Mathematics and the structure of the reality

by Marinus Jan Marijs

The applicability of mathematics to the physical world

It has been said that:

Basic researchers working in pure mathematics often develop fundamental laws, even entire branches of math, without any specific application in mind. Yet, many of these posited laws turn out—sometimes centuries later—to perfectly describe the behaviour of the real world with remarkable precision. This phenomenon was best articulated in the early 1900s by the Hungarian physicist Eugene Wigner as the “unreasonable effectiveness of mathematics.

In ”The Unreasonable Effectiveness of Mathematics in the Natural Sciences” by Eugene Wigner we find the following observations:

The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories…..

I would say that mathematics is the science of skillful operations with concepts and rules invented just for this purpose. The principal emphasis is on the invention of concepts. Mathematics would soon run out of interesting theorems if these had to be formulated in terms of the concepts which already appear in the axioms…..

The preceding discussion is intended to remind us, first, that it is not at all natural that “laws of nature” exist, much less that man is able to discover them (E. Schrodinger, in his What Is Life? (Cambridge: Cambridge University Press, 1945), p. 31, says that this second miracle may well be beyond human understanding).

THE ROLE OF MATHEMATICS IN PHYSICAL THEORIES

The statement that the laws of nature are written in the language of mathematics was properly made three hundred years ago; [It is attributed to Galileo]

It is true, of course, that physics chooses certain mathematical concepts for the formulation of the laws of nature, and surely only a fraction of all mathematical concepts is used in physics. It is true also that the concepts which were chosen were not selected arbitrarily from a listing of mathematical terms but were developed, in many if not most cases, independently by the physicist and recognized then as having been conceived before by the mathematician.

It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them. The observation which comes closest to an explanation for the mathematical concepts’ cropping up in physics which I know is Einstein’s statement that the only physical theories which we are willing to accept are the beautiful ones. It stands to argue that the concepts of mathematics, which invite the exercise of so much wit, have the quality of beauty. However, Einstein’s observation can at best explain properties of theories which we are willing to believe and has no reference to the intrinsic accuracy of the theory. We shall, therefore, turn to this latter question.

IS THE SUCCESS OF PHYSICAL THEORIES TRULY SURPRISING?

A possible explanation of the physicist’s use of mathematics to formulate his laws of nature is that he is a somewhat irresponsible person. As a result, when he finds a connection between two quantities which resembles a connection well-known from mathematics, he will jump at the conclusion that the connection is that discussed in mathematics simply because he does not know of any other similar connection. It is not the intention of the present discussion to refute the charge that the physicist is a somewhat irresponsible person. Perhaps he is. However, it is important to point out that the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena. This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language. Let us consider a few examples.

It may be useful to illustrate the alternatives by an example. We now have, in physics, two theories of great power and interest: the theory of quantum phenomena and the theory of relativity. These two theories have their roots in mutually exclusive groups of phenomena. Relativity theory applies to macroscopic bodies, such as stars. The event of coincidence, that is, in ultimate analysis of collision, is the primitive event in the theory of relativity and defines a point in space-time, or at least would define a point if the colliding particles were infinitely small. Quantum theory has its roots in the microscopic world and, from its point of view, the event of coincidence, or of collision, even if it takes place between particles of no spatial extent, is not primitive and not at all sharply isolated in space-time. The two theories operate with different mathematical concepts the four dimensional Riemann space and the infinite dimensional Hilbert space, respectively. So far, the two theories could not be united, that is, no mathematical formulation exists to which both of these theories are approximations. All physicists believe that a union of the two theories is inherently possible and that we shall find it. Nevertheless, it is possible also to imagine that no union of the two theories can be found. This example illustrates the two possibilities, of union and of conflict, mentioned before, both of which are conceivable.

Let me end on a more cheerful note. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning. (Eugene Wigner).

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How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? —Albert Einstein

The most incomprehensible thing about the universe is that it is comprehensible. —Albert Einstein

Richard Hamming:

But it is hard for me to see how simple Darwinian survival of the fittest would select for the ability to do the long chains that mathematics and science seem to require.

If you pick 4,000 years for the age of science, generally, then you get an upper bound of 200 generations. Considering the effects of evolution we are looking for via selection of small chance variations, it does not seem to me that evolution can explain more than a small part of the unreasonable effectiveness of mathematics.

Eugene Wigner:

Certainly it is hard to believe that our reasoning power was brought, by Darwin’s process of natural selection, to the perfection which it seems to possess.

And then there was the story of one the greatest mathematicians of the twentieth century Nobel prize winner Paul Dirac. Dirac, co-inventor of quantum mechanics, is now best known for his research on anti-matter. Paul Dirac was initially extremely hostile to religious views. Heisenberg recollected a conversation among young participants at the 1927 Solvay Conference about Einstein and Planck’s views on religion between Wolfgang Pauli, Heisenberg and Dirac. Dirac’s contribution was a criticism of the political purpose of religion, which was much appreciated for its lucidity by Bohr when Heisenberg reported it to him later. Among other things, Dirac said:

I cannot understand why we idle discussing religion. If we are honest—and scientists have to be—we must admit that religion is a jumble of false assertions, with no basis in reality. The very idea of God is a product of the human imagination. It is quite understandable why primitive people, who were so much more exposed to the overpowering forces of nature than we are today, should have personified these forces in fear and trembling. But nowadays, when we understand so many natural processes, we have no need for such solutions. I can’t for the life of me see how the postulate of an Almighty God helps us in any way. What I do see is that this assumption leads to such unproductive questions as why God allows so much misery and injustice, the exploitation of the poor by the rich and all the other horrors He might have prevented. If religion is still being taught, it is by no means because its ideas still convince us, but simply because some of us want to keep the lower classes quiet. Quiet people are much easier to govern than clamorous and dissatisfied ones. They are also much easier to exploit. Religion is a kind of opium that allows a nation to lull itself into wishful dreams and so forget the injustices that are being perpetrated against the people. Hence the close alliance between those two great political forces, the State and the Church. Both need the illusion that a kindly God rewards—in heaven if not on earth—all those who have not risen up against injustice, who have done their duty quietly and uncomplainingly. That is precisely why the honest assertion that God is a mere product of the human imagination is branded as the worst of all mortal sins.”

Heisenberg’s view was tolerant. Pauli, raised as a Catholic, had kept silent after some initial remarks, but when finally he was asked for his opinion, said: “Well, our friend Dirac has got a religion and its guiding principle is ‘There is no God and Paul Dirac is His prophet.'” Everybody, including Dirac, burst into laughter.

Remarks made during the Fifth Solvay International Conference (October 1927), as quoted in Physics and Beyond: Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86

In 1971, at a conference meeting, Dirac expressed his views on the existence of God. Dirac explained that the existence of God could only be justified if an improbable event were to have taken place in the past:

It could be that it is extremely difficult to start life. It might be that it is so difficult to start life that it has happened only once among all the planets. …Let us consider, just as a conjecture, that the chance life starting when we have got suitable physical conditions is 10^-100. I don’t have any logical reason for proposing this figure, I just want you to consider it as a possibility. Under those conditions…it is almost certain that life would not have started. And I feel that under those conditions it will be necessary to assume the existence of a god to start off life. I would like, therefore, to set up this connexion between the existence of a god and the physical laws: if physical laws are such that to start off life involves an excessively small chance, so that it will not be reasonable to suppose that life would have started just by blind chance, then there must be a god, and such a god would probably be showing his influence in the quantum jumps which are taking place later on. On the other hand, if life can start very easily and does not need any divine influence, then I will say that there is no god.

Later in life, Dirac’s views towards the idea of God were less hostile. As an author of an article appearing in the May 1963 edition of Scientific American, Dirac wrote:

It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better.

Dirac was puzzled by the same fact that baffled Wigner and Einstein: that the human creation of mathematics enables us to understand the way the universe works. Based upon his mathematical view on the universe Paul Dirac came to the conclusion that the way the universe was created pointed in the direction of a creative transcendent intelligent force. His biographer physicist Graham Carmelo in ”Paul Dirac and the religion of mathematical beauty” says:

The words on his gravestone are ‘Because God made it that way…’. He (Dirac) almost certainly used those words.   

The Fifth Solvay International Conference (October 1927)     

                                  From back to front and from left to right :

1. Piccard, E. Henriot, P. Ehrenfest, E. Herzen, Th. de Donder, E. Schrödinger, J.E. Verschaffelt, W. Pauli, W. Heisenberg, R.H. Fowler, L. Brillouin;
2. Debye, M. Knudsen, W.L. Bragg, H.A. Kramers, P.A.M. Dirac, A.H. Compton, L. de Broglie, M. Born, N. Bohr;
3. I. Langmuir, M. Planck, M. Curie, H.A. Lorentz, A. Einstein, P. Langevin, Ch.-E. Guye, C.T.R. Wilson, O.W. Richardson

Fifth conference participants, 1927. Institut International de Physique Solvay in Leopold Par

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The mathematical structure of the universe
Marinus Jan Marijs

The physical universe is not merely described by mathematics, it has a mathematical structure .

While not all structures that exist mathematically exist physically, (such as infinite dimensional spaces, there are numerical systems which are not realised anywhere and so on)
All structures that exist physically seem to have a mathematical structure
(But not all potentialities are actualised)

Our external physical reality is defined by the mathematical structures

The universe is exceptionally ingeniously ordered at all levels

The universe is incredibly fine-tuned, which can’t be a coincidence, there doesn’t seem to be many independently tuneable parameters, some of these parameters need to be tuned extremely precise otherwise a universe without these, properties and complexity wouldn’t  be able to generate and support life.

Mathematics has an incredible capacity to describe reality .

Mathematics is extraordinarily precise .
The description is possible from the micro structure of atoms to the entities on a very large scale. The mathematical  structure  it describes  according to Richard Feynman is comparable to the distance between New York and Los Angeles to an accuracy of less than the thickness of a human hair .
Mathematics makes it possible to know what happened all the way down to 10 to the power of minus 32 of a single second after the beginning of time (the Big Bang)

Einstein’s theory of relativity  has a precision of 10 to the power 14 (measured in relation to the emitted pulses of the signals of a pulsar, which could be timed with extreme accuracy)

Mathematics is a self-consistent system

Mathematics describes fundamental regularities in nature with an extraordinary efficiency  .

Mathematical concepts turn up in connections in  physics and often permit an unexpected accurate description of the phenomena in these connections

Mathematics play a very important role in physics and science in general ,which makes it possible to formulate the laws of nature in the language of mathematics in order for them to be then an apt object for the use of applied mathematics .

“The more deeply we probe the fundamentals of physical behaviour, the more that it is very precisely controlled by mathematics.”
― Roger Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe

Is mathematics discovered or invented?

Mathematics is discovered, long before mathematicians and physicists did found out about mathematical structures and the laws of nature, they were already in operation .
Mathematics has a reality of its own independent of the physical world .

However concepts, symbols and the language of mathematics are generally invented, but relations among them existed before their discovery.

Mathematics is the language of science

Nature itself is the physical manifestation of mathematical laws
Many purely mathematical theories were developed without the aim for any practical use, often without the intention towards describing any physical phenomena.
Mathematicians sometimes develop entire fields study with no application in mind, which often have proven decades or even centuries later to be the framework necessary to explain the structure , the processes and the workings mechanisms of the physical world

Examples:
The number theory of the British mathematician Geoffrey Hardy helped to establish cryptography . Another part of his purely theoretical work became known as the Hardy-Weinberg law in genetics and was awarded with a Nobel prize.

The famous sequence of Fibonacci is found everywhere in nature, from sunflower seeds and flower petal arrangements to the structure of a pineapple.

The non-Euclidian work of Bernhard Riemann in the 1850s, which Einstein used in his General Relativity theory

Mathematical knot theory which was first developed around 1771 to describe the geometry of position and was used in the late 20^th century to explain how DNA unravels itself during the replication process .

Scottish physicist James Clerk Maxwell’s famous equations: not only do these four expressions summarize all that was known of electromagnetism in the 1860s, they also anticipated the existence of radio waves two decades before German physicist Heinrich Hertz detected them.

These examples seem to indicate an objective reality of mathematical objects

Physics research did uncover mathematical regularities in nature of a very high level of order

Brian Greene  : “The deepest description of the universe should not require concepts whose meaning relies on human experience or interpretation. Reality transcends our existence and so shouldn’t, in any fundamental way, depend on ideas of our making.”

Consistency with our “simple universe”

The mathematical  structures can be described by  relatively small equations

Alexander Vilenkin comments that “the number of mathematical structures increases with increasing complexity, suggesting that ‘typical’ structures should be horrendously large and cumbersome. This seems to be in conflict with the beauty and simplicity of the theories describing our world”.
All the fundamental equations of theoretical physics can be written down on one A4 paper .

The laws of nature

The laws of nature can be expressed as mathematical equations which describe physical phenomena to an astonishing degree of accuracy

The laws of nature are mathematical relationships
The laws of nature are absolute, universal, timeless, immutable independent of the states of the world
Universal laws appear to govern our cosmos: a photon 12 billion light-years away behaves just like an photon  on Earth; light in the distant past and light today share the same traits; and the same gravitational forces that shaped the universe’s initial structures hold sway over present-day galaxies. (Quote Vilenkin ).

The laws of nature transcend space and time .

Abstract objects

It is generally presumed that abstract objects have no causal power, for example the number 7 is merely descriptive and has no causal power.
However while the number 7 is an abstract object, abstract objects according to Plato have causal power, which is called platonic instigation, which is a form of active information, something like the concept of a pilot wave by theoretical physicist David Bohm .

George F R Ellis: ”The laws of nature are not descriptive but prescriptive”

There seems to be a tendency towards organisation into the cosmos, galaxies, stars and biology which have a profound order that we wouldn’t expect to emerge if there wasn’t some organising principle behind it.  

A theological  approach:

THAT God has fashioned the physical world on the structure of the mathematical objects that he has chosen, is essentially the view Plato defended in his dialogue  the Timaeus .
God has fashioned the physical world on the mental plan that he had in mind, so that the physical world has the structure that God has conceived prior to creation . This is the view defended by the Jewish philosopher Philo of Alexandria  .

A philosophical  approach :
Mathematical theories apply to the physical world because the structure of the physical world is an instigation of mathematical structures described by those theories.
The physical world exhibits this complex mathematical structure  .
This seems to be some part of a teleological force / aim .

Questions :
The question remains: is every single number a platonic object, or are integers as a group a singular  platonic object, or both?
Single digits or one group
Object or relationship
Object or process
Numerical units or the operation of subtraction or addition
(Integer, whole-valued positive or negative number or 0. … The integers are generated from the set of counting numbers 1, 2, 3,…)

On a higher level different forms of platonic instigation must be guided by teleological principles otherwise the fine-tuning wouldn’t be there .

Mathematical principles underlie, generate the physical world .

Mathematical principles actually exist in some abstract realm.

Why and how     
   
Why does something abstract as mathematics describe reality as we  understand it?

What gives mathematics its explanatory accurate and predictive powers? .

Most of the mathematicians and surely most of the physicists say that mathematical principles are at the foundation of reality, which exists in a platonic world .

The platonic world is the origin of the laws of physics

Where do the laws of nature come from? Theoretical Physicist  Paul Davis :
https://www.youtube.com/watch?v=pj7POKgkJTs 
The universe is ordered in a rational and intelligible way.    35.55

Platonic instigation

Related to platonic instigation:
Undirected processes are not feasible as an explanation, and there is evidence for platonic instigation such as the cumulative data of the problem of the origin of life and of the first replicator which is required to get this whole  biological process started  in the first place.
Further there is the combinational search space problem, to get new protein folds with its rarity of islands of function in the search space, the digital code in DNA with its information problem, where you find patterns like redundancy and distributed information, data compression and error correction and further irreducible complexity
This is just on the biological realm.     

Transcendent mathematical structures influence physical existence by platonic instigation .

The different forms of platonic instigation are engineering physical structures into existence, generating formative principles by non-local resonance ..

According to the Pythagoreans of 5th century Greece, mathematical entities like numbers were universal principles which  were active agents in nature .

Stephen Hawking: “What is it that breathes fire into the equations that describes this universe and gives them a universe to exist?”

To understand platonic instigation, one needs a different conceptual framework that will be unfamiliar from daily life.
A more or less similar situation occurs in quantum theory with the collapse of an abstract, non-localised, non-material wavefunction that materialises on a specific location
The wavefunction according to quantum theory, is abstract and non-localised in such a way that it can be as wide as the whole universe.

Equations are relationships between different elements .

Teleology
Teleology:  directed towards an end goal .
This teleological process “selects” a particular set of equations, and actualises a possibility . This implies a selecting, a guiding principle .

Why are there universal laws of nature at all?
There is an underlying purpose to the universe.

The guiding principles behind nature who structure the physical world by platonic instigation, are themselves structured by teleological processes, who guide the transformation of the cosmos to the goal that the different ontological levels can interact and the ultimate goal that the highest ontological level can interact with the physical world .

An example :
Nucleosynthesis
“In 1946 Fred Hoyle published a paper, on the creation of elements and the synthesis of elements from hydrogen, Hoyle introduced (or at least formalized) the concept of nucleosynthesis in stars, building on earlier work in the 1930’s by Hans Bethe. Stellar nucleosynthesis is the process of nuclear reactions taking place in stars to build the nuclei of the heavier elements, which are then incorporated in other stars and planets when that star “dies”, so that the new stars formed now start off with these heavier elements, and even heavier elements can then be formed from them, and so on.

Hoyle also theorized that other rarer elements could be explained by supernovas, the giant explosions which occasionally occur throughout the universe, whose immensely high temperatures and pressures would be sufficient to create such elements. Remarkably, he had found a way of testing the theory of star formation in the laboratory, and was able to prove his earlier prediction that carbon could be made from three helium nuclei without an intervening beryllium stage. Although his co-worker William Fowler eventually won the Nobel Prize in Physics in 1983 for his contributions to this work, for some reason Hoyle’s original contribution was never recognized.

As part of this work, Hoyle invoked the so-called Anthropic Principle to make the remarkable prediction, based on the prevalence on Earth of carbon-based lifeforms, that there must be an undiscovered resonance in the carbon-12 nucleus which facilitates its synthesis within stars. He calculated the energy of this undiscovered resonance to be 7.6 million electron-volts, and when Fowler’s research group eventually found this resonance, its measured energy was remarkably close to Hoyle’s prediction.

It was also this work that caused Hoyle, an atheist until that time, to begin to believe in the guiding hand of a god (what would later be called “intelligent design” or “fine tuning”), when he considered the statistical improbability of the large amount of carbon in the universe, carbon which makes possible carbon-based lifeforms such as humans.’
https://www.physicsoftheuniverse.com/scientists_hoyle.html

Biological life is essential for a fundamental part of a cosmic transformation.
This transformation takes place when higher ontological energies, level by level are successively activated, bottom up, until energies are activated on the highest ontological level.
This will activate feedbackloops  ( see further )
https://marinusjanmarijs.nl/meaning-of-life/kosmic-omegapoint/

top down from the highest level to the lowest level which will activate and transform all ontological levels.
But as these higher ontological energies are non-physical, what is the function of the physical world in this all?
When higher ontological energies are connected with biological living systems, they develop different capacities which they wouldn’t have developed without the connections . Like the capacity to activate to the two highest ontological levels and which collectively generates a feedback loop .
 
To achieve this, the physical world must be highly ordered, this ordering includes the fundamental regularities, the laws of nature,  mathematical relationships, the fine-tuning of the constants of nature with an extraordinary precision

Hierarchy of platonic forms
Outside space and time
(The realm of pure consciousness)

Platonic  forms         
12  Ultimate cause of being
Monads     Creation     Pure consciousness     Absolute

11  Involutionary   principles
    Particles         Relations       Structures       Field

10 Trans-temporal   principles
Guided          Attractors     Retro-causal    Teleological                                potentialities     patterns      Resonance         principles
 
9 Archetypical   principles
Model         Structuring        Resonance       Field

8  Ontological   principles
    patterns      structuring      stratification      Fractal
                                                                       structures
7  Integrated  principles
    Patterns        structures        Instigation         Field

6  Lateral   complementary principles
 potentialities    structures      differentiation      Field

5  universal order,   rational principles,    logical 
      unit       structure       superstructures       truths
   essences  

4   symmetric   superstructures
Singular       multiple      interacting       Field

3 Abstract  ordering representational principles
  Elements      connections      functions      categories

2 Abstract entities that are related to the qualia.
prototypes     differentiation    correlations     5 main groups
of Qualia             

1 Organising principles of the physical world.
Integers       Sets       Laws of nature       Hilbert space

A Elements     B Relations      C Laws      D Total field
                           by  Marinus Jan Marijs