Creative Mathematics is Not an Oxymoron!

“So if you’re such a math genius, then what’s infinity divided by infinity?”

In sympathy to the taxi driver’s mathematical ignorance, I kindly and rather apologetically replied: “Infinity is not a number. It’s like asking whether beauty can be divided by itself”.

Vexed by his failure to humiliate an MIT student who claimed to be good at math, and only reveal his own ignorance, the driver, whose identity later manifested to a certain Bob, resorted to a petite tirade on “my people”: the mathematicians. “You know what the problem with you math people is? You’re just a bunch of robots. No imagination. No talent. No creativity like them artists and poetry types just playing with your numbers”. Such derogatory allegation was not unusual to the math student. In a rather cool fashion, I gave him an indication of defeat and Bob’s ego was once again cushioned.

It seems that math has borne the stigma by the general public of being a dull, uncreative and impersonal practice and where the computation of numbers and properties of simple shapes occupy some lonely, poor and socially inept man’s life. Pointing this out, the reader will guess that I believe the converse.

I’m writing this essay to assert that even though the pencil is only limited to a certain shade, the mathematics it metaphysically depicts makes The Sistine Chapel look like a kindergarten arts workshop. In my opinion, mathematics is the most creative, sensitive and permanent subject that has and will ever exist. In short, it is flawless!

When non-mathematically inclined people talk about mathematics, they are typically referring to the “elementary mathematics” that was hammered into their heads in elementary and high school. Unfortunately, there is an implication: someone good at math is good at calculating quickly and getting straight A’s in homework problems. Thus, if this were true, then math may be compared to spelling bee contests or a science classes. This is quite understandable because one may ask “what other kind of math can possibly exist?” If one can calculate with numbers, find angles and maybe be able to do a bit of calculus, then what other “math techniques” may one require? Thus there seems to be a lack of curiosity and questioning and instead a need for application. “What good is it to question?” one may find one’s self asking.

The answer is that questioning is very important. Here’s an illustration:

In planar geometry, (the one we al know and love from junior high), it is proved that all angles inside a triangle sum up to one-hundred-and-eighty. What if one considered the sum of angles in a triangle on a spherical shape like a tennis ball or the Earth? It turns out that such a question lends itself to a counter-intuitive answer: the sum of angles inside a triangle do not add up to one-hundred-and-eighty! Now one may question what is the use of this different theory of geometry then? The answer is a lot! This area, called non-Euclidean geometry, has huge applications in physics and if it weren’t for its theory, space traveling and our understanding of the universe would have been a lot more limited. Moreover, Einstein would have never founded his breakthrough in general relativity theory if it weren’t for this non-Euclidean geometry.

Non-Euclidean geometry illustrates that there is a lot more “useful” and “fundamental” math than that taught in high school and they explain the world we live in today. Moreover, by questioning one sees that the fruition of mathematics is unbounded simply by considering the unconsidered and thus, making it such a creative subject. In addition, we may make more use of this example by mentioning that plain geometry has been known to man thousands of years ago dating back to the Greeks. In fact, the entire math taught at high school is at least three hundred years old! It is quite right and reasonable to assume that mathematicians haven’t just been idling for these past three centuries.

I now illustrate the seemingly outrageous claim that math is the most beautiful, permanent and pure study in both arts and science. To do this, I am much obliged to now and then quote the ever so insightful G. H. Hardy’s “A Mathematician’s Apology” for many of my ideas cannot be better expressed than in his words.

It is the making of patterns to express ideas that so closely links a mathematician to an artist and poet. But it is the way these ideas are expressed that distinguishes the mathematician from the other two. Quoting Hardy:

A painter makes patters with shapes and colours, a poet with words. A painting may embody an ‘idea’, but the idea is usually commonplace and unimportant. In poetry, ideas account for a good deal more; but as Housman [a Cambridge professors of literature Hardy meet] insisted, the importance of ideas in poetry is habitually exaggerated.

…A mathematician on the other hand, has no material to work with than ideas, and so his patters are likely to last longer.

…Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. “Immortality” may be a silly word, but probably a mathematician has the best chance of whatever it may mean.

From these quotations, it seems that math is superior in that it is not a language but some collection consisting of pure ideas. Yet one may argue that philosophy also purely embodies ideas. However, I believe that these ideas are expressed through languages not compatible with philosophy, which makes philosophy more susceptible to fallibility and illogical reasoning.

But one may argue that we ‘created’ the circle and square and many other mathematical symbols and objects and so it too is artificial. However, if one only considers these shapes and symbols as models of ideas than one sees that these models are not unique or artificial: they simply communicate logical ideas.

Finally, science seems to be the least beautiful and permanent. By its permanence, I do not mean to say for example that gravity only sometimes works. (This can only happen in Terry Pratchet books). This confusion may arise because, to me, there exist two kinds of science: that of mans’ and the other of nature. History demonstrates to us that our knowledge of nature is specious in that not everything works in nature the way we expect it to. The history of quantum mechanics is a good example because it brought, concepts completely paradoxical to our way of thinking about nature. However, if we look at nature for what it is, then nature is quite permanent for it cannot change itself (loosely speaking). It is obvious that I speak of man’s science. However in math, a prime number is a prime number in any universe or dimension regardless of what ‘nature’ might think. Science may not be so, for it is nature that ultimately is the judge. Consider this: Are apples really fruits? (I leave this for the reader to investigate).

With all this said, a non-mathematician may find my arguments very derogative and offensive, especially if he (or she) infers that I claim his (or her) occupation to be of a worthless and unintelligent practice. However, this is not so. Even though mathematics has many important roles in life, it is not everything and will never be. Mathematics alone will never give a cure to cancer. In addition, even though math may show many fundamental and important ideas, it does not evoke the same kind of emotions one can get from poetry and I believe emotions play a crucial role in our lives and even in mathematics. Furthermore, even though math is the “tool” of science, it is only science that gives us knowledge of what is around us and math alone cannot give us insight into the way nature works.

I leave my reader with a final quote from Hardy:

I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations," are simply the notes of our observations.

G. H. Hardy “A mathematician’s Apology” pg. 84. Cambridge University Press, 1990.