On Mon, Dec 17, 2018 at 12:26 PM Brent Meeker <meeke...@verizon.net> wrote:
> > > On 12/16/2018 10:46 PM, Jason Resch wrote: > > > > On Mon, Dec 17, 2018 at 12:00 AM Brent Meeker <meeke...@verizon.net> > wrote: > >> >> >> On 12/16/2018 9:30 PM, Jason Resch wrote: >> >> >> >> On Sun, Dec 16, 2018 at 9:39 PM Bruce Kellett <bhkellet...@gmail.com> >> wrote: >> >>> On Mon, Dec 17, 2018 at 1:50 PM Jason Resch <jasonre...@gmail.com> >>> wrote: >>> >>>> On Sun, Dec 16, 2018 at 7:21 PM Bruce Kellett <bhkellet...@gmail.com> >>>> wrote: >>>> >>>> >>>>> Are you claiming that there is an objective arithmetical realm that is >>>>> independent of any set of axioms? >>>>> >>>> >>>> Yes. This is partly why Gödel's result was so shocking, and so >>>> important. >>>> >>>> >>>>> And our axiomatisations are attempts to provide a theory of this >>>>> realm? In which case any particular set of axioms might not be true of >>>>> "real" mathematics? >>>>> >>>> >>>> It will be either incomplete or inconsistent. >>>> >>>> >>>> >>>>> Sorry, but that is silly. The realm of integers is completely defined >>>>> by a set of simple axioms -- there is no arithmetic "reality" beyond this. >>>>> >>>>> >>>> The integers can be defined, but no axiomatic system can prove >>>> everything that happens to be true about them. This fact is not commonly >>>> known and appreciated outside of some esoteric branches of mathematics, but >>>> it is the case. >>>> >>> >>> All that this means is that theorems do not encapsulate all "truth". >>> >> >> Where does truth come from, if not the formalism of the axioms? Do you >> agree that arithmetical truth has an existence independent of the axiomatic >> system? >> >> >> No. You are assuming that arithmetic exists apart from axioms that >> define it. >> > > I am saying truth about the integers exists independently of any system of > axioms that are capable of defining the integers. > > >> There are true things about arithmetic that are not provable *within >> arithmetic*. >> > > It's unclear what you mean by "within arithmetic". > > >> But that is not the same as being independent of the axioms. Some axioms >> are necessary to define what is meant by arithmetic. >> > > You need to define what you're talking about before you can talk about > it. > > > But mathematical objects are completely defined by their axioms. > Are they? Two is a mathematical object. One of the properties of two is the number of primes it separates. For example "3 and 5", "5 and 7", etc. If mathematical objects are completely defined by their axioms, then shouldn't this property be defined and known for two? Yet we don't even know the answer to this question, we don't know if it is infinite or finite. It might even be that no proof exists under the axioms we currently use. > There is no possibility of ostensive or empirical definition. That's the > strength of mathematics; it's "truths" are independent of reality, they are > part of language. > > But in any case, the axioms don't define arithmetical truth, which is my > only point. > > > No, but they define arithmetic, without which "arithmetical truth" would > be meaningless. > Was the physical universe meaningless before Newton? 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 everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
