Psychological "Line" of Mathematics

I asked my class to participate in a simple brain exercise. Select an object: a car, a plant, a ball, a chair, and imagine this object in your mind. Then, think of two of them. I then asked them to think of three, four, five, six…. All of the students could not go past six, and, in fact, it became much harder to imagine six objects.

I did this exercise to teach them that, although their abilities in maths are vastly different from each other, their fundamental development and cognitive abilities when it comes to dealing with real numbers are not. More importantly, I wanted them to understand that fitting numbers inside our conceptual abilities is painfully and shockingly difficult for any small number of objects.

Furthermore, what this demonstrates is that if we are finite beings, bounded by limitations, then there is obviously going to be a cloud of mystery that surrounds how we see the material. What is realized in popular scientific culture is just how cloudy our perceptions really are.

Our limited perceptions lead to common brick walls we just can’t break or get over, both temporally and spatially. Temporal limitations have led completely to the Big Bang and the unexplained miracle that is the beginning of the Universe. Spatial limitations in our analysis have encountered quantum mechanical effects once we scale to smaller and smaller fields of dimension. The universe holds many mysteries we just can’t fully comprehend, and that includes the mental capturing of just a few simple objects.

We can’t store objects in the simple sense. Therefore, it could be necessary to dismiss the notion that perception is a form of storage and that dismissing mathematical intellect as another form of internal storage may also be redundant. What makes you think your perception is from you? You and all your whims, flaws, needs and desires. What makes you certain that your perception is in you and not a gripped extension of you?

What makes human beings unique is our ability to extend our power to tools that allow us to extend our perceptions. What if the same could be true with the emergence of the mathematical intellect? This functional idea that I am proposing could link our human perception with the modern populous notion of modern mathematics, from which it has largely been disconnected.

The populous, new-atheist ideas make a few presumptions about the surrounding world:

(a) it is material, and

(b) it is right (and our deviation from that is wrong).

In the material sense, it is made up of objects, physical things, and the connections between those objects are purely physical and chemical interactions, which science has been describing in ever-increasing detail over the span of human history. It is right, meaning that it is objectively true; we can extract correct behaviour through what is happening by the laws of nature and by what is occurring naturally.

Although presumption (b) has its flaws in the moral sense—that we simply cannot link the world to what we ought to be right about—(a) is the presumption I am most attending to. As a scientist, I was trained and enticed into believing that the material was all there is—that objects were the world. But I soon realized that heartbreak, disaster, and failed relationships couldn’t simply be described by the shuffling and moving of objects, physical or chemical. After teaching maths myself, I don’t even believe you could describe or predict such events. Even if we could describe such events, who would hold the ethics behind such equations that connect love with familial disaster or happy endings? There is a mystery beyond the physical that shapes how we behave, a world that we have indeed explored through plays, art, literature and culture, and yet it is still so invisible that we cannot grasp it.

I am not by any means alluding to an equation that can describe the invisible. In fact, I completely object to the New Religious claim that the so-called “gaps” in our scientific understanding can be filled with the spiritual. Would that mean scientific progress is just our human ability getting closer to the spiritual? What would a human world do with spiritual or divine responsibility if we could even hold it? Science is not the progress or power to progress our entities into divinity. Let’s leave the spiritual in the spiritual where it belongs and not take the Enlightenment to its totalitarian end.

So there are two worlds through which complicated human beings navigate: the visible and the invisible. Both have well-established laws and consequences. Breaking the physical laws has much more immediate consequences: falling out of a building, electrocuting yourself with water and a plug socket, or cooking and burning a nice delicious roast. Breaking the spiritual laws is a little more mysterious. We feel the consequences in the form of our wife suddenly leaving us, being fired from our job, and the feeling when a loved one dies. The movement of the spiritual moves the material, and the movement of the material moves the spirit. It’s difficult to capture these processes even linguistically—mathematically would certainly be impossible. And nor should we capture it mathematically.

Capturing processes mathematically is not the same as realising them physically. As we’ve described, our models, which we layer over our systems, begin to break down if we extend them to different temporal or physical scales. We’ve demonstrated that even capturing physical phenomena, such as the so-called “global warming,” begins to break down in accuracy almost immediately once it’s extended years into the future.

Meanwhile, can we capture the spiritual mathematically? The answer is a solid no. I won’t spend too long highlighting that ridiculous claim: ask anyone to write a set of equations that predict the mystery of love.

Another claim I want to dismiss is “Maths is either right or wrong.” This is commonly repeated in educational circles, mainly to separate the sciences from the humanities because (apparently) people in the humanities are confused morally between right and wrong, and (apparently) people in the sciences are coldly beyond and unconcerned with the trivial grovelling of morals. (No doubt, the absence of scientists in moral and political discussions may have actually led to the vacuum that pervades our modern culture—but that isn’t the concern of this article.)

Mathematics is supposedly a non-biased judge, a blinds-eye, with no in-between, unlike the foolish political and social sciences. However, this linguistic trick of right and wrong creates the intellectual space of sides, in which you (being the robust scientist) occupy a side that is right, and anything outside this space is wrong.

Certainly, in education, people make mathematical errors, trips, and mistakes that lead to wrong numbers and wrong answers. But a deviation from the right area would still place you within the right area, just in a different part of the right. Psychologically, a mathematical error is a much more dramatic change to the “side” that you occupy. Rather, the space of “right” is so thin and straight that it is very difficult to move along without entering (or even being invaded by) both territories of the visible and the invisible.

When errors take place, you are immediately derailed off your line of thought, your logical process, and the impact of the material and the spiritual on the individual is the prime cause of the derailing. If the mathematical line of thought is a train going straight, the material and the spiritual are the elements encroaching on the line.

(I notice the same thing with my own students when I mark their work. Their logical perception is a train being pushed and pulled by their social circle, how they learn, what they learn, their spiritual state, and what movements are happening when they solve these problems. There haven’t been many explanations for the psychological significance of problem-solving, but it would be worth taking a closer look.)

If mathematics cannot completely describe the material due to our perceptive limitations, and neither can it describe the spiritual due to how we simply cannot grasp it, what can be described?

One of the benefits of having a line of thinking (a “line of attack”) is that we actually don’t have to occupy the media of the spiritual and the material to have a meaningful existence. In fact, our very humanity has been dictated by our abilities to plough through both spaces, taking inspiration (and hesitations) from both.

It is clear that both spaces are indeed separate. We cannot solve the spiritual with the physical—ask anyone who has ever tried to buy love with money. Neither would you build a skyscraper with the blueprints of freedom as your engineering guide. In essence, that is what being human is about: we move through both the spiritual and the material—what consequently emerges out of that straight line—to its highest and fullest degree— is mathematics and the pursuit of truth.