> On 24 May 2018, at 01:05, John Clark <[email protected]> wrote: > > On Wed, May 23, 2018 at 8:59 AM, Bruno Marchal <[email protected] > <mailto:[email protected]>> wrote: > > > You were changing the mathematical definition of computations given > > independently by Church, Post, Turing, Markov >> >I don't know what definition you're referring to > > > See any (serious) textbook in logic. > > In other words YOU DON'T KNOW.
? > Nobody says "the proof you are wrong is in some unspecified book" if they > have the ability to provide a better retort. I did answer this many times. I have provided definition of computations, and explicit examples, with the combinators, (may be you were not yet in this list), with numbers, with LISP and lambda-expressions, … I am not inclined to repeat this, because it is long to explain, rather subtle (it needs to prove the intensional Church-Turing thesis), and, as we need all the time to describe the computations, someone who want to fake non-understanding will have plenty of word-play to use, so, I prefer to invite you into studying the subject. By the normal form theorem of Kleene all computations can be transformed into the search of solutions of a degree 4 diophantine polynomial. All the subtleties between proof and computations will come from the association of computations with proof of sigma_1 proposition (basically ExP(x, y) with P decidable/total-computable). Then “provable” has been famously translated in arithmetic. Each time I use “[]” it can be seen as an abbreviation of Gödel arithmetical Beweisbar predicate. Note that, with t the boolean constant “true” and f the boolean constant “false”, the non provability of the false, consistency is ~Bewesibar(“f”), or ~Bewesibar(“0 = s(0)”), with the “ “ “ playing the role of Gödel numbers. The Gödel numbering is only the faithful *representation*, and is only one half of the arithmetization (of metamathematics). What you need to understand, is that at the digital substitution level, the arithmetical truth supports *semantically” the relation making some number relation into computations. The people needs to read only one paper, really, which is the Gödel 1931 paper (it exists in Dover Edition). The arithmetisation is done. I might say more if people are interested in the details. Let me quote Gödel What matter in a computation is that at some (relative )instant, some proposition are true, like “the content at place 3 of the register R is 5” , and Gödel show that the truth value making a computation what it is did not exceed sigma_1, as “[]” is itself sigma_1, and sigma_1 complete, and with induction: it is Löbian: by which I mean Turing universal (sigma1 complete) and knowing it, in the weak sense of proving p -> []p for all p sigma_1. > > >> but if it doesn't have something about actually obtaining an answer > then its idiotic, but neither Church, Post, Turing nor Markov were idiots. > > >Contradiction. > > Yes, and therefore we have an indirect proof by contradiction that when > Church, Post, Turing or Markov was talking about computation they were > talking about actually getting an answer. You lost me. > > > we know since about a century, are not physical notion, but purely > arithmetical one. > > BULLSHIT. Gödel missed it, actually, but it was anticipated by Post, who, actually anticipated both Church’s thesis, or better Kleene’s discovery that Church made an incredible thesis which would, unlike provability resists diagonalisation. Gödel confessed it, and made this point clear. For him, it is a miracle that the enumeration of the partial computable functions is closed for diagonalization. I have explained the “prize” of, which is that machine impredictiblity, and its ability to refute all complete theories made of itself. John, if you don’t buy some book and study, you will just look like a liar. Buy the Dover book by Davis “The Undecidable”, which contains the main staring point of Mathematical logic. Buy also the thin book by Tarski “Undecidable Theories”. >> >>>Of course not. I point to some book and paper which provides a >> counter-example. > > >> >> LIKE HELL IT DOES! The damn book can't calculate 2+2 > > > > That joke again. > > Then please let me in on the joke because I don't get it, I don’t see the > humor. Far from being funny I think its a tragedy that in the 21st century > somebody, who no doubt considers himself an intellectual, believes the > ancient Greeks reached the absolute pinnacle of human achievement and that > calculations can be made without matter that obey the laws of physics even > though there is OVERWHELMING evidence to the contrary because Plato said it > could and the authority of Plato is absolute. > > > You also confuse book and its content. > > OK forget the book we'll just deal with its contents. If one day Apple > decided to put the contents of a book about computational theory into their > next iPhone instead of a microprocessor do you think Apple's stock price > would go up or down the next day? Apple would not exist if Turing and other mathematician did not discovered the universal machineries and machines. By a universal machinery, I mean a computable enumeration of the partial recursive functions phi_i, with their domain w_i. One you have a universal machine u in some enumeration phi_i, you have a new machinery: the one related to u, which is phi_u(1, x), ph_u(2, x), etc. It is the “phi_i” of that u. All that assumes no more than the sigma_1 arithmetical truth, for which G* proves p <-> []p <-> []p & p <-> []p & <>t <-> []p & <>t & p. Yet G does not, making eventually the Löbian machine knowing its incompleteness. The “theology” of the machine is computer science minus computer’s computer science. It is Tarski minus Gödel. > > > you confuse computations and description of computations, > > I think a microprocessor can perform a calculation but a description of a > computation such as one in a book, can not. At least we agree on something. > You believe the opposite. No, on the contrary I insist all the time that we must not confuse a computation with a description of a computation. What you miss is that in the arithmetical reality, not only we have the description of the computations, but they are also implemented in virtue of the true arithmetical relation (provable or not by this or that different machine). Let me quote Gödel (footnote 9) of its 1931 paper “In other words the above described procedure provides an isomorphic image of PM (principal mathematical) in the domain of arithmetic, and all metamathematical arguments can equally well be conducted in this isomorphic image”. You seem to take the arithmetical reality like if it was a book, but it is a reality, it can kicks back, eve, so much that you would win 1000,000$ in case you solve the Riemann hypothesis, which, as Turing saw, is equivalent with a pi_1 sentence. Its negation is sigma_1. If a perverse zero exists we will find it soon or later. > I ask anyone who is reading this to explain why I am the one who is confused > about the difference between a computation and a description of a computation > and not Bruno. Explanation is above. You confuse a computation, which is an abstract relation of silencing of states relatively to a universal machine, with either the physical computations, which are particular implementation relatively to physical universal digital number (actually physical approximations of course). > > > Read Gödel 1931. > > Godel's 1931 paper is one of the greatest achievements in thought in the 20th > century, but there is no evidence Godel's 1931 paper can calculate 2+2. But I have never said so. I just remind you that the translation of the UDA in arithmetic is made possible thanks to what is explained in that paper. > And Godel's 1931 paper only exists because the matter between Godel's ears > obeyed the laws of physics back in 1931. Sure, and group theory assumed the existence of shalk and blackboard, because you can’t really explain group theory without them. Stop attributing me statements that I have never said. All what I say is that the arithmetical reality implements, in the standard sense of Church and Kleene all computations. The full arithmetical truth implements much more than that, but with mechanism, the ontology is restricted to the sigma_1 truth, extensionally equivalent with the sigma_1 provability, but intensionally NOT equivalent. For p sigma_1: G* proves p -> []p, and []p -> p. But G (the machine’s 3p self) proves “only” p -> []p (for all sigma_1 p, that is its Turing universality), but the universal (Löbian) machine cannot prove []p -> p, for arbitrary p. Indeed by Löb, she would prove all p and be inconsistant. Note that consistency (~[]f = []f -> f) is a particular case of correctness ([]p -> p). > > > physics emerge from the numbers. > > You've got it backwards, numbers emerge from physics. That would be explaining the simple from the difficult. And that would be difficult, as all known branch of physics assumes the numbers. String theory even assumes some extraordinary convergence criterion so that the sum of all squares is 0, and te sum of all natural numbers is -1/12. Then, we need only one universal machinery. With less than one, we get none, and with any one of them, we get all the others. The human discovery of numbers can be related to physics, but in the metaphysics, assuming only Q, i_e. Logic + 0 ≠ s(x) s(x) = s(y) -> x = y x = 0 v Ey(x = s(y)) x+0 = x x+s(y) = s(x+y) x*0=0 x*s(y)=(x*y)+x assumes much less than any physical theories, and much less than most mathematical theories. > > > You need only a physical reality to get a physical answer > > So you admit it, physics can do something pure mathematics can't. We need a physical reality, yes, but that does not imply that it does not emerge from something non physical, like the inside view of the web of computations emulates by the tiny sigma_1 true arithmetic. Physics is absolutely important, even for stabilising and filtrating consciousness, but it belongs to the “observable”, which is mainly the “first person" betable” , or its invariant of any universal number/machine. That makes physics solid, with a core provided by arithmetic, and modalise by the constraints of self-referential correctness. > > > explain me what is primary matter > > I think you should have asked somebody what "primary matter" is many years > ago before you started insisting in almost ever post that "primary matter" > didn't exist. You make me think that Xeusippes is right. Maybe Plato should have fired Aristotle after all. > > > and how it makes a physical computations conscious > > Turing (maybe you've heard of him) explained how to arrange atoms in such a > way that the laws physics force atoms to make computations Illustrating that computations are not atoms relation. Just that atoms relation can be Turing universal, and implement a universal machine, which is an abstract, immaterial, yet relatively concretisable object, and this with respect to any universal number, be it physical or arithmetical, or set theoretical, or SK-combinatorical, etc. You are the one who seem to have that belief in some primary matter, or that physics is necessarily the fundamental science. I am the conservative skeptical. > and behave intelligently. And if Charles Darwin was right then consciousness > must be a unavoidable byproduct of intelligent behavior. Darwin use Mechanism, but Mechanism extends Darwin in arithmetic, with the physical laws emerging from the invariant of the number’s observable. You see why we have to backtrack to Plato. You continue to behave like a believer in primary matter, which is mainly Aristotle metaphysics/theology. > > > On the contrary, you are using Deutsch idea that computation are > physical, which is not the standard definition given by the Church-Turing > original thesis. > > You are incorrect. The Church-Turing thesis says a human or anything else > can compute something if and only if a Turing Machine can calculate it; and I > think that's true. Then we agree, except that you were just illustrating the contrary using physical notion. Turing “reduced” the human computer as a finite entity, using a pen and as much paper he needs, and which follows simple instructions. He get the mechanist thesis, defended it, but was also naturalist, which might explains why he missed the immaterialist consequence. Only Post saw it, but change his mind, influenced by a pair by Turing. > >> >>>> it would be easy to prove me wrong; just calculate 2+2, you are >> free to use the contents of that paper you were talking about or any other >> paper or anything else, the only restrictions I place is that you are not >> allowed to use matter or energy or to increase entropy when you perform the >> calculation, other than that anything goes. If you successfully accomplish >> my little task I will publicly declare that I have been wrong all these >> years and you have been totally right and is a genius. So what do you say, >> do you accept my challenge? >> >> >>> You ask me again something impossible. >> >> >> Obviously, but ask yourself a question "Exactly why is it >> impossible?". Because physics can do something that mathematics can’t. > > > Assuming Aristotle theology. > > They I guess you are assuming that "Aristotle theology" (whatever the hell > that is) is true because you just said "You ask me again something > impossible”.. I just pointed out that your reasoning was assuming primary matter, and there is no problem with that, except if you believe in both Church thesis, and that you can survive a digital brain transplant. Because in that case, if you still believe in some primary matter, or primary physical laws, you need to explain how they made some computations realer than other, and this without using “self-multiplication” as arithmetic doe that too. About the “impossible”, you seem to play with me. You ask me to transform some immaterial into something material, but the point is that there is no thing material, just numbers’ dream (computation supporting stable Löbian numbers) which win relative statistical game in arithmetic, which plays the role of a sort of block mindscape. > > >>You may have reasons why physics can do something that mathematics can't > and they may be good reasons or they may be bad reasons it really doesn't > matter because for whatever reason the undeniable FACT remains physics can do > something that mathematics can’t. > > >FACT? > > Yes I think so unless you can show me a FACT to the contrary, like > something that can calculate 2+2 without using matter or energy or producing > entropy. > > >you will not see ONE paper of physics which assumes primary matter, > > First of all you don't know what "primary matter" means. And second of all > you see neither the acceptance nor the rejection of "primary matter" in > modern physics papers because it turned out Leibniz's distinction between > primary and secondary matter was not scientifically useful. Leibniz was a > great man full of ideas, but this idea went nowhere. > > > There are many level of Turing universality in a modern computer. Only > the bottom one is directly physical. > > So our entire universe could be a huge video game run on some alien > supercomputer, and the alien's world could also be a video game run a > mega-alien mega-supercomputer, but eventually at the bottom level you've got > a real computer made of matter that obeys the laws of physics. Why? Only because you have not yet understood what mathematician means since Turing, Post, Kleene. In Brussels, the only critics on the math part was of the type, too much easy, explains too much the bases, and delirious for physics. The physicist find the math part non comprehensible, but the physical part too much easy. Here you show that you have no idea of what a computation is. You invoke a physical reality (which looks like a choice of a particular universal number) and you decide that only the computation made by that number is real. But then you need to give it magical power, to make a universal machine having a different experience when involved. You have a gift to criticise parts where my worst opponents did not even think of ever criticising, if you look at the reports. I am aware it can seem shocking … for people having the Aristotelian faith. But both the quantum evidence, and GPodel’s theorem favours mechanism which leads to a very elegant metaphysics, testable by physics, and logic. I have to go. Bruno > > >> this is a thought experiment and the premise is there is no error in the > ZFC based proof that says a number with certain properties can not exist and > there is no error in a computer that produce a number that has those exact > same properties, the question is "which one do you believe?". Neither of us > said we'd believe the axioms over the computer because neither of us is > insane. > > > Then what you assume is simply that ZFC is inconsistent, > > In those unusual circumstances, which I don't expect to happen, that would > be the only sane think to do, and that's why I was dumbfounded when you said > "if ZFC gives a not too long proof that I can understand, I will believe more > ZF than the computer". > > >> How can you prove an axiom is true? You can’t, > > > Indeed. But you are the one claiming that physicalism is a fact, when it > is an axiom. > > I think it is a safe axiom because there are no known facts that contradict > it an plenty of facts that support it, however if you supply me with an > example of something that can calculate 2+2 without using matter of energy or > producing entropy I will reject that axiom without hesitation. You can never > prove an axiom is true but you can prove an axiom is false, and yet you said > even if there was a fact showing the ZF axioms were wrong "I will believe > more ZF than the computer". > > > If I can prove that an axiom A is false, then I can prove the axiom ~A, > > > You can't prove axiom ~A is false if axiom ~A is not false, that is to say > if no fact exists that could contradict it; or at least you'd better hope you > can't prove it because if you can then you're entire logical system is > worthless. > > 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 [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. 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