> On 9 Apr 2018, at 22:10, Brent Meeker <meeke...@verizon.net> wrote: > > > > On 4/9/2018 7:10 AM, Bruno Marchal wrote: >> >>> On 8 Apr 2018, at 19:42, agrayson2...@gmail.com >>> <mailto:agrayson2...@gmail.com> wrote: >>> >>> http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHAL.pdf >>> <http://iridia.ulb.ac.be/%7Emarchal/publications/SANE2004MARCHAL.pdf> >>> >>> On 3,) Arithmetic Realism (AR), why is the statement "1+1=2", equivalent to >>> the Goldbach conjecture, or the inexistence of a bigger prime, or the >>> statement that some digital machine will stop? >>> Please take each item in list separately? Goldbach conjecture? Inexistence? >>> Why stopping? And why can't the statement "1+1=2", just mean the symbols on >>> the left should be taken to mean the symbol on the right? TIA, AG >> >> I did not claim that those statement are equivalent, I state only that they >> are true of false independently of me. >> >> Then when and if proved they will be automatically equivalent in the weak >> sense of classical mathematical logic, where all (known) truth are >> equivalent. >> >> The arithmetical realism is just the idea that a (closed) proposition is >> either false or truth in the standard model (N, 0, +, *). > > But that depends on what "true" means. Whether it refers to fact, a > theorem, or a convention of language.
Arithmetical truth can be defined in “usual mathematics”. If you agree with the concept of limit in analysis, although it is too long to do it here, you can define arithmetical truth by induction (of course you need more than arithmetic to do that). The whole branch of logic known as “Model Theory” studies all such notion of truth, and usually "true in the model (N,0, +, *) is judged non problematic. To be honest, with mechanism it remains problematic, but for subtle “mechanist” reasons. If you believe that 2+2= 4, you do have the right notion of truth. Or better, if you believe that a conjecture like Goldbach or Rieman hypothesis (which is provably a conjecture in arithmetic) you have this notion. That the machine i stop on input j is neither conventional, nor necessarily a theorem of PA, or of ZF, etc. If truth was conventional, there would be no need to promise 1000,000 dollars to anyone solving Riemann hypothesis. > >> >> We can limit realism to the sigma_1 sentences, which can be shown equivalent >> with the statements saying that a digital machine stops or does not stops in >> arithmetic. >> >> 99,9 % of the mathematicians are mathematical realist, which means that they >> believe that the excluded middle principle is valid in very large part of >> math, like set theory, analysis, etc. > > But valid =/= true. Valid is “true in all models of the theory”, and for theories which are “complete” (in the sense of the completeness theorem, not incompleteness”, that is equivalent with “syntactically provable” in that theory. > It means it prevserves a presumption of true in the premises… Only with that completeness result (Gödel 1930, simplified by Henkin later). > but not that it is the only possible inference rule that does so. Of course. > >> >> Arithmetical realism is doubted only by ultra-finitist, who believe that >> there is no infinities at all, not even at the meta-level. > > No. It is doubted by many mathematicians. My mathematician friend Norm > Levitt used to say, "Mathematicians are Platonists Monday thru Friday and > Nominalists on the weekends.” I have never met a mathematician doubting that Goldbach conjecture or Riemann hypothesis is non sense, nor any parents who take they kids out of school when they are told that there is no biggest prime number. I did met mathematicians which doubt that the cantorial conjecture (like the continuum hypothesis) makes sense, but none for the weak arithmetical realism used in Mechanist Philosophy/theology. As I explained once, Mechanism entails even the a form of ultra-finitism for the ontology. Like with Nelson work, better to not add even the induction axioms to the base theory, and sees them as tools by the “Löbian observer” living in the consequence of Robinson arithmetic. Bruno > > Brent > > -- > 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 everything-list+unsubscr...@googlegroups.com > <mailto:everything-list+unsubscr...@googlegroups.com>. > To post to this group, send email to everything-list@googlegroups.com > <mailto:everything-list@googlegroups.com>. > 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 everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
