Why higher mathematics?

Some basic questions

What is the use of mathematics, and why study it in the first place?

The usual responses to these questions are that:

  1. Mathematics has wide ranging applications in the sciences and in engineering.
  2. Mathematics teaches you to think logically.

I am not sure, however, these responses go far enough. My own view is that mathematics is a language, and fluency in it is an essential skill for the understanding of the world. Mathematics is everywhere, as patterns like the Koch snowflake below remind us.

science space fractal mathematics fractal geometry
The Koch snowflake, an early example of a fractal. Fractals have found applications ranging from modelling veins in your hand, to clouds, and to the stock market.

A mathematical view

Ignoring the mathematical nature of our world is short-sighted. Computer scientists know this historical lesson well: mathematics was part of their toolkit from the beginning.

The mathematician Alan Turing invented the first computer (although some scholars think it was Charles Babbage in the 1830s).  During World War II, Turing built his German code-breaking computing machine during World War II. At the time, no one had a clue that such technology would lead to the computers we know of today. Modern computers all rely on the basic conceptual architecture that Turing introduced. We now use our laptops, tablets, and phones to conduct commerce, communicate, and socialize on an unprecedented global scale.

Turing’s code-breaking machine (1943).

Without mathematics, we aren’t going to make much sense of the universe. But it isn’t a panacea.  Literature, music, and the visual arts give us another a platform to explore reality. We are moved by the Mona Lisa or latest pop song in way different and more immediate than the the Langlands program in number theory or Perelman’s proof of the Poincaré conjecture.

Mathematics doesn’t have to exist in competition with the arts or any other discipline. Rather, it complements them, making their study richer and our human experience broader. Mathematical biology is a good example. We are witnessing unprecedented application of mathematics to topics like the spread of disease and the study of protein networks. The figure below show a recent mathematical model for interventions in the spread of the Ebola epidemic in West Africa. A step towards the cure of certain diseases such as cancer might rely on the analysis of biological data using complex mathematical models. We can only speculate what will occur when mathematics synthesizes more fully with these and other fields. 

Rivers CM, Lofgren ET, Marathe M, Eubank S, Lewis BL. Modeling the Impact of Interventions on an Epidemic of Ebola in Sierra Leone and Liberia. PLOS Currents Outbreaks. 2014 Oct 16

Jenkins v Gowers

Simon Jenkins created buzz recently in the UK by questioning the role of mathematics in the British school curriculum.  Apparently, he has criticized STEM education in the UK for many years in his columns. Jenkins piece led to a response from the prominent Cambridge mathematician Tim Gowers. Gower’s thesis is that it’s not about what is taught in mathematics education, it’s about how it is taught. He wants students to think rather than manipulate meaningless symbols on paper.

Tim Gowers

No surprise: I wholeheartedly agree with Gowers. Our students at all levels (elementary to university) should be exposed to mathematical problem solving. Students need to learn how to think, as my undergraduate algebra professor Günter Bruns would say.

2004 Gunter Obituary_Page_1
The late lattice theorist Günter Bruns (1928-2002).

Our own university curriculum is biased towards rote learning, especially in first year mathematics courses. I love teaching Calculus and Linear Algebra, but I wish I could introduce these subjects differently than the mainstream way. I would much rather teach our first-year students about how matrices are used in Google’s PageRank algorithm, say, than have them practice dozens of Gauss-Jordan eliminations.

The price of progress

Bruns told me once that mathematics is a by-product of an affluent society. There is undeniable wisdom in that; if we are concerned about things like finding food or adequate shelter, then noncommutative algebraic geometry seems irrelevant.

What Bruns didn’t reference in his comment is that we can’t progress as a culture without new advances in mathematics. We likely won’t cure diseases, explore space, or even tackle climate change without a greater understanding of mathematics. We also need parallel advances in the sciences, engineering, humanities, the arts, medicine, to face new and old challenges.

While mathematics is not the only important thing in education, it is an essential part of it. Ignoring that makes your world view more narrow. We must not allow ourselves to be cut off from a deeper understanding of the beautiful mathematical language inherent in nature.

Anthony Bonato

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