Mathematics has a reputation for objectivity. But without real-world infinite objects upon which to base abstractions, mathematical truth becomes, to some extent, a matter of opinion—which is Simpson’s argument for keeping actual infinity out of mathematics altogether. The choice between V=ultimate L and Martin’s maximum is perhaps less of a true-false problem and more like asking which is lovelier, an English garden or a forest? ... However, the field of mathematics is known for its unity and cohesion. Just as ZFC came to dominate alternative foundational frameworks in the early 20th century, firmly embedding actual infinity in mathematical thinking and practice, it is likely that only one new axiom to decide the fuller nature of infinity will survive. According to Koellner, “one side is going to have to be wrong.” scientificamerican/article.cfm?id=infinity-logic-law&print=true
Posted on: Wed, 04 Dec 2013 20:16:55 +0000
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