[FOM] Panel on "Inconsistency Robustness in Foundations of Mathematics" at IR'14 (http://ir14.org)
Carl Hewitt
hewitt at concurrency.biz
Sat Jun 7 13:04:05 EDT 2014
Panel on "Inconsistency Robustness in Foundations of Mathematics" at IR'14 (http://ir14.org)
This panel will discuss current issues in the history and practice of avoiding and repairing inconsistency in powerful mathematical systems grounded in an ongoing saga including the following:
* Perhaps the first foundational crises was due to Hippasus "for having produced an element in the universe which denied the...doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios." Legend has it because he wouldn't recant, Hippasus was literally thrown overboard to drown by his fellow Pythagoreans.
* Frege expressed despair that his life work had been in vain when he received Russell's letter with its revelation of the paradoxical set of all sets that are not members of themselves.
* Fearing that he was dying and the influence that Brouwer might have after his death, Hilbert [1928] fired Brouwer as an associate editor of Mathematische Annalen because of "incompatibility of our views on fundamental matters" (i.e. Hilbert ridiculed Brouwer for challenging the validity of the Principle of Excluded Middle).
* According to Church [1934]: "in the case of any system of symbolic logic, the set of all provable theorems is [computationally] enumerable... any system of symbolic logic not hopelessly inadequate ... would contain the formal theorem that this same system ... was either insufficient [theorems are not computationally enumerable] or over-sufficient [that theorems are computationally enumerable means that the system is inconsistent]...This, of course, is a deplorable state of affairs... Indeed, if there is no formalization of logic as a whole, then there is no exact description of what logic is, for it in the very nature of an exact description that it implies a formalization. And if there no exact description of logic, then there is no sound basis for supposing that there is such a thing as logic." What is the way out of this fundamental paradox?
* Dana Scott [1967] claimed that "there is only one satisfactory way of avoiding the paradoxes: namely, the use of some form of the theory of types." But exactly which theory of types should be used? Russell's ramified theory of types is generally regarded to be a failure.
* A paper presented at this conference challenges the validity of one of the most famous results of mathematical logic: namely, Gödel's result that mathematics cannot prove its own consistency. Currently there is an overwhelming consensus among professional working logicians that Gödel proved that mathematics cannot prove its own consistency if mathematics is consistent. But could they be wrong? Was Wittgenstein after all correct that Gödel's proof is erroneous because inconsistency results from allowing self-referential sentences constructed using fixed points for an untyped grammar of mathematical sentences?
In each of the above cases, means were devised to avoid known inconsistencies while increasing mathematical power. What lessons can be drawn and how should they affect practice?
Join us in an exciting discussion.
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