Defining mathematics. According to Wikipedia in English, in the March 2014 version, the answer to “What is Mathematics?” is
Mathematics is the abstract study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.
Mathematicians seek out patterns (Highland & Highland, 1961, 1963) and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.
None of this is entirely wrong, but it is also not satisfactory. Let us just point out that the fact that there is no agreement about the definition of mathematics, given as part of a definition of mathematics, puts us into logical difficulties that might have made Gödel smile.Footnote 1
The answer given by Wikipedia in the current German version, reads (in our translation):
Mathematics […] is a science that developed from the investigation of geometric figures and the 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 by logical definitions using logic for their properties and patterns.
This is much worse, as it portrays mathematics as a subject without any contact to, or interest from, a real world.
The borders of mathematics. Is mathematics “stand-alone”? Could it be defined without reference to “neighboring” subjects, such as physics (which does appear in the English Wikipedia description)? Indeed, one possibility to characterize mathematics describes the borders/boundaries that separate it from its neighbors. Even humorous versions of such “distinguishing statements” such as
“Mathematics is the part of physics where the experiments are cheap.”
“Mathematics is the part of philosophy where (some) statements are true—without debate or discussion.”
“Mathematics is computer science without electricity.” (So “Computer science is mathematics with electricity.”)
contain a lot of truth and possibly tell us a lot of “characteristics” of our subject. None of these is, of course, completely true or completely false, but they present opportunities for discussion.
What we do in mathematics. We could also try to define mathematics by “what we do in mathematics”: This is much more diverse and much more interesting than the Wikipedia descriptions! Could/should we describe mathematics not only as a research discipline and as a subject taught and learned at school, but also as a playground for pupils, amateurs, and professionals, as a subject that presents challenges (not only for pupils, but also for professionals as well as for amateurs), as an arena for competitions, as a source of problems, small and large, including some of the hardest problems that science has to offer, at all levels from elementary school to the millennium problems (Csicsery, 2008; Ziegler, 2011)?
What we teach in mathematics classes. Education bureaucrats might (and probably should) believe that the question “What is Mathematics?” is answered by high school curricula. But what answers do these give?
This takes us back to the nineteenth century controversies about what mathematics should be taught at school and at the Universities. In the German version this was a fierce debate. On the one side it saw the classical educational ideal as formulated by Wilhelm von Humboldt (who was involved in the concept for and the foundation 1806 of the Berlin University, now named Humboldt Universität, and to a certain amount shaped the modern concept of a university); here mathematics had a central role, but this was the classical “Greek” mathematics, starting from Euclid’s axiomatic development of geometry, the theory of conics, and the algebra of solving polynomial equations, not only as cultural heritage, but also as a training arena for logical thinking and problem solving. On the other side of the fight were the proponents of “Realbildung”: Realgymnasien and the technical universities that were started at that time tried to teach what was needed in commerce and industry: calculation and accounting, as well as the mathematics that could be useful for mechanical and electrical engineering—second rate education in the view of the classical German Gymnasium.
This nineteenth century debate rests on an unnatural separation into the classical, pure mathematics, and the useful, applied mathematics; a division that should have been overcome a long time ago (perhaps since the times of Archimedes), as it is unnatural as a classification tool and it is also a major obstacle to progress both in theory and in practice. Nevertheless the division into “classical” and “current” material might be useful in discussing curriculum contents—and the question for what purpose it should be taught; see our discussion in the Section “Three Times Mathematics at School?”.
The Courant–Robbins answer. The title of the present paper is, of course, borrowed from the famous and very successful book by Richard Courant and Herbert Robbins. However, this title is a question—what is Courant and Robbins’ answer? Indeed, the book does not give an explicit definition of “What is Mathematics,” but the reader is supposed to get an idea from the presentation of a diverse collection of mathematical investigations. Mathematics is much bigger and much more diverse than the picture given by the Courant–Robbins exposition. The presentation in this section was also meant to demonstrate that we need a multi-facetted picture of mathematics: One answer is not enough, we need many.