> On 17 Dec 2018, at 19:26, Brent Meeker <[email protected]> wrote: > > > > On 12/16/2018 10:46 PM, Jason Resch wrote: >> >> >> On Mon, Dec 17, 2018 at 12:00 AM Brent Meeker <[email protected] >> <mailto:[email protected]>> wrote: >> >> >> On 12/16/2018 9:30 PM, Jason Resch wrote: >>> >>> >>> On Sun, Dec 16, 2018 at 9:39 PM Bruce Kellett <[email protected] >>> <mailto:[email protected]>> wrote: >>> On Mon, Dec 17, 2018 at 1:50 PM Jason Resch <[email protected] >>> <mailto:[email protected]>> wrote: >>> On Sun, Dec 16, 2018 at 7:21 PM Bruce Kellett <[email protected] >>> <mailto:[email protected]>> 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.
No, they are not. All theories containing a bit of arithmetic have an infinity of non isomorphic models/realities. > 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. The mathematical truth is independent of language and theories. It is like physics, except that it asks for searching inward, and then testing ideas and theory by dialog with other people. Invoking a physical reality as real, is like invoking god as real. That just explains nothing, even if in some case it works FAPP. > >> 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. We choose some axioms, because we intuit some truth. The axioms only scratch the surface of that truth. The idea that the axioms defines our object of study does not make sense for theories rich enough to define universal machine. The reality behind is too big. Bruno > > Brent > >> >> If they don't, then formalism, nominalism, fictionalism, etc. all fall, and >> what is left is platonism. >> >> 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] >> <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] > <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.
