Cause and Effect

Prime Numbers

Most of us know that a prime number (often referred to as simply a "prime") is a whole number which can only be evenly divided by itself and unity; famous examples include 2, 3 and 5. Some of us may even know that there are infinite number of primes, but it is assured that most of us don't know why. The answer is an example of some of the most beautiful mathematics in the history of the subject; the proof is short and simple, but requires a highly imaginative and non-intuitive tool. We owe it to Euclid of Alexandria, the "father of geometry" and one of the greatest minds in recorded history. First of all, we must familiarize ourselves with the method of logic used. Technically, all mathematical proofs employ deductive logic, but there are a number of varieties; this particular example is called proof by contradiction, or reductio ad absurdum ("reduction to an absurdity" in Latin), in which the opposite of the desired result is assumed or imagined. Then a deductive argument, starting with this initial and opposite claim, is built to show a contradiction (an absurd statement establishing the falsehood of some obviously true fact), which would imply that the original assumption was false, and therefore that the opposite statement (the desired claim!) must be true. For Euclid, this means assuming there are not an infinite number of primes; so there must be a prime such that all numbers larger than it are composite (mathematicians call non-primes "composite numbers" because they are always products of primes; they are "composed" of primes). Euclid then employs his genius and obliges us to imagine a special number; we are to imagine the result of multiplying all of the primes (since there are a finite number, according to our assumption) together and adding 1 to the result. Consider this special number: if we try to divide it by any number on our (finite) list of primes, there will always be a remainder of 1. This is because the product of all the primes is itself a multiple of each prime, so adding 1 will simply create an extra remainder (because 1 isn't divisible by any of the primes). However, this number also must be composite since it is (much) larger than the largest prime; that was our assumption. So this is a composite number that is not divisible by any primes? Ridiculous! The definition of a composite number is one that is divisible by at least 2 primes. Here lies the contradiction. Suddenly, all of the deduction falls into place, and the desired result comes out: there must be an infinite number of primes. Elegance radiates from the "special number" step: mathematics is, to most, a sort of reasoning which includes starting from the most general principles and thinking in the abstract to create a giant "tree of truth" which includes all possible combinations of the fundamental principles. Instead, Euclid uses one very specific and very powerful example to quickly and easily prove much more general and beautiful results. Who would think that the simple and specific could so greatly influence the truths of infinity?