An Important Mathematical Oversight

The original intention for this website was to encourage public awareness of an historical medical crime, one that has remained a tightly-kept British state secret now for more than five decades. The matter is of enormous public interest, not least because the motivation behind the crime itself was that of advancing scientific research into areas that would come to provide the seminal knowledge behind much of the technological progress of the last half-century. My investigation into the matter inspired a parallel enquiry into some of the fundamental principles that underpin that scientific and technological impulse.

There are therefore two principle concerns of this website, and if there is acknowledged to be a substantive connection between them, that has inevitably to do with late 20th Century developments in science and information technologies, and more broadly with the idea of an burgeoning technocracy – the suggestion of a growing alliance between corporate technology and state power – one that might be judged to have atrophied the powers conventionally assigned to liberal-democratic institutions. This link therefore serves as a segue to emphasise the equal importance, to my mind, of what is going on in the X.cetera section of the site, so that that section should not appear, from the point of view of the other, as some kind of afterthought.

X.cetera is concerned with a problem in mathematics and science to do with the way we think about numbers. As a subset of the category defined as integers, elements in the series of the natural numbers are generally held to represent quantities as their absolute, or ‘integral’, properties. It is argued that this conventional understanding of integers, which is the one widely held amongst mathematicians and scientists adopting mathematical principles, is the cause of a significant oversight with regard to changes in the relations of proportion between numerical values, i.e., when those values are transposed out of the decimal rational schema into alternative numerical radices such as those of binary, octal, and hexadecimal, etc.

On the page: The Limits of Rationality it is argued that the relations of proportion between integers are dictated principally by their membership of the restricted group of characters (0-9) as defined by the decimal rational schema; and that corresponding ratios of proportion cannot be assumed to apply between otherwise numerically equal values when transposed into alternative numerical radices having either reduced (as in binary or octal, for instance) or extended (as in hexadecimal) member-ranges.

This is shown to be objectively the case by the results published at: Radical Affinity and Variant Proportion in Natural Numbers, which show that for a series of exponential values in decimal, where the logarithmic ratios between those values are consistently equal to 1, the corresponding series of values when transposed into any radix from binary to nonary (base-9) results in logarithmic ratios having no consistent value at all, in each case producing a graph showing a series of variegated peaks and troughs displaying proportional inconsistency.

These findings are previously unacknowledged by mathematicians and information scientists alike, but the import of the findings is that, while the discrete values of individual integers transposed into alternative radices will be ostensibly equal across those radices, the ratios of proportion between those values will not be preserved, as these ratios must be determined uniquely according to the range of available digits within any respective radix (0-9 in decimal, 0-7 in octal, for instance); one consequence of which of course is the variable relative frequency (or ‘potentiality’) of specific individual digits when compared across radices. This observation has serious consequences in terms of its implications for the logical consistency of data produced within digital information systems, as the logic of those systems generally relies upon the seamless correspondence, not only of ‘integral’ values when transcribed between decimal and the aforementioned radices, but ultimately upon the relations of proportion between those values.

Information Science tends to treat the translation and recording of conventional analogue information into digital format unproblematically. The digital encoding of written, spoken, or visual information is seen to have little effect on the representational content of the message. The process is taken to be neutral, faithful, transparent. While the assessment of quantitative and qualitative differences at the level of the observable world necessarily entails assessments of proportion, the digital encoding of those assessments ultimately involves a reduction, at the level of machine code, to the form of a series of simple binary (or ‘logical’) distinctions between ‘1’ and ‘0’ – positive and negative. The process relies upon a tacit assumption that there exists such a level of fine-grained logical simplicity as the basis of a hierarchy of logical relationships, and which transcends all systems of conventional analogue (or indeed sensory) representation (be they linguistic, visual, sonic, or whatever); and that therefore we may break down these systems of representation to this level – the digital level – and then re-assemble them, as it were, without corruption. Logic is assumed to operate consistently without limits, as a sort of ‘ambient’ condition of information systems.

In the X.cetera section I am concerned to point out however that the logical relationship between ‘1’ and ‘0’ in a binary system (which equates in quantitative terms with what we understand as their proportional relationship) is derived specifically from their membership of a uniquely defined group of digits limited to two members. It does not derive from a set of transcendent logical principles arising elsewhere and having universal applicability (a proposition that, despite its apparent simplicity, may well come as a surprise to many mathematicians and information scientists alike).

As the proportional relationships affecting quantitative expressions within binary are uniquely and restrictively determined, they cannot be assumed to apply (with proportional consistency) to translations of the same expressions into decimal (or into any other number radix, such as octal, or hexadecimal). By extension therefore, the logical relationships within a binary system of codes, being subject to the same restrictive determinations, cannot therefore be applied with logical consistency to conventional analogue representations of the observable world, as this would be to invest binary code with a transcendent logical potential that it simply cannot possess – they may be applied to such representations, and the results may appear to be internally consistent, but they will certainly not be logically consistent with the world of objects.

The issue of a failure of logical consistency is one that concerns the relationships between data objects – it does not concern the specific accuracy or internal content of data objects themselves (just as the variation in proportion across radices concerns the dynamic relations between integers, rather than their specific ‘integral’ numerical values). This means that, from a conventional scientific-positivist perspective, which generally relies for its raw data upon information derived from discrete acts of measurement, the problem will be difficult to recognise or detect (as the data might well appear to possess internal consistency). One will however experience the effects of the failure (while being rather mystified as to its causes) in the lack of a reliable correspondence between expectations derived from data analyses, and real-world events.

So that’s some of what X.cetera is all about.. If you think you’re ‘ard enough!

[ PDF version ]

[ PDF version ]

[ PDF version ]

[ PDF version ]


Download my 167-page
report: Special Operations
in Medical Research

[pdf – 1.88MB]:

Download my Open Letter to the British Prime Minister & Health Secretary
[pdf – 363KB]:

The Limits of Rationality
(An important mathematical oversight)


Radical Affinity and
Variant Proportion in
Natural Numbers


Mind: Before & Beyond Computation

Dawkins' Theory of Memetics – A Biological Assault on the Cultural

Randomness, Non-
Randomness, & Structural Selectivity


Complaint to the General Medical Council

The following letter of complaint was sent to the GMC on 2 February 2012, including the submission of my report and copy of my first MRI scan. In retrospect it was probably ill-advised of me to have complained to the GMC in the first instance, before submitting any complaint directly to Guy’s & St Thomas’ NHS Trust (GSTT), who had failed to disclose the evidence revealed in the MRI scan. However, in view of the serious criminal nature of the historical allegations over events in 1967, which were not the original responsibility of GSTT, at the time it seemed proportionate to do so. I had no expert corroboration of the evidence, and therefore I doubted that I would get any satisfactory response from a complaint to GSTT, as it seemed unlikely that GSTT would voluntarily admit to its own cover-up. I expected that the GMC would have the resources to independently assess the evidential value of the MRI scan; and, in view of the additional evidence contained in my report, would therefore be obliged to treat my allegations with some seriousness.

Letter of complaint to the GMC – 22/02/2012 [pdf – 74KB]

This complaint was made before I had access to the documents provided in response to my subject access request to GSTT in November 2012 (re: the title page of this section), and was therefore ill-informed with regard to the details of the alleged cover-up, in particular the identity of the original reporting radiologist at St Thomas’. However, it does highlight the recent failure of Dr Thomasin Andrews at Guy’s Neurology Dept. to disclose her awareness of the significant anomaly revealed in the scan, as well as pointing out the general difficulty in obtaining expert corroboration of that evidence. The GMC’s initial response was to declare the complaint ‘out of time’, as the events of 1967-68 referred to in my complaint and report occurred more than five years previously. There followed a series of three more letter exchanges in which I drew attention again to the recent cover-up of the MRI evidence (indicating wilful negligence on the doctor’s part), as being clearly within the GMC’s five-year limitation period, and stressing the prima facie value of that evidence.

Throughout this exchange the GMC held to the position, as stated in their second letter of 20/03/2012, that: “The Assistant Registrar has read all the information that you have provided and has seen no evidence of the implementation of a cranial device hence the decision to not progress with your complaint.” At no point in this exchange (prior to their final response) did the GMC specifically mention the MRI evidence itself, or state that they had evaluated that evidence and found it to be normal. It was not until their response to my fourth letter, in which I raised the question that it might be the case that the GMC is unwilling to accept the MRI evidence without further written corroboration from a neurological expert, that they explicitly state the reason for their refusal to acknowledge the evidence, i.e., due to the circumstance that: “[W]e are not able to medically evaluate your scans”; despite having stated (paradoxically) in the preceding paragraph: “[T]he Assistant Registrar has considered all of the evidence that you have provided, including the visual MRI evidence”:

Final response from the GMC – 29/03/2012 [pdf – 42KB]

As the MRI component of the evidence (with respect to images 7.13-7.15, as discussed on the title page) included in my submission to the GMC was clearly crucial to the substance of my complaint, why did it take an eight-letter exchange for the GMC to state explicitly that they are unable to evaluate it? It would have been reasonable for them to qualify the statement in their initial letter of 02/03/2012: “From the information that you have provided, we cannot identify any issues that would enable us to conduct an investigation”, with an explanation to that effect, rather than persist in an eight-letter exchange on the basis of an ambiguity, especially as I had in my first letter pointed out the difficulties inherent in obtaining any such expert corroboration of the evidence. It does not seem unreasonable to expect that the General Medical Council should be in a position to evaluate first-hand medical evidence that is presented to it. However, it is clear that the GMC were simply averse to discussing or even acknowledging the element of the MRI evidence, until they were finally pressed upon the subject.

The allegations I have made have far-reaching implications, not only for the NHS, but for the reputation of the medical profession as a whole. Understandably, nobody wants to accept that members of the medical profession, in whose care we must routinely entrust our lives, are prepared to act with such concerted disregard for their patient’s well-being, and under the direction of imperatives having nothing to do with the care of individual patients. It reminds us of a rather unnerving truth: that, in the absence of additional safeguards, the practice of medicine is not an inherently ethical vocation.

In an important sense therefore, it is difficult to see any individual doctor’s corroboration (or lack of corroboration) as being truly independent. It is also the case that any neurological expert, on examining the evidence, would not simply be faced with the option of a correct established medical diagnosis in favour of an incorrect one, but rather of correcting the omission of a report of a serious medical anomaly, with a disclosure of findings that would be completely unprecedented in the history of medical science, originating as I allege they do from a clandestine medical experiment, carried out without any certified medical approval. It is precisely because the items revealed in the first and second MRI scans did not find their way inside my neck by accident, but by an illicit and covert design, that any individual doctor finds him- or herself unable to discuss or disclose them, not wishing to be compromised professionally, and with a view to his or her own personal safety.

While it is perhaps understandable that those with the relevant expertise might have difficulty finding the words suitable to describe these phenomena, it is of a major concern that doctors, by refusing to volunteer any information which might disclose the fact, can succeed in concealing the worst excesses of the medical profession, with no independent authority having expertise capable of overruling that reticence. On each occasion that I have sought medical consultation over my MRI scans, whether through NHS services or privately, I have come up against this ‘regime of silence’ – and this in spite of the fact that the visual evidence is clear and distinct, perhaps even to an untrained eye. As I have already received consultation over my first scan from a neurologist (my dissatisfaction with which was in part the subject of my complaint to the GMC), it seems intentionally evasive and circular that the GMC, in the final sentence of their last letter to me of 29/03/2012, should refer me back to square one – to discuss the matter with my GP!

September 2018

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