[FOM] Panel on "Inconsistency Robustness in Foundations of Mathematics" at IR'14 (http://ir14.org)

Harry Deutsch hdeutsch at ilstu.edu
Mon Jun 9 08:55:06 EDT 2014

On 6/7/14 12:04 PM, Carl Hewitt wrote:
Do you have the reference for the quotation from Church?  Thanks, Harry 

> 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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