| >those ones seem to be easy. There is no number seven. That's a very definitive answer for a complex question. It is not immediately clear at all that there is no number 7. There are a few accounts for how numbers could exist. Let's break down each account: * formalism - Mathematics is a formalized activity undertaken by humans, but is ultimately a language game. This approach is problematic because no formal system is capable of enumerating all truths about Mathematics. Hence, Mathematics is not a formal activity undertaken by humans, because otherwise the methods of its formalism would allow us to enumerate each true statement. * intuitionism - Mathematics is a psychological artifact of minds, but is not a "natural" phenomenon. This approach has problems because it fails to account for (1) the remarkable consistency and success of Mathematics, and (2) leads us to Godel's disjunction. Godel's disjunction states that either there is no algorithm capable of enumerating all true statements that a human mind enumerates, or there are some Mathematical truths that can not be decided. If there is no algorithm capable of enumerating all true statements that a mind enumerates, then the mind is not a computational system -- meaning it can not be reduced to a mathematical model of neurons communicating -- and so the meaning of a psychological artifact needs reappraised. Specifically, numbers being a psychological artifact does not imply that they do not exist naturally. However, if a human mind is a computational entity, then intutionism doesn't work for the same reasons why formalism doesn't work. * Realism - Mathematical objects exist. Given the problems of formalism and intuitionism, this actually is among the most probable of hypotheses. Many Mathematicians fall into the Realist category, and there are good reasons why. |