The Paradox that Breaks Math – from nothing to everything

The world around us runs on math. However, we take for granted the very foundations that build this field. For instance, our understanding of functions in a graphical plane has aided us in various fields for generations. Despite this, we uncover a puzzling paradox that weakens the foundation of higher-level mathematics.

Math is everywhere. Not only does society rely on its concepts, but so does nature. For example, the Fibonacci sequence is found in various structures of nature including flowers and snail shells. It is also used to identify the golden ratio: \( \varphi =\frac{1+\sqrt{5}}{2}\). Though it may seem insignificant, many artists use this to create a visually appealing figure. Leonardo da Vinci’s Mona Lisa follows the golden ratio and thus the Fibonacci sequence. This emphasizes that math is not only used in STEM fields but can be applied in any field, even those that don’t seem related to mathematics, such as painting. For this reason, it is important for mathematicians to refine the concepts of math that build the foundation for higher-level mathematics that many fields rely on.

It is clear that the higher level of math is based on the use of functions. In high school, calculus and linear algebra both utilize functions on n-dimensional planes. Much of engineering is based on the interpretation and analysis of such functions. However, when we take a closer look at the concept of a simple graph, a paradox forms. We define functions as an infinite collection of points. However, each point has a length of zero in all the dimensions of the plane. Thus, somehow, we create something out of nothing. Two-dimensional functions are said to have no thickness which makes sense since points don’t have a thickness either. Unfortunately, that’s where the sensical connections end. If each point has a length of zero, how is it possible to add up an infinite number of zeros to get a nonzero length? 

It wouldn’t make sense for mathematicians to claim that each point has an infinitesimal length. This then implies that the point has become an infinitesimally small line, skewing our basic understanding of functions. Though this paradox sheds light on the flaws of our current understanding of functions, this understanding has brought us thus far. Despite this, I believe that it is important for mathematicians to address this paradox so we can advance further than our set capabilities


Shlok Bhattacharya

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