In the soft realm of cognitive science here is a quote on mathematically modelling consciousness. It is part of a book review from Bernard Baars (UCB, Cog Sci) of Penrose's next work, Shadows of the Mind, 1994 (SOTM):
``3.3 The problem is of course that a mathematical model is only as good as its assumptions, and those depend upon the quality of the evidence. The whole Turing Machine debate and its putative implications for consciousness is in my opinion a great distraction from the sober scientific job of gathering evidence and developing theory about the psychobiology of consciousness (e.g., Baars, 1988; 1994). The notion that the Turing argument actually tells us something scientifically useful is amazingly vulnerable. After all, the theory assumes an abstract automaton blessed with infinite time, infinite memory, and an environment that imposes no resource constraints. The brain is a massively parallel organ with 100 billion simultaneously active neurons, but the Turing Machine is at the extreme end of serial machines. This appears to be the reason why discussion of the Turing topic appears nowhere in the psychobiological literature. It seems primarily limited to philosophy and the general intellectual media.
3.4 Finally, it turns out that all current cognitive and neural models are formal Turing equivalents. That means the mathematical theory is useless in the critical task of choosing between models that are quite different computationally and on the evidence. It does not distinguish between neural nets and symbolic architectures for example, as radically different as they are in practice. But that is exactly the challenge we face today: choosing between theories based on their fit with the evidence. Here the theory of automata is no help at all.
3.5 A small but telling fact about Penrose's book caught my attention: of its more than 400 references, fewer than forty address the psychology or biology of consciousness. But all our evidence on the subject is psychological and, to a lesser extent, biological! It appears that Penrose's topic is not consciousness in the ordinary psychoneural sense, like waking up in the morning from a deep sleep or listening to music. How the positive proposals in SOTM relate to normal psychobiological consciousness is only addressed in terms of a technical hypothesis. Stuart Hameroff, an anesthesiologist at the University of Arizona currently working with Penrose, has proposed that general anesthetics interact with neurons via quantum level events in neural microtubules, which transport chemicals down axons and dendrites. It is an interesting idea, but it is by no means accepted, and there are many alternative hypotheses about anesthetics. But it is a real hypothesis: testable, relevant to the issue of consciousness, and directly aimed at the quantum level.
3.6 Penrose calls attention to the inability of Turing Machines to know when to stop a possibly nonterminating computation. This is a form of the Goedel Theorem, from which Penrose draws the following conclusion: "Human mathematicians are not using a knowably sound algorithm in order to ascertain mathematical truth." That is to say, if humans can propose a Halting Rule which turns out to be demonstrably correct, and if we take Turing Machines as models of mathematicians, then the ability of mathematicians to come up with Halting Rules shows that their mental processes are not Turing-computable.
3.7 I'm troubled by this argument, because all of the cognitive studies I know of human formal reasoning and logic show that humans will take any shortcut available to find a plausible answer for a formal problem; actually following out formalisms mentally is rare in practice, even among scientists and engineers. Human beings are not algorithmic creatures; they prefer by far to use heuristic, fly-by-the-seat-of-your-pants analogies to situations they know well. Even experts typically use heuristic shortcuts. Furthermore, the apparent reductio of Penrose's claim has a straightforward alternative explanation, namely that one of the premises is plain wrong. The implication psychologically is not that people are fancier than any Turing Machine, but that they are much sloppier that any explicit algorithm, and yet do quite well in many cases...''
(http://psyche.cs.monash.edu.au/v2/psyche-2-08-baars.html)
And, it just so happens that in order to escape the darkening glass of America, Bush, Strauss and other musings, I got George Lakoff's `Where Mathematics Comes From' (2000) last Saturday. It attempts to link a variety of simple linguistic and spatial metaphors to the ZF axioms on sets and show how we understand these through these metaphors, and hence understand numbers, counting, and infinity. He does a very nice job but then I am only up to page 69.
On the other hand, I am a little disappointed because Lakoff chose to ignore another possible route of understanding mathematics, mind and world, via spatial symmetries, groups, gravity, motion and the human body. This is my preferred route because it completely dodges the linguistic track (while baring on it) and can be easily linked to neurophysiology, animal studies, biology, AI studies, robotics, and a host of other fields in physical science.
The key trick is to understand that the physical world is relative to biological evolution apriori structured and therefore living things don't need to creat these structures. They just need to adapt to them. We come along and see the miracle adaptations and presume these adaptations somehow `contain' the structures and are part of the organism. In fact there is no need for this containment and it is quite possible what we are discoverying are merely adaptive template fits to externally existing physical systems of order.
This adaptive fiting accounts for the problem that it seems we can only get so much out of a single locus, the organism. And we find it wanting, mysterious, somehow incapable of revealling its ultimate ordering systems. It may be that these systems which always seem to have a certain quality of transcendence don't exist embodied in a single locus in such a circumscriped way. Rather they exist in their primodial first instance and always in the physical processes of the world, which are then given a greater articulation in such fabulously complex developments as evolution and consciousness. Attempting to find structure in either of these latter articulations, is always doomed, because the fields themselves leave out their interlocutor, the physical world and its structures.
Ultimately, this is an argument for a new kind of Biophysics. The point isn't really to apply physics to some biological content. It is rather to study the interplay between existing physical systems and their embedded biological constituents.
One example. Penrose goes off on an explantion of how quantum phenomenon are at work in the molecular construction of micro-tubules inside cells (relates to axion formation and growth). While it is of course interesting and important to understand the physics and chemistry of micro-tubule formation, this sort of misses the point, from a biological view. The important point in micro-tubule formation is that these act as struts, a micro-skeleton to support the cell. The key physics here is found in how the cell has adapted and grows in relation to the forces of gravity and motion.
Chuck Grimes