Think about what a number is--as I understand the
foundations of mathematics,
natural numbers are constructed from successive sets of the
empty set. Thus, the number 0 corresponds to the empty set itself (represented as zero with a slash through it, or simply {}), 1 corresponds to the set of the empty set ({{}}), 2 to the set of that ({{{}}}), and so on. Now, for any natural number you name, I can write down the corresponding set*. I may need more time than I actually have before I die, but it is in principle possible. This is how we get natural numbers without merely assuming their existence.
This is not possible with infinity. Indeed, it is presumably for reasons such as this that
Luitzen E. J. Brouwer and the
intuitionists had problems with modern mathematics. The following quotation is taken from the
Stanford Encyclopedia of Philosophy's entry on
Constructive Mathematics (http://plato.stanford.edu/entries/mathematics-constructive/):
According to {Brouwer's} view and reading of history, classical logic was abstracted from the mathematics of finite sets and their subsets. ... Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of set theory, for which it is justly punished by the antinomies. It is not that such contradictions showed up that is surprising, but that they showed up at such a late stage of the game. (quoted in Kline 1972, p. 2001)
* Note that I'm not trying to claim that the
epistemic justification for our belief in a
mathematical object comes from the ability to write it down, only that if it is possible to use a well-defined method of notation to write something down, that that implies that it is in fact a clear concept. Similarly, the inability to find a consistent operation and clear method of communicating serves as weak,
prima facie evidence that no clear concept answers to the description offered. For more on this, see
JerboaKolinowski's excellent writeup in the
I can divide by zero node.