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**The protean character of mathematics.**
*(English)*
Zbl 0848.00006

Echeverria, Javier (ed.) et al., The space of mathematics. Philosophical, epistemological, and historical explorations. Revised papers from a symposium on structures in mathematical theories, Donostia/San Sebastian, Basque Country, Spain, September 1990. Berlin: Walter de Gruyter. Grundlagen der Kommunikation und Kognition. 1-13 (1992).

The author discusses some ideas from his seminal book “Mathematics: form and function” [Springer, Berlin/New York (1986; Zbl 0660.00019 and Zbl 0675.00001)]. He particularly discusses the thesis that mathematics is protean: “This means that one and the same mathematical structure has many different empirical realizations. Thus, mathematics provides common overarching forms, each of which can and does serve to describe different aspects of the external world. This places mathematics in relation to the other parts of science: mathematics is that part of science which applies in more than one empirical context” (p. 1).

He gives some evidence for this thesis by presenting examples from arithmetic, geometry, and analysis. He hints at unexpected applications and mathematical forms in different contexts, e.g., tensor product spaces, the concept of a group, the Riemann-Roch formula, relations between abstract algebraical concepts and the conformal field theory in theoretical physics, and finally the \(\lambda\)-calculus.

According to the author the intrinsically formal character of protean mathematics gives the reason that mathematics is essentially axiomatic. It follows furthermore that mathematics does not need foundations: “Any proposed foundation purports to say that mathematics is about this or that fundamental thing. But mathematics is not about things but about form. In particular, mathematics is not about sets” (p. 9). The author admits that there are a lot of dead ends in mathematics which can be realized with the help of a “good understanding of the nature of mathematics”. He lists fuzzy topology spaces, the notions of a groupoid and non-toral graphs, the factorization of large numbers.

For the author mathematics is a “tightly connected network of different forms and concepts” (p. 11) in which progress becomes manifest in two complementary tasks: the solution of outstanding problems and the understanding of achieved results. As an example for the long process of mathematical understanding he gives the main steps in the development of Galois theory between the work of Lagrange and 1990.

The author concludes with the wish “May protean understanding prosper!” His conception gives a home for all the recent studies on the rôle of analogy in the development of mathematics. It proposes a philosophy of mathematics based on the still rather vague notion of “the formal”. Although the author gets rid of platonistic, realistic, naturalistic, etc. mathematical objects, he does not prove that mathematics needs no foundations at all, he himself presents one.

For the entire collection see [Zbl 0839.00019].

He gives some evidence for this thesis by presenting examples from arithmetic, geometry, and analysis. He hints at unexpected applications and mathematical forms in different contexts, e.g., tensor product spaces, the concept of a group, the Riemann-Roch formula, relations between abstract algebraical concepts and the conformal field theory in theoretical physics, and finally the \(\lambda\)-calculus.

According to the author the intrinsically formal character of protean mathematics gives the reason that mathematics is essentially axiomatic. It follows furthermore that mathematics does not need foundations: “Any proposed foundation purports to say that mathematics is about this or that fundamental thing. But mathematics is not about things but about form. In particular, mathematics is not about sets” (p. 9). The author admits that there are a lot of dead ends in mathematics which can be realized with the help of a “good understanding of the nature of mathematics”. He lists fuzzy topology spaces, the notions of a groupoid and non-toral graphs, the factorization of large numbers.

For the author mathematics is a “tightly connected network of different forms and concepts” (p. 11) in which progress becomes manifest in two complementary tasks: the solution of outstanding problems and the understanding of achieved results. As an example for the long process of mathematical understanding he gives the main steps in the development of Galois theory between the work of Lagrange and 1990.

The author concludes with the wish “May protean understanding prosper!” His conception gives a home for all the recent studies on the rôle of analogy in the development of mathematics. It proposes a philosophy of mathematics based on the still rather vague notion of “the formal”. Although the author gets rid of platonistic, realistic, naturalistic, etc. mathematical objects, he does not prove that mathematics needs no foundations at all, he himself presents one.

For the entire collection see [Zbl 0839.00019].

Reviewer: V.Peckhaus (Erlangen)

### MSC:

00A30 | Philosophy of mathematics |

03A05 | Philosophical and critical aspects of logic and foundations |