Marie-José Durand-Richard

Logic Versus Algebra: English Debates and Boole’s Mediation

The work of George Boole (1805–64) is often considered a first attempt at mathe-matization of logic. It is often opposed to that of Gottlob Frege (1848–1925), who is regarded as the true founder of mathematical logic. This way of thinking may be considered as the appraisal of the twentieth-century philosopher, logician or mathematician who is well informed about historical developments in this field since the last century. It lays great stress on the formalization of logic but tends to neglect what could now be called the semantic question, despite its importance for Boole with the problem of the interpretation of symbols, and for Frege with the problem of representation by which he tries to understand the relation between propositions and reality.

PLANIMETERS AND INTEGRAPHS IN THE 19th CENTURY. BEFORE THE DIFFERENTIAL ANALYSER

ABSTRACT When reading books on the history of computing, we are accustomed to find informations on arithmetical engines, and on the developments of the relation between logic and mathematics at the beginning of the 20th century. From a standpoint such as this, it is difficult to understand what is going on in scientific calculus between Babbages analytical engine and the computers. This view of history simply neglects the importance of analog instruments. Certain of them, such as planimeters and integraphs, designed for measuring surfaces, materialized the theoretical integral calculus, and gave results even when the calculus did not. First invented in the beginning of the 19th century, they were essentially developed after the theoretical legitimization of the polar planimeter by Jakob Amsler (1823-1912) after 1856. This small and practical instrument quickly spread amongst the engineers of European industrial countries. Integraphs drew the curves giving the area for each point on the outline of the surfaces to be measured. They were manufactured as prototypes rather than on a large scale, and were not used to such an extent as planimeters.More specifically in England, planimeters and integraphs gained in importance with the particular implication of some engineers-physicists. And the initial system of a roller rolling on a cone or a disc was integrated into more complex apparatus. Lord Kelvins (1824-1907) harmonic analyser used several systems (disc-sphere-cylinder) to draw different Fourier components of the periodic movement of the tides, and Kelvin devised a manner to build them in order to resolve differential equations. Several decades later, when technical problems were resolved, Douglas R. Hartree (1897-1958) followed Vannevar Bush (1890-1974) devising the differential analyser.

The formalization of logic and the issue of meaning

This paper is not directly concerned with musical logic, because I am unable to raise pertinent issues concerning it. But, as is clear from the introductory paper by Francois Nicolas, some parallelism is not lacking between musical logic and formalised logic, in terms of their ontological foundations and symbolical calculations. Thus, my purpose is to focus on the questions linked to the separation of a blind formal calculus from its possible interpretations, as they engage our present queries about meaning. I shall first present the specific point of view of an historian of science, a very different matter from our present knowledge of logic. Then, I shall define the periods I have chosen to show how these questions were dealt with historically. I shall go on to analyse how the relationship of logical calculus to the question of its meaning was viewed during these different periods.