Ed. by Lütterfelds, Wilhelm / Majetschak, Stefan / Raatzsch, Richard / Vossenkuhl, Wilhelm
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The Cosmic Fragment: Härte des Logischen Zwangs und Unendliche Möglichkeit
Nachlass discoveries and Wittgenstein’s conception of generality and the infinite
Wittgenstein’s unrelenting criticism of Cantor, Euler, and Gödel falls within a larger strategy to disarm a philosophy of mathematics which relies on completed infinite sets. Because transfinite numbers are seen to resolve the Zeno paradoxes, creating “a paradise from which we shall not be driven”, according to Hilbert, Wittgenstein’s mathematics began to be seen as backwardlooking, particularly in the period of Turing’s work on the Entscheidungsproblem. It is argued that Wittgenstein offered consistent criticism and alternative approaches to paradoxes of the infinitely large and small through a consistent systemic approach to space, time, and generality. The centrality of proof is looked at carefully, with implications for conceptions of time and generality.
Fragment MS 178e which I here, recalling Heraclitus, term the Cosmic Fragment, combined with the correction of a von Wright error, is used to explore these arguments, from both mathematical and Philosophische Untersuchungen (PU) Nachlass exegesis perspectives. The dating of the Fragment impacts on the wider scholarly discussion of the completion of MS 142 and TS 220, and the Fragment’s clustering of concepts is reflected in the significant re-ordering of remarks in the Zwischenfassung, which are carried through to the final version of PU. Affinities between Wittgensteinian mathematics and language games emerge within the textual network of the Cosmic Fragment and the new connections between MS 117 and TS 213.
It is suggested that a proof-theoretic, potentially infinite system such as Wittgenstein’s, as contrasted with a set-theoretic, actual infinite such as Cantor’s and Gödel’s, in certain respects aligns Wittgenstein’s remarks on space, time and generality with Einsteinian general relativity as originally postulated with the cosmological constant. However, Wittgenstein’s conception of time is neither fully Parmenidean nor fully Heraclitean.