Gerhard Heinzmann

One Hundred Years of Intuitionism (1907-2007)

With logicism and formalism, intuitionism is one of the main foundations for mathematics proposed in the twentieth century; and since the seventies, notably its views on logic have become important also outside foundational studies, with the development of theoretical computer science. The aim of the book is threefold: to review and complete the historical account of intuitionism; to present recent philosophical work on intuitionism; and to give examples of new technical advances and applications of intuitionism. This volume brings together 21 contributions by todays leading authors on these topics, and surveys the philosophical, logical and mathematical implications of the approach initiated in 1907 in L.E.J. Brouwers dissertation.

The Foundations of Geometry and the Concept of Motion: Helmholtz and Poincaré

Argument According to Hermann von Helmholtz, free mobility of bodies seemed to be an essential condition of geometry. This free mobility can be interpreted either as matter of fact, as a convention, or as a precondition making measurements in geometry possible. Since Henri Poincare defined conventions as principles guided by experience, the question arises in which sense experiential data can serve as the basis for the constitution of geometry. Helmholtz considered muscular activity to be the basis on which the form of space could be construed. Yet, due to the problem of illusion inherent in the subject’s self-assessment of muscular activity, this solution yielded new difficulties, in that if the manifold is abstracted from rigid bodies which serve as empirical justification of the geometrical notion of space, then illusionary bodies will produce fictive manifolds. The present article is meant to disentangle these difficulties.

Poincaré: intuitionism, intuition, and convention

Arend Heyting in the introduction of his famous volume Mathematische Grundlagenforschung, Intuitionismus, Beweistheorie, speaks of Poincare’s influence on ‘contemporary intuitionists’, i.e., mathematicians who agree with the opinion that (1) Mathematics has not only a formal but also a contentual signification. (2) Mathematical objects are directly grasped by the thinking mind; hence mathematical knowledge is independent of experience. (Heyting 1934, p.3)1

Correspondence Between Evert Willem Beth and Jean Piaget (1951–1955)

Today’s logic has gradually moved away from a natural logic and the scientific community largely accepts logical pluralism. At the same time, psychology has more or less abandoned his references to classical logic in order to theorize its experimental results. Finally, the demand for collaboration between the two disciplines is becoming more and more up to date, without to be realized. As part of an effort to understand the scope of such cooperation, the present study assessed Piaget’s and Beth’s attempt in the early 1950s to bring psychology and logic together by using the approach of genetic epistemology. The edition of their correspondence gave us an insight in the circularity of Piaget’s historical account of genetic epistemology with respect to logic: Piaget observed empirically an isomorphism between the ‘pre-propositional logic’ of the general coordination of actions and a certain part of formal logic. He then established the psychological development of this part of formal logic. But, in fact, the study of the pre-propositional operations expected to be the basis of some intuitive simple logical structures was itself governed by a complex logical structure. The implications of these results for developing a pragmatic version of epistemology are discussed in this paper.

Some Coloured Remarks on the Foundations of Mathematics in the 20th Century

According to the mainstream in the 20th century, the foundations of mathematics were identified with logic and set theory. Indeed, results concerning philosophically most interesting questions are often negative: the first order axiomatic set-theoretical universe is deductively incomplete, inevitably non-standard, and we have no clear idea of what the intended models of set theory are (part I). So, the foundational view of mathematics itself might be suspect. But in the spirit of Poincare, one should look for an other solution. He remarks that the varieties of classical first order theories is unable to deal with the most common modes of mathematical reasoning such as complete induction and model building. For such a purpose, Hintikkas IF-Logic seems to be an adequate way-out.

Philosophical Pragmatism in Poincare

At the beginning Poincare is using the terms ‘intuition’ and ‘analysis’ in order to describe two psychological attitudes invoived in the logic of invention: Riemann and Klein represent the attitude of intuition, Hermite and Weierstrass the attitude of analysis. Later on, but usually without explicit indication, these two terms stand likewise for two theories about the nature of mathematical activity: on the one hand you concentrate on investigations into the conditions governing the construction (intuition) of mathematical objects, on the other hand you try to describe (analyze) domains of already existing objects.

Mathematical Reasoning and Pragmatism in Peirce

In Peirce’s theory of cognition, the pragmatic maxim is the means used by reflection to connect signs with objects. The pragmatic maxim, in a formulation of 1878, taken up again in 1905, reads: “Consider what effects that might conceivably have practical bearing you conceive the object of your conception to have. Then your conception of those effects is the whole of your conception of the object.”1

Objectivity in Mathematics: The Structuralist Roots of a Pragmatic Realism

This paper proposes a reconsideration of mathematical structuralism, inaugurated by Bourbaki, by adopting the “practical turn” that owes much to Henri Poincare. By reconstructing his group theoretic approach of geometry, it seems possible to explain the main difficulty of modern philosophical eliminative and non-eliminative structuralism: the unclear ontological status of ‘structures’ and ‘places’. The formation of the group concept—a ‘universal’—is suggested by a specific system of stipulated sensations and, read as a relational set, the general group concept constitutes a model of the group axioms, which are exemplified (in the Goodmanian sense) by the sensation system. In other words, the shape created in the mind leads to a particular type of platonistic universals, which is a model (in the model theoretical sense) of the mathematical axiom system of the displacement group. The elements of the displacement group are independent and complete entities with respect to the axiom system of the group. But, by analyzing the subgroups of the displacement group (common to geometries with constant curvature) one transforms the variables of the axiom system in ‘places’ whose ‘objects’ lack any ontological commitment except with respect to the specified axioms. In general, a structure R is interpreted as a second order relation, which is exemplified by a system of axioms according to the pragmatic maxim of Peirce.

On the Origins of Mathematics

Sylvestre Huet: Why and how did man start doing mathematics, in particular based on the most elementary mathematical objects such as the point, the line, or the surface? Issues about the relationship between mathematical objects and reality arise from the onset: why and how does one do mathematics, and what is the relationship between mathematical objects and real objects or natural sciences? The informed general public is usually aware that the most basic mathematical notions, including numbers, were difficult constructions: think about the time taken to invent zero or positional number systems, concepts that we now learn as early as primary level... Yet, the invention of the concepts required bright minds, the brightest of the time. The relationship between mathematics and reality, which may seem obvious, is not so at all. Jean Dhombres could expand upon that, in order to begin the discussion by its historical aspect.

Political and Social History of Mathematics Education

Sylvestre Huet: We are now going to explore three major points concerning aspects of “mathematics and society”: education and training, mathematics and industry, and finally the use of mathematics in political debates and generally speaking in the humanities.