On Mon, Dec 17, 2018 at 12:03 AM Bruce Kellett <[email protected]> wrote:
> 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". > > I think I worded that badly. What I mean is given that truth does not come from axioms (since they cannot encapsulate all of it), then where does it come from? Does it have an independent, uncaused, transcendent existence? > 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. > I agree, let us ignore "exists" for now as I think it is distracting from the current question of whether "true statements are true" (independent of thinking about them, defining them, uttering them, etc.). What I am curious to know is how how many of these statements you agree with: "2+2 = 4" was true: 1. Before I was born 2. Before humans formalized axioms and found a proof of it 3. Before there were humans 4. Before there was any conscious life in this universe 5. As soon as there were 4 physical things to count 6. Before the big bang / before there were 4 physical things > > 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', > Isn't this what professors do with physics tests? Ask there students to prove something or determine what some physical law says should happen? Then they grade an item as wrong if the answer given was "false" under the working theory. > 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. > > Would you liken consistency with the data to soundness in a system of axioms? > > >> 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. > > 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 https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
