Notes by RBJ on
The Oxford Handbook of Philosophy of Mathematics and Logic
edited by Stewart Shapiro
Overview
As ever, it is not intended that these notes provide a review or even fair comment. In relation to the first chapter, in which Shapiro provide a concise overview of the topics of the book, I have in mind sketching a similarly concise counter-statement in which my own position is related on most of the topics he mentions and to most of the positions he describes. I might then go on to more substantial statements on some topics.
Ch.2 Apriority and Application
Ch.3 Later Empiricism and Logical Positivism
Ch.4 Wittgenstein on Philosophy of Logic and Mathematics
Comparing and evaluating the answers of these three logicist's answers to just three questions in the philosophy of mathematics.
Ch.6 Logicism in the 21st Century
Ch.7 Logicism Reconsidered
Formalism, from antiquity to the present day.
Ch.9
Ch.10
Ch.11
Ch.12
Ch.13
Ch.14
Ch.15
Ch.16
Ch.17
Ch.18
Ch.19
Ch.20
Ch.21
Ch.22
Ch.23
Ch.24
Ch.25
Ch.26
Ch.1 Introduction
Motivation, or What We are Up To
Shapiro's take on what the Philosophy of Mathematics and Logic involves.
Global Matters
The broader topics in the philosophy of mathematics and logic.
Local Matters
The Skolem paradox, independence results in set theory, incompleteness results, the application of mathematics...
A Potpourri of Positions
A preliminary presentation of some of the main positions treated in greater depth later in the book.
Logic
Ch.5 The Logicism of Frege, Dedekind and Russell
Comparing and evaluating the answers of these three logicist's answers to just three questions in the philosophy of mathematics.
Introduction and Scope of the Study

The three questions considered are:

  • What is the basis for our knowledge of the infinity of numbers
  • How is arithmetic applicable to reality?
  • Why is reasoning by induction justified?

These questions are preferred to the "more usual formulation" in which logicism is presented as rejecting Kant's view that mathematics is synthetic a priori in favour of the view that it is analytic, which was not in fact held by either Dedekind or Russell. Surely however, the logicist thesis from which the name comes, is that mathematics is logic? The impression given here is that the authors have not made a serious attempt to put give a statement of the logicist thesis which is common to the three authors but has made an excuse to consider some other less central issues.

Frege
Frege's Logic and Theory of Classes

From [the Begriffsschrift] we can extract a presentation of second order logic...
a variation on the misdescriptions of Frege's logic which pervade the secondary literature.

This section is devoted to a detailed account of the derivation of a contradiction in Frege's system.

Frege's Analysis of the Natural Numbers
This is an account of how Frege approaches arithmetic via "Hume's Principle".
The Problem of Applicability
This is about the "Julius Caeser" problem, which it is alleged eluded Frege. Personally I find the whole applicability issue a pseudo problem, insofar at least as this is supposed to discriminate between different accounts of the natural numbers. The natural numbers are applicable, we know this from experience. It makes not difference what the numbers are, so long as the whole has the right structure.
Frege's Account of Reasonong by Mathematical Induction
This starts in part III of The Begriffsscrift with the definition of the ancestral of a relation, and shows how Frege gets from there to the principle of mathematical induction over the natural numbers.
Comparison with Dedekind's Chains
There seem to be some technical errors in this story. He begins with a definition of the chain or closure of a set A with respect to a one-one relation f which yields invariably the field of f. Presumably the alleged correspondence does hold, but this account of it is garbled.
A Remark on Frege and Kant

This section concerns Frege's desire to show arithmetic independent of Kantian intuitions. It begine with some discussion of schematization, about which I am unable to comment.

From that discussion the authors then extract some desiderata for an account of number independent of Kantian Intuition, viz:

  • must explain applicability of numerical concepts
  • must explain the reference of arithmetical equations and inequations
  • must recover any piece of arithmetical reasoning (of certain kinds)
all without dependence on intuitions.

I am inclined to reject all of this. The applicability of arithmetic does not depend upon how the number system is established. A wholly satisfactory account of arithmetic need not identify even what the numbers are, let alone what equations and inequations refer to, I have no inkling of why these referents should be thought important. Of course, a definition of the natural numbers will be useful mathematically only if it is done in a context which permits one to do number theory, but this provides no basis for discriminating between alternative definitions, which need only to say enough to yield the right kind of structure.

My reaction is of course based on a twentieth century understanding of how the natural numbers can be defined, and are therefore naturally more in line with Frege's than Kant's perspective, though I would not myself object to the idea that intuition has a role to play, both in coming up with a definition of number and in applying it.

Summary
Assessment
Russell
Russell's Propositional Paradox and Fregean Thoughts
Having discovered the inconsistency of Frege's system when the writing of "The Principles of Mathematics" was complete, Russell sketched out a type theoretic resolution to the paradoxes in Appendix B to that work. In it he mentions a contradiction which is not resolved by this type theory and which has become known as "the propositional paradox". This section gives an account of the propositional paradox and discusses it in relation to Frege's thought.
Dedekind's Account of our Knowledge of the Infinity of the Numbers
Russell's Logicism and the Rejection of Denoting Concepts
Ch.8 Formalism
Formalism, from antiquity to the present day.
1. Introduction: The Formalist Framework
Formalism is a family of views on the nature of mathematics which share a common framework.
2. The Traditional Viewpoint
The position in classical Greece in relation to the divisions of mathematics and the nature of proof are outlined.
3. Medieval and Renaissance Challenges to the Ancient Ideals
Concerning attacks on the constructive ideal by Jesuit commentators of the late Middle Ages and Latin Renaissance, and the defence against these attacks.
4. The Emergence of Formalism
5. Symbolic Formalism
6. Challenges to Formalism

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