Well, a formal axiom system starts with undefined concepts (in geometry, for example, the concept of a point is undefined). It then makes assumptions (axioms) about those undefined concepts and has rules of deduction that allow us to get from one statement to new statements. This process of using the axioms and rules of deduction is called proof within that system.<quoted text>
You seem good with math.
What does "5 is an abstract concept in a formal system" mean, in Redneck terms?
So, for example, in geometry, the concepts of points and lines are typically undefined. Basic axioms are things like 'between any two points there is exactly one line'. Then we can form proofs just like in high school geometry.
Similarly, arithmetic takes the concepts of number, zero, and successor as undefined and assumes some basic properties of numbers (the successor of a number is a number).
Finally, almost all of mathematics has been brought into set theory, which makes the concept of a set as undefined. Typically, the axioms are those of Zormelo-Fraenkl set theory.
The statement I made is that '5' is an object in the formal axiom system about numbers (and, in a different sense, in that for set theory).
Now, formal systems may or may not say anything about *reality*. They are simply internally consistent systems. What a scientist may do, though, is construct a mathematical *model* of some real situation. Whether the model actually works is determined by testing and observation. So, when we count rocks, we are using the formal system about numbers and we have found that system to work in most cases when counting rocks (there are exceptions, of course). But there are other situations where even a basic mathematical result such as 2+2=4 simply gives wrong results in reality.