I will have to be off and on the internet for the next month or so, and won’t be able to keep up with the formal activities on the List. But I have been thinking a lot about the current state of discussion, along with the several bouts of past discussions on topics related to Peirce’s “Kaina Stoicheia” — I tend to call it that so as not to confuse it with the four volumes of The New Elements of Mathematics.
The editors of The Essential Peirce say that “New Elements” was written “as a preface to an intended book on the foundations of mathematics”, but that much already requires our careful reflection in view of the way that Peirce makes normative science, logic included, depend on the strife-born twins of phenomenology and mathematics. With that in mind, foundations of mathematics is a loose enough term that what the editors say may well be true, in one sense or another, but that sense is likely to be radically other than the meaning of thinkers who would reduce mathematics to deductive logic alone.
At any rate, that is a topic for another discussion.
What I’m noticing in my reflections on past and present discussions of these topics is the evident lack of a common language when it comes to the foundation of mathematics, in whatever sense we might have in mind. So I thought it might serve to collect a few notes on the subject from here and there among the canons and allied commentaries on foundations.
I am going to start with excerpts from the now-classic textbook by Raymond L. Wilder, Introduction to the Foundations of Mathematics. This was my first formal introduction to the subject, used in the course that Frank Harary taught at the University of Michigan back in the 1970s.