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.
1 comment:
Do you think it we be good to think if Mathematics as the study of abstraction? It would be a good way to think of how math can be so useful in many areas of study. Would also explain why it is so fundamental. We can say we see the world visually. But maybe we also see the world with thought. This happens a lot in physics, where we predict a particle should exist with given properties. The higgs boson or the top quark are two such examples. We saw them through mathematics before we experienced them.
Maybe it is difficult to define what mathematics studies because it is not physical. It exists within our minds, but it does exist. We can see the effects on how we think and interpret the world. So often people ask how imaginary numbers are valid. They are valid because we have use for them. Is the number zero valid? What about negative numbers or irrational numbers? They are all valid because we have use for them. They are still abstract ideas. And that is what we study in mathematics.
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