> On 5 Mar 2019, at 15:20, John Clark <johnkcl...@gmail.com> wrote: > > On Tue, Mar 5, 2019 at 12:55 AM Russell Standish <li...@hpcoders.com.au > <mailto:li...@hpcoders.com.au>> wrote: > > > The usual meaning of computable integer is that there exists a program that > > outputs it. > > There is no point in arguing over the meaning of a word, but if that is what > you mean then there is a particular form of "computation" that is as dull as > dishwater and of no mathematical scientific or philosophical importance, in > other words if that's what you mean then you're not wrong but you are rather > silly.
No. It concentrates the real difficulty on the functions, and take the primitive has trivially computable. If that is silly, then the Church-Turing thesis is silly, the classical theory of computability is silly, etc. You need a good knowledge of this classical theory of computability to study the more advanced notion of computability on the real numbers. There are many, and there is no Church-Turing corresponding thesis. A function (from N to N, that is what I always mean by a function) is computable if there is a code which compute it. What you are using here is some notion of constructively computability (we might ask you which one). You are saying that a function is computable if we can exhibit an algorithm for it. That means, if not only the algorithm exists, but we can find it in a finite time, and prove it does what is requires. But then you will lost many fundamental theorem in computer science, which sometimes use the existence of programs which existence can be proved to be necessarily not constructive. Some ask more: they want the concept to be decidable, in which case we lost almost all partial recursive function. It makes sense for security concerns, when working in a bank, but is nonsensical in the fundamental matter where we are confronted to non stopping machine, without us knowing if they stop or not. If you are interested in constructive or intuitionist notion of computability, you might read the book by Beeson, which is excellent. A good help is Dummett’s book on intuitionism. But in the “theology of the machine”, this consists in studying only the ([]p & p) modes of self-reference (the mathematical notion of first person, the soul in Plotinus, the owner of consciousness, …) which natural arithmetical interpretation is intuitionist/solipsist. Bruno > > John K Clark > > > > -- > 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.
