Jean Van Bendegem

Paraconsistency And Dialogue Logic Critical Examination And Further Explorations

The first part of this paper presents asympathetic and critical examination of the approachof Shahid Rahman and Walter Carnielli, as presented intheir paper “The Dialogical Approach toParaconsistency”. In the second part, possibleextensions are presented and evaluated: (a) top-downanalysis of a dialogue situation versus bottom-up, (b)the specific role of ambiguities and how to deal withthem, and (c) the problem of common knowledge andbackground knowledge in dialogues. In the third part,I claim that dialogue logic is the best-suitedinstrument to analyse paradoxes of the Sorites type.All these considerations lead to philosophicallyrelevant observations concerning principles of charityon the one hand, and compactness on the other.The first part of this paper presents asympathetic and critical examination of the approachof Shahid Rahman and Walter Carnielli, as presented intheir paper “The Dialogical Approach toParaconsistency”. In the second part, possibleextensions are presented and evaluated: (a) top-downanalysis of a dialogue situation versus bottom-up, (b)the specific role of ambiguities and how to deal withthem, and (c) the problem of common knowledge andbackground knowledge in dialogues. In the third part,I claim that dialogue logic is the best-suitedinstrument to analyse paradoxes of the Sorites type.All these considerations lead to philosophicallyrelevant observations concerning principles of charityon the one hand, and compactness on the other.

Classical arithmetic is quite unnatural

It is a generally accepted idea that strict finitism is a rather marginal view within the community of philosophers of mathematics. If one therefore wants to defend such a position (as the present author does), then it is useful to search for as many different arguments as possible in support of strict finitism. Sometimes, as will be the case in this paper, the argument consists of, what one might call, a “rearrangement” of known materials. The novelty lies precisely in the rearrangement, hence on the formal-axiomatic level most of the results presented here are not new. In fact, the basic results arexa0inspired by and based on Mycielski (1981).

Alternative Mathematics: The Vague Way

Is alternative mathematics possible? More specifically,is it possible to imagine that mathematics could havedeveloped in any other than the actual direction? Theanswer defended in this paper is yes, and the proofconsists of a direct demonstration. An alternativemathematics that uses vague concepts and predicatesis outlined, leading up to theorems such as ``Smallnumbers have few prime factors.Is alternative mathematics possible? More specifically,is it possible to imagine that mathematics could havedeveloped in any other than the actual direction? Theanswer defended in this paper is yes, and the proofconsists of a direct demonstration. An alternativemathematics that uses vague concepts and predicatesis outlined, leading up to theorems such as ``Smallnumbers have few prime factors.

Inconsistencies in the History of Mathematics

In this paper I will not confine myself exclusively to historical considerations. Both philosophical and technical matters will be raised, all with the purpose of trying to understand (better) what Newton, Leibniz and the many precursors (might have) meant when they talked about infinitesimals. The technical part will consist of an analysis why apparently infinitesimals have resisted so well to be formally expressed. The philosophical part, actually the most important part of this paper, concerns a discussion that has been going on for some decennia now. After the Kuhnian revolution in philosophy of science, notwithstanding Kuhn’s own suggestion that mathematics is something quite special, the question was nevertheless asked how mathematics develops. Are there revolutions in mathematics? If so, what do we have to think of? If not, why do they not occur? Is mathematics the so often claimed totally free creation of the human spirit? As usual, there is a continuum of positions, but let me sketch briefly the two extremes: the completists (as I call them) on the one hand, and the contingents (as I call them as well) on the other hand.