I agree about the awful state of fields that don’t have any formal rules at all. However, I’m not concerned about these so much because, to put it bluntly, nobody important takes them seriously. What is in my opinion a much greater problem are fields that appear to have all the trappings of valid science and scholarship, but it’s in fact hard for an outsider to evaluate whether and to what extent they’re actually cargo-cult science. This especially because the results of some such fields (most notably economics) are used as basis for real-world decision-making with far-reaching consequences.
Regarding the role of formalism, mathematics is unique in that the internal correctness of the formalism is enough to establish the validity of the results. Sure, they may be more or less interesting, but if the formalism is valid, then it’s valid math, period.
In contrast, in areas that make claims about the real world, the important thing is not just the validity of the formalism, but also how well it corresponds to reality. Work based on a logically impeccable formalism can still be misleading garbage if the formalism is distant enough from reality. This is where the really hard problem is. The requirements about the validity of the formalism are easily enforced, since we know how to reduce those to a basically algorithmic procedure. What is really hard is ensuring that the formalism provides an accurate enough description of reality—and given an incentive to do so, smart people will inevitably figure out ways to stretch and evade this requirement, unless there is a sound common-sense judgment standing in the way.
Further, more rigorous formalism isn’t always better. It’s a trade-off. More effort put into greater formal rigor—including both the author’s effort to formulate it, and the reader’s effort to understand it—means less resources for other ways of improving the work. Physicists, for example, normally just assume that the functions are well-behaved enough in a way that would be unacceptable in mathematics, and they’re justified in doing so. In more practical technical fields like computer science, what matters is whether the results are useful in practice, and formal rigor is useful if it helps avoid confusion about complicated things, but worse than useless if applied to things where intuitive understanding is good enough to get the job done.
The crucial lesson, like in so many other things, is that whenever one deals with the real world, formalism cannot substitute for common sense. It may be tremendously helpful and enable otherwise impossible breakthroughs, but without an ultimate sanity check based on sheer common sense, any attempt at science is a house built on sand.
I don’t think we have a real disagreement. I haven’t said that more rigorous formalism is always better, quite the contrary. I was writing about objective methods of looking at the results. Physicists can ignore mathematical rigor because they have experimental tests which finally decide whether their theory is worth attention. Computer scientists can finally write down their algorithm and look whether it works. These are objective rules which validate the results.
Whether the rules are sensible or not is decided by common sense. My point is that it is easier to decide that about the rules of the whole field than about individual theories, and that’s why objective rules are useful.
Of course, saying “common sense” does in fact mean that we don’t know how did we decide, and doesn’t specify the judgement too much. One man’s common sense may be other man’s insanity.
Oh yes, I didn’t mean to imply that you disagreed with everything I wrote in the above comment. My intent was to give a self-contained summary of my position on the issue, and the specific points I raised were not necessarily in response to your claims.
I agree about the awful state of fields that don’t have any formal rules at all. However, I’m not concerned about these so much because, to put it bluntly, nobody important takes them seriously. What is in my opinion a much greater problem are fields that appear to have all the trappings of valid science and scholarship, but it’s in fact hard for an outsider to evaluate whether and to what extent they’re actually cargo-cult science. This especially because the results of some such fields (most notably economics) are used as basis for real-world decision-making with far-reaching consequences.
Regarding the role of formalism, mathematics is unique in that the internal correctness of the formalism is enough to establish the validity of the results. Sure, they may be more or less interesting, but if the formalism is valid, then it’s valid math, period.
In contrast, in areas that make claims about the real world, the important thing is not just the validity of the formalism, but also how well it corresponds to reality. Work based on a logically impeccable formalism can still be misleading garbage if the formalism is distant enough from reality. This is where the really hard problem is. The requirements about the validity of the formalism are easily enforced, since we know how to reduce those to a basically algorithmic procedure. What is really hard is ensuring that the formalism provides an accurate enough description of reality—and given an incentive to do so, smart people will inevitably figure out ways to stretch and evade this requirement, unless there is a sound common-sense judgment standing in the way.
Further, more rigorous formalism isn’t always better. It’s a trade-off. More effort put into greater formal rigor—including both the author’s effort to formulate it, and the reader’s effort to understand it—means less resources for other ways of improving the work. Physicists, for example, normally just assume that the functions are well-behaved enough in a way that would be unacceptable in mathematics, and they’re justified in doing so. In more practical technical fields like computer science, what matters is whether the results are useful in practice, and formal rigor is useful if it helps avoid confusion about complicated things, but worse than useless if applied to things where intuitive understanding is good enough to get the job done.
The crucial lesson, like in so many other things, is that whenever one deals with the real world, formalism cannot substitute for common sense. It may be tremendously helpful and enable otherwise impossible breakthroughs, but without an ultimate sanity check based on sheer common sense, any attempt at science is a house built on sand.
I don’t think we have a real disagreement. I haven’t said that more rigorous formalism is always better, quite the contrary. I was writing about objective methods of looking at the results. Physicists can ignore mathematical rigor because they have experimental tests which finally decide whether their theory is worth attention. Computer scientists can finally write down their algorithm and look whether it works. These are objective rules which validate the results.
Whether the rules are sensible or not is decided by common sense. My point is that it is easier to decide that about the rules of the whole field than about individual theories, and that’s why objective rules are useful.
Of course, saying “common sense” does in fact mean that we don’t know how did we decide, and doesn’t specify the judgement too much. One man’s common sense may be other man’s insanity.
Oh yes, I didn’t mean to imply that you disagreed with everything I wrote in the above comment. My intent was to give a self-contained summary of my position on the issue, and the specific points I raised were not necessarily in response to your claims.