Hi Bruno >> If the UD was a concrete one like you ran then it would start to >> generate all programs and execute them all by one step etc. But are >> you saying that because the UD exists platonically all these programs >> and each of their steps exist also and hence, by the existence of a >> successor law they have an implicit time order? >Yes. The UD exist, and is even representable by a number. UD*, the >complete running of the UD does not exist in that sense, because it is >an infinite object, and such object does not exist in simple >arithmetical theories. But all finite parts of the UD* exist, and this >will be enough for "first person" being able to glue the computations. >For example, you could, for theoretical purpose, represent all the >running of the UD by a specific total computable function. For example >by the function F which on n gives the (number representing the) nth >first steps of the UD*. Then you can use the theorem which asserts >that all total computable functions are representable in Robinson >Arithmetic (a tiny fragment of Pean Arithmetic). That theorems is >proved in detail, for Robinson-ile arithmetic, in Boolos and Jeffrey, >or in Epstein and Carnielli. In Mendelson book it is done directly in >Peano Arithmetic.
>It is because our "3-we", our bodies, or our bodies descriptions, are >constructed within these steps. But our first person are not, and no >finite pieces of the UD can give the "real experience". This is a >consequence of the first six steps: our next personal experience is >determined by the whole actual infinity of all the infinitely many >computations arrive at our current state. (+ step 8, where we abandon >explicitly the physical supervenience thesis for the computational one). This “glueing” idea reminds me of David Deutsch’s attempt to explain how time is an illusion in “The Fabric of Reality”. I never have got this one! I can follow your argument but it seems to put a very special status on the ist person experience. You say that our “3-person”/ bodily descriptions are contained as subprograms in the (infinite) programs which collectively provide Observer Moments for them. But I think you saying that our 1-person experience (frog view) is emergent from the collective (infinite) computations which are consistent with this emergent experience which is elaborated in your steps 1-7. It seems to make this ist person experience somewhat mystical as to why it is “experienced” at all. Some people wonder why we cannot see the other worlds in QM but I am often amazed that we experience one at all! Anyway all of what you say seems consistent with the many worlds picture (which it should be). >> Time is not difficult. It is right in the successor axioms of >> arithmetic. I’ll come back to this >> Here again you confirm the invocation of the successor axioms. >Yes. It is fundamental. I cannot extract those from logic alone. No >more than I can define addition or multiplication without using the >successor terms s(-) : >for all x x + 0 = x >for all x and y x + s(y) = s(x + y) >You have to understand that all the talk on the phi_i and w_i, >including the existence of universal number >(EuAxAy phi_u(<x,y>) = phi_x(y)) can be translated in pure first order >arithmetic, using only s, + and *. >I could add some nuances. "To be prime" is an intrinsic property of a >number. To be a universal number is not intrinsic. To define a >universal number I have to "arithmetize" the theory. The theory uses >variables x, y, z, ..., so I will have to represent "to be a variable" >in the theory. The theory "understands" only numbers. I can decide to >represent the variables by even numbers (for example). "Even(x)" can >be represented by "Ey(x = s(s(0)) * y)". So "variable(x)" will be >represented by the same expression. Then I will represent "to be a >formula", "to be an axiom", to be a proof", "to be a computation", >using Gödel's arithmetization technic (which is just a form of >programming in arithmetic). This will lead to a representation of >being a universal number. Where can I find out about this arithmetization technique and what do you mean by a “universal number”? >Now, would I decide to represent the variable in some other way (by >the odd numbers, for example), the preceding universal number will >still be in a universal number (intrinsically), but I will not been >able to see it, or to mention it explicitly. But here, you have to >just realize (cf the first six step of uda) that the first person >experience depends on all universal numbers, in all possible sense/ >arithmetical-implementations. >In particular "you here and now" are indeed implemented in arithmetic >in both the universal numbers based on (variable(x) = even(x), and >variable(x) = odd(x)). *ALL* universal numbers will compete below your >substitution level. >The fact that elementary (Robinson) arithmetic is already (Turing) >universal is an impressive not obvious fact. But it is no more >astonishing than the fact that Conway Game of Life is already Turing >universal, or that the combinators S and K are Turing universal, etc. Could you explain what your definition of Turing Universal is? Sorry but my background reading is not keeping up with my probingl So to sum up so far: (Please correct me on anything if I have got it wrong) We begin with arithmetic realism so 0 and the positive numbers exist in arithmetical platonia. >From the properties of these it is possible to accept the inherent idea of a successor function from which operations (maps), like addition and multiplication can be inferred or proved to similarly exist. >From these can also be inferred (as logical deductions) that many other things exist in the arithmetical platonia including a universal machine and a UD ( these “exist” like pi or root of 2 etc.). Similarly, finite subsets of the outputs of all the infinity of possible programs in UD* can also be inferred to exist. Hence because we are described from 3 person point of view by finite subsets of UD* then we can be said to exist in arithmetic platonia. Now our ist person perceptions, along with 1 indeterminacy arise as a result of the way many consistent extensions of us compete at each of the program steps ( perceived as time increments) which exist because a temporal order is intrinsic - due to there being program steps - but the matching (gluing) of possible consistent extensions is not exact- hence 1-indeterminancy. We can never be sure which sheaf of computations we will go through – however limited in steps that particular sheaf is before the next indeterminate transition occurs. Now it seems that “concrete” reality is an illusion. In a way this points to our world being a kind of mental construct. ( Is platonia a bit like the mind of a supreme sentient being or does the buck stop with the label?). Now I can accept all this quite reasonably apart from the one stage – the so called gluing bit. I’m still stuck with this one. For example suppose I write down on a piece of paper the list 1, 3,2,4,6,5,7a, 7b, 8a, 8b….. These have an intrinsic order namely 1,2,3,4,5,6,7a, 8a….. or 1,2,3,4,5,6,7b, 8b….. However much I jumble them up, this order is still apparent. They could represent steps in a persons experience as he approaches the splitting world at step 7. One branch goes to a (Moscow) the other to b (Washington). I could write down a function or algorithm on the same piece of paper (platonic paper!) which would indicate how the steps were to be traversed in the right order, but I just don’t get how all this can give us the feeling of experiencing each of these steps as observer moments. How do we become dynamically emergent in this static representation. The block universe idea with world lines would have us think that our world was deterministic The block multiverse is also deterministic but gives us i-indeterminacy in the form of branching paths. What I’m struggling with here is how the platonia enables us to perceive the world from defined numbers and relationships. A computer does this in a first person game. The virtual world is presented to the player in some order at different times which gives the illusion of a reality albeit virtual. Program steps give an illusion of “becoming” but the screenshots of a video game collected (in order) don’t have this property of becoming. You see what I am getting at? There seems to be something dynamic missing. Even if you define a function which maps things, the function won’t do anything of itself it’s surely just more numbers on the platonic paper. I’m sorry if I am so stuck with this one and I appreciate your patience. Funny thing is though, I sense it is me that’s got the wrong viewpoint here. I’m not trying to disprove anything. Rather I want get my philosophical thinking correct. If platonia was intrinsically timeful rather than timeless I’d probably feel better. >> Time is not difficult. It is right in the successor axioms of >> arithmetic. If I could grasp this statement of yours it would help. Could you elaborate on how this would be done. A successor function is applied by a (sentient) being. Just writing it down next to its arguments does not seem to me to be the same as applying it. Or knowing what it is to be applied to. Any algorithm on how this would be done is not the same as it being done. Best wishes Nick On Dec 29, 9:55 am, Bruno Marchal <marc...@ulb.ac.be> wrote: > On 28 Dec 2009, at 21:24, Nick Prince wrote: > > > > >> Well, it is better to assume just the axiom of, say, Robinson > >> arithmetic. You assume 0, the successors, s(0), s(s(0)), etc. > >> You assume some laws, like s(x) = s(y) -> x = y, 0 ≠ s(x), the laws > >> of addition, and multiplication. Then the existence of the universal > >> machine and the UD follows as consequences. > > > Ok so the UD exists (platonically?) > > Yes. The UD exists, and its existence can be proved in or by very weak > (not yet Löbian) arithmetical theories, like Robinson Arithmetic. > The UD exists like the number 733 exists. The proof of its existence > is even constructive, so it exists even for an intuitionist (non > platonist). No need of the excluded middle principle. > > > > >> Better not to conceive them as living in some place. "where" and > >> "when" are not arithmetical predicate. The UD exists like PI or the > >> square root of 2. > >> (Assuming CT of course, to pretend the "U" in the UD is really > >> universal, with respect to computability). > > > Fine so the UD has an objective existence in spite of whatever else > > exists. > > It exists in the sense that we can prove it to exist once we accept > the statement that 0 is different from all successor (0 ≠ s(x) for > all x), etc. > If you accept high school elementary arithmetic, then the UD exists in > the same sense that prime numbers exists. > "exist" is used in sense of first order logic. This leads to the usual > philosophical problems in math, no new one, and the UDA reasoning does > not depend on the alternative way to solve those philsophical problem, > unless you propose a ultra-finitist solution (which I exclude in comp > by arithmetical realism). > > > > > > > > >> There is a "time order". The most basic one, after the successor law, > > >> is the computational steps of a Universal Dovetailer. > >> Then you have a (different) time order for each individual > >> computations generated by the UD, like > > >> phi_24 (7)^1, phi_24 (7)^2, phi_24 (7)^3, phi_24 (7)^4, ... > >> where "phi_i (j)^s" denotes the sth steps of the computation (by > >> the UD) of the ith programs on input j. > > > If the UD was a concrete one like you ran then it would start to > > generate all programs and execute them all by one step etc. But are > > you saying that because the UD exists platonically all these programs > > and each of their steps exist also and hence, by the existence of a > > successor law they have an implicit time order? > > Yes. The UD exist, and is even representable by a number. UD*, the > complete running of the UD does not exist in that sense, because it is > an infinite object, and such object does not exist in simple > arithmetical theories. But all finite parts of the UD* exist, and this > will be enough for "first person" being able to glue the computations. > For example, you could, for theoretical purpose, represent all the > running of the UD by a specific total computable function. For example > by the function F which on n gives the (number representing the) nth > first steps of the UD*. Then you can use the theorem which asserts > that all total computable functions are representable in Robinson > Arithmetic (a tiny fragment of Pean Arithmetic). That theorems is > proved in detail, for Robinson-ile arithmetic, in Boolos and Jeffrey, > or in Epstein and Carnielli. In Mendelson book it is done directly in > Peano Arithmetic. > > > > >> Then there will be the time generated by first person learning and > >> which relies eventually on a statistical view on infinities of > >> computations. > > > Is this because we are essentially constructs within these steps? > > It is because our "3-we", our bodies, or our bodies descriptions, are > constructed within these steps. But our first person are not, and no > finite pieces of the UD can give the "real experience". This is a > consequence of the first six steps: our next personal experience is > determined by the whole actual infinity of all the infinitely many > computations arrive at our current state. (+ step 8, where we abandon > explicitly the physical supervenience thesis for the computational one). > > > > >> Time is not difficult. It is right in the successor axioms of > >> arithmetic. > > > Here again you confirm the invocation of the successor axioms. > > Yes. It is fundamental. I cannot extract those from logic alone. No > more than I can define addition or multiplication without using the > successor terms s(-) : > > for all x x + 0 = x > for all x and y x + s(y) = s(x + y) > > You have to understand that all the talk on the phi_i and w_i, > including the existence of universal number > (EuAxAy phi_u(<x,y>) = phi_x(y)) can be translated in pure first order > arithmetic, using only s, + and *. > > I could add some nuances. "To be prime" is an intrinsic property of a > number. To be a universal number is not intrinsic. To define a > universal number I have to "arithmetize" the theory. The theory uses > variables x, y, z, ..., so I will have to represent "to be a variable" > in the theory. The theory "understands" only numbers. I can decide to > represent the variables by even numbers (for example). "Even(x)" can > be represented by "Ey(x = s(s(0)) * y)". So "variable(x)" will be > represented by the same expression. Then I will represent "to be a > formula", "to be an axiom", to be a proof", "to be a computation", > using Gödel's arithmetization technic (which is just a form of > programming in arithmetic). This will lead to a representation of > being a universal number. > Now, would I decide to represent the variable in some other way (by > the odd numbers, for example), the preceding universal number will > still be in a universal number (intrinsically), but I will not been > able to see it, or to mention it explicitly. But here, you have to > just realize (cf the first six step of uda) that the first person > experience depends on all universal numbers, in all possible sense/ > arithmetical-implementations. > In particular "you here and now" are indeed implemented in arithmetic > in bot the universal numbers based on (variable(x) = even(x), and > variable(x) = odd(x)). *ALL* universal numbers will compete below your > substitution level. > > The fact that elementary (Robinson) arithmetic is already (Turing) > universal is an impressive not obvious fact. But it is no more > astonishing than the fact that Conway Game of Life is already Turing > universal, or that the combinators S and K are Turing universal, etc. > > Bruno > > http://iridia.ulb.ac.be/~marchal/- Hide quoted text - > > - Show quoted text - -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-l...@googlegroups.com. 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