> On 17 Dec 2018, at 07:59, Jason Resch <[email protected]> wrote: > > > > On Mon, Dec 17, 2018 at 12:03 AM Bruce Kellett <[email protected] > <mailto:[email protected]>> wrote: > On Mon, Dec 17, 2018 at 4:30 PM Jason Resch <[email protected] > <mailto:[email protected]>> wrote: > On Sun, Dec 16, 2018 at 9:39 PM Bruce Kellett <[email protected] > <mailto:[email protected]>> wrote: > On Mon, Dec 17, 2018 at 1:50 PM Jason Resch <[email protected] > <mailto:[email protected]>> wrote: > On Sun, Dec 16, 2018 at 7:21 PM Bruce Kellett <[email protected] > <mailto:[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?
The axioms, the brains, the machines, the codes, the numbers can only scratch the arithmetical reality. It is big. > > 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 That question will make sense if someone find a law such that n becomes a function of t. We would discover that 2+2=3 below the Planck length. That 0 = 1, 10^(-179) second after the Big Bang, no room for one and e-zero to cohabit there. God divided by zero! To me "“2+2=4” was true” is already a category error (I got your point to help people realising this). Bruno > > > 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] > <mailto:[email protected]>. > To post to this group, send email to [email protected] > <mailto:[email protected]>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <https://groups.google.com/d/optout>. -- 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.
