I never noticed this comment! The absurdities you point out don't exist once you realize that 0/0 is still undefined, just as ∞/∞ is undefined. For example, step 5 of this proof... https://en.wikipedia.org/wiki/Mathematical_fallacy#Division_by_zero ...has not just division by zero, but 0 divided by 0. So, we don't have 2 = 1, just undefined = undefined.
I know you don't think I'm that bright, but at the very least, be aware that I've thought about this for many years while you were doing more useful things!
But then, what do you mean by _reciprocal_ and _division_? According to the usual definitions, 1/0 = ∞ means 0 × ∞ = 1. Therefore, 0/0 = 0 × 1/0 = 0 × ∞ = 1, rather than being undefined.
I suppose anyone sufficiently interested already knows this much better than I do, but there’s a reason the definition of a field (<https://en.m.wikipedia.org/wiki/Field_(mathematics)>) does not require a reciprocal for the identity element of addition; namely, that fields with such a reciprocal are not very interesting or useful.
Let’s suppose such element exists and call it ∞. Then, 0 × ∞ = 1. In any field, 0 is an absorbing element, so it’s also true that 0 × ∞ = 0. Therefore, 1 = 0. Now, for any element a in the field, 1 × a = a, and 0 × a = 0 = 1, so a = 0 = 1; that is, the field has but one element, which, of course, must be the identity element of both addition and multiplication, and its own reciprocal. That’s why what Eric Brown called “absurdities” are actually true in this system.
I think the real lesson in any human interaction is, “I am in charge, so use my terms and concepts or you’ll suffer”, said or implied by whoever is actually in charge.
> According to the usual definitions, 1/0 = ∞ means 0 × ∞ = 1.
No; if that were true, the usual definitions would also imply that 1×0 = 0 means 0/0 = 1, and now we are dividing by zero. The usual way of carrying out math in grade school holds that multiplication by 0 has no legal inverse, in contradiction to that kind of reasoning.
Although there are other ways of handling it, the "divide by 0" function can be treated as non-invertible in the same way that grade school math treats "multiply by 0" as invertible, because it is many-to-one. That is, if any x divided by 0 = complex infinity, then complex infinity times 0 = any x, or 0 × ∞ = undefined. This is nothing special; according to conventional mathematics which avoids surreals and transfinites, many-to-one functions are often treated as non-invertible.
Granted, sometimes many-to-oneedness can be overcome. Say in the case of e^z, with complex number z, where the inverse function is many-valued log(z) where periodicity requires making a branch cut (e.g. https://functions.wolfram.com/ElementaryFunctions/Log/visualizations/5/ ). This kind of overcoming of objections is common, because "you can't do that" is often a barrier to getting a useful result.
Ultimately, I didn't invent the Riemann Sphere or the hyperreals ( https://en.wikipedia.org/wiki/Hyperreal_number ). Arguments that these are "absurdities" are analogous to claims the square root of negative 1 result in "absurdities." OK, but physicists are going to drive right past those objections with the Dirac Delta function and the Schroedinger Equation, and show our entire reality is "absurd" with the discovery of Quantum Mechanics.
> I think the real lesson in any human interaction is, “I am in charge, so use my terms and concepts or you’ll suffer”, said or implied by whoever is actually in charge.
By that line of reasoning I may as well just welcome you to Things to Read, and remind you that I am in charge. I don't really care if you use my terms and concepts, though; I'm just going to keep pointing out that "Annie Ant is on a pilgrimage to the Holy Land, she's 8 miles away, and travels at 1/4 mile per hour, how long does it take her to get there" is a reasonable question and has a reasonable answer, and "If Annie is squashed flat by an apple cart and her speed is reduced to 0, how long does it take her to get there" is just as reasonable, and has just as reasonable an answer.
There might be something wrong with me, but I just don't care. If they say dividing things by zero is impossible, I say fine. They make the rules! If they say dividing things by zero equals infinity, I say fine too. It makes sense, after all, because dividing something by 0.000001 results in a very big number.
That exercise with nines was difficult to understand. It gave me the impression that there are people who actually like mathematical concepts, the same way that I actually like words. People who like to learn new mathematical concepts for the sake of it. That is strange to me.
Before I moved here to the midlatitudes, I didn't care about climate, at all. Sure, CO2 --> global warming --> something something coral reefs die, and that bothered me, but the actual workings of the Earth could not be of less interest.
Then a time came when I had to look around and consider where to move. I started poking and poking, and found this hideous map, Köppen's map of dizzying complexity, where blue means hot, and also cold:
As before, I had zero interest. But this way, Köppen's way, was the standard way of discussing climates across Wikipedia. And because I was considering places to live for long periods, I grit my teeth for a few hours, and learned the meaning of all the little letter codes to try to make sense of things, and determine where good climates could be found. And then - then, like a ray shining forth from the clouds, I Found This:
Legible, comprehensible, and so, so lovely. The reds, the greens, the blues, the shining whites! Trewartha's glorious map called to me, and I listened; what began as hours stretched into years. I gathered data, wrangled with climate officianados, and finally produced my own map, and my own classification system, about which no one cares.
// that's hilarious, I went to all this effort to make sense but in that moment pander to the lowest common denominator as described in that normal distribution above, all while targeting a pet peeve of yours. Hindsight is a painful logic. I guess any mathematical law or expectation works the same way. Parallel lines defined as never meeting would be another.
// takeaway, there is always hope for narcissist "to world", it will just take longer than the expected lifespan of the universe.
My problem with dividing by zero is that it lets you conclude that 2 = 1, and other absurdities.
There are tons of algebraic reductions that smuggle in a division by zero along the way that end up concluding that 2 = 1.
I never noticed this comment! The absurdities you point out don't exist once you realize that 0/0 is still undefined, just as ∞/∞ is undefined. For example, step 5 of this proof... https://en.wikipedia.org/wiki/Mathematical_fallacy#Division_by_zero ...has not just division by zero, but 0 divided by 0. So, we don't have 2 = 1, just undefined = undefined.
I know you don't think I'm that bright, but at the very least, be aware that I've thought about this for many years while you were doing more useful things!
But then, what do you mean by _reciprocal_ and _division_? According to the usual definitions, 1/0 = ∞ means 0 × ∞ = 1. Therefore, 0/0 = 0 × 1/0 = 0 × ∞ = 1, rather than being undefined.
I suppose anyone sufficiently interested already knows this much better than I do, but there’s a reason the definition of a field (<https://en.m.wikipedia.org/wiki/Field_(mathematics)>) does not require a reciprocal for the identity element of addition; namely, that fields with such a reciprocal are not very interesting or useful.
Let’s suppose such element exists and call it ∞. Then, 0 × ∞ = 1. In any field, 0 is an absorbing element, so it’s also true that 0 × ∞ = 0. Therefore, 1 = 0. Now, for any element a in the field, 1 × a = a, and 0 × a = 0 = 1, so a = 0 = 1; that is, the field has but one element, which, of course, must be the identity element of both addition and multiplication, and its own reciprocal. That’s why what Eric Brown called “absurdities” are actually true in this system.
I think the real lesson in any human interaction is, “I am in charge, so use my terms and concepts or you’ll suffer”, said or implied by whoever is actually in charge.
> According to the usual definitions, 1/0 = ∞ means 0 × ∞ = 1.
No; if that were true, the usual definitions would also imply that 1×0 = 0 means 0/0 = 1, and now we are dividing by zero. The usual way of carrying out math in grade school holds that multiplication by 0 has no legal inverse, in contradiction to that kind of reasoning.
Although there are other ways of handling it, the "divide by 0" function can be treated as non-invertible in the same way that grade school math treats "multiply by 0" as invertible, because it is many-to-one. That is, if any x divided by 0 = complex infinity, then complex infinity times 0 = any x, or 0 × ∞ = undefined. This is nothing special; according to conventional mathematics which avoids surreals and transfinites, many-to-one functions are often treated as non-invertible.
Granted, sometimes many-to-oneedness can be overcome. Say in the case of e^z, with complex number z, where the inverse function is many-valued log(z) where periodicity requires making a branch cut (e.g. https://functions.wolfram.com/ElementaryFunctions/Log/visualizations/5/ ). This kind of overcoming of objections is common, because "you can't do that" is often a barrier to getting a useful result.
Ultimately, I didn't invent the Riemann Sphere or the hyperreals ( https://en.wikipedia.org/wiki/Hyperreal_number ). Arguments that these are "absurdities" are analogous to claims the square root of negative 1 result in "absurdities." OK, but physicists are going to drive right past those objections with the Dirac Delta function and the Schroedinger Equation, and show our entire reality is "absurd" with the discovery of Quantum Mechanics.
> I think the real lesson in any human interaction is, “I am in charge, so use my terms and concepts or you’ll suffer”, said or implied by whoever is actually in charge.
By that line of reasoning I may as well just welcome you to Things to Read, and remind you that I am in charge. I don't really care if you use my terms and concepts, though; I'm just going to keep pointing out that "Annie Ant is on a pilgrimage to the Holy Land, she's 8 miles away, and travels at 1/4 mile per hour, how long does it take her to get there" is a reasonable question and has a reasonable answer, and "If Annie is squashed flat by an apple cart and her speed is reduced to 0, how long does it take her to get there" is just as reasonable, and has just as reasonable an answer.
> I am in charge
Duly noted.
Sorry if you suffered.
There might be something wrong with me, but I just don't care. If they say dividing things by zero is impossible, I say fine. They make the rules! If they say dividing things by zero equals infinity, I say fine too. It makes sense, after all, because dividing something by 0.000001 results in a very big number.
That exercise with nines was difficult to understand. It gave me the impression that there are people who actually like mathematical concepts, the same way that I actually like words. People who like to learn new mathematical concepts for the sake of it. That is strange to me.
I like both.
I liked the mathematical concepts more in school, where I was fed them. That made them seem more relevant.
Interest is a strange thing.
Before I moved here to the midlatitudes, I didn't care about climate, at all. Sure, CO2 --> global warming --> something something coral reefs die, and that bothered me, but the actual workings of the Earth could not be of less interest.
Then a time came when I had to look around and consider where to move. I started poking and poking, and found this hideous map, Köppen's map of dizzying complexity, where blue means hot, and also cold:
https://upload.wikimedia.org/wikipedia/commons/d/d5/K%C3%B6ppen-Geiger_Climate_Classification_Map.png
As before, I had zero interest. But this way, Köppen's way, was the standard way of discussing climates across Wikipedia. And because I was considering places to live for long periods, I grit my teeth for a few hours, and learned the meaning of all the little letter codes to try to make sense of things, and determine where good climates could be found. And then - then, like a ray shining forth from the clouds, I Found This:
https://en.wikipedia.org/wiki/K%C3%B6ppen_climate_classification#/media/File:K%C3%B6ppen-Geiger_Climate_Classification_Map.png
Legible, comprehensible, and so, so lovely. The reds, the greens, the blues, the shining whites! Trewartha's glorious map called to me, and I listened; what began as hours stretched into years. I gathered data, wrangled with climate officianados, and finally produced my own map, and my own classification system, about which no one cares.
Interest is a strange thing.
// I find support for my "logic is a hindsight", in Wolfram's words on entropy & thermodynamics at https://www.youtube.com/watch?v=dkpDjd2nHgo
// that's hilarious, I went to all this effort to make sense but in that moment pander to the lowest common denominator as described in that normal distribution above, all while targeting a pet peeve of yours. Hindsight is a painful logic. I guess any mathematical law or expectation works the same way. Parallel lines defined as never meeting would be another.
// takeaway, there is always hope for narcissist "to world", it will just take longer than the expected lifespan of the universe.
Worry not, dear samorzewski. If you hadn't commented, I wouldn't have even connected this to your post on ratios at https://whyweshould.substack.com/p/the-janus-ratio
When I read it, I had a vague sense you might just be talking about division by zero somewhat metaphorically, or for the sake of making a point.
> there is always hope for narcissist "to world", it will just take longer than the expected lifespan of the universe.
LOL