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Theoretical Mathematics

On the Philosophical Significance of the Jaffe-Quinn Debate
  • Michael Stöltzner

Abstract

Answering to the double-faced influence of string theory on mathematical practice and rigour, the mathematical physicists Arthur Jaffe and Frank Quinn have contemplated the idea that there exists a ‘theoretical’ mathematics (alongside ‘theoretical’ physics) whose basic structures and results still require independent corroboration by mathematical proof. In this paper, I shall take the Jaffe-Quinn debate mainly as a problem of mathematical ontology and analyse it against the backdrop of two philosophical views that are appreciative towards informal mathematical development and conjectural results: Lakatos’s methodology of proofs and refutations and John von Neumann’s opportunistic reading of Hilbert’s axiomatic method. The comparison of both approaches shows that mitigating Lakatos’s falsificationism makes his insights about mathematical quasi-ontology more relevant to 20th century mathematics in which new structures are introduced by axiomatisation and not necessarily motivated by informal ancestors. The final section discusses the consequences of string theorists’ claim to finality for the theory’s mathematical make-up. I argue that ontological reductionism as advocated by particle physicists and the quest for mathematically deeper axioms do not necessarily lead to identical results.

Key words

Jaffe-Quinn debate rigour in string theory final theories Lakatos’s philosophy of mathematics, John von Neumann axiomatic method theoretical mathematics mathematical ontology 

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Copyright information

© Springer 2005

Authors and Affiliations

  • Michael Stöltzner
    • 1
  1. 1.University of BielefeldBielefeldGermany

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