Some philosophies of mathematics affect which kinds of proof are valid, and thus affect which mathematical results are valid.
Constructivism states that the only valid mathematical results are constructed ones.
Finitism is a subset of constructivism that states that the only valid mathematical results are those constructed with a finite number of steps.
Here is a simple constructive proof: that the sum of every pair of even numbers is also an even number. An even number n has form 2*k, where k is an integer. Here is the proof. Start with even numbers n1 = 2*k1 and n2 = 2*k2. Add them:
n1 + n2 = 2*k1 + 2*k2 = 2*(k1+k2)
Since the sum of two integers is also an integer, and since 2*(integer) is even, that proves the result.
A non-constructive proof works by contradiction, by showing that if some mathematical result is false, then it results in a contradiction.
Every proof that there is no largest prime number is non-constructive, by showing that the existence of such a prime leads to a contradiction. Here is a rather simple such proof.
Construct the function P
= (product of (first n primes)) + 1
P
is clearly not divisible by any of the first n primes, so the primes that divide it must be greater than p
. Thus, for every purported largest prime, there is always a larger one.
Note that this proof does not require P
itself to always be a prime. It also works for all its prime factors if it is composite. In fact, P
is a prime for n = 1, 2, 3, 4, 5, but P(6) = 30,031 = 59*509, a composite number. Yet its prime factors are nevertheless greater than the first six primes: 2, 3, 5, 7, 11, and 13.
Turning to finitism, it makes anything involving infinite sets to be invalid. For instance, here is a proof that there is the same number of nonnegative integers as positive ones. The proof consists of finding a bijection between the two sets. Here it is:
For every positive integer n, there is a nonnegative integer n-1.
For every nonnegative integer n, there is a positive integer n+1.
When Georg Cantor published his theory of transfinite numbers, one of his colleagues, Leopold Kronecker, rather strongly opposed it on finitist grounds. That theory remained controversial until the early 20th cy.