Bad Math in Economics
I sometimes feel that economists go wrong when they try to over-apply mathematical theory to their problems. At times, they seem to force the issue.
Don't get me wrong. Math is a language beautifully suited to model many things. We all know that math is useful in physics - to the point that people seem to mistake mathematical models of physics as necessary truths as opposed to viewing them as models. It is useful in computer science. It is useful in chemistry - well sort of. At least it is useful in quantum chemistry where representation theory is used. And math is useful in bio as well. Well, ok, maybe not in bio. But you get the point.
And I certainly don't feel that mathematics is limited as a language to model the "hard" sciences. I think math has utility in describing flows of discourse in Foucault's Archaeology of Knowledge. And the logical positivists and metamathematicians were no slouches either.
The problem I have with the way math is employed in economics is that, sometimes there is a tendency to feel that simply because things are stated in mathematical terms, they are more rigorous - hence more right. And in that impulse, researchers (at times) seem to forget that - well - what matters is not the theorem itself but how well the real world issue maps onto the conditions for that theorem to hold. For example, it is not uncommon to read horrible economics papers, exploiting some local phenomenon and accidentally applying the result as some global pheonomenon to comment on larger-scale results.
Making matters worse, it is not uncommon to read papers and analyses that seem to force issues and positions out of topological convenience. The problem is that results are contingent on fairly arbitrary and convenient assumptions at times. Lots of natural settings aren't Hausdorff spaces (ignore this if you have no idea what that means). The common response is, "well, look we aren't trying to say exactly what goes on. we are trying to get a partial analysis - get closer to the truth or get a better idea of how this one factor might contribute, ceteris paribus." Look, that is cute and all - but here is where that fails. Some of these assumptions that are made literally make the conclusion entirely different. It isn't necessarily a continous function - a slight change in the assumptions of your model does not mean that your conclusion is perturbed only by a bit. In fact, there is no reason a priori for us to think that conclusions are so perturbation-robust in the first place in many of these situations. These are assumptions that should be rigorously justified - but usually they are not because ... well ... to put it mildly, the assumptions are made because it is too hard to deal with the detailed problem in the first place.
At the end of the day - it is important to take a lot of the theory with a grain of salt. And it is important to see if they are robust to assumption-perturbation. Only if this is the case, can we utilize the models to conclude stuff about these "partial analyses" in the first place.
Don't get me wrong. Math is a language beautifully suited to model many things. We all know that math is useful in physics - to the point that people seem to mistake mathematical models of physics as necessary truths as opposed to viewing them as models. It is useful in computer science. It is useful in chemistry - well sort of. At least it is useful in quantum chemistry where representation theory is used. And math is useful in bio as well. Well, ok, maybe not in bio. But you get the point.
And I certainly don't feel that mathematics is limited as a language to model the "hard" sciences. I think math has utility in describing flows of discourse in Foucault's Archaeology of Knowledge. And the logical positivists and metamathematicians were no slouches either.
The problem I have with the way math is employed in economics is that, sometimes there is a tendency to feel that simply because things are stated in mathematical terms, they are more rigorous - hence more right. And in that impulse, researchers (at times) seem to forget that - well - what matters is not the theorem itself but how well the real world issue maps onto the conditions for that theorem to hold. For example, it is not uncommon to read horrible economics papers, exploiting some local phenomenon and accidentally applying the result as some global pheonomenon to comment on larger-scale results.
Making matters worse, it is not uncommon to read papers and analyses that seem to force issues and positions out of topological convenience. The problem is that results are contingent on fairly arbitrary and convenient assumptions at times. Lots of natural settings aren't Hausdorff spaces (ignore this if you have no idea what that means). The common response is, "well, look we aren't trying to say exactly what goes on. we are trying to get a partial analysis - get closer to the truth or get a better idea of how this one factor might contribute, ceteris paribus." Look, that is cute and all - but here is where that fails. Some of these assumptions that are made literally make the conclusion entirely different. It isn't necessarily a continous function - a slight change in the assumptions of your model does not mean that your conclusion is perturbed only by a bit. In fact, there is no reason a priori for us to think that conclusions are so perturbation-robust in the first place in many of these situations. These are assumptions that should be rigorously justified - but usually they are not because ... well ... to put it mildly, the assumptions are made because it is too hard to deal with the detailed problem in the first place.
At the end of the day - it is important to take a lot of the theory with a grain of salt. And it is important to see if they are robust to assumption-perturbation. Only if this is the case, can we utilize the models to conclude stuff about these "partial analyses" in the first place.
posted by arun at 5:49 PM
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