Mathematical Tablet, c. 1790 BCE

Two fragments of a 3,800-year-old clay tablet show a collection of geometry exercises and questions in Akkadian script that would have confronted young Babylonian scholars. Here students are asked to calculate the areas of various subdivisions of squares.
The British Museum.

Did you know the Names of Large Numbers

Large numbers are numbers that are significantly larger than those ordinarily used in everyday life, for instance in simple counting or in monetary transactions. The term typically refers to large positive integers, or more generally, large positive real numbers, but it may also be used in other contexts.

Very large numbers often occur in fields such as mathematics, cosmology, cryptography, and statistical mechanics. Sometimes people refer to numbers as being "astronomically large". However, it is easy to mathematically define numbers that are much larger even than those used in astronomy.

வட்டத்தின் சுற்றளவில் தமிழர்கள்!

கணித வரலாற்றில் தமிழருக்கு என்றும் முதன்மை இடம் உண்டு. வட்டதிற்கான சுற்றளவை முதலில் கண்டவர்கள் நாம் என்ற வகையில் பெருமைப்பட்டுக் கொள்ளலாம். வட்டத்திற்கான சுற்றளவை கணக்கதிகாரம் என்ற தொன்மையான நூல் விளக்குகின்றது. இதில் வட்டதிற்கான சுற்றளவை செய்யுள் வடிவில் கூறியுள்ளார்.

கணக்கதிகாரப் பாடல் : 50

...

“விட்ட மதனை விரைவா யிரட்டித்து
மட்டுநாண் மாதவனில் மாறியே – எட்டதனில்
ஏற்றியே செப்பியடி லேறும் வட்டத்தளவும்
தோற்றுமெப் பூங்கொடி நீ சொல் “

விளக்கம்:

விட்டம்தனை விரைவா யிரட்டித்து = விட்டத்தின் இரு மடங்கு = 2r + 2r = 4r (விட்டம் = 2r ); மட்டு நாண் மாதவனில் மாறியே = 4 ஆல் பெருக்கு; எட்டதனில் ஏற்றியே = 8 ஆல் பெருக்கு; செப்பியடி = 20 ஆல் வகு.

வட்டத்தின் சுற்றளவு = ( 4r x 4 x 8 ) / 20 = 32 / 5 r = 2 ( 16/5) r = 2 π r

இங்கு π = 16 / 5 = 3.2 ( இது ஓரளவுக்குத் துல்லியமான தோராயமே ) இன்று நாம் பயன்படுத்தும் வட்டத்தின் சுற்றளவு = 2 π r என்ற சூத்திரத்தை நம் முன்னோர்கள் பல நூறு ஆண்டுகளுக்கு முன்பே அறிந்துள்ளனர் என்று அறியும் போது உண்மையில் நாம் பெருமைப்பட்டுக் கொள்ளலாம்.

The Deepest Uncertainty When a hypothesis is neither true nor false.

.

Cecile G. Tamura

Georg Cantor died in 1918 in a sanatorium in Halle, Germany. A pre-eminent mathematician, he had laid the foundation for the theory of infinite numbers in the 1870s. At the time, his ideas received hostile opposition from prominent mathematicians in Europe, chief among them Leopold Kronecker, once Cantor’s teacher. In his first known bout of depression, Cantor wrote 52 letters to the Swedish mathematician Gösta Mittag-Leffler, each of which mentioned Kronecker.

But it was not just rejection by Kronecker that pushed Cantor to depression; it was his inability to prove a particular mathematical conjecture he formulated in 1878, and was convinced was true, called the Continuum Hypothesis. But if he blamed himself, he did so needlessly. The debate over the conjecture is profoundly uncertain: in 1940 Kurt Gödel proved that the Continuum Hypothesis cannot be disproven (technically speaking, that the negation of the Hypothesis cannot be proven), and in 1963 Paul Cohen proved that it cannot be proven. Poor Cantor had chosen quite the mast to lash himself to.

How is it possible, though, for something to be provably neither provable nor disprovable? An exact answer would take many pages of definitions, lemmas, and proofs. But we can get a feeling for what this peculiar truth condition involves rather more quickly.

Cantor’s Continuum Hypothesis is a statement regarding sizes of infinity. To see how infinity can have more than one size, let’s first ask ourselves how the sizes of ordinary numbers are compared. Consider a collection of goats in a small forest. If there are six goats and six trees, and each goat is tethered to a different tree, then each goat and tree are uniquely paired. This pairing is called a “correspondence” between the goats and the trees. If, however, there are six goats and eight trees, we will not be able to set up such a correspondence: no matter how hard we try, there will be two trees that are goat-free.

Correspondences can be used to compare the sizes of much larger collections than six goats—including infinite collections. The rule is that, if a correspondence exists between two collections, then they have the same size. If not, then one must be bigger. For example, the collection of all natural numbers {1,2,3,4,…} contains the collection of all multiples of five {5,10,15,20,…}. At first glance, this seems to indicate that the collection of natural numbers is larger than the collection of multiples of five. But in fact they are equal in size: every natural number can be paired uniquely with a multiple of five such that no number in either collection remains unpaired. One such correspondence would involve the number 1 pairing with 5, 2 with 10, and so on.

If we repeat this exercise to compare “real” numbers (these include whole numbers, fractions, decimals, and irrational numbers) with natural numbers, we find that the collection of real numbers is larger. In other words, it can be proven that a correspondence cannot exist between the two collections.

The Continuum Hypothesis states that there is no infinite collection of real numbers larger than the collection of natural numbers, but smaller than the collection of all real numbers. Cantor was convinced, but could never quite prove it.

To see why, let’s begin by considering what a math proof consists of. Mathematical results are proven using axioms and logic. Axioms are statements about primitive mathematical concepts that are so intuitively evident that one does not question their validity. An example of an axiom is that, given any natural number (which is a primitive concept), there exists a larger natural number. This is self-evident, and not in serious doubt. Logic is then used to derive sophisticated results from axioms. Eventually, we are able to construct models, which are mathematical structures that satisfy a collection of axioms.

Crucially, any statement proven from axioms, through the use of logic, will be true when interpreted in any model that makes those axioms true.

It is a remarkable fact that all of mathematics can be derived using axioms related to the primitive concept of a collection (usually called a “set” in mathematics). The branch of mathematics that does this work is known as set theory. One can prove mathematical statements by first appropriately interpreting the statement in the language of sets (which can always be done), and then applying logic to the axioms of sets. Some set axioms include that we can gather together particular elements of one set to make a new set; and that there exists an infinite set.

Kurt Gödel described a model that satisfies the axioms of set theory, which does not allow for an infinite set to exist whose size is between the natural numbers and the real numbers. This prevented the Continuum Hypothesis from being disproven. Remarkably, some years later, Paul Cohen succeeded in finding another model of set theory that also satisfies set theory axioms, that does allow for such a set to exist. This prevented the Continuum Hypothesis from being proven.

Put another way: for there to be a proof of the Continuum Hypothesis, it would have to be true in all models of set theory, which it isn’t. Similarly, for the Hypothesis to be disproven, it would have to remain invalid in all models of set theory, which it also isn’t.

It remains possible that new, as yet unknown, axioms will show the Hypothesis to be true or false. For example, an axiom offering a new way to form sets from existing ones might give us the ability to create hitherto unknown sets that disprove the Hypothesis. There are many such axioms, generally known as “large cardinal axioms.” These axioms form an active branch of research in modern set theory, but no hard conclusions have been reached.

The uncertainty surrounding the Continuum Hypothesis is unique and important because it is nested deep within the structure of mathematics itself. This raises profound issues concerning the philosophy of science and the axiomatic method. Mathematics has been shown to be “unreasonably effective” in describing the universe. So it is natural to wonder whether the uncertainties inherent to mathematics translate into inherent uncertainties about the way the universe functions. Is there a fundamental capriciousness to the basic laws of the universe? Is it possible that there are different universes where mathematical facts are rendered differently? Until the Continuum Hypothesis is resolved, one might be tempted to conclude that there are.

https://en.wikipedia.org/wiki/Continuum_hypothesis

https://plato.stanford.edu/entries/continuum-hypothesis/

http://nautil.us/issue/2/Uncertainty/the-deepest-uncertainty

https://www.ias.edu/ideas/2011/kennedy-continuum-hypothesis

http://mathworld.wolfram.com/ContinuumHypothesis.html

https://plus.maths.org/cont…/…/issue47/features/elwes1/index

Hypatia of Alexandria: mathematics against intolerance

This is the story of the first mathematical woman in history, Hypatia of Alexandria

Ancient scholars, as in the case of the Greeks, were broad-minded thinkers; The same person could be a physician, mathematician, astronomer, and philosopher. This seems impossible in our day, with the increase of the degree of specialization and the necessary efforts to be an expert in a certain field.
In the case of Hypatia of Alexandria (IV and V centuries) she developed a scientific work in fields such as mathematics and astronomy. History has been demonstrating the skills of women in science and how there is no intellectual disadvantage of women vis-a-vis men. The gender gap is simply a matter of social roles assigned over centuries to both gender.
Hypatia was greatly influenced in the intellectual world by her father Theon , Greek philosopher and mathematician who was the last director of the Museum of Alexandria . Education provided by his father was a liberal education, knowing Hypatia today as the legendary freethinker before the intoleracia.
Hypatia was a free woman, educated in the neo - Platonic school and leader of the neo - Platonic belief in Alexandria. She never married: despite her beauty and eloquence, she devoted her life to scientific work.
Her research was reflected in numerous manuscripts, such as "Comments Diophantus Arithmetic" . Diophantus was a Greek mathematician who lived during the third century and was considered the father of algebra and arithmetic , whose work focused on algebraic equations and number theory. Its name comes the Diophantine equations . In an edition of this book Diophantus was where Pierre de Fermat wrote his famous phrase:
On the other hand, it is not possible to divide a cube into two cubes, or a double cube in two cubits, or in general a power superior to the square, to infinity, in two powers of the same degree: I have found a truly admirable proof of this affirmation . The margin requirement could not contain it.
Another of her contributions was the edition of "The Elements of Euclid", with the comments of his father Theon, an expert in the Euclidean work. The Elements of Euclid has been more editions the book after the Bible , and includes a complete treatise on geometry.
She also rewrote a treatise on the "Conic" of Apollonius . His reinterpretations simplified the concepts of Apollonius, with a more accessible language and making it a manual easily readable by the interested reader.
Unfortunately, many of the contributions of Hypatia were lost.
Thanks to his correspondence with her student Synesius (later Bishop of Ptolemais), we know many of their other contributions. Synesius of Cyrene shared the taste for mathematics and astronomy of his tutor, but took other courses, becoming the cleric philosopher. Sinesio records the uniqueness of Hypatia as an intellectual. She claims her authorship in the construction of an astrolabe, a hydrometer and a hydroscope.
The astrolabe is an instrument constructed to determine the positioning of the stars in the sky and served as a guide for sailors, engineers or architects to determine distances by triangulation. A curious fact is the use of this instrument by the Muslim sailors, with whom they were guided to determine the position of the Mecca and thus to be able to pray.
Hypatia also stood out for her praises and for being a follower of Neopitagorism and Neoplatonism; She became an eminent professor of mathematics , teaching at home to a select group of aristocrats, both pagan and Christian. Her intelligence gained him the position of counselor of Orestes, prefect of the Roman Empire of the East, exalumno of his.
The special character of Hypatia, with equal treatment of all her pupils, educated from tolerance and rationality, aroused a series of envy that would raise many enemies against her. Like pagan, supporter of the Greek scientific rationalism and influential political personage, friend of Orestes, underwent the intense hostility between Cirilo (Christian fanatic, bishop of Alexandria) and Orestes. The accusations against her of blasphemy and anticristina, for the simple fact of refusing to traicinar her ideals and to leave paganism, propitiated the ambush of the bishop Cirilo, dragging popular masses to brutally assassinate it.
However, Hypatia never proclaimed her dislike of Christianity . Simply, with her open character she accepted any type of disciple, regardless of his religious beliefs.
Hypatia's life was an interesting life. The life of a strong woman, fighting for their ideals and began the study of science in a few centuries when women were denied access to knowledge . Thus it portrayed in the recent film "Agora" , directed by Alejandro Amenabar in 2009, where Hypatia appears preoccupied with Euclid's Elements, Apollonius conic and the heliocentric system of Aristarchus of Samos. In addition, it is presented as a teacher of astronomy, in a class in which raises the questions: Why do the stars fall ?, why only turn from west to east? Why, instead, does the handkerchief fall to the ground? The students respond and Hypatia analyzes their answers and explains from a Ptolemaic point of view:
"The stars do not fall because they are in a circle. On earth they fall because it is the center of the universe.

"
Manuel de León (CSIC, Founder of ICMAT, Royal Academy of Sciences, Royal Canaria Academy of Sciences, ICSU) and Cristina Sardon (ICMAT-CSIC).
https://en.wikipedia.org/wiki/Hypatia
https://www.bbvaopenmind.com/hipatia-de-alejandria-matema…/…
http://www.smithsonianmag.com/…/hypatia-ancient-alexandria…/
https://www.youtube.com/watch?v=GvbBvq6CRkU
https://www.youtube.com/watch?v=uOXKF1mb9Hc

Fibonacci number & Golden ratio ...??

There is a special ratio that can be used to describe the proportions of everything from nature's smallest building blocks, such as atoms, to the most advanced patterns in the universe, such as unimaginably large celestial bodies. Nature relies on this innate proportion to maintain balance, but the financial markets also seem to conform to this 'golden ratio.' Here we take a look at some technical analysis tools that have been developed to take advantage of it.

The Mathematics

Mathematicians, scientists and naturalists have known this ratio for years. It's derived from something known as the Fibonacci sequence, named after its Italian founder, Leonardo Fibonacci (whose birth is assumed to be around 1175 AD and death around 1250 AD). Each term in this sequence is simply the sum of the two preceding terms (1, 1, 2, 3, 5, 8, 13, etc.).

But this sequence is not all that important; rather, it is the quotient of the adjacent terms that possesses an amazing proportion, roughly 1.618, or its inverse 0.618. This proportion is known by many names: the golden ratio, the golden mean, PHI and the divine proportion, among others. So, why is this number so important? Well, almost everything has dimensional properties that adhere to the ratio of 1.618, so it seems to have a fundamental function for the building blocks of nature.

Prove It!

Don't believe it? Take honeybees, for example. If you divide the female bees by the male bees in any given hive, you will get 1.618. Sunflowers, which have opposing spirals of seeds, have a 1.618 ratio between the diameters of each rotation. This same ratio can be seen in relationships between different components throughout nature.

Still don't believe it? Need something that's easily measured? Try measuring from your shoulder to your fingertips, and then divide this number by the length from your elbow to your fingertips. Or try measuring from your head to your feet, and divide that by the length from your belly button to your feet. Are the results the same? Somewhere in the area of 1.618? The golden ratio is seemingly unavoidable.

But that doesn't mean that it works in finance … does it? Actually, the markets have the very same mathematical base as these natural phenomena. Below we will examine some ways in which this ratio can be applied to finance, and we'll show you some charts to prove it!

The Fibonacci Studies and Finance

When used in technical analysis, the golden ratio is typically translated into three percentages: – 38.2%, 50% and 61.8%. However, more multiples can be used when needed, such as 23.6%, 161.8%, 423% and so on. There are four primary methods for applying the Fibonacci sequence to finance: retracements, arcs, fans and time zones.

1. Fibonacci Retracements

Fibonacci retracements use horizontal lines to indicate areas of support or resistance. They are calculated by first locating the high and low of the chart. Then five lines are drawn: the first at 100% (the high on the chart), the second at 61.8%, the third at 50%, the fourth at 38.2% and the last one at 0% (the low on the chart). After a significant price movement up or down, the new support and resistance levels are often at or near these lines. Take a look at the chart below, which illustrates some retracements:

Created Using MetaTrader

2. Fibonacci Arcs

Finding the high and low of a chart is the first step to composing Fibonacci arcs. Then, with a compass-like movement, three curved lines are drawn at 38.2%, 50% and 61.8%, from the desired point. These lines anticipate the support and resistance levels, and areas of ranging. Take a look at the chart below, which illustrates how these arcs do this:

Created Using MetaTrader

3. Fibonacci Fans

Fibonacci fans are composed of diagonal lines. After the high and low of the chart is located, an invisible vertical line is drawn though the rightmost point. This invisible line is then divided into 38.2%, 50% and 61.8%, and lines are drawn from the leftmost point through each of these points. These lines indicate areas of support and resistance. Take a look at the chart below:

Created Using MetaTrader

4. Fibonacci Time Zones

Unlike the other Fibonacci methods, time zones are a series of vertical lines. They are composed by dividing a chart into segments with vertical lines spaced apart in increments that conform to the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, etc.). These lines indicate areas in which major price movement can be expected.

Created Using MetaTrader

Conclusion

These Fibonacci studies are not intended to provide the primary indications for timing the entry and exit of a stock; however, they are useful for estimating areas of support and resistance. Many people use combinations of Fibonacci studies to obtain a more accurate forecast. For example, a trader may observe the intersecting points in a combination of the Fibonacci arcs and resistances. Many more use the Fibonacci studies in conjunction with other forms of technical analysis. For example, the Fibonacci studies are often used with Elliott Waves to predict the extent of the retracements after different waves. Hopefully you can find your own niche use for the Fibonacci studies, and add it to your set of investment tools!

Read more: Fibonacci And The Golden Ratio | Investopedia http://www.investopedia.com/articles/technical/04/033104.asp#ixzz4Uf747hg7

உங்களுக்கு golden ratio என்பதை பற்றி தெரியுமா?
Fibonacci number...??
இவை நம்மை சுற்றி இயற்கையில் ..இயற்கையாகவே அமைந்து உள்ள ஒரு அதிசய கணிதம்..
இவைகள் என்னவென்பதை பற்றி இன்று பார்ப்போம்..

உங்களுக்கு "pi "என்பதை பற்றி தெரிந்திருக்கும்..அதாவது.. அந்த 3.14....
ஆனால் "phi" பற்றி தெரியுமா?
இது கிரேக்க எழுத்துக்களில் 21 வது எழுத்து. கணிதத்தில் இதுதான் " கோல்டன் ரேஷியோ" காண சிம்பல்.

ஒரு பெரிய கோட்டை இரண்டாக பிரிக்கிறீர்கள்.. சரிபாதியாக அல்ல ஒன்னு பெரிது ஒன்னு சின்னதாக... இப்போது அந்த பெரிய கோட்டை சின்ன கோட்டை கொண்டு வகுத்தால் வரும் எண் இருக்கிறதே.. அது அந்த கோட்டின் மொத்த நீளத்தை பெரிய கோடை கொண்டு வகுத்தால் வரும் எண்ணுக்கு சமமாக இருப்பதை ...அந்த விகிதாசாரதைதான் Golden ratio என்கிறார்கள்.
கோல்டன் ரேஸ்யோவை குறிக்க phi யை பயன் படுத்த காரணம்.. முன்பு சொன்ன அந்த விகிதாச்சாரம் phi இன் மதிப்பாகிய 1.6 க்கு நெருக்கமாக இருப்பது தான்.

இந்த phi ஆர்வலர்கள் ஆய்வாளர்கள்
அந்த காலத்தில் கட்டபட்ட.. மர்மம் நிறைந்த கட்டிடமான பிரமிட்.. இந்த கோல்டன் ரேஷியோ படி கட்ட பட்டுள்ளதை கவனித்தார்கள். க்ரேட் பிரமிடு கிசா இருக்கிறதே அதன் ஓவொரு பக்கத்திலும் அதன் நீளம் 786 அடி. மேலும் அதன் உயரம் 481 அடி.. அதாவது இவை இரண்டுக்கும் உள்ள விகிதம் 1.57 .அதாவது phi யின் மதிப்புக்கு மிக நெருக்கமாய்.
இப்படி பிரமிடுகளில் மட்டும் அல்லாமல் பழைய கட்டிடங்கள் பல வற்றில் இந்த கோல்டன் ரேஸ்யோ பயன்படுத்த பட்டிருப்பதை கவனிதார்கள்.

பிபோனாச்சி நம்பர் என்பதை நீங்கள் கேள்வி பட்டிருப்பீர்கள்.

1200 களில் லியனார்டோ பிபோனச்சி
என்ற கனிதவியலாளரால் இது கண்டுபிடிக்க பட்டது.
அதாவது 0,1,1,2,3,5,8,13,21,34, இப்படி எழுதுவது தான் பிபோநாச்சி நம்பர்.. இதில் உள்ள சிறப்பு என்ன என்று கவனித்தால் எந்த ஒரு நம்பரும் அதற்கு முந்தய இரு நம்பர்களின் கூட்டு தொகையாக இருக்கும்.
இந்த எண்களின் மாறும் விகிதம் தான் கோல்டன் ரேஷியோ.... இது phi மதிப்பிற்கு நெருக்கமாக உள்ளது.. அதிலும் எண்கள் பெரிதாக பெரிதாக phi மதிப்பாகிய 1.618 ஐ அதிகம் நெருங்குகிறது.

உதாரணமாக 3 கும்5 கும் இடையிலான விகிதம் 1.666 ஆனால் 13 க்கும் 21 க்கும் இடையிலான விகிதம் 1.625... இன்னும் பெரிய என்னானால்... 144 கும் 233 கும் பார்த்தால் 1.618 இப்படி....

லியர்னடோ டாவின்ஸி தான் வரைந்த ஓவியங்களில் இந்த கோல்டன் ரேஷியோ சமாச்சாரத்தை பயன்படுத்தி இருக்கிறார்.
அவரது லாஸ்ட் சப்பர் ஓவியத்தில் இயேசு தனது சீடர்களுடன் உணவு உண்ணும் காட்சியில் உள்ள மேஜை மற்றும் சில பொருட்களில் இந்த விகிதாச்சாரத்தை பயன்படுத்தி வரையப்பட்டுள்ளது. அவரது மோனோலிசா ஓவியம் கூட இதை பயன் படுத்தி வரைய பட்டது தான்.

இதை மனிதன் பயன்படுத்தி இருக்கிறான் என்பதை இவ்வளவு நேரம் நான் சொன்னது இல்லை ஆச்சர்யம்..
இனி சொல்ல போவது தான் ஆச்சர்யம்..

இயற்கை தனது படைப்பில் இதை தொடர்ந்து பயன்படுத்தி வருகிறது.
ஒரு சூரிய காந்தி பூவில் மையத்தில் சுருள் சுருளாக இருக்கிறதே அந்த டிசைன் அதை உற்று கவனித்தால் கோல்டன் ரேஷ்யோ படி அமைந்துள்ளதை கவனிக்கலாம் .. ஒரு மரத்தில் கிளைகள் அதில் இலைகள் எப்படி எந்த எண்ணிக்கையில் படி படியாக வளர்கிறது என்று பார்த்தால் இந்த விகிதம் தான்.
தாவரத்தில் மட்டும் அல்ல ஒரு நத்தையின் ஓடு இந்த விகித அடிப்படையில் தான் சுருளாக இருக்கிறது.. விலங்குகளின் உடலமைப்பை கவனித்தால் இந்த விகிதம் இருக்கிறது.
அவ்வளவு ஏன் மனிதனின் கைகள் மற்றும் விரல்கள் இந்த விகிதத்தில் தான் அமைந்துள்ளது என்றால் பார்த்து கொள்ளலாம்.

இப்போது இனொன்று சொல்கிறேன் இயற்கை உயிரினங்களில் மட்டும் இல்லை .. பல இயற்கை ஆற்றல்களில் ...படைப்புகளில் இதை வெளிப்படுத்துகிறது.. என்பது இன்னும் ஆச்சரியமானது.
சூறாவளி ஏற்படும் போது அதன் சுழல் கவனித்தால் இந்த விகிதத்தை பார்க்கலாம்..
கடல் அலையை கவனித்தால் இந்த கோல்டன் ரேஷியோ சமாச்சாரத்தை பார்க்கலாம்.
இன்னும் எங்கெங்கே இயற்கை இதை பயன் படுத்துகிறது என்பதை ஆராய மனிதன் சிரிதினும் சிறிதான DNA சுருளை ஆராய்ந்து பார்க்க அந்த சுருளே இந்த விகிதத்தில் தான் அமைந்துள்ளது என்பதை பார்த்து அதிசையித்தான்.. பிறகு பெரிதினும் பெரிதான கேலக்சிகளை அதன் சுருள் அமைப்பை ஆராய்ந்த மனிதன் இந்த பிரபஞ்சமே அந்த கோல்டன் ரேஷியோ வை பயன்படுத்தி அமைந்துள்ளதை கண்டு மேலும் அதிசயித்து போனான்..

இந்த பிரபஞ்சம் அறிவியலால் ஆனது என்று எப்போதும் சொல்லி வந்துள்ளோம்..ஆனால் அது நாணயத்தின் ஒரு பக்கம் தான் அதன் அடுத்த பக்கத்தில் பிரபஞ்சம் முழுக்க.. முழுக்க கணிதத்தால் ஆனது என்ற அதிசயத்தை வியந்து நோக்கி இன்றைய கட்டுரையை நிறைவு செய்கிறேன்.

அன்பு நண்பன் அறிவியல் காதலன் :ரா.பிரபு

Evolution of mathematics traced using unusually comprehensive genealogy database

Most of the world’s mathematicians fall into just 24 scientific 'families', one of which dates back to the fifteenth century. The insight comes from an analysis of the Mathematics Genealogy Project (MGP), which aims to connect all mathematicians, living and dead, into family trees on the basis of teacher–pupil lineages, in particular who an individual's doctoral adviser was.

“You can see how mathematics has evolved in time,” says Floriana Gargiulo, who studies networks dynamics at the University of Namur, Belgium and who led the analysis.
This result just tickles our fancy a bit.
Our belief here is that 'the mathematical sense" is primary in our brain's functioning (prior to logic and language and indeed, as a key component of what we now know as the Salience Newtork and its operation...without which the Salience Network would be inept and/or inadequate.
And, of course, in tandem with that belief we consider that evolution must show that this particular aspect of our brains has emerged as a key factor in the rest of our culture and society's taking shape.
The fundamental area of the brain in which we now have considerable interest is the Intra Parietal Sulcus...and it is that area where all "numerical sense" is centered, not only numbers, but geometry and space and all comparisons of relative magnitude, whether greater or less, brighter or dimmer, louder or more quiet, sooner or later, faster or slower, friendlier or less friendly...and so on...more or less in every way, incluiding how simnilar or not to a past experience to merit a "novelty" response or a danger response or to simply ignore.... is centered there.
http://www.jneurosci.org/content/28/46/11819.short
We can speak about these matters in ordinary language, and say that what we encounter may be somewhere 'along a spectrum", or in a "dimension" or part of a "space" but that is only possible because the sense of number allows us to take experiences and consolidate them within our brains along such a dimension, spectrum or space.
We now recognize the incredible importance of the Salience Network. Organisms must have a way of distinguishing situations in such a manner that what is 'salient" is noticed and attended to and dealt with. But how does an organism realize which moment of experience presents it with salience. Of course, absolute danger is salient and that is recognizable.
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2899886/
But the key in most life situations that result in adaptation and learning and intelligent responses that develop is to understand "novelty' as it is called nowadays...but which tells us when two experiences are different enough to be salient and thus to warrant adjustment and action on our part.
Without the role played by the Intra Parietal Sulcus the immediately surrounding areas..how would "comparisons" be made. Does an organism actually bring to awareness two experiences, two representations or one of each and then do some sort of inventory of features side by side and then make decisions after such a stepwise item by item comparison.
There must be some better way to achieve those "comparisons". That better way evolves to become "measuring" and to conceiving of things experience along spectra of comparison.
https://www.edge.org/…/stanislas_dehaene-what-are-numbers-r…
And thus the Salience Network would not be able to function without this IPS being a kernel aspect of its function and all the resultant connectivity in the brain.
This particular genealogical study revealed 84 distinct family trees with two-thirds of the world’s mathematicians concentrated in just 24 of them. The high degree of clustering arises in part because the algorithms assigned each mathematician just one academic parent: when an individual had more than one adviser, they were assigned the one with the bigger network.
But the phenomenon chimes with anecdotal reports from those who research their own mathematical ancestry, says MGP director Mitchel Keller, a mathematician at Washington and Lee University in Lexington, Virginia. “Most of them run into Euler, or Gauss or some other big name,” he says
http://www.nature.com/…/majority-of-mathematicians-hail-fro…

Five brilliant mathematicians and their impact on the modern world

Cecile G. Tamura

We owe a great debt to scores of mathematicians who helped lay the foundation for our modern society with their discoveries. Here are some of the most important.

Math. It's one of those things that most people either love or hate. Those who fall on the hate side of things might still have nightmares of showing up for a high school math test unprepared, even years after graduation. Math is, by nature, an abstract subject, and it can be hard to wrap your head around it if you don't have a good teacher to guide you.

But even if you don't count yourself a fan of mathematics, it's hard to argue that it hasn't been a vital factor in our rapid evolution as a society. We reached the moon because of math. Math allowed us to tease out the secrets of DNA, create and transmit electricity over hundreds of miles to power our homes and offices, and gave rise to computers and all that they do for the world. Without math, we'd still be living in caves getting eaten by cave tigers.

Our history is rich with mathematicians who helped advance our collective understanding of math, but there are a few standouts whose brilliant work and intuitions pushed things in huge leaps and bounds. Their thoughts and discoveries continue to echo through the ages, reverberating today in our cellphones, satellites, hula hoops and automobiles. We picked five of the most brilliant mathematicians whose work continues to help shape our modern world, sometimes hundreds of years after their death.

http://www.mnn.com/…/5-brilliant-mathematicians-and-their-i…

 

Photo: Wikimedia Commons
Isaac Newton (1642-1727)

Sir Isaac Newton, considered by many to be the greatest scientist of all time. There aren't many subjects that Newton didn't have a huge impact in — he was one of the inventors of calculus, built the first reflecting telescope and helped establish the field of classical mechanics with his seminal work, "Philosophiæ Naturalis Principia Mathematica." He was the first to decompose white light into its component colors and gave us the three laws of motion, now known as Newton's laws. (You might remember the first one from school: "Objects at rest tend to stay at rest and objects in motion tend to stay in motion unless acted upon by an external force.")

We would live in a very different world had Sir Isaac Newton not been born. Other scientists would probably have worked out most of his ideas eventually, but there is no telling how long it would have taken and how far behind we might have fallen from our current technological trajectory.

 

Photo: Wikimedia Commons
Carl Gauss (1777 - 1855)

Carl Gauss. If Newton is considered the greatest scientist of all time, Gauss could easily be called the greatest mathematician ever. Carl Friedrich Gauss was born to a poor family in Germany in 1777 and quickly showed himself to be a brilliant mathematician. He published "Arithmetical Investigations," a foundational textbook that laid out the tenets of number theory (the study of whole numbers). Without number theory, you could kiss computers goodbye. Computers operate, on a the most basic level, using just two digits — 1 and 0, and many of the advancements that we've made in using computers to solve problems are solved using number theory. Gauss was prolific, and his work on number theory was just a small part of his contribution to math; you can find his influence throughout algebra, statistics, geometry, optics, astronomy and many other subjects that underlie our modern world.

 

Photo: Wikimedia Commons
John von Neumann (1903-1957)

John von Neumann was born in Budapest a few years after the start of the 20th century, a well-timed birth for all of us, for he went on to design the architecture underlying nearly every single computer built on the planet today. Right now, whatever device or computer that you are reading this on, be it phone or computer, is cycling through a series of basic steps billions of times over each second; steps that allow it to do things like render Internet articles and play videos and music, steps that were first thought up by John von Neumann.

Von Neumann received his Ph.D in mathematics at the age of 22 while also earning a degree in chemical engineering to appease his father, who was keen on his son having a good marketable skill. Thankfully for all of us, he stuck with math. In 1930, he went to work at Princeton University with Albert Einstein at the Institute of Advanced Study. Before his death in 1957, von Neumann made important discoveries in set theory, geometry, quantum mechanics, game theory, statistics, computer science and was a vital member of the Manhattan Project.

 

Photo: Wikimedia Commons
Alan Turing (1912 - 1954)

Alan Turing a British mathematician who has been call the father of computer science. During World War II, Turing bent his brain to the problem of breaking Nazi crypto-code and was the one to finally unravel messages protected by the infamous Enigma machine. Being able to break Nazi codes gave the Allies an enormous advantage and was later credited by Winston Churchill as one of the main reasons the Allies won the war.

Besides helping to stop Nazi Germany from achieving world domination, Alan Turing was instrumental in the development of the modern day computer. His design for a so-called "Turing machine" remains central to how computers operate today. The "Turing test" is an exercise in artificial intelligence that tests how well an AI program operates; a program passes the Turing test if it can have a text chat conversation with a human and fool that person into thinking that it too is a person.

 

Photo: Wikimedia Commons
Benoit Mandelbrot (1924-2010)

Benoit Mandelbrot landed on this list thanks to his discovery of fractal geometry. Fractals, often-fantastical and complex shapes built on simple, self-replicable formulas, are fundamental to computer graphics and animation. Without fractals, it's safe to say that we would be decades behind where we are now in the field of computer-generated images. Fractal formulas are also used to design cellphone antennas and computer chips, which takes advantage of the fractal's natural ability to minimize wasted space.

Mandelbrot was born in Poland in 1924 and had to flee to France with his family in 1936 to avoid Nazi persecution. After studying in Paris, he moved to the U.S. where he found a home as an IBM Fellow. Working at IBM meant that he had access to cutting-edge technology, which allowed him to apply the number-crunching abilities of electrical computer to his projects and problems. In 1979, Mandelbrot discovered a set of numbers, now called the described by science-fiction writer Arthur C. Clarke as Mandelbrot set, that were "one of the most beautiful and astonishing discoveries in the entire history of mathematics."

Benoit Mandelbrot died of pancreatic cancer in 2010.

The Seventeen equations that changed the course of history

about these wonderful equations that have shaped mathematics and human history:

1. The Pythagorean Theorem

This theorem is foundational to our understanding of geometry. It describes the relationship between the sides of a right triangle on a flat plane: square the lengths of the short sides, a and b, add those together, and you get the square of the length of the long side, c.

This relationship, in some ways, actually distinguishes our normal, flat, Euclidean geometry from curved, non-Euclidean geometry. For example, a right triangle drawn on the surface of a sphere need not follow the Pythagorean theorem.

2. Logarithms

Logarithms are the inverses, or opposites, of exponential functions. A logarithm for a particular base tells you what power you need to raise that base to to get a number. For example, the base 10 logarithm of 1 is log(1) = 0, since 1 = 100; log(10) = 1, since 10 = 101; and log(100) = 2, since 100 = 102.

The equation in the graphic, log(ab) = log(a) + log(b), shows one of the most useful applications of logarithms: they turn multiplication into addition.

Until the development of the digital computer, this was the most common way to quickly multiply together large numbers, greatly speeding up calculations in physics, astronomy, and engineering.

3. Calculus

The formula given here is the definition of the derivative in calculus. The derivative measures the rate at which a quantity is changing. For example, we can think of velocity, or speed, as being the derivative of position - if you are walking at 3 miles (4.8 km) per hour, then every hour, you have changed your position by 3 miles.

Naturally, much of science is interested in understanding how things change, and the derivative and the integral - the other foundation of calculus - sit at the heart of how mathematicians and scientists understand change.

4. Law of Gravity

Newton’s law of gravitation describes the force of gravity between two objects, F, in terms of a universal constant, G, the masses of the two objects, m1 and m2, and the distance between the objects, r. Newton’s law is a remarkable piece of scientific history - it explains, almost perfectly, why the planets move in the way they do. Also remarkable is its universal nature - this is not just how gravity works on Earth, or in our Solar System, but anywhere in the Universe.

Newton’s gravity held up very well for 200 years, and it was not until Einstein’s theory of general relativity that it would be replaced.

5. The square root of -1

Mathematicians have always been expanding the idea of what numbers actually are, going from natural numbers, to negative numbers, to fractions, to the real numbers. The square root of -1, usually written i, completes this process, giving rise to the complex numbers.

Mathematically, the complex numbers are supremely elegant. Algebra works perfectly the way we want it to - any equation has a complex number solution, a situation that is not true for the real numbers : x2 + 4 = 0 has no real number solution, but it does have a complex solution: the square root of -2. Calculus can be extended to the complex numbers, and by doing so, we find some amazing symmetries and properties of these numbers. Those properties make the complex numbers essential in electronics and signal processing.

6. Euler’s Polyhedra Formula

Polyhedra are the three-dimensional versions of polygons, like the cube to the right. The corners of a polyhedron are called its vertices, the lines connecting the vertices are its edges, and the polygons covering it are its faces.

A cube has 8 vertices, 12 edges, and 6 faces. If I add the vertices and faces together, and subtract the edges, I get 8 + 6 – 12 = 2.

Euler’s formula states that, as long as your polyhedron is somewhat well behaved, if you add the vertices and faces together, and subtract the edges, you will always get 2. This will be true whether your polyhedron has 4, 8, 12, 20, or any number of faces.

Euler’s observation was one of the first examples of what is now called atopological invariant - some number or property shared by a class of shapes that are similar to each other. The entire class of 'well-behaved' polyhedra will have V + F – E = 2. This observation, along with with Euler’s solution to the Bridges of Konigsburg problem, paved the way to the development of topology, a branch of maths essential to modern physics.

7. Normal distribution

The normal probability distribution, which has the familiar bell curve graph to the left, is ubiquitous in statistics.

The normal curve is used in physics, biology, and the social sciences to model various properties. One of the reasons the normal curve shows up so often is that it describes the behaviour of large groups of independent processes.

8. Wave Equation

This is a differential equation, or an equation that describes how a property is changing through time in terms of that property’s derivative, as above. The wave equation describes the behaviour of waves - a vibrating guitar string, ripples in a pond after a stone is thrown, or light coming out of an incandescent bulb. The wave equation was an early differential equation, and the techniques developed to solve the equation opened the door to understanding other differential equations as well.

9. Fourier Transform

The Fourier transform is essential to understanding more complex wave structures, like human speech. Given a complicated, messy wave function like a recording of a person talking, the Fourier transform allows us to break the messy function into a combination of a number of simple waves, greatly simplifying analysis.

The Fourier transform is at the heart of modern signal processing and analysis, and data compression.

10. Navier-Stokes Equations

Like the wave equation, this is a differential equation. The Navier-Stokes equations describes the behaviour of flowing fluids - water moving through a pipe, air flow over an aeroplane wing, or smoke rising from a cigarette. While we have approximate solutions of the Navier-Stokes equations that allow computers to simulate fluid motion fairly well, it is still an open question (with a million dollar prize) whether it is possible to construct mathematically exact solutions to the equations.

11. Maxwell’s Equations

This set of four differential equations describes the behaviour of and relationship between electricity (E) and magnetism (H).

Maxwell’s equations are to classical electromagnetism as Newton’s laws of motion and law of universal gravitation are to classical mechanics - they are the foundation of our explanation of how electromagnetism works on a day to day scale. As we will see, however, modern physics relies on a quantum mechanical explanation of electromagnetism, and it is now clear that these elegant equations are just an approximation that works well on human scales.

12. Second Law of Thermodynamics

This states that, in a closed system, entropy (S) is always steady or increasing. Thermodynamic entropy is, roughly speaking, a measure of how disordered a system is. A system that starts out in an ordered, uneven state - say, a hot region next to a cold region - will always tend to even out, with heat flowing from the hot area to the cold area until evenly distributed.

The second law of thermodynamics is one of the few cases in physics where time matters in this way. Most physical processes are reversible - we can run the equations backwards without messing things up. The second law, however, only runs in this direction. If we put an ice cube in a cup of hot coffee, we always see the ice cube melt, and never see the coffee freeze.

13. Relativity

Einstein radically altered the course of physics with his theories of special and general relativity. The classic equation E = mc2 states that matter and energy are equivalent to each other. Special relativity brought in ideas like the speed of light being a universal speed limit and the passage of time being different for people moving at different speeds.

General relativity describes gravity as a curving and folding of space and time themselves, and was the first major change to our understanding of gravity since Newton’s law. General relativity is essential to our understanding of the origins, structure, and ultimate fate of the Universe.

14. Schrodinger’s Equation

This is the main equation in quantum mechanics. As general relativity explains our Universe at its largest scales, this equation governs the behaviour of atoms and subatomic particles.

Modern quantum mechanics and general relativity are the two most successful scientific theories in history - all of the experimental observations we have made to date are entirely consistent with their predictions. Quantum mechanics is also necessary for most modern technology - nuclear power, semiconductor-based computers, and lasers are all built around quantum phenomena.

15. Information Theory

The equation given here is for Shannon information entropy. As with the thermodynamic entropy given above, this is a measure of disorder. In this case, it measures the information content of a message - a book, a JPEG picture sent on the internet, or anything that can be represented symbolically. The Shannon entropy of a message represents a lower bound on how much that message can be compressed without losing some of its content.

Shannon’s entropy measure launched the mathematical study of information, and his results are central to how we communicate over networks today.

16. Chaos Theory

This equation is May’s logistic map. It describes a process evolving through time - xt+1, the level of some quantity x in the next time period - is given by the formula on the right, and it depends on xt, the level of x right now. k is a chosen constant. For certain values of k, the map shows chaotic behaviour: if we start at some particular initial value of x, the process will evolve one way, but if we start at another initial value, even one very very close to the first value, the process will evolve a completely different way.

We see chaotic behaviour - behaviour sensitive to initial conditions - like this in many areas. Weather is a classic example - a small change in atmospheric conditions on one day can lead to completely different weather systems a few days later, most commonly captured in the idea of a butterfly flapping its wings on one continent causing a hurricane on another continent.

17. Black-Scholes Equation

Another differential equation, Black-Scholes describes how finance experts and traders find prices for derivatives. Derivatives - financial products based on some underlying asset, like a stock - are a major part of the modern financial system.

The Black-Scholes equation allows financial professionals to calculate the value of these financial products, based on the properties of the derivative and the underlying asset.

This article was originally published by Business Insider.

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