What is mathematics?

The word Mathematics comes from the Greek word μάθημα (máthēma) meaning "knowledge, study, learning". When we call someone a polymath, we mean that they are knowledgeable in a range of different areas.

Mathematics gives us languages through which we can describe and model the world that we observe. It goes beyond that into the study of abstractions and their properties.

The logical systems underlying the practice of mathematics are themselves the subject of mathematical study and algebraization.

Fundamental concepts are questioned, generalised, used and connected.

Math is an abstract yet a priori (knowledge independent of prior experience) way of expressing and communicating logic using numbers---presenting them, operating on them, manipulating them.

We can use mathematical constructs to describe many things, some common, some not so common. Some fairly representative examples:

• Addition describes the accumulation of objects into a larger group.
  Subtraction describes the removal of objects from a group.
  Multiplication describes the assembly of same-size groups of objects into larger groups.
  Division describes their separation into same-size smaller groups.
• Limits tend to describe the hypothetical forwards motion towards a destination that may or may not exist.

You see it in Zeno's Paradox, for instance, where you move forwards by half the required distance to get to your destination, an infinite number of times (by the main claim of the paradox, you can never get there, but it's an old paradox, because you can get there).

• Derivatives describe how a quality of an object changes with respect to another quality.

For instance, velocity is how something's position changes over time. The derivative is the velocity at each very short, instantaneous moment in time.

• Integrals could describe the accumulation of infinitely thin intervals of adaptive heights that overall determine the areas and/or volumes of normal and abnormal shapes.

For instance, you can find the volume of a donut (also called a torus) if you knew how use the Shell Method with a circle equation.