Re: immune to Cantor's diagonalization
On 02 Dec 2011, at 05:16, Stephen P. King wrote: Dear Bruno, Thank you very much for this explicit remark. It is very helpful for my research. I have some comments and a question. On 12/1/2011 11:34 AM, Bruno Marchal wrote: Hello Stephen, On 01 Dec 2011, at 13:16, Stephen P. King wrote: Hi Bruno, Could you eleborate a bit about how Computability is the only notion immune to Cantor's diagonalization? Cantor proved that infinity is sensible to diagonalization. Given an infinite set, you can find a bigger set by diagonalization. Gödel proved that any effective provability system (theory) is incomplete. Given a theory rich enough to talk on numbers, you can build a richer provability system by diagonalization. Tarski proved that any system of definition will lack the expressive power to define some notion, notably its truth notion. Again, this follows from diagonalization. Diagonalization is a sort of transcendental operation in mathematics. If you have about anything pretending to be a universal notion in the domain, you can diagonalize against it. That is why Stephen C. Kleene was skeptical when Church told him that his lambda-calculus defined a universal notion of computability. At first, it looks like computability is sensible, NOT immune, to diagonalization. Imagine that there is a universal language for computability L. Consider all the computable function defined on N and with value in N, enumerated from their code in that language: f_0, f_1, f_2, f_3, f_4, etc. Let g be defined on n by g(n) = f_n(n) + 1(g is said to be defined by diagonalization) Each f_i are computable function from N to N, so f_n(n) is well defined, and + 1 is obviously computable, so g seems to be computable. But if L is universal, the g should be in the list. So g = f_k, for some k. But then g(n) = f_k(n), given that g = f_k In particular, with n = k g(k) = f_k(k) But by definition of g, we know that g(k) = f_k(k) + 1 By Leibniz identity rule f_k(k) = f_k(k) + 1 And by using the fact that f_k is a function from N to N, we know that f_k(n) is a number for any n, so f_k(k) is a number, and this number can be subtracted at the left and right hand side of the equality above, so 0 = 1. Church's pretension that L is universal seems to be refuted. But that proof is wrong. Do you see why? Take the time to find the mistake by yourself before reading the solution below. -- It seems that g is not necessarily specified by g(k), since k assumes too much. ? What is wrong is that the language L might define more than the computable functions from N to N, but can also define functions from from subset of N to N. In that case, the reasoning just shows that g(k) = f_k(k) is not defined. This shows that an universal machine can crash (run in a loop without ever giving an output), and this necessarily so to be universal. Part of the Non-halting result... OK. It is easier than the non halting. Worse, there will be no effective means (and thus no complete theory of universal machine or language) to decide if some f_i is defined on N or a proper subset of N. If that was the case, we would be able to filter out the functions from a proper subset of N to N from the functions from N to N, and then the diagonalization above would lead to 0 = 1. Let us call a function partial if that function if either from N to N, or from a subset of N to N, and a function is total if it is defined on N. The reasoning above shows that the set of total functions is not immune to diagonalization. But the superset of the partial functions is, and that is a deep strong argument for Church thesis. But it still seems that there is something missing in this conclusion about the Church thesis. It is that for computation all that one needs to consider in a model is the N - N and n /subset N - N maps/functions. The problem of time that I have complained about is part of this problem that I see. Time is not relevant here. We need only the natural numbers. It reminds me strongly of the problem of the axiom of choice, The axiom of choice is not relevant here. This can be made precise: ZF and ZFC proves the same arithmetical truth. I don't use set theory at all. where the existence of unmeasurable sets cannot be excluded inducing such things as the Banach-Tarski paradox. The same problem that occurs in the result you discuss above: there is a function/object that cannot be exactly defined. Which one? You confuse computation and definition. How do we get around this impasse? I don't see an impasse. I explain old and basic uncontroversial notions. What if there is a way to sequester the pathological parts of the function without having to define the function? This is too vague. (Something like this is discussed here
Re: The consciousness singularity
On 01 Dec 2011, at 20:27, Craig Weinberg wrote: On Dec 1, 10:39 am, Bruno Marchal marc...@ulb.ac.be wrote: On 29 Nov 2011, at 18:44, benjayk wrote: Bruno Marchal wrote: I only say that I do not have a perspective of being a computer. If you can add and multiply, or if you can play the Conway game of life, then you can understand that you are at least a computer. So, then I am computer or something more capable than a computer? I have no doubt that this is true. OK. And comp assumes that we are not more than a computer, concerning our abilities to think, etc. This is what is captured in a quasi operational way by the yes doctor thought experiment. Most people understand that they can survive with an artificial heart, for example, and with comp, the brain is not a privileged organ with respect to such a possible substitution. This is the first problem. It's not that the brain has to be privileged to make it impossible to simulate, no organ can be simulated, it's just that it is possible to simulate some of the functions of an organ to the extent that the person as a whole, i.e. the inhabitant(s) of the brain, can't tell the difference. That is your hypothesis. OK. The brain is a totally. different. story. First of all, you could have a crowbar poking through your skull and not know it if the parts of your brain that related to that awareness (and the pain thereof) were damaged, so subjective accounts of success are not reliable. OK. (I insist often on this. It is provable assuming comp, but if you want assume this in your non-comp theory, it is OK). Secondly, objective accounts are also unreliable owing to the privacy of subjectivity. I agree. Finally, the brain being our only source of experience at all, cannot be compared to anything else in the cosmos. That is the neuro hypohesis. I don't need it, or trivialize it with the notion of 'generalized' brain (the portion of the physical reality which need to be simulated for keeping may consciousness unchanged locally). No person has ever existed outside of a brain as far as we know, so we cannot presume that the brain itself or a person can be simulated. I never presume. I assume. It is my working hypothesis. It simply may not work that way at all. Sure. But this can be said for any hypothesis (hypothesis = theory). A person may be a continuity of unreproducible material + semantic happenstance which builds upon itself cumulatively and idiopathically. That is a speculation. That is possible, even in the comp theory. We might be our brain You contradict an old statement you made to me, according to which we own a brain (and are not a brain). and our brain may be much more than it appears to us from the outside or the inside, but there is nothing to suggest that there is a such thing as an arithmetic essence which is independent of physics and is deterministic. What is an arithmetic essence? I avoid essence. It is only through our brain-grounded subjectivity that we believe there is any such thing as pattern or arithmetic. It's just one way that we make sense of our world. OK. With comp, the contrary is true. It is our arithmetic-grounded subjectivity which makes us believe there is such thing as space, matter, brain, etc. It is also not an abstract digital computer (even according to COMP it isn't) since a biological being is physical and spiritual (meaning related to subjective conscious experience beyond physicality and computability). But all universal machine have a link with something beyond physicality and computability. Truth about computability is beyond the computable. So your point is not valid. Just because computational truth is rooted in non-comp doesn't mean that it is the same non-comp as organic subjectivity. What is organic subjectivity, and why would that be non-comp? Here it seems to me that Statis has convincingly explains that adding a non-comp element in matter does not help. I gave other reason (comp makes matter itself non-comp). Neither can they be derived from it. Physicality can be derived. And has to be derived (by UDA). Both quanta and qualia. I don't think qualia can be derived. I don't think a digital machine can know the difference between visual qualia and aural qualia if they yield the same functionality. You assert and reassert your non-comp hypothesis. Are you believing that comp is false? I don't care. I am not interested in debate on what is true or false. It is not my job. Only the geography cannot be derived, but the physical laws can. You might elaborate why you think they can't. Physical laws are a posteriori analytical abstractions based on our shared experiences of concrete physical events. With comp, the notion of concrete physical event is vague, and relative. With non-comp, I don't know, given that you have not given a sufficiently precise theory in which I
Re: immune to Cantor's diagonalization
On 02 Dec 2011, at 13:26, Stephen P. King wrote: Dear Bruno, Please re-read that last post slowly. I fear that you are rushing to a judgement of what I am trying to communicate and completely missing the idea that I am trying to sketch out. On 12/2/2011 6:26 AM, Bruno Marchal wrote: On 02 Dec 2011, at 05:16, Stephen P. King wrote: snip Worse, there will be no effective means (and thus no complete theory of universal machine or language) to decide if some f_i is defined on N or a proper subset of N. If that was the case, we would be able to filter out the functions from a proper subset of N to N from the functions from N to N, and then the diagonalization above would lead to 0 = 1. Let us call a function partial if that function if either from N to N, or from a subset of N to N, and a function is total if it is defined on N. The reasoning above shows that the set of total functions is not immune to diagonalization. But the superset of the partial functions is, and that is a deep strong argument for Church thesis. But it still seems that there is something missing in this conclusion about the Church thesis. It is that for computation all that one needs to consider in a model is the N - N and n /subset N - N maps/functions. The problem of time that I have complained about is part of this problem that I see. Time is not relevant here. We need only the natural numbers. Your notion of time seems to be lifted directly from the ordering of the natural numbers, so in a sense you are taking time as the sequence 1 2 3 4 .. as inducing a ab initio measure of change. The problem is that unmeasurable sets exists and restricting ourselves to tiny little islands of thought is not progressive. We may find solutions to the measure problem by considering more carefully exactly how we define measures. Ordinary notions of measure seem to be based on Platonic notions, definitions given by fiat. Are you familiar with how topos theory treats the notion of a set? The following is from Jonhstone's Topos theory p. xvii cfbedije.png I am considering an evolving universe, not a fixed one. The notion of classical computability is topos independent. Non classical computability is interesting but not relevant for the issue of the classical Church thesis that we need to make sense of comp in cognitive science. The notion of time is also not relevant. The role of topos is comp is explained in conscience et mécanisme. A topos basically is good to modelize a mathematician's mind, not the arithmetical reality. It reminds me strongly of the problem of the axiom of choice, The axiom of choice is not relevant here. This can be made precise: ZF and ZFC proves the same arithmetical truth. I don't use set theory at all. Are not the natural numbers a set and thus have an implicit set theory? Natural numbers are no more set than fortran program. You can represent numbers with set, but this does not make a number a set. Natural numbers admit a simpler first order theory than set or elementary toposes. Just because you did not invoke a set theory specifically does not absolve you from the implicit use of a set theory. I don't, in the theory. I do in the epistemology at the naive level, like engineers, philosophers or like when I do shopping. Set theory is just not relevant for Church thesis. I write a set theory instead of set theory because set theories are Legion! I have seen instances where huge fights have occurred in academia because of people having completely different set theories with which they are interpreting a theory. That is one good reason to not use set theory. I don't believe in sets, actually. But even this remark is not relevant for what we were talking about. where the existence of unmeasurable sets cannot be excluded inducing such things as the Banach-Tarski paradox. The same problem that occurs in the result you discuss above: there is a function/object that cannot be exactly defined. Which one? You confuse computation and definition. When are definitions computable (constructable by recursive functions) and when are they otherwise? This is not a trivial point! That is a reason for keeping those notion apart. How do we get around this impasse? I don't see an impasse. I explain old and basic uncontroversial notions. LOL! It seems that I am more radical in my thinking and you are conservative in yours here. I have never hide that I am an extreme conservative. Modernity has ended in 523. The problem is that these old and basic uncontroversial notions are causing problems for the advancement of physics and our understanding of old problems, such as the mind-body problem. Indeed, I show that those old and uncontroversial notions makes physics a branch of number theory. Just study and criticize the proof,
Re: The consciousness singularity
On Dec 2, 6:58 am, Bruno Marchal marc...@ulb.ac.be wrote: OK. And comp assumes that we are not more than a computer, concerning our abilities to think, etc. This is what is captured in a quasi operational way by the yes doctor thought experiment. Most people understand that they can survive with an artificial heart, for example, and with comp, the brain is not a privileged organ with respect to such a possible substitution. This is the first problem. It's not that the brain has to be privileged to make it impossible to simulate, no organ can be simulated, it's just that it is possible to simulate some of the functions of an organ to the extent that the person as a whole, i.e. the inhabitant(s) of the brain, can't tell the difference. That is your hypothesis. OK. But do you have any ideas about why it might be valid or not? Finally, the brain being our only source of experience at all, cannot be compared to anything else in the cosmos. That is the neuro hypohesis. I don't need it, or trivialize it with the notion of 'generalized' brain (the portion of the physical reality which need to be simulated for keeping may consciousness unchanged locally). I don't think it's a hypothesis though. The brain IS our only known source of experience. We can change our experience by changing our brain and vice versa. The same cannot be said for anything else in the universe, can it? Not saying for sure that our experience could not some day be exported to another medium (although of course I think that medium would need to be isomorphic in substance to a high degree) just that as far as we know now, the brain is incomparable as far as we are concerned. No person has ever existed outside of a brain as far as we know, so we cannot presume that the brain itself or a person can be simulated. I never presume. I assume. It is my working hypothesis. It simply may not work that way at all. Sure. But this can be said for any hypothesis (hypothesis = theory). A person may be a continuity of unreproducible material + semantic happenstance which builds upon itself cumulatively and idiopathically. That is a speculation. That is possible, even in the comp theory. Comp is speculation too. The question is whether it makes sense or whether there are any specific objections from the start. We might be our brain You contradict an old statement you made to me, according to which we own a brain (and are not a brain). You're right. I am of two minds about it, hah. No, I do still think that we own a brain just as we own our lives and both our lives (and maybe our brain and our lives own us too), I'm just opening it up so that if we want to say that we are our brain, we have no objective reason why it isn't so. The interior is the ontological opposite of the exterior so it isn't appropriate to say that we literally are the brain as the brain looks to us from the outside, but figuratively we are the interior of our brain, body, you could even say home or family. We are our capacity to influence and be influenced by our world, and the brain is the gateway to that world. I say figuratively in the sense of multisense realism though - as a concrete realism equal to that of the exterior, just expressed as semantic entanglement through time rather than object relations across space. and our brain may be much more than it appears to us from the outside or the inside, but there is nothing to suggest that there is a such thing as an arithmetic essence which is independent of physics and is deterministic. What is an arithmetic essence? I avoid essence. Ok, what do you want to call it? Computation? What is the identity of a UM made of? It is only through our brain-grounded subjectivity that we believe there is any such thing as pattern or arithmetic. It's just one way that we make sense of our world. OK. With comp, the contrary is true. It is our arithmetic-grounded subjectivity which makes us believe there is such thing as space, matter, brain, etc. Right, but we know for a fact that changes to our brain can impact our pattern recognition capacity. We don't know of anything that is for sure grounded in arithmetic alone as a disembodied entity. Does comp explain why all arithmetic subjects would always appear to be associated with physical systems to other arithmetic subjects? To suggest that arithmetic can simulate physics is one thing, but why does it *have to* generate physics? It is also not an abstract digital computer (even according to COMP it isn't) since a biological being is physical and spiritual (meaning related to subjective conscious experience beyond physicality and computability). But all universal machine have a link with something beyond physicality and computability. Truth about computability is beyond the computable. So your point is not valid. Just because computational truth is rooted in non-comp doesn't
Re: The consciousness singularity
On 12/2/2011 6:22 AM, Craig Weinberg wrote: I don't think it's a hypothesis though. The brain IS our only known source of experience. We can change our experience by changing our brain and vice versa. The same cannot be said for anything else in the universe, can it? I can change my experience by moving to Canada, getting a new wife, or putting brandy in my coffee. Brent -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: The consciousness singularity
Bruno Marchal wrote: On 29 Nov 2011, at 18:44, benjayk wrote: Bruno Marchal wrote: I only say that I do not have a perspective of being a computer. If you can add and multiply, or if you can play the Conway game of life, then you can understand that you are at least a computer. So, then I am computer or something more capable than a computer? I have no doubt that this is true. OK. And comp assumes that we are not more than a computer, concerning our abilities to think, etc. This is what is captured in a quasi operational way by the yes doctor thought experiment. Most people understand that they can survive with an artificial heart, for example, and with comp, the brain is not a privileged organ with respect to such a possible substitution. If YES doctor means we are just an immaterial abstract computer than there is nothing to deduce (our experience already is only related to computations, since we defined as by them). But if YES doctor just means our bodies work *like* a computer (and thus the substitution works, and we already know that this is the case to some extent) then none of the step works because they assume we work exactly 100% like a abstract computer. In actuality we can eg never be sure that teleportation, duplication etc... work as intended, because actual computers are not totally reliable, and actually quantum objects, and not purely digital in an abstract sense (I argue in a more detailed way below). In other words, you are assuming an abstraction of a computer in the argument, which is already the conlusion. The steps rely on the substitution being perfect, which they will never be. I'm probably making it to complicated, because I can't seem to point out the simple fallacy. That's why I'm continuing to give examples of why either YES doctor does not mean what you need it to mean (we are exactly, and only, and always an abstract digital computer) or why you can't assume that the reasoning work. Bruno Marchal wrote: Bruno Marchal wrote: When I look at myself, I see (in the center of my attention) a biological being, not a computer. Biological being are computers. If you feel to be more than a computer, then tell me what. Biological beings are not computers. Obviously a biological being it is not a computer in the sense of physical computer. I don't understand this. A bacteria is a physical being (in the sense that it has a physical body) and is a computer in the sense that its genetic regulatory system can emulate a universal machine. Usually computer means programmable machine, not something that can emulate a universal machine. It seems you are so hooked on the abstract perspective of a computer scientist, that you don't even see the possibility of the distinction abstract computer / actual computer. Bruno Marchal wrote: It is also not an abstract digital computer (even according to COMP it isn't) since a biological being is physical and spiritual (meaning related to subjective conscious experience beyond physicality and computability). But all universal machine have a link with something beyond physicality and computability. Truth about computability is beyond the computable. So your point is not valid. Yes, but then the whole argument does not work, because it deals with something that even according to your conclusion can't be purely computational (actual computers), so you can't assume it works as they should do. COMP does just mean we work enough like computers to make a substitution possible (we say YES to a *functionally* correct substitution), it does not mean that there is any substitution that works perfectly. Bruno Marchal wrote: Neither can they be derived from it. Physicality can be derived. And has to be derived (by UDA). Both quanta and qualia. Only the geography cannot be derived, but the physical laws can. You might elaborate why you think they can't. Frankly I don't believe in absolute physical laws, so we can't derive them. They are just locally valid approximate rules, like swans are white. Bruno Marchal wrote: And no, there is no need for any evidence for some non-turing emulable infinity in the brain. We just need non-turing emulable finite stuff in the brain, and that's already there. I thought you were immaterialist. What is that finite stuff which is non Turing emulable? Matter. It is a form of consciousness that is finite in terms of apparent size and apparent information content but still not computable, because the qualia of matter itself cannot be substituted. I don't believe in primitive matter, but I believe in stuff as a sensation of stuffiness. Bruno Marchal wrote: I really try to understand. Sometimes it seems you argue against comp, and sometimes it seems you argue against the proof that comp entails the Platonist reversal (to be short). Well, actually I am arguing agains both, but relevant to your argument is just that
Re: The consciousness singularity
On Dec 2, 12:28 pm, meekerdb meeke...@verizon.net wrote: On 12/2/2011 6:22 AM, Craig Weinberg wrote: I don't think it's a hypothesis though. The brain IS our only known source of experience. We can change our experience by changing our brain and vice versa. The same cannot be said for anything else in the universe, can it? I can change my experience by moving to Canada, getting a new wife, or putting brandy in my coffee. You can't experience any of those changes without a brain. If I could change your experience by putting brandy in my coffee then you would have a point. I can't though. I can't change anyones experience unless I do something that changes their brain. Craig -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: The consciousness singularity
On 12/2/2011 10:44 AM, Craig Weinberg wrote: On Dec 2, 12:28 pm, meekerdbmeeke...@verizon.net wrote: On 12/2/2011 6:22 AM, Craig Weinberg wrote: I don't think it's a hypothesis though. The brain IS our only known source of experience. We can change our experience by changing our brain and vice versa. The same cannot be said for anything else in the universe, can it? I can change my experience by moving to Canada, getting a new wife, or putting brandy in my coffee. You can't experience any of those changes without a brain. If I could change your experience by putting brandy in my coffee then you would have a point. I can't though. I can't change anyones experience unless I do something that changes their brain. Craig The point is that any change in our experience does change our brain, otherwise we wouldn't experience it. I don't know why that should make it the source of our experience. Actually I could change your experience by putting brandy in your coffee; in fact I'm changing your experience right now as you read these words. Brent -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: The consciousness singularity
On Dec 2, 3:22 pm, meekerdb meeke...@verizon.net wrote: On 12/2/2011 10:44 AM, Craig Weinberg wrote: On Dec 2, 12:28 pm, meekerdbmeeke...@verizon.net wrote: On 12/2/2011 6:22 AM, Craig Weinberg wrote: I don't think it's a hypothesis though. The brain IS our only known source of experience. We can change our experience by changing our brain and vice versa. The same cannot be said for anything else in the universe, can it? I can change my experience by moving to Canada, getting a new wife, or putting brandy in my coffee. You can't experience any of those changes without a brain. If I could change your experience by putting brandy in my coffee then you would have a point. I can't though. I can't change anyones experience unless I do something that changes their brain. The point is that any change in our experience does change our brain, otherwise we wouldn't experience it. That's my point exactly. This is not the case with any other object in the cosmos. A change in our experience does not change a shoe and changing a shoe does not change our experience - it's the brain and only the brain which fits this description, making it the only known source of our experience. I don't know why that should make it the source of our experience. If grape juice comes from a grape and all grapes produce grape juice, is it not fair to say that grapes are the only known source of grape juice? I can't apply the same logic to juice in general because lots of things could be said to be a source of juice, but we don't have a single thing that we can say is having a human experience without a human brain being involved. Actually I could change your experience by putting brandy in your coffee; in fact I'm changing your experience right now as you read these words. Right but you can't change my experience by putting brandy in *your* coffee. Of course your words are changing my experience and my brain because I am able to read them, but I couldn't read them if my brain had no access to them. Craig -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
