I hope this is the right place to ask this question...My question regards the motivations...
I hope this is the right place to ask this question...
My question regards the motivations behind forming numbers and their connection to physical objects. What I am not asking about, is a rigorous formulation of the natural numbers. Instead, I am curious about the perhaps more philosophical step before. Let me explain:
Generally, when we want to construct the integers (and eventually reals) we motivate the task by saying that a number represents a collection of items, addition of these numbers represents the combination of these items and so forth. (I guess this is sounding a lot like set theory as well). My question is how are these fundamental connections between numbers and the physical task of counting rigorously constructed. The way I have written it (the stuff in italics) is riddled with mysterious language such as “collection,” “combination” etc.
Of course in a purely mathematical context, this is unimportant. However, I think it’s still valuable to appreciate how abstract concepts in numbers all still hail from the mundane (yet seemingly hard to define) task of counting things.
*often times when we think of the physical models behind mathematical results, we arrive at this simple notion of “counting objects. Is this really the most precise way we can define this connection" *
For example, we “count” a collection of five apples as, well, five apples...