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# Category Archives: Math. Thought and Its Objects

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

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

## Parsons’s Mathematical Thought: Sec. 48, The problem of the uniqueness of the number structure: Nonstandard models

”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

## Parsons’s Mathematical Thought: Sec. 47, Induction and the concept of natural number

Why does the principle of mathematical induction hold for the natural numbers? Well, arguably, “induction falls out of an explanation of the meaning of the term ‘natural number’”. How so? Well, the thought can of course be developed along Frege’s … Continue reading

Posted in Math. Thought and Its Objects, Phil. of maths
1 Comment

## Parsons’s Mathematical Thought: Secs. 40-45, Intuitive arithmetic and its limits

Here, as promised, are some comments on Chapter 7 of Parsons’s book. They are quite lengthy, and since in writing them I found myself going back to revise/improve some of my discussions of earlier sections, I’m just posting a single … Continue reading

## Parsons again

There’s now a version of my posts on the first five chapters of Parsons book: so the newly added pages are on Chapter 5 of his book, on “Intuition”. I found these sections unconvincing (when I didn’t find them baffling) … Continue reading

## Back to Parsons

Well, “blogging at a snail’s pace” is all well and good, but my posts about Parsons have recently ground to a complete halt. Sorry about that. Pressure of other things. But I’m back on the case, now with the pressure … Continue reading

## Parsons’s Mathematical Thought: Sec. 35, Intuition of finite sets

Suppose we accept that “it is not necessary to attribute to the agent perception or intuition of a set as a single object” in order to ground arithmetical beliefs. Still, we might wonder whether some such intuition of sets-as-objects might … Continue reading

## Parsons’s Mathematical Thought: Secs 33, 34, Finite sets and intuitions of them

So where have we got to in talking about Parsons’s book? Chapter 6, you’ll recall, is titled “Numbers as objects”. So our questions are: what are the natural numbers, how are they “given” to us, are they objects available to … Continue reading

## Parsons’s Mathematical Thought: Secs 31, 32, Numbers as objects

Chapter 6 of Parsons’s book is titled ‘Numbers as objects’. So: what are the natural numbers, how are they “given” to us, are they objects available to intuition in the kinds of ways suggested in the previous chapter? Sec. 31 … Continue reading

Posted in Math. Thought and Its Objects, Phil. of maths
1 Comment

## Burgess reviews Parsons

Luca Incurvati has just pointed out to me that John Burgess has a review of Parsons forthcoming in Philosophia Mathematica, and an electronic pre-print is available here (if your library has a subscription). Burgess is very polite, but reading between … Continue reading