When I wrote “What is Bunk?” I thought I had a pretty good idea of the distinction between science and pseudoscience, except for some edge cases. Astrology is pseudoscience, astronomy is science. At the time, I was trying to work out a rubric for the edge cases (things like macroeconomics.)
Now, though, knowing a bit more about the natural sciences, it seems that even perfectly honest “science” is much shakier and likelier to be false than I supposed. There’s apparently a high probability that the conclusions of a molecular biology paper will be false—even if the journal is prestigious and the researchers are all at a world-class university. There’s simply a lot of pressure to make results look more conclusive than they are.
In the field of machine learning, which I sometimes read the literature in, there are foundational debates about the best methods. Ideas which very smart and highly credentialed people tout often turn out to be ineffective, years down the road. Apparently smart and accomplished researchers will often claim that some other apparently smart and accomplished researcher is doing it all wrong.
If you don’t actually know a field, you might think, “Oh. Tenured professor. Elite school. Dozens of publications and conferences. Huge erudition. That means I can probably believe his claims.” Whereas actually, he’s extremely fallible. Not just theoretically fallible, but actually has a serious probability of being dead wrong.
I guess the moral is “Don’t trust anyone but a mathematician”?
I guess the moral is “Don’t trust anyone but a mathematician”?
Safety in numbers? ;)
Perhaps it’s useful to distinguish between the frontier of science vs. established science. One should expect the frontier to be rather shaky and full of disagreements, before the winning theories have had time to be thoroughly tested and become part of our scientific bedrock. There was a time after all when it was rational for a layperson to remain rather neutral with respect to Einstein’s views on space and time. The heuristic of “is this science established / uncontroversial amongst experts?” is perhaps so boring we forget it, but it’s one of the most useful ones we have.
I guess the moral is “Don’t trust anyone but a mathematician”?
Theorems get published all the time that turn out to have incorrect proofs or to be not even theorems. There was about a decade long period in the late 19th century where there was a proof of the four color theorem that everyone thought was valid. And the middle of the 20th century there were serious issues with calculating homology groups and cohomology groups of spaces where people kept getting different answers. And then there are a handful of examples where theorems simply got more and more conditions tacked on to them as more counterexamples to the theorems became apparent. The Euler formula for polyhedra is possibly the most blatant such example.
So even the mathematicians aren’t always trustworthy.
The Euler formula for polyhedra is possibly the most blatant such example.
Huh? There are no counterexamples to the Euler characteristic of a polyhedra being 2, and the theorem has generalized beautifully. If anything conditions have been loosened as new versions of the theorem have been used in more places.
Huh? There are no counterexamples to the Euler characteristic of a polyhedra being 2, and the theorem has generalized beautifully. If anything conditions have been loosened as new versions of the theorem have been used in more places.
Well, what do you mean by polyhedron? Consider for example a cubic nut. Does this fit your intuition of a polyhedron? Well, since it has genus that is not equal to 1, it doesn’t have Euler characteristic 2. And the original proof that V+F-E=2 didn’t handle this sort of case. (That’s one reason why people often add convex as a condition, to deal with just this situation even though convex is in many respects stronger than what one needs.) Cauchy’s 1811 proof suffers from this problem as do some of the other early proofs (although his is repairable if one is careful). There are also other subtle issues that can go wrong and in fact do go wrong in a lot of the historical versions. Lakatos’s book “Proofs and Refutations” discusses this albeit in an essentially ahistorical fashion.
There are many things called “the Euler formula for polyhedron”, and you’re conflating all of them. Sounds like you missed the point of Proofs and Refutations.
When I wrote “What is Bunk?” I thought I had a pretty good idea of the distinction between science and pseudoscience, except for some edge cases. Astrology is pseudoscience, astronomy is science. At the time, I was trying to work out a rubric for the edge cases (things like macroeconomics.)
Now, though, knowing a bit more about the natural sciences, it seems that even perfectly honest “science” is much shakier and likelier to be false than I supposed. There’s apparently a high probability that the conclusions of a molecular biology paper will be false—even if the journal is prestigious and the researchers are all at a world-class university. There’s simply a lot of pressure to make results look more conclusive than they are.
In the field of machine learning, which I sometimes read the literature in, there are foundational debates about the best methods. Ideas which very smart and highly credentialed people tout often turn out to be ineffective, years down the road. Apparently smart and accomplished researchers will often claim that some other apparently smart and accomplished researcher is doing it all wrong.
If you don’t actually know a field, you might think, “Oh. Tenured professor. Elite school. Dozens of publications and conferences. Huge erudition. That means I can probably believe his claims.” Whereas actually, he’s extremely fallible. Not just theoretically fallible, but actually has a serious probability of being dead wrong.
I guess the moral is “Don’t trust anyone but a mathematician”?
Safety in numbers? ;)
Perhaps it’s useful to distinguish between the frontier of science vs. established science. One should expect the frontier to be rather shaky and full of disagreements, before the winning theories have had time to be thoroughly tested and become part of our scientific bedrock. There was a time after all when it was rational for a layperson to remain rather neutral with respect to Einstein’s views on space and time. The heuristic of “is this science established / uncontroversial amongst experts?” is perhaps so boring we forget it, but it’s one of the most useful ones we have.
Theorems get published all the time that turn out to have incorrect proofs or to be not even theorems. There was about a decade long period in the late 19th century where there was a proof of the four color theorem that everyone thought was valid. And the middle of the 20th century there were serious issues with calculating homology groups and cohomology groups of spaces where people kept getting different answers. And then there are a handful of examples where theorems simply got more and more conditions tacked on to them as more counterexamples to the theorems became apparent. The Euler formula for polyhedra is possibly the most blatant such example.
So even the mathematicians aren’t always trustworthy.
Huh? There are no counterexamples to the Euler characteristic of a polyhedra being 2, and the theorem has generalized beautifully. If anything conditions have been loosened as new versions of the theorem have been used in more places.
Well, what do you mean by polyhedron? Consider for example a cubic nut. Does this fit your intuition of a polyhedron? Well, since it has genus that is not equal to 1, it doesn’t have Euler characteristic 2. And the original proof that V+F-E=2 didn’t handle this sort of case. (That’s one reason why people often add convex as a condition, to deal with just this situation even though convex is in many respects stronger than what one needs.) Cauchy’s 1811 proof suffers from this problem as do some of the other early proofs (although his is repairable if one is careful). There are also other subtle issues that can go wrong and in fact do go wrong in a lot of the historical versions. Lakatos’s book “Proofs and Refutations” discusses this albeit in an essentially ahistorical fashion.
There are many things called “the Euler formula for polyhedron”, and you’re conflating all of them. Sounds like you missed the point of Proofs and Refutations.
Multiple actual historical versions (such as Cauchy’s proof) are wrong.
I’m pretty sure you’re at least half-joking. But just in case, I need to point out that mathematicians are not immune to this kind of thing.
yep, joke.