Find some development which has been important in human history, and then working backwards argue that broader and broader theoretical contributions were valuable insofar as they facilitated this development.
Your previous comments indicate that you have a ready-made formula for rejecting any such example, which is to deny that the theoretical contributions were “necessary”. That is, you will argue that the important development “could have” happened without them. For instance, you addressed the classic example of non-Euclidean geometry paving the way for general relativity by saying that Einstein could have just invented non-Euclidean geometry on his own (!) when he needed it.
Such an argument is difficult to take seriously and seems to me only to indicate that, whatever your past history, you simply have, at the moment, a strong distaste for abstract mathematics. (Or, alternatively, perhaps you’re playing an intense game of Devil’s Advocate with yourself.) That’s fine, tastes differ. But again, that’s hardly enough reason to jump to the conclusion that nobody should be doing things like inventing non-Euclidean geometry.
You’ve characterized my position as motivated cognition in defense of the status quo, but in my view that is both wrong and unfair. It’s wrong, first of all, because I don’t necessarily think the status quo is optimal—in fact I think mathematical research could probably be done a lot more efficiently. I just don’t happen to favor changes in the particular direction that you advocate (basically the adoption of a concreteness heuristic for deciding what’s “useful”). But even more importantly perhaps, it’s unfair: even if my ideal mathematical world looks more like the current one than yours does, the fact remains that we are living in an exceptional period in human history. Most societies have not conducted extensive amounts of “theoretical research”, but instead have occupied themselves almost exclusively with what they found to be “practical” and “relevant” within their local world; your position, rather than mine, is closer to the human default. In my view, the last few centuries of Western civilization are an exceptional instance of people finally beginning to almost start getting things right with respect to this question.
I don’t actually think the issue here is exclusively empirical; I can sense important disagreements that may be better characterized as value differences (conceivably, these may ultimately also reduce to empirical disagreements, but if so it would be at several more inferential steps’ remove). But I will concede that there is a substantial empirical component. On my analysis, what it boils down to is that you think you have a good way of predicting the long-term impact of mathematical work, whereas I don’t think you do, because I don’t think the heuristics you’re using are any more powerful than (or even particularly different from) the most common human default.
you addressed the classic example of non-Euclidean geometry paving the way for general relativity by saying that Einstein could have just invented non-Euclidean geometry on his own (!) when he needed it.
My claim was that Einstein explicitly considered gravity as curvature of space before being aware of any formal developments in non-Euclidean geometry. I inferred that, if Einstein had lived before any formal developments in non-Euclidean geometry occurred, then non-Euclidean geometry would have become an object of study not because it was interesting but because it was important. If you really want to have this argument, then you should describe where you think things would have broken down if non-Euclidean geometry had not already been developed. Would general relativity never have occurred to Einstein? Would the community have scoffed at the idea more than they did? Would the theory have stagnated before the math could catch up?
But lets count this as a point for the home team. Lets suppose that general relativity had been set back a hundred years, and that we had never considered the possibility of gravitational time dilation until we discovered the empirical formula for the time dilation experienced by satellites. I agree that this is worse than what really happened, but its not bad enough to convince me that I should support work on interesting problems. I would be interested in additional examples, even if they were much less compelling.
You’ve characterized my position as motivated cognition in defense of the status quo, but in my view that is both wrong and unfair
I don’t really care whether your position is motivated cognition. I realized my own beliefs were motivated cognition, and now I would like to replace them with beliefs that better reflect reality.
you think you have a good way of predicting the long-term impact of mathematical work,
We are both trying to predict the long-term impact of human activities. You don’t get to abstain from having beliefs. My claim is that concretely motivated problems are a significantly better use of the smartest researchers’ time than interesting abstract problems; your claim is that they aren’t. I came to my belief by observing the historical record, noting that important advances have at least been proximately due to smart people working on concrete problems, and suspecting that the causal connections drawn to more abstract problems are highly tenuous. If you think this reasoning is the explanation for the rate of human progress prior to the 17th century, then you should say so and I can describe at length why I believe you are almost certainly wrong.
Your previous comments indicate that you have a ready-made formula for rejecting any such example, which is to deny that the theoretical contributions were “necessary”. That is, you will argue that the important development “could have” happened without them. For instance, you addressed the classic example of non-Euclidean geometry paving the way for general relativity by saying that Einstein could have just invented non-Euclidean geometry on his own (!) when he needed it.
Such an argument is difficult to take seriously and seems to me only to indicate that, whatever your past history, you simply have, at the moment, a strong distaste for abstract mathematics. (Or, alternatively, perhaps you’re playing an intense game of Devil’s Advocate with yourself.) That’s fine, tastes differ. But again, that’s hardly enough reason to jump to the conclusion that nobody should be doing things like inventing non-Euclidean geometry.
You’ve characterized my position as motivated cognition in defense of the status quo, but in my view that is both wrong and unfair. It’s wrong, first of all, because I don’t necessarily think the status quo is optimal—in fact I think mathematical research could probably be done a lot more efficiently. I just don’t happen to favor changes in the particular direction that you advocate (basically the adoption of a concreteness heuristic for deciding what’s “useful”). But even more importantly perhaps, it’s unfair: even if my ideal mathematical world looks more like the current one than yours does, the fact remains that we are living in an exceptional period in human history. Most societies have not conducted extensive amounts of “theoretical research”, but instead have occupied themselves almost exclusively with what they found to be “practical” and “relevant” within their local world; your position, rather than mine, is closer to the human default. In my view, the last few centuries of Western civilization are an exceptional instance of people finally beginning to almost start getting things right with respect to this question.
I don’t actually think the issue here is exclusively empirical; I can sense important disagreements that may be better characterized as value differences (conceivably, these may ultimately also reduce to empirical disagreements, but if so it would be at several more inferential steps’ remove). But I will concede that there is a substantial empirical component. On my analysis, what it boils down to is that you think you have a good way of predicting the long-term impact of mathematical work, whereas I don’t think you do, because I don’t think the heuristics you’re using are any more powerful than (or even particularly different from) the most common human default.
My claim was that Einstein explicitly considered gravity as curvature of space before being aware of any formal developments in non-Euclidean geometry. I inferred that, if Einstein had lived before any formal developments in non-Euclidean geometry occurred, then non-Euclidean geometry would have become an object of study not because it was interesting but because it was important. If you really want to have this argument, then you should describe where you think things would have broken down if non-Euclidean geometry had not already been developed. Would general relativity never have occurred to Einstein? Would the community have scoffed at the idea more than they did? Would the theory have stagnated before the math could catch up?
But lets count this as a point for the home team. Lets suppose that general relativity had been set back a hundred years, and that we had never considered the possibility of gravitational time dilation until we discovered the empirical formula for the time dilation experienced by satellites. I agree that this is worse than what really happened, but its not bad enough to convince me that I should support work on interesting problems. I would be interested in additional examples, even if they were much less compelling.
I don’t really care whether your position is motivated cognition. I realized my own beliefs were motivated cognition, and now I would like to replace them with beliefs that better reflect reality.
We are both trying to predict the long-term impact of human activities. You don’t get to abstain from having beliefs. My claim is that concretely motivated problems are a significantly better use of the smartest researchers’ time than interesting abstract problems; your claim is that they aren’t. I came to my belief by observing the historical record, noting that important advances have at least been proximately due to smart people working on concrete problems, and suspecting that the causal connections drawn to more abstract problems are highly tenuous. If you think this reasoning is the explanation for the rate of human progress prior to the 17th century, then you should say so and I can describe at length why I believe you are almost certainly wrong.