Perhaps the most fundamental question about Mathematics, as well as the one least explored by mathematicians is the nature of Mathematics. That is, how exactly can one define Mathematics as a field, properly and rigorously? We have rather good definitions about most fields of study. Chemistry is the study of the composition of matter, Biology is the study of living organisms, Medical Science is the study of disease and Sociology is the study of human societies. Mathematics, on the other hand, is a far more difficult field to encapsulate in a simple description like that.
To illustrate this difficulty, consider any simple definition of Mathematics. It cannot be the study of numbers and quantities, since the most fundamental fields of Mathematics (such as Set Theory, Group Theory, Logic and Proof Theory) rarely have much to do with these concepts. Similarly, it cannot be the study of proofs, since application of knowledge, rather than its formulation, is a very large part of Mathematics. Mathematics can be wholly taught without any symbols, but it can also be taught with only symbols. It is both conceptual and descriptive, both theoretical and applied.
So, what is Mathematics? As noted above, a simple definition as a field of study is hard to come up with. But perhaps Mathematics is not a field of study at all. Once we consider this idea, new possibilities arise. Mathematics is, in fact, a philosophy and a way of thinking. It is logic and inference, applied to any abstract or concrete structure, including numbers and quantities. It is the formulation and usage of knowledge about such structures, in a formal language which is shared by all mathematicians. In the end, it is the fruit of all curiosity. Whenever any field is studied in a formal and precise way, whenever a pattern is being sought out and described, whenever new knowledge is sought and created, the methods used are mathematical in nature.
Naturally, the above is too general a definition for a classroom subject; mathematical education is generally far less ambitious in its scope. But next time you are taking a course in Mathematics, or encounter a mathematical question, think of it as a chance for you to learn a new way of thinking; you may enjoy it, despite the amount of apparently irrelevant theory.
