Many people are fascinated by science through books, comics, TV series and science fiction films. Among these little gems of fantasy and curiosity, we find a title that has shaken the hearts of many: the film ‘The Matrix’ and its trilogy.

The Matrix is a film that deliberately questions everything around us, leading the viewer to question – and take on what Descartes called ‘Hyperbolic Doubt’ – the relationship we have with others and the world in general, right down to the relationship we have with ourselves.

In this article, we will go to the basis of The Matrix, describing not only the world that is depicted – real and virtual – but also the fundamental, mathematical concept that underpins the entire film series.

The masterpiece written and directed by Sisters Wachowski

It is a cult, a masterpiece, so profound and solid that it is still discussed and elaborated upon today by old and new fans alike.

The opening is so simple that it strikes the viewer in-depth and captures him within the first few minutes of the film with a simple question: what if everything around us was not real?

The plot recounts the vicissitudes of Thomas Anderson, a hacker with the pseudonym ‘Neo’, whose actions attract the attention of the infamous Agents of the Matrix, who are ready to intervene and quell any anomalies that arise in the ‘perfect’ virtual world.

Neo gets too close to ‘reality’ and ends up discovering that his life up to that point is fiction, a virtual projection of a world that no longer exists. Nevertheless, he embraces the truth and becomes ‘aware’ of the stark reality of which the Matrix is only a part.

Awakening, Destiny, Search of identity. Who are we, and why are we here

The first theme is definitely ‘awakening’, an awareness that makes us aware of what is real and what is not. We can superimpose this theme on the study of nature through science, the discernment between what is real and true from what is not.

Another topic addressed is destiny, understood as a designated and desired role. We talk about the role we create -and that we want to play- which can strongly conflict with the role we have been designated -thus imposed.

The search for identity is the most articulated theme. The classic hero’s journey is taken to extremes and enhanced to the extreme, and the moral choices in the ongoing search for the answer to the question “who are we?” are extremely strong. The change we observe in the protagonist and the characters is clear and defined.

The theme that drives the narrative is Humanity vs. Machines -a detailed version of the generic Man vs. Environment-. From the outset, the conflict that spawned the Matrix is clearly described and defined, along with the role that mankind plays alongside the machines in the plot. The worldbuilding is solid and studied.

The article shows how a virtual space that can be visited and interacted with originates from the concept of the Mathematical Matrix.

Answers that are difficult to accept: our minds are used to seeking meanings and purposes

Entering into a topic that, although it might have a solid scientific basis, is a purely philosophical one, the answer is indeed very peculiar. We could, as it were, live in a simulation. Unfortunately, however, there is no way to answer such a question.

Meanwhile, we could be fooled by the word ‘simulation’. From the earliest times, we wondered about it, but instead of simulation, we used the word ‘dream’.

Today, we consider the idea because we have achieved truly impressive computing power, and we, therefore, wonder if someone has beaten us to it. From a scientific point of view, there is no way to prove this. From a philosophical point of view, we can do some more reasoning, such as using the ‘Occam’s Razor’ technique.

Can the simplest answer help us?

Who knows.

When one ponders the question of the universe’s origin (which science has been able to describe since a certain point in time), one comes to the question of the cause of birth, the cause that started everything. This also embraces the question of consciousness, of which we have first-hand experience (Descartes again said, Cogito ergo sum: I think, therefore I am).

We can say that these two problems are extremely interlinked and have a very strong limitation in our understanding and interpretation, but that they are topics that cannot, at present, be addressed by science. Science is about description and observation.

Science can help narrow the field, but the contribution is only marginalising the problem.

Science cannot give answers that humans seek from an intimate point of view: why are we in the world? Science cannot provide a meaningful answer on this ground. Science can answer with survival and reproduction. Answers that are difficult to accept: our minds are used to seeking meaning and purpose in everything we do and live.

Nevertheless, the question of the plausibility of a simulation of reality has been analysed by various thinkers-scientists and philosophers.

Already in antiquity, many asked the same question, starting with the philosopher Chuang zi or Descartes. Not only them, but also others thought that our experiences were subjective, therefore fallible and possibly linked to some illusory mechanism, as in the film The Matrix.

Nick Bostrom dealt with this question in a rather interesting way “Are you Living in a Computer Simulation?”. Here’s the link.

According to Bostrom, there are three possibilities, and only one can be true:

- No civilisation will ever be able to reach the necessary technological level
- If they do, they decide not to
- We are in a simulation created by someone else.

According to Bostrom, if we can create a simulation, then we will definitely be ourselves in a simulation.

Many scientists have studied this, trying to debate the plausibility of the question with other scientists and philosophers.

As we have already said, the problem cannot be investigated from a scientific point of view. Only philosophy can deal with it.

We have no concrete reason to think that the hypothesis of being or not being part of a simulation is correct or not. A few scientists, however, have tried to come up with some ideas, such as by analysing breaks and anomalies in natural laws.

This hypothesis is not falsifiable. We cannot prove that it is false or true, so it is not possible in any way to treat, confirm or disprove it.

But what if we had more information on the concept of consciousness?

In that case, again, science could contribute by marginalising certain issues, and nothing else. Could an experiment in a simulation really give useful data? Could mathematics and logical reasoning lead us to absurdity?

These are fascinating questions, no doubt about it, but if it were true that we live in a simulation, then would we have finally justified our existence? Or what we think, desire, and want is planned?

How might this fit in with the non-deterministic nature of reality? To date, it is unthinkable to support or not support simulation theory. Still, we can inform ourselves on some related topics such as the nature of consciousness, the inability to generate random numbers and the loop problem. In fact, even assuming we live in a simulation, we would still be unable to answer the fundamental question of life, the universe and everything else.

Matrix comes from Latin and its meaning is ‘Pregnant animal’ or ‘Womb’

The film ‘The Matrix’ has a large lap structure and humans connected to energy converters. They are mentally connected to the Matrix software and live in virtual reality. A simulation.

We discuss together possible connections between the film and Mathematics, investigating some interesting and fascinating deeper meanings.

Could we read the future if we are good at Statistics?

Mathematically, matrices are rows and columns of numbers. They can be added to each other and sometimes multiplied. You may not think you have ever had to deal with a matrix, but you have not: a computer monitor, for example, is actually operated by a matrix.

The trilogy uses the matrix to address the ‘free will and predestination’ debate. A mind, human or otherwise, living in the reality of the Matrix can come to understand the future, the algorithm of the oracle.

The future is like a set of events with their own probability of occurrence, the realisation of a specific event being the consequence of a finite (but also possibly numerous) number of choices. It is a matter of calculating probability and predicting the most probable events.

But it goes even further. If one thinks of an event as defining the current ‘state’ of the future, the possibility of moving to a different state can vary depending on one’s initial state. In statistics, we study an object called a ‘Markov chain’, a process often introduced as a drunkard’s step. In other words, a Markov chain is a system in which the state of the system uniquely determines the probability of a state i at time k at time k – 1 and not by its history. The drunkard, like the process, has no memory, and his next step will depend only on the one he has just taken. Markov chains, therefore, describe a particular stochastic process suitable for modelling systems that have random behaviour in their evolution; they describe random phenomena that evolve as a function of time and have no memory of previous states.

Let’s put it more poetically:*a process is Markov if,** knowing (exactly) the present,** past and future are independent.*

Markov chains are fascinating because they relate Matrices and Probability Theory to computational operations.

Thus, a matrix in a Markov process helps us to identify the probability of certain future events, although it does not tell us with certainty which chain of states we will pass through or which state we will be in after a certain time t.

In The Matrix, this is shown in the last scenes of The Matrix Reloaded, in a dialogue between Neo and the Architect, the algorithm that strives for the system to be as stationary as possible. The TV screens show the protagonist’s possible choices, but as he decides, they synchronise.

Synchronisation represents the ‘ever-increasing probability’ that Neo will do just that particular action.

An introduction of Matrices, Linear algebra and Equation systems

Matrices are mathematical objects that have a wide application in the most diverse fields of science. They may have different meanings and applications, but the matrix theory remains the same.

From an abstract and purely mathematical point of view, matrices are useful tools for finding solutions and operations between them.

In the most general sense, a matrix is a set of elements with a dual alignment. They have m rows and n columns. Matrix A(m,n) is a matrix of dimension mxn.

When m and n are equal, the matrix is said to be square.

This is a 4×4 matrix because it has 4 rows and 4 columns.

This is a generic m×n matrix:

Each matrix element has an index ij, ranging from 1 to the generic element n or m. What does this mean? The index i indicates the row, while j indicates the column. a23 indicates the element in the second row and third column.

In linear algebra, matrices are connected to the solution of linear systems. One of the main problems in mathematics has always been determining the solutions of an equation: are there values that, when substituted for the unknown, make the equality true?

Let’s take a simple case:

2x = 4.

What is that value substituting for the variable x and multiplied by two gives us 4? The value is 2. In fact, 2*2=4.

This is a simple case of an equation, called a linear equation of the first degree, to one unknown. A one-row matrix (2 4) where the first row represents x and the second the known term. Dividing the matrix by two, we obtain (1 2) i.e. x=2. Simple, isn’t it?

When we have several linear equations, and we have to find consistent solutions to all the equations, then we are studying systems of linear equations.

A convenient way to solve a linear system is using the ‘Matrix’.

A linear equation in two unknowns represents a line in the Cartesian plane.

Let us take an example in two unknowns, x, and y.

Putting two linear equations in two unknowns into a system means asking what the solution for which the lines intersect, i.e. have a point in common, is.

The matrix representing this system is

The first column of the matrix is the x-field, the second the y-field, and the third the known term. Subtracting row two from row one we obtain (0 -4 1). This is a very easy way to solve systems of linear equations, don’t you think?

As we can see, in this hypothesis, we have three possible cases:

No solution: the lines are parallel

Only one solution: the lines are incident

Infinite solutions, overlapping lines

In our case, we have a solution: the point is y= -1/4

In Linear Algebra, Matrices are used to compactly write and work with multiple linear equations (a system of linear equations); they are very useful for finding solutions, doing linear transformations, and more.

We might study and discuss linear algebra in a more suitable place. They are very useful and fascinating.

Algorithms and Linear algebra:Neo is a non solvable polynomial equation

The choices of many elements of the Matrix are defined by Markov chains (simplicity assumption). We have an algorithm/transformation matrix that wants to reset the state of the Matrix to restore order (and not only that).

The inhabitants of the Matrix are solvable polynomial equations, whereas Neo is a non-solvable and, therefore, independent equation.

Agent Smith can copy himself into the minds of the inhabitants of the matrix, acting as a matrix operator capable of manipulating the internal elements so as to make all values equal to himself (and according to what the oracle says, the probability of him being able to clone himself into the entire matrix after a certain time is 100%).

Finally, many other aspects of the simulated world are handled by Linear Algebra Matrices. The film’s title is not necessarily directly related to linear algebra. Still, looking at the sprinkling of code in the opening titles or the trailer, it is obviously related to computer matrices, which are very similar to mathematical matrices.

It is clear that the film has many facets and deals with deep and compelling themes. Without the title ‘The Matrix’, it would probably not have had the same effect of astonishment, fascination and curiosity.

Buchanan, Mark. Nexus: Small Worlds and the Groundbreaking Science of Networks. New York: Norton, 2002.

Watts, Duncan J. Small Worlds: The Dynamics of Networks between Order and Randomness. Princeton: Princeton University Press, 1999.

Kris H. Green – What’s in a Name? The Matrix as an Introduction to Mathematics

Amedeo Balbi – Do we live in a simulated reality?

The post Math behind Matrix: do we live in a simulation? appeared first on Murasaki Hiroshi.

]]>Isaac Newton is acknowledged as one of the greatest scientists of all time. But what precisely is it that lies behind such greatness as to make us consider these heroes as almost superhuman as if they were born with physical and intellectual privileges, kissed by the skies, and predestined for grandeur?

Actually, there’s nothing that special. No incredible luck, no predestination. Allow me to tell you why.

Sadly, behind such courageous and determined people, there is often a painful background, studded with many difficulties. Difficult lives, family dramas, wars, or an environment that was unwilling to stimulate and appreciate the talents of someone with a tendency to think outside the box.

In order to understand what lies behind Isaac’s qualities and how we are connected to him, we will not discuss his biography – something that was done thoroughly and in great detail in many books – but we will talk about Isaac’s human side, trying to understand which were the mechanisms that led this young boy who grew up in an English village with an illiterate family, into becoming a member of the Royal Society and being remembered as one of the greatest scientists of all time, despite being considered the fool of the class at school.

He would feel different, alien and excluded for the rest of his life

He was born frail, and immediately orphan of a father who died when his mother Hanna was six months pregnant, during a battle he fought for Charles I against the MPs. He came into our world on a Christmas day, in the sadness of a loss he could not yet realize, in a village where people felt undecided on whether to believe that any fatherless newborn children would be born with supernatural powers, or doomed to die within their first month alive.

Isaac managed to survive, but he was soon abandoned by his mother, who made the forced choice of remarrying a much older man, in order to guarantee herself and her son an inheritance and some financial well-being.

In order to do this, he left his son in the care of his grandmother, who soon sent him to study with friends of the family. It was with them that Isaac began his new life as an apathetic student, marginalized by his fellow classmates and deemed as impatient and unintelligent by his teachers. From then on, he would feel different, alien, and excluded for the rest of his life.

Isaac began to feel a growing desire for retaliation and revenge

Isaac suffered the same anathema.

Considered strange due to his curiosity towards nature, a place where the boy loved to immerse himself in order to escape from the misfortunes of his life, Isaac began to isolate himself from others around the age of thirteen. He took comfort in his own world of thoughts and fantasies.

He was not very attentive at school, he usually lurked at the back desks, did not listen to lessons, and did not commit at all. His teachers considered him to be an inadequate and intolerant boy. Friends simply called him “stupid”. He loved to build small objects, which he then gave to his female peers, receiving harsh judgments from other boys for this. Precisely because of these attitudes, he was even targeted by a bully, among other things the son of the man who housed him. He was forced to fight him, and by a stroke of luck, he managed to subdue his opponent.

From that moment, Isaac began to feel a growing desire for retaliation and revenge. Realizing that the path of violence would not lead him far, he became convinced that he had to beat them on another level. He then began to study a way to ingratiate himself with the teachers and achieve an important role in his class, so that his classmates no longer had elements to discredit him.

The pre-adolescent Isaac Newton began to approach studies with passion, driven by a sense of revenge towards his companions, and not only: he did do it to face his three greatest enemies: the sense of inadequacy, inferiority, and loneliness.

He usually felt alone, misunderstood, attacked, and hated by those he would have liked to have close

Things gradually changed, but Isaac was never able to genuinely enjoy his position at the top of the class. Rather, he continued to isolate himself more and more, as he was afflicted by a very strong social anxiety.

His curiosity led him to ask unusual questions for a boy or a man not familiar with philosophy, and this convinced him that there was no place for him in that world. That he was doomed to loneliness and feeling marginalized.

One day in his home library, Isaac found answers to some of his questions: Why do objects fall? Why do spinning kids have their heads tilted back?

Why does the rainbow always have the same colors? But above all, he realized he was not alone in the world, that he wasn’t the only one having thoughts and curiosities which his companions considered to be quite bizarre.

Isaac found a family of his own in philosophers, thinkers, and scientists, in people similar to him, finally finding his place and his passion: natural philosophy, philosophical thinking applied to the study of nature, a very close sister of physics, which is the study of nature itself.

Thanks to these figures his sadness lessened, but not his social anxiety.

Once he discovered his passion, Isaac became practically unstoppable: he always gave it all in the study and application of the arts he loved so much and soon achieved the recognition he desired, as well as the love of teachers and family friends alike, who now considered him to be the pride of his village. Unfortunately, with his inability to overcome his anxiety, Isaac would soon fall into the trap of his loneliness. He usually felt alone, misunderstood, attacked, and hated by those he would have liked to have close.

Isaac’s mother was so kind to her son as to deny him even the bare minimum financial support he needed to study

The difficult relationship with his mother was a heavy burden for Isaac, who always failed to feel part of a family even as an adult, and ultimately returned to isolate himself in his thoughts, taking shelter in his inner world. However, Isaac’s mother would soon understand her son’s inadequacy in managing their estate, so she sent him back where he came from, back to his studies, much to Isaac’s delight.

Isaac’s mother was so kind to her son as to deny him even the bare minimum financial support he needed, reducing him to a state of poverty. In order to study, Isaac worked in the service of his own luckier peers, cleaning chamber pots and styling the hair of fellow students who were definitely not intellectually superior to him. The boy’s anger allowed him to continue his studies, but without achieving great results. Luckily for him, the university understood the boy’s unease and offered him a scholarship for completing the course.

Even though the war was over and England had decided to return to the monarchy with Charles II, the times that followed were not easy. The arrival of a pandemic forced Isaac to leave the College again and reunite with his hated mother.

During this pandemic, Isaac closed himself to his studies by excluding himself from forced family life, reaching his most important scientific results.

At first glance we could say that Isaac could finally live in an environment favorable to him, surrounded by fellow scientists. It was not like that at all

It is not unusual that the scientific academic world tends to be particularly hostile to speculation and rigid towards new ideas which are not adequately substantiated. Isaac saw in the Royal Society a veritable academic jungle, and in the figure of Hooke, the fiercest of predators. That young boy had entered the Royal Society talking about the splitting of a light beam, and he was rather excited by his discovery.

He understood that the white light beam was not pure – as other scientists intended – but rather the union of all colored beams, which on the contrary were pure and immutable. It so happened that years earlier Hooke had published a book that explained – and intended to prove – the exact opposite, and this put young Isaac in the jaws of a predator, who would then induce him into leaving the Royal Society shortly after and avoid talking about any of his researches or discoveries, almost for his entire life.

Social anxiety, discouragement, and a sense of exclusion had won once again.

The celestial kingdom before Isaak

2,300 years ago, in ancient Greece, Plato and then Aristotle intended to investigate the divine, define it, describe it and then discuss its implications for the earthly world.

Plato wanted to understand the scientific origin of the gods and advised to use astronomy, which was still in its infancy at the time.

For the first time, human beings were daring to try and understand divine behavior (that of celestial bodies), starting a scientific revolution that would reach its climax with Isaac Newton. For the first time, science and religion cooperated, and this would have been a very happy and profitable marriage for both spouses, although the divorce will not be one of the best.

Let’s go in order. Plato and then Aristotle go on to describe the behavior of the celestial world.

The sun and the stars revolve around the earth in a perfect and eternal way. The moon shares this behavior.

The celestial world, located beyond the moon, is an immutable, eternal, and perfect world. It was extremely regular: they would have called it the cosmos, which in Greek means “ordered”.

However, there were five planets (Planet means wanderer, in Greek) that which not follow circular movements and had a disconcerting behavior. Through some forcing and an imaginary ethereal wind that pushed planets, they too were eventually shown in a perspective of celestial perfection.

The celestial world was composed of ether, an incorruptible and eternal element, while the terrestrial one was composed of four elements: fire, air, earth, and water, which were corruptible and explained the behavior of terrestrial things. Everything was contained and moved by spheres, which were all enclosed by the outermost sphere, the Primum Mobile, the primary mover or God.

This meant that the celestial world was immutable and perfect, while the terrestrial one was changeable and corruptible. However, when asked to explain the behavior of comets, Aristotle said that they were not astral or perfect, but terrestrial exhalations that caught fire at high altitudes. He had placed them between the earth and the moon.

Now that science had provided a flattering explanation of the divine, religions began to use it in order to assert themselves. In particular, St. Thomas Aquinas reconciled Christianity with Aristotle’s geocentric universe. God could only be the primum mobile and the planets, stripped of their role as demigods, would have moved on spheres drawn by Angels.

But following the worst European plagues and epidemics, it happened that the church lost many of its most valuable men, and for this reason, those vacancies were filled by men with less virtuous intentions. This unleashed a series of shameful scandals that put the unity of the church to the test. Martin Luther and Copernicus would have pointed the finger at Catholicism: Protestants were born and divine geocentrism was questioned. The church responded harshly, as in 1572, a new star appeared in the sky (despite the sky having to be incorruptible, eternal, and perfect), and five years later a comet reappeared which through parallax was measured to be farther than the moon (another imperfect element was living in the celestial world?!).

The answer to all this was inquisition against the heretics: Giordano Bruno was burned alive, and many scientists began to prefer their own security to the dissemination of their ideas, although they were genuinely interested in reconciling the new Copernican science with the divine.

Kepler (a Copernican and Lutheran, so he was basically public enemy number one) was an astronomer who managed to discover three interesting things about wandering planets.

– The square of a planetary year is a multiple of the planet’s cube distance from the sun, and this applies to all planets. More distant planets have long years, while closer ones have short years.

– Planets did not have a constant speed, but they accelerated and slowed down.

– Orbits were not circles, but ellipses.

Furthermore, Kepler sensed that the planets were being attracted to the sun, almost by some kind of magnetic force. Heavenly perfection was seriously endangered.

Galilei built a rudimentary telescope and observed the Moon and its imperfections (the craters). Due to this, he was found guilty by a court of cardinals and forced to deny those discoveries, retracting them.

The Holy Church’s reaction pushed science away and led to a sharp and painful separation: that special bond would never be mended again. The dispute would never die out, and God would be banned from scientific discourse in the future.

The death of the celestial kingdom and the birth of physics

The death of the celestial kingdom coincides with the death of Isaac’s mother, who had returned to Woolsthorpe to watch over her. In those moments of delicacy and emotion, the young man regretted many of his choices and cursed his stubbornness.

During one of these melancholic evenings, Isaac was deeply immersed in his thoughts as he was spending time in his garden when he realized he had a beautiful moon in front of his eyes. The same one he had seen fourteen years earlier and that had prompted him to ask himself many questions about celestial bodies and centrifugal force.

During one of these melancholic evenings, Isaac was deeply immersed in his thoughts as he was spending time in his garden when he realized he had a beautiful moon in front of his eyes. The same one he had seen fourteen years earlier and that had prompted him to ask himself many questions about celestial bodies and centrifugal force.

His doubts were these: why does an apple placed on a very tall tree fall to the ground, despite the moon not doing the same? The moon doesn’t fall due to centrifugal force, which opposes the gravitational force exerted by the earth.

Isaac imagined being a child with a rope tied around his torso. The other end of the rope would be tied to a pole or a tree. Running forward, this child would have made a circular journey, thanks to the tensioned rope. This rope is the gravitational force, the child is the moon and the pole is the earth.

Not enough, the moon is in perpetual and rectilinear motion, which becomes circular once again thanks to the centripetal force. As we already said, the centripetal force exerted by the rope attracts the child towards the pole, but its rectilinear motion cancels this effect: in the end, it is only the direction that changes, moment by moment.

But what if there was an elephant in place of the baby?

The tree, or the pole, would have been uprooted: so it all depends on the mass of the two elements involved. An elephant could not uproot a tower, but he would with a pole or a tree still young and thin: it’s almost like child’s play.

Certainly, the rope had a role: length seemed to matter. Earlier on, Kepler had discovered that a planet revolves around the sun thanks to a centripetal force.

Not only that, he had discovered that a planetary year squared was worth the cube of its distance multiplied by a constant. Isaac quickly realized that all these things were connected.

Isaac translated his thought into mathematics: a person of mass m attached to a pole with a rope of length d completes one complete revolution in time T.

This is how Isaac Newton went from the rough idea of Eq.1 to his final equation, basically using Kepler’s definition of the Yearly Period of a planet around the sun, and then generalizing it to all kinds of masses. To properly understand Hint 1, read about Vectors. Please keep in mind that Hints 1 and 2 are not about physics, but Linear Algebra.

**(Eq.1)**\overrightarrow{F}= \frac{k_{1}m\overrightarrow{d}}{T^{2}}

But Kepler had found that for planets:

**(Eq.2)** T^{2}= k_{2}d^{3}

Merging them together (a constitution of T in the first equation) Isaac Newton found this new equation:

**(Eq.3)**\overrightarrow{F}= \frac{ k_{1}m\overrightarrow{d}}{ k_{2}d^{3}}

We should remember that

**(Hint 1)**\overrightarrow{d} = ||d||\widehat{d}

so we can divide the magnitude/intensity of the vector from its directional part, that would take us to our equation.

**(Hint 2)**\frac{\overrightarrow{d}}{d^{3}} = \frac{\widehat{d}}{d^{2}}

and \frac{k_{1}}{k_{2}} = k_{3} = k

We can now compose the equation by merging the elements in Eq.3 to get a premature form of Gravitational Equation (as you will see, we only have one mass m):

**(Eq.4)**\overrightarrow{F}= \frac{km}{d^{2}}\widehat{d}

Isaac quickly understood that something was missing: an object falls perpendicularly to the earth, which is spherical. Considering two masses as points, and not thinking about their size, the idea of Gravitational attraction quickly took the scientist to imagine a reciprocal force. That made a lot of sense if we analyze it on a small scale. So now the mass m becomes:

m =m_{1} m_{2}

and k = G, it’s value is constant and equals to:

G \approx 6.67428 \times 10^{-11} m^{3} kg^{-1} s^{-2}

Isaac Newton is an example to follow: however difficult life is, if we keep working, we will get what we want

What we need to understand and appreciate about Isaac Newton is the fact that in the end, he is not at all different from any of us. He was not blessed with luck, nor was he considered a child prodigy or a role model for the family.

Far from it.

Isaac was bullied by his friends, pushed aside by his mother, he never had a real example to follow in terms of a motherly or fatherly figure. He was lonely, oppressed by his fervent curiosity, debilitated by social anxiety, and disarmed by the terror of the pandemic.

It is in human nature (and it also happens with many animals) to judge what is different, fear it, shy away from it or fight it. We all underwent the effects of this evolutionary “injustice”, perhaps even once in our lives.

Despite concepts such as those of gratitude, sharing and collaborations proving to be great evolutionary adaptations for the survival of our species and solidly backing the roots of our own society, everything that falls out of the scope of our habits is instinctively considered as potentially dangerous. All these scientists, artists, thinkers but also innovative, curious or extremely sensible or empathetic persons are well aware of that.

Isaac Newton is an example to follow, as he shows us that however difficult life is and no matter how many misfortunes seem to be magnetically attracted to us, if we’re driven by the right amount of desire for redemption and armed with plenty of patience, not only will we get what we want, but also make an important contribution to the entire world community.

It’s a given that Isaac did not discover the theory of universal gravity only to make his mother happy or for helping the English community, really. He did it for himself and for love of knowledge, in the same way that an artist doesn’t create for receiving applause or for selling his work, but only for the love and sake of art and for the joy of the process.

If we keep on moving nonchalant of any adversities, we too could be Isaac Newton.

Five Equations That Changed the World – Michael Guillen | Amazon Link

Never at Rest – Richard S. Westfall | Amazon Link

On the Heavens – Aristotle | Amazon Link

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]]>There are artistically accurate figures in the panorama of modern physics in the purest sense of art: to communicate a genuine, vivid concept with sincerity and clarity. In the most harmonious way possible, they reach the viewer’s heart to initiate a process of catharsis or pure contemplative admiration.

Albert Einstein was a creative mind outside of any scheme, who sought the truth in the most disparate streets, driven by pure intuition and vocation. A passionate artist, surrounded by the vivid images in his mind that took shape like spots on the canvas.

A Klimt, at times impressionist, with a quick and decisive stroke. For him, the light was the basis of everything together with freedom: if we could describe Einstein’s essence in a painting, we could draw Klimt’s Kiss as the basis for a black hole, around which light turns, trapped. Everything around is distorted as if the entire cosmos were a waterfall converging towards the two.

If for Pascal, man’s nobility comes from passion and ambition, for Einstein, it’s light and freedom for Einstein. From the free and vibrant light, almost like a piece for Mozart’s violin, one slowly reaches Dirac’s pure beauty in this sensory gallery.

According to the Physicist, a beauty illuminates the way by giving a trace on the path to follow: combining quantum mechanics and special relativity to obtain an equation of motion that does not lose its validity at very high speeds that of light.

What does the world “Beauty” means in Physics?

For the painting to have a good foundation, it was essential to solve a big problem, considered by Heisenberg, one of the worst in modern physics: the theory admitted solutions to negative energies, which Dirac considered devoid of physical sense. But the discontinuous transactions from positive to negative energy levels were present in quantum mechanics, and he could not turn away.

Dirac went against the current and identified this “Sea” made of negative energy levels as a set of holes and unobservable electrons. The mathematician Weyl helped the community to think about Dirac symmetry and the fact that these holes could not be considered protons, due to an unequal mass, but anti-electrons (or positrons). Then the antiproton and the idea of a magnetic monopole would explain the quantization of the elementary charge.

This experience transformed Dirac from a cautious theorist to a revolutionary promoter of mathematical beauty: this would have made him the Trotsky of physics, the world’s strangest genius. A brave adventurer and a severe art critic, who mercilessly judged the ugliness of their physics appearance.

Among these, “renormalization” was a lousy job in his eyes, devoid of aesthetic and even ethical bases—something that put quantum field theory in bad conditions, which would even need to be refunded.

When the renormalization proved not only successful but even in perfect agreement with the experimental data, Dirac’s answer was, «It could be correct if it weren’t so bad.»

We must ask ourselves: «What does the word “beauty” mean in physics? »

The Mathematician Hardy, a contemporary of Dirac, said:

«The models of a mathematician, like those of a painter or a poet, must be beautiful. Bind harmoniously, just like colors or words do. There is no place for bad math».

For Einstein too, the theory had to be beautiful and elegant, just as one would expect from Nature’s fundamental description. The experimental results could not influence this opinion. If the theory was simple and elegant, then it had to be correct regardless of observations.

Dirac agreed: «To walk the path of progress, we should start with the purpose of obtaining beautiful equations, together with a solid intuition.»

The experimental data of the first experiments on the motion of electrons were in disagreement with the predictions of special relativity, but this did not stop Einstein from affirming his theory’s superiority and validity.

According to Dirac (and many others), the theory’s beauty would authorize the physicist to overlook the adverse verdict from the experimental results temporarily. However, there is an ugly theory but with a positive empirical outcome.

Those ideas come from some Dirac conversation with his colleagues:

- If the elegant theory is in agreement with the experiments, do not worry.
- If it disagrees with experiments, the experiments must be wrong.
- If the inelegant theory conflicts with the experiments, one can try to refine it to accord with experiments.
- If a rude theory agrees with the experiments; then there is no hope.

According to experiments, an ugly theory is undoubtedly wrong according to this line of thinking: the physicist should not be content with applying mathematical operating rules for the sole purpose of having the predicted and the empirical in agreement.

Dirac’s critique also has a strong message, an invitation to audacity.

Einstein had won because he had fully believed in his theory’s beauty and had not been discouraged as others had. Although the historical period was not the best and Einstein’s character premises kept him from success, his belief in mathematical beauty guided him towards Nature’s purest description.

This beauty is shown when the physics artist lives without prejudice, immediately admitting the fields’ superiority. Discrimination is what prevents him from understanding quantum mechanics and related relativity in practical terms.

When we have a new Einstein or a new Heisenberg, Dirac continues, we will have a unique beauty gift from Nature. This latest evolution foresees the doubt about something that has never been questioned, and that will lead to shedding light on the conflict between determinism and indeterminism.

To attach too much importance to current concepts is undoubtedly a mistake. It will probably be necessary to work on changing images such as space-time, without falling into the error of attributing too much faith to one’s thinking, as with the case of the ether for electromagnetic waves.

If we hadn’t followed this path, we would never pass Bohr to join Heisenberg.

Dirac suggests to us, once more, that the essential ingredient, once again, is undoubtedly beautiful.

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