> On 27 Aug 2018, at 19:54, [email protected] wrote: > > > > On Saturday, August 25, 2018 at 1:11:47 AM UTC-6, Bruno Marchal wrote: > >> On 25 Aug 2018, at 01:15, [email protected] <javascript:> wrote: >> >> >> >> On Friday, August 24, 2018 at 12:25:03 PM UTC, telmo_menezes wrote: >> On 23 August 2018 at 06:31, <[email protected] <>> wrote: >> > >> > >> > On Thursday, August 23, 2018 at 2:01:24 AM UTC, Jason wrote: >> >> >> >> >> >> >> >> On Wed, Aug 22, 2018 at 4:43 PM <[email protected] <>> wrote: >> >>> >> >>> >> >>> >> >>> On Tuesday, August 21, 2018 at 3:22:04 PM UTC, Jason wrote: >> >>>> >> >>>> >> >>>> >> >>>> On Tue, Aug 21, 2018 at 1:16 AM <[email protected] <>> wrote: >> >>>>> >> >>>>> I've been looking at the Wiki article on this topic. I find that I >> >>>>> really don't understand what it is, or why it's important. Maybe a few >> >>>>> succinct words from the usual suspects can be of help. TIA. >> >>>>> >> >>>>> >> >>>> >> >>>> >> >>>> Bruno provided a great definition and background of the Church-Turing >> >>>> Thesis. I will try to answer why it is important and comes up often in >> >>>> our >> >>>> discussion. >> >>>> >> >>>> >> >>>> The Church-Turing thesis says that anything that is computable is >> >>>> computable by any computer. In other words, there is nothing that the >> >>>> computer in your cell phone can't compute, that your laptop or that a >> >>>> super >> >>>> computer (or even a quantum computer) can. It just comes down to >> >>>> having >> >>>> enough time and memory. >> >>>> >> >>>> This is why you don't need to buy a new phone with new hardware every >> >>>> time you want to install a new app. Regardless of the type of CPU in >> >>>> your >> >>>> phone, it can be extended in its power of what it might compute only >> >>>> given >> >>>> some new software. It is in this sense that computers are "Universal", >> >>>> they >> >>>> are universal in the same sense that of a universal remote, or in the >> >>>> sense >> >>>> that a record player is a universal sound imitating device. A record >> >>>> player >> >>>> might emulate the sounds of an orchestra, Britney Spears, whale songs, >> >>>> etc., >> >>>> all it needs is the appropriate record and it can produce the sound. >> >>>> >> >>>> In the same sense, all a Turing Machine (computer) needs to imitate (or >> >>>> emulate) the right program or function is the right software. Because >> >>>> of >> >>>> this, anything that can be described in software, be it a brain >> >>>> emulation, >> >>>> an AI, a virtual environment, a virtual machine or operating system, >> >>>> can >> >>>> never know what hardware is running it, because the Church-Turing >> >>>> thesis >> >>>> says that any computer is capable of running it. >> >>>> >> >>>> This is why if consciousness is computable (the computational theory of >> >>>> mind) we cannot know what is computing us (e.g. we could be in a matrix >> >>>> type >> >>>> simulation for all we know). The other implication is that if >> >>>> computations >> >>>> exist in mathematics (and they do), then we exist within mathematics. >> >>>> Mathematics (or at least the part necessary to describe computations) >> >>>> becomes the fundamental science of what we experience and what is >> >>>> possible >> >>>> to experience or what we may predict about our future experiences >> >>>> (physics). >> >>>> >> >>>> >> >>>> Jason >> >>> >> >>> >> >>> If someone digitizes (emulates) the Mona Lisa, is this equivalent to the >> >>> Mona Lisa? >> >> >> >> >> >> If you digitize a person and put the digitized Mona Lisa before them, it >> >> is equivalent to the real Mona Lisa to that person, at least as far as >> >> they >> >> can tell. >> >> >> >> >> >>> >> >>> Can you write a function which is not computable? AG >> >>> >> >>> >> >> >> >> If by not computable you mean it never returns, then this is easy: >> >> >> >> function foo(): >> >> while (true) >> >> { >> >> // loop forever >> >> } >> >> >> >> There are also programs for which no one knows if they are computable or >> >> not. If you can prove whether or not this function ever completes, you >> >> will >> >> be world famous, and may even earn a million dollars (though I think the >> >> prize has been retracted, it might be oferred again): >> >> >> >> Step 1: Set X = 4 >> >> Step 2: Set R = 0 >> >> Step 3: For each Y from 1 to X, if both Y and (X – Y) are prime, set R = >> >> 1 >> >> Step 4: If R = 1, Set X = X + 2 and go to Step 2 >> >> Step 5: If R = 0, print X and halt >> >> >> >> All you have to prove is the computer either never gets to step 5 or that >> >> it does get to step 5. Mathematicians have been working on a related >> >> problem for 300 years, no one has solved it yet. >> >> >> >> >> >> Jason >> > >> > >> > I was asking about a well-defined mathematical function that can be >> > written >> > in closed form, or possibly as an infinite series. I believe that all such >> > functions are computable. I was not discussing subroutines that might >> > never >> > terminate. If all well defined mathematical functions are computable, why >> > did computability become a big deal? AG >> >> It is not true that all well-defined functions are computable. You >> have already been given examples by Jason and John of well-defined >> mathematical functions that are non-computable. >> >> You seem to confuse "well-defined" with "written in closed form". The >> latter is not even well-defined (heheh) because it hangs on the idea >> of a set of "well-known" functions, and people already have different >> ideas on what that set includes. Having well-known representations >> such as sin(x) or e^x, or even x + y does not magically make the >> related computations non-algorithmic. How do you think you learned how >> to add, subtract, multiply and divide in basic school? Those were >> algorithms. >> >> Well-defined just means that there is a non-ambiguous way to know if a >> given value corresponds to a given input of the function. If I tell >> you to consider the function f, such that its value is zero no matter >> the input, then I gave you a well-defined function in plain English. >> There is nothing magical about notation. >> >> Telmo. >> >> So a "function" must have a well defined domain set, finite or infinite, and >> is not limited to closed forms but includes infinite series and algorithms. >> In such case, an infinite loop, even if it has an initial value, is not a >> function and not computable, whereas all closed forms are computable. >> Agreed? AG > > Infinite loop can be extended into the computable. The problem are infinite > computations getting more and more complex, and with no loop. > > I don't see why an infinite loop can be computable. I see there are > subtleties in this concept. AG
We will come back on this. If there is a loop, it might be dented and the non computable function is not computable only in a superficial manner: it can be extended into something computable, but that will not be the case for most machine’s attribute. But you need to have a some understanding of the diagonal of Cantor. > > I am not sure what you mean by close form. > > Initially, I meant something that can be written down explicitly or > expressly, like a polynomial, and can be calculated for every value in its > domain. I then extended it to infinite series or sums, which converge. I then > accepted that functions could include software algorithms. AG OK. (That is a combinator, actually, but we will come back on this). > > No worry, we will see soon, example of well defined functions which are not > computable. Basically all attributes of universal machine are not computable. > > This seems very UN-intuitive. Incidentally, my computer broke down again due > to a faulty repair two months ago. I am using a library computer which limits > my time and is unavailable on Sunday. It will slow my plan to study some of > your material, including Cantor's proof. AG OK. Actually I will be slowed down too, as the exam period has begun today. > > Finally, a general critical comment on your theory, CMIIAW, that > consciousness causes the illusion of matter and the apparent reality of > events and processes. As I previously noted on this or a related thread, > consciousness seems to be a late comer to the universe. I can understand, but here mechanism will change this, in an admittedly quite counter-intuitive way. We will have: Number ==> consciousness ==> matter/universes ==> human consciousness. > It's easy to identify events and processes that occurred way before any > identifiable consciousness existed, such as the formation of the Sun, Earth, > and Moon. I'd be interested in your comment on this issue. I commend you for > your guts to attempt solve the problem of consciousness, but because of what > I just wrote, I don't believe your theory can explain what I believe you > represent it doing. AG "My theory” is Digital Mechanism. It helps to formulate the mind-body problem, and to suggest a solution which is closer to Plato’s idealism than to Aristotle’s materialism (current paradigm in science and religion today). Bruno > > Bruno > > > > >> >> > -- >> > 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 >> > <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] <javascript:>. >> To post to this group, send email to [email protected] >> <javascript:>. >> 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] > <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.
