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Introduction To Mathematical Philosophy

Bertrand Russell
Dover Publications
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Dennis Lomas
, on

Russell’s mammoth impact in the fields of mathematical logic, philosophy, and mathematical foundations lends interest to his Introduction to Mathematical Philosophy (now more than ninety years old), a minimally technical book which addresses the logical foundations of mathematics. The book affords an opportunity, limited but still significant, to ponder arguments and observations of historical import.

Russell remarks that the book deals “with a body of knowledge which, to those who accept it, appears to invalidate much traditional philosophy, and even a great deal of what is current in the present day. In this way, as well as by its bearing on still unsolved problems, mathematical logic is relevant to philosophy.” (pp. v–vi) In this, and in a few other comments of a similar bent, Russell gingerly places the content of this book within philosophy. The fit may well have seemed awkward at the time.

Russell held that mathematics can proceed in two directions, one toward increasing complexity.

The other direction, which is less familiar, proceeds, by analysing, to greater and greater abstraction and logical simplicity; instead of asking what can be defined and deduced from what is assumed to begin with, we ask instead what more general ideas and principles can be found, in terms of which what was our starting-point can be defined or deduced. It is the fact of pursuing this opposite direction that characterizes mathematical philosophy as opposed to ordinary mathematics. (p. 1)

Russell’s pursuit (which I will not get into) involves laying a foundation for mathematics in logic. No philosopher is clearer than Russell and this book is no exception. Having taken this bow to Russell, I find that the informal treatment of the subject makes for sometimes tedious reading. Today, a treatment of the same material using standard notation would likely be far more compact. In fairness to Russell, a good part of his intended readership consisted (as he observed) of philosophers with little ability in symbolic reasoning for whom an informal treatment was necessary.

Russell’s ideas have generated an enormous amount of fruitful research. Even so, in reading the book, one might ask: has philosophy’s concentration on Russell’s pursuit (and similar pursuits) led to philosophy’s missing anything of substance about mathematics? If so, what? (David Corfield’s Towards a Philosophy of Real Mathematics is relevant here.)

Dennis Lomas has studied computer science, mathematics, and philosophy. 

Preface; Editor's Note

1. The Series of natural numbers
2. Definition of number
3. Finitude and mathematical induction
4. The definition of order
5. Kinds of relations
6. Similarity of relations
7. Rational, real, and complex numbers
8. Infinite cardinal numbers
9. Infinite series and ordinals
10. Limits and continuity
11. Limits and continuity of functions
12. Selections and the multiplicative axiom
13. The axiom of infinity and logical types
14. Incompatibility and the theory of deduction
15. Propositional functions
16. Descriptions
17. Classes
18. Mathematics and logic