##
**Mathematics: Form and function.**
*(English)*
Zbl 0675.00001

New York etc.: Springer-Verlag. xi, 476 p. (1986).

This review reached Zbl MATH while another one had already been in press (Zbl 0666.00019). Because of the extraordinary character of the book Zbl MATH decided to publish this second review:

Engineers and most experimental scientists consider mathematics as the tool that allows them to express their results in the most condensed and accessible way; its usefulness has been tremendously enhanced by the arrival of high speed computers. The kind of mathematics they need deals with readily understandable idealizations of the most common human activities: counting, timing, measuring and moving. The corresponding mathematical theories were systematically developed and perfected, first by the Greek geometers, and then during the period 1500-1800, which saw the birth of algebra, analytic geometry, mechanics, probability, and the most powerful of all: Calculus.

It is only very seldom that an engineer or an experimental scientist will use anything in mathematics which does not belong to what was known in the early \(19^{th}\) century. It is therefore not surprising that most of them are totally unaware of what is common knowledge among the mathematicians of today: namely, that pure mathematics has undergone such transformations that, superficially, modern books on mathematics do not seem to have anything in common with those written 200 years ago.

It is the aim of the well-known author of this excellent book to bridge that gap. His method consists in showing how the investigations of mathematicians of the \(19^{th}\) and \(20^{th}\) centuries naturally led them to subsume the properties of mathematical entities directly connected with sensory experience, under what he calls forms (or structures), namely systems of relations between entirely abstract notions, introduced axiomatically and only submitted to the rules of logic. He does not try to describe in detail the often tortuous path by which these transformations were made, but he endeavours to make as clear as possible their final result and how it connects with their starting point. There are no difficult technical proofs, so the mathematical background he needs does not exceed the knowledge acquired after 2 or 3 years of study at the university.

The first chapter is a rapid survey explaining, chiefly by means of examples, what are the main concepts in modern mathematics: set, order, metric or topological space, group, boolean algebra; there is also a short list of mathematical activities which gave rise to these notions, such as problem solving, analogy, generalization and abstraction.

The following chapters take up one by one and in greater detail the description of the way in which every “classical” part of mathematics has given birth to modern stuctures. Chapter II starts from natural numbers and goes over to a sample of problems in number theory, and to rational numbers, cardinals and ordinals. Chapter III begins with Euclid and how Hilbert refined his axioms, then deals with non euclidean geometry, orientation and groups of rigid motions. In chapter IV, it is shown how the Greek idea of measuring magnitudes in geometry led to real numbers, to their arithmetic constructions and to vectors, analytic geometry and trigonometry, complex numbers, stereographic projection and finally quaternions. Chapter V is devoted to the fundamental idea of function, leading to the modern concept of map, to transformation groups, and finally to abstract groups; Galois theory is sketched, and also the construction of some groups; the recent enumeration of all simple finite groups is mentioned.

Chapter VI starts with the basic concepts of Calculus: integration, born from measurements of geometric magnitudes, and derivatives, growing out of the construction of tangents to curves on one hand, and of the “rate of motion” on the other. Then comes the birth of Calculus with the discovery of the fundamental link between these two notions. This is illustrated by the first triumph of Calculus, Newton’s deduction of Kepler’s laws from the law of gravitation, which is the beginning of the theory of differential equations. A short description follows of the way calculus is now founded on the notions of limit and continuity, with the use of uniform continuity and of compactness. Next come Taylor’s formula, partial derivatives, differential forms and multiple integrals, leading to the various aspects of modern analysis, including the Calculus of variations.

Chapter VII deals with linear and multilinear algebra. Starting with linear transformations between vector spaces, the author rightly stigmatizes the awful tendency of some modern textbooks to bury this simple idea under “a morass of muddled matrix manipulations”. He describes eigenvalues, dual spaces, inner product spaces and the concepts of orthogonal and of self-adjoint transformations in these spaces. Then come bilinearity and tensor products, quotient spaces, exterior algebra and differential forms, and finally the general notion of module and its application to invariant factors of an endomorphism.

Chapter VIII is devoted to differential geometry, beginning with curves and surfaces in 3-dimensional euclidean space, and notions of arc length and of curvature attached to them. This leads to the intrinsic notion of manifold, which starts with Gauss’s studies and from the notion of Riemann surface; it goes on to the general ideas of topological space and of smooth manifold of any dimension, accompanied by its tangent bundle; the climax is reached with Riemannian geometry and the concept of sheaf.

Chapter IX is on mechanics, but this is not a deviation from the general pattern of the book, since Mechanics was strongly tied up with geometry ever since the Greeks, and became as strongly linked with Calculus after Newton. From the fundamental concepts of momentum, work and energy, one goes over to Lagrange’s equations, and then to the Hamilton-Jacobi formulations; there is even a mention of the modern relations between mechanics and symplectic geometry.

Chapter X starts with the scattered results on functions of a complex variable obtained before 1800, but the theory only comes into its own with Cauchy, Riemann and Weierstrass, whose main achievements fill most of the chapter. Emphasis is laid on the growing part played by geometry, and later topology, in the steady progress of what is now called “analytic geometry”, viewed as a twin sister of “algebraic geometry”. Sheaves are even mentioned, bu it is unlikely that the prospective readers of the book will understand their importance without previous knowledge of cohomology.

Chapter XI differs from the previous ones, since its first part is mainly concerned with the problems of “foundations”, almost entirely confined within the narrow circle of pure mathematicians and some philosophers; furthermore, these problems only became part of mathematics in the second half of the \(19^{th}\) century. So the chapter has to start from scratch: hierarchy of sets, axiomatic set theory, propositional and predicate calculus, leading to Gödel’s incompleteness theorem and to the independence of the continuum hypothesis from the Zermelo-Fraenkel axioms. The author is to be commended for his presentation of the concept of “rigor”, against muddle-headed philosophers like Lakatos: “Rigor is absolute and is here to stay”. The second part is a sketch of the main concepts in the theory of categories and functors, created in 1943 by the author and S. Eilenberg. As an application, the unfamiliar notion of “topos” is introduced, and it is shown how, with it, one can replace the usual description of mathematics by sets and maps by another one, in which the relation \(x\in X\) has disappeared and everything is done by means of “arrows”. One can also describe intuitionistic logic by means of a topos, and even interpret the independence proof of the continuum hypothesis as a construction of an elementary topos using sheaf theory.

In the last chapter, the author returns to the material sketched in chapter I, that he can now develop using the examples of chapters II to XI. He sees mathematics as a “network” of formal systems, moved by ideas and nourished by problems. He emphasizes that mathematics essentially differs from the sciences, since it cannot be “falsified”. He therefore thinks that the question “Is mathematics true” is out of place, but there are other features of mathematics which raise meaningful philosophical questions, that he describes and discusses in a few pages.

It is to be hoped that this book will find many readers, who thus may be convinced that pure mathematics did not stop after 1800.

Engineers and most experimental scientists consider mathematics as the tool that allows them to express their results in the most condensed and accessible way; its usefulness has been tremendously enhanced by the arrival of high speed computers. The kind of mathematics they need deals with readily understandable idealizations of the most common human activities: counting, timing, measuring and moving. The corresponding mathematical theories were systematically developed and perfected, first by the Greek geometers, and then during the period 1500-1800, which saw the birth of algebra, analytic geometry, mechanics, probability, and the most powerful of all: Calculus.

It is only very seldom that an engineer or an experimental scientist will use anything in mathematics which does not belong to what was known in the early \(19^{th}\) century. It is therefore not surprising that most of them are totally unaware of what is common knowledge among the mathematicians of today: namely, that pure mathematics has undergone such transformations that, superficially, modern books on mathematics do not seem to have anything in common with those written 200 years ago.

It is the aim of the well-known author of this excellent book to bridge that gap. His method consists in showing how the investigations of mathematicians of the \(19^{th}\) and \(20^{th}\) centuries naturally led them to subsume the properties of mathematical entities directly connected with sensory experience, under what he calls forms (or structures), namely systems of relations between entirely abstract notions, introduced axiomatically and only submitted to the rules of logic. He does not try to describe in detail the often tortuous path by which these transformations were made, but he endeavours to make as clear as possible their final result and how it connects with their starting point. There are no difficult technical proofs, so the mathematical background he needs does not exceed the knowledge acquired after 2 or 3 years of study at the university.

The first chapter is a rapid survey explaining, chiefly by means of examples, what are the main concepts in modern mathematics: set, order, metric or topological space, group, boolean algebra; there is also a short list of mathematical activities which gave rise to these notions, such as problem solving, analogy, generalization and abstraction.

The following chapters take up one by one and in greater detail the description of the way in which every “classical” part of mathematics has given birth to modern stuctures. Chapter II starts from natural numbers and goes over to a sample of problems in number theory, and to rational numbers, cardinals and ordinals. Chapter III begins with Euclid and how Hilbert refined his axioms, then deals with non euclidean geometry, orientation and groups of rigid motions. In chapter IV, it is shown how the Greek idea of measuring magnitudes in geometry led to real numbers, to their arithmetic constructions and to vectors, analytic geometry and trigonometry, complex numbers, stereographic projection and finally quaternions. Chapter V is devoted to the fundamental idea of function, leading to the modern concept of map, to transformation groups, and finally to abstract groups; Galois theory is sketched, and also the construction of some groups; the recent enumeration of all simple finite groups is mentioned.

Chapter VI starts with the basic concepts of Calculus: integration, born from measurements of geometric magnitudes, and derivatives, growing out of the construction of tangents to curves on one hand, and of the “rate of motion” on the other. Then comes the birth of Calculus with the discovery of the fundamental link between these two notions. This is illustrated by the first triumph of Calculus, Newton’s deduction of Kepler’s laws from the law of gravitation, which is the beginning of the theory of differential equations. A short description follows of the way calculus is now founded on the notions of limit and continuity, with the use of uniform continuity and of compactness. Next come Taylor’s formula, partial derivatives, differential forms and multiple integrals, leading to the various aspects of modern analysis, including the Calculus of variations.

Chapter VII deals with linear and multilinear algebra. Starting with linear transformations between vector spaces, the author rightly stigmatizes the awful tendency of some modern textbooks to bury this simple idea under “a morass of muddled matrix manipulations”. He describes eigenvalues, dual spaces, inner product spaces and the concepts of orthogonal and of self-adjoint transformations in these spaces. Then come bilinearity and tensor products, quotient spaces, exterior algebra and differential forms, and finally the general notion of module and its application to invariant factors of an endomorphism.

Chapter VIII is devoted to differential geometry, beginning with curves and surfaces in 3-dimensional euclidean space, and notions of arc length and of curvature attached to them. This leads to the intrinsic notion of manifold, which starts with Gauss’s studies and from the notion of Riemann surface; it goes on to the general ideas of topological space and of smooth manifold of any dimension, accompanied by its tangent bundle; the climax is reached with Riemannian geometry and the concept of sheaf.

Chapter IX is on mechanics, but this is not a deviation from the general pattern of the book, since Mechanics was strongly tied up with geometry ever since the Greeks, and became as strongly linked with Calculus after Newton. From the fundamental concepts of momentum, work and energy, one goes over to Lagrange’s equations, and then to the Hamilton-Jacobi formulations; there is even a mention of the modern relations between mechanics and symplectic geometry.

Chapter X starts with the scattered results on functions of a complex variable obtained before 1800, but the theory only comes into its own with Cauchy, Riemann and Weierstrass, whose main achievements fill most of the chapter. Emphasis is laid on the growing part played by geometry, and later topology, in the steady progress of what is now called “analytic geometry”, viewed as a twin sister of “algebraic geometry”. Sheaves are even mentioned, bu it is unlikely that the prospective readers of the book will understand their importance without previous knowledge of cohomology.

Chapter XI differs from the previous ones, since its first part is mainly concerned with the problems of “foundations”, almost entirely confined within the narrow circle of pure mathematicians and some philosophers; furthermore, these problems only became part of mathematics in the second half of the \(19^{th}\) century. So the chapter has to start from scratch: hierarchy of sets, axiomatic set theory, propositional and predicate calculus, leading to Gödel’s incompleteness theorem and to the independence of the continuum hypothesis from the Zermelo-Fraenkel axioms. The author is to be commended for his presentation of the concept of “rigor”, against muddle-headed philosophers like Lakatos: “Rigor is absolute and is here to stay”. The second part is a sketch of the main concepts in the theory of categories and functors, created in 1943 by the author and S. Eilenberg. As an application, the unfamiliar notion of “topos” is introduced, and it is shown how, with it, one can replace the usual description of mathematics by sets and maps by another one, in which the relation \(x\in X\) has disappeared and everything is done by means of “arrows”. One can also describe intuitionistic logic by means of a topos, and even interpret the independence proof of the continuum hypothesis as a construction of an elementary topos using sheaf theory.

In the last chapter, the author returns to the material sketched in chapter I, that he can now develop using the examples of chapters II to XI. He sees mathematics as a “network” of formal systems, moved by ideas and nourished by problems. He emphasizes that mathematics essentially differs from the sciences, since it cannot be “falsified”. He therefore thinks that the question “Is mathematics true” is out of place, but there are other features of mathematics which raise meaningful philosophical questions, that he describes and discusses in a few pages.

It is to be hoped that this book will find many readers, who thus may be convinced that pure mathematics did not stop after 1800.

Reviewer: J.Dieudonné

### MSC:

00A05 | Mathematics in general |

00A30 | Philosophy of mathematics |

00A06 | Mathematics for nonmathematicians (engineering, social sciences, etc.) |

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

01A60 | History of mathematics in the 20th century |

01A55 | History of mathematics in the 19th century |