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From Foundations to Ludics

Published online by Cambridge University Press:  15 January 2014

Jean-Yves Girard*
Affiliation:
Institut de Mathématiques de Luminy, UPR 9016–CNRS, 163, Avenue de Luminy, CASE 930, F-13288 Marseille Cedex 09, France.E-mail:girard@iml.univ-mrs.fr

Extract

Ludics [1] is a novel approach to logic—especially proof-theory. The present introduction emphasises foundational issues.

For ages, not a single disturbing idea in the area of “foundations”: the discussion is sort of ossified—as if everything had been said, as if all notions had taken their definite place, in a big cemetery of ideas. One can still refresh the flowers or regild the stone, e.g., prove technicalities, sometimes non-trivial; but the real debate is still: this paper begins with an autopsy, the autopsy of the foundational project.

Up to say 1900, the realist/dualist approach to science was dominant; during the last century some domains like physics evolved so as to become completely anti-realist; but this evolution hardly concerned logic.

By the turn of the XXth century mathematics was jeopardised by paradoxes, the most famous of them being due to Russell. Hilbert's reaction was to focus on consistency. But the reduction of paradoxes—and therefore of foundations—to solely the antinomies is highly questionable: indeed, the typical paradoxical artifacts are secret sharers, objects satisfying the formal definitions but far astray from the intended meaning, typically the Peano “curve” which contradicts our perception of dimension. Fortunately, topology has been able to show that dimension m is not the same as dimension n … but just for a second, forget this and imagine consistent mathematics in which balls in any dimension are homeomorphic: what a disaster! This exclusive focus on consistency—not to speak of the strategic failure of the Programme—should explain why logic, especially foundations lost contact with other sciences during last century.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

[1] Girard, J.-Y., Locus solum, Mathematical Structures in Computer Science, vol. 11 (2001), pp. 301506.Google Scholar