Given the old conception of the relation greater than, the proposition that the whole is greater than the part is an immediate consequence. But being greater in this sense is not incompatible with being equal in the sense of one-one correspondence. Some who failed to recognize this formulated invalid arguments against the possibility of infinite quantities. Others who did realize that the relations of equal and greater when so defined are compatible, concluded that such relations are not appropriately taken as (...) quantitative relations, at least, not in general. If suitable quantitative relations were to be defined, there was a possibility of retaining either the traditional definition of greater and finding another concept of equality, or of retaining the concept of equality as one-one correspondence and defining greater than so that it is incompatible with equality in this sense. The former alternative was implicitly taken by Bolzano, who never succeeded in defining suitable quantitative relations. The latter alternative was taken by Cantor, and is the basis of his great success in constructing a mathematical theory of the transfinite. (shrink)

How are social and institutional circumstances linked to the knowledge that scientists produce? To answer this question it is necessary to take risks: speculative but testable theories must be proposed. It will be my aim to explain and then apply one such theory. This will enable me to propose an hypothesis about the connexion between social processes and the style and content of mathematical knowledge.

2 Popper's Logical Definition of Verisimilitude. 3 Popper's Probabilistic Definition of Verisimilitude. 4 Conclusion.

RésuméCet exposé vise à replacer l'ensemble de la pensée mathématique dans le cadre biologique et anthropologique d'une théorie de l'évolution. La pensée mathématique s'est développée chez l'homme sous la nécessité de simuler les processus du monde extérieur, simulation plus fine que celle fournie par le langage usuel. Mais elle présente par rapport à l'infrastructure biologique l'innovation radicale de structures arithmétiques et algébriques fondées sur l'itération indéfinie d'une opération. La pensée algébrique simule en la poussant à l'extrěme l'opposition indéterminisme‐déterminisme sous la (...) forme du couple: variable‐opération. (shrink)

Popper welcomes Tarski's theory of truth as a vindication of the ‘objective or absolute or correspondence theory of truth’: -/- Tarski's greatest achievement, and the real significance of his theory for the philosophy of the empirical sciences, is that he rehabilitated the correspondence theory of absolute or objective truth … He vindicated the free use of the intuitive idea of truth as correspondence to the facts ….

Mathematical objects are divided into (1) those which are autonomous, i.e., not dependent for their existence upon mathematicians’ conscious acts, and (2) intentional objects, which are so dependent. Platonist philosophy of mathematics argues that all objects belong to group (1), Brouwer’s intuitionism argues that all belong to group (2). Here we attempt to develop a dualist ontology of mathematics (implicit in the work of, e.g., Hilbert), exploiting the theories of Meinong, Husserl and Ingarden on the relations between autonomous and intentional (...) objects. In particular we develop a phenomenology of mathematical works, which has the stratified intentional structure discovered by Ingarden in his study of the literary work. (shrink)

In recent years Sir Karl Popper has been turning his attention more and more towards philosophical problems arising from biology, particularly evolutionary biology. Popper suggests that perhaps neo-Darwinian evolutionary theory is better categorized as a metaphysical research program than as a scientific theory. In this paper it is argued that Popper can draw his conclusions only because he is abysmally ignorant of the current status of biological thought and that Popper's criticisms of biology are without force and his suggestions for (...) its improvement are without need. Also it is suggested that Popper's desire to see scientific theory growth as being in some sense evolutionary may have led him astray about biology. And conversely it is suggested that since his claims about biology are not well taken his analysis of theory growth may well bear reexamination. (shrink)