Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.Quantity, structure, space, and change? These words outline a vast field of knowledge — and they are combined with a very narrow, mechanistic, and, frankly, quite boring description of “what mathematicians do”. Should “what mathematicians do” really be a part of the definition?
The definition given by the German Wikipedia is interesting in a different way: it stresses that there is no definition of mathematics, or at least no commonly accepted one. I translate:
Mathematics is the science that developed from the investigation of figures and computing with numbers. For mathematics, there is no commonly accepted definition; today it is usually described as a science that investigates abstract structures that it created itself for their properties and patterns.Is this a good definition, a satisfactory answer to the question “What is Mathematics”? I believe that Wikipedia (in any language) does not give a satisfactory answer. At the same time, and much more importantly, high school curricula do not give a satisfactory answer. Even the famous book by Richard Courant and Herbert Robbins entitled “What is Mathematics?” (and subtitled “An Elementary Approach to Ideas and Methods”) does not give a satisfactory answer.
Perhaps it is impossible to give a good definition in a sentence or two. Indeed, we may claim that there cannot be one single answer that we could be content with: mathematics in the 21-st century is a huge body of knowledge and a very diverse area of study. There are thus so many ways to experience mathematics — the arenas of national and international competitions, and research experiences that range from years spent working in solitude (think of Andrew Wiles, who proved Fermat’s Last Theorem, or Grigori Perelman, who proved the Poincar´e conjecture) to coffee break discussions at conferences to massive collaborations on internet platforms (such as the POLYMATH projects initiated by Michael Nielsen, Timothy Gowers, Terence Tao, and others).
But perhaps the English Wikipedia is right in one aspect — that in approaching the science called mathematics one should look at the people who do mathematics. So what is mathematics as an experience? What does it mean to do mathematics? Supply your own answer and surely you will find the best definition of Mathematics according to your perspective.
Alfore is a reluctant blogger, a lazy poet, and a great daydreamer. He owns a Nobel, Pulitzer, Oscar and a bunch of weirdly named cats.
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