The development of physics and mathematics has gone hand in hand for much of recorded history. Indeed, the development of mathematics as a purely abstract course of study (with no connection whatsoever with the real world) is a relatively recent historical development. Although mathematics was systematized as an axiomatic (and hence abstract) system as long ago as Euclid, the motivation for most of plane geometry and early mathematics was utterly practical - to aid in mundane tasks such as laying out fields, architecture and navigation. These concrete pursuits and applications guided the intuitions that were eventually turned into ``mathematics".
However, it wasn't until the time of Newton that the marriage between physics and mathematics got its fundamental kick in the pants and shifted into high gear. Geometry had existed by then for thousands of years. Algebra (and concepts such as the zero) had been invented by Arabian mathematicians during the flowering of Islamic culture. Following Galileo, observational astronomy had advanced to where a great deal was known about planetary motion from a geometric point of view (thanks to Kepler and his assistant, Tycho Brahe).
Newton took those observations, and with his sheer genius invented simultaneously a new science (modern classical physics) and a new mathematics (calculus). Although calculus was more or less simultaneously invented by Leibnitz (in an alebraic formulation superior to Newton's that was ultimately adopted in favor of Newton's), in both cases it got its impetus from its obvious and immensely successful connection to Newton's new physics.
Calculus is the mathematics of change. In its differential formulation it relates rates of change of pairs of quantities. In the eyes of physicists (and other scientists, economists, statisticians), these quantities are real quantities, with associated measures and units; in the eyes of mathematics they are algebraic abstractions expressed in the form of functions and their associated variables. In its integral formulation, it permits us to add up tiny differential increments of a quantity to get a sum total.
Since the time of Newton, the fundamental mathematical basis of classical physics has changed only a little. To succeed in physics, you need to understand geometry and its functional algebraic partner, trigonometry. You need mad skills in algebra. Above all, you need to be able to work with calculus, abstracting from a physical problem the important quantities, determining a differential or integral relation from the laws of nature and a nonverbal intuitive understanding of ``how things work", and then working through to an algebraic calculus-based solution.
This suffices to describe nearly any system microscopically, but when dealing with lots of particles (where the value of ``lots'' is determined in part by the tools you have to work with, and with modern computers can be quite large indeed) or when dealing with a system whose state is only approximately known it is often necessary to give up a microscopic approach in favor of a statistical approach. Thus we need to add an understanding of probability and statistics to our collection of the mathematics needed by physics. This is also useful when dealing with experimental physics, as a fundamental characteristic of real-world measurements is that they are imprecise, so that there is always a stochastic element to an experimental observation.
Geometry, algebra, calculus, probability and statistics. These branches of mathematics (and their more advanced offspring such as differential equation theory, geometric algebra, vector calculus, analysis) are still the bulk of the language of physics all the way through to the edge of field theory.
This chapter presents a very brief, note-style review of the essential mathematics required to learn the physics in this course.