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


Full MRI Scan Downloads

The links presented on this page are to downloads of the complete MRI scans discussed on other pages on this website. They are included here for reference and verification purposes only. There are 5 scans in total and each is presented in the form of a ZIP archive containing the original MRI scan material as supplied on disc by the respective hospitals.

The first two zip downloads linked below relate to the Brain MRI and MRI Head scans performed at Guy’s & St. Thomas’ NHS Trust – ‘GSTT’ (MRI-1) and UCLH NHS Trust (MRI-2), in 2008 and 2013 respectively.

The third and fourth zip downloads linked further below relate to spinal MRI scans performed in 2015 at the BMI Hospital Blackheath, London (MRI-3); and the Royal London Hospital (MRI-4).

The fifth zip download linked finally below contains an MRI scan of my cervical spine made in July 2020 at a clinic in Serbia (MRI-5), following a flare-up at that time of the symptoms that have affected the region between my thoracic spine and left scapula since June 2015 (for further discussion see the relevant sections of the page: CSPINE MRI Scan (July 2020)).

After downloading each of the zip files, extract the files to a new folder (unique for each scan) before attempting to view the scans.

1. Brain MRI Scan (2008)

The images from this first scan referred to in the title page of this section are images 13, 14, & 15 from the 26-image Series 7 of the scan (coronal sections). I have indicated certain self-evident anomalies in these images which have not been disclosed in the MRI Findings report made by the Radiology Dept. at St. Thomas’ Hospital. There may be further anomalies in other areas of the scan which I remain unaware of.

BRAIN MRI scan – GSTT NHS Trust, 02/10/2008 – – 52.2MB (cloud source)

For an objective comparison of the scan contents with the MRI Findings report produced by St. Thomas’ Hospital Radiology Dept. on 06/10/2008 (as discussed on the title page of this section), a (partially redacted) copy of the Findings report is linked below:

GSTT MRI Findings – 06/10/2008 – 00-GSTT_MRI_Findings_6.10.08.pdf – 89KB

In order to view this scan it is necessary to burn the contents of the file onto CD Rom (not necessary for the later scans)1. The original disc was designed for compatibility with the Windows XP platform, and has limited compatibility with later versions of Windows. For instance, the disc would play on my own installation of Windows 7, but I experienced it failing to play on some installations of that platform. It may be similarly compatible or incompatible with later versions of Windows (for a workaround see Note 1).

To produce a working CD copy of the scan, download the file above and extract the contents to an empty temporary folder. Then burn the contents of that folder directly onto the root of a blank CD, i.e., not including the temporary folder itself, and preserving the directory structure of the files in the folder. Do not modify any of the files or folders following extraction. The disc will autoplay in Windows, or by running the file LaunchPad.exe from Windows Explorer. Alternatively, and to avoid compatibility issues, the scan may be viewed direct from a hard drive or USB drive by following the workaround described in the footnote below.

2. MRI Head Scan (2013)

The images from the second scan I have referred to here are images 51, 52, & 53 from the 128-image sequence located in the top folder named AAH Scout (sagittal sections), in the scan explorer window which opens when running the scan application. There are further self-evident anomalies (items which are clearly of non-biological origin) revealed in these images but left undisclosed in the report made by National Hospital for Neurology & Neurosurgery (NHNN – a part of UCLH NHS Trust). Again, there may be further anomalies in other areas of the scan which I remain unaware of.

MRI HEAD scan – UCLH NHS Trust, 06/03/2013 – – 60.2MB (cloud source)

The radiology report on the second scan made by NHNN was quoted by Dr. Dominic Heaney, Consultant Neurologist at NHNN, in his letter to my GP of 18/03/2013 linked below:

NHNN’s report on the 2nd scan – 18/03/2013 – 02-NHNN_scan-report_18.3.13.pdf – 45KB

I received the copy of the 2nd MRI scan from UCLH Medical Records Dept. two months after Dr. Heaney’s letter of 18 March. After studying the scan I became aware of the apparent anomalies I have referred to above. At this time I was awaiting a response to my complaint against GSTT, so I did not immediately raise an issue with NHNN over their failure to report anomalies from the 2nd scan. However, at the time of referring the complaint against GSTT to the Health Service Ombudsman, I sent the following email to Jill Rayfield, Dr. Heaney’s secretary, on 26/09/2013, attaching copies of the scan images in question, and asking Dr. Heaney to corroborate the fact of the self-evident anomalies.

Email to Dr. Heaney’s secretary – 26/09/2013 – 03-email_Dr.Heaney_26.9.13.pdf – 361KB

I received the following response from Dr. Heaney, dated 03/10/2013, in which he continues to deny the presence of the anomalies (in spite of the fact that they are quite self-evident) and attempts to explain the objects pointed out in my email in terms of “the posterior aspect of the foramen magnum and the lateral mass of C1” – i.e., the large aperture at the base of the skull and the first cervical vertebra – declaring them as “entirely normal”:

Dr. Heaney’s response to my email enquiry – 03/10/2013 – 04-Heaney_letter_3.10.13.pdf – 92KB

See the page: 2nd MRI Head Scan for a detailed accout of the progress of my subsequent complaint against UCLH NHS Trust.

The second scan may also be burnt to disc, or alternatively will play direct from a hard drive by running the file LaunchPad.exe from Windows Explorer. The second scan is compatible with most (if not all) versions of Windows.

TIP: On some computer screens, particularly laptop screens, or in situations of high ambient illumination, images from the second scan will display with improved contrast if one selects: Edit > Reset Window Level to 100% from the menu bar at the top-left of the screen.

2015 Spinal MRI scans

The two following links relate to scans performed in 2015 at the BMI Hospital Blackheath, London (MRI-3), and the Royal London Hospital – ‘RLH’ (MRI-4).

MRI Thoracic Spine – BMI Hospital Blackheath, 23/07/2015 – – 35.2MB (cloud source)

MRI Whole Spine – RLH, Barts NHS Trust, 11/09/2015 – – 42.5MB (cloud source)

These scans were made in response to a set of complex and aggressive problems that developed in the region between my thoracic spine and my left shoulder blade during the summer of 2015, and which I reported to my GP at that time. For further discussion of the emergence of these symptoms and the inadequacies of the assessment made of them by both the BMI Hospital and the Neurology Dept. at RLH during 2015, see the section: History of the Recent Symptoms and the Response from UK Health Services (and subsequent sections) on the CSPINE MRI Scan (July 2020) page.

The BMI Hospital provided no radiology report upon the scan conducted there. For reference, the following textual report on the scan conducted at the Royal London Hospital was extracted from the ‘Reports’ section of the scan accessible from within the scan display itself:

Radiology report from RLH – 30/09/2015 – Barts_MRI_report_30.9.15.pdf – 248KB

Each of the two spinal MRI scans linked above may be viewed on MS Windows platforms by running the scan applications direct from a hard drive, without the need to burn the scans to disc. The files to launch each respective application are AUTORUN.EXE (MRI-3), and run_cdviewer.exe (MRI-4). When launching AUTORUN.EXE from the MRI-3 folder from a hard drive (rather than from a compact disc), it is necessary to first point the fastView Browser to the MRI-3 folder to be able to load the scan images. In the top left of the fastView Browser window, where it says “Source”, click on the “Browse for folder” button (marked “...”) and browse to the MRI-3 main folder to select it as the source for the images.

When viewing MRI-4, it is necessary to select and highlight the single entry in the “Patient CD Index” window which initially opens, before clicking the “OK” button (!)

C-Spine MRI (July 2020)

More recently, having experienced intermittent relapses of the symptoms affecting the region between my thoracic spine and left scapula over a period of five years, in July 2020, while abroad, I experienced a marked flare-up of the symptoms resulting for the first time in inflammation and irritation to the surface of the skin in the area, as well as a local rash developing there. This development was quite alarming, particularly as it was accompanied by exaggerated internal symptoms, and so I sought a private medical examination at a local clinic in Serbia, where I happened to be travelling at the time.

A C-Spine MRI scan was performed on 28 July 2020 at the Zdravlje Plus Clinic, Novi Sad. The contents of the scan disc are available to download below as “”. An English translation of the main content of the radiological report provided with the scan is available in the subsequent link. To view the scan from a hard drive (MS Windows platforms), download and extract the archive and then run SDViewerFree32.exe or SDViewerFree64.exe (according to your PC system type).

C-Spine MRI scan – Zdravlje Plus Clinic, Serbia, 28/07/2020 – – 44MB (cloud source)

Radiology report from Zdravlje Plus Clinic – 29/07/2020 – MRI_CSPINE_29.7.20_report(trans).pdf – 410KB

For a discussion of the contents of the latest scan and report with reference to a selected image from it, see the page: C-Spine MRI Scan (July 2020).

6 January 2024

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  1. There is a workaround to enable viewing MRI-1 without the need to burn the scan to disc. This requires that one views the MRI-1 scan material through the viewing application provided with the MRI-5 scan, and so requires that you also download the MRI-5 scan archive (cloud source) to do so. From a hard drive, after extracting to its own folder, open that folder and run either SDViewerFree64.exe or SDViewerFree32.exe (according to your system type) from there. Assuming that the MRI-1 scan folder is present alongside the MRI-5 folder, close down the initial Open DICOMDIR window that opens by default in MRI-5 and select: File >> Open >> Open Folder from the menu bar at the top left of the screen. You may then browse to the MRI-1 main folder to select it as the source for the images. This will then load the MRI-1 scan images into the MRI-5 application, where they can be viewed direct from a hard drive or USB drive, without the need to burn the scan to disc.[back]