A note on hypotheticals

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?

Another category of tricky cases is when the hypothetical involves impossibilities. 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.) These crop up frequently in modern physics.

Would Olbers' paradox qualify?

Olbers' paradox Basically, if the universe is of infinite size, then because an infinite number of concentric shells can be constructed centred on the earth. More distant shells have more stars, but the intensity of light we experience is reduced in accordance with the inverse square law. As these two effects cancel out, each shell should be of equal brightness, and so as there are an infinite number shells, the sky should by full of light. If the universe is not infinite, it would collapse due to gravity.

The wrong assumption is that universe is static (and an ignorance of relativity)

Another vague, nameless example:

If someone says A and (A => B => C), and someone else disputes A, it does not mean this second person wishes to reenact glorious battles over C.

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.

"The model used in this hypothetical does not meaningfully correspond to reality" seems relevant and not to fall under those categories, though it may count as impossibility. A lot of objections to hypotheticals, from what I've seen, stem from this conceptual problem but people rarely come out and say this bluntly.

On some level, ALL hypothetical situations are impossible. The universe does not contain that configuration. Some are conceptually further from reality than others, but every single one of them is about a model. Something that's too far removed from a real model of the universe is probably not interesting to discuss (cf. pinhead angels, unstoppable force vs immovable object, etc).

I've gotten into the habit of using the word "counterfactual" rather than "hypothetical". It makes it slightly more clear that the discussion is not about the universe, but about a model of an alternate universe that may share some (conceptual) features with this one.

It has proven useful in short-cutting unproductive and frustrating discussions to ask the question "what's the minimal difference between this universe and that one"? In many cases, this question is isomorphic to the question that the thought experiment is intended to illuminate. Good, it's nice to have multiple frames from which to approach it. In other cases, this question exposes massive flaws in the hypothetical, and saves everybody time.

I try to respond to fantasy scenarios in the spirit of the least convenient possible world, but I appreciate it when the author of a scenario spells out the intended dilemma explicitly, or else proofs it against tempting escapes.

An analogy or hypothetical will subtract value If it causes enough distraction. They are dangerously powerful communication, and practically necessary, even though ideally we can just create notation for the actual problem, and reason using that.