Maths SolverThe first time someone showed me a proof, I asked the obvious question: what was the point? The thing being proved was something nobody seriously doubted. The proof was twice as long as the statement. And nothing about everyday mathematics seemed to require this level of formality — you can solve a quadratic equation without proving the quadratic formula first.
The answer, which I came round to slowly, is that a proof is not a “convincing argument.” It is something more specific than that, and the difference is worth understanding even if you never plan to write one yourself.
What separates a proof from an argument
An argument can be very persuasive and still be wrong, especially in mathematics, because mathematics is the field where almost-true is not true at all. The classic example: “every odd number greater than $5$ is the sum of three odd primes” (the weak Goldbach conjecture). For more than two centuries, every odd number anyone checked turned out to satisfy this. Tens of trillions of cases were verified by computer. The conjecture was “obviously” true. Then in 2013, the Peruvian mathematician Harald Helfgott finally proved it — ending two centuries of “obviously true but unproven” with the only thing that mathematicians recognise as ending it: a proof.
This sort of pattern is so common that mathematicians have learned not to trust intuition or numerical evidence by themselves. There are statements in mathematics that hold for the first $10^{43}$ values and then fail. (Skewes’s number is the classic example: a statement about prime distribution that holds up to about $10^{316}$ before any counterexample turns up.) If you just check cases, you can spend a lifetime never finding the one that breaks the pattern, and still be wrong.
A proof rules out this kind of disaster. A correct proof shows that the statement is true for every case at once, by an argument that does not depend on checking individual cases. That is a different kind of certainty from “I checked a lot of examples.”
A worked example: the square root of 2 is irrational
This is the proof I would show first to anyone who has never seen one. It is short, complete, and demonstrates several techniques that recur throughout mathematics.
Statement. $\sqrt{2}$ cannot be written as a fraction of two integers.
Proof. Suppose, for the sake of contradiction, that it can. Then $\sqrt{2} = p/q$ for some integers $p$ and $q$, with $q \ne 0$. Without loss of generality, take this fraction in lowest terms, so $p$ and $q$ have no common factor.
Squaring both sides: $2 = p^2 / q^2$, which rearranges to $p^2 = 2q^2$.
The right side is even (twice an integer), so $p^2$ is even. But $p^2$ is even only when $p$ is even (an odd number squared is odd). So $p = 2k$ for some integer $k$. Substituting back: $(2k)^2 = 2q^2$, which is $4k^2 = 2q^2$, or $q^2 = 2k^2$.
By exactly the same argument, $q$ is even.
But we assumed $p$ and $q$ had no common factor, and now we have shown both are even — they share the factor $2$. This is a contradiction. So our original assumption (that $\sqrt{2}$ could be written as a fraction) must be false.
Therefore $\sqrt{2}$ is irrational. $\blacksquare$
This proof is from Euclid (or close to it). It is a few hundred words long, every step follows logically from the one before, and the conclusion is bulletproof. There is no fraction $p/q$ that equals $\sqrt{2}$, no matter how cleverly you choose $p$ and $q$. You could spend the rest of time trying to find one and you would fail.
The three styles of proof you will see
The proof above uses one of the three standard styles. The others are worth knowing about, since most mathematics uses some combination.
Direct proof
You assume the hypothesis and derive the conclusion through a sequence of valid steps. The proof of the Pythagorean theorem by similar triangles — or the proof that the sum of the angles in a triangle is $180^\circ$ — is direct. Most school-level proofs are direct.
Proof by contradiction
You assume the opposite of what you want to prove, derive a contradiction, and conclude that your assumption must be wrong. The $\sqrt{2}$ proof above is proof by contradiction. So is Euclid’s proof that there are infinitely many primes (assume finitely many, multiply them all together and add 1, derive a contradiction). Contradiction is often the cleanest tool when “direct” would require you to enumerate cases.
Proof by induction
You prove the statement for the simplest case ($n = 1$), then prove that if it holds for any case $n$, it must hold for $n + 1$. This is like setting up a row of dominoes and pushing the first one. Used constantly when you need to prove something for every positive integer at once. The proof that $1 + 2 + \cdots + n = n(n+1)/2$ is the standard introductory induction.
Real mathematics uses combinations and variants of these. Some proofs run to hundreds of pages and use dozens of intermediate lemmas. The proof of Fermat’s Last Theorem is over a hundred pages and uses techniques from a dozen different fields, but the basic structure — assumptions, sequence of valid steps, conclusion — is the same.
Why proofs matter outside mathematics
Most students do not go on to write professional proofs. So why is proof writing a standard part of secondary school maths in many curricula?
The honest answer is that learning to write a proof teaches a specific kind of thinking that is useful in any field where you have to be careful with reasoning. Programming, law, economics, engineering — any discipline where you have to chain statements together and justify each step — uses the same skills that proof writing develops.
Specifically, writing proofs teaches you:
- To distinguish what you have shown from what you have assumed. A surprisingly common error in casual reasoning is to use the conclusion as a hidden premise. Proof writing makes this almost impossible because every step must be justified from earlier steps or from explicit assumptions.
- To notice when a step doesn’t actually follow. “It is intuitively clear that...” is a phrase that does not appear in correct proofs, because intuitive clarity is not a logical justification. Forcing yourself to write proofs trains you to spot the moves that feel right but aren’t actually justified.
- To handle “all” and “some” correctly. A surprising number of real-world arguments confuse universal claims with existential ones. (“All swans are white” is different from “some swans are white,” and the difference matters when one black swan shows up.) Proof writing makes you precise about quantifiers.
This is why even a humanities student who never touches a proof again after school benefits from having written a few. The skill generalises.
What a proof is not
A few things proofs are sometimes mistaken for, and aren’t.
A proof is not a derivation. Deriving the quadratic formula from completing the square is mostly mechanical algebra; it becomes a proof only when you also justify each step (e.g., that the operations performed preserve equality). In most school contexts the distinction is glossed over and “derivation” is used loosely.
A proof is not a computation. Computing $\int_0^1 x^2\,dx = 1/3$ is not a proof of anything; it is the application of a procedure. A proof of the fundamental theorem of calculus, which justifies why the procedure works, is what makes the computation reliable.
A proof is not a diagram. Diagrams can illustrate proofs and make them easier to follow, but the proof is in the words. A beautifully drawn diagram of two parallel lines and a transversal is not, by itself, a proof that alternate angles are equal — the proof is the verbal justification of why the angles must be equal, with the diagram as scaffolding.
The discipline of certainty
Mathematics is the field that has worked hardest at the question “how do we actually know this?” The answer it has arrived at, after several thousand years, is “by writing down a proof.” A proof is, in this sense, the formalisation of careful reasoning — the gold standard against which every other kind of argument is measured.
You do not have to find this exciting. Most people, even most mathematicians, find a few of their proofs deeply satisfying and many of them tedious. But the existence of the proof system is what makes mathematics distinct from every other intellectual activity. A proven theorem stays proven; it does not need to be re-checked, re-tested, or re-justified by every new generation. That is the unusual thing about mathematics, and it is what proofs buy.
Whether you take that as inspiring or excessive is mostly a matter of taste. But it is real, and once you internalise the distinction between “I am pretty sure this is true” and “here is a proof,” you will never quite read mathematical claims the same way again.