Monday, October 24, 2016

Exaggeration is one thing but this is....

This website is about the history of math and lists famous mathematicians. The ones from the 20th century are biased towards logic, but you should go there yourself and see who you think they left out.

There entry on Paul Cohen is... odd. Its here. I quote from it:

His findings were as revolutionary as Gödel’s own. Since that time, mathematicians have built up two different mathematical worlds, one in which the continuum hypothesis applies and one in which it does not, and modern mathematical proofs must insert a statement declaring whether or not the result depends on the continuum hypothesis.

When was the last time you had to put into the premise of a theorem CH or NOT-CH?
I did once here in an expository work about a problem in combinatorics that was ind of set theory since it was equivalent to CH. Before you get too excited it was a problem in infinite combinatorics having to do with coloring the reals.

I suspect that at least 99% of my readers have never had to insert a  note in a paper about if they were assuming CH or NOT-CH. If you have I'd love to hear about it in the comments. And I suspect
you are a set theorist.

Paul Cohen's work was very important--- here we have an open problem in math that will always be open (thats one interpretation). And there will sometimes be other problems that are ind of ZFC or equiv to CH or something like that. But it does not affect the typical mathematician working on a typical problem.

I have a sense that its bad to exaggerate like this. One reason would be that if the reader finds out
the truth he or she will be disillusioned. But somehow, that doesn't seem to apply here. So I leave it to the reader to comment:  Is it  bad to exaggerate Paul Cohen's (or anyone's) accomplishments? And if so,
then why?


  1. Cohen's introduction of forcing into set theory was a pretty big deal. It gave not just models where CH failed (Godel had already given a model where CH held) but a general technique to construct models of set theory pretty much to specification (although different variants required different exact techniques). I am not sure that the authors are exaggerating the importance of his work as much as mistakenly emphasizing the result over the technique.

  2. I'm in the one percent and I'm not set theorist, last I checked.

  3. In mathematics it shows up. If you are working with infinite sets, groups, etc. in lemmas you want to use. That is what mathematics was considered to be till very recently. It is good to keep in mind that combinatorics and study of finite structures was not considered a major worthwhile branch of mathematics on par with algebra, geometry, analysis and number theory up until very recently.

  4. I'm another non-set-theory 1%-er, at least by this measure. Of course, Lance and I are probably counting the same paper :-). But in my experience, CH doesn't come up nearly as often as AC.