What Is Mathematics?

Introduction

Many people encounter mathematics primarily as a collection of calculations. You are given a formula, substitute some numbers, perform several operations, and obtain an answer.

That is certainly part of mathematics. But it is not the whole subject.

At a deeper level, mathematics is the study of precise relationships, structures, patterns, and logical consequences. Mathematicians do not merely calculate answers. They ask questions such as:

• What exactly does a statement mean?
• Why should the statement be true?
• Under what assumptions is it true?
• Can we prove that it is always true?
• Can we construct an example where it fails?
• What other conclusions follow from it?

These questions lead directly to one of the central ideas of higher mathematics: proof.

Mathematics Is More Than Calculation

Consider the statement:

You can verify this by calculation. But mathematics quickly asks a more general question. What happens for arbitrary real numbers \(a\) and \(b\)?

This is no longer a calculation involving two particular numbers. It is a general mathematical statement. The question becomes: Why is this true for every pair of real numbers?

That question requires reasoning about the structure of the real numbers rather than simply performing arithmetic.

Patterns Are the Beginning, Not the End

Mathematics often begins with the recognition of a pattern. For example:

The results suggest:

A mathematician might reasonably conjecture that this formula is true for every positive integer \(n\).

But observing the pattern is not the same as proving it.

What Is a Mathematical Statement?

A mathematical statement is a sentence that has a definite truth value: it is either true or false. For example:

The distinction will become important because mathematical proofs operate on precise statements.

Definitions Give Mathematics Its Precision

Suppose someone says:

"The sequence eventually gets close to zero."

What exactly does "eventually" mean? What exactly does "close" mean? Different people could interpret those words differently. Mathematics solves this problem by introducing precise definitions.

For example, later in this course we will define convergence of a sequence using a statement involving an arbitrary positive number \(\varepsilon\). The definition will eventually take the form:

At first glance this may look complicated. But it illustrates one of the great strengths of mathematics: an intuitive idea can be converted into a statement whose meaning is completely precise.

Definitions Are Not Theorems

This distinction is fundamental. A definition tells us what a mathematical object or concept means. A theorem tells us something that can be logically deduced from definitions, axioms, and previously established results. For example, we might define an even integer by saying that an integer \(n\) is even if there exists an integer \(k\) such that

That is a definition. We could then prove that the sum of two even integers is even. That is a theorem.

Conjectures

A conjecture is a mathematical statement that is believed to be true but has not yet been proved. For example, after examining several examples, you might conjecture that a particular property holds for every positive integer.

A conjecture can turn out to be:

• true and eventually proved;
• true but extremely difficult to prove;
• false and disproved by a counterexample; or
• still unresolved.

This is an important part of mathematical discovery. Mathematics does not begin with a list of facts that everyone already knows. Mathematicians explore, experiment, conjecture, prove, disprove, generalize, and refine.

Theorems

A theorem is a mathematical statement that has been established by a valid proof. For example, the Pythagorean theorem states that for a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\),

The important feature is not merely the equation. The important feature is that the equation follows necessarily from the assumptions defining the geometric situation.

Lemmas, Propositions, and Corollaries

Mathematics uses several names for proved statements.

The distinction is partly organizational. All of these statements require justification when they are presented as mathematical results.

What Is a Proof?

A proof is a logically valid sequence of reasoning that demonstrates that a mathematical statement follows from accepted assumptions and previously established results. Consider the simple statement:

The sum of two even integers is even.

Suppose \(a\) and \(b\) are even. By the definition of evenness, there exist integers \(m\) and \(n\) such that

Therefore,

Since \(m+n\) is an integer, \(a+b\) has the form \(2k\) for an integer \(k\). Therefore \(a+b\) is even.

Why Examples Cannot Prove a Universal Statement

Suppose someone claims:

Testing values might produce many primes. For example, you could check \(n=1\), \(n=2\), \(n=3\), and many more. But no finite collection of successful examples can prove a statement about every positive integer.

To disprove a universal statement, however, a single counterexample is enough. This asymmetry is fundamental:

Mathematics Builds Knowledge in Layers

One of the most important things to understand about advanced mathematics is that new results are built from earlier results. A simplified structure looks like this:

Mathematical Abstraction

As mathematics becomes more advanced, it becomes increasingly abstract. At first, you might study particular numbers:

Later, you study an arbitrary integer \(n\). Then you study sets of integers. Then functions between sets. Then sequences of functions. Then spaces whose elements may themselves be functions.

The objects become more abstract, but the underlying method remains the same: define precisely, reason carefully, prove what follows.

Why Precision Matters

In ordinary conversation, statements are often understood from context. Mathematics cannot rely on that degree of ambiguity. Compare:

"This sequence gets close to 1."

with:

The second statement is much more demanding. It tells us exactly what "gets close" means. This precision allows mathematicians to prove results that would be impossible to establish reliably using informal language alone.

What You Will Learn in This Course

The goal of this 1,000-tutorial sequence is not simply to memorize mathematical facts. It is to develop the ability to think mathematically. The sequence will gradually introduce:

• logic and mathematical language;
• proof techniques;
• sets and functions;
• the real number system;
• sequences and series;
• limits and continuity;
• topology;
• compactness and connectedness;
• differentiation;
• integration;
• uniform convergence;
• metric spaces;
• and eventually more advanced areas of analysis.

The first lessons will assume almost nothing. Later lessons will increasingly assume that you have mastered the earlier material.

The Most Important Habit

There is one habit that will make an enormous difference as you progress through mathematics:

If a theorem is presented, ask:

• What are the assumptions?
• What exactly is being claimed?
• Why are the assumptions necessary?
• What definitions are being used?
• Which earlier results are being used?
• Could the theorem fail if one assumption were removed?
• Can I reproduce the proof without looking at it?

These questions will eventually become second nature. That is one of the major goals of rigorous mathematics education.

Summary

Mathematics is much more than computation. It is a system for describing structures and relationships with precision and establishing conclusions through logical reasoning.

The central building blocks are:

The most important transition in this course is the transition from doing calculations to constructing and understanding proofs. That transition begins with the very next tutorial.