Is Zero Times One Truly Commutative?

[crxyem]

I used the example of an apple just so we had something tangible to relate to, but it could be any type of object

 

[Cane_Toad]

With a few possible caveats, having a property is the same thing as belonging to a set.

I don't know the convention, but I think a property would be more like a variable in a function, the possible values for which might be member or subset of some set.

 

[honestrosewater]

I don't know the convention, but I think a property would be more like a variable in a function, the possible values for which might be member or subset of some set.

Well, the caveats that I had in mind concern whether a property becomes a set or a class and the order of your language (what you can quantify over). But I'm not sure either is of any consequence here.

Your description doesn't narrow down for me what kinds of things can be properties. All an object needs to do in order to be input to a function is to be a member of some set (the domain of the function), i.e., to exist. Or maybe there is something else that I'm not thinking of. Or maybe you are thinking of a variable as a syntactic object.? Either way, I still don't understand what you're saying. Perhaps you could give an example?

 

[guten]

Let A be the ordered set of all quantities of apples, so A = { 0 apples, 1 apple, 2 apples, ... }.
Let Z+ = { 0,1,2,3, ... }, the set of all positive integers.

Abstractly
Let x,y,z in A and a,b,c in Z+. Also x = a apples, y = b apples. a+b and ca have there usual meanings.
Definition: Addition: x + y = a apples + b apples = a+b apples.
Definition: Multiplaction by a scalar: cx = c * (a apples) = ca apples.

Physically
Imagine a pile of apples, called the pool of apples. We get apples from the pool by ordering baskets with certain quantities of apples. We are also given an empty basket.
Definition: Addition of baskets is the quantity of apples in the given basket after the physical act of filling the given basket with the apples of all the ordered baskets.
Defintion: Multiplying a basket by a scalar is the quantity of apples obtained when the scalar number of baskets of a certain quantity are ordered from the pool and then added together.

Special cases
(i) Abstractly: The scalar multiple of 1 and 0 apples is 1*0 apples or 0 apples.
Physically: Order 1 empty basket from the pool and empty this into the given basket. The given basket is still empty, 0 apples.

(ii) Abstractly: The scalar multiple of 0 and 1 apple is 0*1 apples or 0 apples.
Physically: Order 0 baskets with the result that the given basket can not be filled with any quantity of apples hence it remains empty, 0 apples.

Thus by the abstract and physical definitions we have the result: 1*(0 apples) = (0 apples)*1 = 0 apples.