On Monday, December 17, 2018 at 2:06:41 AM UTC-6, Jason wrote: > > > > On Mon, Dec 17, 2018 at 1:50 AM Bruce Kellett <[email protected] > <javascript:>> wrote: > >> On Mon, Dec 17, 2018 at 5:59 PM Jason Resch <[email protected] >> <javascript:>> wrote: >> >>> On Mon, Dec 17, 2018 at 12:03 AM Bruce Kellett <[email protected] >>> <javascript:>> wrote: >>> >>>> On Mon, Dec 17, 2018 at 4:30 PM Jason Resch <[email protected] >>>> <javascript:>> wrote: >>>> >>>>> On Sun, Dec 16, 2018 at 9:39 PM Bruce Kellett <[email protected] >>>>> <javascript:>> wrote: >>>>> >>>>>> On Mon, Dec 17, 2018 at 1:50 PM Jason Resch <[email protected] >>>>>> <javascript:>> wrote: >>>>>> >>>>>>> On Sun, Dec 16, 2018 at 7:21 PM Bruce Kellett <[email protected] >>>>>>> <javascript:>> 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 >
Suppose one has a spigot* program for π, but you know that to compute the 10^10000th (some really big number) bit would take 10 billion years (longer than this universe may be around). True or false?: That bit has a definite value. There is no answer to that question that people will agree on, hence the pragmatist's dislike of the "truth" thing. * [ https://en.wikipedia.org/wiki/Spigot_algorithm ]: "A spigot algorithm is an algorithm for computing the value of a mathematical constant such as π or e which generates output digits in some base (usually 2 or a power of 2) from left to right, with limited intermediate storage." - pt -- 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.
