Bostock is a **Logicist **and wrote some stuff supporting Borkowski in 1980. He references Shapiro quite a bit, who doesn’t return the favour. The main difference in the structures of the two books is that Shapiro looks at **realism, fictionalism **and **structuralism **in the contemporary scene, whereas Bostock looks at **realism, nominalism/fictionalism **and **predicativism. **I’ll take a look at that last section, then.

**Part I: Plato versus Aristotle:**.

__A. Plato__.

1. The Socratic Background.

2. The Theory of Recollection.

3. **Platonism** in Mathematics.

4. Retractions: the Divided Line in Republic VI (509d−511e).

__B. Aristotle__.

5. The Overall Position.

6. Idealizations.

7. Complications.

8. Problems with Infinity.

__C. Prospects__.

**Part II: From Aristotle to Kant:**.

1. Medieval Times.

2. Descartes.

3. Locke, Berkeley, Hume.

4. A Remark on **Conceptualism**.

5. Kant: the Problem.

6. Kant: the Solution.

**Part III: Reactions to Kant:**.

1. Mill on Geometry.

2. Mill versus Frege on Arithmetic.

3. **Analytic** Truths.

4. Concluding Remarks.

**Part IV: Mathematics and its Foundations:**.

1. Geometry.

2. Different Kinds of Number.

3. The Calculus.

4. Return to** Foundations**.

5. Infinite Numbers.

6. Foundations Again.

**Part V: Logicism:**.

1. Frege.

2. Russell.

3. Borkowski/Bostock (see?).

4. Set Theory.

5. Logic.

6. Definition.

**Part VI: Formalism:**.

1. Hilbert.

2. Gödel.

3. Pure **Formalism**.

4. **Structuralism**.

5. Some Comments.

**Part VII: Intuitionism:**.

1. Brouwer.

2. **Intuitionist** Logic.

3. The Irrelevance of Ontology.

4. The Attack on **Classical Logi**c.

**Part VIII: Predicativism:**.

1. Russell and the VCP.

2. Russell’s Ramified Theory and the Axiom of Reducibility.

3. Predicative Theories after Russell.

4. Concluding Remarks.

**Part IX: Realism versus Nominalism:**.

__A. Realism__.

1. Gödel.

2. Neo-Fregeans.

3. Quine and Putnam.

__B. Nominalism__.

4. Reductive **Nominalism**.

5. **Fictionalism**.

6. Concluding Remarks.

Russell is first talked about wrt logicisim, but after his whole freakout about the Vicious Circle Principle what he’s talking about is termed Predicativism. It’s his whole no-class thing.

From http://plato.stanford.edu/entries/philosophy-mathematics/:

As was mentioned earlier, predicativism is not ordinarily described as one of the schools. But it is only for contingent reasons that before the advent of the second world war predicativism did not rise to the level of prominence of the other schools.

[…] Russell [developed] the simple and the ramified theory of types, in which syntactical restrictions were built in that make impredicative definitions ill-formed. In simple type theory, the free variables in defining formulas range over entities to which the collection to be defined do not belong.

Yeah, we’ve seen this before.