What is Mathematics?

To those who do not know Mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty of nature. ... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.
Richard Feynman. 1918-1988.
American physicist.
The Character of Physical Law

Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
Bertrand Russell. 1872-1970
British philosopher, mathematician.
Mysticism and Logic


We know what biology is about; it is about living things. We know what history is about; it is about events in the past. And we know that sociology is about the behaviour of human beings in society. But if you ask a dozen mathematicians what mathematics is about, you are likely to get a dozen answers.

One answer is that mathematics is the study of abstraction, of how we create, manipulate, interpret, and apply abstract structures. Many people think of mathematics as the study of numbers, or of numbers and geometry. Certainly numbers are one of the abstract structures that mathematics considers.  Geometric concepts like points and lines and circles are another. But these are only examples of some sorts of abstract objects that mathematicians study. Mathematicians also consider sets and categories and functions and graphs and much more.

The process of abstraction is a process of thinking about something complicated, but focusing on just one or a few aspects of it, eliminating all the busyness that could distract from the structure we are interested in.  Numbers are the most familiar abstraction: we consider only quantity without it being the quantity of anything – three without it being three apples or three dogs or three lines, just three.  Then we learn to work with the abstract concepts. We can subtract numbers without having to think of removing five oranges from a pile of seven oranges. We finish our calculation, then apply the result to know that we have two oranges left.

As we continue the study of mathematics we find that we are creating abstraction from abstractions. We can, for example, create systems that behave almost like numbers but don’t follow all the same rules. We can create geometries where parallel lines do meet. We can create spaces with five or eleven or even infinitely many dimensions.

All of this takes us far beyond arithmetic, algebra and geometry. This process of abstraction gives us a set of tools for describing almost anything with structure to it. And the world has structure.

Mathematics relies on logic and proof rather than observation as its standard of truth. But even so, it employs observation, exploration, simulation and even experimentation. Surprisingly, even very high levels of abstractions turn out to be useful in the real world. Mathematics reveals otherwise hidden patterns that help us to understand the world around us. Many have commented on the “unreasonable effectiveness” of mathematics in the natural sciences. 

New applications of mathematics are constantly emerging across the entire spectrum of the natural and social sciences, business, medicine, and almost any human endeavour. Advances in computer design, instrumentation and control, medical technology, computer graphics, economics, finance, genetics and geology all rely on the power of mathematics.

Mathematics is offered at KPU at the Surrey and Richmond campuses, with a select offering at the Langley campus. Check out these offerings in the Calendar.