The Black Box Brain II
My last post was about how the brain-as-Turing-Machine doesn't help and in fact, can be said to lose the notion of any objective concept of the word objective itself.
Casting the brain as a Turing Machine (TM) helps us understand our processing a little more, but only if we think of the brain as not a Turing Machine, first. (In other words, we must think ourselves capable of determining non-trivial properties.) Otherwise, whether we are thinking what we think we are thinking would become problematical.
Think of what Im saying here: Even were thinking a precise and quantifiable process , we would never compute the TM: ComputesThinking because we could not ever tell which TMs would halt on the problem. Now, add to that picture, our ignorance of what thinking, or what we would quantify as thinking.
So to observe the brain as TM, the observer must be free from the rules of TMs, to some degree. Otherwise, we assume a bunch of stuff that is not provable: for example, that the brain TM computes the predicate “x is objective”
Of course, we cannot think as concretely about the processing power of the brain without this model. And perhaps, without it, we would flounder in the muddy waters of philosophy’s Consciousness. So whether or not this model fits, we will examine the brain as an undesigned computer.
The “value of limited knowledge” may sound like a confusing phrase. So let me try to explain in the next couple of paragraphs. (Anybody with a good level of understanding of computers, may find the next few paragraphs boring.)
The electronic-circuit-to-bit transformation helps to illustrate this concept of the value of limited knowledge. At the base of all computers, are electrons and their near-random behavior. Yet somehow, through the magic of circuits, those inconsistencies are smoothed out and produce the abstraction known as a bit. Thats precisely the way the machine looks at the product: They are bitsnothing more. They are abstracted above their mere circumstances and taken as something of a different kind.
The application of electronic circuits has an input and an output. We start from circuit technology. What we require from these circuits is that the wiring for a bit stays in the state in which it was put. It needs to change when it is instructed and keep the state until its instructed to change it. That consistency creates the output concept bit. Just like that every layer of technology layered on top of the basic machine, takes in its wild and woolly nature, and hands to other layers a simplified interface of that level.
Thus, you do not need to be an expert atomic physicist to design a circuit. And you do not need the knowledge to design a circuit to design a machine. And you do not need need the knowledge of somebody who designs an instruction set to program in machine code. (See abstraction at Wikipedia.org.)
And so it goes all the way up to you. Because of all the work that was done up to now, we require you only to know your way around windows, and perhaps the URL for Google. With all that you can make your way around the net, feeding your brain its own process.
So the computer is a black box to many and relies on the concept of black box to limit complexity of each task in almost all phases. But somebody knows each part. It’s all written down in a manual somewhere. So you can learn about electrons, and how circuits are made, and how circuits are turned into the abstractions of processors and storage devices (which is about the lowest level most of us Software designers ever tend to think about). And after moving on, you could learn how instructions are designed from bits. How software is composed from higher and higher abstractions.
If you could spend the time, and had the brains and money for it, you could know the computer from the electron to the CSS stylesheet your browser is using to tell it how to display this page. Theoretically, you could know it all. But you dont need to.
Casting the brain as a Turing Machine (TM) helps us understand our processing a little more, but only if we think of the brain as not a Turing Machine, first. (In other words, we must think ourselves capable of determining non-trivial properties.) Otherwise, whether we are thinking what we think we are thinking would become problematical.
Think of what Im saying here: Even were thinking a precise and quantifiable process , we would never compute the TM: ComputesThinking because we could not ever tell which TMs would halt on the problem. Now, add to that picture, our ignorance of what thinking, or what we would quantify as thinking.
So to observe the brain as TM, the observer must be free from the rules of TMs, to some degree. Otherwise, we assume a bunch of stuff that is not provable: for example, that the brain TM computes the predicate “x is objective”
Of course, we cannot think as concretely about the processing power of the brain without this model. And perhaps, without it, we would flounder in the muddy waters of philosophy’s Consciousness. So whether or not this model fits, we will examine the brain as an undesigned computer.
InfiniteMonkeys.com
The brain, unlike the computer, is a black box. Sure, the computer is a black box to some. And sure, Information Technology advances with the value of limited knowledge.The “value of limited knowledge” may sound like a confusing phrase. So let me try to explain in the next couple of paragraphs. (Anybody with a good level of understanding of computers, may find the next few paragraphs boring.)
The electronic-circuit-to-bit transformation helps to illustrate this concept of the value of limited knowledge. At the base of all computers, are electrons and their near-random behavior. Yet somehow, through the magic of circuits, those inconsistencies are smoothed out and produce the abstraction known as a bit. Thats precisely the way the machine looks at the product: They are bitsnothing more. They are abstracted above their mere circumstances and taken as something of a different kind.
The application of electronic circuits has an input and an output. We start from circuit technology. What we require from these circuits is that the wiring for a bit stays in the state in which it was put. It needs to change when it is instructed and keep the state until its instructed to change it. That consistency creates the output concept bit. Just like that every layer of technology layered on top of the basic machine, takes in its wild and woolly nature, and hands to other layers a simplified interface of that level.
Thus, you do not need to be an expert atomic physicist to design a circuit. And you do not need the knowledge to design a circuit to design a machine. And you do not need need the knowledge of somebody who designs an instruction set to program in machine code. (See abstraction at Wikipedia.org.)
And so it goes all the way up to you. Because of all the work that was done up to now, we require you only to know your way around windows, and perhaps the URL for Google. With all that you can make your way around the net, feeding your brain its own process.
So the computer is a black box to many and relies on the concept of black box to limit complexity of each task in almost all phases. But somebody knows each part. It’s all written down in a manual somewhere. So you can learn about electrons, and how circuits are made, and how circuits are turned into the abstractions of processors and storage devices (which is about the lowest level most of us Software designers ever tend to think about). And after moving on, you could learn how instructions are designed from bits. How software is composed from higher and higher abstractions.
If you could spend the time, and had the brains and money for it, you could know the computer from the electron to the CSS stylesheet your browser is using to tell it how to display this page. Theoretically, you could know it all. But you dont need to.
posted by ChristianPhilosopher at 7:30 PM
4 Comments:
The wording of some paragraphs is strange, but I take it that you mean that thinking and computing are different because we do not have a computable algorithm that will allow us to know when any given TM will halt. This actually makes it more like a mind. We do not have an effective procedure for determining what any given mind will think at some time t1 after a given time t.
True - the mind is so complex that we do not understand all of its workings. We do for computers. But that does not entail that we cannot (given enough time and resources) completely understand the mind as well as we understand the workings of a computer. If you want to appeal to some quantum uncertainty in defense of the mind then the same applies to computers at a subatomic level.
Obviously our understanding of the mind increases manifold each year. It seems likely that we'll have quite a good understanding of it in the next 100 years. After all - neuroscience is still in its infancy.
Reply to lumpypeacoat:
“I take it that you mean that thinking and computing are different because we do not have a computable algorithm that will allow us to know when any given TM will halt. This actually makes it more like a mind.”
And I believe I said something similar. TM-analysis is useful and allows us to make concrete insights into processing power. But, at some level, we must be able to make meta-observations. We must simultaneously consider ourselves as a distinct case of machine that computes the predicate “x is useful” without proof (because we are implying usefulness in our analysis of TM-based analysis)--simpy because we cannot know the “TM-number” for my brain or your brain or Von Nueman's brain. Since it is unknown the worse-case idea is that we can random. Since it is a random TM with a random input (whatever coding the predicate happens to be, should the brain be a TM), we have the pessimism that we will never be able to prove non-trivial properties of processes of the brain.
“This actually makes it more like a mind. We do not have an effective procedure for determining what any given mind will think at some time t1 after a given time t.”
True. But most things that we don’t know are alike in that they are not known and are therefore unpredictable. The unpredictability of what is not fully understood is trivial.
Today, theorist most often talk the key of Science being predictability. Success in prediction implies the fitness of the model. Therefore being unable to predict and not knowing are one and the same thing, according to this theory of Scientific value.
Therefore all phenomena that are not understood change unpredictably, because if we could predict their behavior, we would be able to say that we know them.
The way that moderns want to leverage ignorance into a type of knowing continues to confound me. See, one of the things that TM-theory suggests to me is that without knowledge, many problems become undecidable. What bewilders me is how a system acquires knowledge without any to depend on.
If the brain is “equivalent” to a TM (and then what is the algorithm of the predicate “is equivalent”?), it is sophisticated enough to know how to look through code (an algorithm/TM) and figure in many cases whether it will loop forever. In fact the easiest loops be spotted and we must accept--without proof--that the brain computes the the majority of the cases of “x will loop”.
I’m talking about frontiers of knowledge here, I’m not speaking against what I would call heuristics of analysis. It would be pretty odd indeed to suggest that the brain is TM-equivalent because we can regularly predict that we cannot predict the behavior of either. What kind of predictive model is that?
“True - the mind is so complex that we do not understand all of its workings. We do for computers. But that does not entail that we cannot (given enough time and resources) completely understand the mind as well as we understand the workings of a computer.”
That type of pollyana optimism does not cut it for me. When Russell wrote Principles of Mathematics at the turn of the 20th century, he expressed the type of optimism about Mathematical proofs that later made him suicidal upon publication of Goedel’s work. And I believe that Goedel himself went into this work, not desiring to prove incompleteness, but the opposite.
But there is a big difference between understanding a computer that you’ve designed and understanding what you do not yet know to be a computer. The computer we have designed, was designed with a basis of our predictions on how physical media would perform. In fact were we TM-equivalent, the processing power gained in computers are nothing but a subroutine of our processing power. TMs are always handing off processing to sub TMs—that’s how the proof against halting was computed, with a TM that called halt( t, t ) within it. Thus whatever the blackbox of halt( m, i ) was, we constructed a TM that created a contradiction.
“If you want to appeal to some quantum uncertainty in defense of the mind then the same applies to computers at a subatomic level.”
Roger Penrose, in The Emperor’s New Mind argued that brains may use quantum forces in it’s decision making. We consider such “race conditions” a design defect in computers—so they flat out do not use them. Therefore, where Penrose’s model implies that the brain exploits sub-atomic processes, the computer is designed to resist them. So I strongly doubt that we have an equivalent risk or benefit, in Penrose’s model.
But this is no retread of Penrose here. Quantum mechanisms do make the brain harder to decipher, but it does not preclude an equivalent sequence of events. For in fact, if the brain acheives quantum state S the process is either determinant or not. You cannot use quantum processes, unless quantum state S has effect E. Again using Penrose’s model, the brain decides between two states giving each an equal try in a quantum potentialities and decides on potentials based and collapses the state. But how do you decide between two random states, unless different states provided, with reasonable frequency, different outcomes?
Thus in each potentiality path, the brain achieved another state, and the brain picks one for reality. If such a state is determinant, then we do have a state of brain determined by location l1 as opposed to l2. We have all we need to construct a sequential equivalent of processing that actually happened—if we can ever figure out what actually happened. But QM does not make the brain-as-TM impossible in my mind, just extremely hard to verify.
“Obviously our understanding of the mind increases manifold each year. It seems likely that we'll have quite a good understanding of it in the next 100 years. After all - neuroscience is still in its infancy.”
Infants die, and their survivability rate determines the direction of evolution. This is the simplist way, although semi-equivocal, to express my skepticism of this.
"And I believe I said something similar. TM-analysis is useful..."
I'm having a difficult time understanding any of that. Perhpas you're suggesting that minds differ from machines because they of their ability for introspection. I do not see why machines cannot do the same. They can perform self-maintenance tasks, prove theorems about themselves, and use metalanguages to talk about other langauges (i.e. in relative consistency proofs). So as far as introspection is concerned, minds and machines do not seem different.
Also, 'being useful' is an interpreted property. Turing machins do not compute functions representing interpreted predicates like 'is useful'. If we want the machine to simulate minds, then we would likewise provide the machine with interpreted statements so that it could compute values for functions like f(x)={1 if x is useful, 0 otherwise. Then if we gave it as input 'religion', it could compute a 1 or 0 depending on whether it "thought" (was programmed to think that) religion is useful. If we programmed it to think it was useful, then it would output a 1. This is really all people do. Some of us are taught (programmed) that, say, religion is useful. So when asked about it, we might say 'religion is useful'.
"True. But most things that we don’t know are alike in that they are not known and are therefore unpredictable. The unpredictability of what is not fully understood is trivial."
Really? I don't know the electrical conductivity of peanut butter, but that does not mean that it's unpredictable. I could easily find out if I wanted to. There is a difference between solvable unknowns and unsolvable unknowns. Goldbach's conjecture is not readily solvable and hence unknown, but determining the electrical conductivity of peanut butter is solvable though currently unknown.
"Today, theorist most often talk the key of Science being predictability. Success in prediction implies the fitness of the model. Therefore being unable to predict and not knowing are one and the same thing, according to this theory of Scientific value."
That's partially, but not entirely, true. The hallmark of scientific laws is their power to make accurate predictions. But the appeal of science as a whole is its explanatory power. Presumably, the more we understand/know something, the more accurate our predictions of that thing are, but they are obviously not the same. Quantum physics has no explanatory power - it gives us no understanding/knowledge of subatomic particles, but it is damned good at making predictions.
"The way that moderns want to leverage ignorance into a type of knowing continues to confound me. See, one of the things that TM-theory suggests to me is that without knowledge, many problems become undecidable. What bewilders me is how a system acquires knowledge without any to depend on."
Untrue as I've mentioned above. We may not know that P, but P could still be decidable (though currently not solvable). Goldbach's conjecture is an example. It is either decidedly true or false, but at present we do not know which one.
"If the brain is “equivalent” to a TM (and then what is the algorithm of the predicate “is equivalent”?), it is sophisticated enough to know how to look through code (an algorithm/TM) and figure in many cases whether it will loop forever. In fact the easiest loops be spotted and we must accept--without proof--that the brain computes the the majority of the cases of “x will loop”."
Technically, and I don't want to get technical, algorithms do not compute predicates or formulas. They compute functions. The case of predicates is reducible to representing functions of those predicates, but they are not the same. If a TM is computing a partial function that is undefined for certain arguments, it will loop forever and not know that it will loop forever. Not only that, but a universal TM will not know either whether or not that TM will halt. Not only that (!), we will not know whether that TM will halt, just by looking at its machine table. The halting problem applies to minds just as much as it applies to machines.
"That type of pollyana optimism does not cut it for me. When Russell wrote Principles of Mathematics at the turn of the 20th century, he expressed the type of optimism about Mathematical proofs that later made him suicidal upon publication of Goedel’s work. And I believe that Goedel himself went into this work, not desiring to prove incompleteness, but the opposite."
Huh? Gödel's incompleteness theorems do not undermine PM (principia mathematica), they only undermine Hilbert's finitary program. Gödel's 1st incompleteness theorem only shows that no ONE theory of arithmetic is capable of expressing all arithmetical truths. That is not a problem. Because for any mathematical truth P, we can prove it in some theory T+P. While we may be able to construct a truth G in T+P that we cannot prove in T+P, we can prove it in T+P+G, and so on. Every truth is provable in some, but not one, system. That's what intuitionists have been urging all along. (On another note, Gödel did also prove the related completeness theorem, which just proves that all valid/invalid formulas are provable by finite methods.)
"The computer we have designed, was designed with a basis of our predictions on how physical media would perform. In fact were we TM-equivalent, the processing power gained in computers are nothing but a subroutine of our processing power."
There may be a point of complexity at which we will no longer be able to predict with 100% accuracy the future state of some TM, even though it is in fact predictable to an omniscient observer. We are obviously limited in our capacities, and programming a TM, albeit using a finite but extremely large algorithm, might yield a HUMANLY unpredictable TM. There is also something called genetic programming that simulates human ingenuity. And there is also something called a randomizing device. There are many methods of incorporating human unpredictability into TMs.
I said :
"And I believe I said something similar. TM-analysis is useful..."
You replied:
"I'm having a difficult time understanding any of that. Perhpas you're suggesting that minds differ from machines because they of their ability for introspection."
I am not suggesting that minds differ from machines in any distinct manner. But let's take another look at Rice, shall we?
Either the "ability for introspection" is a non-trivial property of the base algorithm our brain processes, or we can not tell whether or no the unknown TM, brain, has a non-trivial property. Therefore, we lack a distinct way to understand, introspection--except intuitively.
To discern that we can determine a characteristic such as "introspection" (or else what are we saying that has any verifiability?) we must have an algorithm performed by our brain which reaches some decision-state corresponding to our conclusion. However, the brain-universal-TM and the discern-imputation-of-introspection algorithm/TM must operate on the input of the brain (however encoded). Whether or not the brain succeeds in running this program, is unprovable from the perspective of TM-theory. We don't know what wffs we are dealing with , we don't know what axiomization we are dealing with. We have so many degrees of freedom on any side that it only makes sense to apply the pattern of anonymous TMs.
The audience for whom I originally intended this discussion are those that tend to be totally ignorant on problems with computability and decidability.
"I do not see why machines cannot do the same."
Yes but that is very different from knowing how they can do the same. But there is a simple reason why they might not be able to do the same. The brain might not be the equivalent of a TM.
Please understand this: This is not a demand that it is not a TM. It is not a a case that it is unreasonable to accept the possibility that the brain is a TM.
And that is why I have repeatedly say that comparing the brain and TMs is useful. But granted an algorithmic process to determine that thinking on the computer equaled thinking in the brain, we do not know that the algorithm (as it is unspecified) would halt on the input of the human brain and details of the computer circuits. Since their encoding is unknown, the input is arbitrary.
Therefore we have an arbitrary TM (the equivalance algorithm) and arbitrary inputs. It fits the undecidability idea very well.
"They can perform self-maintenance tasks, prove theorems about themselves, and use metalanguages to talk about other langauges"
And everything that "they" have done, comes at the explicit ordering of a programmer's brain. If we attribute "sentience" to the human brain (and that is also problematic) then it has been reproduced by design of a sentient TM.
A programmer solves the problem of increasing generality, and then the machine enacts the more general task. Self-maintenance is something that many programmers have realized they needed, and coded heuristics to figure out the problem.
"So as far as introspection is concerned, minds and machines do not seem different."
You have to admit that at least on the current level of technology they seem very different. Hey, I'm a programmer. I'm no luddite. I don't want to halt progress in AI. I would like my house to cook my dinner while I'm away--or tape my favorite shows. But just because a computer might be able to contribute software design analyses, does not mean that we can say in any real sense that it is thinking.
Whatever it is doing is enough. Turing's Test does not impress me and I may get into why. But I don’t think truth values should be determined based on the popularity of a conclusion among a mislead group of people. (Those who think that they cannot be talking to a machine.)
"Also, 'being useful' is an interpreted property. Turing machins do not compute functions representing interpreted predicates like 'is useful'."
Do we? If we do, then you are contending against what you imply elsewhere. If we do not, then I mean, in the sense that we can say "x is useful".
I don't know where you can argue that 1) the brain is similar to a machine, 2) hold it in your brain, 3) trust your brain's computation of that (predicate?) idea, 4) think that either the generalization "x is similar to y" is not being applied in some way, 5) think that there is some amount of accuracy in applying this general case, and then 6) argue that the brain which is equvialent to a TM (your position) does not compute predicates.
What are you doing then, when you say "brain is similar to machine"? Does it have any meaning at all?
I said:
"But most things that we don’t know are alike in that they are not known and are therefore unpredictable."
You replied:
"Really? I don't know the electrical conductivity of peanut butter, but that does not mean that it's unpredictable."
It is before you know it. You can make reasonable guestimates, but then you are predicting on knowledge of the conductivity of similar substances, not on ignorance. If we say "Hmmm. Oil conducts electricity like so, and corn fiber conducts like so, and so therefore, peanut butter can be expected to have x" then we are predicting from knowledge of peanut butter despite our lack of knowledge of the situation.
"I could easily find out if I wanted to. There is a difference between solvable unknowns and unsolvable unknowns."
I'm not saying that anything unknown to an individual is unpredictable. But you are missing a key point. While you introduce a point about the "appeal" of Science in terms of explanations, I'm talking about criteria for imputing scientific truths by predictability. Predictability establishes the "truth" of the model. Therefore, were we able to predict it, we could instantly be said to know it. If something is, given to be unpredictable, as you implied about minds, then we do not know it in that sense. If we compute peanut butter conductivity on known conductivity of oil and plant fiber, then the value of this approximation is established by pbc being reasonably within range of our prediction.
Scientifically, "unpredictable" and "unknown" are equivalent discriptions of the same thing. If we take Popper's word, whether or not a test can be formed to falsify the prediction is the entrance requirement--but it does not imply knowledge. The prediction of the falsifiable hypothesis establishes the explanation of the prediction as knowledge. Therefore, some amount of predictability is required to imply knowledge.
When you said we can't predict what a brain is thinking from t to t1, your pretty much saying we don’t know what it is thinking by applying our knowledge of the state at time t. There is a reasonable equivalence between "we cannot predict" and "we do not know".
Were there predictable behavior, we would say we know.
But your analysis could have been better. Since I assume that when you said that we don't know what the brain is thinking from t to t1, you weren’t saying that you did not know, but somebody else did, or somebody else knows how to spill a jar of peanut butter on a glass table and pass current through it to see what the brain is thinking at t1. Adding the points that went to personal knowledge and ultimately to knowable and unknowable things, brought us far afield.
To wit:
"Success in prediction implies the fitness of the model. Therefore being unable to predict and not knowing are one and the same thing, according to this theory of Scientific value."
You answer:
"That's partially, but not entirely, true."
Well, since I used phrases like "[theorists] most often talk", "implies", "according to this theory of Scientific value" I think I covered the case of partial truth well enough. And those qualifiers preclude against your reading that I implied any "entire truth" into it.
"But the appeal of science as a whole is its explanatory power."
I think you're wrong here. I think the chief appeal of science is the goodies (technology). Here predictability fits well: I expect my computer to come on and it does. I expect it to publish my words to the word, and it does. I expect to be able to program it and sometimes I can make it do some snazzy things.
Therefore the prediction has an intimate relation with my empirical interaction with the technological fruits. However, the consistent phenomena of falling down had many explanations.
In a utilitarian way, I can operate with the abstraction that machinery is, as long as it does what the designer expected when he designed it (because the materials tend to lend a pattern of expectation) and I expected when I bought it. The results matter, the interlacing together of details is only important insomuch as it does what I want.
However, explanation invites us to try to weave invisible threads among the pattern of repitition. The hicks weren't wrong in the frost reqularly appeared on the window pane. They weren't wrong if they were to invent scrapers to remove the frost. It was in explaining it by "Jack Frost" that they embarrass us moderns. And it was entirely that explanation that had no bearing of the treatment of frost on windows.
Once, men of good reason thought that the explanation that all bodies fell toward the center of the universe as a reason why things fell to the ground. It made a certain sense: the ground was where heavy materials tended to cluster, and they were impacted into the surface. And since many of them knew that the earth was a sphere (and they did), they were invoking a 3-dimensional center.
However, that the earth was the center of the earth was immaterial to technology that solved the problem of keeping things from falling and lifting up heavy things to keep them temporarily from center-seeking.
To me, as a phenomenologist, I find the value of narratives dubious--but nonetheless definitely human.
"Quantum physics has no explanatory power - it gives us no understanding/knowledge of subatomic particles, but it is damned good at making predictions."
Thanks for concluding with my point.
I said:
"The way that moderns want to leverage ignorance into a type of knowing continues to confound me. See, one of the things that TM-theory suggests to me is that without knowledge, many problems become undecidable. What bewilders me is how a system acquires knowledge without any to depend on."
You said:
"Untrue as I've mentioned above. We may not know that P, but P could still be decidable (though currently not solvable). Goldbach's conjecture is an example. It is either decidedly true or false, but at present we do not know which one."
Okay, perhaps I didn't make it clear enough. I did not mean that problems could alter their decidability based on the presence or lack of knowledge. That is one reading, but there are others. A problem is either decidable or not. Yet, without intimate knowledge of algorithm itself, we cannot help but treat them in their more general case. Their they fall prey to the Halting problem and Rice's Theorem. That makes it, from a current perspective an undecidable problem--when taken as an arbitrary process with arbitrary input.
"Technically, and I don't want to get technical, algorithms do not compute predicates or formulas."
Then how would we derive formulas? I assume that we use formulas, and I assume that formulas equal a real thing and not just a semantic cover. Therefore in that our brains are TM-equivalents and derive formulas, you must be at least partially wrong here.
"The halting problem applies to minds just as much as it applies to machines."
Perhaps but we don't experience any "looping". Also you said yourself that for a universal Turing machine to determine whether a Turing machine is looping is problematic (which is what I believe I implied in invoking the Halting Problem in the first place). Therefore it should be impossible at this level of detail to determine that the brain is looping--unless the Brain has more processing power than a TM.
But you overstate your case, I believe. It may not be impossible for a given TM to determine that many other TMs are looping, but it is provable that we cannot use an algorithm to identify it. The one thing that Halting addresses is that there is no determinant method to compute all halting verdicts.
The Halting Problem really only rules out one case. It says that a machine cannot determine every case of looping, including itself fed back into itself as data. We break one of these conditions (a minority of cases where looping can be discerned) and we do not have Turing's solution. Now the details of how exactly it only avoids computing itself on the input of itself is not something I would venture to guess. But if Halting( t, t ) causes the problem, then is is possible that Halting( i, j ) is not a problem for some TM where i =/= j.
But regardless of that we lack a foolproof way of determining which TM solves which problem--unless we construct an algorithm with our intention.
"Huh? Gödel's incompleteness theorems do not undermine PM (principia mathematica),"
Nor will you find that I implied that. And furthermore I wasn't speaking of his collaboration with Whitehead (The Principia) but his earlier, solo effort Principles of Mathematics published around 1902.
I did not say that the where Russell proved his case that Goedel unproved it. Much of what Russell said was sound—I have a philosopher's disagreement on transfinite numbers (which Whitehead shared to a certain degree). But in the book that I mentioned, Russell talks about what theories he believes to be the case, what ones he was "close" to proving, and what he has not proven "yet".
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