# Computation and Mathematics

We often think of Mathematics as the art and science of manipulating numbers. This is not entirely wrong but is still quite a limited view.

## 1 + 1 = 2

⬆this is how mathematics and computation looked many centuries ago

This is modern mathematics:

More recently mathematics is about big abstract ideas and then trying to make them useful in the real world (which is desirable but not always possible). The abstract space of mathematics is conceptually larger than anything else we know and only minuscule parts of it find applications in practice. We often find uses for very abstract theories only after long time has passed since initial discoveries.

## Computation

One of the big ideas of mathematics is the notion of computation. Please note that computation ≠ calculation.

Calculation is understood as applying basic operations to numbers.

Computation is all about continuously and reliably manipulating state to achieve the appropriate output / behaviour of the computing system.

Here is one prominent way of describing computation — a State Machine:

I have tried to show that state machines provide a simple conceptual framework for describing computation.

...

The advantage of state machines is that they can be described using ordinary mathematics. Mathematics is simpler and more expressive than any language I know that computer scientists have devised to describe computations. It is the basis of all other branches of science and engineering. Mathematics should provide the foundation for computer science as well.