In the late seventeenth century, a peculiar mystery began. It is a bit of a favorite, as great math legends go. It’s a classic story math professors love to tell.

A man named Pierre de Fermat jotted something down in the margin of a mathematics book. Translated from the original Latin, it said,

*It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers.*

*I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.*

This dense-looking paragraph can be written neatly as a mathematical equation. But we’ll discuss the equation in the next post. The point here is about Fermat’s promise.

In short, Fermat thought he could prove the truth of that mathematical statement. Nobody had proven it before. His argument was “truly marvelous”—but, alas, the margin was too narrow to hold it.

Here begins the mystery. Fermat died without ever writing or speaking about this proof again. His son Samuel found the cryptic scribble after his father’s death.

Now you are likely waiting for the hook. What was so sensational here? If you are unfamiliar with mathematical language, Fermat’s note might look no more interesting than, say, a microwave construction manual.

But in terms of the study of abstract patterns, this idea was HUGE.

When the note was revealed, a race began to find Fermat’s “marvelous” proof. Brilliant mathematicians tried and failed to prove his statement. The problem became known as Fermat’s Last Theorem.

Fermat’s Last Theorem may sound familiar to you. It was in the news relatively recently – just twenty years ago. Fermat died in 1665, but it wasn’t until 1994 that his theorem was finally proved, by British mathematician Andrew Wiles. It was considered a monumental mathematical achievement.

Why?

Or more specifically, what was it that Fermat failed to accomplish?

The key word here is “proof.”

In everyday life, the word “proof” is somewhat ambiguous. Usually it means “enough evidence to convince people of something.” A ten-year-old can “prove” she is athletic by shooting a basketball from mid-court. A defendant in the courtroom can “prove” his innocence by providing a valid alibi. A scientist or historian can present a theory and “prove” it with experiments or artifacts.

In mathematics, on the other hand, there is only one kind of proof—and it is of a completely different nature. A mathematical proof is an argument that follows the rules of logic perfectly. And because it follows the rules of logic perfectly, it is literally impossible to refute. If I prove something mathematically today, then in a thousand years, my argument will still be unbreakable.

This kind of proof is unique to mathematics. In the scientific kind of proof, theories are always at risk, because new information can come along and contradict the old. But math is built from the bottom up. If done properly, it is unbreakable, no matter what future scientists discover.

Its strength comes from its style of reasoning. Math begins with basic assumptions and follows through with the logical consequences of those assumptions. The assumptions are called “axioms.”

Axioms and logic together are the foundation of every mathematical proof. No test tubes, or field samples, or even apples dropping from a tree—just axioms and logic. With just axioms and logic, any true mathematical idea can be formulated and proven.

The name “axiom” sounds formidable, but the ideas are simple. Axioms are un-splittable kernel ideas. They are agreed upon by everyone. One axiom might be, “between any two points, a line can be drawn.”* This axiom may seem pretty obvious. And it is. Axioms are meant to be as simple as possible. This is how they form the framework of a mathematical field.

After axioms are chosen, the framework is set. Then there is only one way forward: logical reasoning.

Mathematicians who are building proofs start by make deductions from the axioms (or, in practice, by work that was already built up from the axioms). Usually, the logical deductions that come straight from the axioms are nearly as simple as the axioms themselves. Those deductions, in turn, can lead to slightly more complex conclusions. In some ways it is simple, like hammering nails into wood to build a house. But in other ways, it is wide open for creativity: the mathematician is creating the blueprint as he builds. Step by step, logical reasoning gives rise to an entire mathematical theory. By the time we get to full-fledged concepts like spheres or integration, axioms are usually long out of sight.

* As a side note, there is one type of concept that is arguably more basic than an axiom: the objects that the axiom talks about. One must assume the existence of points and lines to even begin discussing, for example, geometry. To do arithmetic, one needs to assume the existence of addition and multiplication. Whereas a triangle can be defined in terms of points, lines and angles, a point cannot be defined in simpler mathematical terms. And whereas exponents can be defined in terms of addition and multiplication, addition and multiplication cannot be defined in simpler mathematical terms.

The beauty of this logical structure is that it lies outside subjective experience. Once the axioms are in place, the rules of logic determine the conclusions. One might imagine each mathematical field as a cavern of undiscovered crystals. The cavern is too dark to see. But the crystals are there. Once we have chosen the cavern we wish to explore, we cannot change what the crystals inside it will look like. We can only shed light on the crystals that are already there.

Logic, of course, warrants its own discussion. But here, the important thing is that any two people applying logic to the same idea will never contradict each other, even if they take different particular steps.

Math can be difficult. It can be complex. It can lead to surprising conclusions or even be completely counterintuitive.

But because every step in each theory was taken carefully and logically, we know that no flaw that will undermine the steps the mathematician has taken. No contradictions will be deduced later. No input from the physical world will change it. All of the discoveries within a mathematical field are consistent with each other– and sometimes consistent with entirely different fields as well. The crystals that form in the caverns all sparkle with the same facets, because they all grew through the same process.

When a person sees the consistent ideas grow through pure logic, the effect is, for many, indescribable. It is a pleasure and a joy. It is one of the main reasons people study mathematics.

And proofs are the way to get there.

Fermat promised an idea tantalizing to those who appreciate proofs—a “marvelous” bit of reasoning that would prove the conjecture he’d written. We’ll discuss Fermat’s Last Theorem more in the next post.

Resources:

Wolfram MathWorld

MacTutor

Very interesting! I’d never understood what a mathematical proof is before.

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I had never understood what one could “prove” in math. Now my understanding is that certain ideas can be reasoned out, from basic assumptions. Then those ideas are “proven” because they have been logically established. Is that correct?

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Very nice post!

As a side note to your side note: once you assume the existence of addition, multiplication can be derived – it is just repeated addition (just like powers are repeated multiplication).

Also, as you implicitly indicate, subtraction and division can be derived as “opposites” or inverses of addition and multiplication.

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To get from addition to multiplication, do you need to assume the existence of repetition? Would you be assuming that anyway?

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If I recall (it’s been a while), the way that addition is defined is by repetitively applying the successor function, or successor operation, which adds 1 to a number. For instance, x+5 is attained by applying that function to x five times.

Similarly 5x is attained by adding (as defined above) x to itself for a total of five x’s.

Putting this aside, I wonder if the idea of repetition is a mathematical idea, object or axiom in the way you describe or is it a more basic notion that is implicit in the human mind. What do you think?

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Ah, I see what you mean.

Hmm… I would think that repetition is implicit in the human mind. Do you have any thoughts about that?

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