On Mon, Dec 17, 2018 at 4:30 PM Jason Resch <[email protected]> wrote:
> On Sun, Dec 16, 2018 at 9:39 PM Bruce Kellett <[email protected]> > wrote: > >> On Mon, Dec 17, 2018 at 1:50 PM Jason Resch <[email protected]> wrote: >> >>> On Sun, Dec 16, 2018 at 7:21 PM Bruce Kellett <[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? > You are equivocating on the notion of "truth". You seem to be claiming that "truth" is encapsulated in the axioms, and yet the axioms and the given rules of inference do not encapsulate all "truth". Do you agree that arithmetical truth has an existence independent of the > axiomatic system? > I agree that there are true statements in arithmetic that are not theorems in any particular axiomatic system. This does not mean that arithmetic has an existence beyond its definition in terms of some set of axioms. You cannot go from "true" to "exists", where "exists" means something more than the existential quantifier over some set. Confusing the existential quantifier with an ontology is a common mistake among some classes of mathematicians. There are syntactically correct statements in the system that are not >> theorems, and neither are their negation theorems. >> > > Yes. > > >> Godel's theorem merely shows that some of these statements may be true in >> a more general system. >> > > So isn't this like scientific theories attempting to better describe the > physical world, with ever more general and more powerful theories? > Except that physics is not an axiomatic system, and does not confuse theorems with truth. It is not useful to classify physical theories as 'true' or 'false', even though this is often done in mistaken homage to Popper. The descriptions of the phenomena that physical theories give are either consistent with the data or not -- even adequate descriptions are not necessarily "true" in any sense. > That does not mean that the integers are not completely defined by some >> simple axioms. It means no more than that 'truth' and 'theorem' are not >> synonyms. >> >> > I agree with this. > Good. Bruce -- 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.
