Chapter 141: The Completeness Of Mathematics

By Darryl Penney

Abstract: mathematics may be incomplete as proposed by Godel, but what is it an incomplete part of? When a proven complete mathematics is used, Fermat last theorem appears to be a triviality and this shows that there are mathematical wormholes into the physical that result in a new way of thinking that could be useful in mathematics.

Keywords: Godel; Russell; number theory; Fermat’s last theorem; fractal universe; relativity; creation equation

Preface

The theory is new and came out of fundamental physics and consists of the application of two parts, vertical organisation and horizontal relativity. I used relativity on fundamental physics to hive off social engineering that is the ‘mirror image’ of materials engineering and may save civilisation from itself and the myriad problems facing the world. An application to mathematics may be warranted in the light of Russell’s paradox that might simplify processes within mathematics in the light of some examples given in the section Test Run, below.

Setting the Stage

Mathematics started from the need to count sheep, cows, cloth etc. and could, for simplicity, be considered to be based on an infinite set of integers and it evolved into an organisation that has been expanding for thousands of years, but is this the only mathematics that we should be using because the set of integers is infinite and there could be a simpler system? I believe that there is a simpler mathematics that has also been used for as long as Life has existed that I call the mathematics of concept-context that describes democracy, the market place, the thinking of the mind-brain from the generator of the fractal that we call the universe. This mathematics is based on relativity, concepts and contexts and we use it without thinking about it, because it is thinking and living and is derived from, what I believe is the creation equation of everything. It also says that the universe is a fractal that continues to exist because of certain restrictions: that it must expand, that organisation must always be minimal [Occam’s razor] and energy must always be minimal [principle of least action].

This paper arose from applying relativity to fundamental physics which led to three important concepts, firstly, the mathematics referred above that the mind-brain uses, secondly, what I call general mathematical physics that is the context of a new way of thinking [concept], that I call “New Think”, and thirdly, a hitherto ‘lost’ social engineering that is, I believe, needed to organise our civilisation and solve the current problems of over-population etc. These do not have a direct effect on mathematics, but they do reflect on mathematics’s basic assumptions because mathematics is a member of, what I call, general mathematical physics which contains the top-down of physics, mathematics, philosophy etc. together with the bottom-up organisation of the creation equation. The question is, does the fractal, that is the universe, affect mathematics, and why does Fermat’s last theorem become trivial when using this mathematics?

Mathematics seems to be proud that it is a completely defined and completely isolated organisation that somehow seems to ‘resonate’ with the world at large, so much so, that ‘the Journal of Mathematical Physics defines the field as “the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories”‘ (Wikipedia) However, if Godel considers the incompleteness at the top-end, should mathematics, as a member of a possible set of mathematics that are bottom-up from the physical, consider the existence of ‘worm-holes’ between the bottom-up physical mathematics and traditional mathematics? This idea broaches another paradox that is a restriction of the creation equation that says that concept and context are independent and yet entangled and this implies, in the real world, that generalists and specialists are necessary, but don’t necessarily understand each other.

Consider Pythagoras’ theorem,’the theorem has been given numerous proofs – possibly the most for any mathematical theorem. They are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps and cartoons abound.’ (Wikipedia, Pythagorean theorem)

Consider also, ‘mathematician and philosopher Bertrand Russell (1872 – 1970: “Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.” In other words, Russell saw even our description of the universe through mathematics as being dangerously close to some sort of selection effect.’ (The Equation That Couldn’t Be Solved, Mario Livio, p 250 This is presumably the thinking over the last 100 years when fundamental physics effectively closed down because it was rooted in Newtonian physics and not the physical, but now with a new view presented below [Form of the Universe], the quotation is possibly reversed and fundamental physics determines two new mathematics.

I would offer the following warning, that this paper is an ‘opinion’ piece and is not scientific because it does not have a bibliography and does not build on the peer-reviewed work of others, and that is because it is new and fills a ‘hole’ in our thinking that currently lacks relativity by being top-down only. A further warning is given, from the creation equation, that generalists and specialist are orthogonal and have difficulty relating to the depth and width of papers, like this one, that cross academic disciplines. This latter difficulty has the same source as the debate over the wave-particle duality that is the relativity from the creation equation and cannot be avoided.

The History

‘Godel demonstrated that no complex mathematical system was complete’ (The Man Who Loved Only Numbers, Paul Hoffman, p 117) ‘That mathematics was incomplete and possibly inconsistent was a body blow to those who saw mathematics as the most logical of logical systems . . . . most card-carrying mathematicians still clung to the belief that mathematics was in fact free of contradictions, even though they now knew they could never prove this.’ (p 117) ‘Godel’s incompleteness theorem of 1929 had finally reared its ugly head in a real situation [Continuum Hypothesis]. (p 226) What a ‘pretty pickle’!

Mathematics may be ‘the most logical of logical systems’, but it is entangled with the mind-brain of the mathematician, who is entangled with the universe and the two systems are not the same because mathematics displays ‘organic’ growth, whereas the universe is a fractal. In other words, mathematics is very old and evolved from the counting of sheep, areas of land, volumes of water etc., but where does it fit into the larger scheme of things? Reality is required in the universe, but how does it work? When these questions are answered, physics is found to not actually consider the physical and a new discipline emerges that I call social engineering, a new mathematics that I call the mathematics of concept-context that is the mathematics on which the mind-brain is based and a new way of thinking.

The question to be considered here is, ‘are the systems compatible?’ or not. Mathematicians tried to build mathematics from pure logic, but they forgot to ask ‘who’s logic?’, theirs or the universe’s logic and that question could be a ‘game-changer’ because ‘logicians like Frege were working to shore up arithmetic . . . when he learned from Russell of an unavoidable paradox in the concept of a set of sets . . . The paradox Russell found had “an affinity with the ancient Greek contradiction about Epimenides the Cretan, who said that all Cretans are liars”. (p 115) This ‘led Russell to the idea ”consider the classes that are not members of themselves; . . . . Thus each alternative leads to its opposite and there is a contradiction.”’ (p 116) . . . ‘Try as Frege did, he could not circumvent Russell’s cunning conundrum about the class of all classes.’ (p116) ‘David Hilbert, the leading mathematician of the time . . . said, “with the paradoxes of set theory . . . never will happen again.” Hilbert’s words were taken as gospel’. (p 116) ‘Russell and Alfred North Whitehead responded to Hilbert’s call. Like Frege before them, they tried to build up all of mathematics from first principles in three impenetrable volumes of Principia Mathematica. . . . until young Godel derailed it.’ (p 117)

‘Russell’s paradox is not hard to extend. Take:

A transitive verb <V>, that can be applied to its substantive form.

Form the sentence: The <V>er that <V>s all (and only those) who don’t <V> themselves,

Sometimes the “all” is replaced by “all <V>ers”.

An example would be “paint”: The painter that paints all (and only those) that don’t paint themselves.’ (Wikipedia, Russell’s paradox)

This example shows firstly, that Russell’s paradox can be expressed in words [concepts and contexts] and suggests that number theory might be usefully extended to words, and secondly, shows that the mathematics of concept-context, as derived from the creation equation, might meet number theory in wormholes, where the two meet. This can enhance the range of proofs available, as shown with Fermat’s last theorem, below, which appears because mathematics is a product of parasites that are themselves dependent on the mathematics of concept-context that can be derived from the creation equation. Godel showed that mathematics is incomplete, which worried mathematicians and can be handled by restrictions, but there is huge scope opened up by considering wormholes into a general mathematical physics. Consider: ‘it evolved into the now-standard Zermelo–Fraenkel set theory (ZFC). The essential difference between Russell’s and Zermelo’s solution to the paradox is that Zermelo altered the axioms of set theory while preserving the logical language in which they are expressed, while Russell altered the logical language itself.’ (Wikipedia, Russell’s paradox)

Conclusion: clearly, mathematics is in trouble, but can that be solved by redefining mathematics as part of a wider mathematics in a similar way that Russell’s paradox was removed?

Wormholes

Wormholes are the science fiction writer’s method of overcoming the restriction that the speed of light is relatively slow and makes intergalactic travel inconvenient and provides a ‘quick-fix’ to the impossible, but they do exist, as shown by Pythagoras’ Theorem and Fermat’s last theorem. The square on the sides of Pythagoras’ theorem are there, I believe, because relativity requires that if a measurer measures a side, the universe’s organisation records this, as a relativity, and this shows that Einstein’s equation [E=mc2] is a statement of the creation equation [energy plus organisation equals zero] plus the act of measurement. This shows that wormholes appear in number theory that join with the physical, and number theorists should be aware that they exist, if for no other reason than the ‘waste’ of energy that was afforded by the proof of Fermat’s last theorem. The proof that I give below is trivial and could have been written in the proverbial margin!

The Way In

Assuming that the law of conservation of energy is the bed-rock of physics [energy cannot be created nor destroyed], so, energy must be ‘something’ and considering that our universe was created out of nothing [the simplest surmise], there must be a relativity to energy that I could call organisation, which changes the conservation law to ‘energy plus organisation equals zero’. This ‘word equation’ generates a fractal and uses, what I call, the mathematics of concept-context and it is complete [because we exist], so, to solve mathematics’ problem of completeness, we need only set up a relativity of mathematics and general mathematical physics [plus relativity and restrictions]. The mathematics of concept-context explains the working of the mind-brain, democracy, markets, peer-review etc., whereas general mathematical physics is the context of a new way of thinking that I call “New Think” [concept]

A small digression referring to Russell’s paradox because mathematics is a logical organisation whereas a fractal is the organisation generated by a simple equation and they are orthogonal, in that they are independent [yet entangled] and one is driven by organic growth, whilst in a fractal, everything is similar and not driven by organic growth. A fractal is a different organisation, as Adam Smith [economics] said ‘what is good for the individual is good for the economy’. There appear to be ‘wormholes’ where the two systems come together, as in Pythagoras’ theorem [shows the squares of relativity] that means that difficult problems, when using number theory, such as Fermat’s last theorem, become simple when viewed bottom-up. A number of examples are given below.

The Other Mathematics

There appear to be three mathematics that need consideration, firstly, the traditional mathematics that has evolved to be useful over thousands of years, secondly, the mathematics of concept-context that arises out of the relativity of the creation equation of the fractal universe in which we live and defines, democracy, market-place and our thinking, and thirdly, a composite general mathematical physics [context] that combines the first two and provides a new and complete way of thinking that I call “New Think”.

A fractal is seemingly complex, but is generated from a simple word equation [energy plus organisation equals zero] and so, just what is mathematics? If we are to consider a new way of thinking [“New Think”] as a concept, the necessary context is general mathematical physics that embraces all of mathematics, physics, philosophy etc. In a practical sense it is the de-siloing [context] of academic disciplines [context] and is the only way that the concept-context duality can be handled and this shows that I have done my part and now need a new duality of mathematicians to carry this discussion forward, just as a fractal does.

Conclusion: Russell seems to be the thread to follow, through paradoxes, word equations, sets upon sets etc., but two things are clear, that relativity has brought forward a step to completeness, as shown by the emergence of social science that was hidden by the incompleteness of physics that we need to engineer better people and organisations. Secondly, the universe is necessarily and sufficiently complete, for if Occam’s razor and the principle of least action were violated at any time, the universe must become chaotic with multiple solutions. This answers Godel’s assertion of the incompleteness of mathematics because Fermat’s last theorem is nonsensical in that we can only ask the question because it is true [else the universe would not be here]. Perhaps Godel’s assertion is satisfied by a general mathematical physics that is possibly complete because it deals with top-down and bottom-up organisation, relativity, sundry restrictions and that context implies a new way of thinking [concept, “New Think”] that would not go amiss.

A Test Run

If affordances are the organisation that the environment affords us, based on our intention, which must be behind every measurement, the relativity of that organisation is consciousness and philosophy’s perennial question of ‘when did consciousness start?’ can, I believe, be finally answered. The recognition of any affordance requires consciousness, such as by an organism moving away from heat, and “New Think” uses this same consciousness, but it has the distinction of including bottom-up and being complete [across the disciplines], so “New Think” must be superior and we, as Homo sapiens sapiens attain a higher level of consciousness than the animals.

Absolutes simplify our understanding of the universe because, I believe that the universe is built on absolutes and the law of gravitation is given below as the multiplication [relativity] of two constants [absolute four]. This relationship has never been proven before, that I know of, because ‘this is a general physical law derived from empirical observations by what Isaac Newton called inductive reasoning.’ (Wikipedia, Newton’s law of gravitation) It is hard to believe that over hundreds of years, this simple, but important relationship has not been derived, but I believe that that lack indicates that physics is incomplete in the way that I have described. Similarly, Euler’s equation, below, is another enigma that cannot be answered by mathematics, but like everything in a fractal, it describes, or refers to the creation equation.

There are stories that Einstein imagined riding on a photon, so, the creation equation becomes E/m=i2, on the photon, where ‘i’ is the square root of ‘-1’, E is the energy and m is the organisational form of that energy. Outside of the photon, the third absolute says that the speed of a photon must be ‘c’ to the measurer, so the measurement is ‘c squared’, and, E=mc2. Clearly, Einstein’s derivation becomes obvious from the creation equation, but what is not so obvious is that the universe measures our measuring [relativity], and knows everything so as to give a unique organisational solution. This is shown by Pythagoras’ theorem as the squares on the sides that leads to Fermat’s last theorem.

Consider, ‘In number theory, Fermat’s Last Theorem (sometimes called Fermat‘s conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation a n + b n = c n for any integer value of n greater than 2.’ (Internet) If this is true for n=2, a right angle exists and the above derivation of measurement becomes true [Pythagoras’ theorem] which signals an orthogonalty that is the construction of the universe and that demands that a single unique minimum is required for both energy and organisation [Occam’s razor for organisation and the principle of least action for energy] and that proves that Fermat’s Last Theorem is true, throughout the universe, and the inter-connectivity of academic disciplines also shows why a general interconnected mathematical physics is necessary. This proof is a physics solution, using “New Think” that possibly equates to Andrew Wiles 200 page mathematics proof ‘which threw the entire kitchen sink of complex twentieth-century techniques at the problem’ (p 199)

Conclusion: you may ask why these diverse problems have yielded so easily to “New Think” and that is, I believe, because “New Think” is complete, also, the context crosses academic borders [de-silos], contains relativity and contains both top-down and bottom-up organisation. In other words, the specialist and the generalist are orthogonal in thinking, and both must be used. Social engineering has been hidden by the incompleteness of physics, and now, I believe, social engineering has the legitimacy to demand change in governance, law, social security etc. to prevent the world’s problems, when we have agreed on the form it should take to attain some goal. Christianity, like farming and the industrial revolution started with a small group and perhaps we can do likewise.

Form of the Universe

Relativity is the functioning of the universe and a lack of relativity is the form of the universe and a lack of relativity is easily created [and our understanding of the universe] by the ratios of the dimensions [energy (E), organisation (O), time (t) and length (l)] created by expansion. The five absolutes are firstly, the sum of energy and organisation is always zero [from the creation equation energy plus organisation equals zero], secondly, energy and organisation [dark energy] are necessarily created to balance the necessary expansion [for the creation equation to exist] of the universe [E/t+O/t, all volume], thirdly, the constant speed of light [with respect to any measurer] is l/t (all E and O) and fourthly, gravity [so called quantum gravity] is E/l+O/l (all t). The law of gravitation is:

E(mass1)/l times (for relativity) E(mass2)/l plus O(mass1)/l times (for relativity) O(mass2)/l

Notice that the ‘inverse square law’ is inappropriate [one mass, charge etc. can not exist] and is actually derived from the absolutes and relativity and the ‘+’ in the creation equation stands for all relationships [physical, logical, restrictive, use etc.] between two entities. Further, ‘as with the Schrodinger equation itself, we still have no fundamental way of deriving Born’s rule.’ (Beyond Weird, Phillip Ball, p 41) This is not surprising because Born’s rule requires the same derivation as the law of gravitation. ‘If the amplitude of an electron wavefunction at x is 1 (in some units), and at y it is 2, then repeated experiments to determine the electron’s position will find it at y four times (2×2) more often than at x…. How did Born know this? He didn’t. Again, he “guessed”’. (p 41). In every oscillation between a wave and particle [wave-particle duality], the particle has to reappear somewhere, and it appears with a probability dependant on the square of the amplitude of the wave because, as quantum gravity [absolute (4)] varies inversely as the separation, relativity requires the inverse square law [or the square law, in this case and in Pythagoras’ theorem] and there is obviously relativity between the wave and particle.

‘Heisenberg’s uncertainty principle…. This restriction on precise knowledge does not apply to all pairs of quantum properties. It applies only to some, which are said to be “conjugate variables”. Position and momentum are conjugate variables, and so are energy and time (although the uncertainty relationship between them is subtly different from that between position and momentum) … I have never found an intuitive explanation of what makes two variables conjugate’. (p 150) The universe is created from an orthogonality [independent, but entangled at the origin] of energy [momentum] and organisation [position] and trying to measure an orthogonality [measuring each exactly is the same as between the two] is logically impossible because it is a restriction on the creation equation [independence]. Energy and time, along with organisation, volume and length are dimensions and must be orthogonal so that ratios can uniquely define absolutes.

Fifthly, the role of Occam’s razor and the principle of least action is crucial to the understanding of the functioning of the universe and the latter asks ‘why does light travel in a straight line?’. Newton’s laws of motion say that a photon must travel in a straight line otherwise the laws do not work and so misses out on vital information and is, again, ‘up in the air’. I believe that the answer is that there has to be a unique answer and the only unique answer in every case is the minimum and the organisation that belongs to the minimum energy is the most efficient organisation. I can say this with conviction because if either energy or organisation were not at a minimum, there would be two solutions at the same time and this would cause chaos in the functioning of the universe. This last sentence questions whether our universe is “real”, although derived from nothing is a bit of a difficulty, but then, what or where do we expect it to come from and suggests that it is an organisational solution based on possibilities created by measurement [such as Pythagoras’ theorem]?

If there is a creation equation, as I propose, the universe must be a fractal and everything in it must conform to certain simple rules. Adam Smith was the first to realise this in Economics, where an ‘invisible hand’ works so that what is good for the individual, is good for the economy. Clearly everything shows this form of the universe in its use and as an example, let’s look at Euler’s equation, which is claimed by Mathematics as the enigmatic relationship between the fundamental mathematical quantities pi, e, i, 1 and 0, though what 1 has to do with the others appears a little strange. However, as a description of the physical universe, it makes more sense because it determines the form of the universe [(e to the power i times pi +1) = 0 can be written (e to the power i times pi + e to the power 0) = 0, which is an expression of orthogonality and describes an expanding [e, simple interest expansion] sphere [pi] from 0 symmetrical [i] through the centre]. This ‘subsuming’ is the expected result in a fractal and Euler’s equation appears enigmatic because of the appearance of ‘i’ [the square root of ‘-1’], but it’s appearance becomes obvious due to relativity. Consider the quotation “wavefunctions generally contain ‘imaginary’ numbers – one involving the square root of -1, which is not something that has a physical meaning” (p 53). I am drawing attention to it because it shows the current confused thinking of physics, in that ‘i’ is an operator from quantum gravity [E/l+O/l for all t] because relativity is shown by ‘1’ and ‘-1’ from the inverse square law and that must be generated by ‘i’ and that is why “wavefunctions generally contain ‘imaginary’ numbers” because ‘i’ [and every number] is not only a number [concept], but also an organisation [context] and quantum gravity is the ‘spread’ from the atom [quarks] to gravity [in galaxies].

So, Newtonian physics is a creation of the mind and has nothing to do with the physical until general mathematical physics is used and then it can be seen that additional information is created from measuring organisation. For example Einstein’s Special Theory of Relativity shows that there is a simple relationship between mass, length and time, but this is incomplete because the above says that energy, organisation, length and time are simply related through the ratios that destroy relativity. Einstein was looking at the relativistic changes, whereas the above looks at the things that don’t change [invariants of the universe]. Newton’s laws of motion, Einstein’s theory and Maxwell’s equations all hide organisation and that obscures the picture, for example, magnetism is an organisation that registers the relativity between a charged particle and the measurer as can be seen by its odd behaviour [sign depends on direction, magnitude on speed]. It is also important to realise that the dimensions are independent, but entangled organisationally.

Measuring organisation such as beauty, music, religion, buildings and parades etc. creates energy that we release as laughter, in extreme cases [good joke], dance energy [foot-tapping] or just feeling emotional energy of appreciation [Mona Lisa painting possibly due to the golden triangle ratios]. Thus, social engineering is necessarily orthogonal to material engineering and is the key to controlling our civilisation and preventing a (so far) inevitable break-down. The difficulty with the question of energy shows that Newtonian physics is convenient for us in our world, but does not consider the physical host that we live within [as parasites], and it behoves all good parasites to understand and consider the health of their host, for to kill their host is to die as well. This is a truth that we should seriously consider acting upon because “new Think” is based on truths.

References: no references were given as everything has been derived from first principles.