There have been distractions. So it’s been painfully slow progress on various projects over the last few weeks. In particular, it has been very slow work trying to write the early (pre-category-theoretic, scene-setting) chapters I wanted to put together for the Gentle Introduction. Partly that’s because I have been trying to decide what I actually believe about set-theoretic reductionism/foundationalism/imperialism …
Here though is a fairly recent paper that I’ve found very helpful. And I am prompted to mention it here because I discovered (in talking to a small and very unrepresentative sample!) that word about it hadn’t got around. So — if you are interested in the topic — let me recommend
Penelope Maddy, ‘Set-theoretic foundations’, in Caicedo et al, eds., Foundations of Mathematics, 2017.
Maddy begins, “It’s more or less standard orthodoxy these days that set theory – ZFC, extended by large cardinals – provides a foundation for classical mathematics. Oddly enough, it’s less clear what ‘providing a foundation’ comes to. Still, there are those who argue strenuously that category theory would do this job better than set theory does, or even that set theory can’t do it at all, and that category theory can. There are also those who insist that set theory should be understood, not as the study of a single universe, V, purportedly described by ZFC + LCs, but as the study of a so-called ‘multiverse’ of set-theoretic universes – while retaining its foundational role. I won’t pretend to sort out all these complex and contentious matters, but I do hope to compile a few relevant observations that might help bring illumination somewhat closer to hand.” And, particularly in the first part of the paper, it strikes me that she does an admirable and very judicious job of distinguishing various things that might be meant by talking of foundations here. Though that still leaves me with much to think about.
In another neck of the woods, I am getting back to updating what was the TYL Study Guide, and I have just uploaded a slightly revised version of Chapters 1 to 8. I hope to finish the new Chapter 9 over the coming days.
About fifty years late, I’ve also been enjoying thinking a bit more carefully about the Prior Analytics, and (relatedly) about ways in which the opening chapters of IFL could be significantly improved.
And so it goes … But, as I say, slow progress all round.