What is an abstraction? There are many definitions of abstract thinking, from being free from details, to distinguish the important, to remark on common features, etc. But the requirement for abstraction is to be able to distinguish by own perception among different forms and events and then to connect the distinguished different forms and events into groups (“species”) according to their common features. Advanced abstraction also means the ability (experience) to distinguish in given conditions what is important, what is secondary and what can be neglected. I repeat in given current conditions – for conditions change in the real world. And what was previously the important, determining factor may no longer be, and vice versa what was negligible may become the important, determining factor, or at least a secondary one. Above all, it is good to know, to have the experience that the given conditions change. Therefore, it is impossible to determine what is important, secondary and negligible forever. The real world is still changing more or less differently at different levels. Certainly the conditions (forms and events) change, but the experience of abstraction remains.
It is good to have mathematical formulas or models, but such models or formulas have a limited range of validity. For example, to solve the achievable frequencies for an LC oscillator or the energy for an arbitrarily chosen wavelength of electromagnetic radiation.
The top of abstraction is to determine what will be important in the near future, what will be secondary and negligible. In short, to know, to intuit, how conditions are likely to change in the near future and possibly how they have changed in the past.
The above kinds of abstraction are intended to lead us to an absolute ability which can no longer be called abstraction, but to which abstraction leads – Experience – the ability to create new forms and events and to group these into different species according to the actual need. All people are invited to do this. And not just to distinguish common characteristics or to determine what is currently important or what will be important in the future. But more about that in Art, Craft, Science .
Lawfulness, regularity or periodicity in a set of data is more likely to be found by those people who expect it and know how to look for it, while the most precise observational data alone does not lead to a deeper understanding.
But periodicity or regularities are not perfect, better said, they are not eternal – in short, they are valid for a limited time. Even if they are valid for a very long time, like the orbit of the planets around the sun, that does not mean that the stability of the orbits of the planets will be the same for millions of years to come. There are other laws – like the motion of comets or nuclear forces – that can unexpectedly disrupt the stability of planetary orbits. Not to mention the basic fact that all periodicities and regularities are “grafted” onto a purely chaotic environment. Thus, the most precisely measured “constant” slightly change in the rhythm of vacuum fluctuations . So what is the point? Is it about observation or is it about abstract thinking? It is both! To learn how to abstract what is significant at a given time, at the current time. And at the same time to know the general regularities (laws) that can affect the most stable natural processes. That is, not only to think, but to give place to other observations – perhaps in the field of comets or nuclear forces. All this is called Experience, fine and firm decisive perception.
The motion of comets and even planets is a chaotic motion, see the chaotic fluctuations of the orbits of Earth and Mars due to their precession. A similar situation occurs with Mercury, Venus and Jupiter. In other words, there is chaotic motion in the solar system, but for us on a long-term scale, as opposed to our short-term observations (hundreds of years). This chaotic motion in the solar system does not allow us to predict its evolution over tens or hundreds of millions of years, so the behavior of the universe on a macro scale cannot be predicted at all. And the oppposite to determine the distant past.
Not to mentioned the top of science – the discovery of chaotic matter (quantum) field with its “zero” fluctuations that permeate the entire universe. As we know the chaotic behaviour is completely indescribable. In this chaotic ocean there are certain forms and processes with varying degrees of stability. This degree of stability varies over a great range, from the “stability” of protons, electrons and bound neutrons to the instability of free neutrons and many other elementary particles, as well as the stability of chemical compounds, minerals and organisms. These stable structures are subsequently describable. The extent of their description is determined by the degree of their stability.
Eg: The law of reflection is clear, but for ideal conditions. The angle of reflection is equal to the angle of incidence. But where on earth are the ideal conditions, both for light waves and for material objects. See bounced balls below – only different trajectories
The first picture above shows the bounce of the ball for almost ideal conditions. But the reality is very different in the real world. See gunfights in cities – bouncing bullets are the worst. See the upper two pictures showing the bounce of e.g. a bullet off a road, concrete or stone. The bullet comes in at almost right angles and bounces almost parallel due to irregularities in the materials. or vice versa the bullet comes in almost horizontally and bounces almost at right angles, again the effect of the material – deformation resistance.
It is impossible to describe everything. Every form, every shape, every structure and every process differs to each other. How many are different shapes so many are describing shapes. Where to find describing shapes? For better understanding we must make sets with likewise characteristic.
An infinite number of shapes need to be formed, reduced, grouped into a limited number of shapes that will be similar. Create sets of shapes with given properties – common characteristics while respecting the originality of each shape. See maple leaves, clearly at first glance we identify maple leaf. But if we take a closer look at all the maple leaves we have identified, if we look at them in great detail, we find that all the leaves differ from one another. These sets — called biological species change over time. Not only ontogenetically (development of the organism as an individual from germ to adulthood), but also phylogenetically (evolution of organisms through geological epoch) – see clearly visible differences between present-day plant life and perhaps Devon plant life.
Already here we are offered a simple definition of biological species and indeed all natural objects and processes in the world. Easy to distinguish. The maple leaf differs rapidly from the oak leaf or from the lime leaf, or from other leaves. Similarly, a liquid is clearly different from a solid or from a gas. Whereas determining the type of gas or liquid is already harder. But even here, there are methods to clearly divide oil from water or air from steam. And so we can go deeper. This is where the intensity of observational or dividing methods increases. Distinguish types of oils or degree of moisture in water steam. And one can go even deeper. Eventually we come to the conclusion that there are not two exactly the same oils in the world, or two completely identical types of petrol or exactly the same snowflakes.
And on the other hand, we can generalize. Matter comes in four basic forms – solid, liquid, gas, plasma. The only difference between a solid and a liquid or gas is in the stiffness or degree of freedom of chemical bonds. Solid, liquid and gas are common occurrences of atoms. Stable atoms, non-ionized atoms. And that makes this trio (solid, liquid, gas) radically different from plasma. In plasma, atoms are not stable, but atoms are more or less ionized. With free electrons, that is. In the event of very high temperatures, atomic nuclei may fuse or divide.
The basic physical classification of the world
The base of current physics – omnipresent matter (quantum) field, that completely fills the universe. This field is the supporting chaotic environment for the following effects such as radiation or matter. The radiation has no restmass as opposed to matter
Next, we can divide radiation by wavelengths of radio, microwave, infrared, light, ultraviolet, röntgen, gamma radiation. How to further divide ultraviolet or gamma rays? Hard or soft? We can divide the light radiation into pretty rainbow colors ranging from violet to blue to green to yellow to red. And artists can go even further – to distinguish degrees of saturation. But this resolution is no longer done by the machine, by the measuring apparatus, it’s only done by the man because of his perception.
Let’s get back to matter. Matter is divided into ionized and non-ionized. Ionized matter is plasma. Non-ionized matter is divided into solid, liquid, gas. Furthermore, the solids are divided into crystalline, amorphous. Liquids are divided according to viscosity – oil, water. Gases are divided according to their density. And in this way, we can continue to divide up to a point. And that limit is the originality of each observable shape or process.
The basis of all sciences, the basis of all crafts, the basis of all art, the basis of all human activity is a clear resolution. Without resolution and sorting facts, science cannot exist. Distinguishing is also needed in the craft, as well as in the arts.
We are still left with the question where in the chaotic ocean of matter (quantum) field the regular or quasi-regular structures come from. Why they hold together for some time, why they evolve in spite of the surrounding and internal chaos.
Interesting evolution of the universe – out of the violence and roughness come such fine structures – plants, organisms.
The violence and roughness are meant by primordial states of universe and sharp oscillations in form of matter. Fine structures mean so most gentle organic structure like human brain, plant blossoms, flowers, butterfly pel and so on.
What is time, what is life? It is not possible to separate the origin of life on Earth from that of the Universe. It is not possible to distinguish the evolution of life on Earth from the evolution of matter in the Universe to the conditions for the “formation” of life. In fact, life manifests itself in the beauty of minerals or in structures of glass. However, we can see in the sky similarities of certain cloud structures to graptolite slates, or cracked glass with beetle grooves and much more.
The science is about limits, about bonding shapes, about different shapes and structures with common characteristics – briefly about discretion. Without differentiable shapes there is no science.
Scientists argued about differences among species. Where to put their boundaries. It could be better to think what discreteness in species does mean. What is the meaning of the fact that Continuously Nature is bounded to discreet appearances like species in biology? Biological species do not continuously change from one species to another. E.g., an apple doesn’t change continuously into a pear. See a figure below.
pear apple
Formulated graphically – we can see two curves of frequency distribution of common characteristics of apple and pear. Apple P(M), pear P(N). See next figure below.
It is clear that biological species are the result of quantum behaviour at the micro level.
See a figure below – a “sharp” line spectrum of radiation emitted from heated atoms of various elements such as hydrogen, sodium, potassium, iron, etc. Such “sharp” spectrum has probabilistic distribution of different wavelenghts – marked by greek´s lambda.
To be sure, all chemical elements are a result of the Pauli exclusion principle. Naturally, all substances formed from chemical elements, whether living or non-living, are also a result of exclusion principle. This behaviour is inflated to large dimensions in Nature. Not only biological species, but also species of minerals, liquids, solids and gases. However, water does not continuously change into oil or hydrogen peroxide.
Water is clearly distinguishable from hydrogen peroxide, let alone oil. Just as an apple is clearly distinguishable from a pear. If not at first glance, on a closer analysis we can see a clear difference.
Without quantification, without limitation, without quantum behavior, no recognition is possible. Recognition of different marks. The basis of theory of sets are elements. These elements must be differentiable. Very well differentiable. If there is no differentiability then there is no set, no math, no science either.
See the next figure below – resolving power between two peaks
These two peaks must have a certain minimum distance from each other. Otherwise they cannot be distinguished. In optics we know this as resolving power.
Without resolution there is no measurement, without resolution there are no laws whether mechanical, thermodynamic or natural at all.
The law of conservation of energy is valid only for isolated space like second law of thermodynamics. The sum of total energy before the experiment in a isolated space is equal with the sum of total energy after the experiment. Where there is the isolated system in the Universe? I mean the completely isolated system without any interactions with its surroundings. Such ideal isolated system really does not exist in the world like the ideal gas or like the ideal point or line or cube or anything else. Not to mention, the differences among isolated, closed and open systems. Where boundary conditions may change – e.g. changing an isolated system to a closed system. Where there will only be an exchange of energy without an exchange of matter.
It is impossible to make ideal closed space in Nature. Especially when we don’t know the origin of quantum foam and where did matter came from. E.g. the ideal coordinate system – we pretend rectangular coordinate system like a squared paper. All squares are the same. But the space without matter has no sense. And matter is responsible for curved space including the coordinate system, either. The empty space without visible matter is full of fluctuated elements (particles and antiparticles). The grainy structure – like untuned TV screen .
It is very interesting how several germ cells can develop into such a complicated structure as an organism, whether plant, animal or human. Yes, DNA structure is clear but it also depends on the influence of the environment. So does the universe. From the first few forms, the universe has developed into a very higly complex structures including organisms. And what about the environment of our Universe?
Self-organization theory – the reciprocal movements of a complex system controlled by the laws of non-equilibrium thermodynamics. According to this theory, the system can “spontaneously” organize itself if energy flows through it. In other words, if we have energy differences and a chaotic system among them, and we start the energy flow given by energy differences, then the initially chaotic system will start to organize itself into higher orderly predictive complex structures. This is verified many times not only in thermodynamics, physics, but also in chemistry or biology.
The proven theory in practice many times. But it adds more questions to the origins of life and the origins of the universe than it explains them. On the one hand, the requirement of an energy differences. Secondly, an impulse triggering the flow of energy. Furthermore, the regulation of the flow of energy and, above all, the origin of energy, or what we call energy. We don’t get away with explaining that energy is the ability to do work. Very reduced – energy is given by the frequency of oscillations of electromagnetic waves E = f, when we give Planck’s constant equal to one. The higher the frequency the higher the energy.
Thus, the energy flow occurs between two or more different frequencies. Roughly, we can imagine two compressed springs that differ in rigidity.
It is not enough to have only two or different frequencies, but we have to have very many dissipated particles between them, entities that somehow have to interact with the flow of energy. And here comes the question of the size, number, and proportion of the entities to the size of the original energy difference, that is, the frequencies already. Furthermore, it is a carrier of frequencies. We no longer have ether, but a quantum field full of vacuum fluctuations. So we have pure chaos. Then there is the question about two different frequencies – their origin in an environment like a chaotic quantum field.
Then there is the existence of laws that define the behavior of different frequencies. Why more frequency depresses less one and not the other way around. But it also depends on rigidity. An example from mechanics – however compressed a spring with low rigidity does not overwhelm a very low compressed spring with high rigidity. No matter how compressed the clockspring does not overcome the uncompressed spring of the car’s suspension. So much for distributing the properties of a quantum field. It’s impossible to describe a chaotic quantum field.
In the end, different frequencies can be modeled on clay, and nothing happens. The clay must be living. Only when we start moving the clay will something happen, but if I don’t want a chaotic behaviour I have to issue regulations, set limits. And where the limits and regulations for so-called self-organization come from? Regardless, in an environment like the chaos of quantum fluctuations, nothing can stay stable. Compare this with the dissipative environment of corrosive acid against a piece of cloth. In other words, in an environment like the chaos of quantum vacuum fluctuations, we still have to recover again and again through the creation and annihilation of particles, after that to hold the validity of laws and regules among particles and antiparticles with waves with given laws – let’s call them electromagnetic waves. And then consider classical particles, as so-called frozen energy or better as excitations of the quantum field.
The laws of nature as we know them were not around just after the creation of the universe. Or at the moment of creation. The question is what that moment is and how long it lasts. See the theory of no beginning of the universe, but of the random expansion of one fluctuation of the quantum field into its present cosmic form. So the laws of nature, not only physical, chemical (especially organic chemistry) but biological, sociological were sort of condensed in an initial state of the universe of which we don’t know how long it lasted. We don’t have a scale. The atomic cesium clock as we know it today did not exist. But their idea has been condensed into primordial forms, as have their creators and users. There’s no telling what’s hidden in the quantum field. So let’s keep exploring, so we have something to look forward to.
Actually, the whole world as it is, including all biological species and the beauties of nature through the history, including all works of art, music, all of that was condensed in the primordial beginning? The origin of the jet loom, any inventions, musical or artistic works, ideas, technological processes, etc. This was all condensed in the early days of the universe? Or else? I had forgotten the idea that our universe was about the mass of a bag full of potatoes at the beginning, and the rest was built up during the expansion. The law of conservation of energy is right – the negative energy of gravity versus the present enormous mass (energy) of the universe. So it balances out like a scale. My remark is not mocking, diminishing the degree of knowledge reached. Because one has to formulate an idea, law or equation based on the available facts and then see how it agrees with further observations. We formulate new facts at the risk of making incorrect conclusions. But that’s progress, we know which way is wrong and we try differently.
The reality or our ideas with models? Which do we prefer? The reality of the universe gradually revealed by us, which guides us. Revealed at the cost of a painful search followed by immense joy from understanding and applying new knowledge.
Simplification, mathematical abstraction has its own limits. The limits of the actual observable state. It is not possible exactly to describe the curve of a tree trunk.
Some simplification is possible – an approximation to the ideal state. This is followed by a mathematical model. See closer and note the small bud on the tree trunk. A next whole branch can grow from it, or there was a branch. It doesn’t matter. But it doesn’t matter that this branch can influence the whole tree and therefore the whole situation. Just as from the smallest quantum the whole universe can arise – hardly then to be neglected! Let us return to the bud. The bud is included in the orange curve shown in the image below.
Still, we can’t determine what will be or what was without further observation. Let us finish by noting that the actual reality is so multivariate with unpredictable influences that it is impossible to realistically describe the past or future based on the current state more than corresponds to multiples of the current state. Nothing against idealized models, but they are only models with limited validity. It is not possible to make long-range predictions based on such models that don’t work out, and then blame Reality (or G-d) for not being fair. Our superficial doing is not fair.
Our predictions based on idealized models have decreasing value into the future without further observation, without further correction. So does the deduction of what was in the past.
Reality is consistent, without contradictions. But our knowledge is limited. There will always be contradictions in our knowledge. Precisely because our knowledge is limited, and always will be, there will be contradictions no matter how slight or how great. It’s a bit like Gödel’s incompleteness theorems.
Briefly and roughly – K. Gödel proved in 1931 that for every infinite set of logical statements S it is possible to construct by the method of this set S such statements T whose truth cannot be proved or disproved by the means of the set S. At the same time, Gödel proved that it is always possible to supplement the original set S with a new set of statements U so that the new extended set S1 = S + U gives the possibility to prove or disprove the truth of all statements T. Unfortunately, in the expanded set S1 there will be new undecidable statements T1 whose truth cannot be proved or disproved by the means of the set S1. Thus, we must again expand the already expanded set S1 with a new set of statements U1 so that the newly expanded set S2 = S1 + U1 gives the possibility to prove or disprove the truth of all statements T1, but …….. and so we can continue indefinitely.
In spite of the above, there is the certainty of the level of knowledge that has been reached, a repeatable experience that continues to grow if we want it to be so.
How to discover and formulate natural laws? For example, the law of gravity. Only by observing of stabilized processes. Based on observation of planetary orbits and further abstraction. The planets are replaced by ideal points of certain mass. Or the size of the planet is unimportant in relation to the distance from the sun. We can solve the orbital periods with sufficient accuracy. But the situation is quite different in the case of dense nebulae, forming planets – planetisimals. Differently shaped masses. See Fig. below
Abstraction in this case is absolutely impossible. Sure, we can divide the shape of matter into infinitesimally small objects. These objects are subject to mutual attraction according to discovered laws on the basis of so-called ideal bodies. Differential calculus helps us only in the case of clearly definable shapes. In the case of purely random shapes which, moreover, change randomly, there is nothing to do but observe. See Fig. below
Conclusion: It is quite impossible to determine the age of the solar system based on a derived law of gravity from the current “stabilized” shape of the planets. There is a contradiction here. It is absurd to determine the orbital period during the formation of the planetesimal of the later Earth and to measure the entire formation period of the solar system by this period on the basis of the hypothesis of the origin of a stabilized solar system from a dense primeval nebula.
So how do we determine the formation period of the solar system and, moreover, using our ideal tropical year? Use a scale other than gravity. Use a quantum scale. Use the decay of atomic nuclei. To use a half-life value of atomic nuclei to decay. The half-time is the time it takes for 1/2 of radiocative nuclei to decay. We obtain the rate of decay. Roughly written – the tropical year or parts of it (day, hour, second) are compared with the value of the half-life of suitable radioisotope. We can then determine the formation time of the solar system expressed in years. And years as we know them today are completely useless in the time of formation of planetesimals.
There has been a change. Instead of gravitational effects, we measure TIME using quantum effects – the decay of atomic nuclei. But that’s not such an ideal solution. Better written it is an ideal solution, but it is not realistic, it does not correspond to reality. In short, we’ve replaced one scale, gravitational time, with a second scale, quantum time. Missing connection between gravity and quantum. Missing the unified theory supported by experiment.
Try to solve gravitational effects (trajectory, time, velocity, etc) on the arbitrary selected “ball” in the case of the following environment – See Fig below
Especially if such environment is indefinable changing. To use gravitational laws? Impossible! To use quantum mechanics? Impossible, also.
A very fundamental question. The limits of the law of gravity in a quantum field. When did the law of gravity begin to work? What conditions must be satisfied for it to start work? The meaning of the function of the law of gravity was hidden in the primordial forms of the universe? Or is the existence of the law of gravity purely random? And under different conditions of primordial forms, would there be a different law, or different laws? Primordial forms of the Universe like DNA in biology? So much current different forms (chem. biol. species) pressed in primordial forms? Where did the law of conservation information came from? The law of conservation of differentiability, such so plenty variability in species, etc. Hidden properties or laws of matter? In other words, later properties of matter are hidden in the primordial forms? Or on the contrary, the forms and laws of matter are the result of further evolution – the influence of random fluctuations of primordial forms – disturbances and inhomogeneity of vacuum. By matter I mean stabilized resp. renewed excitations of the quantum field in the form of quarks, elementary particles. Moreover, the law of gravity needs inertial mass – bound mass (energy), such as particles with so-called rest mass.
By the way, polar bond in the plasma state is not possible. So are other chemical bonds – covalent, ionic, etc.
Allowed combinations of chemical bonds – e.g. methane CH4, but HC4 is not allowed. The allowed and disallowed states are determined by the “discrete” arrangement of the electron shell of the atoms – Pauli exclusion principle along with Planck’s constant. And again another question. When quantum laws began to exist – thus the discrete arrangement of matter?
Imagine a omnipresent quantum field – chaotically arising and disappearing vacuum fluctuations. Everything changes over time. But Time has not yet been defined! Time has meaning only with matter. By matter we mean the exciations of the vacuum field into various wave forms – quarks, electrons, photons, shortly particles that are characterized by mass. These grouped excitations of the vacuum field called matter particles (quarks, electrons, photons) are constantly being renewed. These particles are free or bound. Free particles like photons do not have a so-called rest mass. Bound particles have rest mass. Bound particles are grouped into atoms and these into molecules – the basis of all matter in the universe. Time, or the sense of time, as we perceive it, arises only with the formation of matter, respectively with renewed excitations of the quantum field in the form of the bound particles, i.e. particles with rest mass. Photons do not have time – all their energy is free. Electrons, protons, neutrons, i.e. particles with rest mass are characterized by what we perceive and measure as time.
Very interesting question – thickness of the biosphere on Earth.
Not to mention the very big empty surrounding universe, but very limited space of our Earth. So much empty space relative to such a small area of the globe. In terms of the occurrence of life – the surrounding empty universe and such a small area of our Earth. So much empty space in relation to such a small area of the globe. The thickness of the biosphere is estimated to be around 8 to 10 km. A few kilometers in the sea and about 5-6 km in the air to the mountains. Living organisms are found within this range – Flora and fauna. But fauna – animals depend on fauna – plants or algae. The fact that the eagle can live at high altitudes is thanks to plants. Or rather, thanks to the smaller birds and raptors that depend on plants. It’s the same with deep-sea fish. Their food are smaller fish that feed on plankton or algae.
Let’s do a thought experiment. We have a thin non-translucent membrane that covers the variously textured surface of the Earth and the surface of the oceans and seas, lakes. We place this membrane a few millimeters above the surface of the land and waters. What’s going to happen? All plants, algae and lichens disappear. Many animals survive even at high altitudes. Likewise, a lot of deep-sea fish won’t notice the change. So the 8 km biosphere will still be inhabited. But after a few days or weeks, life will disappear – the biosphere will be no more.
The same would happen if we were to place the supposed impenetrable membrane a few mm below the surface of the oceans, seas and variable textured surface of the earth. On the surface no change, but plants, algae would not exist after a few days. As a result of the food chain, there will be plenty of animals both on land and in the seas for several weeks.
Conclusion: It is well known that all life on earth depends on green plants. But green plants have stem, roots and leaves. In short, these plants combine two completely different environments. And they can only exist in a combination of these two environments – air and solid or water in hydroponics or aeroponics.
The transition of the two different environments is a necessary condition for the formation of the plant and the algae. The transition of two different properties is the source of some interesting processes. The source of the effects is the transition. Regardless of the area then affected. See electronics, the transition between different semiconductors. An almost infinitesimal transition if we reduce millimetres to micrometres.
Mathematics has its own limits. These limits arise from the very nature of mathematics. For mathematics to exist there must be distinguishable events in global space and time. These different events are products of local changes in global space and time. Try to mathematicaly describe the image below – really hard work or absolutely impossible when the shapes change continuously 
which means that local different events (shapes) must be distinguishable to each other for a long time. Only then can maths get to work – to differentiate events, to sort and name different events into sets that have common attributes. Sets are grouped different events with common attributes. After that mathematics is able to count these sets – to add them, to subtract them, to multiply them, to divide them, to solve equations along with graphical analysis, derive, integrate, solve differential equations and much much more operations. Mathematics predicts totaly identical sets or subjects or processes or anything else. Otherwise it’s not possible, it wouldn’t be mathematics. See the basic equations 1 + 1 = 2. But we know that Nature is alive and ever-changing and that no two events are absolutely identical. The same is valid for a group of events called sets.
Briefly to remember – Maths is the result of human abstract thinking. See below
In the upper image there are six or eight differentiable shapes. Human abstraction has been to condense these shapes into so-called ideal points. The ideal point has 0(zero) size. But zero size is impossible in the world. Just like the ideal line or curve or plane. These idealities are only our abstract projection. The real world has a foggy (probabilistic) structure. Very evident in the particles that make up our ordinary world.
Not to mention contradictions – it’s hard to construct anything of a certain size from zero size – whether a segment on a line or something else. There is no chance to connect ideal point with the ideal line. Btw, the ideal line seems to be like the ideal point if we look at it in the direction of its axis. As well as the plane seems to be like the ideal line if we look at the ideal plane in the direction of its edge in case it was possible.
So ideal points or ideal lines or ideal systems, yes – why not, but they must be subordinated to the real facts of nature around us. As well as to respect changes in the base units, which are certainly not ideally constant.
What is accuracy? The greatest number of oscillations? The more oscillations the greater the accuracy? In the same way we have to verify the movements of the billiard balls after N-collisions. Tt is not enough to predict that after 1000 collisions the balls will be placed so and so. However, the calculation accuracy is limited. After N collisions between the balls, the next direction of the balls movements will be in the range of +- 180 degrees. Pure chaos. Not mentioned the accuracy of Planck constant and Gravity constant is limited, like the speed of light. And it doesn’t help if we make them fixed values.
Do the scales or rulers change as well? The rotation time of the Earth or the frequency of oscillation of the cesium atomic clock. We only expect constant rulers over the time history. Only in the long period average we suppose identical fluctuations. But neither organisms nor nature itself are the identical. Each organism of the same species is slightly different to each other. But over geological epochs the changes among organisms are very extreme. See the differences between the Devonian flora and the plants of today.
How to validate, how to measure ever-changing events, shapes, structures or forms at all? E.g. to use base form. Let´s have a very good example – the topography, resp. the mapping. The common basis of the mapping effort there is a triangle. The best tool how to measure and validate the earth´s surface is the use of trigonometry in geodesy.
To accurately measure a single length on the earth’s surface, perhaps on the order of x hundred metres. And then just measure from both ends of the marked length two angles to the top of the highest mountain, which is x tens of kilometres away.
But not only geodesy but also in astronomy – to find the distance of the nearest stars by using twice the radius of the orbit around the sun (see astronomical unit). See the definition of parsec.
The triangle is the best shape in the world. Other shapes like squares, rectangles and polygons can be constructed from trinagles and not vice versa. Let´s write down some interesting properties of triangles. See below
There are three sides and three angles. The longest side is called the hypotenuse. There is a first rule – the sum of distances of two sides must be greater then the distance of the hypotenuse. There is also a next rule – the longest side (hypotenuse) of a triangle is always opposite the largest angle. The smallest side of a triangle is always opposite the smallest angle, and the middle side of a triangle is always opposite the middle angle. If all sides are equal then all angles are also equal. Or if two sides are equal then two angles are also equal. See below
There is a special triangle called right-angled triangle. This triangle is the result of a rotation of the hypotenuse in the unit circle.
Triangles must be submitted to real world and not vice versa. Imagine situation of part of Nature – rocks with valleys and to measure them, their their retreat or approach and to estimate how were conditions in early stages, without verification. Similarly, mathematics must be subordinate to Reality and not vice versa. See the image below – Worldwide Geodetic Network
The location of the continents is changing. Where to find a fixed point or fixed distance in the network shown above? What to choose as the basic unit of distance? Perhaps the longest distance on land? But even that changes, although less in comparison to the drift of the continents. We have the locations of the continents millions of years ago. See A. Wegener. The connection of South America with Africa, etc. The evidence is clear – paleontological findings and geological formations. But how to verify the location of pangea or gondwana? This can only be roughly estimated. There is no verification like in the case fo upper network, where we can use satellites, lasers or radars to verify and correct the base distances.
So we discovered that the shape of the earth is indefinable – from a distance like an ideal sphere, up close like a geoid flattened at the poles, and very close up like a pear-shaped hemisphere with the northern hemisphere smaller than the southern hemisphere.
Not to mention that the earth’s surface is constantly changing, the dynamic effects of the lithosphere on the earth’s crust, etc. There is no chance to find an absolute fixed point or distance on the Earth.
The purpose of the following highly simplified model is to replace the Earth’s geoid with a flat “geoid” or a flattened ellipse with shifting and changing surfaces (continents). In short, we will deal with a trigonometric network in an environment of local and global changes. So-called rubbersheet geometry.
… to be continued
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