Zero divided by Zero =

[Gary Childress]

Mathematicians seem to have an unfair advantage in this conversation. It seems like they get to frame the debate. What if we don't want to agree with them? Then what do we do?

You don't have to do anything more, once you have disagreed, you've already done what you set out to do.

OK. That's good. Just chalk me up as categorically disagreeing with any mathematics used in this thread.

[Skepdick]

Any answers?

It depends on the meaning of "divide".

The standard way to explain division to a child is to ask "what number when multiplied by x gives you y?"
What number when multiplied by 0 gives you 0? And the answer is... all of them.

Of course any given answer is (ultimately) a product of Mathematical convention, and the current convention insists that division by 0 is undefined, but in the realm of formalized mathematics and automated theorem provers we are working with total (computable) functions and we cannot allow for undefined behaviour so rather than leaving x/0 as undefined we need to chose an answer. Any answer.

And the most pragmatic convention is x/0 = 0.

https://xenaproject.wordpress.com/2020/ ... ory-a-faq/

[nemos]

And the most pragmatic convention is x/0 = 0.
https://xenaproject.wordpress.com/2020/ ... ory-a-faq/

Of course, anything can be agreed, and in theory you can even implement a function that contains only specifically defined answers. But it is unlikely to reveal the laws of nature, which is really what mathematics is supposed to do.
If you look at the graph of the function x/y(where y->0) = 0(if y=0), you will see that at first the function smoothly tends to infinity, or any upper bound number you define, and then suddenly jumps to 0 creating an ugly singularity. The graph for a function that only accumulates singular points will be even uglier, unless its purpose is to create a reproduction of an artist's painting or something like that.

I am not really clear on the underlying motive of this topic at the moment. For if it is claimed that division does not mean mathematical division, and is not defined in any other sense, then what common denominator should such a theme lead to ?
This seems like some form of protest, hopefully not against mathematics?

By the way, I don't know what age of child was in question, but I would start explaining division to him like this:
if there were 2 apples in one basket and we divide them equally into 2 baskets, how many apples will be in each basket ? Or,
if we split one apple in half into 2 equal parts, what would we get ?

[Skepdick]

Of course, anything can be agreed, and in theory you can even implement a function that contains only specifically defined answers. But it is unlikely to reveal the laws of nature, which is really what mathematics is supposed to do.

That's not a view I share. Mathematics is just rules/laws for manipulating abstract objects. That some people happen to use this mental faculty as a model of nature and that this even works - that's just a crazyly-useful coincidence.

If you look at the graph of the function x/y(where y->0) = 0(if y=0), you will see that at first the function smoothly tends to infinity, or any upper bound number you define, and then suddenly jumps to 0 creating an ugly singularity.

Sure. And? Are you presupposing an axiom/expectation like "all functions are smooth/continuous/infinitely differentiable" or something? You sounds more like a physicist than a mathematician.

Mathematicians don't divide by 0 in practice, so the fact that you are facing a singularity is only because you've done something prohibited anyway.

I am not really clear on the underlying motive of this topic at the moment.

I thought it was pretty clear from the accompanying blog post? The formalization of Mathematics using computers.

For if it is claimed that division does not mean mathematical division, and is not defined in any other sense, then what common denominator should such a theme lead to ?
This seems like some form of protest, hopefully not against mathematics?

Here is the relevant quote from the blog post (that you didn't seem to read):

Mathematicians don’t divide by 0 and hence in practice they never notice the difference between real.div and mathematical division (for which 1/0 is undefined).

If you never actually invoke the function with a 0-denominator (which people don't do **in practice**) the two functions are pragmatically equivalent.

So, yeah - I guess you could call it a protest. Against idealism and in favour of pragmatism.

By the way, I don't know what age of child was in question, but I would start explaining division to him like this:
if there were 2 apples in one basket and we divide them equally into 2 baskets, how many apples will be in each basket ? Or,
if we split one apple in half into 2 equal parts, what would we get ?

Which would be a false analogy, Wouldn't it?

And even if you were to put the 2 apples into NO baskets you'll still have 2 apples.

2 apples in 2 baskets is 2 apples e.g 1 apple per basket
2 apples in 1 basket is 2 apples e.g 2 apples per basket
2 apples in 0 baskets is 2 apples e.g 2 apples per NO basket

[nemos]

... You sounds more like a physicist than a mathematician...

Physics or mathematics, in my opinion, is just a means to an end. At one time, I chose physics. If I had to make a choice right now, it would probably be math.
Because the model of physical space (and physicists work directly with a model that includes all current knowledge and of course leaves room for the unknown. Again, this is just my opinion, I'm not going to argue.) is only a specially defined mathematical space, with relevant boundary conditions.
But by changing the means, I certainly would not change my original goal.

If the movement does not have a specific direction, e.g. thermal motion, no matter how intense it is, the average motion over time will be 0. And even a relatively small directional component on the background of thermal motion, such as the motion of charged particles in an electric field, can do impressive work. Therefore, the direction is decisive, not the means.

No Martini, no party.
No "baskets" no apples.
In general, it is difficult for children to understand the meaning of zero. Also historically zero appeared relatively late, already after the negative numbers, as a need to deal with the 1-1 situation.

[accelafine]

[Skepdick]

In general, it is difficult for children to understand the meaning of zero.

That's only true because we keep insisting that 0 is a number and attempt to give it a Mathematical/abstract/formal treatment.

In practice, children already understand the concept in English which is a far more powerful computational model than Mathematics. So just teach them about monads.

x/y is a number. Maybe.
It's Either a number or it's Nothing.

https://en.wikipedia.org/wiki/Monad_(fu ... ple:_Maybe
https://hackage.haskell.org/package/cat ... ither.html

[nemos]

... So just teach them about monads ...

https://en.wikipedia.org/wiki/Monad_(fu ... ple:_Maybe
https://hackage.haskell.org/package/cat ... ither.html

It doesn't leave me feeling that when it comes to children, we understand things a little differently.
When I talked about baskets, I had in mind children between the ages of 5-7. What age children were you talking about when you pointed to monads? In which class should they start learning it?

[Skepdick]

It doesn't leave me feeling that when it comes to children, we understand things a little differently.
When I talked about baskets, I had in mind children between the ages of 5-7. What age children were you talking about when you pointed to monads? In which class should they start learning it?

By the time they are 7 and they speak English they sure know how to use the English terms "nothing" and "maybe".

You have 2 apples. Maybe you have a basket. Maybe you don't.

How many apples can you put in a basket you don't have.

2 apples / 0 baskets = ???

[nemos]

... By the time they are 7 and they speak English they sure know how to use the English terms "nothing" and "maybe"...

At the moment, I have doubts about whether we belong to the same species, because children of my species at the age of 7 something like this

... So just teach them about monads ...

https://en.wikipedia.org/wiki/Monad_(fu ... ple:_Maybe
https://hackage.haskell.org/package/cat ... ither.html

will not be able to learn.

However, conversations with members of other species can still be quite engaging.

... Maybe you have a basket. Maybe you don't ...

Why, assuming the basket may not be there, do you still distribute the apples by leaving them? If you have chosen a direction, you should be braver, accepting that not only the baskets may not be there, but also the apples.
And if there is neither basket nor apple, then what are we talking about here?

By the way, when the math teacher at school gave you exercises with conditions, you still claimed that these conditions may or may not exist, and therefore it is not mandatory to follow them?

[Skepdick]

And if there is neither basket nor apple, then what are we talking about here?

0/0.

Obviously.

By the way, when the math teacher at school gave you exercises with conditions, you still claimed that these conditions may or may not exist, and therefore it is not mandatory to follow them?

The conditions always seemed to have alternative interpretations such that more than one possible solution existed

[alan1000]

Not many people know about it.

Apart from being an an excellent mathematician Schlechtfunkel also created the famous, Bavarian pastry. I've been known to down 2 or 3 Schlechtfunkels when I'm very depressed.

You may not know that he was also a talented musician. In the 1920s and 30s he would often be found performing in some Bavarian tavern or other, with his life long friend, Otto Schleiman, Together they formed the very popular folk duo of Schleiman and Schlechtfunkel. They were best known for their rendition of Brücke über die unruhige Donau.

Yes, I remember it well. Happy days. For many years, I had the 78rpm record, but I accidentally sat on it one day and broke it.

[alan1000]

0 ÷ 0 = ∞ ÷ ∞

This equation is mathematically correct, provided 0 is understood as Robinson's h, and not as the natural number. The completed equation should, of course, read 0 ÷ 0 = ∞ ÷ ∞ = ∞.

[wtf]

0 ÷ 0 = ∞ ÷ ∞

This equation is mathematically correct, provided 0 is understood as Robinson's h, and not as the natural number. The completed equation should, of course, read 0 ÷ 0 = ∞ ÷ ∞ = ∞.

What is "Robinson's h?" Pretend I have a detailed understanding of Robinson's hyperreal numbers.

The hyperreals are a field. That means for any nonzero hyperreal h, h/h = 1.

Ok your turn.