Its sort of true that the correct distance function depends on your values. A better way to say it is that different distance functions are appropriate for different tasks, and they will be “better” or “worse” depending on how much you care about those tasks. But I dont think asking for the “best” metric in this sense is helpful, because you dont have to use the same metric for all tasks involving a certain space. Sometimes you want air distance, sometimes travel times. Maybe you have to decide because youre computationally limited, but its not philosophically relevant.
With that in mind, my attempts at two of your examples. The adversarial examples first, because its the clearest question: I think the problem is that you are thinking too abstractly. I dont think there is a meaningful sense of “concept similarity” thats purely logical, i.e. independent of the actual world. The intuitive sense of similarity youre trying to use here is propably something like this: Over the space of images, you want the propability measure of encountering them. Then you get a metric where two subsets of imagespace which are isomorphic under the metric always have the same measure. That is your similarity measure.
Counterfactuals usually involve some sort of propability distribution, which is then “updated” on the condition of the counterfactual being true, and then the consequent is judged under that distribution. What the initial distribution is depends on what youre doing. In the case of Lincoln, its propably reasonable expectations of the future from before the assasination. But for something like “What if conservation of energy wasnt true”, its propably our current distribution over physics theories. Basically, whats the most likely alternative. The mathematical example is a bit different. There lot of ways to conclude a contradiction from 0=1, but its very hard to deduce a contradiction from denying the modularity theorem. If you were to just randomly perform logical inferences from “the modularity theorem is wrong”, then there is a subset of propositions which doesnt include any claim that is a dircet negation of another in it, that your deductions are unlikely to lead you out of (it matters of course, in what way it is random, but it evidently works for “human mathmatician who hasnt seen the proof yet”).
Its sort of true that the correct distance function depends on your values. A better way to say it is that different distance functions are appropriate for different tasks, and they will be “better” or “worse” depending on how much you care about those tasks. But I dont think asking for the “best” metric in this sense is helpful, because you dont have to use the same metric for all tasks involving a certain space. Sometimes you want air distance, sometimes travel times. Maybe you have to decide because youre computationally limited, but its not philosophically relevant.
With that in mind, my attempts at two of your examples. The adversarial examples first, because its the clearest question: I think the problem is that you are thinking too abstractly. I dont think there is a meaningful sense of “concept similarity” thats purely logical, i.e. independent of the actual world. The intuitive sense of similarity youre trying to use here is propably something like this: Over the space of images, you want the propability measure of encountering them. Then you get a metric where two subsets of imagespace which are isomorphic under the metric always have the same measure. That is your similarity measure.
Counterfactuals usually involve some sort of propability distribution, which is then “updated” on the condition of the counterfactual being true, and then the consequent is judged under that distribution. What the initial distribution is depends on what youre doing. In the case of Lincoln, its propably reasonable expectations of the future from before the assasination. But for something like “What if conservation of energy wasnt true”, its propably our current distribution over physics theories. Basically, whats the most likely alternative. The mathematical example is a bit different. There lot of ways to conclude a contradiction from 0=1, but its very hard to deduce a contradiction from denying the modularity theorem. If you were to just randomly perform logical inferences from “the modularity theorem is wrong”, then there is a subset of propositions which doesnt include any claim that is a dircet negation of another in it, that your deductions are unlikely to lead you out of (it matters of course, in what way it is random, but it evidently works for “human mathmatician who hasnt seen the proof yet”).