Nature is Not Mathematical

Pythagoras

This piece wouldn’t have been called ‘Who Says Nature is Mathematical?’ if it weren’t for the many other similar titles which I’ve seen. Take these examples: ‘Everything in the Universe Is Made of Math — Including You’, ‘What’s the Universe Made Of? Math, Says Scientist’ and ‘Mathematics — The Language of the Universe’. Indeed from Kurt Gödel to today’s Max Tegmark (who says that the “physical universe is mathematics in a well-defined sense”) and ontic structural realism, mathematical Pythagoreanism (or derivations thereof) seems to be in the air.

The first thing to say is that the claim that “nature is mathematical” hardly makes sense. It’s not even that it’s true or false. Taken literally, it seems almost meaningless. So it must be all about how we should interpret such a claim.

Some of the applications of Gödel’s theorems, etc. to physics, for example, simply don’t seem to make sense. They verge on being Rylian “category mistakes”. This bewilderment is brought about in full awareness of the fact that there would hardly be physics without mathematics. Indeed that’s literally the case with quantum physics and everything which goes with it.

However, when the English theoretical physicist and mathematician John D. Barrow asks us whether or not “the operations of Nature may include such a non-finite system of axioms” (as well as when he replies to his own question by saying that “nature is consistent and complete but cannot be captured by a finite set of axioms”), it can still be a philosophical struggle to see the connection between mathematical systems (or Gödel’s metamathematics) and Nature.

Strictly speaking, Nature isn’t any “mathematical system of axioms” and it doesn’t even “include” such a thing. Mathematics is applied to Nature or it is used to describe Nature. Sure, ontic structural realists and other structural realists (in the philosophy of physics) would say that this distinction (i.e., between mathematics and physics) hardly makes sense when it comes to physics generally and it doesn’t make any sense at all when it comes to quantum physics. However, surely there’s still a distinction to be made here.

Similarly, Nature is neither consistent/complete nor inconsistent/incomplete. It’s what’s applied to — or used to describe — Nature that’s (in)complete/(in)consistent. Again, certain physicists and philosophers of science may think that this distinction is hopelessly naive. Yet surely it’s still a distinction worth making.

This phenomenon is even encountered in contemporary philosophy of logic.

The philosopher Graham Priest, for example, mentions the world (or “reality”) when he talks of consistency and inconsistency. When discussing the virtue of simplicity he asks the following question:

“If there is some reason for supposing that reality is, quite generally, very consistent — say some sort of transcendental argument — then inconsistency is clearly a negative criterion. If not, then perhaps not.”

As it is, how can the world (or Nature) be either inconsistent or consistent?

What we say about the world (whether in science, philosophy, mathematics, logic, fiction, etc.) may well be consistent or inconsistent (we may also say — as with Spinoza later — that the world isn’t “beautiful” or “ugly”). However, surely the world itself can neither be consistent nor inconsistent.

Thus within Graham Priest’s logical and dialetheic context, claims of Nature’s consistency or inconsistency don’t seem to make sense. That must surely also mean that inconsistency is neither a (as Priest puts it) “negative criterion” nor a positive criterion when it comes to Nature itself.

Spinoza vs. Anthropocentrism or Anthropomorphism

What some philosophers of science and physicists are doing seems to contravene Baruch Spinoza’s words of warning about having an anthropocentric or anthropomorphic (though that word is usually applied to non-human animals) view of Nature.

Spinoza’s philosophical point is that Nature can only… well, be. Thus:

“I would warn you that I do not attribute to nature either beauty or deformity, order or confusion. Only in relation to our imagination can things be called beautiful or ugly, well-ordered or confused.”

Spinoza says Nature simply is. All the rest is simply (in contemporary parlance) human psychological projection.

There’s even a temptation to contradict Galileo’s well-known claim about Nature. Thus:

“Nature is written in that great book which ever is before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.”

Surely we must say that Nature’s book isn’t written in the language of mathematics. We can say that Nature’s book can be written in the language mathematics. Indeed it often is written in the language of mathematics. Though Nature’s book is not itself mathematical because that book — in a strong sense — didn’t even exist until human beings began to write (some of) it.

Yet perhaps I’m doing Galileo a disservice here because he did say that “we cannot understand [Nature] if we do not first learn the language and grasp the symbols in which it is written”. Galileo is talking about understanding Nature here — not just Nature as it is in itself.

Nonetheless, Galileo also says that the the “book is written in mathematical language”. Thus he’s also talking about Nature as it is in itself being mathematical. He’s not even saying that mathematics is required to understand Nature. There is, therefore, an ambivalence between the idea that Nature itself is mathematical and the idea that mathematics is required to understand Nature.

The Mathematical Description of Disorder

Another point worth making is that if mathematics can describe random events or chaotic systems (which it can), then it can also describe just about everything. What I mean by this is that it’s always said that mathematics is perfect for describing (or explaining) the symmetrical, ordered and even “beautiful” aspects of Nature. Yet, at the very same time, if I were to randomly throw an entire pack of cards on the floor, then that mess-of-cards could still be given a mathematical description. The disordered parts of that mess would be just as amenable to mathematical description as its (accidental) symmetries.

Similarly, if I were to improvise “freely” on the piano, all the music I played could be given a mathematical description. Both the chaos and the order would be amenable to a mathematical description and a mathematical explanation. Indeed a black dot in the middle of Sahara desert could be described mathematically; as can highly-probabilistic events at the quantum level. It’s even possible that mathematicians can find different — or even contradictory — symmetries in the same phenomenon.

In a similar way, some of the mathematical studies of Bela Bartok’s late string quartets have found mathematical patterns and symmetries which the composer was almost certainly unaware of. (See this example.) True, Bartok was indeed aware of the golden ratio and other mathematically formalisable aspects of his and other composers’ music. Nonetheless, the analyses I’m referring aren’t really formal in nature. They’re more like micro-analyses of the notes; and they serve, I believe, little purpose. Now there can indeed be interesting formal aspects and symmetries in music which the composers themselves weren’t aware of. Yet, at the same time, a mathematician may still gratuitously apply numbers to specific passages of music in the same way he could do so the same to my mess-of-cards.

Nothing?

The very idea of nothing (or nothingness) is hard - or even impossible - to conceive or imagine. This means that (at least for myself) it fails David Chalmers' idea of conceivability.

David Chalmers (the well-known Australian philosopher) claims that if something is conceivable; then that entails that it's also – metaphysically - possible. The problem with this is we can distinguish conceivability from imaginability. That is, even if we can't construct mental images of nothing (or nothingness), we can still conceive of nothing (or nothingness). I, for one, can't even conceive of nothing (or nothingness).

But can other people conceive of nothing? Do we even have intuitions about nothing or about the notion of nothingness?

So how can we even name or refer to nothing? (We shall see that Parmenides might have had something here.) There's nothing to hold onto. Yet, psychologically speaking, thoughts about nothing can fill people with dread. There's something psychologically (or emotionally) both propelling and appalling about it. And that's why existentialists and other philosophers – with their taste for the dramatic and poetic - found the subject of nothing (or at least nothingness) such a rich philosophical ground to mine. (See if you can wade through Jean-Paul Sartre's Being and Nothingness.)

The very idea of nothing also seems bizarre. It arises at the very beginning of philosophy and religion. After all, how did God create the world "out of nothing"? Did God Himself come from nothing? Indeed what is nothing (or nothingness)?

Not surprisingly, then, Giacomo Casanova (1725–1798) - in conversation with a priest - had this to say on the subject:

“… while the earth, suspended in air, stood firmly at the center of the universe that God had created out of nothingness. When I said to him, and proved to him, that the existence of nothingness was absurd, he cut me short, calling me silly.”

However, John the Scot - or Johannes Scotus Eriugena (c. 815–877) - had previously maneuvered his way around this problem by arguing that God is actually the same thing as nothingness; at least in the context of the question: “How did God create the world out of nothing?” Does this mean, then, that God created the universe out of Himself, not out of nothing?

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Some philosophers use the technical term “non-being” as a virtual synonym for the word “nothing”. (That may be true of the words; though what about the “thing” - nothing?) Having said that, since the notion of nothingness is itself either bizarre or unimaginable, then perhaps the word “nothing” is a technical term too.

Thus the term “not-being” also has its own problems:

i) What is being?

ii) How can there be non-being?

Parmenides

The Greek philosopher Parmenides (5th century BC) based his philosophy of nothingness primarily on logical arguments. Though, as we shall see, this is a prima facie reaction to Parmenides' position.

As soon as the subject was treated scientifically or empirically, however, it can be said that Parmenides' extreme and seemingly absurd position began to fade away.

Parmenides argued that there can be no such thing as nothing for the simple reason that to name it means that it must exist. And nothingness (unlike a stone or a proton) can't exist. This position was resurrected - if in modified form - in the 20th century by philosophers like Bertrand Russell and Willard van Orman Quine. The former obliquely supported it; whereas the latter rejected it. (See later.)

Parmenides' argument is more complete than it may at first seem. Not only is nothing/ness an abstraction to reject; so too is the existence of historical facts or history itself. The possibility of change is similarly rejected.

These are his basic positions (i.e., it's not an argument) on nothing:

i) Nothing doesn't exist.

ii) To speak of a thing, is to speak of a thing which exists.

iii) When one speaks of “nothing”, one speaks of it as if it is something which exists.

In the positions above nothing has been spoken of (it has been named). Therefore, by Parmenides' own light, either nothing must exist or he had no right to speak of it.

What about the events in the past or the past itself? The positions are very similar.

i) If we can't speak of (or name) nothing,

ii) then we can't speak of (or name) things or events of the past.

iii) Such events or things don't exist.

iv) Therefore when we refer to them, we're referring to nothing.

Here again there are references to nothing; which Parmenides warns us against.

What about change, which Parmenides similarly rejects? This rejection of change is strongly connected to his rejection of the past. The argument is this:

ia) If the past doesn't exist,

ib) then only the present exists.

iia) And if only the present exists,

iib) then there can be no change from past to present (or present to future).

iii) Therefore there can be no change at all.

Logical Form and Content

At the beginning of this piece it was mentioned that scientific or empirical philosophers rejected Parmenides's ostensibly pure logical arguments. Aristotle is one example. Indeed he goes further than a mere philosophical rejection. He wrote:

"Although these opinions seem to follow logically in a dialectical discussion, yet to believe them seems next door to madness when one considers the facts."

Nonetheless, Parmenides does seem to be on fairly safe ground. After all, Roy A. Sorenson defines a paradox

“as an argument from incontestable premises to an unacceptable conclusion via an impeccable rule of inference”.

Similarly, Roger Scruton says that paradoxes

“begin from intuitively acceptable premises and derive from them a contradiction – something that cannot be true”.

In other words, it might well have been the case that Parmenides used arguments which are both logically valid and sound. Or, as Aristotle put it, his “opinions seem to follow logically in a dialectical discussion”. It's only when we concern ourselves with semantic (or otherwise) content - rather than logical validity and soundness - that problems arise.

So Parmenides doesn't have it quite so easy. It's also the case that there are logical arguments against his logical arguments. For a start, Parmenides arguments aren't – in actual fact - purely logical in nature. (That is, they aren't purely formal.) This is the case in the simple sense they also involve content. After all, he refers to the “past”, “things”, “change”, the “present” and whatnot. If his arguments had only used variables, propositional letters and other logical symbols (as autonyms), then he'd have been on much safer ground. As it is, his positions - even if they are backed up with logical arguments – are also philosophical (or ontological) in nature.

Leucippus on the Void

One way in which science impacts on Parmenides' position is when it comes to the notion of the void.

Is the void “non-being” or is it something else? Why was the void seen as being “the opposite of being”?

Leucippus (early 5th century BC) - being a naturalist or at least a proto-naturalist - was the first to argue that the void is a thing. Nonetheless, it's a thing without also being a "body with extension" (to use Cartesian terminology).

If the void is non-being, then it throws up many problems. Leucippus , for one, realised that there could be no motion without a void. However, if the void is nothing, then how can something move in it? How can something move in nothing? Or how can some thing move in something which is not a thing?

Leucippus decided that there is no void if it is seen as nothing. Instead we have an “absolute plenum”. This is a space which is filled with matter. And nothing can't be filled with anything – especially not matter. Nonetheless, that didn't solve the problem of motion because the plenum was also seen - in Leucippus's day - as being completely full. Thus how could there be motion within it? Leucippus opted for the solution that there are many plenums; which presumably meant that objects can move from one plenum to another plenum. Democritus (circa 460 BC – 370 BC) seems to have taken this idea of multiple plenums further. He believed that the void exists between things or objects.

Prima facie, the idea of multiple plenums sounds similar to the idea of multiple spaces. However, the idea of a multiplicity of plenums was seemingly contradicted when Isaac Newton propagated the idea of absolute space – as opposed to (relative) spaces (i.e., in the plural).

Science and Empiricism

Aristotle - being a great empiricist and scientist - offered the obvious (in retrospect!) solution to Parmenides's ostensible paradoxes. He simply made a distinction between things which are made of matter and things which aren't made of matter. The latter includes space. In other words, space isn't non-being or even a void. It is, instead, a “receptacle” which acquires objects or in which objects can move.

Bertrand Russell – over two thousand years later - also offers us a good take on this.

Russell - also as an empiricist - started with observed data. He observed motion! From his observation of motion, he then constructed a theory. This is unlike Parmenides; who, when he observed motion, disregarded it for philosophical and logical reasons. In other words, for the Greek philosopher, logic and philosophy trumped observation.

Russell and Quine on Nothing

Bertrand Russell - in his 1918 paper 'Existence and Description' - believed that in order for names to be names, they must name – or refer to - things which exist. Take this remarkable passage:

“The fact that you can discuss the proposition 'God exists' is a proof that 'God', as used in that proposition, is a description not a name. If 'God' were a name, no question as to existence could arise.”

That, clearly, is fairly similar to Parmenides's position on the use of the word “nothing”. Russell's argument, however, is very different. Personally, I don't have much time for it. It seems to have the character of a philosophical stipulation. It's primary purpose is logical and philosophical. Russell, at the time, was reacting to the “ontological slums” (as Quine put it) of Alexius Meinong. However, this semantic philosophy (as I said) simply seems like a stipulation (or a normative position) designed to solve various philosophical problems.

As for Quine, he has no problem with the naming of non-beings or non-existents (though non-being and non-existence aren't the same thing). In his 1948 paper, 'On What There Is', he firstly dismisses Bertrand Russell's position. Quine, however, puts Russell's position in the mouth of McX and uses the word “Pegasus” rather than the word “God”.

Quine wrote:

“He confused the alleged named object Pegasus with the meaning of the word 'Pegasus', therefore concluding that Pegasus must be in order that the word have meaning.”

Put simply, a name can have a “meaning” without it referring to something which exists (or even something which has being). Quine unties meaning from reference; whereas Russell only thought in terms of reference (or, at the least, he tied meaning to reference).

Parmenides, of course, makes similar mistakes (as we've seen). He didn't think that a name could have a meaning without the thing being named also existing or being. However, we can speak of something that doesn't exist because the naming of such an x doesn't imply its existence. Though - in homage to Meinong (as well as, perhaps, to the philosopher David Lewis) - Russell would have asked us what kind of being the named object (or thing) has.

Thus Russell's theory is an attempt to solve that problem by arguing that if a named x doesn't exist (or have being), then that name must be a “disguised description”. (In the case of the name “Pegasus”, the description would be “the fictional horse which has such and such characteristics”.)

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So, as we've seen, nothing (or nothingness) is a difficult notion to grasp. Yet philosophers throughout the ages have had a good stab at it. The problem is (as ever with philosophers) that they've said very different things about it. Then again, nothing (or nothingness) also perplexes physicists and cosmologists; as indeed it does the layperson. Perhaps it's precisely because there's nothing to grasp in the first place that the notion has thrown up so many absurdities and surprises.

Does all this therefore mean that anything goes when it comes to nothing or nothingness?

Roger Penrose's New Physics of Consciousness (2)

As a non-scientist, I've often little idea if statements or arguments about quantum mechanics and its influence on the brain and consciousness are true. I sometimes understand what's being said. However, that understanding alone doesn't in and of itself tell me whether or not what's being said is true (or correct). A scientist may tell me that p is true or correct; though is that enough?

Perhaps I should have faith in the science instead. The problem is, faith in which scientific theory and in which scientists? And what are the “criteria of theory-choice” for the layperson; never mind for the scientist?

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Roger Penrose's New Physics?

We can ask if the new physics which Roger Penrose demands - for a science of consciousness - is physics at all. It's of course the case that physics has often taken off in radically new directions. Nonetheless, that doesn't mean that every radically new direction is to be accepted or will prove to be successful or productive. And it doesn't mean that every new direction will remain securely in the domain of physics.

So what's the nature of Penrose's new physics?

Take one example: Penrose's position that the very-small scale and the large scale may be physically connected. Traditionally it was thought that what applied to the very-small scale doesn't apply to the large or very-large scale. What's more, it is primarily quantum gravity that accounts for this phenomenon.

How does this apply to the brain and consciousness?

The quantum happenings in the microtubules (the very-small scale) are said to affect the brain as a whole (the large scale) and thus be responsible for consciousness. Again, it's quantum gravity that connects these quantum happenings in the microtubules within the brain as a whole.

Yet it's often said - often by philosophers - that quantum happenings in the brain have negligible effect on the brain as a whole. As a consequence, critics conclude that quantum happenings in the microtubules can't be the source of consciousness. However, quantum happenings do indeed have an effect on the large scale; as Penrose himself makes plain here:

“The very existence of solid bodies, the strengths and physical properties of materials, the nature of chemistry, the colours of substances, the phenomena of freezing and boiling, the reliability of inheritance — these, and many other familiar properties, require the quantum theory for their explanations.”

True, these “solid bodies”, etc. may “require the quantum theory for their explanations”; though that doesn't automatically mean that they have effects which can be observed or which are in any way substantive. Thus all this doesn't also mean that the impact of quantum mechanics on the macro-world is important or even mildly important. It simply means that quantum mechanics is a part of the whole picture. So, in the sense of supplying a complete picture - then, yes, of course quantum theory will be required. However, this question still remains: 

In what precise ways do quantum happenings effect macro-objects, macro-events and macro-conditions?

Despite this (possibly) quasi/neo-Kantian or quasi/neo-phenomenalist account of the nature of quantum happenings on Penrose's cricket balls, the point may still be missed. (Penrose himself mentions the lack of quantum-mechanical effects on cricket balls.)

In any case, as a possible consequence of quantum-mechanical effects on macro-objects, macro-events and macro-conditions, Penrose finishes off by saying something that was - perhaps - spurring him on all along. Thus:

“Perhaps, also, the phenomenon of consciousness is something that cannot be understood in entirely classical terms. Perhaps our minds are qualities rooted in some strange and wonderful feature of those physical laws which actually govern the world we inhabit... Perhaps, in some sense, this is ‘why’ we, as sentient beings, must live in a quantum world, rather than an entirely classical one... Might a quantum world be required so that thinking, perceiving creatures, such as ourselves, can be constructed from its substance?”

Again, the question remains:

Is it the case that microtubules, quantum gravity and quantum mechanics provide the answers to all Penrose's questions about consciousness?

Quantum Gravity

Roger Penrose puts his position on quantum gravity and quantum mechanics in the following way:

“This change is to play its role when quantum mechanics becomes appropriately united with general relativity, i.e. in the sought-for theory of quantum gravity. Most physicists do not believe that quantum theory needs to change when it is united with general relativity.”

More relevantly to our discussion, Penrose was and is still is deeply aware – and evidently so - of the arguments against his position as it's applied to the brain and consciousness. Indeed when he wrote The Emperor's New Mind in 1990, he didn't have a position on microtubules. (Microtubules aren't mentioned in this well-known book.) Even neurons and neurotransmitters only get four mentions.

This is what Penrose also had to say about the opposition's position on the brain and quantum mechanics:

“... they would argue that on a scale relevant to our brains the physical effects of any quantum gravity must be totally insignificant! They would say (very reasonably) that although such physical effects might indeed be important at the absurdly tiny distance scale known as the Planck length — which is

10 35 m, some 100000000000000000000 times smaller than the size of the tiniest subatomic particle — these effects should have no direct relevance whatever to phenomena at the far far larger ‘ordinary’ scales of, say, down only to 10 12 m, where the chemical or electrical processes that are important to brain activity hold sway.”

Indeed Penrose went further by saying that his detractors would say that not even ordinary gravity (as it were) could affect the brain. Thus:

“Indeed, even classical (i.e. non-quantum) gravity has almost no influence on these electrical and chemical activities.”

Penrose concludes with the following sceptical words (as offered by his opponents):

“If classical gravity is of no consequence, then how on earth could any tiny ‘quantum correction’ to the classical theory make any difference at all? Moreover, since deviations from quantum theory have never been observed, it would seem to be even more unreasonable to imagine that any tiny putative deviation from standard quantum theory could have any conceivable role to play in mental phenomena!”

In the above, Penrose placed his cards down on the table when he said that “deviations from quantum theory have never been observed”. (Though the word “observed” may be controversial here.) Nonetheless, this is the crux of Penrose's position:

i) Quantum gravity (or the “structure of space-time”) may have an impact on quantum mechanics.

ii) Therefore quantum gravity may affect the nature of the brain and consciousness.

All the above is expressed in the following:

“.. I am not concerned so much with the effects that quantum mechanics might have on our theory (Einstein’s general relativity) of the structure of space-time, but with the reverse: namely the effects that Einstein’s space-time theory might have on the very structure of quantum mechanics.”

This is, of course, “an unconventional view-point” (or it was in 1990!). It's “unconventional that general relativity should have any influence at all on the structure of quantum mechanics!”. More basically, Penrose believed that “the problems within quantum theory itself are of a fundamental character”.

Penrose concluded – again, in 1990 (27 years ago) - by saying that “any putative quantum gravity theory would surely be very remote from the phenomena governing the behaviour of brains”.

Quantum Gravity and Microtubules

At its most basic, gravitation and general relativity aren't integrated in quantum theory. That alone makes Penrose's views on consciousness speculative. Penrose himself says that quantum theory doesn't play a part in (his) “objective state reduction”. It's here where quantum-gravitational effects come to the rescue – at least as far as Penrose is concerned.

Why the pressing need to square (or couple) gravity with quantum mechanics? Some argue that that a classical system can't be squared (or coupled) with a quantum system. (Though that would of course depend on what the word “coupling” or “squaring” means.)

So why does Penrose see his position on quantum gravity as important in relation to brains and their microtubules? According to Penrose, it's in the microtubules where quantum states can become “reduced” by gravitational influence. This isn't the case with the large-scale brain as a whole – the “classical environment”. Or at least with the brain as a whole (or its parts) taken as also being large-scale.

What's happening here is a strange a connection between the very-small scale and the large scale. In Penrose's view, the brain (or its microtubules) are linked to fundamental spacetime geometry and thus also to quantum gravity.

Max Tegmark (1999/2000), among many others, has questioned not only the relevance of quantum states in the brain as a whole (or in the brain's separate parts); but also in the case of the very small microtubules. Put simply, quantum happenings can't last long enough to have an important role in consciousness. (The consensus seems to be on Tegmark's side on this; rather than on Penrose's.)

More technically and in terms of quantum happenings, Tegmark believes that even in the microtubules (never mind the brain or its other large-scale parts), quantum superpositions within the microtubuli would - or could - never last long enough to bring about consciousness (or any other significant mental phenomena). Nonetheless, Tegmark's position has in turn - rather predictably - been the subject of criticism. S. Hagan and Stuart Hameroff (2002), for example, have backed up the position of Penrose by saying that such quantum superpositions could (or can) last long enough to have significant large-scale effects – at least in theory. Indeed there's been substantive research – i.e., outside the brain and consciousness - into interacting spins and entanglement which suggests that superpositions can have a longer lifespan.

The connections between the very-small scale and the large scale have just been stressed. Confusingly, this phenomenon runs parallel with the need for quantum happenings within “systems” to remain isolated from the outside of the aforesaid systems. More specifically (in terms of the brain and consciousness), Hameroff and Penrose do believe that microtubules are isolated from the rest of the brain. Due to the spiral structure and shape of microtubules, the quantum happenings within them remain isolated from the outside world (i.e., from the rest of the brain and outside the brain itself). However, if this is the case, how is the quantum very-small scale connected to the brain's large scale? According to Penrose, it's quantum gravity which links the quantum happenings in the microtubules with the brain to bring about consciousness.

These supposed connections between the very-small scale and the large scale are part of the problem.

In detail, the philosophers Rick Grush and Patricia Churchland have argued that microtubules can't pass on their quantum happenings to each other (or from one neuron to another). That would be needed, for example, in order to explain “the unity of consciousness”. More concretely, how are the quantum happenings in the microtubules (the very-small-scale happenings) passed onto the large-scale happenings in neurons, neurotransmitters and neuromodulators?

To bring this grand theorising down to earth, let's cite one of Penrose and Hameroff's own examples of this.

Hameroff once believed that when a subject is rendered unconsciousness by anaesthetics, that the subject's microtubules were responsible for this. As a consequence, Hameroff made a strong connection between these microtubules and consciousness. In other words, if microtubules can turn consciousness off; then surely they can (or must) also turn consciousness on. The problem is that this specific theory about anaesthesia was shown to be false. Microtubules don't cause unconsciousness.

More specifically on anaesthetics. It's agreed that in certain cases of anaesthetically-induced unconsciousness the microtubules are involved. However, this isn't true in all cases. Indeed anaesthetics can bring about unconsciousness without involving microtubules. It's also the case that microtubules can be damaged - or effected in other ways - without bringing about unconsciousness.

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Some Additional Science Stuff: Nonlocality, Etc.

Quantum coherence (at least in terms of microtubules, the brain and consciousness) is the idea - or reality - that there can be “subatomic cooperation”. In other words, there's a possibility of communication between subatomic elements within the brain. (Let's not get too pedantic about the word “communication” here.)

Thus it's the case that nonlocality can account for subatomic particles “knowing each other”; even though they're separated by large distances. Can we move from that fact to the possibility of objects (yes, which are made up of particles) knowing each other at large distances? More specifically, can we say that different microtubules (embedded in neurons) can know other microtubules at large distances? (Let's not get too pedantic about the word “know” here.)

In terms of microtubules, that could mean that communication occurs between microtubules because of quantum superpositions. Thus neurons themselves may be connected by virtue of the aforesaid microtubular connections. Indeed that neuronal connectivity rides on the back of quantum microtubular connectivity. More specifically, quantum gravity (according to Penrose) may cause microtubular collapse: it is each collapse that may be responsible for each basic act (if “act” is the correct word here) of consciousness.

Despite these speculations about microtubular connectivity also causing inter-neural connectivity, it's still argued that there's a problem with the theory that happenings in single synapses have an impact on the happenings of, say, neural assemblies.

If there is microtubular nonlocality, then there's also microtubular entanglement. Or, more precisely, the subatomic particles which make up microtubules are involved in entanglement. If we move away from the brain and their microtubules, it's certainly the case that nonlocality and entanglement are part of the same picture and both have been demonstrated in many experiments. What hasn't been conclusively demonstrated is what Penrose and Hameroff have to say on, well, microtubular nonlocality and entanglement. (That's if we can say that it's the microtubules -, rather than their subatomic parts - which become entangled.)

It's also been said that entanglement and nonlocality may be applicable to communication systems and to other phenomena not directly related to the human brain. Anton Zeilinger, for one, has shown - in his laboratories - that entangled particles can indeed be reconstituted in different places. And, if this is the case with artificial experiments involving laboratories, then surely it's even more likely to be the case when it comes to the brain's microtubules... Or, indeed, is it less likely?