In my view, Penrose's theory that computation could not explain human thought was based on the flawed idea that there exist problems that humans could solve which no computer could. I prepared the following to offer my explanation for why this is an unsupported supposition:
------------- In general, modern theories of consciousness react to the consequences of the Church-Turing thesis in one of three ways, which has yielded the following three philosophical camps: Non-computable physicists Weak AI proponents Computationalists Believe human thought involves physical processes that are non-computable, and therefore conclude that it’s impossible to replicate the behavior of a human brain using a computer. Believe the behavior of the human brain can be replicated by computer, but assume such a reproduction, no matter how good, would not possess a mind or consciousness. Believe the behavior of the human brain can be replicated by a computer, and assume that when the reproduction is sufficiently faithful, it possesses a mind and conscious. Descartes reason for rejecting mechanism was that he believed no machine could be engineered with enough complexity to account for human language and thought. Yet today, it is nearly universally accepted (among biologists, physicists, cognitive scientists and philosophers) that such machines are possible, for unless human beings violate the laws of physics, we ourselves are examples of such machines: machines that can think. Yet, not everyone agrees that the behaviors exhibited by we “human machines” can be replicated by the kind of machines Turing described. It may be that some element of human reasoning relies on a physical process that is not computable. For example, something that is infinite in complexity or fundamentally nondeterministic. The English mathematician and physicist Roger Penrose falls into this camp. He has argued that human thought cannot be emulated by any algorithm or computer. This argument, known as the Lucas-Penrose argument, is based on the observation that there are certain problems which no computer is able to solve. A famous example of which is the halting problem, first described by Turing in his 1936 paper. The halting problem is: given an arbitrary computer program, determine whether that program will, if started, ever complete (halt). In the same paper, Turing proved that regardless of the approach (algorithm) employed by the computer, there would always exist some programs for which it was impossible to answer the halting question. The British philosopher John Lucas and Roger Penrose both considered this limitation of computers to be clear evidence of the superiority of man over (Turing) machines. It further inspired Penrose to propose, in his 1989 book The Emperor’s New Mind, that non-algorithmic (not Turing emulable) processes must be involved in the brain. This would have to be in order to explain why humans could solve these problems while computers could not. Yet, all currently known laws of physics (with the notable exception of wave-function collapse) are deterministic, and can be modeled by Turing machines. This led Penrose to propose that the currently known laws of physics are inadequate to explain the workings of human thought, and that some undiscovered physics holds the key to understanding the workings of the mind. Penrose, together with the anesthesiologist and consciousness researcher Stuart Hameroff, speculated that the brain may harness quantum mechanical effects, such as wave-function collapse, or somehow makes use of (the presently not well understood) quantum gravity to achieve its unique reasoning abilities. However, Lucas’s and Penrose’s argument contains an overlooked assumption: that humans were capable of solving the halting problem. Certainly, a human programmer or team of programmers can examine a program and figure out whether or not it will ever complete. Even if it is a very complex program, given enough time, or a skilled enough team, a determination can eventually be made. Or so they believed. As it turns out, it is quite easy to write a computer program such that whether or not it ever completes depends on some, as of yet unknown, result in mathematics. For example, in number theory the Goldbach conjecture hypothesizes that every even number greater than 2 can be represented as the sum of two prime numbers. Despite countless efforts by mathematicians since 1742, and even an offered prize of $1,000,000 to anyone who could prove or disprove it, the truth or falsehood of the conjecture remains unknown. A program based on the Goldbach conjecture might look something like this: Step 1: Set X = 4 Step 2: Set R = 0 Step 3: For each Y from 1 to X, if both Y and (X – Y) are prime, set R = 1 Step 4: If R = 1, Set X = X + 2 and go to Step 2 Step 5: If R = 0, print X and halt This simple program searches for a counterexample to the Goldbach conjecture. If the Goldbach conjecture is false, this program (given enough time) will disprove it by printing an even number that can’t be represented as the sum of two primes and entitle its discoverer to everlasting fame. As of 2014, every even number up to 4,000,000,000,000,000,000 has been checked, and so far no exceptions to the rule have been found. But the fact that an exception hasn’t been found doesn’t mean there isn’t one. There remain quite a few numbers left to try. Yet if no exceptions exist, this program will go on forever, stuck in an infinite loop in which the computer searches for a number that doesn’t exist (as fruitlessly as trying to find an even number that ends in 5). We see that if the Goldbach conjecture is false, the program finds an exception and halts, but if it’s true then the program loops forever and never reaches the halted state. With this example we see that despite the simplicity of the program, determining whether or not this program halts has yet to be solved by any person. And since proving that it halts or doesn’t is equivalent to solving the Goldbach conjecture, there were a million dollars available to anyone who could have proved whether or not the above program halts. Yet, despite being offered for two years (between 2000 and 2002) the prize went unclaimed. Lucas and Penrose assumed the impossibility of the halting problem didn’t apply to humans, but as it happens, it’s impossibility applies equally to humans as it does to computers. In 1931, the Austrian logician Kurt Gödel proved that within any given mathematical framework some mathematical statements cannot be proved or disproved. When a program’s continued operation depends on statements that are unprovable under some mathematical framework, then solving the halting problem is equally impossible for humans as it is for any computer when working within that framework. So the halting problem’s impossibility does apply to humans, and there never was any evidence to suggest we possess special, inherently non-computable, mental abilities or that there remains some undiscovered non-computable physics that gets put to use by our brains. Penrose’s idea that human thought requires an undiscovered physics was based on an invalid assumption. Accordingly, this theory has remained a fringe view, and has attracted criticism from computer scientists and physicists. The artificial intelligence researcher Marvin Minsky said, “[Penrose] tries to show, in chapter after chapter, that human thought cannot be based on any known scientific principle.”, and elaborated, “one can carry that quest too far by only seeking new basic principles instead of attacking the real detail. This is what I see in Penrose’s quest for a new basic principle of physics that will account for consciousness.” Max Tegmark also tested the feasibility Penrose’s and Hameroff’s idea that the brain might harnesses wave-function collapse as part of its operation. Tegmark determined that particles in the brain can remain in a state of superposition for at most one 10 trillionth of a second. This time period is billions of times shorter than the fastest neurons can fire and thus the effect could have no meaningful impact on the behavior of neurons in the brain. Ongoing work by IBM's Blue Brain Project, which has the stated goal of reverse engineering the mammalian brain, appears to further refute the idea that what the brain does is not computable. The director of the Blue Brain Project, Henry Markram believes that he and his team have created the first biologically accurate model of a small section of a rat’s brain. Applying simulated electrical impulses to their neural model, the simulated neurons began to respond just like a real neural circuit: clusters of connected neurons fired together, the cells wired themselves together and large scale patterns of activity flashed across the entire model. This model did not include any quantum mechanical effects, (nor any unknown physical laws), yet its behavior matched the observations of groups of real neurons. The positive results of the Blue Brain Project suggest that the brain operates according to presently known physical laws, all of which are computable. Given this, together with the realization that the halting problem is generally insoluble (not just insoluble for computers), we can, with some confidence, reject the hypothesis that human thought requires non-computable physics. In doing so, two possibilities remain: weak AI and computationalism. Jason On Tue, Apr 4, 2017 at 9:47 AM, David Nyman <da...@davidnyman.com> wrote: > I've been thinking about the Lucas/Penrose view of the purported > limitations of computation as the basis for human thought. I know that > Bruno has given a technical refutation of this position, but I'm > insufficiently competent in the relevant areas for this to be intuitively > convincing for me. So I've been musing on a more personally intuitive > explication, perhaps along the following lines. > > The mis-step on the part of L/P, ISTM, is that they fail to distinguish > between categorically distinct 3p and 1p logics which, properly understood, > should in fact be seen as the stock-in-trade of computationalism. The > limitation they point to is inherent in incompleteness - i.e. the fact that > there are more (implied) truths than proofs within the scope of any > consistent (1p) formal system of sufficient power. L/P point out that > despite this we humans can 'see' the missing truths, despite the lack of a > formal proof, and hence it must follow that we have access to some > non-algorithmic method inaccessible to computation. What I think they're > missing here - because they're considering the *extrinsic or external* (3p) > logic to be exclusively definitive of what they mean by computation - is > the significance in this regard of the *intrinsic or internal* (1p) logic. > This is what Bruno summarises as Bp and p, or true, justified belief, in > terms of which perceptual objects are indeed directly 'seen' or > apprehended. Hence a computational subject will have access not only to > formal proof (3p) but also to direct perceptual apprehension (1p). It is > this latter which then constitutes the 'seeing' of the truth that > (literally) transcends the capabilities of the 3p system considered in > isolation. > > If the foregoing makes sense, it may also give a useful clue in the debate > over intuitionism versus Platonism in mathematics. Indeed, perceptual > mathematics (as we might term it) - i.e. the mathematics we derive from the > study of the relations obtaining between objects in our perceptual reality > - may well be "considered to be purely the result of the constructive > mental activity of humans" (Wikipedia). However, under computationalism, > this very 'perceptual mathematics' can itself be shown to be the > consequence of a deeper, underlying Platonist mathematics (if we may so > term the bare assumption of the sufficiency of arithmetic for computation > and its implications). > > Is this intelligible? > > David > > -- > 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 everything-list+unsubscr...@googlegroups.com. > To post to this group, send email to everything-list@googlegroups.com. > Visit this group at https://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/d/optout. > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. 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