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But this is my point - the definitions are the way they are because it's a good model of counting (and other counting-like things). They weren't invented in a v

But this is my point - the definitions are the way they are because it's a good model of counting (and other counting-like things). They weren't invented in a vacuum. Similarly the fact that multiplication and division are inverses is a property of this model. Conceptually you can imagine splitting and copying groups of objects quite independently of one another (and which one you view as fundamental is really a post-hoc choice). In general these days we mostly see clean mathematical abstractions because all the scaffolding has already been removed by mathematicians past. And as a result people come to believe that this is how mathematics is done. But always there is an initial period of exploration (which eventually gets forgotten) as people try to work out how to axiomatise the various systems they are interested in.

ggu7hgfk8j 2y ago
At the end of the day it's about practicality. Contrast 0^0 with 0/0. Both of them are considered undefined, but the former is often defined "locally", e.g in a given textbook or article to have the value 1. That's not true for 0/0 because it's not been found to be useful.
- bubblyworld 2y ago
  Indeed, 0^0 is often defined to be 1 in combinatorics textbooks, for instance, because in those contexts that's a useful/meaningful choice. In measure theory it's often useful to augment your reals with positive/negative infinity. In projective geometry it's sometimes meaningful to allow division by zero (to counter one detail of your comment). In nonstandard analysis you would consider infinitesmals to be valid numbers, and in game theory you might consider stuff like the surreals which are yet another to view the familiar numbers, with different laws. You can say we're talking about many different systems here, and that's true in a formal sense. I'm just pointing out that these formal systems come from somewhere, and mathematics is really about the thought-stuff underlying them. You should be willing to bend a rule here and there if it is truer to the concepts you care about - statements like "division by zero is undefined" should never be taken as absolutes. (but of course this is just my personal philosophy of mathematics, take it with a pinch of salt)
- fph 2y ago
  There definitely are contexts in which it makes sense to define 0/0 = 0. Example: given a vector x and an approximation y to it, you wish to compute the "componentwise error" delta, i.e., the minimum positive number delta such that each entry of y is within relative error at most delta from the corresponding entry of x: (1) abs(y_i - x_i) <= delta * abs(x_i) for each i. This number can be computed as (2) delta = max (abs(y_i - x_i) / abs(x_i)) over each i. If you wish to allow zero entries in your vectors while keeping the equivalence between (1) and (2), you have to define 0 / 0 = 0.

ggu7hgfk8j 2y ago

At the end of the day it's about practicality. Contrast 0^0 with 0/0. Both of them are considered undefined, but the former is often defined "locally", e.g in a given textbook or article to have the value 1. That's not true for 0/0 because it's not been found to be useful.

bubblyworld 2y ago
Indeed, 0^0 is often defined to be 1 in combinatorics textbooks, for instance, because in those contexts that's a useful/meaningful choice. In measure theory it's often useful to augment your reals with positive/negative infinity. In projective geometry it's sometimes meaningful to allow division by zero (to counter one detail of your comment). In nonstandard analysis you would consider infinitesmals to be valid numbers, and in game theory you might consider stuff like the surreals which are yet another to view the familiar numbers, with different laws. You can say we're talking about many different systems here, and that's true in a formal sense. I'm just pointing out that these formal systems come from somewhere, and mathematics is really about the thought-stuff underlying them. You should be willing to bend a rule here and there if it is truer to the concepts you care about - statements like "division by zero is undefined" should never be taken as absolutes. (but of course this is just my personal philosophy of mathematics, take it with a pinch of salt)
fph 2y ago
There definitely are contexts in which it makes sense to define 0/0 = 0. Example: given a vector x and an approximation y to it, you wish to compute the "componentwise error" delta, i.e., the minimum positive number delta such that each entry of y is within relative error at most delta from the corresponding entry of x: (1) abs(y_i - x_i) <= delta * abs(x_i) for each i. This number can be computed as (2) delta = max (abs(y_i - x_i) / abs(x_i)) over each i. If you wish to allow zero entries in your vectors while keeping the equivalence between (1) and (2), you have to define 0 / 0 = 0.