Straggler writes:
Let me ask - Does the number pi exist? Is it only a number that has meaning to the decimal point that we require it to have meaning to? Or does it mean more?
That's a good question. Certainly for most human purposes, we could restrict to some large finite number of decimal places and we'll never need to know any number beyond that precision.
Pi and e and other such irrational numbers are better behaved than most, in the sense that they have a well-defined formula for calculating them. Take the more extreme case of the uncomputable numbers, numbers for which there is no algorithm for computing their digits. In fact virtually all numbers are uncomputable.
The reason we need all these numbers is not because we'll ever use them. The reason is because we need to do calculus. In order for calculus to work in all its glory, you need a number system with a certain collection of axioms. Those same axioms imply the existence of a vast sea of uncomputable numbers.
This is the kind of thing that splits mathematicians on the objective nature of mathematics. An awful lot of mathematicians I have met simply view these numbers as a necessary evil in order to get calculus, that the numbers are just a bunch of formal junk.
As for what I think about mathematics and objective reality, it's a difficult issue. I'm by no means sure of my opinion.
Considering I can easily just write down a set of axioms and explore that system, I don't think every mathematical system has a relation to reality. For example even chess is a formal system with axioms (its rules).
I do think there is something objective about two things:
(a) Some mathematical systems have a very strong relation to reality.
(b) I think that the most subjective part of maths is the axiom system. For example chess, could have had different rules. The most objective part, I think, is the consequences of those rules/axioms. So even though the rules of chess are arbitrary, given those rules the 2006 World championship game, for example, was always (in some sense) possible outcome. It "existed" somewhere in the space of all consequences of those rules.
Although in truth, I don't really know what I'm talking about.
As for objective reality in general, that's an even more difficult question. I'll take a toy model for consideration. Imagine there is a world containing a red box and three beings Alice, Bob and Carl. Also Carl has malfunctioning senses, he perceives the box as yellow. That is the objective truth of this world. I'm not even sure of how the inhabitants would obtain a definition of objective reality in this toy world.