In 1931, a young 25 year-old logician from what is now Brno in the Czech Republic, published the first of two theorems which became landmarks in 20th century mathematics. Gödel's Incompleteness Theorems together stated that with respect to any given formal and consistent mathematical system, either its completeness or its consistency could be proved within its own terms but not both simultaneously, and each only at the expense of the other. In other words, either it could be proved consistent, but hence incomplete; or complete, but hence inconsistent.
Gödel's work has been widely extrapolated beyond its limited and specific terms of reference, perhaps because it appears to offer valid comment on the nature of proof in general. It has been invoked in support of many ideas up to and including skepticism – ie. if every system of thought is necessarily incomplete and cannot be proved consistent within its own terms, then nothing demonstrated by the terms of that system can be regarded as proved. While its support for such sweeping conclusions is limited at best, what it does hint at is the inherently self-referential nature of any system, which it does (rather neatly) by using a self-referential statement within its own logic. (See also Blog and Further Reading below.)
Around the same time as Gödel was working on his logical theorems, physicists such as Max Planck, Albert Einstein, Niels Bohr, Werner Heisenberg, Erwin Schrödinger, Paul Dirac, Max Born, John von Neumann, Wolfgang Pauli and others were turning the 150 year-old Laplacian fantasy of the deterministic predictability of existence into the world of probablities and potentialities that is Quantum Mechanics. Einstein's relativity theories started the ball rolling by rejecting the classical notion of absolute concepts of time and distance. 11 years after Einstein's General Theory of Relativity, Heisenberg's Uncertainty Principle demonstrated that both the speed and position of a particle cannot be determined simultaneously – either can be determined, but only at the expense of the other, thus paralleling Gödel's Theorems – and that the act of observation itself has an impact on what is being observed. Einstein himself stopped short of the full implications of quantum theory, unwilling to relinquish the notion of determinism with his famous statement "God does not play dice". In 1935, he published a paper which was the result of a thought experiment between himself, Boris Podolsky, and Nathan Rosen. The EPR Paradox, as it came to be known, was devised to show the incompleteness of quantum mechanics as a physical theory, but was ironically demonstrated experimentally in the early 1980s, proving the validity of the counter-intuitive proposition of quantum entanglement which Einstein had perceived as inconsistent with his Special Theory of Relativity.
So what does all this tell us and how is it relevant to a discussion about the nature of scientific enquiry?
Essentially it implies that all our models of the fundamental nature of reality are unverifiable as complete and consistent within their own terms; they are self-referential, relative and contingent. The universe is not predictable. God does play dice. And what's more, the entirety of existence is tangled up with everything else in the entirety of existence. So ultimately, everything (and hence everyone) is a manifestation of God ... playing dice.
As Professor Stephen Hawking said in his public lecture Does God Play Dice? "Although quantum mechanics has been around for nearly 70 years [nearer 80 now], it is still not generally understood or appreciated, even by those that use it to do calculations. Yet it should concern us all, because it is a completely different picture of the physical universe, and of reality itself."
This is an important point, because much of science – particularly medical science – dealing with the macroscopic world amenable to our senses and direct experience, still behaves as if absolute objectivity, internal consistency, linear logic and deterministic predictabiliy are cornerstones of the fundamental "reality" under consideration.
As Hawking goes on to point out in the same lecture, "Many scientists are like Einstein, in that they have a deep emotional attachment to determinism. Unlike Einstein, they have accepted the reduction in our ability to predict, that quantum theory brought about. But that was far enough. [...] I feel these scientists have not learnt the lesson of history. The universe does not behave according to our pre-conceived ideas. It continues to surprise us."
It's probably reasonable enough to assume that the vast majority of scientists have an emotional attachment to at least some aspect of the models and theories they work with (most likely those fundamental assumptions we're largely unconscious of and least inclined to question). Perfectly natural, and of itself, neither good nor bad. The problem arises when the emotional component is not recognised for what it is. In particular, when it's mistaken for dispassionate rational assessment or some absolute "truth". At its weakest, emotional attachment has a habit of acting as a selective filter on the evidence accepted in support of any given viewpoint. At its strongest, it appears capable of swaying the outcome in any situation on which it has a bearing.
"Belief" is a rationale that's frequently invoked to undermine the objective basis, and hence "reality", for the postulated efficacy of complementary medical interventions – with some justification – but those who employ it invariably miss the necessary corollary of the argument. Which is that it's equally applicable to conventional techniques. The placebo effect doesn't magically kick in the minute someone visits a complementary, as opposed to conventional, therapist. In fact, if its strength is in any way proportionate to the strength of acceptance and belief in the underlying therapy, one would expect the placebo effect in conventional medicine to be a far greater factor than it is in complementary modalities! Indeed, attempts to measure its impact in the treatment of chronic pain have rated it at around 35% in that context.
In a February 2000 article in the UK Guardian on the rise of complementary medicine in the UK entitled Healing in Harmony, Jerome Burne wrote,
"Medicine is both an art and a science, but science has been firmly in the driving seat for 40 years. The arrival of CM practitioners may allow some of the more intangible aspects of the healing profession to re-emerge into the light, such as the power of the doctor's own belief. "When I was starting out as a doctor, my professor told me about a new migraine drug," recalls Marshall Marinker, professor of general practice at the University of London. "I prescribed it to a number of my patients, and it worked brilliantly. Many were completely cured. But then I began thinking about its mechanisms and how to design a trial, and it somehow stopped working so well. I never again got such good results as when I totally believed in it. I don't think you can measure that sort of thing in clinical trials, but it is absolutely vital to the way medicine should work.""
Yet medical science continues to insist, ever more stridently, on "evidence-based medicine". Again, of itself, no bad thing, even if a little unbalanced. But exactly what constitutes acceptable evidence on which to base that medicine is another matter. A certain amount of double standards appear to apply. What is acceptable for theoretical frameworks which fall within the purview of the broad consensus does not appear to be consistent with what is acceptable for theoretical systems outwith that consensus. Trials investigating complementary medical treatments are routinely subjected to far greater scrutiny than is applied to those testing conventional medications. Fair enough perhaps from some angles, but look hard enough and you'll find flaws in the vast majority of randomised controlled trials. And the more positive the conclusions are in favour of a system which appears to undermine the foundations of the biomedical model (eg. homeopathy), the harder the trials are scrutinised. This leads to the suspicion that what's really at issue here has precious little to do with the scientific pursuit of any underlying "truth" inherent in any particular modality, and far more to do with the conservation of a world view which is comfortable and familiar to its adherents. In other words, one in which they have a substantial degree of emotional investment. Tolstoy (quoted left) hit the nail on the head.
This is nowhere more apparent than in the arguments for and against the various complementary modalities which appear regularly in the media. You don't have to look far to find them. A recent (March 2005) example from the UK Guardian is a case in point. In an article entitled Believe in Facts not Fads, Dick Taverne writes,
"A Mori poll this week shows that eight out of 10 Britons think science makes a good contribution to society. Yet paradoxically, only four out of 10 think themselves very or fairly well informed about science. Is this why people ignore scientific evidence in much of their daily lives?
"Consider the popularity of alternative medicine. Some 40% of GPs now offer complementary therapies as part of their services and many are qualified homeopaths. In fact homeopathy is nonsense on stilts. It was founded on the proposition that "like cures like", for which there is no evidence, and since this involves treating patients with toxic substances that can poison them, a second law was invented, the law of infinitesimals. This states that the more a substance is diluted, the greater the benefit. A dilution to the power of 30 is standard, that is, one part to 1,000,000,000,000,000,000,000,000,000,000. None of the original substance remains and any success is achieved through the placebo effect.
"Organic food is another case in point. It is promoted by intensive propaganda from the Soil Association, assisted by supermarkets that benefit from its premium prices, and its virtues are accepted by restaurateurs and lifestyle magazines. [...]"
A week later, a letter from a Dr Ian Francis was published in response. He writes,
"The science community needs honest and impartial advocates in debates about scientific credibility. On the evidence of his article (Believe in facts not fads, March 17), Dick Taverne is neither.
"He is being intellectually dishonest by mixing up something that all scientists agree about (the pointlessness of homeopathy) with something that is the subject of valid scientific debate (the effects of GM crops on natural ecosystems, for example). [...]"
Dr Francis has an excellent point in drawing attention to the need for honesty and impartiality, but completely fails to see his own lack of it. Both writers are confusing personal opinion with "fact" and are attempting to pass off those opinions as something concrete and objectively, even "scientifically", determined. Both are apparently sincere in their beliefs – so accusations of dishonesty are perhaps inappropriate – but each has selectively taken the evidence that supports their feelings and ignored the rest.
The emotional foundation of their arguments is betrayed in their choice of language – "In fact [x] is nonsense on stilts", "It was founded on the proposition [y] for which there is no evidence", "He is being intellectually dishonest", "... something that all scientists agree about", etc. Note the judgemental terms (nonsense, dishonest) and prevalence of bold (but unsupported) statements couched in absolutes (no evidence, all scientists) – both good indicators for emotionally-based arguments. Logically, the idea that anyone would pluck a proposition out of thin air based on no evidence whatsoever – and build their life's work and a complete medical system on it – is somewhat unlikely, to say the least. One might disagree about the interpretation of the evidence or whether the evidence was valid in the context of the proposition, but to say there is none has an irrational edge to it. The classic "... something that all scientists agree about" is a delight. The list of things that all scientists agree about would probably fit on the back of a postage stamp. And not a commemorative issue at that. Alright. Perhaps I exaggerate slightly there. A postcard? Regardless, a quick glance at the research on homeopathy in the annals of conventional medicine alone will instantly show Dr Francis' statement to be factually incorrect.
The fact of the matter, as far as homeopathy and most other contentious issues are concerned, is that there is no acceptably conclusive modern evidence either way, and certainly none which satisfies all parties involved in the debate. While a large number of scientists cannot reconcile something like homeopathy with the way they see the world and hence dismiss it as "nonsense on stilts" no matter what evidence comes with it, a growing number are working at incorporating it into their world view and looking for mechanisms which might provide a basis for understanding its apparent action. The jury is out, but it's self-evident that the more heated the debate becomes, the more we are dealing with emotional attachments and less with anything genuinely "scientific".
Of course, Gödel's Theorem alone should allow us to predict this kind of scenario with respect to any system of thought. If no system can be demonstrated as simultaneously complete and consistent within its own terms, then those who stand outside that system will invariably be able to identify its incompleteness, inconsistencies and self-referential nature ... while, naturally, they are simultaneously blind to the incompleteness, inconsistencies and self-referential nature of their own system of beliefs, which in turn will no doubt be pointed out to them from across the other side of the debating floor. And what keeps the more heated fringes of the debate going is as much the attachment of both sides to determinism, a concept which has been demonstrated as invalid in any objective and absolute sense for the best part of a century.
The Map is not the Territory but the Territory is the Map.
If we can stop taking positions for just one moment, the nature of what we're looking at here should start to become clear. What we take for "reality" at any particular time is simply a hegemony of a particular way of seeing things, not reality itself. This is pretty obvious when looking at world views to which we ourselves don't subscribe. Less so when the spotlight is turned on our own. From the outside, there is little to distinguish the present hegemony in terms of validity from any other world views preceding it or co-existing with it. What makes it different is the subjective experience of existing within it as if it were "true", because it does indeed appear to be true.
How can we explain this?
All systems of thought are inherently self-referential, or, in different terminology, positive feedback systems. Their recursive nature appears to, in effect, create – or become a function of – a "spin" (a process already recognised in terms and figures of speech like "to spin a yarn", "put a positive spin on it", "spindoctor"). Anything which spins around an axis creates rotational forces in and around it. It creates a vortex. And to understand the nature of the vortex, we can look to the models of the largest ones so far believed to exist. The black holes in outer space.
A black hole is a vortex within the spacetime continuum so enormous, with such huge gravitational forces, that not even light can escape. The plane at which those forces reach sufficient strength for nothing to be able to escape them is known as the "event horizon". So any thought system could be imaged as a vortex with its own "event horizon", beyond which the thought system becomes entirely self-referential and self-confirming. The "spacetime curvature" created around a thought system acts as a gravitational field, pulling in and entraining events to its particular axioms (ie. what it takes to be self-evidently "true") and thereby confirming the validity of those axioms whether they are recognised as being to some degree self-made and self-maintained or not. Yet, because of quantum entanglement – the fundamental correlation of the entirety of existance – all such systems remain open to intrusions from other systems at their "event horizons". In other words, the universe will continue to surprise us.
This model has the virtue of being able to accomodate both Einstein's (and others') conviction that some degree of determinism has validity and Hawking's equally valid perspective of the fundamental unpredictability of existence. Perhaps God has a set of dice but only brings them out on Sundays.
If, collectively and individually, the nature of our interface with reality is to a large degree subjectively determined and maintained (see quote left), this does not mean that what we observe is not real. It's the spin we put on it – ie. the map we make of it; the interpretation we place on it, the understanding that we derive from those interpretations, and the manner in which that understanding impacts our ongoing interaction with it – which is largely our own creation, consensual or otherwise. Yet the spin, the map, also becomes a force of containment for the reality we perceive within our own personal event horizons, hence our perception that reality is how we perceive it to be.
Yet since we are also a part of that reality as well as interfacing with it, each and every one of us necessarily encapsulates a sense of the ultimate truth of it. We're all part of the hologram. In being so, everyone's expression of their experience of it, no matter how well spun, has a bearing on the whole. Nobody is entirely wrong, despite the "general relativity" of their position, and nobody is entirely right either.
In this context (once we've arrived back on Earth again), it would seem that much of the debate about such things as the validity of complementary medical techniques – from both sides of the therapeutic divide – is as much an impassioned discussion about the merits of competing football teams as it is "science". Perhaps Dick Taverne should have taken a long hard look in the mirror before giving his article the title Believe in Facts not Fads, but then by another token he's merely confirming the self-referential nature of all thought systems ...
So where do we go from here?
Logic did not stop with Gödel's Incompleteness Theorems. If a system could not be proved to be simultaneously complete and consistent in its own terms, then what conditions were necessary in order to prove its completeness and consistency? Gentzen's Theorem provided the answer. Paraphrased, it states that "It is possible to prove both the consistency and completeness of a formal system, but only in a proof theory with induction of strictly greater order." Like Gödel's Theorems which preceded it, Gentzen's Theorem, while being highly specific in its terms and application, also appears to lend itself to valid comment on the nature of proof in general, and can be extrapolated in broad terms to systems and situations beyond formal logic.
The implications here for the "proof" of complementary medical systems are important. If Gentzen's conclusions are valid in this context, then in order to adequately prove or disprove any complementary medical system, it is necessary to use a proof theory of a greater order than any of the individual systems. Conventional medicine, being of a comparable order to complementary modalities despite its hegemony, cannot therefore be used to prove or disprove any other system of medicine, and vice versa. (No wonder then at the lack of conclusive results from the various attempts to prove or disprove complementary modalities from the perspective of the biomedical model, and that we're left with the enduring image of warring factions on the football terraces.)
What would be the characteristics of a theoretical framework of a greater order? One perhaps capable of modelling all of these divergent systems simultaneously, consistently, without bias or contradiction, and accounting for their co-existence? Such a framework only needs to prove itself consistent in order to prove the validity of any of the lower order systems (though of course it would need a framework of a greater order still to prove its own completeness and consistency). The outline of the vortex model which I've introduced in this essay comes from the perspective of a potential candidate for that greater order framework. One which I intend to describe in more detail in the next essay in this series.