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Bulletin of the American Mathematical Society

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Making proofs without Modus Ponens: An introduction to the combinatorics and complexity of cut elimination
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by A. Carbone and S. Semmes PDF
Bull. Amer. Math. Soc. 34 (1997), 131-159 Request permission


Modus Ponens says that if you know $A$ and you know that $A$ implies $B$, then you know $B$. This is a basic rule that we take for granted and use repeatedly, but there is a gem of a theorem in logic by Gentzen to the effect that it is not needed in some logical systems. It is fun to say, “You can make proofs without lemmas” to mathematicians and watch how they react, but our true intention here is to let go of logic as a reflection of reasoning and move towards combinatorial aspects. Proofs contain basic problems of algorithmic complexity within their framework, and there is strong geometric and dynamical flavor inside them.
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Additional Information
  • A. Carbone
  • Affiliation: IHES, 35 Route de Chartres, 91440 Bures-sur-Yvette, France
  • Address at time of publication: Mathématiques/Informatique, Université de Paris XII, 94010 Creteil Cedex, France
  • Email:
  • S. Semmes
  • Affiliation: Department of Mathematics, Rice University, Houston, Texas 77251
  • Email:
  • Received by editor(s): July 3, 1996
  • Received by editor(s) in revised form: December 1, 1996
  • Additional Notes: The first author was supported by the Lise-Meitner Stipendium # M00187-MAT (Austrian FWF) and the second author was supported by the U.S. National Science Foundation. Both authors are grateful to IHES for its hospitality.
  • © Copyright 1997 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 34 (1997), 131-159
  • MSC (1991): Primary 03F05; Secondary 68Q15
  • DOI:
  • MathSciNet review: 1423203