# Concerning Pi, again

For Pi Day this year (2018), I will provide some results related to this interesting mathematical constant. These results are mostly inspired or taken from answers I gave at quora.com about  $\pi$ or about similar mathematical topics.

The millionth decimal digit of π is 1 (verified with Mathematica).

The 10 millionth decimal digit of π is found to be 7, and the 100 millionth decimal digit of π is 2.

The billionth decimal digit of $\pi$ (in base 10) is 9 (verified with Mathematica).

The 2 billionth decimal digit of π is found to be 0 (this result takes a longer time to compute with Mathematica).

Here are some (repeated) number sequences or numeric strings found among the first 2 billion decimal digits of π.

The numeric string 777777777 appears at the 24, 658, 601 st decimal digit of π :

9919408245759718530477777777724846769425931046864

The numeric string 111111111 appears at the 812, 432, 526 th decimal digit of π :

2891450444990691713511111111100399876875718824885

Here are also two numeric strings from 1 to 9 in increasing order and decreasing order:

The numeric string 123456789 appears at the 523, 551, 502 nd decimal digit of π:

7260489917323889207212345678922486448188070486710

The numeric string 987654321 appears at the 719, 473, 323 rd decimal digit of π :

5221398663241526793698765432160793011913242320510

The numeric strings above can be calculated or found with the help of the following link or web page:

Irrational Numbers Search Engine

The numerical value of $\pi^{\pi}$ to 1000 decimal digits is equal to:

36.46215960720791177099082602269212366636550840222881873870933592293407436888169990462007987570677485436814688343670070542736699139359264431565675267180230917777595737242260530320050233549595161382594571885422222305402433199779769167302876444780028452117394296018175249159350019492001619423210110480018557258718860782819839215304503453543238476218257664861595609057280314341958390400811991506636066295817900302292747422204210046403709493285441101884797707466358510710362803891181156618083260884536505255311095948029552909133361385823497120761861157606574436205295895657736468959837126404885207348833917602169536002174958035720670509318770633086434935593202425189496008880555048213388792769304015639745480898380866639428337794250284522113741878602793251848366623602214652151453276098545038540482041645516919082097210265379423765817354600472953993840487842116366153365057093066926223775915023204727672609958737278566633593210689698807508602353552267490321670730929373240451651980750157959708689483469068

Two expressions involving π and infinite sums:

Representation of π in continued fraction form:

The sum of π and e, the base of natural logarithms, is equal to:

$\pi + e=\displaystyle \sum _{k=0}^{\infty } \frac{(3 k)^2+1}{(3 k)!}+2 i \ln \left(\frac{1-i}{1+i}\right)i$

The letter i  represents the imaginary unit of complex numbers.

Another expression involving π, e, and an infinite product:

$\pi = \displaystyle 2 e \prod _{k=1}^{\infty } \left(\frac{2}{k}+1\right)^{(-1)^{k+1} k}$

And here is an identity relating the Golden Ratio, π, e, and the imaginary unit i:

$\displaystyle \varphi=e^{i\pi/5}+e^{-i\pi/5}=\frac{1+\sqrt{5}}{2}$

The value of π can be deduced from the identity above:

$\displaystyle\pi =5 i \ln \left(\frac{1}{2} \left(\varphi -\sqrt{\varphi ^2-4}\right)\right)$

# Concerning the relationship between science and philosophy

This post consists of the  elements of an answer I wrote at quora.com ; the question there was: “Is philosophy the top of all kinds of sciences?”

I think it would be convenient to distinguish between the general term “science”, referring to the state or fact of knowing, or to knowledge acquired by study and learning, and the modern meaning of “science”, mostly referring to mathematics and to the exact sciences using the rules of the scientific method (astronomy, physics…).

Philosophy and science were not separate in Antiquity.

In the original sense, philosophy meant the love, study, or pursuit of wisdom, or the knowledge of things and their causes, theoretical as well as practical.

Pythagoras was a mathematician, and at the same time it is said that he was the first one to call himself a philosopher, or “lover of wisdom”.

Plato was a philosopher who recommended the knowledge and the study of geometry. In The Republic, Plato thought that the best ruler was the king-philosopher.

Aristotle studied nature and wrote works about physics, biology, logic, etc, from a philosophical point of view.

According to the OED:

“In the Middle Ages, ‘the seven (liberal) sciences’ was often used synonymously with ‘the seven liberal arts’, for the   group of studies comprised by the Trivium (Grammar, Logic, Rhetoric) and the Quadrivium (Arithmetic, Music, Geometry, Astronomy).”

The expression Natural philosophy was frequently used for centuries :

“Natural philosophy or philosophy of nature (from Latin philosophia naturalis) was the philosophical study of nature and the physical universe that was dominant before the development of modern science. It is considered to be the precursor of natural science.

From the ancient world, starting with Aristotle, to the 19th century, the term “natural philosophy” was the common term used to describe the practice of studying nature. It was in the 19th century that the concept of “science” received its modern shape with new titles emerging such as “biology” and “biologist”, “physics” and “physicist” among other technical fields and titles; institutions and communities were founded, and unprecedented applications to and interactions with other aspects of society and culture occurred. Isaac Newton‘s book Philosophiae Naturalis Principia Mathematica (1687), whose title translates to “Mathematical Principles of Natural Philosophy”, reflects the then-current use of the words “natural philosophy”, akin to “systematic study of nature”. Even in the 19th century, a treatise by Lord Kelvin and Peter Guthrie Tait, which helped define much of modern physics, was titled Treatise on Natural Philosophy (1867).

In the last few centuries, alchemy separated from chemistry, astrology separated from astronomy, and there was also a certain separation between philosophy on one side, and mathematics and the exact sciences on the other side.

Mathematics became progressively the most prominent and the essential scientific discipline, it is acknowledged as the language of science and of the physical world.

Philosophy is nowadays often regarded as a reflection, view or study of the general principles of a particular branch of knowledge, or activity. There is a philosophy of science, philosophy of mathematics, philosophy of education ,etc.

Some theories or views related to epistemology (which is concerned with the general theory and the study of knowledge) and philosophy, such as rationalism, empiricism, and positivism, share a number of principles with the scientific approach to events and phenomena.

(Source of the image above: Wikimedia Commons)

A scientist or a physicist can also be a philosopher. Important thinkers can be philosophers and create philosophical systems, but modern philosophers must take into account the advances, discoveries and theories in modern science. A historical example would be Immanuel Kant elaborating his philosophical system and philosophical ideas at the end of the eighteenth century in light of and in relation to the exact sciences known at that time, especially Euclidean geometry and Newtonian classical physics and mechanics.

# A poem I wrote years ago

I was fifteen- soon to be sixteen- years old ; I had been reading (important) books about science, physics, philosophy , and other similar topics,  and all those ideas in my head intermingled and inspired me to write a poem involving particle physics and particle collisions and combining elements of science and philosophy .

I wrote the poem in French , using the French alexandrine poetic meter of twelve syllables, but I didn’t follow the poetic rules very closely.

I will provide the final version of the poem here , with a line by line English translation. Different people have different tastes and opinions , I hope it will be liked .

The hydrogen-1 atom mentioned in the title of the poem is also called “protium” , but this last word is not much used in French. Protium is the most common hydrogen isotope, having one proton ( and one electron) and no neutrons.

A proton is supposed to be talking or telling the story in the poem . I think I was a little inspired by the poem ” Le Bateau ivre ” by Arthur Rimbaud .

Here it is :

Bombardement d’atomes par un proton d’hydrogène 1H
Bombardment of atoms by a proton of protium 1H

Synchrotrons , canons à électrons, cyclotrons
Synchrotrons, electron guns, cyclotrons

Soyez prêts, particules, deutons, neutrons, hélions
Be prepared, particles, deutons/deuterons, neutrons, helions

En attendant que les hommes préparent les canons
Until men prepare the guns

Le moment est arrivé, l’appareil frappe
The moment has come, the apparatus strikes

Dans son coeur vidé moi, le proton j’attrape
In its emptied heart I , the proton take

Le coup et je vais croiser les atomes en grappe
The blow and I go meet the atoms in clusters

Je fuis dans l’espace et le temps calculables
I flee in computable space and time

Ma vitesse est vertigineuse, incroyable
My speed is vertiginous, incredible

Non pas celle de la lumière, infranchissable
Not that of light, insurmountable

C’est le lieu de la relativité impie
It is the place of impious Relativity

Masses, longueurs, lois de la physique varient
Masses, lengths, physical laws vary

Ma trajectoire déterminée sera suivie
My particular/determined path will be followed

Par d’autres microcosmes malheureux
By other unfortunate microcosms

Le trajet est terminé, le choc a eu lieu
The journey is over, the shock/collision occurred

Je donne la vie à de nouveaux corps heureux
I give life to new happy/fortunate bodies

Quanta de matière utilisés pour la paix
Quanta of matter used for peace

Dans le monde de la science un pas est fait
In the world of science a step/discovery has been made

L’humanité en marche en connaît les bienfaits
Humanity in motion/advancing knows the benefits (of this discovery)

# Question: what is the square root of 36 ? – Part Two

I will continue with answers and results equal to the square root of 36 ( originally answered at quora.com ). This time the results are mostly related to physics.

With physics one has to take into account the units and the corresponding dimensions of the equations and of the constants.

6 and the square root of 36 are dimentionless numbers , so the result must be dimentionless .
If the result is a simple fraction with numerator and denominator , then the units usually cancel out.
In other cases when one deals with logarithms one should multiply with the inverse dimensions to get a dimentionless result.
In one or two results where I didn’t look for the inverse units  I multiplied the equation with a quantity I called (U) ,  which represents the inverse of the units by which one should multiply the result to get a dimentionless number.

Here are the results :

One possible way to explain what I have done here is the following:
If some people , living on an isolated fictitious island or on an another hypothetical planet , attached a great importance to and had a fixation on  the square root of 36 (or the number 6) for one reason or another , and got accustomed to the use of 6 as a fundamental constant ,  unit or number , then they would have likely  tried to construct a system of measurement  based on the number 6 , and  to express physics and math formulas ,equations , constants and rules in relation to 6.

After all , 6 or $\sqrt{36}$ is equal to :

• The floor of  $2 \pi$ :
$2 \pi \approx 6.2831853071795864769$ ;$6=\lfloor 2 \pi \rfloor$
• $\frac{1}{60}$ of the circumference of a circle in degrees.
• It is also  one tenth of 60 seconds which make up a minute , one tenth of 60 minutes which make up an hour , one fourth of 24 hours which equal a day on Earth , one half of  12 months which make up a year , etc.
• A peculiar ‘hexacentric’ system , so to speak.

Or this can be seen as a (creative) exploration of or exercise in advanced math and physics in order to express many equations , formulas and constants in relation to the number 6 (or $\sqrt{36}$ ) .
Or whatever.

Apologies to Isaac Newton , Leonhard Euler , Bernhard Riemann , Einstein , Stokes , Coulomb , Avogadro , Lagrange , and others (wherever they may be) , for playing around with their equations , formulas , constants , and/or functions.

Does the future of humanity depend on answering what is the square root of  36 , or not?
Have philosophers from Antiquity to the present overlooked this fundamental question , which goes beyond the Kantian categories of space and time set out in his Critique of Pure Reason , and beyond Nietzsche’s Beyond Good and Evil , ushering the transmutation of all values and a defining moment for a new era  in the history of Humankind?
It’s just a square root , for common sense’s sake (or is it?).

Anyway , enough philosophizing.

Here are ( $3=\frac{\sqrt{36}}{2}$ ) more answers to $\sqrt{36}$ , this time with images :

$\sqrt{36}$  is equal to :

The number subjected to a geometric rotation in the following image (done with Mathematica and some Photoshop) :

The number expressing the power and the coefficients in the equation of the curve in the polar plot below :

The number expressing the degree of the root  and the power of the variables in the 3D plot below :

The rotated number  and the polar plotted curve in the first two images  above seem to exhibit symmetry.
Symmetry is an very important property in science , math , physics , equations , nature , and wherever it is found.

Online sources and reference works related to what I have written in this answer can be found in my pages about Science booksand philosophy books in this site/blog.

Some other online sources:

http://en.wikipedia.org/wiki/Category:Mathematics

http://en.wikipedia.org/wiki/Category:Physics

# About languages I know and languages I’m getting to know

I have a problem. Could be a problem , or not.The thing is , I read a lot , and in more than one language.

At present I know three languages very well , and I’ve read literary books and various kinds of books in these languages.But I have also other books and coursebooks for learning other languages , some are in print form and others are ebooks along with online learning material I’ve downloaded . Audio files are generally  helpful while learning a new language.The three languages I know best are English , French and Arabic . I’ve studied and learned other  languages intermittently , depending on the circumstances or on how much free time I had.

For example , a few years ago I met a Russian girl and I had a relationship with her , and that gave me the incentive to learn Russian.Then I met a German girl , and that encouraged me to start learning German using a coursebook I had bought two years earlier.I have a Spanish language coursebook as well , so I’ve studied recently the first few chapters of this book.Of course the Spanish language is interesting and useful since it is spoken not only in Spain but in most of Central and South America and in many countries around the globe.

I have additionally a Portuguese language coursebook waiting to be read ,  and a Chinese language textbook for beginners and a Chinese language learning software I haven’t used yet . Now  the Portuguese and Spanish languages are more or less related to each other , but learning Chinese could be more difficult as it is a non-alphabetic language and uses characters such as logograms , pictograms and ideograms . I’ll have to see if or when I will have enough time to start learning these languages.

Moving to another group or type of languages , I’ve been using and working with the Mathematica software and the (Wolfram) Mathematica programming language for more than fifteen years , and I have occasionally used in the past programming languages such as QBasic and the BASIC-like language of the TI-92 Plus calculator. I have also studied web programming for a certain period of time  , and I got to know well a number of programming (and scripting ) languages. These include HTML , PHP (and the MySQL database management system which uses the SQL query language) and JavaScript.

However , I have to say that among the languages I know the most important one is a language called Mathematics.This is the essential and fundamental language of the exact sciences , and according to good old Galileo , the mathematical language is the one in which the great book of Nature and the Universe is written.

And if you are of the opinion that Mathematics is not as important as other languages for communicating and connecting people , think again.
From Astronomy and Astronautics to Physics and Chemistry to electronic devices and components (which use disciplines such as solid-state physics , circuit and signal analysis , differential and difference equations , Fourier analysis , etc) , to computers (which use fields such as Boolean algebra , binary logic ,  and combinatorics)  , to Biology , Psychology , Economics and business studies , Mathematics and applied math are indispensable to any  study or research in science and engineering . Programming languages are more or less related to applied , discrete and computational math and to mathematical logic. Even the social and human sciences use math and statistics to  get a reputation for exactness , precision and scientificity .

# Some more thoughts about education

Children who have the possibility to finish school at an early age and to start their higher education at a very young age are often called child prodigies or gifted children.They are
frequently treated as curiosities or rare people with an acute intelligence and uncommon abilities .These kids may have special aptitudes but there is a plausible explanation for their situation .To put it simply , either they have fast stages of intellectual development and they were noticed and helped by their parents , family, teachers and/or professionals in order to have an accelerated education , skip grades and go to college at a very young age (probably 9 or 11 or 12) , or they have  average (or slightly higher than average) developmental stages and were also taught , helped and trained by their  parents , teachers and education professionals in ways which allowed them to finish school early and to enter the university precociously.

I think  the way these precocious or gifted children are taught ,  accelerated education and the opportunity to complete one’s studies early and to start university studies at a younger age ought to be given and extended progressively to all children .This way fast learners will not be left behind or neglected , and generation after generation more and more kids and young people will have faster stages of growth and will be able to assimilate more information at younger ages . When lowering the age of entry to the university , some youngsters could  start higher education at the age of 11 or 13 , but the minimum age could be set at 10 or 9 .

Moreover , a smaller , restricted acceleration could be applied  to higher education and university studies . For example , becoming an engineer requires four years of study in certain countries and five years in other countries . The five-year programs could be condensed into four years by appropriate methods such as adding hours to the four years of study , condensing some courses and/or adding summer courses . Medical studies usually take eight or ten years or even longer to be completed . By applying the convenient acceleration and condensation of courses and studies the eight years of study could be reduced to seven and the ten to nine and so forth , without loss of knowledge or qualification for the future doctor or medical practitioner. Similarly predoctoral and master’s degree studies could be reduced from five to four years or from six years to five.

Hence in the long run everybody would benefit from this reform and acceleration of education , and the gap between youngsters considered to be average or normal and child prodigies would be less wide ,  which I consider to be  a good thing , providing more educational equality and efficiency .

A short remark concerning one aspect of education: Nowadays young people are generally becoming sexually aware (and sometimes active) at younger ages compared with older generations , and this fact should be taken into consideration in education.

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I’ll add a final remark here : The general system  of education followed in a large number of countries today was formed about two or three centuries ago . In a country such as France  a major reform in education took place , coinciding with the period of the French Revolution , and with the  time when the first  successful endoatmospheric human flights were made using
hot air balloons  . One thing to note is that the Montgolfier brothers , who flew the first balloon , were elevated to the nobility as a reward . Nowadays the old aristocratic nobility
titles are gone in France , so people are instead rewarded by  honorary degrees ,which can be considered as  contemporary nobility titles . Many of the thinkers and scientists who
contributed to the reform of  the education system in France two centuries ago , such as Condorcet and Laplace , had acquired old nobility  titles (such as ‘marquis’) , titles related
to the ‘Ancien Régime’ , and were forming the new system of instruction which would provide the new (education related)  nobility titles in France and elsewhere . Perhaps nobility
titles in one form or another will frequently or regularly succeed one another with time , but there should always be room for reassessment , open-mindedness , change and reform . Wilhelm von Humboldt witnessed and was influenced by the French Revolution . He attempted to reform and reorganize the Prussian and German educational system , and founded the University of Berlin in 1810 . In the following decades of the nineteenth century , educators in the United states and other countries emulated and implemented the Prussian education system.
The rest of the  nineteenth and most of the twentieth centuries witnessed generally lesser scattered reforms in education , the elaboration of educational and psychological theories , and were the scene of  a progressive worldwide adoption of the educational ideas and institutions developed in European nations before , during and after the Enlightenment period and the  French Revolution .This epoch  coincided with the development of aerodynamics,
aeronautics and heavier-than-air aircraft , and with the beginnings of astronautics and space exploration .

The following idea is worth considering: While the existent educational system , with its diplomas and degrees and its requirements to study or act in a certain way and to start
higher education at the average age of eighteen , has worked well and has been sufficient for people who move or travel on planet Earth and in its surroundings, I think the reform of
the educational system  I have proposed and written about here , i.e accelerating education , skipping  or condensing  grades and lowering the age of entry to college or the university , will
prove to be important , more efficient  , and necessary in this era of planetary globalization , and more so  in the future , as humans in the age of space travel  intend or attempt to go to Mars and  others planets , traveling in the solar system beyond the Earth-Moon system and the immediate vicinity of planet Earth.

The notion of educational level is usually related to the educational system or framework within which it is used . For example , there are three cycles (sometimes called ‘levels’) in
higher education . Levels are considered as corresponding to diplomas and degrees ( i. e. High School diploma , Bachelor’s , Master’s degree , PhD) awarded by the educational
establishments , so that people with higher degrees are considered to have a better or higher level . Sometimes the word level is also used (especially in selective schools or
educational institutions) in the sense that students with higher grades or marks have a higher or better level than those with lower or average grades.

It has been shown over the years and decades   by numerous success stories that students with excellent grades ( ‘A’ students in the American education system) don’t necessarily
achieve better in life than average (or ‘C’) students . Not to mention the many  examples of college dropouts who were successful and made important , notable achievements
compared with other people who finished their higher education and got university degrees but accomplished less.

I have written in earlier posts  about kids or young people who have  fast stages , states or phases of  intellectual development and are often misunderstood , cannot adapt to the existing education system and even  fail in it.These young people may   also be called fast learners or fast receivers (of information or knowledge) . I think one possible explanation for the
achievements of average students or dropouts as compared with ‘A’ students  is that a number of these average students and dropouts are fast learners . Their personal  stages ( or
cycles ) of development are faster than and incompatible with the stages or cycles acknowledged by the educational system . Therefore they don’t follow the system well and don’t study as required by the current curricula  , and tend to fail or leave the system  altogether.

Sometimes people pass through a period of time (probably a few years) during which they study a lot and have high grades , then this period ends and they start reading a lot
instead of studying and have lower or average grades . This period takes place at a younger age for some people and ends earlier for them , while it can last longer for others . That’s
why there are people who have high grades at school and then become average ‘C’ ( or even ‘D’) students at the university , whereas others go on and have high grades for a long time
and can continue studying and having degrees till they are thirty or forty years of age . The fast  learners in these cases are the ones who stop studying a lot or stop succeeding in the
education system at a younger age .

It can be seen by the analysis I’ve made above that ‘level’ and ‘educational level’ are expressions that are used abusively in relation to the current educational system . They can lead
to discriminatory attitudes and do not reflect the real potentials , aptitudes or abilities of students and young people.

In the third cycle of higher education it is usually not required of students to study a lot but to read a lot and to do personal research for about three years in order to prepare and
obtain their PhD degree . This is one proof that in the existing system the final , more advanced stage or cycle takes place when reading and research , not studying and getting good grades ,
become the essential requirements , and although this is commonly expected from students at ages over 20 or 25 or 30 , fast learners reach  this stage at younger ages , maybe at 18 or 15  or
younger . Therefore accelerating  education and lowering the age of entry to the university is necessary to take into account the fast ones and their abilities.

Students and pupils who have high grades or a high academic achievement generally get certain privileges and are accepted before others as they pass the  entrance examinations of high
schools or universities . They are  preferred by cram schools and schools which apply academic selection. In a country such as France an example of selective schools would be
the ‘Classes préparatoires aux grandes écoles’  , and in England the grammar schools.
By the arguments stated above , and since this period of having high marks is one that lasts for a certain time and then passes away ,
it would be preferable if people with high grades are given less privileges , the important thing being to pass the exams and tests and succeed in the educational system ,even within a reformed system where educational acceleration is applied .

When or if (young) people have good grades and high academic achievement , then good for them , they can be proud of what they
have done but can also keep it for themselves . I remember I was the top of my class at school and I had high grades for about seven years before the age of fifteen , and I was at that
time very proud of having such very good results , but then things changed and I began reading more than studying and started having average grades . So one could be happy
about one’s good grades and tell others about it , but there is no need to brag a lot about it and no need to be given additional privileges.

# About ‘The Cleft’ and Doris Lessing.

I’ve had the intention of  reading The Cleft by Doris Lessing  for over  a year  , but it so happened that I started reading it a few days after I sadly learned she had passed away.

It’s a story about the origins of humanity told by a Roman historian ,supposedly taken from old forgotten records.
Lessing claimed that she was inspired by a scientific article which said that female humans came first before males.Some would consider this as an inversion of the biblical creation narrative, but anyway it’s a good story to read , although  scientific evidence for it is doubtful or lacking.The story tells that first women were lonely creatures living by the sea who used to give birth to baby girls by some sort of parthenogenesis , until one girl gave birth to a baby boy.The women  were perplexed  and got rid of that first baby boy , but then more boys were born and with time they formed a tribal group of their own . Some of the women  went to meet them and got to know them despite the ban from the older females , so afterwards babies were  born as a result of sex and a normal pregnancy.With time (perhaps a long time) the females and males ( who are called respectively ‘clefts’ and  ‘squirts’, imagine why) learned to know each other better and lived together , and started exploring their surroundings.

Since I’m writing about Doris lessing I’ll take this occasion to say something related to her and to the education post I wrote earlier.
According to her biography , Doris Lessing left school at 14 years of age , and was self-educated from then on.This is an example of a famous and important writer who had to be an autodidact for a (large) period of her life and who won many important literary prizes , including the Nobel prize for Literature , without having a formal higher education or a university degree.
It’s not the (existing) educational system that makes the great writer (or great man or woman). More often than not  great thinkers or intellectuals are the ones who create , manage or reform educational systems , either by getting personally involved or by influencing that creation or transformation through their actions , teachings and writings.
Doris Lessing may have received an honorary degree at an older age , but this happened after she had to be self-educated and had to struggle and publish many books and novels in order to be recognized as a significant writer.

I’m going to mention suppositional events that could have been or could have happened , but in any case I want to note that  if (for example) Lessing had had  the possibility as a young girl to receive a convenient accelerated education and to go to college or to the university at the age of ten or eleven , she would have left formal education at the same  age of 14 but with a degree  in literature instead of no degree at all.This could be applied to many bright and able youngsters who get stuck in the educational system because of their fast intellectual development or have to leave it early for different reasons , and   I think that is one of the advantages of an adequate accelerated education.

I ‘m not comparing myself to a writer such as Doris Lessing , but I couldn’t help thinking that if  Lessing or a writer of her importance (and her educational background) had had the idea to call that school like I have done , and had told them only that he or she had a very good teaching experience  and very good knowledge of the English language but no degree , they would have rejected her or would have probably hung up.

Degrees and academic credentials are useful and important but sometimes it depends within which educational system or framework they are given , and too much insistence on diplomas alone without taking other variables and factors ( experience , the age factor in education  being  relevant ones among others)  into account may turn out to  be counterproductive and inadequate.

Allow me to present to those who refuse to believe these successive improvements of the human species an example taken from the sciences where the march of truth is the
safest, where it can be measured more accurately . These elementary truths of geometry and astronomy which had been in India and in Egypt a secret doctrine on which
ambitious priests had founded their empire, were in Greece at the time of Archimedes and Hipparchus , vulgar knowledge taught in public schools. In the last century, it
took only a few years of study to know everything that Archimedes and Hipparchus could have known , and today two years of the teaching of a teacher go beyond what
Liebniz  or Newton knew. Let us  meditate this example, and seize this chain that extends from a priest of Memphis to Euler , and fills the immense distance which
separates them ; let us observe at each epoch the genius ahead of the current century, and  mediocrity up to what he had discovered in the one that preceded it , we shall
learn that nature has given us the means to save time and spare attention, and there is no reason to believe that these means may have an end .
These words were written by Condorcet two centuries ago in the 1790s in his book “Cinq mémoires sur l’instruction publique” (Five papers on public education).
Liebniz  and Newton were scientists who lived about a century before Condorcet’s time , so for our period of time they could be replaced by scientists such as Einstein , Schrodinger and others from the twentieth century.Nowadays young people learn the (updated) scientific theories of Newton (about calculus, gravitation , etc) during their secondary studies or as beginning college/university students.Einstein’s theories and relativity and quantum physics are taught in introductory courses at the secondary
‘level’ ( I use the word level between quotations because I think it has a meaning related to the existing educational system and it should be analyzed more thoroughly)  and taught in detail during the college or university studies.

We can infer that in a few years or decades  the scientific theories taught in higher education will be more and more understood by younger people and taught to students at a younger age.Of course these scientific theories could be changed and reformed or replaced by newer theories or discoveries , and the more recent or difficult theories will be taught at an older age , and so on.
One thing or fact to be taken into consideration is that the change and progress in scientific knowledge , information or theories should be accompanied by a reassessment of and a change in the educational system , framework or methods within which this information is given or taught.
The schools and especially the  universities and established higher education institutions nowadays follow educational practices , methods or systems that originated two or three centuries ago in countries such as France , Great Britain or Germany , obviously because these countries were the most advanced during that time.Many colleges and universities in the United States also follow these European models , and many countries that were French , British or European colonies  have also copied or followed these models.Hence a global education system surfaced where most countries and nations follow or imitate these educational models.
When young people of a similar age are placed in one same class , they may not all think , study or behave the same way.There are students who are average , slow ,or fast learners . The fast ones are sometimes called gifted , talented or precocious , having exceptional aptitudes or abilities.In most cases this means that at their age ( let’s call it age X) they think or behave in a way in which other students will think and behave at a later age (for example at the age X+2 or X+5 etc ).

Suppose we have a classroom where most students are approximately 10 years old.A fast child or student would be someone who thinks , behaves , studies and understands things at age 10 in a certain way.At that  same age of 10 the average or not so fast students would be thinking and behaving as the fast one did when he was perhaps 8 or 7 years old.Conversely these average students would think , behave and study the same way as the fast kid is doing when he’s 10 years old  but for them this will happen  at the age of 13 or 15 or probably 18…
To use an expression from Piaget we could say that fast kids pass through faster  stages of  (intellectual or cognitive ) development.
When students of  a similar age are put in the same classroom without any differentiation it is usually the average learners who are taken into consideration and who succeed.Both the slow and fast  ones would suffer and have bad results .The  fast or ‘gifted’ ones would be placed in classrooms  and in an educational system which they cannot adapt to and which is not adapted to their capabilities.Consequently they are misunderstood and have their needs neglected.They might even fail like slow students in such detrimental educational settings and conditions ,  but for completely  different and opposite reasons.The situation could become somewhat disturbing and nightmarish when these fast learners , who have more advanced thinking , behavior and  stages or phases of intellectual development than all of their classmates ,find themselves forced to resort to cheating or to exam fraud in this education system  which doesn’t notice them , ignores them or doesn’t treat them well .At the end nothing works for them and they may realize  it’s better for them  to leave the system , even without a (final) degree .

At a global international scale,  students who follow the existing national educational curricula and go through all the years of primary and secondary (often compulsory) education end up going to colleges and universities at the average age of 18.This age may be the current characteristic age taken for granted as the age of the start of higher education studies, but this is one of the concepts or factors that ought to be reformed and changed by accelerating education and lowering the age of entry to institutions of higher learning.
Today new and alternative theories and methods of education are emerging , from academic acceleration and skipping grades to homeschooling and unschooling to online education and e-learning , etc.

Autodidacticism and self-education can always be valuable and useful ways to acquire knowledge. Most great scientists , thinkers and innovators had to be autodidacts at some period of their lives.Even if they learned from teachers and studied in recognized educational institutions , they had sometimes to read books and learn about new concepts and theories which were not a part of the official curricula in order to create important original ideas and theories on their own .Public libraries (and the more recent online libraries) and college or university libraries which allow people who are not registered students or staff to borrow books are noteworthy and helpful places for educating oneself , increasing one’s knowledge and doing personal research , more or less independently of the existing educational system.

Apart from becoming autodidacts or unschooled , I think that young people and students should be taught in a new way  and be given the opportunity to finish their secondary studies and start college or university studies at a younger age.For example , a child or kid could be homeschooled and finish high school at a younger or lower age , then he or she could follow the formal education system and go to college or the university at that lower age in order to complete his or her higher (post-secondary) education.
Thus one efficient educational method is to teach children and youngsters by using a combination of (accelerated) homeschooling and  formal instruction.