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 which 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 numbers when numerical 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 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!


Download my 165-page report
[pdf – 1.9MB]:

Download my Open Letter to the British Prime Minister & Health Secretary
[pdf – 612KB]:

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


Radio-Frequency Radiation & Microwave Communications: Research Papers

Some of the conclusions made explicit in the document Special Operations in Medical Research (the 'report') are supported by reference to research papers on the subject of the effects of exposure to radiofrequency radiation on human physiology and behaviour. These references have revealed aspects of this subject which have more general relevance, beyond the specific concerns of the report, especially with respect to the non-thermal effects of RFR (or 'electromagnetic radiation' – EMR) on human biochemistry; that is, in consideration of our (relatively recent) newfound dependency on mobile communications technology. This page attempts to present a synopsis of those research papers, which I feel may be of some public interest.

"We can hear RFR delivered in pulses. An explanation for this 'hearing' effect is that it is caused by thermoelastic expansion of the head of the 'listener'. In a classic paper by Chou et al. [1982], it was stated that " hears sound because a miniscule wave of pressure is set up within the head and is detected at the cochlea when the absorbed microwave pulse is converted to thermal energy." The threshold of hearing was determined to be approximately 10 microjoule/gm per pulse, which causes an increment of temperature in the head of one millionth of a degree centigrade!"

Neurological Effects of Radiofrequency Electromagnatic Radiation, Henry Lai.

"We have natural EMR-based communication systems in our brains, hearts, cell and bodies. External natural and artificial EMR resonantly interacts with these communication systems altering hormone balances and damaging organs and cells [...] When EMR interferes with the EEG this is communicated to the body by neurotransmitters and neurohormones, including the serotonin/melatonin system. EMR reduces melatonin. Melatonin is vital for the health of the Immune System, the Brain, The Heart and every cell, because it is the most potent naturally produced antioxidant."

Health Effects of Electromagnetic Radiation, Dr Neil Cherry.

"Blood mononuclear cells from healthy donors were exposed to 1300 MHz pulse-modulated microwaves for 1 hour. The average SAR was 0.18 W/kg. A variety of proliferative and immunoregulatory properties were examined.

The results indicated that the RF radiation influenced the monocyte-dependent immunoregulatory mechanisms responsible for the initiation of the immune response. The exposure significantly increased the production of interleukin-ß and decreased the concentration of its antagonist IL-1ra.

[...] The authors conclude that the immunostimulatory effects of 1300 MHz pulsed microwaves preferentially affect the immunogenic function of monocytes in vitro. Further research is needed to confirm these findings and to extend them to in vivo situations."

Immunotropic effects in cultured human blood mononuclear cells pre-exposed to low-level 1300 MHz pulse-modulated microwave field, Dabrowski et al.

"An important question regarding the biological effects of RFR is whether the effects are cumulative, i.e., after repeated exposure, will the nervous system adapt to the perturbation and, with continued exposure, when will homeostasis break down leading to irreparable damage? [...]

Existing results indicate changes in the response characteristics of the nervous system with repeated exposure, suggesting that the effects are not 'forgotten' after each episode of exposure.

[...] we found that a single episode of RFR exposure increases DNA damage in brain cells of the rat. Definitely, DNA damage in cells is cumulative. Related to this is that various lines of evidence suggest that responses of the central nervous system to RFR could be a stress response [Lai, 1992; Lai et al, 1987a]. Stress effects are well known to cumulate over time and involve first adaptation and then an eventual break down of homeostatic processes when the stress persists."

Neurological Effects of Radiofrequency Electromagnatic Radiation, Henry Lai.

Stress responses and immunoregulatory stimulation – experimentally identified as two areas of probable health effect from exposure to sustained radiofrequency radiation. Recent research into the effects of RFR/EMR on human biochemistry have identified an important concern for the non-thermal, or 'micro-thermal' effects, in addition to the thermal (direct tissue-heating) effects from the proximate use of mobile telephones – the latter which have conventionally been measured as 'SARs' (specific absorption rates).

"[A] new situation arose when significant part of general population [became] exposed chronically (much longer than previously investigated durations of exposures) to non-thermal microwaves from different types of mobile communication including GSM and UMTS/3G phones and base stations, WLAN (Wireless Local Area Networks), WPAN (Wireless Personal Area Networks such as Bluetooth), DECT (Digital Enhanced Cordless Telecommunications) wireless phones. It should be anticipated that some part of population, such as children, pregnant women and groups of hypersensitive persons could be especially sensitive to the non-thermal microwave exposures. It is becoming more and more clear that the SAR concept that has been widely adopted for safety standards may not be useful alone for the evaluation of health risks from microwaves of mobile communication. How the role of other exposure parameters such as frequency, modulation, polarization, duration, and intermittence of exposure should be taken into account is an urgent question to solve. Solving this question would greatly benefit from the knowledge of the physical mechanisms of the non-thermal microwave effects. The understanding of mechanisms for the non-thermal microwave effects is far away from comprehensive. [...]

The data about the effects of microwaves at super low intensities and significant role of duration of exposure in these effects along with the data showing that adverse effects of non-thermal microwaves from GSM/UMTS mobile phones depend on carrier frequency and type of the microwave signal suggest that microwaves from base stations/masts can also produce adverse effects at prolonged durations of exposure and encourage the mechanistic in vitro studies using real signals from base stations/masts. Further investigations with human primary cells under well controlled conditions of exposure, including all important parameters as described above, are urgently needed to elucidate possible adverse effects of microwave signals that are currently used in wireless communication, especially in new technologies such as UMTS mobile telephony."

Non-thermal Biological Effects of Microwaves, Belyaev, I.

Although non-thermal effects may be cumulative, they are not proportionate to the level of exposure or to the degree of any direct tissue-heating effect. They are non-linear effects, which means that they are not experimentally measurable in the way that thermal effects are (that is, by measuring the internal heat-increase of dummy heads). They are also present environmentally, due to the proximity of cell towers, and the general distribution of mobile devices. These effects have been largely ignored by telecoms providers, when advising on the relative safety of mobile communication devices.

The human body has been described as an "electromagnetic instrument of great and exquisite sensitivity". There is what has been referred to as an "oscillatory similitude"*, between certain frequencies of microwave radiation, and the specific frequencies of electro-chemical processes at the cellular-electrolytic level, so that corresponding microwave frequencies may interfere with and disrupt those physiological processes.

It is not so much the intensity or proximity of a source of microwave emission which is relevant to a non-thermal or micro-thermal effect, but its frequency, which does not vary with intensity. In this way low-intensity microwave radiation, at an appropriate frequency, may interfere 'synergistically' with critical regulatory bodily functions. When we consider this, in the light of recent research, as a problem which may accumulate biologically, the potential effects upon the population, and upon children in particular, can neither be quantified nor fully anticipated.

* How Exposure to GSM & TETRA Base-station Radiation can Adversely Affect Humans, Dr. G J Hyland (full details below).

Further references:

Balmori, A., Electromagnetic pollution from phone masts. Effects on wildlife, Pathophysiology 16, 2009, pp.191-199:

Belpomme, D., Professor of Clinical Oncology, University of Paris-Descartes, Clinical and biological description of the electromagnetic field intolerance syndrome (EMFIS), 8th National Congress on Electrosmog, Berne, 2011:

Belyaev, I., Non-thermal Biological Effects of Microwaves, Microwave Review, Vol.11/2, Nov. 2005:

Belyaev, I., Non-thermal biological effects of microwaves: current knowledge, further perspective and urgent needs, Workshop: "Do sinusoidal versus non-sinusoidal waveforms make a difference?" Zurich, Switzerland, Feb. 2005:

Belyaev, I., Risk Assessment of Chronic Exposures to Non-Thermal Microwaves from Mobile Communication, Stockholm University, Department of Genetics, Microbiology and Toxicology:

Cherry, N., Health Effects of Electromagnetic Radiation. Evidence for the Australian Senate Committee, Lincoln University Christchurch, Po Box 84, Canterbury, New Zealand:

Cherry, N., Probable Health Effects Associated with Base Stations in Communities: The Need for Health Surveys (Short Version), Lincoln University Christchurch, Po Box 84, Canterbury, New Zealand: [last accessed 30/03/2019].

Dabrowski M.P.; Stankiewicz W.; Kubacki R.; Sobiczewska E.; et al, Immunotropic effects in cultured human blood mononuclear cells pre-exposed to low-level 1300 MHz pulse-modulated microwave field, Electromagnetic Biology and Medicine vol.22, no.1, pp.1-13, 2003.

De Pomerai D. I.; Dawe A. S., Brette Smith & John Worthington School of Life and Environmental Sciences, University of Nottingham, Nottingham NG7 2RD, UK. Non-thermal Biological Effects of Microwave Radiation on Caenorhabditis Elegans: [last accessed 30/03/2019].

European Parliament, Directive 1999/5/EC of the European Parliament and of the Council of 9 March 1999 on radio equipment and telecommunications terminal equipment and the mutual recognition of their conformity: [last accessed 30/03/2019].

Hyland, G. J., How Exposure to GSM & TETRA Base-station Radiation can Adversely Affect Humans, Associate Fellow, Department of Physics, University of Warwick, Coventry, UK; Executive Member-International Institute of Biophysics, Neuss-Holzheim, Germany, August 2002:

Hyland, G. J., The Physiological and Environmental Effects of Non-Ionising Electromagnetic Radiation, Publisher: European Parliament Directorate General for Research, Directorate A, The STOA Programme, University of Warwick, UK International Institute of Biophysics, March 2001:

Hyland, G. J., Physics and biology of mobile telephony, Department of Physics, University of Warwick, Coventry, UK, and the International Institute of Biophysics, Neuss-Holzheim, Germany:

Lai, H., Neurological effects of Radiofrequency Electromagnetic Radiation, Bioelectromagnetics Research Laboratory, Department of Bioengineering, School of Medicine and College of Engineering, University of Washington, Seattle, Washington, USA:

Microwave News, Changes in Protein Folding: A Nonthermal RF Mechanism. Is There a Risk of Neurological Disease?, Microwave News, May/June 2003: [last accessed 30/03/2019].

Moulder, J., Professor of Radiation Oncology, Static Electric and Magnetic Fields and Human Health (FAQ), Medical College of Wisconsin. Milwaukee, Wisc., U.S.A.: [last accessed 30/03/2019].

Murphy E., PhD. (facilitator), EMF Effects on Cellular Calcium, EMF Science Review Symposia Theoretical Mechanisms and In Vitro Research Findings (1997):

ORGALIME, Conference on the Application of the Precautionary Principle to EMF, Draft speaker's notes of Philippe Portalier, 24 February 2003, Luxembourg:

ORGALIME & WEM Position, Coherence of the EU policy with regard to the protection from exposure to electromagnetic fields (EMF), Paper 1: Annexes & Recommendations, 13 December 2002.

Royal Society of Canada, A Review of the Potential Health Risks of Radiofrequency Fields from Wireless Telecommunication Devices, An Expert Panel Report prepared at the request of La Société royale du Canada, March 1999. ISBN 920064-68-X:

Sandyk R.; Anninos P.A.; Tsagas N., Magnetic fields and seasonality of affective illness: implications for therapy, Department of Psychiatry, Albert Einstein College of Medicine/Montefiore Medical Center, Bronx, NY 10461. Source: International Journal of Neuroscience, June 1991: 58(3-4):261-7: [last accessed 30/03/2019].

Sivani, S.; Sudarsanam, D., Impacts of radiofrequency electromagnetic field (RF-EMF) from cell phone towers and wireless devices on biosystem and ecosystem – a review, Biology and Medicine, v.4(4), pp.202-216, 2012:

Sri Nageswari K., Biological Effects of Microwaves and Mobile Telephony, Professor & Head, Department of Physiology, Govt. Medical College & Hospital, Prayaas, Building Sector 38, Chandigarh, India: [last accessed 30/03/2019].

World Health Organisation, Electromagnetic Fields and Public Health: Mobile Telephones and Their Base Stations: [last accessed 30/03/2019].

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