> On 11 May 2018, at 23:32, John Clark <[email protected]> wrote: > > On Fri, May 11, 2018 at 12:18 PM, Bruno Marchal <[email protected] > <mailto:[email protected]>> wrote: > > > If you started with the basic axioms of number theory and proved the > Goldbach Conjecture is true, and you were convinced you had not made an error > in the proof, and then the next day a computer found a huge even number that > was NOT the sum of 2 primes, would you: > A) Conclude that there must be something wrong with the basic axioms of set > theory. > Or > B) Conclude that computers can’t be trusted because for some unknown reason > all computers always make an error when making that particular calculation. > If its A then you are tacitly giving the laws of physics the right to > determine truth from falsehood because those laws determine how the machine > operates. If you choose B then madness awaits because your brain also > operates according to those very same laws. > > How so? > > I'm surprised I have to spell this out. > > > In A, no physical assumption is used. Only the axioms of Number Theory. > > A computer needs to know nothing about number theory and it assumes nothing.
I use computer for “universal Turing machine”. That notion assumes (and is Turing-equivalent with (very) elementary arithmetic). > Computers are made of matter that obeys the laws of physics, Physical computer. But I do not assume a primary physical reality, given that I have shown that we have to justify it from arithmetic (or Turing equivalent). > when the voltage on one of the inputs of the microchip is positive physics > orders it will do one thing and when the voltage is negative it will order it > to do something different. By picking A you are in effect saying you have > looked at the pattern of voltages physics told the microchip to have and you > have interpreted that pattern to to be a even number that is not the sum two > prime numbers, and you believe what physics is telling you even if the axioms > of Number Theory says such a number can not exist. I have not assume physics anywhere, so this does not make any sense. > By picking A you are saying physics is more trustworthy than any set of > axioms could be, if there is a contradiction between the two it is the axioms > that need to give way not physical law because althoughphysics can be weird > it has no self contradictions, but man made axioms can. Physical theories can be contradictory. And physical reality is not an assumption available at the start. > > > I guess you know that Gödel’s second incompleteness theorem shows that if > a machine or a theory is consistent, > > A real machine will NEVER operate contrary to the laws of physics, In which theory. You are using implicit metaphysical assumption. You take fro granted physicalism, which beg the question I am addressing. > but a set of axioms will ALWAYS be inconsistent or incomplete or both. If a theory is inconsistent, it is obviously complete. You can’t have both inconsistency and incompleteness. > It could be that the the current axioms of number theory are not strong > enough to prove or disprove Goldbach, so if the laws of physics ever tell us, > by way of a computer, that there is a even number that is not the sum of 2 > primes then it would be wise to add the negation of Goldbach as a new axiom > because physics is the ultimate arbiter about what is true and what is not. If a physical computer (assumed to have no bugs in it) can find an even number not being the sum of two prime, then that number can be find by very elementary arithmetic, and this happens in arithmetic by the embedding of metamathematics in arithmetic (Gödel). Bruno > > John K Clark > > > > > > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected] > <mailto:[email protected]>. > To post to this group, send email to [email protected] > <mailto:[email protected]>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <https://groups.google.com/d/optout>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
