In defense of mathematics [excerpt]

Once reframed in its historical context, mathematics quickly loses its intimidating status. As a subject innately tied to culture, art, and philosophy, the study of mathematics leads to a clearer understanding of human culture and the world in which we live. In this shortened excerpt from A Brief History of Mathematical Thought, Luke Heaton discusses the reputation of mathematics and its significance to human life.

Mathematics is often praised (or ignored) on the grounds that it is far removed from the lives of ordinary people, but that assessment of the subject is utterly mistaken. As G. H. Hardy observed in A Mathematician’s Apology:

Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people really interested in mathematics than in music. Appearances suggest the contrary, but there are easy explanations. Music can be used to stimulate mass emotion, while mathematics cannot; and musical incapacity is recognized (no doubt rightly) as mildly discreditable, whereas most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity.

The considerable popularity of sudoku is a case in point. These puzzles require nothing but the application of mathematical logic, and yet to avoid scaring people off, they often carry the disclaimer “no mathematical knowledge required!” The mathematics that we know shapes the way we see the world, not least because mathematics serves as “the handmaiden of the sciences.” For example, an economist, an engineer, or a biologist might measure something several times, and then summarize their measurements by finding the mean or average value. Because we have developed the symbolic techniques for calculating mean values, we can formulate the useful but highly abstract concept of “the mean value.” We can only do this because we have a mathematical system of symbols. Without those symbols we could not record our data, let alone define the mean.

Mathematicians are interested in concepts and patterns, not just computation. Nevertheless, it should be clear to everyone that computational techniques have been of vital importance for many millennia. For example, most forms of trade are literally inconceivable without the concept of number, and without mathematics you could not organize an empire, or develop modern science. More generally, mathematical ideas are not just practically important: the conceptual tools that we have at our disposal shape the way we approach the world. As the psychologist Abraham Maslow famously remarked, “If the only tool you have is a hammer, you tend to treat everything as if it were a nail.” Although our ability to count, calculate, and measure things in the world is practically and psychologically critical, it is important to emphasize that mathematicians do not spend their time making calculations.

The great edifice of mathematical theorems has a crys­talline perfection, and it can seem far removed from the messy and contingent realities of the everyday world. Nevertheless, mathematics is a product of human culture, which has co-evolved with our attempts to comprehend the world. Rather than picturing mathematics as the study of “abstract” objects, we can describe it as a poetry of patterns, in which our language brings about the truth that it proclaims. The idea that mathematicians bring about the truths that they proclaim may sound rather myste­rious, but as a simple example, just think about the game of chess. By describing the rules we can call the game of chess into being, complete with truths that we did not think of when we first invented it. For example, whether or not anyone has ever actually played the game, we can prove that you cannot force a competent player into checkmate if the only pieces at your disposal are a king and a pair of knights. Chess is clearly a human invention, but this fact about chess must be true in any world where the rules of chess are the same, and we cannot imagine a world where we could not decide to keep our familiar rules in place.

Mathematical language and methodology present and represent structures that we can study, and those structures or patterns are as much a human invention as the game of chess. However, mathematics as a whole is much more than an arbitrary game, as the linguistic technologies that we have developed are genuinely fit for human purpose. For example, people (and other animals) mentally gather objects into groups, and we have found that the process of counting really does elucidate the plurality of those groups. Furthermore, the many different branches of mathematics are profoundly interconnected, to art, science, and the rest of mathematics.

In short, mathematics is a language and while we may be astounded that the universe is at all comprehensible, we should not be surprised that science is mathematical. Scientists need to be able to communicate their theories and when we have a rule-governed understanding, the instructions that a student can follow draw out patterns or structures that the mathematician can then study. When you understand it properly, the purely mathematical is not a distant abstraction – it is as close as the sense that we make of the world: what is seen right there in front of us. In my view, math is not abstract because it has to be, right from the word go. It actually begins with linguistic practice of the simplest and most sensible kind. We only pursue greater levels of abstraction because doing so is a necessary step in achieving the noble goals of modern mathematicians.

In particular, making our mathematical language more abstract means that our conclusions hold more generally, as when children realize that it makes no difference whether they are counting apples, pears, or people. From generation to generation, people have found that numbers and other formal systems are deeply compelling: they can shape our imagination, and what is more, they can enable comprehension.

Featured image credit: Image by Lum3n.com – Snufkin. CC0 public domain via Pexels.