Introduction
This book is intended to organize my thoughts on mathematics and to provide a reference for the concepts that I have learned.
What is Mathematics?
It is very difficult give a precise definition of mathematics. It is a very broad field that encompasses many different areas of study. For this reason, I will give the definition that I like the most. First i want to define what mathematics is not.
- Mathematics is not only about numbers: A lot of people think that mathematics is only about numbers. This is not true. For instance, logic, abstract algebra, topology, and geometry are areas of mathematics and they can be studied without using numbers.
Although, mathematics is not only about the things that are listed above, it is also about them. So, what is mathematics? Mathematics is an approach to understand the world around us by modeling them. And these models are actually abstractions of the real world. This is why mathematics is so powerful. It is not about the real world, it is about the models of the real world.
For this reason, I believe that the most important property of mathematics is abstraction. Here are some quotes that support this idea:
Mathematics is the art of giving the same name to different things.
Henri Poincaré
Mathematics is the art of reasoning about quantitative relations between abstract structures.
Paul Erdős
Mathematics is the science of patterns.
Keith Devlin
Mathematics is the science of patterns, and we study patterns in the abstract.
Ron Aharoni
This definition, I think captures the essence of mathematics. Another approach that I follow generally to define a thing
What is abstraction?
Abstraction is a process of removing details from a concept in order to focus on the essential properties of that concept. Lets look some examples of abstraction:
A word apple is an abstraction of the concept of an apple. We are removing the details of the apples and focusing
on the essential properties of an apple. Lets imagine that we have two apples in front of us. One of them is red and
the other one is green. We can say that the red apple and the green apple are different. But, if we abstract the
concept of an apple, we can say that they are the same and both of them are apples. This is the power of abstraction.
Definitions
A definition is where everything starts. It is the most basic concept in mathematics. A definition is a statement that gives the meaning of a term. It is a way to give a name to a concept. Without giving names to concepts, it would be very difficult to talk about them. For this reason, definitions are very important in mathematics. If you think, you could not grasp a topic you should first check the definitions of the terms that are used in that topic. Without understanding the definitions, it is impossible to understand the topic. You should really understand the definitions.
Lets look some examples of definitions:
A natural number is either zero or a successor of a natural number.
A number is said to be even if it is divisible by 2.
A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.
Propositions
A proposition is a statement that is either true or false.
Axioms
An axiom is a statement that is assumed to be true without a proof. Axioms are the foundation of mathematics. They are the building blocks of a mathematical structure. Axioms are the starting point of a mathematical theory.