I propose a definition of mathematics:
Mathematics is the deductive analysis of structures
- Deductive, because empirical data does not generate mathematical results; only logical deduction does so.
- Analysis, because mathematicians tease out the consequences of the definitions of structures, rather than merely admiring, or using, or playing with them.
- Structures because … well, this is the question: what do mathematicians study?
What do Mathematicians Study?
The OED definition of mathematics has
“The abstract deductive science of space, number, quantity, and arrangement, including geometry, arithmetic, algebra, etc., studied in its own right (more fully pure mathematics), or as applied to various branches of physics and other sciences (more fully applied mathematics).Shorter OED 2007
but this is clearly an ad hoc list, as the “etc” acknowledges. Are there limits to what mathematicians study? If so, they must change every time a new area is invented.
But every branch of mathematics straightforwardly has this in common : it analyses a particular structure, or family of structures (that is, a structure of structures), and deductively analyses it, that is to say draws out its properties and relationships to other structures.
There are favoured structures. Numbers, of course. Then the Euclidean plane, which is the structure of lines and points on a flat surface. These favoured structures define the familiar major areas of mathematics–number theory, algebra, geometry, analysis. “Progress” in mathematics divides into, on the one hand, discovering new things about known structures; and on the other hand choosing new structures to study.
New structures may be chosen precisely for the light they shed on old ones: complex numbers, for instance, shone a new light on algebra, as did topology on geometry. Whenever a new structure is found to have interesting properties it becomes, inevitably, a part of mathematics and even, if there is work enough in exploring it, is dubbed a branch of mathematics, which is perhaps the ultimate mathematical status.
The virtue of naming structure as the subject of mathematics is that it becomes easy to say whether something is or is not mathematics: anywhere there is a structure that can be analysed deductively, there is a subject of mathematics. The ad-hoc element of the definition is banished.
And it reminds us not to be surprised every time a new branch of mathematics opens up. If it moves, or even if it doesn’t, it’s fair game to a mathematician.