About Numbers and Mental Calculators

21 Dec

My mind to me a kingdom is. – Epictetus.

The man does not contain the mind: the mind contains the man. – Socrates.

Everything possible to be believed is an image of truth. – William Blake.

Try not become a man of sucess but a man of value.- Albert Einstein.

From the stone age to the foundations of megalopolis and modern science, it was the use of numbers that allowed us to make that leap of civilization. Numbers help us at various levels. We use them to count objects, and a lot of other tasks such as to make statistics FN[1], to gamble, to tell what time it is now. Besides these utilities, numbers shaped the way we think the world. Einstein said that numbers are the “symbolic counterpart of the universe”, and according to Lord Kelvin  “When you cannot measure…your knowledge is of a meagre and unsatisfactory kind” FN[2].

But the man that more importance gave to numbers was Pythagoras. He went to Egypt to learn the basis of a new science, a science unknown to the Magi of Persia, to the Chaldeans, to the Gymnosophists of India, to the Celts, to the Druids, to the so called Barbarians. Pythagoras spent a long time in Egypt (Thales and Sidon, two of the Seven Wise Men of Greece, went before him)  before to settle down in the south of Italy (in a region of Italy called at that time Grecia major) and to found a sect called the Italics [4].

He went to Egypt departing from Samos, with Letters of Recommendation written by Polycrates to Amasis II, at the time King of Egypt, and a close friend of Polycrates. The court of Amasis was settled in Heliopolis, in northern part of the Nile Valley [5]. We must say that Egyptiens at that time very rarely shared their sciences with foreign people, and even with their countrymen they only allowed them to go though their studies after a long period of probations. Hence, Pythagoras was sent to Memphis, and then to Diospolis, where the Egyptiens hoped Pythagoras would be discouraged with the rigorous rules imposed upon the novitiate. But Pythagoras did not abandon its goal, and was even circumcised.

School of Pythagoras in Croton. Image credit: sciencephoto.com

After 25 years in Egypt, Pythagoras went to Babylon, where he learned from the Magus Zoroaster. After that, he went to Crete and then to Sparta, aiming to instruct himself in the Laws of Minos and Lycurgus, whose States had a reputation of the best governed in the world (where our leaders learn today?…). The beauty of the following text is a testimony of the character of this great men and an example for all of us [4], living now in a world ruled by markets, consuming goods, led by corrupted politicians to an expected dramatic outcome, and dramatically poor in spiritual values:

After his long absence, he [Pythagoras] found Samos in a condition very different from that in which he had left it. Polycrates, who had made himself absolute Master of it, as we hinted before, puff’d up with his prosperity, exerciced a tyrannic power over the oppressed Samias. Pythagoras hated injustice too much, and was too fond of right and equity, to undergo the yoke of a tyrant. He preferred a voluntary banishment to the slavery that threatned him, and went to find out a place of refuge, where he might preserve liberty, the most precious of all treasures, and for the preservation whereof all men, who have known its worth, have ever sacrificed all their, other goods, and even their life.

So, Pythaforas left Samos and travelled thorough the Peloponesus, stopping in Phlius, where an intelligent prince Leo, governed. These two men had long conversations and, Leo surprised with his wisdom, asked to Pythagoras: “What profession you follow?”, Pythagoras answered, “None; I am a philosopher”. And Leo replied, “What is to be a philosopher? What is the difference between a philosopher and any other men”. Pythagoras gave him the following answer [4]:

That this life might well be compared with the Assembly that was held every fourth year at Olimpia, for the solemnity of the Games celebrated there; for, as in that Assembly there are some that come in search of glory, and the crowns that are rewards of the victors; others in search of gain, by buying or hiring out of merchandises; and a third sort more noble than the two former, come thither neither for profit nor applause, but only to enjoy the pleasure of beholding that wonderful spectacle, and to see and know what passes in it: so we in like manner, leaving our country, which is heaven, come into this world, as into a place of public meeting. There, some toil after glory, others for profit, and small is the number of them, who contemning and trampling under foot all manner of avarice and vanity, apply themselves to the study of nature: these left are they, added he, whom I call philosophers. And as in the solemnity of the games there is nothing more noble than to be a spectator whitout any interfere or selfish end, so likewise in this life, the contemplation and knowledge of nature are infinitely more valuable than all studies and applications.

This extraordinary explanation of what is to be a philosopher can be resumed in this manner: Man was created to know and contemplate. 

Pythagoras was the man that loved numbers, that recognized in numbers representative value and spiritual values. He was a philosopher, mathematician, musician and religious leader. I am not sure if he was a mental calculator in the sense we give nowadays, but he was for sure one being that gave a decisive twist on human history. From his ideas sprang the modern mathematics and physical sciences, eagers of quantitative responses.

Maybe for historical reasons, namely the attempt from the Hebrews to get independent from the Greek aculturation, Jews do not count their own people. There is a reference to this in Samuel II, Chapter 24, where King David was punished by God when he took a census of his people. According to the Jews, ascribing a number, a finite value to a Jew, which is supposed to have a spiritually infinite value, is wrong doing. When the Jews needed to be counted at that time, each one contributed with a coin, and then coins were counted (see here).

Also, long-term studies have shown that not every people developed the notion of number the same way. At least, these are the recent findings of a French team of cognitive neuroscientists led by Pierre Pica, from the University of Paris. This team studied the Mundurukú, a tribe of Indians living in central Brazil. Mundurukú lacks vocabulary for numbers beyond 5, and counting above numbers 4 or 5 needs the attribution of names to each number, otherwise counting becomes an impossible task.

If you ask a Mundurukú with six children, “how many kids do you have?”, he will say, “I don’t know.” That’s what Pierre Pica recounts after his almost half a year living period among them. Amazingly, when returned to Paris, Pica realized, “When I come back from Amazonia, I lose sense of time and sense of number, and perhaps sense of space” [6,7].

Mundurukú named the numer 1 by the word pug; number 2, by xep xep, but above numbers the name were not consensual, although the majority had a specific name:

1 pug

2 xep xep

3 ebapug

4 edadipdip

5 pug pogbt (literally means handful)

Pica noted that the number of sylabs of each word was equal to the number itself, but after number four this rule was not followed, as if they could not count up to five but only to four. Pica said, “This is amazing, since it seems to corroborate the idea that humans possess a number system that can only track up to four exact objects at a time” [6]. Another fascinant discovery of this team, the Mundurukú do not distribute numbers on a linear scale.

On Fig. 2 it is shown how Mundurukú distributes  numbers. The intervals between the numbers start large and progressively decrease as the numbers increase in a kind of logarithmic scale. However, when American adults were given this test, they placed the numbers at equal intervals along the line, considering numbers are equally distants from each other.


It is interesting to note that the logarithmic approach is not exclusive to Amazonian Indians, in fact we all are born with that sense of numbers. In the kindergarten, children have the same notion of numbers logarithmically distributed. When they start studying at school, this natural intuition is lost, their minds are already culturally framed to “see” numbers linearly.

The intuition of a logarithmic scale is more appropriate when we live in nature. We need to make fast decisions when facing enemy, to quickly estimate how many enemies are there in front of us, or which tree has more fruits than others. When we look to a set of trees, each one distant of the other one hundred meters, the size of the trees, as seen by us, follows a logarithmic scale, is not linear. Quite paradoxically, we perceive time logarithmically: yesterday seems longer than the days before. Also, as we get older, time seems to run faster (although quite strangely I start to hear from my younger students the same impression…)

Then we can go further in our thinking, and to argue that the madness of our current so-called “civilizations” is also based on this distortion of our minds, printed by the “school”, just invented for the purpose of formatting our minds. Therefore, we are framed to obey, to accept the “political” directives of our  governments, enacted by lazy and corrupted people paid by outlaw corporations, without being able to imagine that other solutions (and probably much better) are still possible. No wonder that our planet is suffering with our immaturity and lack of ability to understand natural phenomena with mostly an exponential character (in fact, mathematically the inverse of the logarithmic scale behavior).

But, what to think about our common use of numbers. Normal people do count numbers from 1 up to 10 without troubles, but common mathematical operations are usually done with the help of a calculator. The truth is that our brain has limited capacities. When adding decimal numbers, our brain can process 12 bits (binary digit) per second; when counting individual objects we can process 3 bits per second(!); but our visual processing module can easily process 10,000,000 bits per second. In fact, only a limited number of the population possess the extraordinary calculation skills, an outstanding capacity in doing mental calculations. Win Klein was one of those, while working at CERN. Among this class of people, we may also include Daniel McCartney, Salo Finkelstein, Alexander Aitken and Daniel Tammet.

The existence of this kind of individuals naturally induces us to think about how they achieved their high level of attainment, how they succeeded in mastering their minds. Contrary to the general view present in our societies, the understanding the need of a healthy body in a healthy mind is not as simple as it seems. Besides the interest that each of us has in developing a healthy and vigorous (musculated) body, we also have interest in developing our “physical education” [1]. This means to have delicacy of hearing (if you live in the jungle you need to recognize the slightest rustling in order to survive), the quality of sight (some blind persons know colors by touch, see movie below), the skill of hands and general motor skills of our bodies (athletes do need them,…and we, altogether), the sensitiveness of odor and taste (so important to wine enologists).

But if the impressions made by the exterior on the sense-organs, the nerves and brains seem to assure the certainty of the information we got from the world, the ensuing mental calculations and the body behaviour may be somehow confusing. When we say that our senses deceive us, in fact what we miss is the “art of interpretation”.

“Trained perceptions are the very foundation of all mental power. – W. R. C. Latson, in Secrets of mental supremacy.

The problematic of the mind-body is well illustrated with the Helen Keller’s extrarodinary case. She was blind, deaf and mute. Keller did not have the innate ability and the genetic basis for interpreting the symbols that we have today, we people with normal faculties. The role of her teacher was crucial, because she discovered that Helen could learn the meaning of “water” making water flow through her hands while writing on them the word “water”. This technique applied to Helen opened the whole world of the language and understanding and the story should be read by everyone [2,3], since it is an extraordinary example of the power of will in transforming our physical and intelectual capacities.

The slow evolution that have shaped the brain and the primitive number processor that we all possess inside our brain, allows us to use current multiplication algorithm, the concepts of square root, complex numbers and the geometrization of physics. The evolutionary processes that shaped the brain are also at the basis of the generation of the delicate mechanism of vision, the hummingbird’s wings or the intricate mechanism of the ants body.

Areas of the brain. Image credit: dhushara.com

In order to train the mind we must develop:

  • Memory (you may have all kind of powers and unusual experiences, but unless you have the ability to retain them in your memory, you will have no effective use of it all );
  • Power of associating memories (it has no use to have stock all collection of knowledge, data and experiences, if you are unable to associate, analyze, compare, constrast  all that in order to be capable to write , have nothing to say);
  • Imagination and judgment (this means that  all previous faculties are not enough, unless you have imagination, the power of recombine certain memories in a new pattern, with a new ideas and mental images);
  • Concentration (the entire mind must be engaged upon one task, and this power of mind needs the assistance of a well developed will, which represents in fact the difference between strong men and women and the cowards, between the honored and the unnoticed, between the masters and the serfs, in fact, in what actual “civilization” seems to divide society…)

Hummingbird's wings are a marvellous product of mother nature. Im. credit: animals.nationalgeographis.com

Probably Platon and Aristotles although brilliant minds as they were, missed a mode deep understanding of nature if they had an observatory. But «in the last years of Pericles, the Athenian people passed a decree by which all who propagated astronomical theories were made liable to impeachment» (see p. 9, ref [4,5]).

We must bear in mind that to acquisition of knowledge is done only through our bodies, by experiencing our bodies.

The Astronomer is a painting of the Dutch painter Johannes Vermeer in 1668, and it is frequently shown in books as an illustration of the movement of thought itself, giving an illusion of the concentration on a challenging task. The man which is portrayed is apparently Antonie van Leeuwenhoek, a man who was a tradesman and a scientist (should scientists return to this new form of living?…), the founder of microbiology.

Several techniques may speed up mental calculations. As you may notice, these techniques are merely cultural inventions with the purpose to help us to handle numbers (remember, we all are excellents to count from 1 to 4…). And very good books were written to improve our abilities [1,2].

For example, suppose you want to multiply two numbers with the second digit below five by 11, such as 32×11. We sum mentally 3+2, which gives 5, and insert 5 between 3 and 2.

What about 85 times 11, 85×11? Apply the same technique

Astronomer, paiting by the Dutch Johannes Vermeer. Image credit: http://www.essentialvermeer.com

In the following video you can learn to multiply with your fingers. It is amazing too.

In a world in turmoil, with unsecure foundations at a civilizational level, it is certain that this extraordinary capacity of mankind, the creative and constructive thinking, must be enhanced in our society. One requisite is certainly, to start talking more about the qualities of mankind, and not only dwelling about our defaults, our inabilities, which is too much frequently vehiculated by media, tearing down the dignity of the human beings, and putting serious threats to his freedom.

So, finally Epictetus was right: my mind to me a kingdom is (read, in addition Ref. [9]).

Dear Readers, I hope to join you in January. Until then,

Enjoy Christmas time!


[1] Secrets of mental supremacy, by W. R. C. Latson (The Elizabeth Towne, Holyoke, 1913)

[2] The Faith of Helen Keller-The Life of a Great Woman, With Selections From Her Writings (Hallmark Editions, Kansas City, 1967)

[3] The Story of My Life, Helen Keller (Doubleday & Company, New York, 1905)

[4] The Life of Pythagoras with his Symbols and Golden Verses, by M. Dacier (Jacob Tonson, 1707) -> One of the best books on Pithagoras

[5] Pythagoras: His Life, Teaching, and Influence, by  Christoff Riedweg and Steven Rendall

[4] Oswald Spengler, The Decline of the West (Alfred A Knopf, New York, 1927)

[5] Physical Science: Men and Concepts, by  Guy C. Omer (Heath, 1962)

[6] Alex Bellos, The Way We See Numbers, The Guardian 01/04/2010

[7]Pierre Pica et al., Exact and Approximate Arithmetic in Amazonia Indigen Group, Science 306, 499 (2004)

[8] Individual differences in arithmetic, Ann Dowker (Psychology Press, New York, 2005)

[9] Creed of Epictetus


FN[1] The history of statistics begin with the oldest civilizations, Egypt (maps of the country and construction of the pyramids, c. 3050, and registration by the police of all heads of families…, c. 600), Judea (census of the population), China (c. 2300, with a description of the provinces by Yuking), Persia (messenger mail(!), survey roads), But the first fundamental manuscript on this matters was written by Sebastian Muenster, entitled Cosmographia. to know more, read History, Theory, and technique of Statistics, by August Meitzen (American Academy of Political and Social Science, Philadelphia, 1891)

FN[2] Lord Kelvin created the absolute scale of temperature, enhancing the study of Thermodynamics and Statistical Physics.

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