What is mathematics? Counting. Counting what? Counting what is distinguishable to our perception. We perceive the world around us and inside us. We perceive distinguishable shapes in nature, the intensity of feelings and much more. We express these distinguishable shapes, structures and feelings in some way. And we do this again with distinguishable shapes and structures. And this in three forms. Gestures, voice and image. Gestures have remained and language has been developed from the voice. And from pictures, writing has evolved over time. And then came mathematics. To determine quantity. To determine the quantity of similar objects, shapes, structures or sensations. Self-similarity – the basis of abstraction. To abstract from details to common characteristics. See the difference between an apple and a pear (though every apple and pear is a non-repeatable original).
The problem with mathematics, like language, is that it calculates, distinguishes only and only existing objects, shapes, structures, processes or feelings. Roughly speaking, it is behind development. Behind the creative abilities. Mathematics, like language, does not calculate, does not name ideas, that which arises, that which is born in the mind, or that which is realized. Whether new objects, shapes, structures, processes – we call all this craft, artistic or scientific activity. Mathematics, like language, cannot tell us what of the “pile” of ideas and intentions will be useful in the future. For this, there must be experience. Experience that cannot be calculated primarily, experience that counts secondarily. An experience whose results can then be counted, analyzed, evaluated, predicted over a period of time before an unexpected event, idea, trend, process, object, etc. In short, there is a novelty. This novelty then greatly disrupts or cancels all the most complex calculations, evaluations, analyses and predictions.
We are at the meaning of mathematics, the meaning of language. The meaning is to lead us through our experience to ever greater or gently perception, to distinguish the previously indistinguishable and not to settle for the currently distinguishable, perceptible and calculable.
How many different curves can there be in the world. How to determine the degree of their differentiation. What will be the standard of difference. Who is going to judge?
How many mathematical curves can there be, describable, compressible curves like y=f(x), where y is a dependent variable that depends by a functional prescription f(x) on the independent variable x. See sin(x), tg(x), log(x), … , sin(2x) + log(3x), … , etc. Are there more indescribable curves than mathematically describable ones, perhaps by the most complicated rules? And what about the complexity of the prescription? Wouldn’t it be better to give a table of values for y and x in a given interval?
See the laws of physics or the mathematical theory of sets using the concept of a set as a well-defined differentiable objects. See a pottery – the pot is or the pot is not. A moment ago the pot was not, now it is, and after a while it will not be again. It is the same in biology, in the beginning there is no apple, after a while there is an germ, then a growing apple, then a fully grown apple, and finally there is no apple again.
Generalized – in the beginning there is nothing, virtually chaotic quantum foam, which looks like nothing, after a while a part of the quantum foam inflates like a balloon and further expands and inside the expanding universe are created particles, stars, chemical elements, minerals and rocks, also the earth is created with all apples, organisms and people, who want to know the meaning of life.
To realize this in the world – there must be for some time worldly fixed forms and structures incl. processes among them. Shapes, structures and objects must be stabilized for some time – so called unchanging or quasichanging shapes and structures – see the evolution of biological species throughout the history of the Earth. There were no apples in the primordials. But there were trilobites, graptolites, etc. But I don’t see them around anymore. Sorry, I can see them, but only their fossils.
How to count the conceptual (existing only in potter´s mind) designs of the pots or pots that are not yet finished or those that are on the potter’s wheel or those that are being broken? And this is the way with everything in nature.
In short, mathematics is useful for describing and classifying visible, perceptible, distinguishable structures. It is good to be familiar with the limits of mathematics. Mathematics cannot express what arises, what the unexpected arises. This is a surprise even for mathematicians. Certainly after a new object arises, mathematicians can describe its characteristics and classify the newly arisen object into the existing classification of objects. Further, mathematicians (mathematics alone cannot do this) can determine the object’s future evolution with some degree of probability based on observations of the object’s changes. But that’s all. Mathematics is not almighty. It doesn’t create new, it doesn’t destroy old, it just observes, describes and predicts.
In other words – to practice mathematics requires a stable distinguishable shapes, structures, processes and changes among them in the world for some time.
How to describe changes in shapes and structures? The described shapes and structures must be stable for some time. See biological species. A ladybird is a ladybird for as long as it is. So we have time to describe it. And if we don’t, another ladybird is born, and it goes on like this for thousands of years. But trilobites, for example, have been on the earth for about a hundred million years, but they’re no more. The same story will be valid for the appearance of ladybirds in the world. But let us cheer ourselves up with the thought that instead of ladybirds there will be another species of beetle, which our descendants will again have time to describe.
Back to descriptions of shapes and structures. We know that these must not change so often before we can describe them. In other words, shapes must not change faster than our ability to describe them. There is a slip between the change of the described shape and the change of the describer (usually a human, sometimes an automati). Description of slow changes by other faster changes. Both change, but the described changes are slower than the describing changes. If there were no noticeable difference, it would be impossible to describe.
What is the purpose of this? The purpose is to have growing experience through our perception of what is happening around us and why it’s happening. For which mathematics is the best tool. In short, to stimulate our senses with the use of mathematics. Mathematics cannot replace our way of learning knowledge and experience.
Mathematics, like computers, is the best and most wonderful tool ever created for learning about the real world. There is no better helper, no better tool than mathematics together with computers. But on the other hand, there is no worse master than mathematics with computers. A good servant, the best servant, but the worst master. The master was, is and should remain a human being.
See the mathematics part – combinatorics is closely linked to the number and distribution of otherwise indistinguishable points. As a substitute for indistinguishable points, we can mention electrons, we also cannot differentiate between them. More points or electrons does not mean that all possibilities could be realized. See the entropy from 2nd law of thermodynamics. There must be some initial differences in energy levels.
Whatever originated in the world can’t be divided into exactly the same parts. Whatever came into being is the original, whatever structure is unique. Unique is a whole that includes all subsets, each of which is unique. It is not possible to construct two equal segments and then connect them at exactly twice the length and it is equally impossible to split one segment into two equal halves. Splitting the continuum requires irrationality.
The equation 1 + 1 = 2 has no meaning in the real world. There are no the same shapes, structures, elements, subjects or anything else. Yes, we use this equation like model. But we must know this is only the model, our approximation.
In the same way we could pretend the ideal straight line if we see a very fine piece of polished metal surface. The same with the ideal ball if we see very pretty polished balls into gears. If we go closer then we see something like mountainous landscape – surface roughness profile that vibrates in response to the thermal movement of molecules and atoms. If we go more closer then we are able to see foggy appearance of particles as excitations of quantum field.
See also long-distance action. There has to be physical contact of the bodies. That’s why once upon a time centuries ago gravity was hard to understand, and later electrical and magnetic forces. How it is possible to act through empty space without direct physical contact of the bodies. Much later, we recognized that the contact of the bodies itself also took place at a distance — Pauli’s exclusion principle applicable to electrons in atomic shells. When we take a closer look at the contact of the bodies, we see only and only the deformation of the electromagnetic fields of atomic shells. Like approaching two magnets with the same pole facing each other.
The law of inertia for material bodies through the history
a brief recapitulation:
Aristotle – limited motion of bodies
Newton – unlimited motion of bodies
Einstein – „limited“ motion of bodies in spacetime
The history of extrapolation. Extrapolation from observations of natural processes at a given level of natural processes.
Newton’s first law: A matter body at rest remains at rest or a matter body in motion, remains in motion at constant linear velocity unless an external force acts on it.
The law of Inertia. The result of the first law of motion is an escape velocity: Every body accelerated to the escape velocity will move away until infinity.
According to physical textbooks, the Newton first law of motion is an brilliant extrapolation of our experience.
Every body remains in a state of rest or straight forward motion when it is not forced by external forces to change this state. In the real world, where there are always some forces (resistive, frictional, … , gravitational), the body will stop after some time or will stop at infinity in the case of an escape velocity. In other words, the so-called ideal case is not possible in the world. It is possible to reduce the forces resisting motion, but not to cancel them. However, even in the so-called ideal state, the definition (without external forces) is meaningless. There is an contradiction in the case of the ideal state. The misunderstanding is at the very beginning – the level of observable natural processes.
It follows from the very nature of a material body – for wherever there is a body or bodies, there are always external forces (gravity – deformation of spacetime). There is no possible state of having material bodies without external forces. The external (gravitational) forces are the results of the deformation of spacetime and they are related to every material body.
As we know today, bodies are only in the universe along with time and space. There are also forces interacting with each other body as the result of space and time – see gravitational force. For there is no space or time without matter and vice versa. There is no matter, no body without space and time.
And so the meaning of the Newton 1st law of motion in the light of space-time bound to matter (bodies) does not make sense also for so-called the ideal state. There will always be a gravitational force acting on a body, e.g. in intergalactic space, and such force always influences the body. The material body itself is an indivisible part of space and time. The material body as we perceive and measure it is an excitation of the omnipresent quantum field.
Thus the ideal state without all acceleration no forces would act on the body, is impossible. The nature of the contradiction is in the very beginning. That it is a body. If it’s not a body, that’s different. But we don’t know and can’t describe the motions of non-bodies. Where there are bodies, there is always force action and the impossibility of movement without limits. It follows necessarily from the nature of bodies. Even if there is one single body, it will necessarily gravitationally affect itself in motion.
There always be forces among bodies. Yes, in a case of an escape velocity, there is no chance to stop accellerated body, but such body will be still influenced by gravitational field of the first body and, no doubt, self-gravitational field.
There is another point of view. The body is the source of a gravity force. Such force curves the spacetime.
Bodies, by the very nature of bodies, cannot move freely through spacetime without forces.
There is an equation – the body means gravitational forces and gravitational forces mean the body or bodies. Without bodies there are no gravitational forces and vice verse. Without gravitational forces there are no bodies, either.
Let’s imagine a thought experiment – there are only two bodies in the universe, that have been accelerated to escape velocity from each other – so they will move away to infinity, as we calculated from the equations of classical mechanics.
But the reality in light of Einstein’s equations of general relativity is different – see the following
Two bodies accelerated from each other to the level of the escape velocity do not mowe away until infinity, but these two bodies will deform spacetime. The conclusion – these two bodies with escape velocity do not mowe away, but they will follow the main curve of spacetime. In an ideal state the main curve will be the main circle of an sphere. The result – two accelerated bodies will meet after a long time. The time is given by (growing) mass of two bodies and their escape velocity from each other.
This is not a negation of Newton’s law, but a specification of it by knowledge that Newton did not have in his time. In other words – every model, every theory, is valid for a certain range of natural processes, and it is impossible to establish a universal formula. Especially if we don´t know every process in Nature, plus every process is unique – there are only common characteristic – like appearance of hexagons in snowflakes.
The age-old human tempation of applying abstract models or ideas from a given level of natural processes to the whole of nature or the universe and then being surprised that nature or the universe does not work according to them.
However, models are good, proven models. Yes, they are good and repeatedly tested, but at a given level, at a given scale of natural processes. Moreover, a model is a model – i.e. a simplified, abstracted description – so it has limited validity even at the original level from which the model was created.
Go back to bodies:
The bodies have to be together in spacetime (spacetimematter). There is no chance for matter bodies to escape from each other to infinity. There is an exception if there is external force (power) above all bodies. Sometimes called dark energy.
The greater the mass of the bodies and the greater their escape velocity, the greater the deformation of spacetime and the smaller the radius of the main circle and the shorter the time for the bodies to meet. And vice versa – the smaller the mass of the bodies and the smaller their escape velocity, the smaller the deformation of spacetime and the longer the time it takes for the bodies to meet.
In a rough analogy, it is like sailors on two ships in an earthly ocean, drifting apart into “infinity” until they meet on the other side of the globe.
See below – there are articels on the subject of this section. If you want, download the pdf file.
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