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MovedTo CarlHewitt-StandardIoT
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In a competitive race to the bottom, many Internet companies depend on ever increasing consumer surveillance in order to better target advertising. However, a nation's security depends on limiting surveillance of their citizens by foreign intelligence agencies enabled by companies domiciled in other nations. The presumption is that intelligence agencies have access to all information in datacenters of foreign-domiciled companies.  Consequently, a nation’s security requires that its citizens’ sensitive information not be accessible in datacenters of foreign-domiciled companies. Furthermore, every imported IoT device (cell phone, refrigerator, car, insulin pump, TV, climate-control system, etc.) is going to have to be certified not to have a backdoor available to a foreign intelligence agency. Thus US industry faces the crises that its current IoT business model is about to become illegal.

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Inconsistency robustness is information system performance in the face of continually pervasive inconsistencies---a shift from the previously dominant paradigms of inconsistency denial and inconsistency elimination attempting to sweep them under the rug.

Inconsistency robustness is a both an observed phenomenon and a desired feature:
• Inconsistency Robustness is an observed phenomenon because large information-systems are required to operate in an environment of pervasive inconsistency.
• Inconsistency Robustness is a desired feature because we need to improve the performance of large information system.

This volume has revised versions of refereed articles and panel summaries from the first two International Symposia on Inconsistency Robustness conducted under the auspices of the International Society for Inconsistency Robustness (iRobust http://irobust.org). The articles are broadly based on theory and practice, addressing fundamental issues in inconsistency robustness.

The field of Inconsistency Robustness aims to provide practical rigorous foundations for computer information systems dealing with pervasively inconsistent information.

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See link for retrospective on extremely successful symposium.

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To: Foundations of Mathematics<fom@cs.nyu.edu>

Dear Monroe,
 
In Computer Science, we need very strong foundations for mathematics so that our computer systems are not handicapped.  Consequently, having the Use Theorem rule is highly valuable. For example, the Use Theorem rule is essential to Natural Deduction and is used in many mathematical proofs including the proof of consistency of mathematics.  We would have inconsistency if Gödel’s result held that mathematics cannot prove its own consistency if it is consistent. It is important not to have inconsistencies in mathematical foundations of Computer Science because they represent security vulnerabilities.
 
Of course, since there are uncountably many propositions (e.g. one for each real number), it is not possible to code them using natural numbers.
 
Large cardinals are not fundamental to Computer Science and consequently theories of large cardinals do not belong in the mathematical foundations of Computer Science.  On the other hand, being able to reason about theories is of fundamental importance.  The existence of nonstandard models of first-order Peano axioms (with infinite numbers) represents a severe defect for using these axioms in the mathematical foundations of Computer Science.
 
Cheers,
Carl
 
PS.  I have attached the article with the above results that will be presented at IR’14.
 
From: Monroe Eskew [mailto:meskew@math.uci.edu]
Sent: Monday, June 09, 2014 11:50
To: Carl Hewitt
Subject: Re: [FOM] Panel on "Inconsistency Robustness in Foundations of Mathematics" at IR'14 (http://ir14.org)
 
Dear Carl,
 
I remember perplexing myself with a similar argument during my first quarter in grad school.  The resolution is that the “Use Theorem” rule as you call it is not valid in the kinds of systems to which Godel’s theorem applies.  In a first-order system, we refer to propositions indirectly by coding them as natural numbers.  The faithfulness of the coding to its intended meaning may fail in nonstandard models.  This is of course a meta-statement and cannot be expressed within the first-order system.  Godel’s argument generates many concrete examples of statements P such that the statement, “There is x coding a proof of #P” does not imply P modulo the axioms.  
 
If \Phi(x) is the formula for there is no proof of sentence x in PA, and \sigma is a sentence is given by the fixed point lemma, so that PA proves "\sigma iff \Phi(#\sigma)", then PA does not prove “If #\sigma is provable then \sigma.”
 
If you think PA is a bad example because it is weak, note that Godel’s theorem applies to very strong systems such as ZFC + large cardinals.  You can do a lot of math there, some would say all.
 
Monroe

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To: Foundations of Mathematics<fom@cs.nyu.edu>

Dear Marcin,
 
Thanks for your remark and your question.
 
I agree that Gödel’s proof has been very well checked for correct reasoning.  However, things could still go wrong:
      1)  Gödel’s proof makes assumptions that are undesirable for Computer Science.  For example, the assumption that propositions are countable is inconsistent with there being a distinct proposition for each real number such that the proposition holds for only that number.
       2) Gödel’s proof makes assumptions that result in contradiction. For example, the proof assumes the existence of self-referential sentences constructed using fixed points on an untyped grammar for mathematical sentences. Simply checking the proof for correctness will not reveal a contradiction in the assumptions.
       3) There is a simple proof of the consistency of mathematics which contradicts the thrust of Gödel’s original result that powerful systems like Principia Mathematica cannot prove their own consistency.
 
The IR'14 conference paper maintains all three of the above do in fact hold.
 
With respect to your question, see page 32 of the following:
              Morris Kline. Mathematical Thought from Ancient to Modern Times Oxford University Press, 1990.
 
Cheers,
Carl
 
-----Original Message---
From: fom-bounces@cs.nyu.edu[mailto:fom-bounces@cs.nyu.edu] On Behalf Of Marcin Mostowski
Sent: Sunday, June 08, 2014 15:55
To: Foundations of Mathematics
Subject: Re: [FOM] Panel on "Inconsistency Robustness in Foundations of Mathematics" at IR'14 (http://ir14.org)
 
Dear all,
 
I have two things related to the message by Carl Hewitt, one remark and a question.
 
Remark:
 
You write: "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?"
 
You have two isomorphic reasonings. One about numbers and one rather logical. The first one was very well checked and we have very good reasons to accept it as a correct reasoning. The claim that the second one is incorrect seems to be a simple mistake. I do not see any use of discussing the issue.
 
Question:
 
You write: "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."
 
Have you any good historical reference to the story? I agree that it was one of crucial points in our intellectual history.
 
Marcin Mostowski
 
 
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