How to Read Mathematics

Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs.

Mathematics is written in a language of its own, with a few hundred symbols, a few dozen stock phrases and a logic stricter than any lawyer’s. None of it is hard, but it is dense: $H_n(X)=\ker {\partial }_n/\mathrm{im}{\partial }_{n+1},$ the formula this whole book is travelling towards, packs a paragraph of meaning into a single line. This chapter is your phrasebook. Nobody learns a phrasebook by heart: read it once, then come back whenever a symbol or a sentence stops you.

What a page of mathematics is made of

A page of a novel is a stream of sentences. A page of mathematics is built from a handful of standard blocks, each with a label, and each with a different job. Knowing the job tells you how to read the block.

The words “theorem”, “proposition” and “lemma” describe the importance of a result, not its kind: all three are true statements with proofs. A conjecture is a different kind of sentence altogether: nobody knows yet whether it is true. In this book each block lives in its own coloured box, so you can always see what kind of sentence you are reading. Some boxes are the book’s own: key ideas, intuition, warnings, history notes, and questions to pause and ponder. The conjecture boxes describe open problems from every part of the subject together with what is known about them, and the Open Problems page collects them all.

Definitions are membership tests

A mathematical definition is not a vague description; it is a test. Each clause is a box to tick, and an object qualifies exactly when every box is ticked. Here is a definition you have known since childhood, written the way a mathematician would write it.

Run some objects through the test. Is $6$ even? Yes: $6=2\cdot 3,$ and $3$ is an integer. Is $0$ even? Yes, $0=2\cdot 0$ — something people often doubt, but the definition settles it. Is $-4$ even? Yes, $-4=2\cdot (-2).$ Is $7?$ Here the test gets interesting. The equation $7=2k$ does have a solution, $k=3.5,$ but $3.5$ is not an integer, so $7$ fails. It fails on a single clause, “for some integer”, and that shows you what the clause is for: without it, every number would be even. Non-examples that almost qualify are the quickest way to see what each part of a definition is doing.

There is one convention you must know. Read literally, “$n$ is even if $n=2k”$ only says that numbers of the form $2k$ are even; it does not seem to say that every even number has this form. But it is meant both ways. As Richard Hammack puts it in Book of Proof, “it is an almost-universal convention that definitions are phrased in the conditional form, even though they are interpreted as being in the biconditional form.” In a definition, “if” means “if and only if”.

Theorems have hypotheses and a conclusion

Most theorems have the shape “if [hypotheses], then [conclusion]”. The hypotheses are what you are allowed to assume; the conclusion is what you get. You can take a theorem apart this way even when you do not understand its words. For example, a theorem from topology says:

You do not yet know what “continuous” or “compact” mean (that is §2.1 Spaces and Continuity), yet you can already say: there are two hypotheses, “$f$ is continuous” and “$X$ is compact”, and one conclusion, “$f(X)$ is compact”. Before you can use a theorem, you must check all of its hypotheses. Halmos’s questions are worth asking every time: is each hypothesis really necessary, and where does the proof use it?

Conjectures: mathematics in the making

Every theorem was once a guess. A guess that has been stated precisely, tested and shared, but not yet proved, is a conjecture, and conjectures are where much of mathematics happens: they tell a field what to try to prove next. George Pólya put it this way: “mathematics in the making resembles any other human knowledge in the making. You have to guess a mathematical theorem before you prove it; you have to guess the idea of the proof before you carry through the details” (Pólya 1954, vol. I, p. vi). There is also a saying among mathematicians, often repeated and never traced to a source, that ranks the work: a mathematician proves theorems, a good mathematician makes conjectures, and a great one creates definitions. It is unfair to proofs, which are the only way a guess becomes knowledge. But it is right that knowing what to prove is much of the art, and that the deepest contributions are often new ways of seeing, written down as definitions.

Bernhard Riemann made both kinds. In 1854 he gave a lecture defining what we now call a manifold, a space that looks like flat space near each of its points (Riemann 1854); this book uses that definition from §2.4 Manifolds and Surfaces on. In 1859, in an eight-page paper on counting primes, he guessed that the zeros of a certain function all lie on one line in the plane (Riemann 1859). That guess, the Riemann hypothesis, is still the most famous open problem in mathematics.

A conjecture can end in four ways.

• Proved. Pierre de Fermat wrote, around 1637, that $x^n+y^n=z^n$ has no solutions in positive whole numbers when $n\ge 3.$ Andrew Wiles announced a proof in 1993; a gap was found, and in 1994 he closed it, with a final step done together with Richard Taylor (Wiles 1995; Taylor & Wiles 1995).
• Disproved. In 1769 Leonhard Euler conjectured that a fifth power cannot be a sum of fewer than five fifth powers, and similarly for higher powers. In 1966 Leon Lander and Thomas Parkin found, by computer search, that $27^5+84^5+110^5+133^5=144^5$ (Lander & Parkin 1966).
• Undecidable. Some questions cannot be settled by the usual axioms of mathematics at all. The first famous example is about sizes of infinity, and it appears in §1.1 Sets and Functions.
• Still open. Like the Riemann hypothesis, and like the conjecture in the box below.

Lately a conjecture can also meet a new kind of fate: a claimed proof found by a computer. In 2026 AI systems produced proofs, or claimed proofs, of several well-known open problems, some of which appear in boxes later in this book. A claim is not yet a theorem. It becomes one when experts have checked it, or when it has been verified line by line by a proof assistant, a program that accepts a proof only if every step follows from the rules; and even then mathematicians want to understand why it is true. The boxes in this book say, for each problem, how far that checking has got.

Here is one of the oldest open conjectures, simple enough to explain to a child, and the first of the conjecture boxes in this book.

The logic of mathematical sentences

Everyday language lets its small words wobble. A parent who says “if you don’t finish your vegetables, there is no pudding” is heard by every child as a promise of pudding if they do, although, strictly, nothing of the kind was said. A proof cannot afford that wobble, so mathematics gives each small word one exact meaning.

First, the raw material. A statement is a sentence that is definitely true or definitely false. “$7$ is a prime number” is a true statement; “$2+2=5”$ is a false one. “$x>3”$ is not yet a statement — its truth depends on $x$ — but it becomes one as soon as $x$ is known, or as soon as we say “for every $x”$ or “for some $x”$ in front of it (the subject of the next section). Statements are combined using a handful of small words, and those are what we pin down next.

And, or, not

• “$P$ and $Q”,$ written $P\wedge Q,$ is true when both $P$ and $Q$ are true. (The symbol $\wedge$ looks like an A, for “and”.)
• “$P$ or $Q”,$ written $P\vee Q,$ is true when at least one of them is true — including when both are. This is the inclusive or. In everyday speech, “tea or coffee?” usually means one or the other, not both; in mathematics “or” always allows both.
• “not $P”,$ written $\neg P,$ is true exactly when $P$ is false.

Negating “and” and “or” follows two rules called De Morgan’s laws: “not ($P$ and $Q)”$ means “not $P,$ or not $Q”$ — at least one fails — and “not ($P$ or $Q)”$ means “not $P,$ and not $Q”$ — both fail. If someone says “it is not true that I speak French and German”, they might speak neither, or only one.

If … then: an implication is a promise

The most important little word in mathematics is “if”. The statement “if $P$ then $Q”,$ written $P\Rightarrow Q$ and read “$P$ implies $Q”,$ is best understood as a promise: whenever $P$ happens, $Q$ will happen too. A promise is broken in exactly one situation — when $P$ happens and $Q$ does not. In every other situation the promise is kept.

The last two cards are the ones people find strange. If it does not rain, the promise “if it rains, I bring an umbrella” cannot have been broken, whatever I did. In logic, “not broken” counts as true. A statement $P\Rightarrow Q$ whose hypothesis $P$ is false is called vacuously true. So “if $2+2=5,$ then the Moon is made of cheese” is a true statement — a silly one, but true.

This convention is not a trick; it is exactly what mathematics needs. Consider “every element of the empty set is purple”. Is it true? It says: for every $x,$ if $x$ is in the empty set, then $x$ is purple. Since nothing is in the empty set, the “if” is never satisfied, and the promise is never broken. So the statement is (vacuously) true — and so is every statement of the form “every element of the empty set has property such-and-such”. This will matter: the empty set, and the empty shape, have all properties of this kind.

There are many ways to say $P\Rightarrow Q$ in words, and you will meet all of them:

• if $P,$ then $Q;$ $Q$ if $P;$ $Q$ whenever $P;$
• $P$ implies $Q;$ $P$ only if $Q;$
• $P$ is sufficient for $Q;$ $Q$ is necessary for $P.$

“$P$ only if $Q”$ trips everyone up at first: it means that $P$ cannot happen without $Q,$ which is the same promise as $P\Rightarrow Q.$ “You may board only if you have a ticket” rules out boarding without a ticket; it does not promise that a ticket gets you on board.

Converse and contrapositive

Every implication $P\Rightarrow Q$ has two famous relatives:

• its converse, $Q\Rightarrow P,$ which swaps the two sides;
• its contrapositive, $\neg Q\Rightarrow \neg P,$ which swaps them and negates both.

An implication and its contrapositive are always both true or both false: “every $P$ is a $Q”$ and “anything that is not a $Q$ is not a $P”$ are the same promise, seen from opposite ends. The converse is a different promise altogether, and it can fail even when the original holds. Confusing a statement with its converse is one of the commonest errors in reasoning, in and out of mathematics. The child with the vegetables made it: “finish and you get pudding” is the contrapositive of “no pudding means you didn’t finish”, which is the converse of what the parent said. Now try the next puzzle before reading its explanation. Most people get it wrong the first time.

If and only if

When a promise holds in both directions — $P\Rightarrow Q$ and $Q\Rightarrow P$ — we write $P⟺Q$ and say “$P$ if and only if $Q”,$ often shortened to “$P$ iff $Q”.$ It means $P$ and $Q$ are true in exactly the same situations: they are equivalent, and each is both necessary and sufficient for the other. To prove an “iff”, you prove two implications, one in each direction; proofs often label them ($\Rightarrow )$ and ($\Leftarrow ).$

Quantifiers: for all, there exists

A sentence like “$x^2\ge 0”$ becomes a statement once we say which $x$ we mean. Two symbols do almost all of this work.

• $\forall ,$ read “for all” or “for every” (an upside-down A, for all). “$\forall x\in \mathbb{R},x^2\ge 0”$ says every real number has a non-negative square. True.
• $\exists ,$ read “there exists” or “for some” (a backwards E, for exists). “$\exists x\in \mathbb{Z},x^2=4”$ says some integer has square $4.$ True: $x=2$ works (so does $-2).$ “There exists” means at least one.
• $\exists !,$ read “there exists exactly one”. “$\exists !x\in \mathbb{R},x^3=8”$ is true (only $x=2),$ but “$\exists !x\in \mathbb{R},x^2=4”$ is false: there are two.

The phrase “$\in \mathbb{R}”$ (“in the real numbers”) names the domain the variable ranges over, and it matters: “$\exists x,x^2=2”$ is true over the real numbers ($x=\sqrt{2})$ but false over the fractions, as we prove below.

Proving and disproving

Each quantifier tells you exactly what a proof or a disproof must do:

• To prove “for all $x,$ …”: take an arbitrary $x$ (“let $x$ be any real number”) and argue for it without using anything special about it. Checking examples, however many, is not a proof.
• To disprove “for all $x,$ …”: find a single counterexample, one $x$ for which it fails.
• To prove “there exists $x,$ …”: exhibit one $x$ that works.
• To disprove “there exists $x,$ …”: show that every $x$ fails — which is a “for all” statement.

Order matters

When a sentence has several quantifiers, their order changes its meaning completely. Compare: “everyone has a mother” and “there is someone who is everyone’s mother”. In symbols, with $x$ and $y$ ranging over people:

The first is true; the second is absurd. The difference is who chooses first. In the first, $y$ is chosen after $x,$ so it may depend on $x$ — each person has their own mother. In the second, one $y$ must be chosen before seeing any $x,$ and it must work for all of them.

This is best understood as a game. You play $\exists :$ your job is to find the values that make the statement true. A Skeptic plays $\forall :$ their job is to choose values that make it false. The quantifiers, read left to right, say who moves when. The statement is true exactly when you have a strategy that always wins.

Negating a statement, mechanically

To disprove a statement, you prove its negation. So you need to be able to write the negation of a complicated statement, and do it reliably. The good news is that it is completely mechanical: write “not” in front, then push it inward one step at a time using a handful of rules. Each time “not” passes a quantifier, the quantifier flips:

(The symbol $\equiv$ here means “says the same thing as”.) “Not everyone passed” means “someone did not pass”; “nobody passed” — “there is no one who passed” — means “everyone failed”. When “not” passes “and”, “or” or “if … then”, it uses De Morgan’s laws and the broken-promise rule $\neg (P\Rightarrow Q)\equiv P\wedge \neg Q.$

The shapes of proofs

A proof is an argument that would convince a careful, sceptical reader, in which every step follows from definitions, from results already proved, or from logic. Henri Poincaré put the division of labour in a single line: “it is by logic one demonstrates, by intuition one invents” (Poincaré 1908, Book II, ch. 2). Finding a proof is an act of imagination; writing it down is an act of logic. Most proofs you will meet in this book have one of six shapes, and recognising the shape is half of understanding the proof.

Direct proof

To prove $P\Rightarrow Q:$ assume $P,$ and reason forward until you reach $Q.$

Notice the move in the first line: we replaced the word “even” by its definition. Unpacking definitions is the first step of almost every proof.

Proof by contrapositive

To prove $P\Rightarrow Q,$ it is enough to prove the contrapositive $\neg Q\Rightarrow \neg P,$ since the two say the same thing. This is useful when “not $Q”$ gives you something concrete to work with.

Trying to prove this directly is awkward: knowing that $n^2$ is even tells you little about $n$ itself. The contrapositive turns the problem around into something easy. Invariants in topology work exactly this way.

Proof by contradiction

To prove a statement, assume it is false and derive something impossible. Since logic never leads from truths to impossibilities, the assumption must have been wrong.

So the argument is more than two thousand years old. Look back at the proof and you will find that clash of odd and even at its heart.

Double inclusion

To show that two collections of objects are the same, show that everything in the first belongs to the second, and everything in the second belongs to the first. For instance: the numbers that are sums of two consecutive integers are exactly the odd numbers. First direction: $m+(m+1)=2m+1$ is odd. Second direction: an odd number $2k+1$ equals $k+(k+1).$ Both inclusions hold, so the two collections coincide. In §1.1 Sets and Functions this becomes the standard way to prove two sets equal.

Induction

To prove that a statement holds for every whole number $n=1,2,3,\dots ,$ it is enough to prove two things: the base case, that it holds for $n=1;$ and the inductive step, that whenever it holds for some $n,$ it also holds for $n+1.$ Think of an infinite row of dominoes: the first one falls, and each falling domino knocks over the next. Then all of them fall.

“Well-defined”

The last shape is not a proof of a theorem but a check on a definition. Sometimes we try to define something by a rule that seems to depend on a choice. The rule is well-defined if it gives one and the same answer whatever choice is made. For example, “the numerator of a fraction” is not well-defined: $\frac{1}{2}$ and $\frac{2}{4}$ are the same number, but their numerators are $1$ and $2.$ Yet “double the numerator” is well-defined, although it looks at the numerator too: $\frac{1}{2}$ becomes $\frac{2}{2}$ and $\frac{2}{4}$ becomes $\frac{4}{4},$ and both answers are $1.$ Using a choice is allowed; the answer must not depend on it. Such checks are everywhere once we start “declaring things the same” in §1.2 Equivalence and Quotients, and every map between homology groups must pass one.

A dictionary of symbols

Every symbol is an abbreviation for words, and every formula can be read aloud as a sentence. When you meet a formula, try reading it out loud, slowly; if you cannot, find the symbol that stops you. The dictionary below lists the most common symbols in this book, with how to say them and where they are explained (the small § numbers).

Greek letters and their jobs

Mathematicians run out of Latin letters quickly, so Greek ones are everywhere. Each tends to have a typical job, which gives you a head start: a $\gamma$ is probably a path, a $\sigma$ probably a simplex, an $\epsilon$ probably small. Here is the whole alphabet in order, with a common English pronunciation and the job each letter does in this book. The few letters this book never needs are there too, with the work they do elsewhere.

One symbol, several meanings

A few symbols do several different jobs. Context always decides, but it helps to be warned:

Sameness, and which way the arrows go

“The same” is the most important and most slippery phrase in mathematics. The equals sign is only the strictest of several sameness symbols, and choosing the right notion of sameness is often the whole art of a subject. Topology is the art of choosing a very generous one.

The equivalence sign $\sim$ will be the hero of §1.2 Equivalence and Quotients, where we learn how to declare things the same and what happens when we do. The symbol $\cong$ is introduced properly in §1.3 Groups (isomorphic groups) and again in §2.1 Spaces and Continuity (homeomorphic spaces), and $\simeq$ in §2.3 Homotopy.

Subscripts push forward, superscripts pull back

Here is a reflex worth building now, long before it is needed. Suppose $f:X\to Y$ is a function. Two kinds of things can be transported along it, and they go in opposite directions.

• Points go forward. A point $x$ of $X$ is carried to the point $f(x)$ of $Y.$ So is anything built from points: a set, a path, a loop.
• Measurements come back. If $g$ assigns a number to every point of $Y$ — a temperature, say — then $g\circ f$ assigns a number to every point of $X:$ to measure at $x,$ go to $f(x)$ and read $g$ there. The function $f$ went from $X$ to $Y,$ but the measurement travelled from $Y$ to $X.$

How to read a mathematics book

Mathematics is not read like a novel. A page can take as long as a chapter of fiction, and that is not a sign that something is wrong. Here is some advice that generations of mathematicians have found useful.

• Fight it. Paul Halmos’s advice, from the epigraph, continues with a list of questions to ask of every result: “Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?” You now know what every word in those questions means.
• Examples first. When you meet a definition, stop and build an example, and a non-example. When you meet a theorem, test it on the simplest case you can think of, and on a silly one (the empty set, a single point).
• Keep a pencil. Redo each computation. Draw the pictures. Fill in the steps the author skipped (“clearly” and “it is easy to see” mean “check this yourself”).
• Statement before proof. Make sure you understand what a theorem says, and why anyone would want it, before worrying about why it is true. On a first reading it is fine to skip a proof and come back.
• Don’t miss the big picture; make the idea your own. These two pieces of advice come from Shai Simonson and Fernando Gouvêa’s essay How to Read Mathematics. Ask where a section is going, and try to say each idea in your own words.
• Being stuck is normal. Every mathematician spends most of their time not understanding something. Being stuck is not a sign of failure; it is what thinking feels like. Move on, and come back.

The book itself is built to help you read it this way. Dotted underlined words open a definition when you hover or tap them; links like §1.2 Equivalence and Quotients take you to where an idea is explained; every exercise has a hidden solution, which you should open only after an honest attempt; and every figure is something to play with, not just to look at.

A first look at the destination

Here, finally, is the formula at the centre of this book: the definition of homology. In §0.1 The Shape of a Question you saw what its pieces mean. Now read it the way a mathematician does: symbol by symbol, aloud. You are not expected to understand it yet — only to read it. Step through it, or hover over any symbol.

Read in words, the formula says: the holes of a shape are its cycles (pieces with no loose ends), where two cycles count as the same if they differ by a boundary, and a cycle that is itself a boundary counts as no hole at all. Every word in that sentence will be given an exact meaning, a picture, and a set of examples you can play with. By the end of Part III, $H_n(X)=\ker {\partial }_n/\mathrm{im}{\partial }_{n+1}$ will read as naturally as $2+2=4.$

Exercises

Summary

Further reading

• Book of Proof, Chapters 2 and 4–10 — Richard Hammack free

  Logic, quantifiers and every proof shape in this chapter (direct, contrapositive, contradiction, induction), explained slowly with many exercises. The best free companion to this chapter.
• How to Prove It: A Structured Approach — Daniel J. Velleman

  Teaches proofs by analysing the logical form of statements — exactly the skill of the quantifier and negation sections, developed in depth.
• Reading Mathematics (Mathematical Communication) — MAA Mathematical Communication project free

  A gathering of advice on reading mathematics, including Simonson and Gouvêa’s essay “How to Read Mathematics” (“don’t miss the big picture”, “make the idea your own”).
• Proofs: A Long-Form Mathematics Textbook — Jay Cummings

  A friendly, chatty introduction to proofs that shows the “scratch work” behind each one — how proofs are found, not just how they are written.
• There’s more to mathematics than rigour and proofs — Terence Tao free

  The three stages of mathematical education — pre-rigorous, rigorous, post-rigorous — and why rigour should sharpen intuition, not replace it.
• On proof and progress in mathematics — William P. Thurston free

  A great mathematician on what it means to understand mathematics, and the many ways people think about the same idea.