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Introduction to Mathematics

"A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas."

G. H. Hardy, 1940

The definition for mathematics is not as clear-cut as one might think.

  • As the study of numbers - this is only partially true. Mathematics is not just about numbers, but also about shapes, patterns, and relationships. Many things deal with non-numerical concepts, such as geometry, topology, logic, and set theory.
  • As a deductive science - this is also only partially true. Mathematical reasoning is not just deductive, but also inductive, sometimes even abductive. Many things deal with inductive reasoning, such as statistics, probability, and combinatorics. Rather than being "pure" or "applied", mathematics is a spectrum of both, and various fields of mathematics can lie anywhere on this spectrum.
  • As the "science of formal systems" (Haskell Curry in 1951) - this does not capture the full essence of mathematics. Mathematics is not just about formal systems, but also about intuition, creativity, and discovery. Pure math without intuition is soulless.

However, the definitions aside, there are some common themes in mathematics:

  • Mathematics is about abstracting away the unnecessary details and focusing on the essential properties of objects. The first thing you learn, numbers, are abstractions of the real world. The number 1 is not a physical object; you can't say "give me 1". Instead, numbers, are abstractions of counting, and they have properties that are useful in various contexts.

    One advantage of abstraction is that it allows us to study a wide range of objects with similar properties. For instance, the same differential equations that describe the motion of a pendulum can also describe the flow of heat in a metal rod.

  • Mathematics is logical. It is about making precise arguments and drawing conclusions from them. You start with a set of axioms, which are assumed to be true, and then use logical reasoning to derive theorems. The axioms are the starting point, and the theorems are the conclusions. The process of deriving theorems from axioms is called proof. However, mathematics is not just about proving theorems; estimation, approximation, and conjecture are also important.

  • Mathematics is about patterns and structures. It is about recognizing patterns in the world and using them to build structures. For instance, the Fibonacci sequence is a pattern that appears in many natural phenomena, such as the branching of trees and the arrangement of leaves on a stem. The concept of a group is a structure that appears in many areas of mathematics, such as number theory, geometry, and algebra.

Generally, mathematics is a logical, abstract, and creative discipline that studies the world in a precise and structured way.

Discovery Process​

As previously mentioned, mathematics mixes together intuition and formulation. The intuition is the visual and intuitive interpretation of the concept, while the formulation is the more formal description that shows how the concept can be generalized and computed.

The general discovery process for mathematics is;

Specific Example → Intuition → Mathematical Model → Wide-Ranging Applications

This shows the advantage of abstraction. By abstracting things into their fundamental properties, we can apply the same concepts to a wide range of problems.

For instance, when Newton was studying the motion of the planets, he developed the concept of calculus to describe the motion of objects in general. Today, calculus is used everywhere, from physics to economics to computer science to just about every field you can think of.

One shortcoming of the modern education system is that it usually does this process in reverse. It starts with the formulation. This is probably partly responsible for the large distaste for mathematics. It's like learning to paint by starting with the chemical composition of paint.

Fields of Mathematics​

Mathematics is a vast field, with many different branches and subfields. Some of the main branches of mathematics include:

  • Algebra, which studies the properties of mathematical objects such as numbers, vectors, and matrices.
  • Analysis, which studies the properties of functions and their derivatives and integrals.
  • Geometry, which studies the properties of shapes and spaces.

These are just a few of the many branches of mathematics. Each branch has its own set of concepts, techniques, and applications, yet they are all interconnected in some way.

Another way to categorize mathematics is by its level of abstraction:

  • Pure mathematics, which studies abstract concepts and structures for their own sake.
  • Applied mathematics, which uses mathematical tools and techniques to solve real-world problems.

Pure and applied mathematics are not mutually exclusive; they are two ends of a spectrum. Many areas of mathematics lie somewhere in between, using abstract concepts to solve practical problems.

One particular saying I like is that "pure mathematics explores the world of mathematics, while applied mathematics explores the world with mathematics". It perfectly illustrates the difference between the two, and how they are both important in their own right.

Resources​

The main resources I used to create these notes are:

These are all excellent resources for learning (esp. self-learning) mathematics.