Works Cited

For an extreme example, those neurons of the cerebellum known as Purkinje cells have about 80,000 excitatory synpatic endings.

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The second point is that, at least in my own opinion, parallel classical computation is very unlikely to hold the key to what is going on with our conscious thinking. A characteristic feature of conscious thought (at least when one is in a normal psychological state, and not the subject of a 'split-brain' operation!) is its 'oneness'--as opposed to a great many independent activities going on at once.

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Suppose that a photon arrives at the retina, having been previously reflected off a half-silvered mirror. Its state involves a complex linear superposition of its striking a retinal cell and of its not striking a retinal cell and instead, say, travelling out of the window into space. When the time is reached at which it might have struck the retina, and so long as the linear rule U of quantum theory holds true (i.e. deterministic Schroedinger state-vector evolution...), we can have a complex linear superposition of a nerve signal and not a nerve signal. While this impinges upon the subject's consciousness, only one of these two alternatives is perceived to take place, and the other quantum procedure R (state-vector reduction...) must have been effected... Thus, according to the viewpoint I have been putting forward, the procedure R could have been already effected well before we perceive the flash of light, or not, as the case may be. On this viewpoint, our consciousness is not needed in order to reduce the state-vector!

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...consider the ruthless process of natural selection. View this process in the light of the fact that, as we have seen in the last chapter, not all of the activity of the brain is directly accessible to consciousness. Indeed, the 'older' cerebellum--with its vast superiority in local density of neurons--seems to carry out very complex actions without consciousness being directly involved at all. Yet Nature has chosen to evolve sentient beings like ourselves, rather than to remain content with creatures that might carry on under the direction of totally unconscious control mechanisms. If consciousness serves no selective purpose, why did Nature go to the trouble to evolve conscious brains when non-sentient 'automaton' brains like cerebella would seem to have done just as well?

...why should natural selection bother to favour such a race of individuals, when surely the relentless free market of the jungle should have rooted out such useless nonsense long ago!

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One view that I have heard expressed is that awareness might be of an advantage to a predator in trying to guess what its prey would be likely to do next by 'putting itself in the place' of that prey. By imagining itself to be the prey, it could gain an advantage over it.

It may well be that there is some partial truth in this idea, but I am left very uneasy by it. In the first place, it supposes some pre-existing consciousness on the part of the prey itself, for it would hardly be helpful to imagine oneself to 'be' an automaton, since an automaton--by definition unconscious--is not something that is possible to 'be' at all!...

The idea alluded to above seems to relate to a point of view about consciousness that one often hears put forward, namely that a system would be 'aware' of something if it has a model of that thing within itself, and that it becomes 'self-aware when it has a model of itself within itself... Despite the claims that seem to be frequently made, the real issues concerning awareness and self-awareness are, in my opinion, hardly being touched by considerations of this kind. A video-camera has no awareness of the scenes it is recording: nor does a video-camera aimed at a mirror possess self-awareness.

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It has, indeed, been an underlying theme of the earlier chapters that there seems to be something non-algorithmic about our conscious thinking. In particular, a conclusion from the argument in Chapter 4, particularly concerning Goedel's theorem, was that, at least in mathematics, conscious contemplation can sometimes enable one to ascertain the truth of a statement in a way that no algorithm could... Indeed, algorithms, in themselves, never ascertain truth! It would be as easy to make an algorithm produce nothing but falsehoods as it would be to make it produce truths. One needs external insights in order to decide the validity or otherwise of an algorithm... I am putting forward the argument here that it is this ability to divine (or 'intuit') truth from falsity (and beauty from ugliness!), in appropriate circumstances that is the hallmark of consciousness.

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Why do I say that the hallmark of consciousness is a non-algorithmic forming of judgements? Part of the reason comes from my experiences as a mathematician. I simply do not trust my unconscious algorithmic actions when they are inadequately paid attention to by my awareness. Often there is nothing wrong with the algorithm as an algorithm, in some calculation that is being performed, but is it the right algorithm to choose, for the problem in hand?

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One may start from some axioms, from which are to be derived various mathematical propositions. The latter procedure may indeed be algorithmic, but some judgement needs to be made by a conscious mathematician to decide whether the axioms are appropriate.

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To my way of thinking, there is still something mysterious about evolution, with its apparent 'groping' towards some future purpose. Things at least seem to organize themselves somewhat better than they 'ought' to, just on the basis of blind-chance evolution and natural selection. It may well be that such appearances are quite deceptive. There seems to be something about the way that the laws of physics work, which allows natural selection to be a much more effective process than it would be with just arbitrary laws.

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We must first consider the possibility that different mathematicians use inequivalent algorithms to decide truth. However, it is one of the most striking features of mathematics (perhaps almost alone among the disciplines) that the truth of propositions can actually be settled by abstract argument! A mathematical argument that convinces one mathematician--providing that is contains no error--will also convince another, as soon as the argument has been fully grasped. This also applies to the Goedel-type propositions.

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Poincaré describes, first, how he had intensive period of deliberate, conscious effort in his search for what he called Fuchsian functions, but he had reached an impasse. Then:

"...I left Caen... The incidents of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go to some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea: I should not have had time, as upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for convenience sake, I verified the result at my leisure."

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Aesthetics in the arts is a sophisticated subject, and philosophers have devoted lifetimes to its study. It could be argued that in mathematics and the sciences, such criteria are merely incidental, the criterion of truth being paramount. However, it seems to be impossible to separate one from the other when one considers the issues of inspiration and insight. My impression is that the strong conviction of the validity of a flash of inspiration (not 100 per cent reliable, I should add, but at least far more reliable than just chance) is very closely bound up with its aesthetic qualities. A beautiful idea has a much greater chance of being a correct idea than an ugly one.

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A striking example is given vividly by Mozart:

"When I feel well and in a good humor, or when I am taking a drive or walking after a good meal, or in the night when I cannot sleep, thoughts crowd into my mind as easily as you could wish. Whence and how do they come? I do not know and I have nothing to do with it. Those which please me I keep in my head and hum them; at least others have told me that I do so. Once I have my theme, another melody comes, linking itself with the first one, in accordance with the needs of the composition as a whole: the counterpoint, the part of each instrument and all the melodic fragments at last produce the complete work. Then my soul is on fire with inspiration."

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To speak of 'Plato's world' at all, one is assigning some kind of reality to it which is in some way comparable to the reality of the physical world. On the other hand, the reality of the physical world itself seems more nebulous than it had seemed to be before the advent of the SUPERB theories of relativity and quantum mechanics... The very precision of these theories has provided an almost abstract mathematical existence for actual physical reality. Is this in any way a paradox? How can concrete reality become abstract and mathematical? This is perhaps the other side of the coin to the question of how abstract mathematical concepts can achieve an almost concrete reality in Plato's world.

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Of course mathematicians sometimes make mistakes. It seems that Turing himself believed that this was where the 'loophole' to the Goedel-type arguments against human thinking being algorithmic lay. But it seems unlikely to me that human fallibility is the key to human insight!

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It seems to me that the fact that animals require sleep in which they appear sometimes to dream (as is often noticeable with dogs) is evidence that they can possess consciousness. For an element of consciousness seems to be an important ingredient of the distinction between dreaming and non-dreaming sleep.

Roger Penrose, The Emperor's New Mind