I was recently studying the calculus (well integral calculus, to be specific) and found out something really cool about myself and others too. Well, just to give you a little background, integral calculus is a branch of the calculus that concerns itself with finding irregular areas under curves. While reading the myriad of symbols (including the famous snake-like ‘summa’), I realized something really amusing: here I was reading calculus which is often considered mankind’s greatest achievement, and when I really really thought about it, I found out that I did not understand a immensely simple concept in the sense that I couldn’t really define it. This concept actually is a Grade 2/3 concept: area ! I asked a couple of my friends and they also gave me pretty vague answers – answers along the line of how big something is or how many square meters something occupies. But that isn’t a precise definition at all, so I thought a little about it. Well, here’s what I thought about!
In most secondary schools and even high schools, a formal definition of area (or length or volume) is rarely given; it is often taken as a given that students, being born humans, are hard-wired to understand these concepts. Of course, that is not the same as saying that a definition of a ‘unit’ of area is not given; if you jog your memory down memory lane, I’m sure you can remember some high school teacher giving a definition of a ‘meter-squared’ or a ‘centimeter squared’. But that still does not answer the question of what is area, a term we use so freely when buying lengths of fabrics, quantifying geographical size of countries or boasting of our sizeable real estate ownership. Clearly, very few people need a precise definition of area as they might need of any other entity or word such as ‘pedantic’ or ‘lugubrious’. Therefore, it is apparent that a definition of area (or length or volume) is a theoretical and philosophical one. Some mathematicians have identified certain axioms or basic prerequisites that the so called ‘area function’ should satisfy. These include basic requirements such a non-negative output, ability to be added, etc. But identifying axioms does not neatly fit the notion of a ‘definition’. Therefore, what is area? To answer we must first consider what is length? But again to answer, we must consider what a point is? And that is what we shall try to answer (only briefly and unsuccessfully).
A point, as anyone with a particularly strong memory will remember, is one of the famously ‘undefined’ terms of Euclidean geometry which also include line and plane. Unfortunately, I won’t be able defy the insight of Euclid, Archimedes or any of the great Ancient Greek philosophers and try to define a point. And to those who think a point may be easily depicted simply by taking a pen and applying downward pressure on it, I can only say that anything drawn on the paper will exist in two dimensions and therefore won’t fit the bill for a point. A point can only exist in our minds and nowhere in the real world. Something which I think gives mathematics its mysterious abstract beauty. I believe a point is dimensionless. Is it simply nothing? Might be. I’ll take our inability to define a point as an instance of the mysteries of mathematics. The interesting portion of defining area comes when ones tries to define length in terms of a point, taking a dimensionless point as a ‘given’.
I think that when we speak of length we speak of the number of times a dimensionless entity (a point) is multiplied out in a given direction to form something in one dimension- a line. I mean that a finite accumulation of dimensionless entities leads to a line segment. The magnitude of which (the length) is the number of times the dimensionless entity is accumulated along a given direction. This ‘definition’ of length seems edgy at best and illogical at worst. For example, how does multiplication of a dimensionless entity (a point or a ‘zero’) lead to a definite length? Does multiplication even make sense in this context? Clearly, any definition of an abstract and indefinite entity such as length or volume is bound to be fraught with inconsistencies. Nevertheless, let us continue in our definition of length and move to area.
Again, I think of the quantity of ‘area’ in terms of ‘accumulations’-this time of line segments. Having established that length is a measure of an entity existing in the first dimension, we can proceed to think of area in a manner analogous to how we thought of length. To understand, our finite minds will again have to simulate or at least model a line as a stroke of a fine pen on paper, though any representation on paper will be existing in at least two dimensions if not the third. Anyways, we can again think in terms of accumulations of some definite line segment of constant magnitude. Area is then the multiplication or accumulation of a given line segment (finitely or infinitely) along another line segment of definite length. This intuitively results in a smooth, continuous figure in two dimensions. (in this case, a square or a rectangle). Area is then a measure of the finite number of dimensionless points existing in the second dimension.
Let us try to move further and try to define a volume in terms of a finite area again by ‘accumulations’. An entity of definite ‘area’ existing strictly in the second dimension is multiplied and then accumulated along a given line segment a possibly infinite number of times to result in a smooth, continuous solid of definite volume. Volume may then be thought of as a measure of the number of dimensionless points existing in a given solid. Again, the key to ‘defining’ a notion of volume was ‘accumulations’ along a given first dimension line segment. Could this be applied to volume again to result in a measure of some higher dimension? A fourth dimension, perhaps?
I have tried doing it in my head and found that it results again in a third dimension entity. Using the ‘Accumulation’ idea and multiplying along a first dimension line segment, we again end up with a third dimension solid. In other words, the 1-D line segment fails to unlock the door to the fourth dimension as it did the others. Could this be a hint that there is no fourth dimension? Maybe so. But wait, nowhere in the real world were we able to accurately depict a line or a point. Maybe our finite mind has finally reached its limits. Using our brains and the ‘accumulation notion’, we cannot imagine a fourth dimension. Does it mean that it does not exist? It might not exist for us, but empirically it might, able to be visualized only by some higher power. Indeed, it is certainly on the list of the mysteries of mathematics.