Monday, April 02, 2007

Paul J. Cohen dies

Set theorist and Fields medal winner Paul J. Cohen of Stanford University died at age of 72:
When I was an undergrad, people would say that mathematical logic and set theory was very important for philosophy. As a guy who thought that physics was much more important, I got interested in that field and found all these theorems and logical glitches with the axiom systems more entertaining than expected although not as profoundly philosophically important as theoretical physics. ;-)

The name of Paul Cohen appeared quite frequently. He wrote the book "Set theory and continuum hypothesis". Is the hypothesis true? Today, I view these questions as "unphysical". They are definitely "untestable" and more generally, they can't influence anything that is causally connected to natural sciences. They always deal with some kinds of "infinite", "infinitely fine", or otherwise physically pathological objects. ;-)

It's a matter of convenience whether you accept some of these axioms although deeply inside my soul, it is still hard to get rid of the feeling that statements such as the continuum hypothesis or the axiom of choice must be either correct or wrong. But nowadays, I view these feelings as religious feelings.

Does anyone have the same feelings?

But scientifically speaking, I am a complete agnostic about these matters and have no bias about them. For example, it may be elegant not to believe the axiom of choice because then you might believe (or add the axiom) that all sets have a measure. That would also be nice, wouldn't it? You can only prove the existence of unmeasurable sets with the axiom of choice.

Also, I believe that it is plausible that all "normal" mathematical assertions that you would ever invent - such as the Riemann Hypothesis - must be provable or disprovable, and all statements that are neither are "artifically doctored" and form an uninterestingly small universality class in the landscape of mathematical assertions.

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