3 ms·

Huh, this is... not the terminology I am used to, as a mathematician? My experience is that what C++ is calling a "weak order", and what you are calling a "wea

Huh, this is... not the terminology I am used to, as a mathematician? My experience is that what C++ is calling a "weak order", and what you are calling a "weak order", is what in math is called a "pre-order" or a "quasi-order". Meanwhile, the distinction you are making between strict and nonstrict is just ignored (except by constructivists), since they're equivalent ways of talking about the same thing. Indeed I'm not sure why you included both because, well, they're just two ways of talking about the same thing (again, unless you're a constructivist). Really (assuming classical logic) there's just 4 possibilities here: 1. Total order (what they're calling a "strong ordering") 2. Total preorder (what they're calling a "weak ordering") 3. Partial order (which they are also calling a "partial ordering") 4. Partial preorder (which they don't account for)

mehrdadn 7y ago
You're agreeing with me? "Preorder" is just a (better-known?) synonym for "non-strict weak order" [1] [2], so we don't disagree there. "Strict" vs. "non-strict" I merely included because C++ comparisons return strict orders. I thought it was worth including, but feel free to ignore it. So just as you pointed out in your own list, and just as I've been saying, "strong total order" isn't the terminology (every total order is strong), and neither is "weak total order" (it means "total preorder" which is... just a total order). Like you said, they should say "total" order, "preorder" (or "weak" order as I said), "partial" order. The "strong" and "weak total" stuff is just something they seem to have invented in contradiction with the established mathematical terminology for... no reason/gain? [1] https://en.wikipedia.org/wiki/Weak_ordering#Total_preorders https://en.wikipedia.org/wiki/Weak_ordering#Total_preorders [2] http://fitelson.org/roberts_measurement_theory.pdf#page=56 http://fitelson.org/roberts_measurement_theory.pdf#page=56
- Sniffnoy 7y ago
  Well, I disagree that the article is confusing on this point. They say quite explicitly what they mean. I agree that the terminology is annoying, because they should just match the existing math terminology, and it's also annoying that they didn't account for partial preorders. I'm not sure whether I agree that the terminology is confusing; on the whole I think it isn't. The reason it's not confusing is that it sufficiently different from usual math terminology so as not to interfere -- i.e., the terminology doesn't actually disagree at any point, it's not incompatible. Like, concepts get reinvented all the time and you just kind of have to get used to things having multiple terms, and be ready to translate unusual terminology into standard terminology, even as of course you should do what you can to reduce this happening. As long as you don't end up in a situation where one word means two different things, there's not really confusion, just different terminology. And, as I said above, I definitely disagree that the article is confusing on this point, because they're very explicit about what they mean.
  - mehrdadn 7y ago
    I said the terminology is confusing, not the article... > the terminology doesn't actually disagree at any point, it's not incompatible It does disagree though? I thought I just explained this... I'll do it again. "Total orders" are by their very definition in math always 'strong' -- their entire point is to use equality as the equivalence relation. That's what distinguishes them from weak orders, which can have other equivalence classes. So a "weak total order" makes no sense -- if a weak order is total, it's by definition the same as a 'strong' total order. That's in direct contradiction with their terminology.
    - Sniffnoy 7y ago
      I mean, that's like saying "A ring by definition is always associative, so saying 'a ring without associativity' is contradictory". Neither ordinary language nor technical terminology work that way; not every phrase is compositional in such a straightforward way, including in mathematics. I mean, a "fake X" is not an X. In mathematics the word "weak" plays a similar role; e.g. in differential equations a weak solution need not be a solution; it's easy to find more examples. Similarly when one talks about a "non-Y X" or a "X without Y" where ordinarily Y is part of the definition of X. The result is not an X, but it's still (usually) clear what's meant, and these are still the terms that get used. Anyway, the point is, new terms someone is defining have to be considered on the whole. An actual incompatibility, like I said, would be giving an existing phrase a new meaning. E.g. if they had referred to total preorders as partial orders, that would be an incompatibility.
      - mehrdadn 7y ago
        > An actual incompatibility, like I said, would be giving an existing phrase a new meaning. E.g. if they had referred to total preorders as partial orders, that would be an incompatibility. This is what they're doing!! They're calling pre-orders "weak total orders". That's a direct incompatibility... "weak total order" already means "total order", just like "total weak order" already means "total pre-order" which already means "total order". "Weak" and "total" are both adjectives, and they're interchangeable. If you asked anyone (who unlike you had already heard of the term "weak order" before) that's exactly how they would interpret it. Nothing in it would be left open for interpretation since everything is defined. Whereas in your solution example, unlike here, the phrase "weak solution" didn't already mean something else, and there was no existing term for the concept either. And unlike in "ring without associativity", they use the word "total" in "weak total order" to mean nothing. They could've taken it out and "weak order" would've meant exactly what they meant. If you want to make a comparison, what they're doing would be like using "a nonnegative positive solution" to mean "a nonnegative solution" (rather than "a positive nonnegative solution"). Or using "an fractional real number" to mean "a fractional complex number" rather than a "real fractional number" (i.e. real fraction). Which would be completely nuts.

mehrdadn 7y ago

You're agreeing with me? "Preorder" is just a (better-known?) synonym for "non-strict weak order" [1] [2], so we don't disagree there. "Strict" vs. "non-strict" I merely included because C++ comparisons return strict orders. I thought it was worth including, but feel free to ignore it. So just as you pointed out in your own list, and just as I've been saying, "strong total order" isn't the terminology (every total order is strong), and neither is "weak total order" (it means "total preorder" which is... just a total order). Like you said, they should say "total" order, "preorder" (or "weak" order as I said), "partial" order. The "strong" and "weak total" stuff is just something they seem to have invented in contradiction with the established mathematical terminology for... no reason/gain? [1] https://en.wikipedia.org/wiki/Weak_ordering#Total_preorders https://en.wikipedia.org/wiki/Weak_ordering#Total_preorders [2] http://fitelson.org/roberts_measurement_theory.pdf#page=56 http://fitelson.org/roberts_measurement_theory.pdf#page=56

Sniffnoy 7y ago
Well, I disagree that the article is confusing on this point. They say quite explicitly what they mean. I agree that the terminology is annoying, because they should just match the existing math terminology, and it's also annoying that they didn't account for partial preorders. I'm not sure whether I agree that the terminology is confusing; on the whole I think it isn't. The reason it's not confusing is that it sufficiently different from usual math terminology so as not to interfere -- i.e., the terminology doesn't actually disagree at any point, it's not incompatible. Like, concepts get reinvented all the time and you just kind of have to get used to things having multiple terms, and be ready to translate unusual terminology into standard terminology, even as of course you should do what you can to reduce this happening. As long as you don't end up in a situation where one word means two different things, there's not really confusion, just different terminology. And, as I said above, I definitely disagree that the article is confusing on this point, because they're very explicit about what they mean.
- mehrdadn 7y ago
  I said the terminology is confusing, not the article... > the terminology doesn't actually disagree at any point, it's not incompatible It does disagree though? I thought I just explained this... I'll do it again. "Total orders" are by their very definition in math always 'strong' -- their entire point is to use equality as the equivalence relation. That's what distinguishes them from weak orders, which can have other equivalence classes. So a "weak total order" makes no sense -- if a weak order is total, it's by definition the same as a 'strong' total order. That's in direct contradiction with their terminology.
  - Sniffnoy 7y ago
    I mean, that's like saying "A ring by definition is always associative, so saying 'a ring without associativity' is contradictory". Neither ordinary language nor technical terminology work that way; not every phrase is compositional in such a straightforward way, including in mathematics. I mean, a "fake X" is not an X. In mathematics the word "weak" plays a similar role; e.g. in differential equations a weak solution need not be a solution; it's easy to find more examples. Similarly when one talks about a "non-Y X" or a "X without Y" where ordinarily Y is part of the definition of X. The result is not an X, but it's still (usually) clear what's meant, and these are still the terms that get used. Anyway, the point is, new terms someone is defining have to be considered on the whole. An actual incompatibility, like I said, would be giving an existing phrase a new meaning. E.g. if they had referred to total preorders as partial orders, that would be an incompatibility.
    - mehrdadn 7y ago
      > An actual incompatibility, like I said, would be giving an existing phrase a new meaning. E.g. if they had referred to total preorders as partial orders, that would be an incompatibility. This is what they're doing!! They're calling pre-orders "weak total orders". That's a direct incompatibility... "weak total order" already means "total order", just like "total weak order" already means "total pre-order" which already means "total order". "Weak" and "total" are both adjectives, and they're interchangeable. If you asked anyone (who unlike you had already heard of the term "weak order" before) that's exactly how they would interpret it. Nothing in it would be left open for interpretation since everything is defined. Whereas in your solution example, unlike here, the phrase "weak solution" didn't already mean something else, and there was no existing term for the concept either. And unlike in "ring without associativity", they use the word "total" in "weak total order" to mean nothing. They could've taken it out and "weak order" would've meant exactly what they meant. If you want to make a comparison, what they're doing would be like using "a nonnegative positive solution" to mean "a nonnegative solution" (rather than "a positive nonnegative solution"). Or using "an fractional real number" to mean "a fractional complex number" rather than a "real fractional number" (i.e. real fraction). Which would be completely nuts.