On Mon, Dec 17, 2018 at 1:50 AM Bruce Kellett <[email protected]> wrote:
> 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. > Whether or not we can or determine it is irrelevant. The point is it is out there, waiting to be determined, like the 10^100th bit of Pi is either definitely "0" or definitely "1"; we just don't know which yet. > > > 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. > > So where does that put arithmetical truth in relation to the 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! > > But the question I am asking is "What are they dependent on (if anything)? > > >> 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. > > So would you say it was false before it was asked and the terms defined? Was the 10^100th bit of Pi set only at such time that Pi was defined, or did it have a set value before humans defined Pi? 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.
