OK, maybe if we look at some other definitions of equality we can get a grip on it? In set theory, you say that two sets are equal if they’ve got the same elements. How do you know the elements are the same i.e. equal? You just know.
You are misunderstanding the axiom of extensionality, which states that two sets A and B are equal if both (1) every element of A is an element of B and (2) every element of B is an element of A. This does not require any nebulous notion of "they've got the same elements", and is completely unrelated to the concept of equality at the level of elements of A and B.
By the way, the axiom of extensionality is an axiom rather than a definition; in set theory equality is treated as an undefined primitive, axiomatized as a notion of equality as in first order logic. This is important because if A and B are equal according to the axiom of extensionality, then that axiom implies that A is in some collection of sets C if and only if B is in C.
But if you enrich the category with some more discriminating maps, say distance preserving ones, then the sphere and cube are no longer equal. Conversely, if you reduce the category by removing all the isomorphisms between the sphere and the cube, then they are no longer equal.
Actually you have just described the same thing twice. There are actually fewer distance-preserving maps than there are continuous ones, and restricting to distance-preserving maps removes all the isomorphisms between the sphere and the cube.
So if the climate is moving out of the optimal temperature for the species, it might make sense for you to produce more females, because they are a lower risk strategy?
This seems confused to me. In general, males are more risk-seeking than females because (inclusive) fitness is not a linear function of successfulness at endeavors, with the function being closer to linear for males and more like linear-with-a-cutoff for females. But males and females are still both perfectly risk-neutral when measured in the unit of fitness, since that follows from the definition of expected fitness which is what needs to be greater than average in order for a mutation to propagate throughout a population.
I would expect that if a species has more females than males in some circumstances, then either it is because females are cheaper to raise for some reason, or else that it is due to a fact of biology that the DNA can't really control directly.
There are some repeated paragraphs:
Elaine nodded. “Tell me, suppose that instead you had a hundred times as many wolves captured, and brought to those forests for release—what would happen then?”
Elaine looked a little surprised, before her face went expressionless again. “Yes, that’s so. Like you said, there’s no Magic powerful enough to directly oppress the farmers and shopkeepers of a whole country. So we’re not looking for a straightforward curse, but some new factor that has changed Santal’s balancing point.”
Let's talk about a specific example: the Ultimatum Game. According to EY the rational strategy for the responder in the Ultimatum Game is to accept if the split is "fair" and otherwise reject in proportion to how unfair he thinks the split is. But the only reason to reject is to penalize the proposer for proposing an unfair split -- which certainly seems to be "doing something conditional on the other actor’s utility function disvaluing it". So why is the Ultimatum Game considered an "offer" and not a "threat"?
Yeah, but what does "purposefully minimize someone else’s utility function" mean? The source code just does stuff. What does it mean for it to be "on purpose"?
It all depends on what you mean by "sufficiently intelligent / coherent actors". For example, in this comment Eliezer says that it should mean actors that “respond to offers, not to threats”, but in 15 years no one has been able to cash out what this actually means, AFAIK.
Here's Joe Carlsmith making the second argument: https://joecarlsmith.com/2022/01/17/the-ignorance-of-normative-realism-bot
Depends on how dysfunctional the society is.
Yes. If f and g are in the original category and are inverses of each other, the same will be true of any larger category (technically: any category which is the codomain of a functor whose domain is the original category).