Problems in the philosophy of mathematics. Proceedings of the International Colloquium in the Philosophy of Science, London, 1965, Volume 1.

*(English)*Zbl 0155.33603
Studies in Logic and the Foundations of Mathematics. 47. Amsterdam: North-Holland Publishing Comp. xvi, 241 p. (1967).

This readable and informative volume contains the papers an philosophy of mathematics that were presented to the International Colloquium in the Philosophy of Science, held in July 1965 at Bedford College, London. There are nine papers, each followed by the principal contributions made to the subsequent discussion. The authors all give priority to philosophical aspects of their topics, and formal reasoning is kept to a minimum. Nevertheless, the collection is also of substantial logical and metamathematical interest, and several of the papers help significantly to make other more technical publications more accessible by explaining their setting. The individual papers, and their contents, are as follows.

1. Á. Szabó: “Greek dialectic and Euclid’s axiomatics”. The author argues that the axiomatic method used by Euclid, and the method of indirect proof in particular, derive from the dialectic of the Eleatics.

2. A. Robinson: ”The metaphysics of the calculus”. The author examines the historical development of the calculus, and the part which philosophical ideas played in it. He uses non-standard analy-sis to make his argument clear.

3. F. Sommers: ”On a Fregean dogma”. ’This paper is more for logicians than for mathematicians. It is concerned with a way of formulating syllogistic reasoning without quantifiers.

4. A. Mostowski: “Recent results in set theory”. This paper, which gives a survey of much recent work in set theory, deals particularly with strong axioms of infinity, and with the construction of models of set theory, as in the work of Cohen.

5. P. Bernays: “What do some recent results in set theory suggest?”. In this short paper, the author argues that Cohen’s independence proof does not apply to set theory as such, but to a certain type of axiomatization of set theory, and he discusses the relevance of this fact to the formalist philosophy of mathematics.

6. S. Körner: “On the relevance of post-Gödelian mathematics to philosophy”. Some philosophical questions are considered in the light of metamathematical results concerning non-categoricity and independence – in particular, the relation between mathematics and experience.

7. G. Kreisel: “Informal rigour and completeness proofs”. The author maintains that it is not adequate to regard mathematics as based solely on arbitrarily adopted formal axioms and rules of procedure. An essential element in it is ‘’informal rigour’, whereby new primitive notions are arrived at by analysis and idealization of notions already existing in an intuitive form. He illustrates his theme by considering: (1) the classification of mathematical concepts according to the order of the language needed to define them; (2) the predicate \(Val\) of intuitive logical validity, and its relations with the predicates \(V\) of validity in all set-theoretic structures and \(D\) of formal derivability; (3) a proof of the intuitionist principle \(\sim\forall[\sim\sim\exists x(\alpha x=0) \supset \exists x(\alpha x=0)] \) with the aid of a formal concept that is an idealization of Brouwer’ s thinking subject; and (4) the question whether standard or non-standard models are more fundamental.

8. L. Kalmar: ”Foundations of mathematics - whither now?”. The author puts forward the view that it is no longer sufficient to look upon mathematics as an abstract deductive science; and that future work on its foundations may well bring in the relation of mathematics to empirical knowledge.

9. J. A. Easley, jun.: ”Logic and heuristic in mathematics curriculum reform”. The author considers the controversy in America between the advocates of reformed curricula for school mathematics and their critics, and he relates the issues involved to possible attitudes towards the philosophy of mathematics. He proposes his own compromise between the opposing factions.

1. Á. Szabó: “Greek dialectic and Euclid’s axiomatics”. The author argues that the axiomatic method used by Euclid, and the method of indirect proof in particular, derive from the dialectic of the Eleatics.

2. A. Robinson: ”The metaphysics of the calculus”. The author examines the historical development of the calculus, and the part which philosophical ideas played in it. He uses non-standard analy-sis to make his argument clear.

3. F. Sommers: ”On a Fregean dogma”. ’This paper is more for logicians than for mathematicians. It is concerned with a way of formulating syllogistic reasoning without quantifiers.

4. A. Mostowski: “Recent results in set theory”. This paper, which gives a survey of much recent work in set theory, deals particularly with strong axioms of infinity, and with the construction of models of set theory, as in the work of Cohen.

5. P. Bernays: “What do some recent results in set theory suggest?”. In this short paper, the author argues that Cohen’s independence proof does not apply to set theory as such, but to a certain type of axiomatization of set theory, and he discusses the relevance of this fact to the formalist philosophy of mathematics.

6. S. Körner: “On the relevance of post-Gödelian mathematics to philosophy”. Some philosophical questions are considered in the light of metamathematical results concerning non-categoricity and independence – in particular, the relation between mathematics and experience.

7. G. Kreisel: “Informal rigour and completeness proofs”. The author maintains that it is not adequate to regard mathematics as based solely on arbitrarily adopted formal axioms and rules of procedure. An essential element in it is ‘’informal rigour’, whereby new primitive notions are arrived at by analysis and idealization of notions already existing in an intuitive form. He illustrates his theme by considering: (1) the classification of mathematical concepts according to the order of the language needed to define them; (2) the predicate \(Val\) of intuitive logical validity, and its relations with the predicates \(V\) of validity in all set-theoretic structures and \(D\) of formal derivability; (3) a proof of the intuitionist principle \(\sim\forall[\sim\sim\exists x(\alpha x=0) \supset \exists x(\alpha x=0)] \) with the aid of a formal concept that is an idealization of Brouwer’ s thinking subject; and (4) the question whether standard or non-standard models are more fundamental.

8. L. Kalmar: ”Foundations of mathematics - whither now?”. The author puts forward the view that it is no longer sufficient to look upon mathematics as an abstract deductive science; and that future work on its foundations may well bring in the relation of mathematics to empirical knowledge.

9. J. A. Easley, jun.: ”Logic and heuristic in mathematics curriculum reform”. The author considers the controversy in America between the advocates of reformed curricula for school mathematics and their critics, and he relates the issues involved to possible attitudes towards the philosophy of mathematics. He proposes his own compromise between the opposing factions.

Reviewer: G. T. Kneebone

##### MSC:

00B25 | Proceedings of conferences of miscellaneous specific interest |

00A30 | Philosophy of mathematics |

03-06 | Proceedings, conferences, collections, etc. pertaining to mathematical logic and foundations |