Epistemology versus Ontology: Essays on the Philosophy and by Peter Dybjer, Sten Lindström, Erik Palmgren, Göran Sundholm

By Peter Dybjer, Sten Lindström, Erik Palmgren, Göran Sundholm (eds.)

This booklet brings jointly philosophers, mathematicians and logicians to penetrate vital difficulties within the philosophy and foundations of arithmetic. In philosophy, one has been focused on the competition among constructivism and classical arithmetic and the several ontological and epistemological perspectives which are mirrored during this competition. The dominant foundational framework for present arithmetic is classical good judgment and set concept with the axiom of selection (ZFC). This framework is, notwithstanding, weighted down with philosophical problems. One vital substitute foundational programme that's actively pursued this present day is predicativistic constructivism in response to Martin-Löf sort conception. linked philosophical foundations are which means theories within the culture of Wittgenstein, Dummett, Prawitz and Martin-Löf. what's the relation among proof-theoretical semantics within the culture of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or different bills of meaning-as-use? What can proof-theoretical analyses let us know concerning the scope and boundaries of positive and predicative mathematics?

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Extra resources for Epistemology versus Ontology: Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf

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34 For example, the concept of the number 5 is instantiated in an image of 5 dots. Moreover, Kant says, when we think of a number (be it small or large) we are not so much thinking of such an image, as of a rule for producing images Mathematical certainty is also called evidence, as intuitive knowledge is clearer than discursive knowledge. e. evidence, however much the judgement may otherwise be apodictically certain. i. ’) [A734/B762] But it is not a term that Kant actually uses often. 32 ‘Einige wenige Grundsätze, welche die Geometer voraussetzen, sind zwar wirklich analytisch und beruhen auf del Satze des Widerspruchs; .

That general qualification is absent from his later work; in the light of the considerations in the present paper, that seems, conceptually if not historically as well, to be no coincidence. 24 That Brouwer here describes a sequence of nested intervals, and not of rationals, is not essential to the question at hand. 18 8 M. van Atten We call such an indefinitely proceedable sequence of nested . . intervals a point P or a real number P . We must stress that for us the sequence . . itself is the point P .

Hrsg. von der Königlich-Preussischen Akademie der Wissenschaften zu Berlin, 1902–. Kritik de reinen Vernunft. Second edition. Hartknoch, Riga, 1787. Edition used: W. ), Suhrkamp, Frankfurt, 1974. English translations of AA are my own; those of A and B are taken from N. Kemp Smith’s translation Immanuel Kant’s Critique of Pure Reason, St. Martin’s Press, New York, 1965. van Atten, M. 2007. Brouwer meets Husserl. On the phenomenology of choice sequences. Dordrecht: Springer. Beiser, F. 2008.

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