I happened to notice that the Horizon programme on BBC2 TV this evening was about "Infinity and Beyond". Hoping to learn of some new research I tuned in but was sorely disappointed. The programme was aimed at about the intellectual level of a five-year-old. The commentary was given by an Aleister Crowley lookalike who was filmed in murky black and white endlessly walking up stairs and reappearing again, Escher-like, on the bottom landing, and making pompous and portentous-sounding statements and poses. Half-way through he even renamed Georg Cantor "Gregor".
All the usual elementary illustrations of infinity were included, such as Hilbert's Hotel, Cantor's diagonal argument and monkeys typing Shakespeare, followed by speculation about whether the universe might be infinite. There was one chap who didn't believe in infinity, but all he could say was that there was a largest number, but no-one knows what it is, and it is followed by zero.
My argument for finitism runs as follows: It is true that we can generate symbols for numbers in a systematic manner using the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and the positional convention, but this does not mean that the set of all such symbols 'exists' already until we actually construct it. Nor does the mere construction of a symbol, such as n+1 for a number imply that the number 'exists' in this sense.
The mathematical term 'finite' applies to sets of things and the numbers of things in those sets: a set is said to be finite if it has the sensible property that it cannot be placed in one-to-one correspondence with a part of itself; a number is finite if it describes the size of a finite set. The Finitist maintains that all sets, and therefore all numbers, are in fact finite.
In order to introduce infinity into mathematics it is necessary to postulate that it exists, or to assume some other axiom that implies this, for instance Peano's axiom that every number has an immediate successor. Further the properties of infinities depend on the axioms that are chosen. For example Paul Cohen proved that, under the usual axioms for arithmetic, it is impossible to say whether there is an infinity between that of the integers and the real numbers.
On the other hand the properties of finite sets and numbers are a matter of physical fact, at least within the 'realisable' realm, where they can be applied to material objects. Statements about 'all' numbers, such as Goldbach's conjecture, may not be realisable.
What do we mean by saying that something 'really exists'? The simplest definition is that something exists if it is material, that is if it has measurable mass. On this basis it might be argued that 'ideas' like numbers do not exist since they are immaterial. But are they? Ideas exist in the minds of people, and presumably therefore they exist materially in the form of electrical or chemical energy in the brains of those who think about them. By Einstein's equation, E = mc², anything that has energy has corresponding mass. So if mathematician's brains really contained the infinite set of all whole numbers they would have infinite mass and implode into a black hole!
By a similar argument, the universe is finite in mass, since if it were infinite there would be infinite gravitational force at every point in the universe (a version of Olbers' paradox).
Even if we discount the argument by weight, so long as we accept that ideas exist in the form of electrical or chemical configurations in the brains of thinkers, there can still only be a finite number of ideas in existence, certainly of human ideas, held by human beings, because there is only a very finite number of human beings extant, and their brains contain only finite numbers of neurons.
EDIT: In contrast to the puerile "Horizon", Melvyn Bragg's "In Our Time" on Radio 4 this morning was an adult-level programme about "unintended consequences" in mathematics, on how ideas developed purely out of mathematical interest later prove to have practical consequences: such as prime number theory in cryptography, complex numbers in alternating electrics, and non-euclidean geometry in relativity. Why does TV have to dumb-down, while Radio does ideas so well?