Emil L. Post 1936 "Finite Combinatory Processes. Formulation 1." - from the concluding paragraph:
"The writer expects the present formulation to turn out to be logically equivalent to recursiveness in the sense of the Gödel-Church development. Its purpose, however, is not only to present a system of a certain logical potency but also, in its restricted field, of psychological fidelity. In the latter sense wider and wider formulations are contemplated. On the other hand, our aim will be to show that all such are logically reducible to formulation 1. We offer this conclusion at the present moment as a *working hypothesis*. And to our mind such is Church's identification of effective calculability with recursiveness.8 Out of this hypothesis, and because of its apparent contradiction to all mathematical development starting with Cantor's proof of the non-enumerability of the points of a line, independently flows a Gödel-Church development. The success of the above program would, for us, change this hypothesis not so much to a definition or to an axiom but to *natural law*. Only so, it seems to the writer, can Gödel's theorem concerning incompleteness of symbolic logics of a certain general type and Church's results on the recursive unsolvability of certain problems be transformed into conclusions concerning all symbolic logics and all methods of solvability." Footnote: 8 "Cf. Church, lock. cit, pp. 346, 356-358. Actually the work already done by Church and others carries this identification considerably beyond the working hypothesis stage. *But to mask this identification under a definition hides the fact* that a fundamental discovery in the limitiations of mathematicizing power of Homo Sapiens has been made *and blinds us to the need of its continual verification*." Effective calculability; Post seems to insist with the incredibly clear and simple Formulation1, is "merely" intuitive notion. I know Church wasn't too happy with this. Continual verification, ok. PGC On Thu, Jul 10, 2014 at 11:46 PM, Quentin Anciaux <allco...@gmail.com> wrote: > > Le 10 juil. 2014 23:40, "meekerdb" <meeke...@verizon.net> a écrit : > > > > > On 7/10/2014 2:08 PM, Quentin Anciaux wrote: > >> > >> > >> Le 10 juil. 2014 22:46, "meekerdb" <meeke...@verizon.net> a écrit : > >> > > >> > On 7/10/2014 1:21 PM, Quentin Anciaux wrote: > >> >> > >> >> > >> >> > >> >> > >> >> 2014-07-10 21:56 GMT+02:00 meekerdb <meeke...@verizon.net>: > >> >>> > >> >>> On 7/10/2014 12:19 PM, Quentin Anciaux wrote: > >> > >> >> No we need computation to relate a physical instantiation to it > (that's how we can say two != computers compute the same thing, it's > because they relate to the same computation), > >> > > >> > > >> > But that's not true. I have a differential equation integrator in my > computer and it could be going through exactly the same states in two > different instances; one computing heat transfer in a disc brake and other > computing diffusion of pollutant in a pond. So there is not a one-one > mapping either way. > >> > >> that's not possible... if they compute different thing the state > machine is different. > > > > > > Only when it's printing the headers of the columns of the numbers. > > if they are different computations they don't go through the same > states... QED What you said is simply false. > > > > 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. > > To post to this group, send email to everything-list@googlegroups.com. > > Visit this group at http://groups.google.com/group/everything-list. > > For more options, visit 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 http://groups.google.com/group/everything-list. > For more options, visit 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 http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
