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I am looking for any suggestions for research-level survey articles that expose Russell's type theory (in the context of his philosophy of mathematics). Of particular interest are:

  1. Russell's reasons behind introducing his theory of types - what was his dissatisfaction with existing systems (that of Frege for example). Is the notion of a type theory generally plausible in the philosophy of mathematics?
  2. The role and merit of the Axioms of Infinity and Reducibility - to what extent is their use justified, and have recent developments shed light on their role in Russell's philosophy and more generally as axioms in mathematics.
  3. Russell's reasons for ramifying his theory of types - was the ramification well-motivated and was it successful?

I would be grateful for recommended reading, particularly if there are any authors who publish extensively on Russell's mathematical philosophy and cover all of these topics, or if any survey articles exist.

Many thanks.

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  • You can start with SEP entries on Bertrand Russell, Principia Mathematica, The Notation in PM, Propositional Function, Russell's Paradox and Type Theory; then I strongly suggest you Nicholas Griffin (ed), The Cambridge Companion to Bertrand Russell, Cambridge UP 2003.
    – Mauro ALLEGRANZA
    Apr 2, 2014 at 7:44
  • Then you can go on with more "technical" studies : Bernard Linsky, The Evolution of Principia Mathematica: Bertrand Russell's Manuscripts and Notes for the Second Edition, Cambridge UP 2011, Gregory Landini, Russell's Hidden Substitutional Theory, Oxford UP 1998, on the historical side, and Peter Andrews, An introduction to mathematical logic and type theory: to truth through proof, Kluwer Academic Publishers, 2nd ed 2002 and Fairouz Kamareddine & Twan Laan, A Modern Perspective on Type Theory, Kluwer Academic Publishers, 2004.
    – Mauro ALLEGRANZA
    Apr 2, 2014 at 8:02

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Alasdair Urquhart comes to my mind when thinking about authors mainly dealing with Russell's theory (or theories) of types. See his contribution to the Cambridge companion mentioned by Mauro.

Apart from that Urquhart has written an excellent article about Russell's reasons for ramification, which also gives a nice description of Russell's earlier type-theoretical development: A. Urquhart: "Russell's Zigzag Path to the Ramified Theory of Types". Russell: the Journal of Bertrand Russell Studies. 8 (1988). There are some free copies out there in the web. Here is (a legal) one: http://digitalcommons.mcmaster.ca/cgi/viewcontent.cgi?article=1415&context=russelljournal

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