Philosophy Of Science

"It was, of course, a lie what you read about my religious convictions, a lie which is being systematically repeated. I do not believe in a personal God and I have never denied this but have expressed it clearly. If something is in me which can be called religious then it is the unbounded admiration for the structure of the world so far as our science can reveal it."

"A knowledge of the existence of something we cannot penetrate, our perceptions of the profoundest reason and the most radiant beauty, which only in their most primitive forms are accessible to our minds - it is this knowledge and this emotion that constitute true religiosity; in this sense, and this alone, I am a deeply religious man."

"I believe in Spinoza's God who reveals himself in the orderly harmony of what exists, not in a God who concerns himself with the fates and actions of human beings."

: "My religion consists of a humble admiration of the illimitable superior spirit who reveals himself in the slight details we are able to perceive with our frail and feeble mind."

"WHAT WE CAN KNOW OF THE WORLD CONFORMS TO CERTAIN SYNTHETIC A PRIORI CATEGORIES, WHICH ALTHOUGH WE RECOGNIZE BY EXPERIENCE DO NOT ARISE FROM EXPERIENCE. THESE CATEGORIES (E.G. SPACE AND TIME) ARE LAID DOWN BY THE MIND AND TURN SENSE DATA INTO OBJECTS OF KNOWLEDGE"

At the beginning of the last century, there were just two seemingly small clouds, which obscured our otherwise impressive view of the physical world. Most of physics was believed to have been well understood, there just the small problem of the inability to detect the luminiferous ether and the statistical anomaly in black body radiation. These ‘small’ clouds on the horizon did however grow to gigantic proportion, casting a gloomy shadow over the physics community which resulted in two great revolutions viz. Relativity and Quantum theory.

The inability to detect the ether was an indication that our understanding of space and time was flawed and eventually culminated in Einstein's general theory of relativity, which not only discarded the notion of an absolute space and universal time but also demonstrated that gravity was a manifestation of a curvature in the space-time continuum. Matter tells space and time how to warp and these in turn determine how test particles move (along geodesics of maximised space-time intervals, or proper time). Henceforth the laws of physics were no longer valid under Galilean transformations but instead had to be Lorentz covariant and Maxwell’s equations of electromagnetism, were unintentionally the first to be written in this form.

The ultraviolet catastrophe that had occurred in the theoretical study of black body radiation was yet another example of where ‘a conflict was brought to light’. Using the two pillars of classical physics (Maxwell’s electromagnetism and Newtonian mechanics), it was possible to predict the distribution of frequencies emitted by a perfect black body radiator. However the embarrassingly infinite amount of radiation that is predicted at the shorter end of the spectrum, could only be avoided if we introduce a quantised (instead of continuous), emission by the atoms that were in thermal equilibrium with their radiation. The energy is therefore released in discrete amounts that were proportional (via Planck’s constant) to the frequency. Planck himself was troubled by the implications of his own radical alteration but far worse was yet to come. Einstein’s paper on the photoelectric effect showed that it was not that thermal oscillations emitted discrete amounts of electromagnetic waves but that light (and other radiation) has an inherently particulate nature. This view was emphatically reinforced by Compton’s electron scattering experiment, which demonstrated that X-ray quanta possessed the same kind of momentum that is associated with particles in collisions.

A further challenge to classical physics came from the fact that Rutherford’s alpha scattering experiment showed that atoms consist of a positive nucleus orbited by electrons. This in turn conflicted with electromagnetic theories since such accelerating charges should continuously emit radiation and spiral into the nucleus. At the same time De Broglie had suggested, that entities such as electrons, could be regarded as possessing wavelike characteristics. Indeed J.J. Thomson was awarded the Nobel Prize for demonstrating that electrons are particles while his son later received the same honour for showing that they were waves!

Also experimental data regarding the change in the specific heat of materials as they are cooled to very low temperatures, can only be explained by using the phonon model of lattice vibrations. Indeed it was the partial success of Einstein’s earlier attempt at such a quantum description of specific heat, that was particularly influential in causing the widespread acceptance of quantum theory. However it did take the first quarter of a century before the downfall of classical physics was finally complete and the early efforts of Einstein, Planck and De Broglie were shown to be a manifestation of the far more sinister reality of Schroedinger’s wavefunction and Heisenberg’s uncertainty principle.

At the dawn of a new century (indeed a new Millennium) the main cloud that we face in physics, is the inability to reconcile General Relativity (and hence gravity) with Quantum Theory. It may not be fully appreciated for quite some time, as to how big a revolution in our understanding this will require, although there are indications that it will be considerable. On a wider front, the other great challenges will include an understanding of the human mind (particularly consciousness) and what constitutes life, as opposed to other complex chemical systems.

The last century did leave us with the legacy of the nuclear age, one that initially promised a Utopia but did so much to disappoint. The new century could unleash a far greater jeopardy, since the bio-technological time bomb that we are in danger of releasing, may be far more perilous than the nuclear bomb that we have learnt to live with so far. There is consequently an even greater need for the general public to become more educated in science, but without becoming professionally indoctrinated

According to Raven, the dilemma in which honest students found themselves, arose out of the need to maintain their ‘religious experience or moral responsibility’ whilst retaining their scientific integrity. In his analysis ‘an agnostic humanism or an authoritarian supernaturalism seemed an obvious answer to their dilemma. ie. either a release from ecclesiastical dogma, faith in progress and a belief in man being the measure of all things or the unconditioned acceptance of theology in dictating a moral code and the spiritual reality beyond science.

James Maxwell provides a good example of how such an ‘honest student’ came to terms with the task of advancing the frontiers of scientific while maintaining the spiritual truths of his religion. Maxwell was imbued with a deep mystic piety, characteristic of his native Gallaway and adopted a theological basis to nature that involved a deep romantic appreciation. This is particularly exemplified in his formation of his statistical laws of thermodynamics, in which he realised that the only satisfactory way of explaining the observed facts was to implement an epistemology of acausal chance but at the same time he retained a belief in ontological determinism.

Such a nuance resulted directly from his personal need to reconcile science with religion. At that time there was abhorrence towards deterministic materialism and its proclivity for atheism, but there was also a reaction against Darwin’s theory, which deemed that chance rather than design was the cause of evolution. To have accepted the materialistic ontology of Lucretius, with its violation of atoms and deification of chance would have sacrilege the omnipotence of God. Whilst replacing the ontological randomness of Lucretius with the apparent randomness of Laplace, he regarded the macroscopic world as one of ‘chance and change’ in which his epistemology is akin to the certainties of the census taker rather than to Laplacian probabilities which was steeped in determinism. Such a stance was not only pleasingly detrimental to the epistemology of determinism but also succeeded in maintaining an ontology that was amicable to a teleological view of God

In order to emphasis the distinction between the apparent randomness due to our inability to know and ontological certainty, Maxwell invented a being (unfortunately dubbed ‘Demon’ by Kelvin), who could violate the Second law of thermodynamics due to his omnipotence, thus showing the law to be of statistical rather than of absolute nature. We now know that the entropy associated with information makes any such proposal a theoretical impossibility but this does not diminish Maxwell’s GEDANKEN, since to him the Demon was analogous to God and hence not subject to materialistic constraints – being a creature of divinity, he was not limited to energy requirements. Likewise, being faced with the concept of a universe which was dissipating under the Second Law, he again related to a divine being to whom the universe is always ordered and controllable and from which he can always extract work.

The debate over free will versus determinism had been a source of contention between the Calvinists (whose hellfire attitude denied the former) and the Armenians whose moral teachings were based upon volition – the child being the father of the man, rather than the wretched being eternally damned. Maxwell, having allegiance towards the latter, was consequently averse to the use of determinism as applied to man. Being aware of the transitory nature of scientific theories, he did not want to attach them to religious ideas (whose knowledge to him was absolute) since it ‘may help to keep the hypothesis above ground long after it ought to be buried and forgotten.’

Whilst desiring consonance between science and his personal belief, he was not prepared to make religious appeasement’s which he knew would become an embarrassment when that particular theory became obsolete. He realised the approximate nature of different scientific theories and was likewise tolerant to the various religious denominations, felling equally at home in Baptist, Presbyterian or Anglican churches. In this way he avoided having to interpret religious scriptures in terms of the prevailing scientific view. He believed that behind the empirical realities of nature and the various sectarian postures of religion, lies a deeper level of reality and did not wish any of his conceptions to be ‘carried beyond the truth by a favourite hypothesis.’

His most monumental work, that of the unification of electrostatatics, magnetism and light had also been driven by his conviction for a creative element of the universe – a conviction which inspired, guided and justified his scientific endeavors. He attached great significance to both inner experience as well as insights from the external world – what he described as the ‘Two Gateways to knowledge. His radical evangelical beliefs had both directed him in life and supported him in death.

Being born a Jew it was taken for granted that Emile Durkheim would become a Rabbi like his father. However during his student day he became more secular but with a strong bent towards moral reform and he tended to take a silent agnostic view upon whether religious beliefs did have a divine basis outside its social function. His personal ideology was somewhat stoical, believing that "efforts and even sorrow are more constructive to the spiritual progress of the individual rather than pleasure or joy."

Durkheim's approach relied heavily on scientism (particularly biologism) by which categories of science are extrapolated into other fields of knowledge, such as ethics and theology. Although believing in the autonomy of social facts, without needing to resort to reductionalist explanations, his functionalism did rely upon biological analogies. Durkheim was a socialist for whom Homo Faber had the power to drastically influence his own environment. Natural selection did not therefore play such a powerful role upon society as it had done with the development of life itself -- modes of expression depending upon cultural rather than genetic inheritance. In his view morality is only justified in its social acceptance by the majority, taboos and rituals included and in some respect his tolerant treatment of all religions is somewhat disconcerting as he states " The most barbarous and most fanatic rites and strangest myths translate some human need of social life...."

For Durkheim there is no conflict between religion (ads he views it) and science with which he attempts to explain it -- the very act of which presupposes it. Religion fills the void left by science, with speculation and the resulting conflict between scientific reason and religious faith often results in the outpouring of idiosyncrasies. He states "Science is fragmented and incomplete, it advances but slowly and is never finished but life cannot wait". The theories which are destined to make men live are therefore obliged to pass science and complete it prematurely." He therefore regards both the religion and science as attempting to satisfy the need to connect things with each other in order to obtain a greater understanding and even views scientific logic as being of religious origin, although the former purges superfluous elements and incorporates a spirit off criticism. Scientific thought is therefore regarded as a more refined form of religious thought, from which it has developed both of which are ultimately founded upon faith. Whereas religion claims the right to go beyond science, Durkheim personally eschews such metaphysical beliefs but instead regards religion as playing a crucial and unavoidable social role.

The Danish physicist Neils Bohr, pioneer of modern atomic theory, was an incarnation of altruism. During the war, he had assisted refugees escape Nazi tyranny and had even donated his Nobel Prize to the Finnish war relief, while after he was instrumental in establishing the ‘Atoms for peace’ conferences. His mother was of Jewish descent but his upbringing was that of a traditional Protestant. During his youth he was an avid reader of his compatriot Soren Kirkegaard, a religious thinker, regarded as the originator of Existentialism (Leap of Faith=Quantum jumps). Kierkergaard, had reacted against the wave of intellectualism that had threatened to engulf Christian belief at that time and in addressing the paradox of Jesus necessarily being the embodiment of both men and God, he wrote ‘When subjectively (inwards) is the truth, the objectively defined becomes a paradox and the fact that the truth is objectively a paradox shows in turn that subjectivity is the truth’ It is quite possible that Bohr reflected upon this view when he came to grips with the Quantum postulates, in which the question of whether an electron (or a photon) is a particle or a wave is meaningless and leads to a paradox when viewed objectively, being only answerable in terms of the subjection description of a quanta. [Kierkigaard had similarly shifted the focus of attention away from the contemplation of who God is to how God relates to us –in other words how we interact with him].

Independently of Heisenburg’s Uncertainty principle, Bohr had formulated his ‘framework of Complementarity’(1927), a new feature of natural philosophy which means a radical revision of our attitude as regards physical reality’. Bohr maintained that any distinction between observing systems and the observed object are arbitrary made by the description of that interaction. The anti-realism implication of recent developments (viz matrix mechanics) had become a continual torment to Bohr. ‘How can our lord possibly keep this world in order?’ Bohr could have kept our classical framework and merely discarded the relativistic interpretation but in fact he did just the opposite, what he rejected was not realism but the classical version of it, which he replaced with his complementarity – not a theory or a principle but an epistemological framework!

Bohr thus obtained spiritual solace from his new found realism, in which certain dynamic attributes (eg position and momentum) are contextual, while the static attributes of mass, charge and intrinsic spin are innate. However the philosophical framework which he tried which he tried to instigate was intended to extend beyond the domain of quantum theory but unfortunately it often became incarcerated by if not mistaken for the Critical positivism of the Copenhagen interpretation, which simply endorsed which simply endorsed the unavoidable use of conjugate pairs of variables, for which precise information was simultaneously denied. Bohr believed that his ‘epistemological lesson of complementarity’ had potential in solving many age old dilemmas such as mechanism versus vitalism in biology, free will vs. determinism in psychology, or nature vs nurture in anthropology; he dreamt of a ‘conceptual framework’ which could be the ‘unity of knowledge.’

It was during the ensuing debate with Einstein that the full implications of what Bohr admonished became apparent- a favourite quotation of his being Schiller’s Sayings of Confucius ‘Only wholeness leads to clarity/and truth lies in the abyss.’ Einstein’s argument involving the EPR experiment appeared to demonstrate that an electron did indeed posses more information than was extracted by an observation (thus demonstrating that quantum theory failed the completeness criterion.) However Bohr’s counter was his belief that the whole was greater than the sum of the parts. Although some earlier influences towards such beliefs can be traced back to discussions with his father Christian, brother Harold and the philosopher/friend Neils had been considerably influenced by eastern religious beliefs. Having spent some time touring the orient (in particular China) when knighted, he chose the Taoist symbol of Yin and Yang as his coat of arms1 this pristine symbol of harmony represents a holism that is very similar to that enshrined in Bhor’s complementaity framework.

Up until then western philosophy had tended to find reality in substance, whilst oriental philosophy had found it in relation. Bohr wrote ‘isolated systems are abstractions, their properties being definable and observable only through their interaction with other systems.’ Just as Yin complements Yang, so does position complement momentum as does energy with time. Although Bohr was reticent upon ontological maters, he in effect had made science more spiritual and was quite likely struck by the similarities between empirical scientific endeavour and primordial eastern belief, in which space, time and causality are like a glass through which the absolute is seen; in that absolute there being neither space, time or causality [Heisenburg had likewise become deeply impressed and guided by allegorical eastern beliefs, during his stay in India.] This dynamic interplay of Yin and Yang as exemplified in the Uncertainty principle or the complementarity of a wave-packet is even more apparent in the latter more successful theory of Quantum electrodynamics by which even a vacuum is a fluctuating continuum of ephemeral particles.

Bohr was one of the first to realise that physics and Eastern beliefs were coterminous in their attempt to understand reality beyond our immediate senses. He tried to point a way in which the idea of complementarity could throw light upon many aspects of human life and thought, even venturing comments upon its relevance to art, music and religion and the age old problem arising from the fact that ‘man both acts and is the spectator of his actions in the drama of existence.’ His life long consolidations of complementarity (during which he wrote more than 20 varied papers) had probably therefore been inspired at first by the religious teachings of Kierkergaard and later by oriental beliefs which emphasis harmony, wholeness and rhythmical interplay. This would have endorsed a conviction for the personal subjectivity of God and the moral belief that man cannot insulate himself from his deeds or his environment, his actions having repercussions that may ultimately affect his own wellbeing. For Bohr, this perhaps would have been a fitting atonement for the loss of causality.

During the Weimar republic after the first world war, Spengler's book " Decline Of The West" became very influential amongst intellectuals. The outlook of the Weimar milieu, was ne-oromantic and existentialist; it tended to build a future out of its past, returning to the romanticism of Schiller and Goethe. Spengler Idealism was hostile towards the ideology of the exact sciences and his philosophy influenced many including Weyl, Brouwer, Heisenberg and Schrodinger. As a result, intuition rather than logic became favorable in mathematics, while destiny rather than causality tended to dominate in science. The two main defenders of the traditional view were Hilbert, who defended the primacy of logic in the foundation of mathematics and Einstein who defended the primacy of causality in physics. However, the Spenglerian ideology of revolution triumphed both in physics and mathematics, since Heisenberg discovered the limits of causality in atomic phenomena and Godel the limits of formal deduction and proof in mathematics! However eventually the vision of Spengler became irrelevant, since Chemists who had never heard of him could use QT to calculate covalent binding energies, while the discoveries of Godel did not lead to a victory of intuitionism but rather a recognition that no single scheme of mathematical foundations has a unique claim to legitimacy

Dilemma is an old Greek word, which means that one is faced with two possible choices, non-of which is acceptable! Consider the riddle of the prisoner who is sentenced to death. The judge decrees that the last statement of the condemned man shall decide the method of execution. If his last statement to the executioner is true he will be beheaded if it is false he will be hanged – the choice is therefore in the hands of the prisoner. On the morning of his execution he whispers his final words to the executioner, who being faced with a dilemma, decides to release him. What were the prisoner’s words? [The answer is below]

A well known dilemma in Physics is ‘what happens when an irresistible force meets an immovable mass.’ The solution is actually quite straightforward once one realises that the phrases need to be clarified. Referring to Newton’s Second law F=MA, there is no such thing as an ‘immovable mass’ (object), since even the smallest of forces can move the largest of masses, although the acceleration would be minute. Also the concept of an ‘irresistible force’ needs to be clarified, since as before, even the smallest of forces produces movement (acceleration) on the largest of masses, if there is no opposing force to balance it. Hence if a force is to be considered irresistible it must be confirmed that it is not possible anywhere in the Universe to provide another force of equal magnitude to oppose it.

There is a similar dilemma in religion, which questions the existence of God, since he is supposed to be both omnipotent and benevolent. However Evil exist either in spite of his efforts (which precludes him being omnipotent) or because of him (hence he is not benign.) The inability to resolve the dilemma in its present context indicates that the very concept of God needs to be clarified. Therefore, if one asks if God exists, the possible answers could be ‘Yes’(Deism), ‘No’(Atheism), ‘its not possible to decide’(Agnosticism) or that the question (as it stands) has no meaning. There are examples in physics in which questions have no meaning, such as “is an electron a particle or a wave.”

St. Augustine once stated that God did not create the universe at a given moment in time but rather that time was created with the universe. This showed considerable foresight, in that it is only this century that physicists have firm evidence of the big bang, in which space and time were created along with all the matter/energy of the universe. However, there have been those who posit that only matter and motion exist and recently, even some physicists have doubts about whether time actually exists. Immanuel Kant believed that what we can know of the world, is subject to certain synthetic a priori categories, that are laid down by the mind, which although we recognize by experience, do not arise from experience. Space and time were cited as such examples, which act rather like filtering lenses by which the brain turns sense data into objects of knowledge.

The physicist P. C. W. Davies remarked that we may not fully understand time until we understand the human mind and even Einstein originally believed that space and time are modes by which we think rather than conditions in which we live! [He once mischievously said that if you place your hand on a hot metal plate, time moves slowly but when you are sitting next to a pretty girl, it goes quickly – that’s relativity]. His later theory of General Relativity (GR) did however give a more ontological reality to space-time, which supplanted the primacy of gravitational mass. Quantum theory (QT) was however difficult to reconcile with the space-time of special relativity and this was only achieved by the introduction of spinor mathematics and necessitated a field theory. The space-time of GR is still incompatible with QT, since spinor manifolds do not avail themselves to Riemannian technique and more importantly, the unitary groups that dictate interactions in QT are not compatible with the unimodular group of relativity (unless one resorts to supermanifolds).

As physical theories have progressed our notion of time has been continually revised. The Galilean/ Newtonian view, is that time is absolute and that everyone can agree upon what events are simultaneous, no matter where they occur. This is what is now regarded mathematically, as an example of a fibre bundle i.e. a bundle whose base space is time and whose fibres are that of Euclidean 3-space. Minkowski demonstrated that space and time are inextricably linked, forming a 4-dimensional manifold as illustrated by Einstein’s 1905 paper on Special Relativity. By 1916, an extension to a general (non-inertial) co-ordinate system showed that time is slowed down (and space is warped) by gravitational mass. More recently Penrose has developed theories using projective twister space (rather than space-time) which he believes is a fundamentally better manifold for structuring physical laws. Hawking on the other hand believes that QT is the way forward, although he does utilize an imaginary time co-ordinate.

There have been several speculations as to the possibility of time travel, most of which rely on the notion of warped space time. Special Relativity is itself completely consistent with time travel into the future (moving clocks run slower**) but traveling into the past could lead to paradoxes. If a man can travel into the past he could change events (e.g. kill his grandfather) which would prevent him from ever being born in the first place. In 1949 Kurt Godel found a solution to Einstein's field equations, in which a static universe would be stable providing it rotated sufficiently fast (the centripetal acceleration is in balance with the gravitational attraction of the whole mass in the universe). He found that such a universe would result in a curving around of space-time in such a way that traveling in closed loops, would not only change your displacement but would also allow you to travel backwards in time. Since a rotating inverse is not the case, the next best option is inside a spherical black hole (which are believed to be quite common), where the space and time coordinates also become interchanged. However there is no way of escaping from a spherical black whole once you have passed the event horizon. There may however be a way to circumvent this problem, if one could create a rotating cylindrical shaped black hole, whose rotation would allow you to escape from being dragged down into the singularity, but such structures may not be stable. Also there are worm holes that allow distant regions of space to be locally connected by a short cut through space and which also suggest the feasibility of backwards time travel, however these structures are also believed to be fundamentally unstable for any would be time traveler.

Yet another possibility arises from the study of cosmic strings; when two such entities come together it can be shown from topological considerations that it may be possible to travel backwards in time by moving in a circular path around the 2 strings. However such cosmic strings have not yet been identified in the universe. Also any such time travel machine would only allow you to travel back to the time in which the devise was first built - this is often cited as the reason why we have not been visited by a more advanced civilization from our future. As already mention backward time travel can lead to dilemmas. For example imagine that one day a girl physicist discovers a new solution written by Einstein, which describes the workings of a feasible time machine. Having the technology to produce such a machine, she travels back in time to visit Einstein and shows him the solution which he then records (and is subsequently discovered by the girl). The paradox then arises as to who actually discovered the solution! Finally an interesting scenario has been put forward concerning the rapid expansion in computer technology.

Future civilizations will have sufficient computer power (especially with the advent of quantum computers) to create perfect virtual reality universes, which they will be able to run backwards and forwards in time. [It has been recently demonstrated that even installing simple rules on a computer program, can cause the evolution of quite organised systems, which are then able to evolve and survive on their own.] Indeed they would quite easily be able to run millions of such universe in parallel, each of which permits time travel for those in control. We then arrive at the disturbing realization that, statistically speaking, ours is more likely to be one of these virtual universe set in the past, rather than being the actual real universe which exists only for this one future civilization (also we are clearly not sufficiently advanced to be that universe).

The fact that time appears to flow in a forward direction is also a peculiar property and is enshrined in the 2^nd law of thermodynamics. This statistical based law is not however the only physical phenomena that indicates an ‘arrow of time’. Certain sub atomic events involving the weak interaction are known to be time asymmetric (specifically charge parity CP violation) as is the collapse of the quantum wavefunction itself and indeed decoherence theory. [The collapse of the wavefunction may be an effect of, an as yet unknown theory of quantum gravity, which itself is probably time asymmetric as well as non local.] Also in cosmology the universe expands rather than contracts with the forward flow of time.Other notions have built upon the block universe approach to time that exists in relativity. As with space, time is regarded as an axis that can be traversed in both directions and some have proposed ideas that are based upon action potentials moving backwards in time from the future, to influence the present.

One of the most popular approaches in fundamental physics, is that of string theory, in which the Lagrangian, is that associated with the surface being swept out by a string, rather than that of a point like particle which, as it moves through time sweeps out a curve. However instead of the 4 dimensions of space-time we have (according to the most recent version), 11 dimensions, which became compactified during the spontaneous symmetry breaking that occurred in the early universe. Hence not only sub atomic particles but time itself (and also space), is comprised of the dynamics/topology of these strings.

An increasing number of physicists have therefore started to take the view, that it is our (inadequate) concept of time that may be responsible for some of the intractable problems that face QT and GR.[Particularly relating to the asymmetric collapse of the wavefunction but also a QT of gravity implies quantised space-time!]. Indeed J. Barbour believes that time does not exist but is merely an illusion. He has a tentative theory, which involves stationary cosmological wave functions (akin to that used in the Wheeler-DeWitt equation) acting upon a configuration space of the whole universe (referred to as Platonia). Of course in this cosmic wavefunction, we must also include the human mind but I am still not convinced that this will be sufficient to completely justify Barbour’s claim of ‘The End of Time’ in physics. As Hume would say, how can something that exists as a series of states (the Nows), be aware of itself as a series? (cf. Relativity section under Quantum Loop Gravity, which also eliminates the use of 'a background time along which everything flows')

The concept of both time and God are very useful/important to civilisation, however it is possible that something as seemingly self-evident as time, may not actually exist after all.

** This time dilation effect has been accurately verified on many occasion. Specifically for man it has been estimated that the worlds most experienced cosmonaut has reduced his aging by one fiftieth of a second as a result of all his time spent orbiting the Earth. This calculation was based on the speed at which he has been traveling (producing a minute slowing down in the passage of time) and also the reduced gravitational field compare to that at the actual surface of the Earth (this causes an even smaller speeding up of the flow of time).

Both Wallace and Mivart provide examples for Young's axiological perspective of Victorian naturalism, in which science
and religion shared many of the same values. In particular they wished to obviate any materialistic interpretations which Darwin'stheory might attach to evolution (a term which Darwin himself never used in his "Origin of Species")Wallace being cofounder of the theory, supported the mechanism of natural selection, although due to his egalitarian beliefs, he did not wish the theory to dictate a Laissez-faire attitude but instead advocated a benevolent welfare society, which was not subject to materialism. He believed that although the theory explained the physical evolution of species, it did not fully account for
the emergence of mankind. To him, Darwinism was not incongruous with a spiritual dimension of human existence - on the
contrary its shortcomings in this area lends support to human existence - on the contrary its shortcomings in this area lends support to a belief in the spiritual development of man which is independent of natural selection. Mivart on the other hand proposed that Darwin's theory contained many anomalies and that it only described a subordinate action of a more 'divine natural law which had been laid down in the beginning by the creator'. Being a Catholic, he was more sensitive to to the claims of Darwinism, which conflicted with his metaphysical beliefs regarding design and creation. Such deference regarding the church was not however shared by Wallace, who did not specifically seek to reconcile evolution with theology. Although not a Christian,Wallace's dualism did extend to a belief in the spiritual world which transcended the epistemology of Darwin, who himself was reluctant to engage in any metaphysical speculation as to the origin of the soul or that of life itself. Wallace did not therefore dispute the material evidence that different species had evolved from a common origin along separate paths in which natural selection was a deciding factor but he did make human beings an exception to the theory, whose spiritual development he attributed to the intervention of an 'Unseen Universe.'

Survival of the fittest was a viable theory in explaining the large diversity of species and a great deal of evidence of its
occurrence had been collected but Wallace also realised its inadequacy and proposed that the very notion of natural selection can be used reprehensibly when applied to mankind's 'intellectual and moral nature', since the development of these in man's ascent, cannot be correlated to any such external pressures and thus requires another prime motive. His conviction that the spiritual nature of man evolved along disparate route to natural selection was, an attempt to compromise science with belief. Wallace's spiritualism offered an alternative to the excretory 'heat death' scenario and attached importance to moral values; our happiness and spiritual progress depending on the way we conduct ourselves in the material world, which exists for this sole purpose --death being merely a transition from this material existence to the spiritual life. His view is contrary to the theological dogma that man being the created in communion with God had fallen into sin by Adam's disobedient act. Instead he regards original sin as being the animal nature, inherent in brutes, away from which our spiritual perfection is evolving, by means which are independent of natural selection. Wallace argues that the successive development of primitive savages is completely independent of their cultural growth, since in Darwin's theory it is variations that are immediately useful for survival, which become maintained and not characteristics that will be of some use later (Mivart applies a similar reasoning when criticizing physical discrepancies) and in this context there seems to be stimulus for the growth in mathematical, musical or artistic faculties in primitive savages, let alone animals. He stresses that 'no creature can improve beyond its necessities for the time being' and although beneficial mutations may arise, the occurrence of genius occupies such a small percentage of the population, Wallace believes it unlikely to have much of an impact upon the evolution of mental aptitude, since intellectual development had not been prone to to natural selection and such mutations would consequently become swamped by populous. "We have to ask therefore, what relation the successive stages of the mathematical faculty had to the life or death of its possessors."

Mivart however being a deist, concentrated his attack upon the inability of Darwinism to explain all the biological variations that occur in nature. He is thus trying to illustrate that another (divine) force, other than natural selection is necessary to account for such phenomena as adaptive radiation and convergence of species. Citing the absence of certain transitional fossils he criticises the continuous variations of Darwinism and instead advocates a punctuated evolution (salutation) which is not subject to environmental pressures. In his 'Genesis of Species' he states "Another difficulty seems to be the first formation of the limbs of higher animals.........how are the preservation and development of the first rudiments of the limbs to be accounted for ---such rudiments being, on the hypothesis in question, infinitesimal and functionless?"

Mivart specifically mentions discrepancies with regard to the development of baleen in the mouth of the whale and the
peculiar provision of the young kangaroo in relation to its sucking habits, maintaining that the utility of certain factors which differentiated related species, was either non existent or became existent only when the difference had become fully developed. In the case of the baleen system although obviously useful to the whale, any intermediate stage in its evolution could not have been beneficial (he does not concede that any such embryonic organs may have had a different use from the completed version.) Mivart is evidently attempting to diminish the status of Darwin's theory by illustrating that it is insufficient to explain the prime cause for the diversity of life forms and indeed Darwin's pre-genetic theory did lack any viable explanation as to the mechanism of heredity and mutation. In respect of chance variation and natural selection -- which he regarded as having only a secondary control -- he proposes that 'an internal power or tendency is an important if not the main agent in evoking the manifestation of new species on the scene of realised existence' although his reasoning is somewhat tautological. He also uses fossils/geological evidence upon which Darwinism is based in order to highlight its technical shortcomings (at that time plate tectonics and continental drift were not established) and there were considerable evidence to illustrate that certain transitions amongst species were not smooth but more catastrophic or punctuated. Ironically Mivart's attempted mediation, resulted in him being 'excommunicated' from both his catholic and scientific brethren.

Wallace also uses the idea of a sudden insurgence amongst life forms but confines his dissertation to that of the mental spheres. He mentions several examples from the anthropology of civilisation in which certain cultural upheavals took place which were in no way related to the need for survival of the Lamarckian influence of the environment. He writes "the barbarous conquerors of the east, Timurlane and Gengkis Khan did not owe their success to any superiority of intellect" but despite this there
are associated cultural revolutions which have occurred throughout the ages independent of natural selection.The musical
development made by the Hindu and Egyptians, Greeks sculpture and geometry, the renaissance, Enlightenment and many more cultural plateaux are regarded by him as an example of unexplained bursts of human intellect, suddenly appearing upon society without any prerequisite.[He does not consider them to be merely bi-products of intelligence factors, which are subject to selection]

However it is not only in civilised western man that Wallace produces examples, he also believes that there is evidence of an unseen agent in pristine tribes. He is quite prepared to accept that man had descended from primitive animals but that natural selection only caused the impetuous for physiological change. Wallace refutes the existence of certain mental qualities (humour,moral conscience, sense of beauty,religious revelation etc.) in animals or that they can be developed to that extent in humans by means of natural selection, since no such stimulus exists for their production. He does not give credence to the belief that cunning and ingenuity, required for adaptation can inherently give rise to such qualities. Ethics and altruism in his view are not agents of survival and he does not consider that co-operation amongst individuals leads to a more viable fitness as a group. Instead special reverence is given to abstract faculties which he considers far removed from the genetic imperative associated with the struggle for existence. These faculties he claims are almost non existent in savages but appear spontaneously in civilised races. Another mode of attack that Wallace uses is to illustrate that just because a theory provides a reason for certain events to have taken place and is accurately descriptive of the way things are, does not necessarily mean that it is complete in itself. As an example he aptly states Lyell's original belief that the Earth had been sculptured solely by the upheavals and depressions of land and the denudation of wind and rain, until the study of glaciers showed that other agents were responsible for certain relief
features of the Earth. Both Wallace and Mivart do however share common ground upon their views of morality-- to which they attach a transcendental significance -- as neither believe that natural selection could be responsible for its emergence in man. Wallace tells us that "the love of truth, the delight in beauty, the passion for justice ........are the workings within us of a higher nature which has not been developed by means of a struggle for material existence". Mivart also finds it inconceivable that natural selection could produce "from the sensation of pleasure and pain experienced by brutes, a higher degree of morality than was useful" .[ In this respect he like Wallace is dissident towards the biblical teachings of original sin.] Instead he proclaims that the germs of human morality do not exist in primitive animals and is therefore evident in humans as a result of some higher process than natural selection, while Wallace accedes to its existence to some extent in lower animals but regards its elation to the height of human morality, as proof of a spiritual mechanism at work. Unlike Darwin, they are both contentious towards the skepticism of the Sophists who regard moral values as merely human convention. They conjecture that there is no natural selection for morality and as Mivart puts it "no stream can run higher than its source" implying that even if the usefulness criteria of evolution has developed into what is today regarded as morally righteous " we see that the very fact of an act not being beneficial to us makes it more praiseworthy" thus dispelling the utility of formal morality. Unlike Wallace he regarded the maternal morality exhibited by animals as being disparate to the formal morality of human beings and that natural selection alone is unable to bridge the gap.

They both relate to the three phases or modes of existence viz. inorganic (unconscious), organic (conscious) and spiritual (intellect) or in Mivart case the physical, hyperphysical and supernatural and both relate to the belief that an almighty force is involved in their transformation or interaction. Wallace states "These three distinct stages of progress from the organic world of matter and motion up to man, point clearly to an unseen universe -- to a world of spirit." Mivart also concludes "there is and can be absolutely nothing in the physical sciences that forbids [us] to regard these natural laws as acting with divine concurrence" although he only vaguely speculates as to the underlying ontology behind the "subordinate action of natural selection."

To summarise therefore, both advocate evolution and each aspire to a reconciliation between Darwinism and their own
personal beliefs. Whereas Wallace demonstrates the inapplicable use of natural selection regarding mankind's intellectual growth, Mivart projects the technical defects of Darwin's theory and its inability to coherently account for the differences in related but separate species. However they both rely upon ethical arguments to reinforce their position and claim that human personality has not developed by means of natural selection, since many attributes occur very rarely and are not related to survival. Mivart's religious beliefs upon creation and those spiritual convictions held by Wallace are centred upon a teleological view which contends Darwin's monopoly of evolution and his explanation of all life on Earth as being " products of the blind eternal forces of the Universe."

Idealism and Realism in Physics

It is often said that the Holy Grail of physics is to find a single equation that can explain the whole of physical phenomena - an equation that can be written across a T-shirt. The fact that the world's greatest minds have been unable to achieve this is a testament as to how complex this mathematical structure must be - if such an equation actually exists. However, we can at least begin to speculate upon the nature of any such equation.
The equations that are already known to accurately describe fundamental physical laws are rather esoteric, while those theories that are yet more ambitious (and speculative) employ concepts that are even further removed from everyday experience. String theory for example, replaces the geometry of points with that of extended one-dimensional objects and with its 2nd Revolution (M-Theory), employs higher dimensional p-branes. These all require invoking extra dimensions of spaces that become compactified under symmetry breaking. Supersymmetry, requires the introduction of (anti-commuting) grassman variables in order to produce 'unreal' fermionic dimensions (which are measured in supernumbers), in addition to the real (bosonic) dimensions. An alternative strategy for a Theory Of Everything (TOE) is Twister theory, which is based on spinor mathematics and relies greatly on sheaf cohomology, neither of which bears any resemblance to Euclidean space. So how are we to interpret the underlying nature of any theories written in such abstract mathematics?
We could possibly gain something from Plato's perspective. He believed that there were three distinct worlds, namely the Physical (real) world, the Spiritual world and the Ideal world - the latter being the more superior. The Ideal world is (as its name implies) perfect and contained for example exact circles, to which only approximate copies exist in the Physical world. The whole of Euclidean geometry (the main branch of maths in those days) relied on constructions using only the circle and straight line. [The three great-unsolved mathematical problems of antiquity, namely the squaring of the circle, the trisection of an angle and the doubling of a cube, were examples of the limitation of these two particular Platonic tools. It was another two millennia before Galois theory showed the impossibility of these quests.]
In some ways maybe the new fundamental theories should be regarded as being of a Platonic Ideal, rather than physically real. Indeed quantum theory describes the world in terms of complex wave functions, from which we extract physical observables (eigenvalues) by means of hermitian operators. Complex numbers have an imaginary component and we do not therefore attach physical reality to the wave function itself (which only represents a probability amplitude). But what about strings and higher dimensions?
Two great equations of the early part of the last century were Einstein's field equation in General Relativity and Dirac's equation in Quantum Theory. [Both these pass the T-shirt test, although their solutions are often lengthy] One of the significant features that they share is that they are both coordinate free. In other words we can accept the equations as accurately predicting the physical world but there is no special privilege given to any particular coordinate system in which they are expressed. The equations are like perfect sculptures over which we could draw any contour lines, non of which would be as important as the statue itself. The tensor nature of the former does not give uniqueness to an observer's space and time. Likewise, the contact transformation which forced Dirac's to produce his acclaimed equation, also removes the importance of any particular reference system. It suggests that the underlying nature of these theories exist in a Platonic context and that it is their particular (and often inexact) solutions what we experience in our real (material) world
Even the solutions that these equations give can seem bizarre, although many have been tested to be true. Dirac's equation correctly predicted antimatter and the fact that an electron rotated through 360 degrees would become inverted, while the so-called twin paradox [sic] in special relativity has also been demonstrated. In general relativity the Cosmological Principle is invoked in order to find a possible description for the universe at large i.e. the universe is assumed to be isotropic and homogeneous. This produces solutions that depend upon the present mass density, which if sufficiently large (Omega value greater than unity) would produce a closed universe which although finite in size is unbounded. However if the total value of Omega* is less than unity the universe will be open and negatively curved (every point will in space be a saddle point - impossible to visualise but otherwise quite acceptable). However what I do find uneasy is that this implies an infinite mass, which must accompany the infinite space of a universe, that has existed since the big bang and has been continually expanding into-- infinity!!! [I am not aware of this specific issue being adequately addressed, maybe some readers can provide enlightenment.]. The situation is somewhat reminiscent of Cantor's Transfinite numbers, but the insight of pure mathematics into the Ideal world is one thing, while the reality of the physical (material) world is another.
In summary, although such equations are of primary importance, they may appear to exist in a Platonic Ideal world (which allows such peculiar constructs), while their solutions are manifest in the physical world. Any attempt to give physical reality to such structures as strings, twisters and supermanifolds, seem to me to be ill fated. The philosophy of Idealism may not appeal to many pragmatic scientists but as the mathematics that underlies physical theories becomes more abstract and aloof, I feel it is perhaps the most suitable viewpoint.

*Aside
It is believed by many that the value of Omega is actually unity, since this avoids the above problem, while Hawking's work on quantum cosmology relies upon the universe being closed. Theoretical reasons therefore need to be given for such an unlikely balancing act, as well as an explanation for the apparent missing mass. Recent proposals suggest values for a cosmological constant that would significantly lower the required density of matter and explain the observed '5th force' of repulsion that is volume dependent . [However the introduction of a cosmological constant, may ruin essential symmetries in string theory.]
Indeed some theories invoke a cosmological constant that was initially very large and has decreased continually as the universe expanded. This has the benefit of removing the need for inflation that accompanied the spontaneous breaking of the Higgs field (a mechanism responsible for separating the strong force from the electo-weak force during the very early universe). It does however require the speed of light to become progressively reduced during this expansion and for the possibility of successive universes being seeded within one another producing a fractal structure! Recent observations on distant galaxies seem to suggest that the coupling constants of the fundamental forces have indeed changed over the history of the universe, which would imply that the laws of physics have not remained the constant as the universe expanded (for example the mass of the proton and neutron would have been different in the distant past).

Since my previous article (Idealism and Realism in Physics), I have become aware of recent developments concerning our understanding of the big bang in relation to M theory. The raison de etre of M theory is that by adding an extra dimension, we find that the 5 contending string theories (and supersymmetry) become a network of equivalent mathematical models when viewed from this 11 dimensional manifold. Our universe is then explained in terms of 2 dimensional membranes (rather than 1dimensional strings) and other higher dimensional manifolds (unfortunately termed p-branes) are also possible. New developments in M theory invoke parallel membranes, which help explain several intriguing phenomena, such as the weakness of the gravitational interaction [Gravity is viewed as emanating from this parallel membrane into the 11 dimensional manifold and since we only experience its effect in our 4 dimensions of space time, its strength is greatly diluted.] Of particular importance is the recent discovery that the big bang itself may be explained in terms of the collision of 2 such membranes. This allows us to avoid the embarrassment of the singularity at the instant of creation and allows us to talk meaningfully about physics before the big bang.
In some ways these new developments seems to undermine the viewpoint of my previous article, since these parallel membranes and higher dimensional p branes, are supposed to have existed before our physical universe and could therefore be considered as more real! Indeed an infinity of universes similar to our own but subtly different, are also believed to co-exist. [This however is in a different context to the 'many worlds' interpretation of basic quantum mechanics but both invoke the concept of a multiverse]. Even other more exotic universes are allowed (corresponding to higher dimensional p branes), in which the laws of physics are completely different from our own. My view however, is that although they provide exiting possibilities, what is actually being discovered is the properties of a new 21st Century mathematics, which may or may not refer to the real universe. Also even if it does provide a Theory Of Everything, it does not mean that because it predicts the possibility of other universes that they must actually exist.
By means of a simple analogy, the graph of the quadratic equation y = x^2 +1 does not actually cut the X-axis and therefore does not have any real solutions for when y = 0 but does have imaginary solutions of x =+-i . We can therefore talk meaningfully of the mathematical properties of the roots even though the intersection of the graph with the X-axis does not actually exist. Likewise there is a danger of assuming that these visualised membranes physically exist, which may or may not be the case. If one takes the Positivist outlook of Bohr and Hawkings then the importance of a physical theory lies in its ability to accurately describe the universe and one cannot speculate as to the ontology behind the theory.

A, B, OR C ?
Although Newton's inverse square law became the first formulation of a gravitational force, it is not the only classical description of gravity. For example we could express the gravitational law in terms of a field in which each point in space could be allocated a gravitational potential. Alternatively we could predict the motion of a body by utilizing the Maupertuis Principle of Stationary Action (analogous to Fermat's principle of Stationary Time regarding geometric optics). Each of these descriptions is equally correct mathematically but describes gravity from 3 different viewpoints.
The advent of 'Modern' physics arose out of the revolutionary concepts of relativity and quantum theory. Relativity theory can also be viewed in three equivalent ways. From a kinematic analysis, by accepting an ultimate limit to the speed of communication in order to justify causality, we are forced to re-evaluate our understanding of simultaneity and hence also time, length, and mass. Historically, Einstein realized the conflict between Newtonian mechanics and Maxwell's laws of electromagnetism and developed the special theory by boldly overthrowing the former. A third method of approach would be to start with the postulate that matter and energy are intrinsically the same. Consequently as we increase the kinetic energy of an object its mass increases (relative to us) and this again leads up to the scenario of relativity.
Whereas the old quantum theory was born out of Planck's postulate that energy could only be radiated in discrete amounts, the new quantum theory (especially field theory) has its roots formulated around Heisenberg's Uncertainty Principle. It became evident that the four dimensional quantity termed 'action' was discontinuous and to speak of any quantity smaller than Planck's quantum of action was meaningless. Again we can look at three different viewpoints in order to develop our model but this time they are not all equivalent
Firstly we could take Einstein's interpretation which accepts that there is a fundamentally lower limit to the accuracy with which we can make 'incompatible' measurements (e.g. energy and time) but that there is an objective reality beyond these limitations even though it cannot be perceived directly. The statistical nature of the wave function ('psi'), in his view, merely expressed the indeterminate aspect of the apparatus interacting with the object and did not doubt the absolute reality of the object. Alternatively we could also in the Uncertainty Principle by postulating that although there is a hidden reality to an object, there is also an inherent tendency for all objects to fluctuate in their paths within the quantum of action, thus giving it an undulating uncertainty. The Maupertuis Principle would then be an approximation to particle propagation, in the same way that Fermat's Principle is an approximation for light (neither of them is applicable to wave phenomenon). Just as General Relativity refines classical gravity, so Quantum Theory refines the Maupertuis Principle.
Thirdly there is the orthodox (Copenhagen) view, that all we can be sure of are observable values and that the subject is inexorably interwoven with the object. In other words there is no objective reality beyond the constraints of our measurements and the way in which we carry out our observations affects the reality of an object. It is therefore meaningless to ask whether an electron is a particle or a wave. Einstein once asked a friend if he really believed the moon only existed when he looked at it. Ironically his mentor Ernst Mach regarded science as merely an "economy of thought"; all we can ever do is make observations but not complete explanations regarding ontology. Science to him was nothing more than empirical facts grouped together under uniformity and conservation laws.
Each of the viewpoints support the Uncertainty Principle but there is a profound difference in their implications. The first two interpretations, which allow varying degrees of objectivity, do not stand up to recent experimental evidence (c.f. John Bell's Inequality and Alain Aspects experiment). Instead it seems that 'knowing' has an almost mystical effect on the universe at the microscopic level. Just as mind is the ghost of the body, the quantum has become the ghost of the atom. Current understanding seems to indicate a deeper meaning to the concept of relativity, in which knowledge is not absolute but conditioned by standpoint and circumstance. Sages of ancient India taught that the mind is the centre of the Universe but whereas modern psychologists tend to use a reductionism approach to the mind, modern physicists are seriously contemplating an holistic approach, reminiscent of Taoism, forced upon them by the outcome of Quantum Theory. [ Bohr adopted the Yin Yang symbol for his coat of arms!]
As a consequence of the apparent validity of the orthodox interpretation of Quantum, we unavoidably return to the dilemma of 'instantaneous action at a distance' - that which Einstein was happy to ameliorate with his General Theory of Relativity.

Interestingly in the limit of "c" approaching infinity, Einstein's equation approaches the Newtonian prediction of an inverse square law and gravitational force is proportional to the product of the masses, while in the limit of Planck's constant approaching zero, Quantum predictions agree with classical theory. [Indeed it is the sameness of "h" which make QT effects unobserved except on atomic scales and the largeness of c which makes relativity effects unimportant in everyday circumstances]

Leibniz considered if all proposition is decidable within logic (i.e. completeness), while much later Hilbert asked if axiomatic logic can be formalized in a consistent way. Between 1910 and 1913 Russell and Whitehead produced the 3 volume Principia Mathematica, in which they claimed to have reduced all of mathematics to a unified system of axioms from which all theorems of mathematics could be derived, just as Euclid had attempted to do for geometry. Hilbert however was skeptical and challenged mathematicians to prove rigorously that Russell and Whitehead's program had succeeded.. This question was settled in 1931 by the theorems of Kurt Godel, who demonstrated that in a system of sufficient complexity, such as the theory of numbers, there must exist statements that cannot be proved either true or false. [As a corollary there must also be true statements hat cannot be proved].Consider the following statement "This theorem cannot be proved". If this statement is false, then it can be proved so we have a contradiction i.e. an inconsistency. However if the assertion is true we have a statement that cannot be proved, hence incompleteness. Godel's theorem does something similar to this for formalized systems such as arithmetic by using Godel numbers to encode axiomatic statement. He allows statement and numbers to refer to themselves and by a process of diagonalization allows statements that are true of their own Godel number. Hence if we code that an arithmetical statement similar to "This theorem cannot be proved" which refers to its own Godel number we arrive either at a contradiction (if the statement can be proved) or an incompleteness (if it cannot be proved the assertion is true but cannot be proved). Turing also posed a problem on decidability (called the Halting problem) , by considering a universal machine that could run all programs (similar to today's PC's which can run on any valid operating system). Now when you run a program, either it stops and spits out an answer or it goes on for ever. He asked whether it would be possible to decide in advance on whether a given problem could be solved by a particular program (set of algorithms) in a finite amount of time. Turing showed that the Halting problem is undecidable. To do so he played much the same game as Godel, by assuming that the halting problem is decidable, Turing showed (c.f. proof below) that you could construct a program that stops if and only if (IFF) it does not stop (this contradiction therefore shows that the assumption is false). So Turing's halting problem is a similar example of Godel's undecidability.

+ - * / ( ) = 1 2 3 4 5 6....x y z .......etc corresponds to the following digits

where 2,3,5.7 is the sequence of primes and the powers 12, 1, 15 ,7,9,9 are the codes of the symbols 4,+,7,=,1,1 of the string

In this way we associate with each string a code which will be a whole number. Thus if the code is 720 we can uniquely factorise this as 720 = 2^4*3^2*5^1 and the symbols whose codes are 4, 2, 1 are / - + hence 720 is the code for the string / - +

Is the set of all extraordinary sets, itself ordinary or extraordinary? This is Russell's paradox which can be reframed in terms of a country in which each library has an index of all its books. Some libraries also include the actual index book itself (i.e.the index book is an ordinary set), while in other libraries, the index book does not mention itself (an extraordinary set). Now the national museum contains a master index book, which lists all the index books of those libraries that do not contain themselves (i.e. a list of all the extraordinary sets). The question therefore arises as to whether this master index book should contain its own title. If it does not, then it cannot be said to contain all those index books that do not contain themselves, whereas if it does contain itself it cannot be an index only of books that do not contain themselves? This discovery by Russell devalued 10 years of Frege's work on the reduction of arithmetic to set theory.

Wittgenstein said that a sentence cannot refer to itself; all sentence can do is say what it means .The following are examples of self referential sentences, which therefore lead to paradoxes. Similar 'strange loops' can be found in Godel's incompleteness theorem, Richard's paradox, Cantor's Paradox, Bach's fugues, Escher's drawings and in Turin's universal computer (halting problem) and is present in Russell's criticism of Frege's set theory (the set of all sets that do not contain themselves)

The word “long” is heterological (since the word is actually short in length) whereas the word “short” is not heterological (since the word “short” is short). Is the word “heterological” itself heterological?

"The least integer not describable using less than 19 syllables" has only 18 syllables. (this is Berry's paradox)

"I cannot imagine the world existing without me" {A statement which illustrates the impossibility of perceiving personal non existence}

Richard's paradox assigns supposes a list of all real numbers between 0 and 1. It is then possible by a diagonal slash process to define a number on the list (by taking the nth number on the new list to be one greater than the nth digit of the nth number on the list). This new number however is clearly a finite definition that would satisfy the condition for being a member of the list and yet differ from every member of it.

Cantor 's diagonal theorem shows that any set has strictly more subsets than it has members This leads to a paradox in that for an all inclusive infinite set, every subset of such a set would be a member of it but due to his diagonal theorem, every set has strictly more subsets than it has members; there is thus no largest cardinal number. Cantor proved (using a diagonal slash technique) that there are more real numbers between any two integers than there are whole numbers. The infinite number of natural (whole numbers) is called Aleph zero while the infinity of real numbers (including decimals irrational numbers and transcendental numbers) is the cardinal c = 2^Aleph zero. The continuum hypothesis asserts that c = Aleph one (i.e. Aleph one = 2^Aleph zero, however Cohen has proved that this statement is independent of the other axioms of set theory and we can add on the axiom that the continuum hypothesis is true (or false) without making the system inconsistent. Cantor also showed that there are more transcendental numbers than rational numbers since if the number of Transcendental numbers was also Aleph zero then the total number of reals would be Aleph zero +Aleph zero =Aleph zero, which is false since we know that the number of reals is c =2^Aleph zero> Aleph zero.

To summarize, what Godel did in producing his First incompleteness theorem was to find a statement G£ in the formal language of £ that expresses the mathematical sentence "G£ is not provable from £". In other words G is to represent the self referential sentence "This sentence is not provable from £". Hence G£ must be true but not provable, since if it were false it leads to a contradiction, so it must be true and therefore to be consistent, it asserts its own unprovability. Godel achieved this in two steps; first he found a way of assigning a code number to each sentence in the language of £ and then by a method of diagonalization he provided a way of making sentences in the language of £ that refer to themselves. If we try to write out G£ in English we get an infinite sentence: "£ cannot prove that £ cannot cannot prove that £ cannot prove that £ cannot prove that ...... . . . .In other words we say that G£ . . . IFF. . . .(£ cannot prove that G£). His second theorem states that if we demand consistency then the theorem that states this cannot be proved. Other central negative results of logic include Tarski's theorem on the undefinability of truth (that arithmetical truth is not arithmetically definable)and Church's theorem on the undecidability of logic (that arithmetic is not decidable). Godel's theorems devalued much of Russell's work, which tried to reduce arithmetic to logic (as initiated in his Principia) and also showed Turin that his initial optimism of designing an algorithmic machine that could decide if conjectures were true or false (e.g. the 4 colour problem, Fermat's last theorem or Goldbach's conjecture) was unattainable.

By focusing on provability rather than on truth, Godel's sentence avoids the absurdity of the liar's paradox (" this sentence is false"). If formal arithmetic is consistent, meaning that only true statements can be proven, then Godel's statement must be true. If it were false then it could be proven, contrary to the consistency! Furthermore it cannot be proven, because that would demonstrate just the opposite of what it asserts, its unprovability. Moreover Godel showed that if the consistency of the formal system could be demonstrated inside the system itself, then the informal argument just given could be formalized and the formalized version of the statement " THIS STATEMENT IS UNPROVABLE" would itself be proven, thereby contradicting itself and demonstrating the inconsistency of the system.

There is no consistent, complete, axiomatizable extension of Q. (Q being the Peano 'arithmetical' axiom system)

In other word Godel became famous for proving that you couldn't prove everything that is true, even in such an apparently simple subject as arithmetic. In effect, he showed that, it is not possible to prove that all true statements in arithmetic can be proved (even its own consistency).

Let £ be an axiomatic set theory that is a normalization of an ordinary arithmetic A. Inside S we have symbols from which we construct strings and the axioms of S tell us how we are to manipulate strings. Hence 2+2=4 is both a formula in A and a string in S. In particular strings that involve a numerical variables 'n' are termed signs. Now every sign can be labeled by a Godel number that are arranged in order, and let R(n) is the nth sign. Hence every sign is equal to some R(n) for a suitable choice of n.

Let [R(n),n] represent the string which substitutes a variable n into the sign labeled R(n). Now if it is NOT possible to prove the string [R(n),n] in £, for a particular value of n, then we include it in a Cantor set K. For example the string n + 6 =0 is not provable for n=2 so n is an element of K.

The statement, S that a particular value of n is a member of K, can itself be given a Godel number . Now, by a process known as diagonalization**, it is possible for a Godel number to represent a statement that is true of its own Godel number!!! [Diagonalization of a A will be a sentence that says that A is true of its own Godel number- or more precisely, the diagonalization will be true IFF (if and only if) A is true (in £) of its own Godel number. So [R(n),n] can be a statement labeled R(n) which is about S as implied by the ~ provability of [R(n),n] where ~ means negative (of provability). Hence this statement R(n) can be about the Godel number R(n) itself! With this particular situation, if we consider the string [R(n), n] it can be shown that it is not provable in £ but also it will be shown that NOT [R(n),n] is also not provable, in other words it is undecideable. To summarise, R(n) can be a statement which states the fact that the non provability of a string means that the particular number it refers to then becomes a member of a set. The paradox then arises when we let that be true for the Godel number R(n) itself (i.e. the Godel code for the sign itself becomes the number for which the statement is true).

**(This is reminiscent of that introduced by Cantor in his theory of transfinite numbers, in which a new number is obtained from a diagonalization technique and this can then be introduced into the set, ad infinitum)

G(n) . . . . . . .. . . . . . . . . . . .. . . Statement about a number (which might be true(provable) or false depending on the number)

. . . . . . . . . . . . . . . (Sign S).. = ..n is an element of set K and if ~Prov[G(n),n] (i.e.if G(n) cannot be proved for n)

Diagonalization is a process which says of a sign S that "S is true of its own godel number", or more precisely, the diagonalization will be true IFF (if and only if) S is true of its own Godel number. Hence we could code any statement and give it a corresponding Godel number and then decide if the statement would still be true if the variable were replaced by its own Godel number; if so then we have successfully diagonalized the statement and we can express this fact by labeling it with the originall Godel number. Let (by the process of diagonalization), the Godel number R(n) now represents a statement that is true of its own Godel number i.e. we represent Sign S by the Godel number R(n) and obtain;

R(n).. . . . . . . . . . . . . . . . . .. .S . = .Statement that n is an element of set K as implied by ~Prov [R(n),n]

It therefore seems legitimate to ask whether the (new) statement, which is represented by [R(n),n], is true, in other words can we prove [R(n),n]. We then can provide the following proof of incompleteness;

If {1} can be proved then it means that n is a member of K, which because of diagonalization, implies (by definition of K), that {1} is not provable for some n -- hence {1} asserts its own unprovability. However neither is statement {1} disprovable, since if the negative of {1} (which is written in statement {2}), is provable, this implies the negative of 'being a member of K' in other words n is not a member of K. Because of diagonalization, this in turn means that [R(n),n] is provable for some of n, which contradicts (2) and since £ is assumed consistent it follows that {2} is not provable in £. Hence the assumption that the negative of {1} is provable is false and therefore {1} is neither provable or disprovable. i.e. it is undecidable! The statement [R(n),n] can therefore be regarded as asserting its own unprovability. Hence in mathematics we have to abandon the dream of being able to create a machine that will operate a computer program capable of churning out theorems which have been proved to be true. Instead we have to rely on the ingenuity and creativity of mathematicians to decide upon the validity of Fermat's last theorem or the four colour problem. [Note that {1} cannot be false since as we have just shown, that would means that R(n) can be proved and hence{1} is true, which is a contradiction, so we are left with the alternative that {1} is true which means that R(n) cannot be false since that would be a contradiction, in other words R(n) must be true. Also note that{1} is true if it cannot be proved, hence we arrive at the only alternative, that R(n) can be true if it is not provable.

To recapitulate, by coding axiomatic statements in terms of Godel numbers, we allow the possibility of numbers expressing statements about themselves. For example if a statement is made about a number, this statement is itself designated as a Godel number and the possibility then arises that this statement represented by the number, refers to the number itself. We can test the provability of the Godel number that represents a given statement about a specific value of a variable (e.g. 2 + x =7; if x=9 ) and in particular (by means of diagonalization), a Godel number can refer to a string which is true of its own Godel number. A Godel number can then be given to a statement that relates to a negative outcome of another statement (i.e. unprovability, as in relation to set K), which is coded by the same Godel number, and we can then demonstrate the unprovability of the overall statement. In summary we have constructed a string viz. [R(n),n] that expresses its own unprovability! Hence in any formal system we will have theorems, which although true, are not provable within the systems and even if we amend the axioms and enlarge the system so as to be able to encapsulate these known theorems, this very act will produce more unprovable theorems! Some people have speculated beyond this precise formalism to less rigorous systems, such as the human Genome. For example there certain degenerate diseases that a given human phenotype is prone to, but some would suggest that even if we identify the aberrant genes and make alterations to the Genome that guarantees immunity to particular ailments, this very act will itself ensure that other (new) self destructive biological effects would become extant. To continue this analogy, we can regard systems as record players which work well for most records but the very design of the player means that there is always a record that when played, will cause the player to self-destruct. However any modification to the player's system that prevents this happening with such a record, will automatically produce changes that will ensure that it is capable of being destroyed by another, as yet unknown record (and so on ad infinitum).

For example in the above, S is the statement that n is an element of a set K when ~Prov[R(n),n]. Now if we let S be given the Godel number R(n) we can show that the statement [R(n),n] cannot be proved since this would imply that there is a number that is a member of the set K which in turn implies a Godel number, that itself asserts that the statement [R(n),n] cannot be proved, so we arrive at a statement that asserts its own unprovability!. What's more we cannot disprove the statement, since by asserting the negative statement, it implies that n is not a member of K and this in turn means that R(n) is provable for some of n, which contradicts the negative of the statement viz ~[R(n),n]. Hence the assumption that ~[R(n),n] is provable is false and therefore [R(n),n] is neither provable or disprovable. i.e. it is undecidable!

If A is a formula in the language of arithmetic that contains just the variable x free then the diagonalization of A will be a sentence that says that 'A is true of its own Godel number', or more precisely, the diagonalization will be true IFF (if and only if) A is true of its own Godel number.

Prov£[m,n] is true if and only if (IFF) m codes up a proof from £ of the sentence coded up by n.(i.e. m proves n)

The First incompleteness theorem is proved by finding a sentence G£ in £ that expresses the mathematical sentence "G£ is not provable from £". . G£ must therefore be true but not provable. [The reason is that if this statement can be proved true, we have a contradiction i.e an inconsistency. However if it cannot be proved, the assertion is true but cannot be proved, hence incompleteness].

** . G £ ... . . . ... . ..if and only if... . . .. ..... .~E(m)[Prov£[m,<G£>] . . . . . . . . . . . . . .{1}

Note that if {1} is false then we obtain the statement E(m)[Prov£[m,<G£>] is true.

We will now demonstrate that when the above statement is true there is no proof that codes for the statement i.e. G£ asserts its own unprovability..

G£ states that no natural numbers m of a certain kind exist (namely those that code a proof for <G£>), so it seems legitimate to ask whether G£ is a true or false statement about the natural numbers. Note that G$ is true if it is not provable from £, so it seems that either G£ is true and not provable by £. or that G£. is false and provable by £. Now if we assume that £ is consistent, then we can rule out the second option and conclude that G£ is true but not provable in £, which is Godel's First Incompleteness Theorem. [If there were a proof of G£, then the meaning of the statement that G£ actually asserts, namely that there is no proof, would be false, so G£ would have to be false as an arithmetical proposition which implies that our formal system is so inconsistent as to allow false propositions to be proved. Thus it must be true that there is no proof of G£, which is exactly what G£ is stating, hence we have a true statement which has no proof within the system. Also since we have just established that G£ is true, then ~G£ must be false and therefore we cannot prove ~G£ to be true, otherwise we will again have a system that proves false propositions]

We have therefore created a particular self referential statement that, in order to be consistent, cannot be proved and hence the statement that ' it cannot be proved ' is true but in turn cannot be proved. So in order to have consistency G£ has to be true, but G£ states that the statement about itself cannot be proved, which is true and so we have a statement that says of itself that it is not provable even though it is true. In other words if G£ is true, it implies that G£ is not provable for some m, hence it asserts its own unprovability. Therefore we cannot prove something that is known to be true (namely the statement that proclaims that this diagonalization is not provable). [If G£ can be proved false, then not G£ is provable, which means that G£ is provable for some m, which is a contradiction and we therefore lose consistency.] Hence there are certain statements in arithmetic that cannot be proved nor disproved i.e. they are undecidable! So if we insist on consistency then we must have incompleteness but if we try to demand completeness we cannot have consistency. As we shall see below, an example of G£ is the statement that £ is consistent i.e. CON(£) as this leads to the conclusion that £ is consistent only if it is not provable, which is Godel's Second Incompleteness theorem

Note that {1}cannot be false since as we have just shown, that would mean that G£ can be proved for some m and hence {1} is true, which is a contradiction, hence we are left with the alternative that {1} namely ~E(m)[Prov£[m,<G£>] is true, which means that G£ cannot be false since that would be a contradiction, in other words G£ must be true.. Also note that{1} is true if it cannot be proved, hence we arrive at the only alternative, that G£ can be true if it is not provable Statement {1} is an assertion that diagonalisation (which is assumed true by its definition), when applied to a particular (true) statement regarding its own provability, is not provable, in other words we have a truth that cannot be proved. It is therefore true that we cannot prove that this diagonalization is provable, in other words we have a truth that cannot be proved. [If we state that this particular diagonalization is possible to prove i.e. the negation of {1}then we arrive at a contradiction.] Hence the statement that says that "its own diagonalization cannot be proved" is true but undecidable i.e. it cannot be proved nor disprove. If we try to write out G£ in English we get an infinite sentence: "£ cannot prove that £ cannot cannot prove that £ cannot prove that £ cannot prove that ...... . . . .In other words we say that G£ . . . IFF. . . .(£ cannot prove that G£).

If Z is consistent then the statement that says this is not a theorem of Z ( In other words Z cannot prove its own consistency, or £ is consistent if and only if £ cannot prove CONsistency(£). Z being the axiomatic system of arithmetic)

Let T be the string [R(n),n] which as we have just shown asserts its own unprovability, and let Z be any formula in £ which asserts the consistency of £. We therefore want to prove that Z cannot be proved in £. Godel's first theorem reads 'if £ is consistent then, then T is not provable in £'. We can therefore express this in £ as:

'T is not provable in £' is just T itself, because T asserts its own unprovability, so Godel's first theorem written in £ takes the form of

If we could prove Z in £ then this would enable us to prove T. However we know that T cannot be proved, hence Z cannot. Since Z asserts the consistency of £, it is not possible to prove Z consistent within £, which is his second theorem. [The consistency of £ could be an example of Godel's First theorem, since even if £ is consistent it is not provable i.e. R(n) (or its equivalent G£) above, becomes CON(£) which asserts its own unprovability].

Alternatively stated, we know that if £ is consistent then T is true. In this way we can show that there is a proof from £ of CON (£) implies T. Now if £ could prove CON (£) as well, then we could apply Modus Ponens and obtain a proof from £ of T, which is impossible (since T asserts its own unprovability). Therefore £ cannot prove CON(£). The main point is that such axiomatic systems cannot prove their own consistency without having to 'step out of the system' and if we do this, how do we know that this system in turn is consistent, unless we again move to a higher system in order to check its consistency and so on ad infinitum. Likewise in Godel's First theorem, if we produce a sufficiently rich system of axioms, there will always be true statements within that system that cannot be proved and whatsmore, if we identify these statements and add suitable axioms that will ensure that these can now be proved, then we will inadvertantly introduce new true theorems, that in turn are undecidableand and so on . . ..Some people use this to 'prove' that we will never completely understand the universe because we are part of it (some cite Heisenberg, uncertainty as an example of this) or that we cannot completely understand the mind (and therefore our thinking/theories will be limited) because we cannot step outside of it. However Godel's theorems do not necessary relate to such ideas and indeed it can be shown that some smaller formal systems such as the axioms of geometry are consistent [These axioms are not sufficient to produce a complete system of arithmetic for example, however the integers without the operation of multiplication can be consistently formalized but the system is then restricted i.e.incomplete]. Godels theorem may have some bearing on Law however, in that it demonstrates that it is not possible to have a legal system that is guaranteed to dispense the law in a way that is instinctively just at all times (in other words we need a judge to intuitively interpret given situations). Hence given a particular system of law (which itself varies considerably throughout the world), it is always possible to find a scenario that will lead to a verdict that would intuitively appear to be unjust. [We often attempt to override these occurrences by appealing to another, usually higher court, but as already emphasized, the consistency of even this may be questionable] In the same way that quantum mechanics provided limitations on what we could demand regarding determinism, Godel's theorem demonstrates the limits of formalism. Indeed Einstein's theory of general relativity contains its own incompleteness in that it predicts the inevitability of a singularity (where its own laws of physics break down) at the centre of a black hole for a collapsing star beyond a certain mass.

It is difficult to actually find interesting, examples of undecidable statements but a good candidate is known as "P = NP", which asks whether every class NP question is actually class P. in other words, if an answer to a question can be checked in polynomial time, can it always be found in polynomial time. At first sight the answer to P = NP seems to be no, since finding an answer to something ought to be harder than checking it once someone has found it. Yet nobody has been able to prove or disprove it, and it may in fact be undecidable

Suppose we make a list of all real numbers between 0 and 1. It is then possible by a diagonal slash process to define a number that is not on the list (by taking the nth number on the new list to be one greater than the nth digit of the nth number on the list cf. example below*). This new number however is clearly a finite definition that would satisfy the condition for being a member of the list and yet differ from every member of it. Using this diagonal slash technique Cantor proved that there are more real numbers (R) between any two integers (Z) than there are whole numbers. The infinite number of natural (whole numbers) he called Aleph zero, while the infinity of real numbers (including decimals irrational numbers and transcendental numbers) is the cardinal c = 2^Aleph zero (which from Cantors diagonal theorem below, is greater than Aleph zero itself).* The continuum hypothesis asserts that c = Aleph one (i.e. Aleph one = 2^Aleph zero), however Cohen has proved that this statement is independent of the other axioms of set theory and we can add on the axiom that the continuum hypothesis is true (or false) without making the system inconsistent. More generally, Cantor suggested the Generalized Continuum Hypothesis in which for all values of x, 2^Aleph x = Aleph x+1.

Aleph 0 is therefore the lowest infinity (transfinite number) and is associated with the natural numbers N

The number of even numbers is also Aleph 0 as is the number of odd numbers, since both these sets can be put into one-to-one correspondence with each other, so;

Also, Aleph 0*Aleph 0 = Aleph 0 . . . . (since the cartesian cross product of the two can be placed in a grid, each unit of which contains a pair of numbers that can be put into one-to-one correspondence with the same size grid labeled with the natural numbers. By moving around the grid in a cyclic manner, if we keep filling in the pairs of numbers in the first grid and a corresponding natural whole number in the second grid, we will just fill in the upper right hand quarter of each infinite grid ).

Now the cardinality of the real line is c = 2^Aleph 0. We can therefore assert that the cardinality of the number of points on a plane is c*c = 2^Aleph 0*2^Aleph 0 = 2^Aleph 0 +Aleph 0 = 2^Aleph 0 = c , and likewise the cardinality of the number of points in a 3 dimensional space is also just c =2^Aleph 0

Cantor 's diagonal theorem shows that any set has strictly more subsets than it has members. The number of subsets (including the null set and the whole set is equal to 2 ^k where k is the number of members. [This can be obtained by considering that each subset is one of a combinatorial series kC0, kC1, kC2 .....kCk and that these form a typical Pascal triangle in the binomial series, whose sum of the coefficients in each line are 2^k. viz. 4, then 8 then 16 etc,for the binomial expansions of squared cubed and quartic etc. More generally if we have C choices of colour and k elements, we obviously have C^k combinations of colouring each element and if we limit this to just black and white, - where black represents an omission - we obtain a total of 2^k possible 'subset' combinations]. Cantor then showed using a diagonal method shown below**, that k is always less than 2^k even for infinite sets. This leads to a paradox in that for an all inclusive infinite set, every subset of such a set would be a member of it, but due to his diagonal theorem, every set has strictly more subsets than it has members; there is thus no largest cardinal number.

*The proof that the cardinality of the real line is 2^aleph zero, is obtained by considering subdividing the unit length real line into 2 an infinite number of times, so as to obtain a small enough interval of [0,1] namely 1/2^n . We need to carry out this an infinite number of times, in order to home in on an actual real number, hence the cardinality of c is 2^ Aleph zero

**The proof that k< 2^k is obtained by arranging k sets of (1,0) and by choosing one member in turn from each of the k sets, so as to produce a series of new sets Sn, each containing k members which are either 1's or 0's. For example consider the following list of Sn sets

etc, etc down to Sk, so that the width of the sequence of horizontal 0's and 1's is equal to k which is also the height of the vertical column i.e. we have 'k rows and k columns'. So the numbers of s rows equals k, while the number of combinations of 0's and 1's (which also has k members) must be 2^k ( since we have a choice of two digits from each k set). Now by employing the diagonal slash technique we alter each of the diagonal numbers highlighted in red (1 becomes 0 and 0 becomes 1). This ensures that we obtain a new sequence since the first member will be different from that of S1, the second member will be different from that of S2 and so on. Hence we have demonstrated that there are members of the set described by 2^k that are not counted within the list k i.e. k < 2^k even for infinite sets

Cantor also showed that there are more transcendental numbers than rational (and algebraic) numbers. The proof resides in the construction of a function h(f) which is defined as the height of a polynomial f(t), being equal to the sum of the moduli of all the (integer) coefficients of f(t)

so for f(t) =a0+ . + . + ant^n . . . . . . .This defines t as an algebraic number

Now there are h integers that are <=h, hence the number of polynomials is <<h^h , Consequently, there is a finite number of polynomials over the integers of given height h. In other words each integer h is associated with a finite number of polynomials and hence algebraic numbers - that is they are Z countable. Since we have proved above that R is uncountable and we now find that the algebraic numbers are countable then we have demonstrated that transcendental numbers exist! Now if the number of Transcendental numbers was also Aleph zero then the total number of reals would be algebraic (countable ) numbers + transcendental numbers = Aleph zero +Aleph zero =Aleph zero, which is false since we know that the number of reals is c =2^Aleph zero> Aleph zero. In other words the number of transcendental numbers must exceed the number of algebraic (e.g rational) numbers. [The three great mathematical problems of antiquity viz. the trisection of an angle, the doubling of a cube (the Delian problem), and the squaring of the circle, are all impossible to achieve by compass and ruler and the last of these relies upon 'pi' being a transcendental number and so crown 3000 years of mathematical effort!]

It is easily proved that the square root of a prime number is irrational (indeed one of the Pythagorians showed this for SqRt 2). More generally for a prime number p we can prove by contradiction that SqRt p is irrational:

assume that SqRt p = a/b . . . . for some integers a and b. For simplicity I will show this specifically for the prime 5 although this is valid for any prime just by substituting the 5 for the variable p

hence a^2/b^2 = [SqRt c x 5]/ [SqRt d x5] . . . which is a contradiction! . . . (where SqRt 5 = SqRt [c/d]

NEXT what about the n^Rt [2], is it possible for this to be rational. Assume that n^Rt 2 = a/b

But this would contradict Fermat's Last Theorem (see below), which has recently been proved valid for n>2

Hence n^Rt2 is irrational. This approach also shows that even for n=2 there are no Diophantine solutions such that a^2 +a^2 = b^2, for 2 integers a and b

The final piece in this proof was provided by Andrew Wiles and can be summarized by the following developments. Assuming that there was a Diophantine equation which did violate Fermat's theorem, it was possible to recast such an equation, in a form known as an elliptical equation (y^2 =x^3 +ax^2 +bx^ + c). It was then eventually shown that such an equation could not be a modular form. Now the T-S conjecture claims that all elliptical equations are modular form. Hence if the T-S conjecture could be proved, then it must be true that such an imagined equation cannot exist i.e. such an equation which violates Fermat's theorem cannot exist. This T-S conjecture (or at least a portion of it that was crucial to Fermat's theorem) was provided by Wiles.

For natural numbers n and integers b,a , the n th Root of [(b/a)^n + 1] is irrational

The main thrust is in demonstrating that if the n thRoot of any integer Q is rational, then n th Roots of (Q+_1) must be irrational [In other words for any consecutive integers Q and Q+1, it is not possible for both of these to have n th roots that are both rational.numbers.]

IF the n th Root of [(b/a)^n + 1] is rational, then so is n th Root[(b/a)^n + 1] x a

thus d^n + e^n = c^n, (where d=qa and e=bq), which contradicts Fermats theorem which has recently been proved by Andrew Wiles. Hence n th Root [(b/a)^n + 1] must be irrational

Now we have an infinite number of rational numbers (a/b) which yield an infinite number of irrational numbers for each power n. If we make 'n' to be one of the infinite number of primes, then we can form an infinite series of irrational numbers each of which is a different prime root of a rational number. [Using prime numbers is not necessary but it does increase the chance of the number being transcendental rather than just irrational]. From this set of cardinality aleph 0, we can obtain by combination under addition, all the (irrational) subsets of the individual irrational numbers. Now the cardinality of such subsets is 2^Aleph 0, which is greater than aleph 0. Hence we have demonstrated that the cardinality of ALL irrational numbers (including those that are transcendental) is greater than that of all the rational numbers (which is only Aleph 0) and is equal to the cardinality of the reals i.e. 2^Aleph 0. Most of these irrational numbers will in fact also be transcendental, which also have a cardinality of Aleph 0, since the number of algebraic numbers rational + irrational) is in fact only Aleph 0. This new set (whose elements are formed as subsets, from a combination of the sum of the original irrational numbers), provide a means of producing 'most' of the transcendental numbers, but it is not apparent how we can distinguish these from the ordinary irrational numbers (such as the original irrational elements).

If we denote the ordinality of the infinity of the Natural numbers as w (corresponding to the cardinal value of aleph zero), then we find that unlike finite numbers we no longer have commutivity of the operations of arithmetic viz

. 1+ w = w . . . . . . . . but w + 1 = w + 1

2*w = 2+2+2+2+.... = w. . . .but w*2 = w + w . .i.e w*2 is two omegas placed next to each other, which gives the ordinal w+w but2*w is omega twos placed next to each other which makes an ordered set with ordinal number w!

We can proceed to higher ordinals such as 0,1,2,3,w,w+1,w+2,w*2,w*2+1,w*2+2,w*w,w*w*w(i.e.^3), w^4, w^w What about the first ordinal a such that w*a =a? well if we take w^w it is evident that w*w^w changes nothing, so a=w^w. [In other words w*w^w=w^(w+1) = w^w]. now the firs ordinal such that w^a =a is called epsilon zero e0, which must have the following form w^w^w^w^w^w.. . . . . . . . evidently putting such a symbol in the exponent position over an omega does not change anything, since a stack of omegas 1+w high is the same as a stack w high A better way of describing this is by the operation of tetration (tetra for 4,since it is the next logical progression after add, multiply and exponentiation); for example 4 tetration 2 means 2^(2^(2^2))=2^2^4= 2^16 =64,536 while 3 tetration10 is 10^(10^10) = 10 ^10 billion, which is a one followed by ten billion zeros!. Hence we describe e0 as being equal to w tetration w. But we needn't stop there since we could go to pentration of w, that is; w tetration w tetration w tetration w tetration w tetration w. . .etc. . etc. Now if we define 2 sets as having the same cardinality if there is a one-to-one map between them then we can define Aleph one as being the first ordinal with cardinality greater than w (which has cardinality of Aleph zero ---- it is in one-to-one with the Natural numbers). we can than proceed to Aleph2, Aleph3, Alephw, Aleph(w+)1, Alephw^w, Aleph(Alephw) and eventually we arrive at a number theta % such that % =Aleph% and one way of obtaining this is by having an infinite stack of tetrated Alephs (i.e. a pentration of Aleph) viz. Aleph tetration Aleph tetration Aleph tetration Aleph tetration . . etc From this we can then proceed to Aleph%+1,Aleph%+w, Aleph%+1, Aleph%+Aleph w and so on without end until we arrive at the absolute infinity, Capital OMEGA which is by definition indescribable and inconceivable

Turing Halting problem

1. To prove that all mathematically true statements could be proved i.e. the completeness of mathematics

2. To prove that only true mathematical statements could be proven i.e. the consistency of mathematics.

3. To prove the decidability of mathematics, that is the existence of a decisive procedure to decide the truth of falsehood of a mathematical statement.

The first two of these were disproved by Godel, while the third was disproved by Alan Turing (a method which although different, also relied upon Cantor's diagonal slash technique). Alonzo Church also disproved the third challenge using a completely different approach. Godel had shown that there are propositions '&' (say), such that neither & nor -& is provable , (and as a consequence, there is no proof of consistency of a formal system 'K' within that system). Turing on the other hand showed that there is no general method which tells whether a given formula '$' is provable within a formal system 'K', or what amounts to the same, whether the system consisting of K with -$ adjoined as an extra axiom is consistent

Let Tn(m) represent the result of a Universal Turing machine that can imitate the nth Turing machine carrying out a computation on the number m. (e.g. a particular operating system trying to use a computer program to decide if a particular even number fails Goldbach's conjecture and cannot be split into the sum of 2 primes). Some of these Tn(m) will produce a numbered result, while others will run for ever (we will denote the latter by Tn(m) =& ). Now, assume that there is such a universal machine H(n,m) that can determine if an answer can be produced. In other words H 'decides' whether or not the nth Turing machine acting on the number m eventually stops. For example let the machine output 0 if it does not stop and 1 if it does. Next we use H to eliminate all those combinations of n and m that do not halt and replace the output with a 0 and then allow Tn to act on m only if H(n,m) =1. Thus our new procedure is given by Q(n,m) = Tn(m) * H(n,m) and produces a table of infinite dimensions, which must contain every computable sequence; something along the lines of

Finally we apply the slash diagonal technique in which we add 1 to each diagonal element (i.e. the sequence 1,2,1,3,7 in red, becomes,3,2,4, 8) to obtain a number which cannot be in the list, thus demonstrating a contradiction which shows that our original assumption about the existence of H must have been false. In other words assuming H exists, there is some Turing machine number, say k, for the algorithm (diagonal process) 1 +Q(n,m), so we have

whereas if Tk(k) does not stop (so H(k,k) = 0 ) we have another inconsistent result that 1 + 0 = & (i.e. it stops if and only if, it does not stop!). Hence it is not possible to construct a Universal machine, that could decide in advance whether there is a number which contradicts (say) Goldbach's conjecture (which would in effect prove or disprove such conjectures without actually needing to find such numbers or proofs!).

Indeed Godel's result follows directly from Turing's (although historically they were derived the other way round) For Turing showed that there are some true statements that are not recursively enumerable and we can can express a Godel statement in this form. So for any formal system F we can encode a Godel statement G(F), which is not provable by a Universal Turing machine. So even if F is believed to yield only true statements, G(F) must escape the net cast by F despite the fact that we must conclude that G(F) is a true Turing statement. What Godel and Turing's theorems tell us, is not that there are unprovable mathematical propositions but rather that whenever we lay down axiomatical rules of proof beforehand (and if we accept that those rules are trustworthy), then we are provided with a new means of access to certain mathematical truths, that those particular rules are not powerful enough to derive!

Galileo was reported to have uttered these words shortly after he was forced to recant his support for the Copernican' view, of the sun being at the centre of the solar system. He is regarded as the grandfather of physics and in actual fact the main reason for Galileo facing the Spanish inquisition, was due to his book being interpreted as being rather sarcastic towards the Pope. Indeed it was difficult for laymen of the time to accept the fact that the Earth actually rotates, since this seemed to go against everyday experience of spinning objects. (We had to await Foucault's pendulum for the simplest experimental demonstration of the Earth's rotation). Much of the story that has passed down is apocryphal but Galileo was forced to spend the remainder of his old age under household arrest. Part of his legacy is that of rotational dynamics, and in particular that of non inertial frames of reference, which produce peculiar effects and are relevant to both general relativity and quantum theory!

The principles illustrated below have many applications including gyroscopic compass and gyro-boat stabilizers