Tag Archives: Mathematical depth

Re: Paper and slides on indefiniteness of CH

Dear Harvey,

I am certainly interested to see the papers coming out of this workshop. Of course you are looking for something going well beyond ordinary observations, and perhaps you are not there yet.

Thanks for your interest.  You’re quite right that one welcome outcome would be ‘something going well beyond ordinary observations’, but this workshop was just preliminary field work for an investigation of that possibility.

All best,
Pen

Re: Paper and slides on indefiniteness of CH

Dear Harvey,

I realize you’re talking about ‘good mathematics’ rather than ‘deep mathematics’, but I hope you won’t mind if I take the liberty of transposing your remarks to the context of the workshop.

I have never heard mathematicians talk directly about what good mathematics is in any kind of generality.

Actually, for example, Urquhart presents some interesting quotations from people like Timothy Gowers and Terence Tao.

In talking separately to different kinds of mathematicians over the years, it is obvious that there is a huge amount of disagreement about how to evaluate mathematics, what it’s purpose it, what it means, and so forth.

Sure.  And it may turn out that ‘depth’ is just a term people use to mean ‘I like it’.  But that isn’t entirely clear yet.  In fact, many of the ideas you touch on were discussed at the workshop as potential symptoms or features of depth.  For example …

If the problem has resisted solution for a very long time, and it is known that some mathematicians with very strong reputations worked on it and failed to solve it, then mathematicians will generally regard the solution as extremely good mathematics.

This looks more like a symptom than a feature.

the solution uses considerable machinery … that promises further solutions to further problems.

This looks like ‘fruitfulness’, which may or may not be connected to depth.

There is a major premium paid for interactions between areas of mathematics – particularly if the interaction is unexpected.

This last is perhaps the suggestion most uniformly embraced.  The group pondered the problem of what counts as an ‘area’ of mathematics — is this something inherent in the math, or is it just a reflection of our human ways of dividing things up?  There’s also the question of ‘unexpected’ — is this just a matter of what we humans happen to be surprised by or is it a sign of something more fundamental in the math itself?

All best,
Pen

Re: Paper and slides on indefiniteness of CH

Thanks for doing this, Pen. The topic seems to me clearly an important but difficult one. But it seems to me fine if depth overlaps other properties such as fruitfulness, interconnectedness, unifying in surprising ways, etc. It also seems fine (compatible with an objectivist view  that at least a great deal of non-constructivist mathematics is definite in classical truth-value) if depth  both has objective significance and is interest-dependent. I look forward to seeing the volume.

Best,
Geoffrey