Tuesday, December 13, 2011

The nonexistence of mathematical graphs

In many of our Mathematics textbooks or notes, we see these objects called "graphs". They are basically geometric figures that exist in metric space (with certain axes) and are well-defined, and what governs their behavior is a mathematical equation. Two examples of such graphs with their corresponding mathematical equations are shown below:
As you can see, these graphs are perfectly visible. In fact, nothing seems to be a problem here. Everything's smooth... 

But things are not as simple as they seem. In fact, there exists a deep flaw in this and all outlooks of graphs. I will be explaining to you why all graphical representations of mathematical equations are wrong.

Firstly, I want you to take out a piece of paper (best if it's recycled; save the Earth!), and draw an x and y axis, something like in the picture above. Then, using only conventional wisdom, I want you to draw points that represent the co-ordinates (1,1), (2,1) and (1,2). Just to let you know, a theoretical point in 2 dimensions is a geometric representation of a set of 2 real numbers. Using only conventional wisdom, one should come up with something like this:

Now, let's zoom in on the point (1,1):
This point looks like a shaded circle, with its center at (1,1). It has a radius of epsilon (3 inverted laterally), such that epsilon is greater than zero. This has to be true. If epsilon was equal to zero, then we would not be able to see the point. If epsilon was negative, things wouldn't make sense. Then, assume epsilon equals to 0.1. The circular point would then encompass all possible real numbers between 0.9 and 1.1 on each axis. This basically means that this point does not represent a single set of 2 real numbers or a single co-ordinate, but encompasses an infinite number of them. Thus, we have now arrived at a contradiction. We started off by saying that this point we drew represents a single set of 2 real numbers or a single co-ordinate in 2 dimensions, but now we have discovered it does not. We have discovered that the point encompasses a range of co-ordinates.

Because of this contradiction, the only logical conclusion we can make is that the point we have drawn is wrong. So what now? Using the ideas above, we can tell that epsilon's existence entails the existence of more than one co-ordinate being encompassed by the point. Thus, the only logical conclusion would be that epsilon cannot exist. Only then would the point represent a single co-ordinate. Nonexistence is defined by the number zero. Thus, epsilon must be equal to zero to accurately represent a single co-ordinate geometrically. 

But now another problem arises. By having a zero radius, a point is no longer a point. In other words, a point no longer exists visually; it is invisible. This may seem trivial, but this reality has great implications. Since continuous graphs like the one right above consist of a succession of these invisible points, the direct result is that any graph governed by a specific mathematical equation must also be invisible in metric space. This doesn't make any sense! But it's true.

In other words, all graphical representations of mathematical equations are theoretically incorrect, because one should not even be able to see a graphical representation of a mathematical equation in the first place.

QED

This is really bizarre. One question that arises here is that if graphs are in reality invisible, then what do visible graphs (that we see in documents) represent?

Even though graphs in reality are supposed to be invisible, the co-ordinates that comprise them do have a certain orientation and position in metric space. We cannot deny that. That's how metric space is defined. Thus, even though the graphs cannot exist visually, they do have some sort of a virtual existence, with a virtual shape and orientation. The visible graphs that we see in our daily lives actually represent approximations of the true "invisible" graphs. The visible graphs are not perfectly correct or precise (hence their visibility), but they provide us with an idea of how the true co-ordinates of a graph are distributed in metric space. The picture they provide us with is not perfect, but it's sufficient for us human beings because the margin of error is relatively low; much lower than what would seem perceivable or largely hindering to us.

Another question that could arise is that why do we even need these visible graphs in the first place? Why is it a need for us to commit a "mathematical crime" by bringing something mathematically unlawful into existence?

From the point of view of a mathematical purist, we actually do not need these graphs. Mathematics at its purest manifests as algebra. Algebraic problems can be solved by algebraic means, and these algebraic means characterize the purest means of Mathematics. I'll give you a simple example and solution of such an algebraic problem:



This problem was solved easily by algebraic means. Theoretically, all algebraic problems can be solved by algebraic means. There is actually no need for figures like graphs whatsoever. However, algebraic means to solve algebraic problems may not always be practical. Sometimes, solving an equation by algebra may be close to impossible. An example is shown below:


In such scenarios, graphs become very useful. Graphs help us by telling us the number of roots of the equation, the polarity of the roots (positive or negative) and also the approximate values of roots. Once again, the graphs may not be perfect, but the error inherent in the graphs is small enough for us human beings to neglect. The graphs corresponding to the equations above are shown below:


Using the graphs, we can know that there exists only one negative root to the equation, which is between -1 and -1.5, about -1.3.

This is one reason why we use such visible graphs: for the purpose of practicality.

Another possible reason why we like to use these graphs is because of their structural beauty. When you see a graph like the one below, you can't help but be struck by its beauty.


Perhaps we like visible graphs because they remind us of the structural beauty of the world that we live and thrive in. In the attempt of mapping the physical beauty of our world onto the inanimate, axiomatic mathematical world, we somehow forgo all of our mathematical purism. Humanity is pervasive isn't it?

Picture source: http://www.mathsrevision.net/gcse/pages.php?page=24
Math codes (LaTeX) source: http://www.codecogs.com/latex/eqneditor.php

4 comments:

  1. Interesting post!
    I think sometimes graphs are used out of convenience, because some people are just more visual than others. Personally I prefer working with graphs more than equations. Even when I don't have a graph I would visualize one. It's a habit :)

    ReplyDelete
  2. Thanks for commenting Thu :)

    Glad to know that you like math.

    ReplyDelete
  3. you have definitely proved that all those in school who could not get marks in graph based questions were actually most knowledgeable in the field because they drew invisible graphs
    anu masi

    ReplyDelete