Some Coloured Remarks on the Foundations of Mathematics in the 20th Century

  • Gerhard Heinzmann
Part of the Logic, Epistemology, And The Unity Of Science book series (LEUS, volume 1)


According to the mainstream in the 20th century, the foundations of mathematics were identified with logic and set theory. Indeed, results concerning philosophically most interesting questions are often negative: the first order axiomatic set-theoretical universe is deductively incomplete, inevitably non-standard, and we have no clear idea of what the intended models of set theory are (part I). So, the foundational view of mathematics itself might be suspect. But in the spirit of Poincaré, one should look for an other solution. He remarks that the varieties of classical first order theories is unable to deal with the most common modes of mathematical reasoning such as complete induction and model building. For such a purpose, Hintikka's IF-Logic seems to be an adequate way-out.


Mathematical Reasoning Abstract Entity Intended Model Axiomatic Method Elementary Arithmetic 
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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Gerhard Heinzmann
    • 1
  1. 1.Department of Philosophy, University of Nancy 2Laboratoire de Philosophie et d'Histoire des Sciences – Archives Henri PoincaréFrance

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