Defined mathematics

Summary: Mathematics often begins with definitions; however, it is much more difficult to succinctly describe the whole of mathematics.

For most students, the subject of mathematics is studied on an almost daily basis from kindergarten through high school. But if asked, “What is mathematics?” the majority might struggle to formulate a meaningful answer, perhaps saying that mathematics is “arithmetic” or “algebra.” But saying that the essence of mathematics is arithmetic is akin to saying that the essence of chemistry is the periodic table; mathematics and chemistry are both so much more. Not only may students have difficulty describing what mathematics is, but even professional mathematicians struggle to provide a succinct, convincing description of the nature of their subject of expertise.

In his delightful 1940 essay, A Mathematician’s Apology, G. H. Hardy (1877–1947) spends about 150 pages presenting a passionate case for the meaning, essence, and importance of mathematics and the professional mathematician. Along the way, he offers many keen insights into the nature of mathematics:

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.… The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.

This quote may appear odd; most people do not view mathematics as a creative endeavor, much less one that can be rightly considered “beautiful.” Often, students who learn the subject view the discipline as one that is rigidly bound by rules, one in which there is always one right answer, and perhaps even that there is only one right path to follow. But mathematicians have a decidedly contrary viewpoint. Faced with an interesting problem to solve, the mathematician strives to have his full cadre of creativity flowing, perhaps asking: “What unusual approach might I take to solve this problem, one that nobody else has yet considered?” “How might I alter the problem to a new, related one that I might be able to solve first?” and “Is there new language or notation that I might introduce that makes the problem easier to understand or similar to another problem that is already well understood?”

More than this, as Hardy’s quote alludes, mathematics is about more than individual problems; rather, it involves the study of patterns. If mathematicians can solve one particular problem, they are next interested in knowing if their methods extend to solving an entire collection of related problems. If a theorem can be proved to explain a wide class of situations, is it possible to extend the result to include even more possible scenarios? In this way, mathematics and mathematicians seek to recognize, understand, and explain patterns. Some of these patterns occur in the world around us; others may be purely theoretical. Once a pattern is understood or explained, mathematicians wonder if they have found the best explanation. What is “best”? While that is somewhat a matter of individual taste, most mathematicians agree that the best mathematics is clear, brief, and elegant. In Hardy’s words, “the ideas… must fit together in a harmonious way.” It usually takes a great deal of creative insight (creative thinking, creative writing, and creative problem solving) to make the ideas fit together in a harmonious way.

Philosophers on Mathematics

Philosophers have argued for centuries, even millennia, over the nature and meaning of mathematics. There are entire schools of thought—referred to with names like “intuitionism,” “logicism,” and “formalism”—that seek to explain what mathematics is. However, each somehow comes up short. Perhaps mathematics itself is simply too big to describe with a formal philosophical system. Some parts of mathematics do rely on our intuition and understanding of physical happenings in the surrounding world; other aspects of the subject rely considerably on the foundations of logic; and part of mathematics grows from the formal rules that seem to many to lie at its very core. But no one of these perspectives encompasses the entire subject nor satisfactorily describes its essence. Nor does any one of these perspectives fully answer the question of where mathematics exists. Is it embedded in the surrounding world, or is it a mental construct that rightly belongs to humanity’s collective brain?

Nobel prize–winning physicist Eugene P. Wigner (1902–1995) was one scientist who recognized the beauty, power, and harmony of mathematics while still being somewhat mystified by its nature. In his own 1960 essay on the question of what mathematics is, titled “The Unreasonable Effectiveness of Mathematics,” Wigner tells the story of a statistician sharing with a friend his work in analyzing population trends. In one of his key formulas, the symbol π arises. The friend asks, “What does that symbol represent?” The statistician notes that, as usual, the symbol is the familiar π associated with circles—the ratio of a circle’s circumference to its diameter. The friend is incredulous, for how can the relationship between a circle’s circumference and diameter have anything to do with how a population is distributed?

This story that opens the essay illustrates Wigner’s broad point: mathematics is unreasonably effective, with abstract mathematical ideas emerging in remarkable and surprising places. As a scientist trying to understand the workings of the physical universe, Wigner was particularly mystified by how well mathematics helped him describe observable phenomena. In his words, “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation for it.” He goes on to argue that somehow the very nature of mathematics, even though it is abstract and a mental construct, leads the way in describing the surrounding world and that somehow this is indicative of a deeper truth. He concludes the essay by observing, “the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.”

To the pure mathematician, mathematics may be a quest to recognize, understand, and explain abstract patterns that arise in considered ideas. To the applied mathematician, mathematics may be a language that aptly describes patterns that emerge in some sort of physical reality. Somehow, it is the same mathematics in both cases, and the subject seems not to care whether or not it is used for abstract or applied purposes. The history of mathematics is filled with stories that show how mathematics emerges from the mental doodling of interested people, only later to find rich connections with other areas of mathematics itself, and then finally to spectacularly describe some deep physical reality.

As an example, the Greeks (c. 350 b.c.e.) came to know a beautiful number with marvelous abstract properties, today called the “golden ratio”

This number, approximately 8/5, arises naturally from considering line segments or rectangles that can be divided in ways that are self-similar and possesses a wide variety of interesting geometric and numeric properties. Roughly 1500 years later, people in India (c. 1150) first encountered the so-called “Fibonacci numbers”: (1, 1, 2, 3, 5, 8, 13, 21, 34,…), which come from starting with a pair of 1s, and then adding the preceding two numbers to create the next. Spectacular patterns and relationships exist among the Fibonacci numbers, and mathematicians have been fascinated with them since. An early observation showed that the ratios of consecutive Fibonacci numbers (5/3, 8/5, 13/8, 21/13,…) forms a sequence of numbers that converges to

Much later, near the end of the twentieth century, mathematicians and biologists came to understand the apparent role that both Fibonacci numbers and the golden ratio play in explaining seed distributions in plants, such as coneflowers and sunflowers: the golden ratio appears to be the constant angle at which seeds are “born,” and the relationships the golden ratio enjoys with the Fibonacci numbers help explain why this phenomena occurs, which one can better understand when the seeds in the flower are numbered.

What is Mathematics?

There is a great deal of delightful reading one can pursue to learn more about the nature of mathematics. Such investigation will help each person decide individual answers to the question “What is mathematics, really?” For the novice mathematician, Steven Strogatz has written the quintessential modern sequence of essays on the topic, essentially in the form of a blog for the New York Times. Strogatz, a prominent applied mathematician who has done groundbreaking work in the field of dynamical systems, begins with the wonderful 2010 essay “From Fish to Infinity,” where his overall goal for the series is to be “writing about the elements of mathematics, from pre-school to grad school, for anyone out there who’d like to have a second chance at the subject—but this time from an adult perspective. It’s not intended to be remedial.” For the reader with a bit more mathematics background, one can consider the American Mathematical Society’s Online Feature Column, a monthly column that takes a look at accessible mathematical research and (often) its applications. To begin, the interested reader might read David Austin’s immensely popular 2006 explanation of Google’s PageRank algorithm, “How Google Finds Your Needle in the Web’s Haystack.” For a more historical view, it is hard to beat the marvelous writing of Professor William Dunham in his 1990 book Journey Through Genius, which surveys some of the great theorems of mathematics.

An encyclopedia entry is a tiny start to describing the essence of mathematics. Each person must read, explore, think, and investigate to seek understanding of what mathematics really is. It is a beautiful example of the depth and complexity of mathematics itself that so many different perspectives on the subject ring true and that each person can find something unique in the subject.

Bibliography

Austin, D. “How Google Finds Your Needle in the Web’s Haystack.” American Mathematical Society. http://www.ams.org/featurecolumn/archive/pagerank.html.

Dunham, W. Journey Through Genius. Hoboken, NJ: Wiley, 1990.

Hardy, G. H. A Mathematician’s Apology. Cambridge, England: Cambridge University Press, 1940.

Naylor, M. “Golden, √2, and π Flowers: A Spiral Story.” Mathematics Magazine 75, no. 3 (June 2002).

Wigner, E. “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Communications in Pure and Applied Mathematics 13, no. I (February 1960). http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html.