Because ‘mathematical depth’ is a notion that hasn’t received much philosophical attention, this workshop was largely exploratory. Maybe it would help to say a bit more about how it was organized:
This workshop will bring together mathematicians and historians and philosophers of mathematics to try to get a preliminary sense of whether or not the notion of ‘mathematical depth’ can play a useful role in our understanding of the nature and practice of mathematics. Speakers have been encouraged to present an example or examples of concepts, theorems, subject areas that they think qualify as deep, or as not deep, and to lay out the particular mathematical features of those examples that lead them to make those judgments. Then, in discussion, we aim to do several things:
- See if there is agreement on which examples are deep and not deep.
- See if there are commonalities in the kinds of features cited in defense of depth and non-depth assessments in the various examples.
- Ask whether depth is or isn’t the same as fruitfulness, surprisingness, importance, elegance, difficulty, fundamentalness, explanatoriness, etc.
- Ask whether depth seems likely to be an objective feature or something essentially tied to our interests, abilities, and so on. (Even natural science is tied to our interests and abilities, in that we might be drawn to certain areas of inquiry by our interests, hampered or helped by certain of our abilities, etc. The question is whether depth is tied to our interests and abilities in some more fundamental way.)
One possible outcome would be: this is a non-starter. Another would be: this is worthy of further study.
So, yes, Stillwell largely gave examples, which were then most helpful in the discussions. Arana’s talk had more analytical content in pursuit of just one example. Gray talked about Gauss’s notion of depth. Lange discussed some very interesting ‘Martin Gardner level’ examples of comparative depth. Urquhart considered a range of things mathematicians have said, in print or online, on the topic. And so on. But if you want a sense of the real consolidating work that was done, you might dip into the discussions, which gathered steam over the two days.
On the other hand, if you’re interested in something more polished, you might wait for the special issue. It’s projected to include papers by Arana, Gray, Lange, Stillwell and Urquhart based on their talks, and a substantial editorial Foreword and Afterword that summarizes and explores the high points of the discussions.
Again, this is only a preliminary investigation, beginning from cases and asking what, if anything, can be gleaned from them. There’s no final answer on offer here — at best it’s a start.
PS: An aside to Sy — I believe my concerns about your notions of ‘truth’ and ‘the concepts of set and set-theoretic universe’ are independent of my own ‘radical’ views on these matters.