Micaela Simon

2022-06-21

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...

### Answer & Explanation

Tianna Deleon

The best way to motivate counting in the way you are talking about is to consider it in terms of the concepts of "Bijection" and "Abstraction".
A bijection is pairing between different classes of things. For example if you had a pack of dogs where you gave each dog in that pack its own unique name. You can uniquely refer to that dog by it's name and when seeing that dog you can recall that dog's name.
Abstraction is the removal of the specifics to create a model whose rules are more universal. So for example consider the act of herding sheep past a gate. As each sheep passes though the gate you place a stone in a bag. Again you are creating a bijection between stones and sheep. However there is nothing really identifying one sheep from another (or one stone from another).
Further more when you use this method of using stones to record other items such as bushels of grain, etc the rules of how the stones operate don't change. This allows us to take the next step and abstract away the stones and just use the rules.
This turns out super useful if you are the government of an agrarian culture that has just developed agriculture and needs a way to record how much food you have in your store houses.

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