Intuition, iteration, induction

TL;DR

The paper compares Parsons' and Brouwer's views on mathematical intuition, arguing that Brouwer's perspective can be justified with detailed analysis using Husserl and Brouwer's ideas, and that Parsons' discussion hints at Brouwer's notions.

Contribution

It provides a detailed justification of Brouwer's claims on intuition using Husserl and Brouwer, and shows how Parsons' discussion can be extended to align with Brouwer's views.

Findings

Brouwer's claims on intuition can be justified with Husserl's and Brouwer's ideas.

Parsons' discussion contains elements that can be developed into Brouwer's notions.

The analysis clarifies the philosophical debate on mathematical intuition.

Abstract

In Mathematical Thought and Its Objects, Charles Parsons argues that our knowledge of the iterability of functions on the natural numbers and of the validity of complete induction is not intuitive knowledge; Brouwer disagrees on both counts. I will compare Parsons' argument with Brouwer's and defend the latter. I will not argue that Parsons is wrong once his own conception of intuition is granted, as I do not think that that is the case. But I will try to make two points: (1) Using elements from Husserl and from Brouwer, Brouwer's claims can be justified in more detail than he has done; (2) There are certain elements in Parsons' discussion that, when developed further, would lead to Brouwer's notion thus analysed, or at least something relevantly similar to it. (This version contains a postscript of May, 2015.)