On Tue, Jan 21, 2014 at 3:30 PM, meekerdb <[email protected]> wrote:
> On 1/21/2014 8:13 AM, Jason Resch wrote: > > Why would you want to do that? It seems like an unnecessary extra axiom > that doesn't have any purpose or utility. > > > It prevents the paradoxes of undeciability, Cantor diagonalization, and it > corresponds more directly with how we actually use arithmetic. > > > I'm not sure it helps. What you may gain from avoiding paradoxes makes > many of our accepted proofs false. E.g. Euclids proof of infinite primes. > Or Euler's identity. Most of math would be ruined. A circle's circumference > would not even be pi*diameter. > > Would this biggest number be different for different beings in different > universes? What is it contingent on? > > > You're taking an Platonic view that there really is an arithmetic and > whether there's a biggest number is an empirical question. I'm saying it's > an invention. We invented an system in which you can always add 1 because > that was convenient; you don't have to think about whether you can or not. > So to use this same line of reasoning, would you say there is no definite (a priori) fact of the matter of whether or not a given program terminates, unless we actually build a machine executing that program and observe it terminate? If that is the case, when is it determined (for us) that a certain program terminates? Is it when the first being anywhere in any universe tests it, when someone in our universe tests it, when someone in our past light cone tests it, when you test it yourself or read about someone who did? Would it ever be possible for two beings in two different universes to find different results regarding the same program? If not, then what enforces this agreement? > But if it leads to paradoxes or absurdities we should just modify our > invention keeping the good part and avoiding the paradoxes if we can. > Peano's arithmetic will still be there in Platonia and sqrt(2) will be > irrational there. But the diagonal of a unit square may depend on how we > measure it or what it's made of. > Does this instrumentalist approach prevents one from having a theory of reality? 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 post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
