Category Archives: Math. Thought and Its Objects

Encore #10: Parsons on intuition

Just yesterday, Brian Leiter posted the results of one of his entertaining/instructive online polls, this time on the “Best Anglophone and German Kant scholars  since 1945“. Not really my scene at all. Though I did, back in the day, really love … Continue reading

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Encore #9: Parsons on noneliminative structuralism

I could post a few more encores from my often rather rude blog posts about Murray and Rea’s Introduction to the Philosophy of Religion. But perhaps it would be better for our souls to to an altogether more serious book … Continue reading

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Parsons, the whole story, at last

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

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Parsons’s Mathematical Thought: Sec 55, Set theory

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

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Parsons’s Mathematical Objects: Sec. 54, Arithmetic

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

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Parsons’s Mathematical Objects: Secs 52-53, Reason, "rational intuition" and perception

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

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Parsons Mathematical Thought: Sec. 51, Predicativity and inductive definitions

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

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Parsons Mathematical Thought: Sec. 50, Induction and impredicativity, continued

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

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Parsons’s Mathematical Thought: Sec. 50, Induction and impredicativity

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

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Parsons’s Mathematical Thought: Sec. 49, Uniqueness and communication, continued

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

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