On Wed, Aug 28, 2019, 11:36 AM John Clark <[email protected]> wrote:
> On Tue, Aug 27, 2019 at 7:45 PM Jason Resch <[email protected]> wrote: > > *> You can write a program that outputs the string "2 + 2 = 5", but you'll >> never find a program that outputs a proof of 2 + 2 = 5 in any consistent >> and sound system of axioms.* >> > > Even if your system is consistent there is no way you can prove its > consistent while remaining within the system, and if you go outside the > system to prove it then you've just kicked the problem upstairs and you > can't prove the meta system is consistent. And soundness means any formula > that you can derive from axioms, that is to say prove, is true. So if you've > got a error free proof that 2+2=4 how do you know it's true, how do you > know 2+2 isn't 5? You're going to need a independent method of > determining the truth of that and there is only one way to do it, with > physics. > You put 2 hydrogen atoms on a scale and note it reads about 2, you put 2 > more on and it reads about 4. You never get exact integers but physics > tells us that 2+2=4 is a good approximation of the truth, so we make sure > our axioms and rules of inference produce that. > > From day one when we started to construct our mathematics we've tried to > make it consistent with physics but there have been a few bumps in the > road. It turns out that is some places (where spacetime is flat) Euclid's > fifth axiom is true but in other places (where spacetime is curved) it's > not true. And both the Axiom Of Choice and the Continuum Axiom involve > infinity and physics has no use for infinity so physics doesn't care if > those axioms are true or not, so there is no way to independently determine > their truth, so stuff based on them are the equivalent to mathematical > Harry Potter stories. > Is this to say you agree? If you disagree with anything I said that missed it. It is true we never access truth, but the same dilemma haunts us in physics. Physics never tells us our theories are true, only that they haven't been refuted so far. That's how the bgs work with building axioms. We can hope they approximate the truth, and we hope to find better axioms in the future. Have I changed your mind regarding what Minsky said regarding possible programs? Regarding axiom of choice, I believe it's independent of ZFC, but that doesn't imply it's independent of another more powerful system. Jason -- 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 view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/CA%2BBCJUjdjRHS2w9VFywCAE%2BAwT3HShm52zXTMA3%2BH4Recg%2Brxg%40mail.gmail.com.
