I think this is a slightly different issue. In Magic there’s a concept of “strictly better” where one card is deemed to be always better than another (eg Lightning Bolt over Shock), as opposed to statistically better (eg Silver Knight is generally considered better than White Knight but the latter is clearly preferable if you’re playing against black and not red). However, some people take “strictly better” too, um, strictly, and try to point out weird cases where you would prefer to have the seemingly worse card. Often these scenarios involve Mindslaver (eg if you’re on 3 life and your opponent has Mindslaver you’d rather have Shock in hand than Lightning Bolt).
The lesson is to not let rare pathological cases ruin useful generalizations (at least not outside of formal mathematics).
The lesson is to not let rare pathological cases ruin useful generalizations (at least not outside of formal mathematics).
By the way even in formal mathematics (and maybe especially in formal mathematics), while pathological cases are interesting, nobody discards perfectly useful theories just because the theory allows pathologies. For example, nobody hesitates to use measure theory in spite the Banach-Tarski paradox; nobody hesitates to use calculus even though the Weierstrass function exists; few people hesitate in using the Peano axioms in spite of the existence of non-standard models of that arithmetic.
Nitpick: I would consider the Weierstrass function a different sort of pathology than non-standard models or Banach-Tarski—a practical pathology rather than a conceptual pathology. The Weierstrass function is just a fractal. It never smooths out no matter how much you zoom in.
I agree that the Weierstrass function is different. I felt a tinge of guilt when I included the Weierstrass function. But I included it since it’s probably the most famous pathology.
That being said, I don’t quite understand the distinction you’re making between a practical and a conceptual pathology. The distinction I would make between the Weierstrass and the other two is that the Weierstrass is something which is just counter-intuitive whereas the other two can be used as a reason to reject the entire theory. They are almost antithetical to the purpose of the theory. Is that what you were getting at?
However, some people take “strictly better” too, um, strictly, and try to point out weird cases where you would prefer to have the seemingly worse card. Often these scenarios involve Mindslaver
Ahh, that would do it. The enemy being the one who uses the card would tend to make inferiority desirable in rather a lot of cases.
I think this is a slightly different issue. In Magic there’s a concept of “strictly better” where one card is deemed to be always better than another (eg Lightning Bolt over Shock), as opposed to statistically better (eg Silver Knight is generally considered better than White Knight but the latter is clearly preferable if you’re playing against black and not red). However, some people take “strictly better” too, um, strictly, and try to point out weird cases where you would prefer to have the seemingly worse card. Often these scenarios involve Mindslaver (eg if you’re on 3 life and your opponent has Mindslaver you’d rather have Shock in hand than Lightning Bolt).
The lesson is to not let rare pathological cases ruin useful generalizations (at least not outside of formal mathematics).
By the way even in formal mathematics (and maybe especially in formal mathematics), while pathological cases are interesting, nobody discards perfectly useful theories just because the theory allows pathologies. For example, nobody hesitates to use measure theory in spite the Banach-Tarski paradox; nobody hesitates to use calculus even though the Weierstrass function exists; few people hesitate in using the Peano axioms in spite of the existence of non-standard models of that arithmetic.
Nitpick: I would consider the Weierstrass function a different sort of pathology than non-standard models or Banach-Tarski—a practical pathology rather than a conceptual pathology. The Weierstrass function is just a fractal. It never smooths out no matter how much you zoom in.
I agree that the Weierstrass function is different. I felt a tinge of guilt when I included the Weierstrass function. But I included it since it’s probably the most famous pathology.
That being said, I don’t quite understand the distinction you’re making between a practical and a conceptual pathology. The distinction I would make between the Weierstrass and the other two is that the Weierstrass is something which is just counter-intuitive whereas the other two can be used as a reason to reject the entire theory. They are almost antithetical to the purpose of the theory. Is that what you were getting at?
Ahh, that would do it. The enemy being the one who uses the card would tend to make inferiority desirable in rather a lot of cases.