On Tue, May 14, 2019 at 3:47 PM 'Brent Meeker' via Everything List < [email protected]> wrote:
> > > On 5/14/2019 9:10 AM, Jason Resch wrote: > > > > On Mon, May 13, 2019 at 4:46 PM 'Brent Meeker' via Everything List < > [email protected]> wrote: > >> >> >> On 5/13/2019 8:50 AM, Jason Resch wrote: >> > But then what is arithmetical truth? We have no label for it. It >> > cannot be derived from or defined by labels. >> >> And it depends on the model. > > > Saying truth depends on the model is like saying facts about something > depend on what you are talking about. > When I said arithmetical truth, it should be clear the model is > arithmetic, and so arithmetical truth are the facts concerning arithmetic. > > > But which arithmetic? There is more than one model of Peano's axioms for > example. But , you say, I mean the natural numbers model of > arithmetic...but the natural numbers are something hypothesized from > empirical observation. > I see a 100% analagous situation to the natural sciences: "There's more than one model of gravitation for example. But, you say I mean the gravitation of our universe...but gravitation is something hypothesized from empirical observation." Axiomatic systems are just like theories in the sciences. They attempted to systematize what is out there. But we can never be sure our models correctly reflect the reality. We can only hope to improve our models over time to become more powerful in what they can explain. > > > >> Which is why it's undefinable within the >> system. > > > Could you clarify this point? > > > There is more than one model of PA and "true" is relative to the model. > I think you mean "provable" is relative to the model. In Newton's gravity you could "prove" something about the expected orbital velocity of Mercury in that model. It just wouldn't be true when compared to reality. > > >> And also why it's not the same as the "true" in "It is true >> that snow is white." >> >> > How is it different? > > > Snow is defined ostensively, as are the natural numbers. > Do we agree that the true properties of the natural numbers are objective? If so no need to debate this any further. Jason > But what mathematicians (like Goedel) prove theorems about is the > axiomatic system. That's why Bruno makes the point that provability is > well defined but truth isn't (in mathematics). > > Brent > > > 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%2BBCJUgQZz3nO%2BOKaiWZrtmbVivC8E_0BtwfjhW7hm9PjRoZ_Q%40mail.gmail.com > <https://groups.google.com/d/msgid/everything-list/CA%2BBCJUgQZz3nO%2BOKaiWZrtmbVivC8E_0BtwfjhW7hm9PjRoZ_Q%40mail.gmail.com?utm_medium=email&utm_source=footer> > . > > > -- > 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/c8ccb7d1-0417-4718-aae2-484716798892%40verizon.net > <https://groups.google.com/d/msgid/everything-list/c8ccb7d1-0417-4718-aae2-484716798892%40verizon.net?utm_medium=email&utm_source=footer> > . > -- 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%2BBCJUhqBMhAbbO9z4225FsqOUpSqTvK2JvQ7rbYenHAKU5aQA%40mail.gmail.com.
