Is Zero Times One Truly Commutative?

[crxyem]

Well here's the discussion that always arises with my father and I.

It has to do with the multiplication of 1 and 0 and the Commutative Law

Given: In ordinary (real number) arithmetic

$$x * y = y*x$$

The argument is that
$$0*1 = 0$$ and $$1*0=0$$ should not be Commutative.
in a purely mathematical sense this seems very logical. But, in a philosophical sense of tangible objects it does not.

For example.
I have nothing or zero apples and I multiply this by 1 apple, the result I still have nothing as there was nothing to multiply. This logic is acceptable.

I have one apple and I multiply it by zero apples, which seems to possesses some logic. Although this is supposed to lead to a result of having zero apples. But this is not the case the result is I still have my original apple. No one took it away synonymous with subtraction.

So what are your thoughts ?

 

[honestrosewater]

The two operations that you describe, as mathematical operations, are equally mathematical. And the two sets of philosophical ideas that you have about the operations are equally philosophical.

You just have two different operations.

 

[-Job-]

When i think of A * B, i think of it as producing a B number of As.
From this perspective 1 * 0 should yield 0. The result of 1 * 0 is zero ones, which is 0.

 

[honestrosewater]

Well here's the discussion that always arises with my father and I.

It has to do with the multiplication of 1 and 0 and the Commutative Law

Given: In ordinary (real number) arithmetic

$$x * y = y*x$$

The argument is that
$$0*1 = 0$$ and $$1*0=0$$ should not be Commutative.

Oh, also, did you want to set 1*0 != 0*1, or would you be happy setting 0 = 1? You could leave * commutative that way and have 1*0 = 0 = 0*1 = 1.

Or perhaps I missed the point and you are seeking "the right way", or the way that makes the most sense? I don't see why there must be one best way.

 

[out of whack]

I have one apple and I multiply it by zero apples, which seems to possesses some logic. Although this is supposed to lead to a result of having zero apples. But this is not the case the result is I still have my original apple. No one took it away synonymous with subtraction.

So what are your thoughts ?

First, if you multiply apples then the result is not apples but apples squared, whatever this might be. In both cases you will end up without any apple squared at all. Well, this makes sense since I have never seen an apple squared. To correct this, you need to get rid of one of these "apple" units on one side of the multiplication. You have four possibilities:

0 * 1 apple = 0 apple
0 apple * 1 = 0 apple
1 * 0 apple = 0 apple
1 apple * 0 = 0 apple

Now your problem with the last case is solved: you have one apple and you multiply that fact by zero, which essentially nullifies the fact (as if someone ate your apple) and you end up with no apple.

But that's not exactly the nature of your problem. The problem is that you disagree with the interpretation that multiplying something by zero is equivalent to subtracting the same value so that you end up with nothing. But that's just a matter of accepting the definition or not. Multiplying by 0 indeed, by common understanding of this operation, means that you end up with nothing, in the same right as multiplying by 1 means that nothing happened. If you disagree with this then you should not use multiplication but use something else that you will define for yourself. It's pretty straightforward in the end, either you accept the common understanding of the operation or you create one of your own. But then nobody else will understand what you are doing until you publish your new rules.

I can't believe I wrote so much on this subject...

 

[crxyem]

I myself accept the mathematical proofs (Riemann sphere, l'Hôpital's rule) that exist to prove such cases that you can divide by zero or at least rationalize using zero in the denominator.

I like the argument of "out of whack" has stated that when you multiply say $$x * x$$ the result is $$x^2$$ and I myself have never seen a square apple either.

So trying to be less structured and mathematical and more philosophical during a conversation was a more interesting question, what does it mean to multiple something by zero

 

[honestrosewater]

I myself accept the mathematical proofs (Riemann sphere, l'Hôpital's rule) that exist to prove such cases that you can divide by zero or at least rationalize using zero in the denominator.

I don't know what you are referring to here, but whether or not some operation is defined for some input is just a matter of definition. There is nothing to prove in the usual sense.

I like the argument of "out of whack" has stated that when you multiply say $$x * x$$ the result is $$x^2$$ and I myself have never seen a square apple either.

You wouldn't see a square apple because squares are 2-dimensional. You might see a cube-shaped apple someday, though. I would think they would be more efficiently transported and stored.

But anyway, I don't think that was even the intended word. "Squaring" there was meant as an operation, not a geometric figure (a quadrilateral with sides of equal length or whatever).

So trying to be less structured and mathematical and more philosophical during a conversation was a more interesting question, what does it mean to multiple something by zero

Perhaps this kind of activity is super and useful in its own right, but it is not philosophical thinking. Philosophical thinking requires, at least, some rigor and precision. What you are doing seems more like some kind of free-flow superficial word- or concept-association.

 

[JonF]

We know 0 + 0 = 0 by the definition of zero

We then know 0*1 + 0*1 = 0 by the definition of one

We know that (0+0)1 = 0 by the law of distribution

But since 0+0 = 0 by the definition of zero we know then know (0)*1 = 0.

The 1*0=0 case is similar symmetric


The properties that 0*1=0 follows by logical necessity from distribution. And if you throw distribution out the window, you are talking about a different type of operation than “+” in the sense it is used combine apples.

 

[honestrosewater]

We know 0 + 0 = 0 by the definition of zero

We then know 0*1 + 0*1 = 0 by the definition of one

You have it already right there. I assume that, by the definitions of 0 and 1, you mean as the additive and multiplicative identities, respectively. You can have right-identities (x1 = x) and left-identities (1x = x). For the most familiar number systems, it is both, or two-sided (1x = x1 = x).

 

[JonF]

You have it already right there. I assume that, by the definitions of 0 and 1, you mean as the additive and multiplicative identities, respectively. You can have right-identities (x1 = x) and left-identities (1x = x). For the most familiar number systems, it is both, or two-sided (1x = x1 = x).

Yup normally, but he was essentially asking for a reason why identities are commutative. Not all algebras have THE identity, some have a left-sided and a right-sided which are not equal. My proof showed that as long as we have the distributive law 1*0=0*1 by necessity.

 

[honestrosewater]

Yup normally, but he was essentially asking for a reason why identities are commutative.

Haha, my bad. I totally missed the point there.

(By the bye, I think you have a typo, if you want to fix it: "(0)*1 = 1")

 

[berkeman]

First, if you multiply apples then the result is not apples but apples squared, whatever this might be. In both cases you will end up without any apple squared at all. Well, this makes sense since I have never seen an apple squared. To correct this, you need to get rid of one of these "apple" units on one side of the multiplication. You have four possibilities:

0 * 1 apple = 0 apple
0 apple * 1 = 0 apple
1 * 0 apple = 0 apple
1 apple * 0 = 0 apple

Now your problem with the last case is solved: you have one apple and you multiply that fact by zero, which essentially nullifies the fact (as if someone ate your apple) and you end up with no apple.

But that's not exactly the nature of your problem. The problem is that you disagree with the interpretation that multiplying something by zero is equivalent to subtracting the same value so that you end up with nothing. But that's just a matter of accepting the definition or not. Multiplying by 0 indeed, by common understanding of this operation, means that you end up with nothing, in the same right as multiplying by 1 means that nothing happened. If you disagree with this then you should not use multiplication but use something else that you will define for yourself. It's pretty straightforward in the end, either you accept the common understanding of the operation or you create one of your own. But then nobody else will understand what you are doing until you publish your new rules.

I can't believe I wrote so much on this subject...

Yes, but you saved me the time of writing out, which has value (at least to me right now). I owe you one...

apples squared

 

[honestrosewater]

We know 0 + 0 = 0 by the definition of zero

We then know 0*1 + 0*1 = 0 by the definition of one

We know that (0+0)1 = 0 by the law of distribution

But since 0+0 = 0 by the definition of zero we know then know (0)*1 = 1.

Hm, something is still not right here. You want to prove that a left-identity for multiplication combined with some other assumptions makes your left-identity two-sided, right? And then do the same for a right-identity? Did I miss something? You don't appear to have done that.

You didn't specify your premises, so I'll just list a bunch a see what gets used. I might try later. I looked quickly and the way I saw didn't work.

Assume:
There exists some 0 and 1 such that, for all x, y, and z,
a) (x + y) + z = x + (y + z)
b) (x * y) * z = x * (y * z)
c) 0 + x = x
d) x + 0 = x
e) 1 * x = x
f) x * (y + z) = (x * y) + (x * z)
g) (y + z) * x = (y * x) + (z * x)
h) x * 0 = 0
i) 0 * x = 0
j) For all x, there exists some -x such that x + -x = 0
k) For all x != 0, there exists some x^-1 such that x * x^-1 = 1.

Prove:
For all x, x * 1 = x

 

[JonF]

nope, that wasn't what i intended. All I proved was the special case of 1*0 = 0*1 as long as we have distribution. I didn't prove in general 1*x=x*1 because that wasn't the question.

your assumptions kinda undermine his question because he is essentially asking: do the left and right identity need always be the same? If there is a left identity must there be a right identity? Are 1*0 and 0*1' necessarily equal (where 1 is the left identity and 1' is the right identity)? I only addressed the last question.

Also note: a few of your assumptions can be derived from other assumptions you make and others aren't necessary for the conclusion you seek...

 

[honestrosewater]

nope, that wasn't what i intended. All I proved was the special case of 1*0 = 0*1 as long as we have distribution. I didn't prove in general 1*x=x*1 because that wasn't the question.

Oh, I guess I misunderstood your explanation.

your assumptions kinda undermine his question because he is essentially asking: do the left and right identity need always be the same? If there is a left identity must there be a right identity? Are 1*0 and 0*1' necessarily equal (where 1 is the left identity and 1' is the right identity)? I only addressed the last question.

Also note: a few of your assumptions can be derived from other assumptions you make and others aren't necessary for the conclusion you seek...

Heh, that's why I said:

I'll just list a bunch and see what gets used.

(I was actually trying to find out what precisely you meant by "the definition of one", etc.)

You're right, though, that I didn't see that interpretation of the OP's question. I thought they wanted * to not be commutative generally. Thanks.

 

[honestrosewater]

Sorry, this is still eating at my brain. Something seems superfluous to me again.

Are 1*0 and 0*1' necessarily equal (where 1 is the left identity and 1' is the right identity)?

If 1*x = x and x*1' = x for all x, then 1*x = x*1' as they're both equal to x, without the need for distribution or restriction to x = 0. Or where am I confused?

 

[Hurkyl]

some have a left-sided and a right-sided which are not equal.

I disagree.

1_left = 1_left * 1_right = 1_right

 

[crxyem]

I thought they wanted * to not be commutative generally. Thanks.

Thanks for the discussion on the topic, I generally can handle a good philosophical debate until it comes to debating math, I have a terrible memory for rules and theory of mathematics. Yet I can solve a Laplace Transform without trouble.

I was just stating that the equation 0*1 = 1*0 , shouldn't be commutative, because the left doesn't necessarily equal the right, which is my fathers argument and I've always pretty much accepted the fact. But if somehow I could prove his case it could be quite enlightening.

Sorry, this is still eating at my brain. Something seems superfluous to me again.
If 1*x = x and x*1' = x for all x, then 1*x = x*1' as they're both equal to x, without the need for distribution or restriction to x = 0. Or where am I confused?

Wow, I never looked at the equation that way before but it makes sense to. After the discussion/arguments I've seen. I can now bring this up with my father again and really prove that it is in fact the same solution.Also when I was talking about l'Hôpital's rule I was referring to it's use with limits and indeterminate form which allows a solution to be found when dividing by zero

 

[honestrosewater]

Thanks for the discussion on the topic, I generally can handle a good philosophical debate until it comes to debating math, I have a terrible memory for rules and theory of mathematics. Yet I can solve a Laplace Transform without trouble.

Heh, you're not the only one. I find myself sometimes lost as to where to start a mathematical argument or even make sense of a definition. The problem is usually just in getting started, and I think it might be needing to assign meaning to enough of the parts so that you can actually work in a model. (That might be why I sometimes try to resort to mechanical computation.) Some researchers have proposed two different styles of thinking, one that works more step-by-step and one that works more like building a web. It seems sometimes that I prefer web-style and mathematics is often presented step-style. The research was actually on differences between males and females. I could try to find it again if anyone is interested. Anywho...

I was just stating that the equation 0*1 = 1*0 , shouldn't be commutative, because the left doesn't necessarily equal the right, which is my fathers argument and I've always pretty much accepted the fact. But if somehow I could prove his case it could be quite enlightening.

Well, by the usual definition, that specific equation is true if * is commutative (which I guess is why I jumped to that), so you would have to make * non-commutative or at least change the definition to include this one exception.

I imagine that if it makes sense to you, you can find the definitions that work. You only have to worry about your theory being inconsistent, and if you aren't mistaken in your mental model of the system, then inconsistency isn't a problem. You can't imagine an inconsistent system any more than you can imagine a square circle. Well, that's maybe saying it too loosely, but you get my drift, right?

Wow, I never looked at the equation that way before but it makes sense to. After the discussion/arguments I've seen. I can now bring this up with my father again and really prove that it is in fact the same solution.

I liked Hurkyl's even better.

 

[JonF]

I disagree.

1_left = 1_left * 1_right = 1_right

eh, i musta be remembering something wrong from my group theory course. I was almost certain that the professor said there were algebras with left sided and right sided that were distinct but your proof is valid so i must be confused or forgetful.

 

[nabuco]

(just a comment on something irrelevant... as if the rest of the thread were )

if you multiply apples then the result is not apples but apples squared, whatever this might be [...] I have never seen an apple squared

Actually you probably have seen apples squared. For instance, 4 square apples would look like this:






Same as four square feet. Funny how trivial things are not always obvious.

 

[Hurkyl]

(just a comment on something irrelevant... as if the rest of the thread were )



Actually you probably have seen apples squared. For instance, 4 square apples would look like this:






Same as four square feet. Funny how trivial things are not always obvious.

That looks like 16 apples to me.

 

[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

 

[Hurkyl]

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

"apple" isn't a measure of volume: it's a measure of apples.

 

[out of whack]

That looks like 16 apples to me.

Yes, me too. I would like to see just one apple squared to see how it differs from a plain apple. Hey, maybe it's trivial and obvious...

 

[honestrosewater]

Yes, me too. I would like to see just one apple squared to see how it differs from a plain apple. Hey, maybe it's trivial and obvious...

Perhaps you can generalize the concept of dimension.

Or, hm, this is admittedly a bit out there, but couldn't you order objects by their appleness?

 

[out of whack]

couldn't you order objects by their appleness?

A bit out there is fine in this thread!

 

[honestrosewater]

Well, you laugh, understandably, but I don't know how to come up with something new and have all the details worked out immediately.

Maybe what the OP, or their father, is thinking about is some type of object that differs from the already familiar objects or structures of mathematics like numbers, sets, spaces, algebras, etc. Or maybe it's familiar and just not recognizable yet. Who's to say that following the idea through won't lead to something that makes perfect sense?

Like the OP said, these are tangible objects. So maybe you think of the operations on them in terms of whether they obey some kind of conservation laws, multiplication being one operation that conserves whatever, tangibility. That seems in line with the thinking of the OP, and it makes sense to me that someone would be irritated by a perceived violation of a conservation law.

 

[Cane_Toad]

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

Whee, a hyperapple! Does apple * apple = apple * Ritalin?

 

[nabuco]

That looks like 16 apples to me.

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

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

I don't get it.

 

[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?).