On Mon, Dec 17, 2018 at 5:59 PM Jason Resch <[email protected]> wrote:
> 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? > I don't know what that would mean. I don't think the truth of arithmetical statements comes from some underlying consistent model in which the axioms are "true". How do you determine the truth of the Godel sentence in some axiomatic system? Only by going to some more general system, not by reference to some underlying model. Do you agree that arithmetical truth has an existence independent of the >>> axiomatic system? >>> >> Since truth does not equal 'theorem of the system', there is a sense in which this is true. But it does not mean that the truth of any syntactically correct statement is independent of any axiom set. > >> 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.). > True statements are true by definition! > 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 > "2+2=4" is a tautology, true because of the meanings of the terms involved. So its truth is not independent of the formulation of the question and the definition of the terms involved. 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.
