[FOM] Solution (?) to Mathematical Certainty Problem
Harvey Friedman
friedman at math.ohio-state.edu
Sun Jun 22 14:20:07 EDT 2003
Reply to Lindauer 1:06PM 6/21/03.
>How does this avoid the possibility that the underlying hardware has
>a systematic but humanly undetectable flaw? Your system is good
>for those mathematical problems which people can see are correctly
>solved (e.g. "Looks right to me."). What about ones with large
>numbers or minute fractions where our intuitions tend to become
>muddled?
>
>For instance, is:
>
>123890164571029365109343456822478103948560192834750160594382741092845610
>923874109348564127893561024938750162938740213964501928734016503945817205
>961029387401928465019284730196509817230498162054938712043986102983570194
>283605981720398461095879042387501285630981712734891026509382741092856192
>837401928560192873410986598475192837409182605349857109238456019283741757
>283495621039874016501923874106539487120945610923874016520394871034650192
>834750165049871045610294857023946850198273409283645091823745092836405958
>712094651412845609387410926519283793685198237401938465067586405986985697
>459834659182734011001010109183274509236450179231010192939292929383883274
>929562973415610274938561029748374938759825691720394810365019283740165027
>493845682739481056029837492794723848472956105017514374589230645598127431
>752893468573940298364359827340659103847502638459713049623984752093861098
>273457171717172172717717727371727173727712171113123890164571029365109343
>456822478103948560192834750160594382741092845610923874109348564127893561
>024938750162938740213964501928734016503945817205961029387401928465019284
>730196509817230498162054938712043986102983570194283605981720398461095879
>042387501285630981712734891026509382741092856192837401928560192873410986
>598475192837409182605349857109238456019283741757283495621039874016501923
>874106539487120945610923874016520394871034650192834750165049871045610294
>857023946850198273409283645091823745092836405958712094651412845609387410
>926519283793685198237401938465067586405986985697459834659182734011001010
>109183274509236450179231010192939292929383883274929562973415610274938561
>029748374938759825691720394810365019283740165027493845682739481056029837
>492794723848472956105017514374589230645598127431752893468573940298364359
>827340659103847502638459713049623984752093861098273457171717172172717717
>727371727173727712171113
>
>Prime or not? If not, what is the next highest prime?
>
>We can imagine a long text file, a log of all the prime numbers less
>than THAT and the proofs that they are the prime numbers, and then
>the next prime number, listed, and if it is identical with THAT, we
>MIGHT be able to check it by looking at it (I for one can't
>memorize numbers more than 10 digits reliably... being a little
>dumb). So in order to verify that THAT number is not in the list,
>we'd have to use a computer, but that computer would be subject to
>the same problem, that we couldn't verify that it was talking about
>THE SAME NUMBER.
>
>And even if some savant COULD memorize such numbers, how would we
>verify their magical ability to memorize numbers and so on?
>
>The point is simple, what improvement over the ordinary "looks like
>a proof to me" is your method?
>
There seems to be no problem. See the postings of Jones 7:31AM
6/21/03, Jones 5:36PM 6/21/03, Wiedjk, 8:17PM 6/21/03, Vestergaard
11:07AM 6/22/04.
Intuition plays no role in this. Perhaps you doubt if any
hardware/software system, no matter how low level, can be truly
verified?
For example, I want to build a system that can tell whether or not a
given input string of length 1 billion is a palindrome. I.e., is the
same as its reverse. And I want to verify that system. Custom built
hardware. You doubt if this can be done?
I wouldn't trust unaided humans to tell me whether or not a given
input string of length 1 billion is a palindrome.
The idea is to reduce proof checking to things like this.
Harvey Friedman
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