The key challenge here is to come up with a set of intuitive arguments which uniquely specify a particular definition/metric, exactly like a set of equations can uniquely specify a solution. If our arguments have “many solutions”, then there’s little reason to expect that the ad-hoc “solution” we chose actually corresponds to our intuitive concept.
Maybe I’m missing something in the post, but why is this the case? Isn’t it arbitrary to suppose that only one possible metric exists that fully ‘solves’ the problem?
Good question, I was hoping someone would ask this. There’s some subtleties here that I didn’t want to unpack in the post.
Sometimes, I use a formula to specify things other than points. Like, I could use a formula to specify a line (e.g. y = 3x+2) or a sphere (x^2 + y^2 + z^2 = 1). These equations have “more than one solution” in the sense that there are many points which satisfy them. However, I’m not actually trying to specify one particular point; I’m trying to specify the whole set of points which satisfies the equation (i.e. the line or the sphere). And the equations do fully specify those sets of points.
In general, any set of equations fully specifies some set of solutions (possibly the empty set).
The interesting question is whether the set-of-solutions-specified actually matches our intuitive concept. If not, then we have no reason to expect that the set-of-solutions will generalize in the ways we expect our intuitive concept to generalize.
Now let’s go back to the idea of ad-hoc-ness. Suppose I give some intuitive argument that my concept should satisfy the formula x^2 + y^2 + z^2 = 1. But I also think that the concept-I-want-to-specify is a circle, not a sphere; so this formula alone is not sufficient to nail it down. If I were to arbitrarily choose the circle given by the equations (x^2 + y^2 + z^2 = 1, z = 4x—y), then that would be an ad-hoc specification; I have no reason to expect that particular circle to match my intuitive concept.
Then there’s the question of why I should expect my intuitions to nail down one particular circle. That’s something which would have to have an intuitive argument in its own right. But even if it’s not picking one particular circle, there is still some set of answers which match my intuition (e.g. a set of circles). If we want our formula to generalize in the cases where we intuitively expect generalization (and fail to generalize in the cases where we intuitively expect failure of generalization), then we do need to match that set.
This argument generalizes, too. Maybe someone says “well, my intuitions are fuzzy, I don’t expect a sharp boundary between things-which-satisfy-them and things-which-don’t”. And then we say ok, we have mathematical ways of handling fuzziness (like probabilities, for instance), we should find a formulation for which the mathematical fuzz matches the intuitive fuzz, so that it will fuzzily-generalize when we expect it to do so. Etc.
Maybe I’m missing something in the post, but why is this the case? Isn’t it arbitrary to suppose that only one possible metric exists that fully ‘solves’ the problem?
Good question, I was hoping someone would ask this. There’s some subtleties here that I didn’t want to unpack in the post.
Sometimes, I use a formula to specify things other than points. Like, I could use a formula to specify a line (e.g. y = 3x+2) or a sphere (x^2 + y^2 + z^2 = 1). These equations have “more than one solution” in the sense that there are many points which satisfy them. However, I’m not actually trying to specify one particular point; I’m trying to specify the whole set of points which satisfies the equation (i.e. the line or the sphere). And the equations do fully specify those sets of points.
In general, any set of equations fully specifies some set of solutions (possibly the empty set).
The interesting question is whether the set-of-solutions-specified actually matches our intuitive concept. If not, then we have no reason to expect that the set-of-solutions will generalize in the ways we expect our intuitive concept to generalize.
Now let’s go back to the idea of ad-hoc-ness. Suppose I give some intuitive argument that my concept should satisfy the formula x^2 + y^2 + z^2 = 1. But I also think that the concept-I-want-to-specify is a circle, not a sphere; so this formula alone is not sufficient to nail it down. If I were to arbitrarily choose the circle given by the equations (x^2 + y^2 + z^2 = 1, z = 4x—y), then that would be an ad-hoc specification; I have no reason to expect that particular circle to match my intuitive concept.
Then there’s the question of why I should expect my intuitions to nail down one particular circle. That’s something which would have to have an intuitive argument in its own right. But even if it’s not picking one particular circle, there is still some set of answers which match my intuition (e.g. a set of circles). If we want our formula to generalize in the cases where we intuitively expect generalization (and fail to generalize in the cases where we intuitively expect failure of generalization), then we do need to match that set.
This argument generalizes, too. Maybe someone says “well, my intuitions are fuzzy, I don’t expect a sharp boundary between things-which-satisfy-them and things-which-don’t”. And then we say ok, we have mathematical ways of handling fuzziness (like probabilities, for instance), we should find a formulation for which the mathematical fuzz matches the intuitive fuzz, so that it will fuzzily-generalize when we expect it to do so. Etc.