If you are responding to a hypothetical that tests a mathematical model, and your response doesn’t use math, and doesn’t hinge on a consciousness, infinity, or impossibility from the original problem domain, your response is likely irrelevant.
Right.. I’ll give a few more examples from math. Say you’re arguing that calculus is a lie because deriving dy/dx clearly involves division by zero. In this case, you’re getting ‘emotionally involved’. You’re focusing on the notation dy/dx and all sorts of things about the existence of infinitesimals and division by zero. But that impossibility doesn’t exist in the original theory, because (standard) calculus is founded on limits and not division by zero or infinitesimals. The infinities and infinitesimals aren’t part of the original model which you’re arguing against
Likewise, if you’re arguing that ZFC is inconsistent by Russell’s paradox, because you can construct peculiar but plausible sounding sets which imply contradictions, you’re making the same mistake. You’re being emotionally involved with your naive/primitive concept of a ‘set’, whereas the theory in question (ZFC) doesn’t even allow you to construct such sets.
The above arguments are less common, but I have heard them. A more common argument concerns the Axiom of Choice, and goes a little something like this:
To me, the strongest argument in favor of AC is one if the many equivalent statements: if A_i is a family of non-empty sets then the cartesian product of the A_i is non-empty.
I pulled that from the math subreddit where it was posted a few days ago, and it’s a fairly common argument. But the commenter has become emotionally involved with day-to-day sets and Cartesian products. What would the product of an uncountable collection of uncountable sets even look like? Once one refers to the formal, very abstract definition, it should be clear that we have absolutely no right to expect anything about it’s emptiness or nonemptiness, because the intuition and emotional involvement are replaced by formal abstraction. The things which one assumes exist aren’t actually there in the original theory (ZF).
It’s possible to accidentally construct a hypothetical that makes an assumption that isn’t valid in our universe. (I think these paradoxes were unknown before the 20th century, but there may be a math example.
The paradoxes falling out of the geocentric model, maybe?
But the commenter has become emotionally involved with day-to-day sets and Cartesian products.
Not necessarily. At least, not necessarily more so than anyone becomes “emotionally involved” when deciding on axioms to use in a mathematical theory.
AC is after all independent of ZF. So of course no argument in favor of it can be constructed literally on the basis of the “original theory” (ZF). Saying that the Cartesian product of nonempty sets ought to be nonempty is an aesthetic statement about what the rules of the game should be, not a mistaken inference from the axioms of ZF.
Because of the independence, we logically have a “right to expect” anything we want (AC or not-AC). The choice is a matter of taste, and the tastes of the mathematical community evidently incline toward AC.
Note how different this is from your two previous examples regarding derivatives and Russell’s paradox. Those involved outright logical errors; whereas in the case of AC, the commenter is making a legitimate aesthetic argument.
Right.. I’ll give a few more examples from math. Say you’re arguing that calculus is a lie because deriving dy/dx clearly involves division by zero. In this case, you’re getting ‘emotionally involved’. You’re focusing on the notation dy/dx and all sorts of things about the existence of infinitesimals and division by zero. But that impossibility doesn’t exist in the original theory, because (standard) calculus is founded on limits and not division by zero or infinitesimals. The infinities and infinitesimals aren’t part of the original model which you’re arguing against
Likewise, if you’re arguing that ZFC is inconsistent by Russell’s paradox, because you can construct peculiar but plausible sounding sets which imply contradictions, you’re making the same mistake. You’re being emotionally involved with your naive/primitive concept of a ‘set’, whereas the theory in question (ZFC) doesn’t even allow you to construct such sets.
The above arguments are less common, but I have heard them. A more common argument concerns the Axiom of Choice, and goes a little something like this:
I pulled that from the math subreddit where it was posted a few days ago, and it’s a fairly common argument. But the commenter has become emotionally involved with day-to-day sets and Cartesian products. What would the product of an uncountable collection of uncountable sets even look like? Once one refers to the formal, very abstract definition, it should be clear that we have absolutely no right to expect anything about it’s emptiness or nonemptiness, because the intuition and emotional involvement are replaced by formal abstraction. The things which one assumes exist aren’t actually there in the original theory (ZF).
The paradoxes falling out of the geocentric model, maybe?
Not necessarily. At least, not necessarily more so than anyone becomes “emotionally involved” when deciding on axioms to use in a mathematical theory.
AC is after all independent of ZF. So of course no argument in favor of it can be constructed literally on the basis of the “original theory” (ZF). Saying that the Cartesian product of nonempty sets ought to be nonempty is an aesthetic statement about what the rules of the game should be, not a mistaken inference from the axioms of ZF.
Because of the independence, we logically have a “right to expect” anything we want (AC or not-AC). The choice is a matter of taste, and the tastes of the mathematical community evidently incline toward AC.
Note how different this is from your two previous examples regarding derivatives and Russell’s paradox. Those involved outright logical errors; whereas in the case of AC, the commenter is making a legitimate aesthetic argument.
By “paradox” I mean something stronger than “the model is inaccurate”. I mean a hypothetical where all possible answers seem to be wrong.