Proofs

In the natural sciences, truth is established by empirical means, involving observation, measurement, and (the gold standard) experiment. In mathematics, truth is determined by constructing a proof—a logically sound argument that establishes the truth of the statement.

The use of the word “argument” here is, of course, not the more common everyday use to mean a disagreement between two people, but there is a connection in that a good proof will preemptively counter (implicitly or explicitly) all the objections (counterarguments) a reader might put forward.

When professional mathematicians read a proof, they generally do so in a manner reminiscent of a lawyer cross-examining a witness, constantly probing and looking for flaws. Learning how to prove things forms a major part of university mathematics. It is not something that can be mastered in a few weeks; it takes years. What can be achieved in a short period, is gain some understanding of

What it means to prove a mathematical statement?

Why mathematicians make such a big deal about proofs?

Constructing or reading a proof is how we convince ourselves that some statement is true. I might have an intuition that some mathematical statement is true, but until I have proved it—or read a proof that convinces me—I cannot be sure. But I may also have need to convince someone else. A proof of a statement must explain why that statement is true.

In the first case, it is generally enough that my argument is logically sound and I can follow it later. In the second case, where I have to convince someone else, more is required: the proof must also provide that explanation in a manner the recipient can understand i.e they also have to succeed communicatively

Most proofs in mathematics can be read and understood by any professional mathematician, though it can take days, weeks, or even months to understand some proofs sufficiently to be convinced by them. Examples given to university mathematics students can usually be understood with at most a few hours’ effort.

Fermat’s Last Theorem is an unusual example. The great French mathematician Pierre de Fermat conjectured (stated in belief) in the seventeenth century - "that for exponents n ≥ 3, the equation x n + y n = z n has no whole number solutions for x, y, z ". This finally proved in 1994 when the British mathematician Andrew Wiles constructed a long and extremely deep proof. Though most mathematicians lack the detailed domain knowledge to follow Wiles’ proof themselves, it did convince the experts in the field of analytic number theory. As a result Fermat’s ancient conjecture is now regarded as a theorem. It is also popularly known as Fermat’s Last Theorem.

Proving a mathematical statement is much more than gathering evidence in its favor

The great mathematician Leonard Euler stated his belief that every even number beyond 2 can be expressed as a sum of two primes. This property of even numbers had been suggested to him by Christian Goldbach, (and became known as the Goldbach Conjecture). It is possible to run computer programs to check the statement for many specific even numbers, and to date (July 2012) it has been verified for all numbers up to and beyond 1.6 × 10 raised to 18 (1.6 quintillion). Most mathematicians believe it to be true. But it has not yet been proved. All it would take to disprove the conjecture would be to find a single even number n for which it could be shown that no two primes sum to n.

Incidentally, mathematicians do not regard the Goldbach Conjecture as important. It has no known applications or even any significant consequences within mathematics. It has become famous solely because it is easy to understand, was endorsed by Euler, and has resisted all attempts at solution for over 250 years

There is no particular format that an argument has to have in order to count as a proof

Whatever you may have been told at school, there is no particular format that an argument has to have in order to count as a proof. The one absolute requirement is that it is a logically sound piece of reasoning that establishes the truth of some statement. An important secondary requirement is that it is expressed sufficiently well that an intended reader can, perhaps with some effort, follow the reasoning. (In the case of professional mathematicians, the intended reader is usually another professional with expertise in the same area of mathematics; proofs written for students or laypersons generally have to supply more explanations.)

This means that in order to construct a proof, you have to be able to determine what constitutes a logically sound argument that convinces not just yourself, but also an intended reader. Doing that is not something you can reduce to a list of rules. Constructing mathematical proofs is one of the most creative acts of the human mind, and relatively few are capable of truly original proofs. But with some effort, any reasonably intelligent person can master the basics.

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