Peano's contributions to logic are surveyed under several headings. His use of class abstraction is considered first, together with his recognition of the distinction between membership and inclusion. Then his strategy of notational inversion is appraised. Finally, class abstraction is considered again, from ontological points of view; and Peano's achievements are compared with Frege's.

Wilfred Sieg. Relative Consistency and Accesible Domains.

First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic. The last quarter of the 19th century witnessed a remarkable growth of interest in the foundations of arithmetic. This work analyses both the reasons for this growth of interest within both mathematics and philosophy and the ways in which this study of the foundations of arithmetic led to new insights in philosophy and striking advances in logic. This (...) historical-critical study provides an excellent introduction to the problems of the philosophy of mathematics - problems which have wide implications for philosophy as a whole. This reissue will appeal to students of both mathematics and philosophy who wish to improve their knowledge of logic. (shrink)

The object of this article is the study of possibilities and tendencies arising from the use of symbolic language including signs, characters, and symbols in mathematics. Five aspects are discussed: compactness and simultaneity, problem‐solving and generalizations, heuristics and progress, mechanisms and calculations, formalism. This is done primarily by looking at three disciplines, which at the same time are of fundamental importance to theoretical physics: classical algebra, calculus, and vector analysis.Mathematical achievements and statements by eminent mathematicians from antiquity to the present (...) (Archimedes, Leibniz, Newton, Euler, Gauß, Hilbert) show the importance of adequate symbolism for the development of mathematics. The corresponding philosophical discussions (Hobbes, Condillac, Degérando, Apelt, Cassirer, among others) are controversial. (shrink)

Because Bertrand Russell adopted much of the logical symbolism of Peano, because Russell always had a high regard for the great Italian mathematician, and because Russell held the logicist thesis so strongly, many English-speaking mathematicians have been led to classify Peano as a logicist, or at least as a forerunner of the logicist school. An attempt is made here to deny this by showing that Peano's primary interest was in axiomatics, that he never used the mathematical logic developed by him (...) for the reduction of mathematical concepts to logical concepts, and that, instead, he denied the validity of such a reduction. (shrink)

Positivists identify science and certainty and in the name of the utter rationality of science deny that it rests on speculative presuppositions. The Logical Positivists took a step further and tried to show such presuppositions really no presuppositions at all but rather poorly worded sentences. Rules of sentence formation, however, rest on the presuppositions about the nature of language. This makes us unable to determine the status of mathematics, which is these days particularly irksome since this question is now-since Abraham (...) Robinson-one that mathematicians cannot ignore. Since mathematics is the paradigm of a logical discourse, logic must offer a system adequate enough to serve mathematics. This fact makes it difficult to avoid making question-begging moves in both mathematics and logic. We must therefore view the rationality of logic as partial and hope it is stepwise improvable. The theory of rationality thus turns to be the major presupposition of logic, and one which has ample metaphysical background to it. The very supposition, basic to all logic, that language is divisible into form andcontent is under suspicion-mathematics perhaps belongs to neither. (shrink)