by Darryl Penney
Abstract: science and mathematics are shown to be incomplete and limited to special cases when compared to general mathematics/organization leading to enigmas that have been unsolvable until now. In particular, social problems become amenable through the mathematics of concepts because of the numericalization of context, also, physics becomes complete with the realization that there is a physical and logical space that comes about through entanglement. Parasitic Life uses mind-space to infiltrate itself into the physical world and a simple example of these spaces is given in a possible partial explanation of the Pauli Exclusion Principle that allows visualization of relativity using waveforms and a more complete definition of general mathematics/organization is given. A method is given to legitimise mathematics by acknowledging that it is a special case of the mathematics of concepts and general mathematics that are derivable from the dimensions of a probability space, and a proof of the completeness of general mathematics is given.
‘But what is science, that we should place such confidence in it? Science, according to John Ziman’s admirably straightforward definition, is nothing more than public knowledge. And the word public indicates that we must limit ourselves to knowledge about which there is some sort of consensus.’ (The Material World, Rodney Cotterill, p 7) Unfortunately, this definition makes it very difficult to introduce new thinking and leads to an ‘upheaval’ system that occurs when a new generation thinks differently to the old generation. Science has been ‘Newtonian’ for the last several hundred years and, I believe that new thinking is needed because a number of enigmas have stubbornly remained, in spite of new concepts, such as relativity and quantum mechanics and the context of these concepts have not been recognized for what they are.
I am suggesting that these concepts of relativity and quantum mechanics are ‘tips of icebergs’ that are fundamental aspects of a measuring space [(a+b)=1] that I call relevance and questing, and in the latter, why should it be odd that if you seek a particle, you get a particle and vice versa for a wave. In other words, if you seek a top-down guess-work science, that is what you get, and the system constrains you to this view and that explains the enigmas of gravity, electron spin, diffraction, speed of light etc.
If mathematics and science are failing us, as I believe that they are, the problem is to determine exactly where the problem lies and how to fix it, and I have been concerned at the number of enigmas that mathematics and science have left ‘dangling’ as they push on into new discoveries and so, I am going to use general mathematics/organization to pinpoint the problem. From chapter 86, ‘I defined general mathematics to be ‘the mathematics of concepts and the four axioms of Life that we, as parasites, have built on the measurement/entanglement of our probability universe’ and this addition of the four axioms links the context of the mind/brain into the physical world.’ This definition of general mathematics/organization is expanded in the conclusion and contains the addition of three spaces that have evolved with life over 3,000 million years, and these spaces are physical-space, logical-space and mind-space.
Relevance and questing are two of the search axioms and are derivable from a measuring space, whilst the others are forward planning (from Life) and elegance, from the Golden ratio. The most important, in the physical world, are relevance and questing and if these are not taken into account there is deep trouble, and mathematics and science are in deep trouble because, from the first paragraph, both mathematics and science rely on public knowledge, and questing is looking at all possibilities. If you do not consider all possibilities, science and mathematics are little better than the musings of the Ancient Greeks, so let us look at some enigmas.
Diffraction has been an enigma for hundreds of years and signifies that problems exist in basic physics and the lack of resolution of these long-standing enigmas is causing problems in recent research in the modern world and it is unfair for the sins of the past to be carried on into the present. These problems should have been promoted for fixing, and not ‘swept under the carpet’. ‘By the last decades of the twentieth century, physics has probed the natural world in unprecedented scope and at scales ranging from the subatomic to the astronomical. Yet entire categories of readily visible, everyday phenomena remain stubbornly inexplicable.’ (Physics in the 20th Century, Curt Suplee, p 152)
‘Among the most problematic were certain kinds of physical systems with multiple parts in motion. And none was more infuriating than the apparently random behavior of moving fluids such as water or air. That kind of unpredictability seemed to make no sense: such systems are made up of individual macroscopic units – droplets or molecules – each of which is obliged to follow strictly deterministic Newtonian rules of force, motion, and position. Yet their collective properties often become chaotic as they change over time.’ (p 152)
‘Although every single water molecule is governed by inviolable laws, and its condition at any given moment is knowable in theory (at least within the uncertainty constraints of quantum mechanics), the aggregate motion remains unpredictable.’ (p 152) Unfortunately, Newtonian mechanics cannot predict diffraction, so how can it predict molecular flows adequately? There is a force/entanglement that is missing and this must have been suspected for hundreds of years. So, what is this entanglement?
In case it is thought that some small ‘tweak’ will ‘fix’ these modern-day problems, I will give an example that describes the missing parts in a more ‘vibrant’ manner, but it is a fundamental lack, I believe, that combines the problems of diffraction, chaos and electron spin and that latter problem will be considered here. However, the full story is told in the expansion/quest of the fifth dimension and the general mathematics/organization, below.
‘In the Stern-Gerlach experiment… there was, however, an annoying problem with the idea. It was almost certainly impossible in any conventional physical sense. For electrons to spin at the rate required to generate the magnetic quanta observed, they would have to be moving faster than the speed of light. So they were not really spinning on an axis, even though they clearly behaved as if they were; in the same way, physicists were coming to realize, electrons weren’t really revolving around the nucleus, even though their apparent angular momentum could be quantified. In those cases, and many more to come, scientists gradually became accustomed to the idea that, in the quantum world, objects had properties for which there were no visualizable physical counterparts.’ (p 77)
The last sentence is, I believe, taking the easy way out by saying that something is unvisualizable, but then, the statement is true because it is not recognised that there are visualizable logical counterparts that we do not make use of. In other words, I believe that there are three spaces, physical, logical and the mind-space that are not being fully recognized and further, the ‘visualizable counterparts’ are all around us, but unrecognised and they are relevance and questing that are properties of a measuring/probability space. In fact, science has not decided just what are the properties of our space, and scientists are totally concerned with measuring the various effects in our space without offering a general theory/solution.
‘After Galileo’s death scientific thought gradually veered around to the idea of the sun-centred solar system. In 1992, after more than three and a half centuries, the Vatican officially reversed the verdict of Galileo’s trial.’ (The Little Book of Scientific Principles, Theories and Things, Suendra Verma, p 28) Could it be that science is still wary of commenting about our space/universe? Our space must be a simple space [Occam’s razor] and the Michelson-Morley experiment indicates a probability/measuring space.
I have defined general mathematics to be ‘the mathematics of concepts and the four axioms of Life that we, as parasites, have built on the measurement/entanglement of our probability universe’
Further, from chapter 81, ‘If we unfold (a+b)=1 in a probability space, as above, including Life, we get:
concept/forward-planning/quest/relevance/beauty/context.
If we unfold (a+b)=1 in a probability space, as above, excluding Life, we get:
measurement/quest/relevance/entanglement.
Notice that forward-planning is a dimension specific to Life and necessary for the predator/prey basis of iteration and the four axioms are immediately obvious in the above.’
The last dozen lines are incomprehensible to mathematics [the mathematics of concepts includes mathematics as a special case] and physics cannot handle measurement/quest/relevance/entanglement [bottom-up versus top-down]. Questing [including quantum mechanics] at the present time is like asking the Delphi Oracle a question, but getting back the correct answer that we cannot understand because our view of the world is incomplete. As an example, electric and magnetic fields are extremely important to modern life and are the fundamental method of transmission of energy [photon], but we cannot even answer the question of why the speed of the photon is constant [the dimensions space to time is constant for all energies], let alone the enigma of why every observer sees the speed of light as constant, irrespective of their motion [Michelson-Morley experiment and our universe is a measuring space].
The answers are ‘blowing in the wind’ all around us, but are invisible because of our limited view of science and mathematics, but like an iceberg, science and mathematics have a ‘tipping point’, and it is not far off. To summarize the above, mathematics is a counting space (a+b) for sheep etc., physics has no clear idea of the space that is the universe [more than (a+b), but less than (a+b)=1], whereas a measuring space (a+b)=1 quests to (a+b)=1 [physical space, physical entanglement], (a and b)=1 [logical space, logical entanglement] and ‘+/and’ [mind space]. I believe that mathematics and physics need all of the physical, logical and mental spaces to fully understand the parasite that is Life that has evolved new spaces within our physical universe.
The basic reason for this state of affairs is that physics has allowed enigmas to remain unexplored over hundreds of years and one of the worst omissions is diffraction. Newton believed in the corpuscular theory of light and Huygens in the wave theory and both studied diffraction, but Newton ignored the bending of light that occurred when light passed through a small aperture (in contradiction to his first law of motion) and Huygens explained the phenomenon, but not why it occurred [the wavelets are probabilities]. So, what do I believe really happened? I have fully outlined the reasons in chapter 75, so I can move a little faster here.
In a probability/measuring space (a+b)=1, world P, where a and b are measurement/record, there are no absolutes, as Plato found, there is only measurement/record and entanglement. Compare this to a counting space (a+b) that mathematics is based on, and it will be seen that the former is much ‘richer’ in possibilities and, I believe quests the universe, where quest is the motive power behind generating the Mandelbrot series (and others) expansion and considers all possibilities. World O is our world where we use Newtonian units/rules of force, motion, and position etc., and these were derived from the predator/prey contests that required speed and distance cognisance for safety. They are not the units of the physical world P.
From chapter 86, ‘let us unfold (a+b)=1, where unfolding is following the questing [of quantum mechanics, evolution, business etc.]. The fifth dimension (a+b)=1, in this simple form produces five operators (ignoring, for the moment, the fact that the speed of light is an absolute and must be constant, that the energy per unit of space is constant and the law of conservation of minimum energy):
(a) measurement/entanglement that are local physical and independent/orthogonal (a+b)=1 [classical local action and reaction of matter that provides expansion of the universe, reflection, diffraction of light and water waves],
(b) measurement/entanglement that are universe-wide logical and independent/orthogonal (a and b)=1 [conservation of (zero) energy across the universe, gravity, creation of space/mass/energy/time through the Lorentz contraction],
(c) measurement/entanglement that are local physical and independent/orthogonal, (a+b) [local/personal appreciation], and
(d) measurement/entanglement that are universe-wide logical and independent/orthogonal (a and b) [universal/reality-wide appreciation]
(e) measurement/entanglement that links the operators ‘+’ and ‘and’ together in the fundamental relationship that links the physical/logical/organizational [mind-space, conservation of minimum energy]’
Now, returning to diffraction, Newton used the attraction of gravity to explain the motion of the planets and knew that it was a weak force that acted over large distances, and was not applicable to the spread of light in diffraction. Obviously there is an attraction between light and the body of the aperture (or repulsion between photons) to cause the spreading, but what is it? I believe that (a+b)=1 is physical energy, but (a and b)=1 is an entanglement. It can be seen from these two equations that there is a physical and logical entanglement between a and b that, in physical space form measurement/entanglement and in Life’s mental space becomes the concept/context of the mathematics of concepts.
This entanglement answers the problem that light leaves a point source evenly around the source compared to statistical mechanics that says that it is due to the numbers of photons that the distribution is equal [principle of least action]. In other words, physical-space needs logical-space to fully describe nature to itself and Newtonian physics uses only the physical and the four search axioms that link the mind to the physical world. The reason that Newtonian physics can (partially/incompletely) describe physics in terms of the physical is the mind-space ‘+/and’ that enables an emphasis toward the physical [for example, momentum and force are forms of energy].
The ‘+’ and ‘and’ signify another field/operator that is an orthogonality of physical and logical that simply means that everything contains a physical and a logical part/component. I call ‘+/and’ a mind-space because, I believe, that it is responsible-for/used-by Life to evolve a thinking mind (see chapter 86). In world P, it is the link between the physical and logical and provides what is missing in our current physics and is the principle reason that enigmas have remained unsolved/unsolvable for hundreds of years. In other words, I believe that the more logical parts of physics become enigmas because physics is Newtonian and ignores the logical part, or considers it as physical.
The assumption that our universe is a mathematical probability space has paid great dividends, but it contains a logical inconsistency along with all mathematics, including general mathematics/organization in that the four search axioms are needed for the condition that the sum of every point in the space equals one. This follows the general assumption that the law of conservation of energy requires the conservation of the energy of the Big Bang and this requirement has to ask, what mind is doing the calculation or what physical process is doing the calculation. This paragraph suggests that there needs to be a fundamental change in the way that we look at energy and that comes from a bottom-up look at the photon that will be investigated at a later date.
Science has traditionally used top-down ‘guesses’ helped by peer-review etc., and that system is not, I believe, ‘up to the job’ and I propose a bottom-up approach. Hence, electron ‘spin’ is possibly a logical necessity (to be investigated below), diffraction is local entanglement and that entanglement is of interest, above, and lies between laminar flow and turbulence. Newtonian physics denies entanglement and cannot answer questions of why things bounce, reflectivity of images in mirrors etc. Reflectivity in metals is thought to result from free electrons on the surface of the metal, but is that not a physical explanation of local entanglement?
The problem of electron spin is not difficult if you include the possibility of logic. Now, it is a quest/theory/explanation that ‘Niels Bohr … realized that the packaging of energy into Plank’s quanta automatically leads to stable electron orbits in an atom. And he made a bold assumption: when an electron is orbiting around a nucleus it does not radiate energy, even though it is describing a curved path.’ (The Material World, Roger Cotterill, p 37) ‘Louis de Broglie (1892-1987) attempted to reconcile the Bohr model of atomic orbits with wave-particle duality by suggesting that only those orbits which comprise a whole number of wavelengths around their orbital paths are permitted.’ (p 37)
My contribution/triviality is that a standing wave is relativistic in that there is no way to tell which way it is going unless you measure it. This logic means that two standing waves can occupy the same space/orbit, with each going in different directions [total is standing still], but it says something more, that it will occur because it can occur through local entanglement. To repeat, this means that every orbit can contain two electrons presumably because of logic! I have mentioned this before, that everything must be examined because it is simpler and requires less energy to fill the orbits from the bottom up, and in the case of boron, carbon and nitrogen to fill the single 2p shells before doubling up in the case of oxygen (p 52). Further, that minimum energy must be used due to Occam’s razor or the possibility occurs that a logical discrepancy might occur.
The magnitude of the spin, referred to above, as being physically impossible, simply means that it is not simply physical. That does simply explain the Pauli exclusion principle that two electrons can reside in each space/orbit and that two identical electrons necessarily need an entanglement that allows them to share an orbit and, that they must fill from the bottom up. Put another way, that using a rule/principle that more than one electron can share an orbit is an enigma, but by using logic and by accepting entanglement, we can visualize the reason for two electrons being able to share the same orbit.
In conclusion, this derivation uses general mathematics/organization and it is apparent that any general mathematic/organization must include the three spaces that Life has evolved within itself, physical-space, logic-space and mind-space, where a distinction of world O and P must be made at all times. Clearly, mathematics and science need to change in the ways that I have outlined within these constraints. So a definition of general mathematics/organization consists of:
(1) the mathematics of concepts, and in particular, the orthogonality of concept and context and the necessity of numericalization of context,
(2) the four search axioms (forward-planning, questing, relevance and elegance) that are derived from Life, the probability space and the fifth dimension, or from common sense,
(3) recognising physical-space, logical-space and mind-space as consisting of both world O and P.
The above is effectively the conclusion (concept), but what of conclusion (context) that is equally important, and is contained in a few quotations. ‘Practitioners of a complicated technique are likely to revel in its complications, once they have mastered them. The same occurs fairly often in modern research. It’s unfortunate, because the essence of good mathematics is to penetrate to the heart of a problem: a solution alone is not the ultimate goal.’ (The Problems of Mathematics, Ian Stewart, p 36)
‘In scientific research, one might expect new ideas from elsewhere, new techniques, new phenomena, to be seized upon and exploited. Actually, this only happens if they don’t have to cross the traditional boundaries between subjects. (p 78) ‘In the mid-1600s Antoine Arnauld argued that the proportion –1 : 1 = 1 : -1 must be nonsense: “How can a smaller be to a greater as a greater is to a smaller?” The dangers of verbal reasoning in mathematics could hardly be plainer’ (p 119). I might point out that verbal reasoning is the realm of the mathematics of concepts.
These quotations show the difficulty of introducing something new, as well as the limitations of a mathematics that is devoid of a mathematics of concepts that, as above, allows social problems to be handled. Mathematics is a special case that we evolved (world O) and ‘numbers are so closely allied to certain aspects of the natural world that we tend to think of them as something unique and almost physical. It is only when they are analysed more deeply that it becomes clear that they are an invention of the human mind – a method whereby our brains can model aspects of Nature. They are not Nature herself.’ (p 36)
This quotation is saying that we derived mathematics to model Nature, but we have done such a poor job that we have neglected the social sciences because, as above, mathematics cannot handle ‘verbal reasoning. Nature can be modelled and can be modelled via concepts if we use the mathematics of concepts that is written in the dimensions and is applicable to everything in the universe. In other words, mathematics could become part of Nature if it is admitted that mathematics is a special case of the true mathematics of Nature, that is, the mathematics of concepts, and general mathematics, when we are included.
Finally, it is common knowledge that Godel [there are true statements in arithmetic that can never be proved (p 214)], Turing [certain very natural questions have no answer whatsoever (p 214)] as well as myself complaining that mathematics is a special case, demands a proof that general mathematics is complete. It is also common knowledge/wonderment that there is a smearing of disciplines “Mark Kac – who considered himself an applied mathematician – has said ‘Miraculous as it may seem, fibre bundles, homotopy, and Chern classes are becoming as much parts of physical terminology as instantons, gauge fields, and Lagrangians are becoming part of the mathematical one.” (p 225) It is also common knowledge/wonderment that pure mathematical ideas become applicable at a later date, ‘George Boole’s ideas on mathematical logic, developed in the 1850s for no good practical reason, turned out to be just what the electronic engineers of the forties and fifties needed to build computers.’ (p 226)
So, I will use general mathematics to simply prove the ‘why’ of the quotations, above, that everything is linked, that general mathematics is complete and we can measure and record anything that we can envisage. The first step is to highlight the four search axioms [forward-planning, questing, relevance and elegance] that link us [as parasites] with the physical world and point out that nothing will happen without forward planning [press start]. The fifth dimension of a probability space [our universe] is (a+b)=1 [obvious simplification of the space] and there is no absolute solution [Plato’s problem] for a [measurement] and b [record/observer].
A probability space is a questing/measurement space that does two things [by definition] and that is, firstly, quests any measurement asked of it in terms of the question asked [wave or particle in quantum mechanics] and secondly, determines its relevance [conservation of minimum energy]. Hence (a+b)=1 shows entanglement (a+b) for any a and b and always returns an answer [the entanglement is universe-wide] when quested. The universe exists, as do we [Descartes] and it only exists because it is logical/organizational and has not entered chaos, where chaos is a non-returnable state caused by a singularity, such as those that the Lorentz transformation inhibits [logical and physical singularities].
We [parasites] evolved new and different spaces that I call physical-space, logical space and mind-space, and in each case there exists world O [our] and world P [physical/probability] because they are used in different ways. ‘There are without doubt areas of mathematics that will never be useful to anybody else. If we could identify those areas in advance, it would be an excellent idea to abandon research into them. But there is a catch. An item of no discernible purpose today may be just what’s needed in the science of tomorrow.’ (p 226)
This question of relevance is twofold, firstly, that ‘pure’ research returns rewards, but, secondly, what is the opportunity cost of using a defective model of the real world, as with mathematics. General mathematics contains every solution that we need [by completeness] and we do need these solutions to the social problems facing the world today [over-population, global warming etc.].
As the fifth dimension is a dimension [as are space-time], and general mathematics is derivable from the fifth dimension [only] by questing the three spaces [physical, logical and mental], it must be complete. There are no boundaries to questing and the four search axioms define any mathematical advance that may be applicable physically, or if we can invent something that is workable [Mandrake effect in chapter 86]. The above proof depends on the dimensions of a probability space, and so, the first step is to decide if our universe is a probability space, and this is long overdue. However, it is only now that I believe that our universe is not a probability space, but it is a way in, and this will be taken up at a later date.
William Thomson (Lord Kelvin) ‘once said “Science is bound by the everlasting laws of honour to face fearlessly every problem that can be presented to it.”’ (The Little Book of Scientific Principles, Theories and Things, Surendra Verma, p 92)
References: all quotations are fully referenced in the body and earlier chapters can be found on darrylpenney.com if required.
Chapter 75: The Nature of Life and Logic, Newton Laws of Motion, Reflection and Diffraction.
Chapter 86: How the Mind Works, Evolution in Mind-space, the Placebo/nocebo Effect Requires Two Parts, Combating Chronic Pain, Why Eastern and Western Medicine are Similar, Unfolding the Fifth Dimension and the Law of Conservation of Minimum Energy.
Chapter 81: Parasites in Probability Space, General Mathematics, Logic, Measurement, Organization, the Four Axioms of Measurement that Link the Mind/brain to Mathematics and the Dimensions of a Probability Space, Life as a Possible Sixth Dimension, the Why of Godel’s Incompleteness Theorems and the Goal of Explaining Everything by a Single, Elegant, Unified Equation is Attained.
darrylpenney 