Why Division by Zero is a Bad Idea

[fresh_42]

You could use rings with zero divisors, where division cannot be defined also for non-zero elements. It may be insightful.

Yes. I even thought about adding my example with $\mathbb{Z}_4$ and $V_4$ but I didn't want to make it complicated and write a mathematical essay. This would have been over at the $1=0,$ or instead with the explanation of my favorite argument $0\not\in \mathbb{F}^*.$ A discussion would then become pure mathematics about zero-divisors, rings, hyperreals, etc. Those words occurred, but not as a main topic. I only wanted to bring those arguments on the table that I have read here throughout the years when people come and try to make sense of $1/0,$ especially setting it $\infty .$

 

[martinbn]

I don't think his assumption that $1 \neq 0$ is any more "hidden" than yours. Both of you are simply relying on the obvious fact that in order to have a useful number system at all, different numerals, such as $0$ and $1$, should refer to different (i.e., unequal) numbers.

In fact @symbolipoint doesn't even mention the number $1$ in his post; the number he picked is $10$. So his assumption is actually that $10 \neq 0$, not that $1 \neq 0$. Where in your article do you explicitly state that you are assuming that $10 \neq 0$? Wouldn't be better to just acknowledge that both of you are assuming that numbers that aren't $0$, um, aren't $0$? And that that assumption is fine because you have to make it to have a useful number system at all?

There is a bit more to it. I am guessing that for @fresh_42 1 and 0 stand for the neutral elements with respect to multiplication and addition (not just the names of two elements, so if the elements are different you choose different notations). And the question whether they are different, or the assumption that they are, is meaningful.

 

[fresh_42]

I don't think his assumption that $1 \neq 0$ is any more "hidden" than yours. Both of you are simply relying on the obvious fact that in order to have a useful number system at all, different numerals, such as $0$ and $1$, should refer to different (i.e., unequal) numbers.

In fact @symbolipoint doesn't even mention the number $1$ in his post; the number he picked is $10$. So his assumption is actually that $10 \neq 0$, not that $1 \neq 0$. Where in your article do you explicitly state that you are assuming that $10 \neq 0$? Wouldn't be better to just acknowledge that both of you are assuming that numbers that aren't $0$, um, aren't $0$? And that that assumption is fine because you have to make it to have a useful number system at all?

Useful, meaningless, impossible, unusual. Those words can hardly be argued upon. They reflect an opinion. That's why I argued against their use. It is an implicit judgment ahead of the argument. It is called polemic in German. But I did not write what we call a pamphlet, I wanted to do math. This discussion here was polemic from the start and still is, despite some user's efforts to make it a scientific discussion.

 

[PeterDonis]

I am guessing that for @fresh_42 1 and 0 stand for the neutral elements with respect to multiplication and addition (not just the names of two elements, so if the elements are different you choose different notations).

That's true, but it just raises the question of why, if that was the meaning @fresh_42 was using for $1$ and $0$, he didn't say so explicitly. If one is going to criticize others for making hidden assumptions, one should not make them oneself.

And the question whether they are different, or the assumption that they are, is meaningful.

For the general case, yes. But I see nothing in the article until point #1, near the end of that section, that even raises the general case. When I read the start of the article, certainly through point #10, I see $1$ and $0$ being used to denote those integers (which is of course what a lay person is going to think those notations mean), which is much more specific than just "the neutral elements with respect to multiplication and addition", and for which $1 \neq 0$ is obvious.

 

[PeterDonis]

Useful, meaningful, impossible. Those words can hardly be argued upon. They reflect an opinion. That's why I argued against their use.

And that's fine, but it's not what I am objecting to. I am objecting to you criticizing @symbolipoint for making a "hidden assumption" that $1 \neq 0$ (and I've already commented about the "hidden" part, which I don't think is justified), when you yourself make the same assumption. If your objection was that "meaningless" is an opinion, you should have limited your objection to that.

Also, if "meaningless" is an opinion and doesn't help discussion, why did you yourself use it? You said @symbolipoint's comment was "meaningless". Again, you criticize others for doing the same thing that you yourself do. Why is it wrong for others but not for you?

 

[martinbn]

That's true, but it just raises the question of why, if that was the meaning @fresh_42 was using for $1$ and $0$, he didn't say so explicitly. If one is going to criticize others for making hidden assumptions, one should not make them oneself.For the general case, yes. But I see nothing in the article until point #1, near the end of that section, that even raises the general case. When I read the start of the article, certainly through point #10, I see $1$ and $0$ being used to denote those integers (which is of course what a lay person is going to think those notations mean), which is much more specific than just "the neutral elements with respect to multiplication and addition", and for which $1 \neq 0$ is obvious.

Yes, if the context was the whole numbers, it is not even an assumption, there $1$ and $0$ are different.

 

[fresh_42]

And that's fine, but it's not what I am objecting to. I am objecting to you criticizing @symbolipoint for making a "hidden assumption" that $1 \neq 0$

Sure? I haven't seen any assumptions. I could only guess.

Again, you criticize others for doing the same thing that you yourself do. Why is it wrong for others but not for you?

Do not reverse cause and effect, please. I answered at the level I had been dragged down to. It was a personal from post #3 on: "I haven't read it [the article] but it [the title] is meaningless!" Hello? How else do you read this as a provocation?

 

[PeterDonis]

Sure?

I don't know what you mean. Here is what you explicitly posted:

Sorry, but your comment is meaningless. It contains a lot of hidden assumptions and "meaning" is not quantifiable, it is an opinion, not a fact. By what right do you make assumptions like $1\neq 0$?

You call his post "meaningless" and say he is making hidden assumptions. What are you not "sure" about?

Do not reverse cause and effect, please. I answered at the level I had been dragged down to. It was a personal from post #3 on: "I haven't read it [the article] but it [the title] is meaningless!" Hello? How else do you read this as a provocation?

If the issue was that he didn't read your article, a simple response like "You say you haven't read the article. Please do so. It addresses the issues you are raising." And then stop.

We moderators routinely tell people on these forums not to take responses personally, not to get "dragged down to" what we perceive to be a lower level that someone else is at, and to stick to the topic of discussion. It seems to me that we ought to be able to take our own advice.

 

[fresh_42]

I don't know what you mean. Here is what you explicitly posted:

I posted:

Sorry, but your comment is meaningless. It contains a lot of hidden assumptions and "meaning" is not quantifiable, it is an opinion, not a fact. By what right do you make assumptions like 1≠0?

I kept it unspecified. "like" clearly indicated that I was guessing.

You call his post "meaningless" and say he is making hidden assumptions. What are you not "sure" about?

"meaningless" was a tit-for-tat response. I am not sure about what he did assume and what not. $1\neq 0$ is again a hidden assumption. We cannot know what he assumed. That's what I am not sure about. My attempt with "like" should have transported this. Sorry, if my German understanding of linguistic tools does not match the English usage.

If the issue was that he didn't read your article, a simple response like "You say you haven't read the article. Please do so. It addresses the issues you are raising." And then stop.

We moderators routinely tell people on these forums not to take responses personally, not to get "dragged down to" what we perceive to be a lower level that someone else is at, and to stick to the topic of discussion. It seems to me that we ought to be able to take our own advice.

I do not dare to imagine the situation if I had given a "insulting other members" warning with its 4 points or a thread ban. Stop measuring with two scales, varying by the situation and the argument that suits you.

 

[PeterDonis]

I do not dare to imagine the situation if I had given a "insulting other members" warning with its 4 points or a thread ban.

I don't see any ground in @symbolipoint's post for any such warning, or for your interpretation of what he said as a "provocation". Making a comment without having read the article is premature, sure, but that's not a personal attack on you. It's just someone making a premature comment, for which the obvious response is just what I said before.

 

[symbolipoint]

The difference is that @symbolipoint said "meaningless" based on hidden assumptions. I criticized a) that "meaningless" is nothing an argument can be based on, and b) that his assumptions are not stated; mine were.

I made no conscious assumptions. 1 is a number. 0 is a number. 1 is not the same as 0.

 

[weirdoguy]

It is a judgment. It also says, that people who ask this, have a meaningless question. This is patronizing.

Nonsense. That may be in german (it is in polish), but it is not in english. Maybe you should ask people whose first language is english, before forcing on us you understanding of this word. And yet another thread (besides some other posts throughout the years) of yours where your way of discussing does not suit your mentor badge.

 

[Greg Bernhardt]

We're going to close this for now to let the discussion mood settle. tbd