Is Zero Times One Truly Commutative?

[Cane_Toad]

It might be worth pointing out that an "apple" is not a unit of measure. It is a complex thing. It's an object (a class, really, but the OP's apple was an object) with properties, and certain operations are not supported, i.e. you only get to apply scalar multiplication to type "apple".

Now if you'd like to continue the fun, I'd be curious whether all the same wildness applies when manipulating objects (classes, categories, functions?) instead of UOMs. My math isn't up to dimensions of functions.

 

[Hurkyl]

I don't understand. Where's the difference:

4m*4m=16m^2
4apples*4apples=16apples^2

I don't get it.

Neither of those relate to the picture you drew. You drew 4 rows with 4 apples each: 4 * (4 apples) = 16 apples. You did not draw (4 apple) rows with 4 apples each.

 

[honestrosewater]

It might be worth pointing out that an "apple" is not a unit of measure. It is a complex thing. It's an object (a class, really, but the OP's apple was an object) with properties, and certain operations are not supported, i.e. you only get to apply scalar multiplication to type "apple".

Are you thinking of an apple as a vector? Why so? My thinking would be to just start with a set of (atomic) "tangible" objects and sort them. Perhaps then it's just that you cannot, say, compare apples and oranges.

It might also be worth noting that a property is just a relation.

Now if you'd like to continue the fun, I'd be curious whether all the same wildness applies when manipulating objects (classes, categories, functions?) instead of UOMs. My math isn't up to dimensions of functions.

There are general ways of treating mathematical objects. I can't think of anything that can't be defined as a structure. What do you want to do to them? (And what do you mean by dimensions of functions, arity?)

 

[Cane_Toad]

Are you thinking of an apple as a vector? Why so?

It was just the first analogy that came to mind about what a valid multiplication for an apple might be.

My thinking would be to just start with a set of (atomic) "tangible" objects and sort them. Perhaps then it's just that you cannot, say, compare apples and oranges.

What set? Apples, oranges, and pears?

Why is it necessary to have dissimilar objects to restrict the valid operations on an apple?

It might also be worth noting that a property is just a relation.

?? I thought they were distinct concepts, with properties forming the foundation for a structure of relationships.

There are general ways of treating mathematical objects. I can't think of anything that can't be defined as a structure. What do you want to do to them? (And what do you mean by dimensions of functions, arity?)

I was referring to square apples, and:

Office_Shredder: Notice four feet squared is not capable of being measured in feet (in fact, you double the dimension). Four apples squared would actually require six spatial dimensions I believe, as each apple in fact constitutes a three dimensional object, hence squaring it would double the dimension

I guess I was thinking of an object space(objects instead of points?) which would means its dimensions are described in terms of objects, but I don't know the terms. I'm just starting to wander around this stuff now, so Hilbert, function, etc. spaces are all unfamiliar.

 

[honestrosewater]

What set? Apples, oranges, and pears?

The set = your domain, which I was taking to be a set of "tangible objects", whatever that might end up meaning. I am thinking of them as mathematical objects that, incidentally, someone might find useful for, say, counting apples.

Why is it necessary to have dissimilar objects to restrict the valid operations on an apple?

It's not necessary to do so. I thought you were perhaps wanting to give objects some internal structure in order to distinguish different types. I was just saying that you can achieve the same thing by sorting them first and defining your relations and operations on the sorted sets. But evidently, I misunderstood, so it's not important.

?? I thought they were distinct concepts, with properties forming the foundation for a structure of relationships.

I don't know what that means.

If S is a set, an n-ary relation R on S is just a subset of Sⁿ. I've seen the terms "property" and "predicate" used more often in logic and philosophy, but you can switch back and forth between them for the most part. With a few possible caveats, having a property is the same thing as belonging to a set. Consider the set O of all organisms. (Just grant me that this is a set.) To say that an organism f has the property of being female is just saying that there exists a subset F of O and f is in F.

I was referring to square apples, and:

I guess I was thinking of an object space(objects instead of points?) which would means its dimensions are described in terms of objects, but I don't know the terms. I'm just starting to wander around this stuff now, so Hilbert, function, etc. spaces are all unfamiliar.

Oh. I haven't really understood any of that, so I don't know what to say there.

I've been using "object" generically, so to me a point is an object, numbers are objects, everything is an object.

I thought the problem was basically just looking for a slightly different way of counting (in which case what does it matter what you're counting?).

 

[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.
definition: 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.