Category Archives: Math. Thought and Its Objects

I have been blogging on and off for quite a while about Charles Parsons’s Mathematical Thought and Its Objects, latterly as we worked through the book in a reading group here. I’ve now had a chance to put together a … Continue reading →

The final(!) section of Parsons’s book is one of the briefest, and its official topic is about the biggest — the question of the justification of set-theoretic axioms. But, reasonably enough, Parsons just offers here some remarks on how the … Continue reading →

How does arithmetic fit into the sort of picture of the role of reason and so-called “rational intuition” drawn in Secs. 52 and 53? The bald claim that some basic principles of arithmetic are “self-evident” is, Parsons thinks, decidedly unhelpful. … Continue reading →

Back to Parsons, to look at the final chapter of his book, called simply ‘Reason’. And after the particularly bumpy ride in the previous chapter, this one starts in a very gentle low-key way. In Sec. 52, ‘Reason and “rational … Continue reading →

The final section of Ch. 8 sits rather uneasily with what’s gone before. The preceding sections are about arithmetic and ordinary arithmetic induction, while this one briskly touches on issues arising from Feferman’s work on predicative analysis, and iterating reflection … Continue reading →

Suppose we help ourselves to the notion of a finite set, and say x is a number if (i) there is at least one finite set which contains x and if it contains Sy contains y, and (ii) every such … Continue reading →

Here’s the first half of an improved(?!?) discussion of this section: sorry about the delay! Parsons now takes up another topic that he has written about influentially before, namely impredicativity. He describes his own earlier claim like this: “no explanation … Continue reading →

In sum, then, we might put things like this. Parsons has defended an ‘internalist’ argument — an argument from “within mathematics” — for the uniqueness of the numbers we are talking about in our arithmetic, whilst arguing against the need … Continue reading →

Parsons now takes another pass at the question whether the natural numbers form a unique structure. And this time, he offers something like the broadly Wittgensteinian line which we mooted above as a riposte to skeptical worries — though I’m … Continue reading →

”There is a strongly held intuition that the natural numbers are a unique structure.” Parsons now begins to discuss whether this intuition — using ‘intuition’, of course, in the common-or-garden non-Kantian sense! — is warranted. He sets aside until the … Continue reading →