A Simple Structure

The perception of mathematics in society is that it is an activity that takes place only in a classroom, with pen and paper, involving some or other calculation. Although activities such as these can certainly be considered as mathematical, in terms of the beauty that they capture, it is very far from the complete picture.

At its core, mathematics is a form of bounded creative thinking.

It may sound paradoxical to talk about bounded creativity, but as humans, we experience a certain ecstasy from observing patterns and order. In poetry we are bounded by language and words, we are satisfied when a piece of writing is both meaningful and obeys certain rules. In music, we are restricted to the world of sound, and a song is particularly pleasing when it possesses elements of novelty, while still conforming to existing musical structures. Another criterion we sometimes have for the beauty of an artwork is that it must not yield all its content instantly. We appreciate it if there are various layers of meaning, that only become apparent upon extensive reflection.

Mathematics is no different. It lives in the world of thought, and it is bounded by the rules of logic. What is interesting about this restriction, is that every human seemingly possesses (to some degree) the ability to decide whether or not certain thoughts are logical, without ever having been taught what the rules of logic are. An example of a “rule” that everyone intuitively understands, is what is referred to formally as “modus ponens”, it involves taking 2 statements, and forming a third:

“Whenever the sun rises, a new day follows” could be the first statement, and

“The sun has risen” could be the second.

Then most humans would agree that the most natural statement to conclude from these would be:

“A new day is on its way!”

What most humans don’t realise however, is that in reaching this conclusion, they have engaged in mathematics! And this is not said as a joke, or as an endearing motivational comment. The process of reaching the above conclusion is genuinely much closer to the true nature of mathematics, than the manipulation of numbers on a sheet of paper. The rule that describes a conclusion like this underlies not only all of mathematics, but it is so deeply engraved into our minds that we cannot even imagine an existence where it is not true.

In terms of layered meanings, and the reluctance of a thought to reveal its true nature, mathematics trumps all other art forms in versatility: There are mathematical thoughts such as the one we looked into above, that are simple enough for a young child to understand, and then there are lines of thought that are complicated enough that it may take someone a lifetime to fully grasp. It goes even further, with thoughts that are so complex, that it requires the joint effort of humans over many generations to uncover their true meaning.

A sad truth about the status of mathematics today, is that the first exposure many children have to it, is being forced to pretend that they have understood some line of thought that took humanity decades to fully understand. It is not surprising then that some people have an aversion to the subject years later; It represents a time in their life when they felt inadequate for not achieving something that was framed as being simple, when it was incredibly complex.

This leads me to say that the way to change the outlook of society on mathematics is not to attempt to frame it as something fun (which it certainly is) or as something simple (which it certainly isn’t) but to have compassion when introducing it to people for the first time and to help them realize that in fact, they already are a mathematician, and we are only helping them uncover something that they already possess inside.