly the ratio of a circle’s circumference to its diameter?  Furthermore, and more to the point of our focus here, how can we actually know such things are actually true and come to use them?  Questions like these are exciting and make one wonder at the nature of our universe and our very conception of how things work.  This is the reason why at once the topic seems both strangely unique and difficult.

Where does this leave us?  Perhaps expecting the subject to be as popular as others is missing the mark somehow.  Appreciating mathematics requires trying things out, asking intuitive questions, and seeing how others have come to solve such problems.  At a distance, numbers and symbols on a page have no feeling or life to them.  “Why should we care?  Someone else has figured this out who is better suited to such things.  This holds no relevance to me.”  The fatal flaw in this intuition is that math isn’t merely a means to an end—it’s an end unto itself.  Appreciating where sine and cosine come from, or how we can actually calculate distances in outer space, is a wholly self-sustaining magic.  Perhaps even more magically, taking numbers and concepts that we merely observe or discover actually leads to principled descriptions of completely different phenomena.

Furthermore, something here that should not be overlooked is that the language of math in many ways accurately describes the physical universe.  This seems incredible, given that at some level this description of the world was simply brought into existence through thought.  The sheer magnitude of applicability to the natural world is perhaps the most amazing thing to be appreciated.  In fact, there’s a famous essay titled “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” published in 1960 by physicist Eugene Wigner that describes this concept in the following way:

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.


There are many things in life that, to appreciate in a profound and serious way, require effort.  Enjoying a beautiful tone on the violin is considerably enhanced once one realizes just how difficult that project can be.  The same sort of reasoning can be applied here.  Understanding and proving to oneself the self-consistent beauty and logic of mathematical concepts is nothing short of exhilarating and even exalting.

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