aesthetics of mathematics

Following on from today's session, here is a version of the proof of the infinity of primes:

1.Each number is either prime or not prime (ie composite)
2.All composite numbers can be broken down into prime factors (eg 154 = 2 × 7 ×11) – this is the fundamental theorem of arithmetic
3.Let’s suppose that there is only a finite number of primes
4.Let’s call the members of this set: a, b, c, d, and so on, up to z, which would be the largest prime
5.Multiply all the members of the set of all primes together: a × b × c × d × …. × z – let’s call this number P, the product of all primes
6.Add 1 to make P + 1
7.P + 1 must be composite since it is larger than the supposedly largest prime
8.So, according to statement 2, there must be a prime number – let’s call it p – that divides P + 1
9.But for p to be a member of the set of primes a, b, c, d… z it would have to be also a divisor of P, and no number larger than 1 can divide two consecutive natural numbers
10.Hence for any set of prime numbers there will always be at least one more prime outside the set; therefore the number of primes must be infinite – QED

Do you see how some people can sense elegance or beauty in such a construction? More broadly, does mathematics have anything in common with the arts? Historically, mathematics has sometimes been regarded as an art...

Please also note that the powerpoint presentation that I showed you today can be found on the local network in the folder at K:\Staff_To_Students\IB Subjects\Core\TOK.