>>12
That's the ultimate problem: A symbol is MEANINGLESS without MEANING.
I think that's what
>>1 (or yours?) problem is: you're forgetting that a string of symbols can be interpreted and has a meaning, or to say it more clearly: semantics. Math talks about relations and behavior of very abstract objects, that is, of truth about general objects and structures - if you added real-world characteristics to them, the truth would remain unchanged, the only thing that sets them apart are very simple general properties and the rest is irrelevant. To put it another way, finite numbers will always have the same properties given the standard interpretation of arithmetic, regardless of whatever other properties you ascribe to those numbers (such as beings rocks, bits, atoms, whatever) - math can be seen to describe these general timeless relations, the syntax is less important as long as it can be learned and someone can interpret it, if you wanted to, you could do all your inferences and syntax in lisp, despite that most mathematicians tend to use more 'natural' styles that they're now used to (I'm not claiming that the syntax they use is that good, just that it works).
>>40
I know that you're probably an ultrafinitist, but theorem provers/formal systems can easily talk about things they can't reach. Infinity for me is just a way of talking about uncomputable properties, which computable objects(such as programs) can have, for example, wether a program halts or not, or more general convergence-related properties. Infinity is not a meaningless concept, even if you never directly touch it, but it's always needed when you want to go meta(such as about 'all' programs or 'all' functions). Since you seem very pragmatic and don't like to talk about such general things, maybe math isn't for you (not talking about syntax here, but about semantics), however these 'unreachable' concepts are important and they lead to deeper truths which will nevertheless hold, regardless of what you think about 'infinity' or wether you want to contemplate it at all.