The limitations of mathematics 

Being able to use and conceptualize math is one of the fundamental skills that people use every day. Despite how concrete it seems, a lot of math we use for complex physics is actually uncertain and based on probability. Illustration by Anna Iorfino/The Daily Campus

Having chosen torment by quantum mechanics as one of my fall semester courses, I stared down the barrel of the Schrodinger equation on the first day of class. What kind of  mathematical whirlwind had I signed up for? I was quickly frustrated by the core concepts of this subject. Every math course taken up to then saw me able to solve problems with an exact solution. But the word “exact” is a completely alien concept in quantum mechanics. Despite the wide acceptance of quantum mechanics being the most successful branch of physics ever produced (having the capacity to describe subatomic interactions to extremely high precision), the entire theory is contingent on statistical probability. As if to solidify my frustrations, our class was introduced last week to the Heisenberg uncertainty principle. This principle expresses our inability to precisely measure a particle’s position and its momentum simultaneously, as the information you seek on one is inversely proportional to the information gathered from the other. How could it be that the most acclaimed theory describing the cogs and gears of our universe is unable to define something with absolute certainty? Let alone describing several properties of a single entity with certainty. The fundamental deficiencies of analytical solutions when attempting to describe our observations puts mathematics in its rightful place: merely the best attempt for three-dimensional beings trying to interpret a higher dimensional world. 

The depths of human conception and ingenuity, while incredible, are often too romanticized. Mathematics is a prime example of this. The other side of the argument reasons that mathematics are intrinsic to nature. If the universe ceased to exist tomorrow, the laws underpinned by the order of mathematics would still stand, meaning it’s up to us to discover its workings to get a complete picture of the eternal fabric of space, time and all of its constituents. Is this anything but the most narcissistic appraisal of the human mind? Are we incapable of accepting the possibility that there is a universe that exists beyond the perception of human observation and experience? We can leave that to the philosophers, of course, since there are no empirical truths that can be derived from such thought-provoking questions. What we can do is continue to interpret the world as we perceive it, because that’s exactly what the universe is to us. 

The limited brainpower of human beings compel us to seek the creation of mathematical models as approximations of reality. When these models fail, they are subject to a revision process and then a newer model, or an entirely different branch of mathematics takes its place. When Isaac Newton wrote “Philosophiæ Naturalis Principia Mathematica,” his universal law of gravitation predicted the gravitational interaction between massive objects to an accuracy of 10 to the power of negative seven (.0000001), until Einstein developed the special and general theory of relativity which predicted the same interaction to an accuracy of 10 to the power of -14 (.00000000000001). These extraordinary feats that continuously agree with observations have allowed humans to acquire knowledge exponentially, with new technological heights being reached in progressively smaller intervals. But perhaps there will come a time when Einstein’s theory is replaced by something else entirely, and we continue attempting to close the gap on zero uncertainty. 

Unfortunately the paradox of observation, and using mathematics to reason what we observe, will always pose the greatest challenge to filling the gaps of our knowledge. Consider the simple thought experiment: What is a chair made of? A reasonable answer would be wood fibers, which are made up of cells, composed of molecules, consisting of atoms, comprising protons and neutrons, which in turn are organized into collections of quarks and gluons. At this depth, we are unable to even recognize these “particles” as individual entities, rather a mathematical probability density cloud where the particle likely exists. Even then, one would be perfectly reasonable to ask what quarks and gluons are made up of, and so on. The universe is like an infinite onion; we keep trying to peel back the layers to find the core, only to find more layers. The picture that consistently emerges is that all mathematical models of the physical world break down at some point. What’s left when we are unable to go any further? Albert Einstein once stated that it is entirely possible that behind the perception of our senses, worlds are hidden of which we are unaware.  

If we can accept that mathematics is invented, rather than discovered, we can start asking deeper questions, be more daring and motivated to create further change. I recall asking my professor in one of the first lectures “Is there any way for us to know where a particle was an instant before we measured it?” My question was met with a wry smile, and the shrug of a shoulder. “So basically, there are two schools of thought…” he went on to say. It was at this point where I had to accept that not all questions were going to have an answer. Nor could certain fundamental questions about the physical world ever be answered by mathematics. For now, we use it as a product of the human mind, inherently tailored for the human mind.  

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