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An Evidence Based Solution to Russell's Paradox, Grounded in Cosmology
Truth-spirit, enlighten and guide our research
Oh Mary conceived without sin, pray for us who have recourse to you
The laws of thought are reworded into a paradoxical form using Ordinary Language Philosophy. Rather than amending them, a solution is hypothesised and proven philosophically, which identifies the whole and the null, as two faces of the same cosmos. Such philosophical proof is transposed into a scientific hypothesis. The cosmos is considered as the sum of the observer with the observed universe. This adds a physical domain to astrophysics. The cosmos would be curved and static, while the universe would be the virtual vision of a Big Bang and its virtual expansion in a flat space. This would be due to the flatness of the observer's light cone. Current empirical evidence is applied to a construction of such hypothesis and provides empirical evidence to the defined equality between the whole, the null and the cosmos. The characteristics of such cosmos result as those of the idea of God. Such cosmos also has characteristics identical to those of Russell's paradox. The observer would as such live inside Russell's paradox, and a solution to the paradox consists in considering the paradoxical rewording of the laws of thought as valid.
A SOLUTION TO RUSSELL'S PARADOX
A simple formulation of Russell's paradox is in the form of the barber's paradox (Falletta, 1983). A remote village has only one barber, who shaves only those who avoid shaving themselves. Who shaves the barber? If the barber shaves himself, then he should refrain from performing his work, therefore he should refrain from shaving. If the barber refrains from shaving himself, then he should perform his work and shave.
In approaching the search for a solution to such paradox, a paper (Benazzo, 2010) presented, at the international conference organised by the Department of Philosophy of the University of Ibadan, on 22-24 November 2010, is summarised, rewritten and developed by targeting paradoxes. A specific philosophical approach is applied, that of Ordinary Language Philosophy. A definition of such approach may be found in Ludwig Wittgenstein (1972, 116) when he affirms: "When philosophers use a word - 'knowledge', 'being', 'object', 'I', 'proposition', 'name' - and try to grasp the essence of the thing, one must always ask oneself: is the word ever actually used in this way in the language which is its original home? What we do is to bring back words from their metaphysical to their everyday use." Professional philosophers use language in particular ways to answer big and/or very specific questions in life. In general, most people search for answers to the big questions in life and by doing so are philosophers, it may be said amateurs philosophers. These adapt professional and technical philosophical language to everyday use in their daily lives.
Millions of people with diverse backgrounds, interests, fields of competencies, ways of life, ideals, ages, and so on, have shaped an agreed language with all its nuances and variations. In order to draw from the broadest knowledge, in these pages the meaning of words tends to be considered in its broadest sense; that which is in use by the largest number of people. Such approach draws from collective wisdom. This has been investigated for example by Émile Durkheim as collective consciousness. Ordinary language philosophy draws in reverse from the result of such adaptations. Here below, the Ordinary Language Philosophy approach is used applying it back to metaphysics by targeting the broadest meaning of concepts.
Russell's paradox (Russell, 1903) stems within Russell's tentative to avoid misinterpretations, in the principles of mathematics. He wished to systematise language with formal propositions that would establish unambiguous logical forms, either true or false. Russell's paradox challenges such an attempt, generating ambiguous forms. Russell, in collaboration with Alfred North Whitehead, has tackled this by restricting the notion of set in order to prevent the paradox from occurring. While this has generated other problems elsewhere, on the other side it respects the laws of thought, which Russell (1967) defines as:
"(1) The law of identity: 'Whatever is, is'
(2) The law of contradiction: 'Nothing can both be and not be'
(3) The law of excluded middle: 'Everything must either be or not be'"
A question posed here is: Are the laws of thought always unambiguous? If there are special cases in which the laws of thought become ambiguous, then there may be cases in which certain paradoxes characterised by certain types of ambiguity acquire a legitimate standing. If that were the case, there may be at least a solution to Russell's paradox that legitimates it with particular significance rather than requiring constructing logic in such ways as to prevent its occurrence.
Classes constitute the basis of logic and of mathematics; the formulation of Russell's paradox concerning classes goes as follows:
A type of class includes itself as element; another type of class excludes itself from its elements. Concerning the second type of class, does the overarching class comprising each class that excludes itself from its elements also exclude itself from its elements?
If such a class excludes itself from its elements then it has the same characteristics of its elements, therefore it includes itself as its element. If it includes itself as its element, then it lacks the basic characteristic of its elements, therefore it excludes itself from its elements.
Such circularity characterises the ambiguity of Russell's paradox, the absence of a clear truth or falsehood.
The laws of thought may be considered as the fundamental tools that allow dissipating ambiguity, the building blocks of clarity of thought. So now the laws of thought are considered in particular.
The laws of thought are used to consider everything, in order to improve the knowledge of everything. Does 'everything' include also dreams and spiritual perceptions, meditative feelings? 'Everything' bases its meaning on 'thing'. This is mainly used as a pronoun. An example is the ostensive use of 'thing' in "Look at the thing there!"
The antecedent is unnecessary because the others actually see the 'thing' with their eyes. In most cases the 'thing' becomes evident from the context; it is deducted, without need of either antecedent or pointing to an ostensive 'thing'. Many times 'thing' is used when the exact noun is unknown or momentarily forgotten. In such a way, the listener may recall the word or understand the denoted concept. While a pronoun usually needs an antecedent, 'thing' is mainly used as pronoun without antecedents. An abstract identity, an idea, a quality may well feature a context that clarifies the specific 'thing' recalled by the pronoun use of the word. This may happen as well as with the more tangible occurrences. On the other hand, when considering meditative feelings, spiritual perceptions, dreams, then the discourse implies a high degree of subjective approach. In such cases a general context and other related information may well be insufficient to infer what 'thing' is pointing to.
Then the antecedent needs to be used before using the word 'thing' as pronoun. The use of the word 'thing' becomes then quite redundant and may be used to highlight, to remark, rather than for its main use of pronoun without antecedents. There may as such be a lot of debate whether 'thing' covers all the occurrences on which the thought may hover or investigate specifically. As the laws of thought should be applicable by everybody and both for objective and subjective argumentations, then a substitute of the word 'thing', one less open to such debate, is looked for. What would be main characteristics of 'everything' and 'nothing' to be maintained by their substitutes?
A useful aspect of 'everything' is to cover both on one hand each single one of all occurrences, considering each in its particular singularity, and on the other hand the whole set of all occurrences in its entirety.
'The whole' is the part of the previous sentence that characterises this second use of the word 'everything'. Bertrand Russell (1903) asserts that there is an infinite number of wholes. Each of such wholes should be marked as 'a whole' then. Conversely, the whole comprising all occurrences, all cases, is a single one and may be indicated as 'the whole'. 'Everything' is then substituted with 'the whole'. 'Nothing' also contains the word 'thing'. It may be denoted as the absence of 'everything'.
As 'everything' is substituted by 'the whole', 'nothing' may be substituted by the absence of the whole. 'The null' implies also the concept related to zero in numbers and may be said to indicate zero occurrences, absence of any occurrence. When a document is said to be null and void, null is intended in the sense that the absence of such document leaves the situation exactly unchanged. 'The null' is as such used to indicate the complete absence of 'the whole', and thus its opposite. Concerning the individual aspect of 'everything' that considers each occurrence, 'thing' may be substituted with 'occurrence' that has been used in the paragraphs above. 'Occurrence' comprises meanings such as event, experience, thing, episode, case, instance, appearance, presence. The scope of occurrence communicates less controversially both objective and subjective instances.
Such substitutions are now applied to Russell's formulation of the concerned laws of thought. The individual aspect of each one of all occurrences is first rendered:
(2) The law of contradiction: 'No occurrence can both be and not be'
(3) The law of excluded middle: 'Each occurrence of the whole must either be or not be'
With 'occurrence', the laws of thought maintain their conventional wisdom aspect.
Next, the all comprehensive aspect is rendered:
(2) The law of contradiction: 'The null can both be and not be'
(3) The law of excluded middle: 'The whole must either be or not be'
With 'the whole' and 'the null', the laws of thought breach conventional wisdom. The null can both be and not be, while in conventional wisdom the null is not. The whole must either be or not be, in other words, it can also not be, while in conventional wisdom the whole can only be.
One could decide to change such laws in order to fit conventional wisdom, to cancel the 'be' for 'the null' and 'not be' for 'the whole'. Such an approach would be inconsistent with the Ordinary Language Philosophy targeted at investigating into collective wisdom. The formulation of the laws of thought generates from centuries of human endeavours. The respect of such embedded knowledge leads to another approach, that or keeping the laws of thoughts as here reworded and look for a solution elsewhere. Frege and Wittgenstein, in different ways, were both underlying the presence of language traps.
Each was tackling them in different ways and for different objectives. Frege considered them as defects and wanted to resolve them using a formalised language or notation (Hanfling, 2000). Frege (1979) used this to better relate mathematics to concepts, for clarifying meanings and logical relations. Wittgenstein's approach was rather to make sense of language traps by considering more the ordinary language (Wittgenstein, 1972). The approach here used is to consider the traps as rather puzzles, profound knowledge far away from conventional wisdom and embedded in ordinary language. Solutions to such puzzles may then both clarify meanings and logical relations on one side, while at the same time keep philosophy in close contact with ordinary language.
What possible solution would make sense and give logical relations to the above reformulation of the laws of thought?
A philosophical hypothesis is made, that the whole and the null are the same, albeit seen from different perspectives (Benazzo, 1998). How could this be? Considering the physical environment, and the cosmos, all the energies and the forces with their time space coordinates would annihilate partially together, and completely altogether into the null when the last atom and the same observer are also added. As an example, three observers are considered. One stays at a distance, observing the other two who stand next to each other. Each of these stretches a hand towards the other so that the two hands touch facing each other.
The two would exert a force on the hands, such that the two hands stay still somewhere between them. They execute a kind of frozen, 'give me five' gesture. Each one of these two would feel a force from the other hand and would exert a force. The far away observer conversely, may well think that the other two are pretending to push their hands as the observation shows a null horizontal force. The two forces cancel out. The observer actually observes a null resulting horizontal force. Considering the whole cosmos, a complete annihilation of the cosmos into the null would be unobservable by the observer, as the observer is part of that cosmos. The observer may only observe the cosmos minus the observer self, i.e. minus the physiological processes of the observation. As the observer studies empirically the universe, a differentiation is made between the cosmos and the universe: cosmos = universe + observer of the universe.
The ancient Greek philosophers considered the cosmos in different ways, which had a thread that could be summarised in defining the cosmos as a complete, orderly harmonious system. The whole is complete as it includes all occurrences. It is in a way orderly as it includes the opposites of occurrences. The discourse about harmony would need another article just for itself. The whole may be considered to well adapt to the definition of the cosmos. In physical terms the cosmos needs to include all occurrences, also the manifestation of physiological processes such as dreams, perceptions, meditations, spiritual experiences, also the physiological act of observation. In such a perspective the cosmos would be concurrently the whole and the null, the whole as comprising all occurrences, the null as the complete aggregation of all the energies and forces and time-space dimensions of all the occurrences, altogether. Such philosophical argumentation allows the paradoxical rewording of the laws of thought to have a reasonable and coherent explanation within a coherent method referred to Ordinary Language Philosophy.
The reasonability and coherence provide philosophical proof of equality between the whole and the null, as two aspects of the same cosmos. Such proof lacks support by the laws of thought, as the laws of thought themselves are subject to proof. It implies a certain functioning of astrophysics and cosmology. Empirical evidence of such functioning would confirm the philosophical proof. Physically, the perspective in which the null and the whole feed into each other concurrently requires that such equality occurs beyond time. This is incompatible with the Big Bang paradigm, which corresponds to a creation in time from the null into a universe. Is there an alternative paradigm supported by empirical evidence, in which the whole and the null are concurrent? Such a paradigm is presented in another article in Tidningen Kulturen: Benazzo, Piero, 2010, "Curved Cosmos Seen as Virtually Flat in the Universe:
A Scaling Agreeing with Empirical Evidence", week 51. In short, an additional physical domain is added to the observed universe: the cosmos as defined above. The above philosophical proof is transformed into scientific hypothesis: first that the Big Bang reads the empirical evidence applying it to the observed universe, which results as flat and expanding. In addition to this, when the observer is added to the universe, the empirical evidence applied to such cosmos would instead provide for a curved and static cosmos, as the original Einstein's cosmology. The expanding universe would be a virtual effect due to how the observer would be obliged to see the curved static cosmos through the flat light cone from the past. The Big Bang would as such be a virtual reading, while the alternative topology would correspond to the actual physical coordinates actually experienced when travelling through the cosmos.
The mentioned article provides a geometrical representation of such a paradigm. Calculations of main characteristics of such topology are in agreement with empirical evidence, the same empirical evidence that supports the Big Bang paradigm. More details about the verification of such construction and the calculations may be found in the article mentioned. In such a paradigm, the time and space axes tilt gradually proportionally to the distance from the observer, and this tilting goes through a 360 degrees tilting until the coordinates are back to the starting point of the observer. To each region of the cosmos corresponds as such on average another region of the cosmos with a comparable quantity of matter and energy which has inversed space and time axes arrows, such that the energies of the two regions tend to cancel out. Einstein had theorised an average quantity of matter equal to zero (Einstein, 1961).
In addition, the geometrical construction implies also communication channels that act beyond time, instantaneously, between the distant regions. This conforms to empirical evidence that has proved that the future may select among different pasts, such as the famous experiment by Wheeler (Davies, 2006). Through such channels, the future would affect the present and the present would affect the past. Through such instantaneous channels, the matter of a region would act locally as matter, while it would interact as if it was anti-matter with the matter at inverted dimensions coordinates, those at the farthest distance of the cosmos. This would occur for the whole cosmos, in the aggregate of the observer plus the observed universe, resulting as such transcendental to the observer. These channels would bring as such the whole to annihilate, beyond time and transcendentally. The whole and the null are considered as two perspectives of the same cosmos.
One may also reflect upon whether at least some occurrences are beyond the cosmos so considered. This in particular may be asked relative to the idea of God. Can God stay outside such cosmos? With a cosmos considered as the whole, comprising even the last particle, the last dream, the last perception, etc. whatever is considered is part of the whole; only a null occurrence may stay beyond. As such, if God would reside outside the whole, then the whole would be a partiality, as there would be God beyond it. God then needs to be either the whole itself, or be internal to the whole. Can a God internal to the whole be omnipotent over the whole? Some religions consider this possible. Christianity for example believes Christ as incarnating such God. Even such incarnation is considered to incarnate a transcendental, i.e. invisible, God. Likewise, the cosmos considered as the whole is transcendental as it adds the observer to the observed universe, such that the observer is unable to observe directly such aggregation.
Beyond the whole, the null is inexistent and in itself lacks any derivation, it is characterised by absence of derivation. Any occurrence other than the null derives from other occurrences, both in terms of the physical interactions in the observed universe, and in terms of definitions in the mind of the observer, such as ideas, dreams, perceptions, feelings. In line with Parmenides, any occurrence other than the null is considered to exist (Thanassas, 2007), in other words, to derive from other occurrences, to be definable. The perfect way to define the null was for Parmenides to avoid talking about it (Thanassas, 2007). Talking about its inexistence would risk giving it some existence. As such, a wooden table that is destroyed is usually considered to stop existing in the physical universe observed by the observer in a particular time. On the other hand, in the above paradigm, as the cosmos is considered beyond time and space, in line with some empirical evidence (Benazzo, 2010), such physical table results as always existing in the whole of the cosmos.
This would occur in a time that is in the past with respect to the one considered by the observer. The destroyed table as such becomes absent in physical terms in the observed universe, even if it remains present in the memories of the observer and in physical terms in another time and space of the whole cosmos. There is a correspondence in such analysis between 'deriving from' and 'existing'. In this light, only the null lacks existence in itself, it lacks derivation. This latter corresponds to the idea of God, considered without derivation, from which all the rest derives. Those who believe in God, however, consider God as existing rather than inexisting. When Parmenides is talking about the null to indicate that the best way to define it is to avoid speaking about it, he still talks about the null. Is it possible to talk more about the null without degrading its definition? As an example, it is possible to define it as the absence of the whole, as the absence of any definition, of any occurrence, of any atom, of any thought, with the absence of language, with the absence of the observer to consider it, with also the absence of the concept of absence, and also the absence of the concept of concept. If the null was only inexistent, then it should be completely indefinable. It is though definable in opposition to the all encompassing aggregation of what is definable. This is coherent with the null considered as derived from the complete annihilation resulting from the aggregation of all the partialities of the whole.
The whole encompasses all potentialities, all possible cosmos conformations, all possible universes; it conforms to the idea of multiverse in astrophysics, of the possibility of infinite universes with infinite different combinations of fundamental physical laws. The comprehensiveness of all potentialities gives omnipotence to the whole. The whole as such corresponds to the idea of God considered omnipotent. The null, on the other hand, results as null-potent. The whole derives from the complete and all encompassing aggregation of all its components, and as such it exists. Either the null or the whole on its own gives an incomplete account of the idea of God. If God was just the whole, then it would derive from other occurrences, losing the idea of complete independence. If God was just the null, then it would be null-potent and inexistent, rather than omnipotent and existent. God considered concurrently as both the whole and the null would comprise all the main characteristics of the idea of God: existence, omnipotence, independence. The only independence in such a paradigm is that of the null, free from any derivation.
Independence and existence would as such be two opposites. God considered as concurrently the whole and the null would then result as the supreme being on the side of the whole, and as the humblest being on the side of the null, so humble as to reach inexistence, absence of being. Such humbleness would be completely independent. No one single occurrence of the whole, or aggregation, may shake it. A change somewhere in the universe necessarily implies a change in the observer, as the two together would annihilate in the unshakable null. Conversely, a change mastered by the observer implies a change in the observed universe, again because their aggregation would yield the null. The force exerted by such inexistence, such complete absence, such null-potency, results equivalent to its other face, the force of omnipotence. Given such corresponding characteristics on one side between the all-encompassing whole and God, on another side between the independence of the null and God, and on another side among the whole, the cosmos and the null, there is then an implied equality among all the four concepts. Each of the four is considered as transcendentally adding the observer to the observed universe (the whole = the null = cosmos = God = (observer + universe)).
According to conventional wisdom, laws of thought want to pare away any paradox, while the above described paradigm provides an evidence based cosmos that is itself a paradox. In it the existing whole annihilates into the inexistent null, concurrently. From this, any whole can be derived by splitting the null into energies and forces coupled with corresponding anti-energies and anti-forces. The observer as such lives inside a paradox, even if normally the observer is unable to experience physically a paradox, like concurrently climbing and descending a staircase. Are there associations between such paradox and Russell's paradox? In Russell's paradox formulation of the barber that shaves only those that refrain from shaving themselves, the paradox stems from the village being remote from other inhabited areas. If the village was near another village with another barber, the barber could resolve the paradox by getting shaved in the nearby village. The paradox happens because the situation is closed and self sufficient.
In the formulation of the overarching class that comprises each class that excludes itself from its elements, the fact that such class refers its definition strictly to its elements closes the class onto itself generating such paradox. The cosmos coinciding with the whole and the null, leaving null occurrence beyond itself, closes itself. The first law of thought may then be reworded as: 'Whatever is part of a whole is. Whatever is a completely self sufficient and closed-onto-itself whole, both has and lacks existence." The null can be contained in any class (Russell, 1903), so it is contained also in the whole. The null is also derived above as equivalent to the whole. The null then contains itself as one of its elements. The whole is a class that excludes itself from its elements.
The cosmos as both the null and the whole then both contains itself as one of its elements and excludes itself from its elements, paradoxically. Its structure and nature is identical to the class constituting Russell's paradox. Russell's paradox may be formulated in many ways, what characterises it is the structure. The observer as such lives inside Russell's paradox. Such paradox acquires then a standing which needs to be validated. The condition to make it valid, when dealing with the whole and the null, is to accept the above special exception to the laws of thought, derived by rewording. Its acceptance resolves the paradox. When the cosmos is considered also as God, then its formulation as Russell's paradox may be considered as one of the names of God. For the other Russell's paradox formulations, like the barber's paradox, the condition to validate them is to consider such formulations as language representations of the discussed transcendental concurrency of the whole and the null. The null and Russell's paradox, which breach conventional wisdom, concur in such formulation to the foundation of life, the foundation of the so defined cosmos in which the observer lives.
The process starts from a rewording of the laws of thought, implying a paradox. This is resolved philosophically by identifying an identity among the whole, the null and the cosmos. Its characteristics correspond to the idea of God. The identity results grounded on empirical evidence in astrophysics, by adding a physical domain to conventional cosmology: the expanding flat universe would be a flat virtual vision, through the flat light cone, of a static curved cosmos derived adding the observer to the observed universe.
The null concurrently stems as the cosmos itself and as contained in it, as in any class. The null aspect of the cosmos would contain itself as one of its elements, while the whole side of it would exclude itself from its elements, concurrently. The observer would live inside such cosmos equating to Russell's paradox. This null-whole paradox results thus as one of the names of the so defined God. Russell's paradox is resolved by such empirically validated rewording of the laws of thought.
Further research may be done on implications of the reworded laws of though on scientific discovery; on how different paradoxes relate to Russell's paradox, on other related implications in philosophy and other sciences.
I am grateful to Sim Smiley for her mother tongue proofreading, with which she spotted language issues. Any remaining issue is due to the author.
Pictures: Maurits Cornelis Escher
I am grateful to Sim Smiley for her mother tongue proofreading, with which she spotted language issues. Any remaining issue is due to the author.
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