Tag Archives: Men of Mathematics

The House of Wisdom

Dear lovers of knowledge,

                A few days back when I was researching for my next book( “Pilgrims of Episteme”, expected in 2013) I came across some very interesting information. I will share that with you in this VERITAS.

                As you are aware, the Greeks had a great tradition of science, mathematics and philosophy. People like Socrates, Aristotle, Euclid, Plato etc – there is a long list of Greek greats in these fields. In the second century BC after the Punic and Macedonian wars, the Greek city states declined and the Romans became the dominant power in the Mediterranean.  The Romans were very powerful in terms of military and had a great political system. However they were weak in Science, Arts, Philosophy etc. So they admired the Greeks for these things. On the other hand the Greeks considered themselves much superior in terms of culture and knowledge but they feared the military might of the Romans.

                After the Punic wars Greek culture became very popular amongst the Romans. They learnt the Greek language, copied their architecture, even identified their gods with the Greek gods. The good thing that the romans did was to preserve the Greek culture, language, philosophy and science. The Romans did not add anything to Greek thought but did their best to preserve it. So Greek science and philosophy continued to progress under Roman rule.

                In the 4th and 5th century AD the Roman empire declined. Also there was a lot of conflict between the new Christian religion and the older pagan religion. This resulted in the destruction of the Great Library at Alexandria in 391 AD( See VERITAS: Loss of a Great Library, 27 July 2000, http://unvarnished-veritas.posterous.com/the-loss-of-a-great-library). Rome was attacked and looted several times by Gothic tribes between 409 and 476 AD. One of the tribes, the Vandals were extremely ruthless and destroyed just about everything. The word vandalism is derived from this tribe. In 476 AD the emperor of Rome abdicated and the Roman empire finished. Europe descended into the dark ages. The knowledge of Greek was lost, the Pope acquired great powers and the tradition of science and questioning came to an end.

                The eastern part of the Roman empire became the Byzantine empire and it is here that the Greek language and culture continued to flourish to some degree. However the Byzantines could not add much to the great works of the Greek.

                So the Romans and The Byzantines cannot be considered the real inheritors of Greek science and mathematics.  To find the real succesors of the great Greeks we will have to look elsewhere- a different time and a different place.

                Lets now talk about Baghdad in the 8th century AD.  At that time the Abbasid Caliphate was in power. The Abbasid Caliphs had come to power in 750 AD and in 762 AD shifted their capital to a new city: Baghdad. The Abbasids were influenced by the Hadith( sayings by Prophet Mohammed) such as “The Ink of the Scholar is more precious than the blood of the martyr” and “Seek knowledge from the cradle to the grave” and they started making Baghdad a centre of learning and knowledge. The third Caliph of the Abbasid Calipate, Harun al-Rashid formed Bayt Ul-Hikma or “The House of Wisdom”.  Harun al-Rashid was a great patron of science, music and art. We all know him from the book “A Thousand  and one Nights” also known as “Arabian Nights”.  Many stories in this book involve Harun al-Rashid and his vizier, Jafar.

  The House of Wisdom became the centre of what is known as the Translation movement. The Abbasids had formed a library at the House of Wisdom and they were obsessed with getting every book in the world translated into Arabic and kept in the library. First the Persian texts were translated. Then the focus shifted to Greek texts. Harun al-Rashid’s son, Al-Mamun who ruled from 813 to 833 took the House of Wisdom to great heights. Al-Mamun was hugely influenced by Greek thought. It is said that when he was a young man he once dreamt of Aristole. In the dream Aristotle came to him and spoke about philosophy and politics. Under Al-Mamun’s rule, the works of Plato, Aristotle, Galen, Eulid, Archimedes – every great Greek philosopher, mathematician, physician were translated into Arabic.

                Now the translation of works of Philosophy, Science, Mathematics and medicine is not an easy task. The translator has to be knowledgeable about these subjects to  be able to translate them well. Al-Mamun was not just a collector of books, he was also a collector of the best minds. So everyday, translators, scribes, scientists, mathematicians, physicians met at the House of Wisdom to translate works by the great Greeks. The House of Wisdom became so famous that scholars from around the world came to join it.

                Now what happens when scholars get together under one roof? New ideas, new thoughts emerge. So what started as a Translation Movement soon became a thriving place where original work was carried out in science, mathematics and other subjects.  The scholars at the House of Wisdom extended the works of the great Greeks and took science and mathematics forward.

                Another thing that happened at the House of Wisdom was that texts from India and China were also translated. The scholars were able to combine the works from the east and the west to make great leaps in knowledge. Let me take an example: the Greeks and Romans could not multiply the way we can. Yes, Aristotle, Plato, Euclid, Archimedes etc could not multiply as easily as a school kid can in today’s age. They used to add repeatedly. Why? Because they did not have a decimal system: Imagine trying to multiply XCVIII with DCCVII! Now lets convert it to another notation: 98 X 707. Now it becomes so simple. Why? Because the decimal system gives each place a value- it is a positional system. A positional system requires the number 0. The Arab scholars got this system from India and were quick to realize how superior it was over the Greek-Roman system. Al-Khwarizimi wrote “On the calculation with Hindu numerals” in 825 AD. Al-Kindi wrote a 4 volume subject on the same subject in 830 AD.  

                Now that was one example. The House of Wisdom was the centre of world science and mathematics at that time. Let me give you a few more examples: The word Algorithm comes from Al-Khwarizimi who as we saw earlier wrote the book on Indian number system. Al-Khwarizimi described a step by step way of solving algebraic equations- we call it Algorithm these days. Talking about algebra- the word algebra comes from Al-Khwarizimi ‘s book Kitab Al-jebr. The words Alchemy, Chemistry, Almanac, Amber, Alcohol, Alkali, Azimuth come to us from Arabic and are all associated with some major contributions towards science or mathematics by the scholars at the House of Wisdom.

                The works of many greek philosophers and scientists would have been unknown to us if they were not carefully translated and preserved by the scholars at the House of Wisdom. Lets take the case of Galen, the great Greek man of medicine. The 3 million words that he wrote are available to us only because the Arabs translated them and kept them so carefully. Mohammed al-Razi was known as the “Galen of the Arabs”. He used Galen’s works as a basis and then  wrote a 23 volume text on medicine. He formed a fuction between Persian, Greek and Indian systems of medicine.

  The Islamic Golden Age of Science lasted from 750 Ad to 1258 AD- 5 centuries of scientific progress by the greatest minds of the day based on the great works of Greeks, Indians and Chinese. But then, as it typically happens, lust for power destroys love for knowledge. Baghdad was conquered in the year 1258 by the Mongols led by Hulgau Khan who was the grandson of Genghiz Khan. The House of Wisdom was destroyed. The Mongols took the books from the House of Wisdom and threw them into the Tigris to form a bridge on the river so that they could cross it!

                Friends, from 476 AD to the 14th century, Europe was in dark ages. It was a place of orthodox religion, superstition and ignorance. The centre of science, philosophy, culture and mathamatics was in Baghdad, in the House of Wisdom. When Renaissance began in Italy in the 14th century, it was based on what the Arabs had preserved. The House of Wisdom prepared Europe for Renaissance. Without the House of Wisdom, Renaissance would not have happened, or would have been delayed by several centuries.

                In the end I will leave you with a thought by Al-Kindi, a mathematician and a scholar from the House of Wisdom: “We ought not to be embarrassed of appreciating the truth and of obtaining it wherever it comes from, even if it comes from races distant and nations different from us. Nothing should be dearer to the seeker of truth than the truth itself.”

 

 Like you, a lover of Knowledge

Kanwar

 

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Go wondrous creature, mount where science guides

go measure earth, weigh air, state the tides,

instruct the planets in what orbs to run

correct old time, regulate the sun

======================================================================

 

Read my latest book, “Shadows of Lost Time”. http://shadows-of-lost-time.posterous.com/

The Copernicus of Geometry

 

     A few weeks back we had discussed the method of mathematical proof called “Reductio Ad Absurdum”. We had also learnt that Euclid used this method in several proofs. In that VERITAS I had mentioned that Euclid made a wrong proof of the fifth postulate or the parallel postulate.

        In today’s VERITAS we will discuss the parallel postulate and also read the story of the person who dared to challenge this postulate.

My inspiration for this VERITAS is E.T Bell’s book “Men of Mathematics”.

I can safely bet that anyone who reads this book cannot escape from falling in love with mathematics and a desire to become one of them!

      The parallel postulate can be stated in various ways. The simplest definition is this: ” if there is a straight line and a point X not on that line then there is only one line that can pass through X and never intersect the given line( even if they are both extended to infinity).

This line is parallel to the given line”. This looks pretty obvious if you draw lines on a piece of paper. Euclid tried to prove this but was never really satisfied. So he used it as an axiom. That was about 300 BC. No one challenged this for about 2200 years. And then came Lobatchewsky, the Copernicus of geometry ….

       Nicolas Lobatchewsky was the son of a minor government official. He was born on November 2, 1793 in Russia. He did his schooling at Kazan. He entered the Kazan University in 1807. In 1811 he obtained his degree from the university. In 1813 he was appointed to a teaching post in that University.

      The Kazan University was a new and a small one. And it was in a small town. Lobatchewsky professorship in the university was not restricted to teaching mathematics. There was little order in the university and in the town. So Lobatchewsky did nearly everything : he taught Mathematics, Physics, Astronomy. He was inchange of the laboratories, he was inchange of the library, he had to supervise the elementary schools in the area, he would even dust the books and catalog the books in the library himself. So he did just about everything.

In 1827 Lobatchewsky was appointed the rector of the university.

That was one of the top posts there. But that did not deter Lobatchewsky from doing everything. He would still take a mop and clean the lab.

When the government decided to modernize the university by making a few new buildings, Lobatchewsky was incharge of overseeing the project. But Lobatchewsky could not just oversee! He had to be in it! So he learnt architecture and designed the buildings himself! The university was his life!

         But mathematics was his first love. In 1823 he produced a work that would challenge the Great Euclid himself. He created a new geometry in which the parallel postulate is not valid. And this new geometry was fully self consistent( or complete). This geometry later came to be known as non-Euclidian geometry. Euclid’s geometry was the geometry of flat surfaces, Lobatchewsky’s geometry was the geometry of curved surfaces.

      To get an idea of geometries other than the euclidian lets take an example of our own curved earth. When we travel between 2 distant places the shortest path is not a straight line but a curve. ( Remember the last time you flew from Jalandhar to Southall the TV in the aeroplane showed you a curved path). The shortest distance between two points on the earth is called a geodesic. How is this defined: Imagine a flat plane that passes through the 2 points and cuts the earth and passes through the centre of the earth. The line that this plane makes on the surface of the earth is called a geodesic.

      The geometry on a sphere is different from euclidian geometry. On a plane surface ( euclidian geometry) two lines will either be parallel, or cross each other at exactly one point. On a sphere 2 geodesics will either be parallel or cross each other at TWO points. There are several other differences.

      On a sphere the sum of the angles of a triangle is always more than 180 degrees.

      And there are several other geometries: the hyperbolic geometry and the geometry on a pseudo-sphere. On a pseudo sphere there are exactly 2 lines which pass through a point and are parallel to a line. Then there is Riemann’s geometry in which 2 angles of a rectange are obtuse. Riemann’s geometry was used by Einstein in his General theory of Relativity in which he showed that gravity curves space-time.

      So Lobatchewsky’s genuis was in challenging a 2200 year old axiom. He opened the gates for several types of geometries. He showed that Euclid’s geometry was only a special case.

        But his genuis as a mathematician was never recognized in his lifetime. In 1846 he was charged with mismanagement in the university and was asked to leave. He loved the university, it was HIS university.

This broke him. His health suffered and he became blind. In 1856 he died a sad and broken man.

      Physicists are now realizing that all of Physics and all the laws that govern the Universe may be manifestations of the Geometry of space time.

      A few poetic words on geometry by Emily Dickenson:

 

      Best Witchcraft is Geometry

      —————————

 

      Best Witchcraft is Geometry

        To the magician’s mind —

        His ordinary acts are feats

        To thinking of mankind.

 

Kanwar

 

|======================================================|

  Go, wondrous creature! mount where Science guides:                 

  Go, measure earth, weigh air, and state the tides:                 

  Instruct the planets in what orbs to run,                           

  Correct old time and regulate the Sun;                             

|======================================================|

Women of Mathematics : Sophie Germain

 

Marie Curie worked hard for 10 years. She sacrificed her life for Science. In the end she was rewarded with a Nobel Prize.

  Have we not heard such tales of Scientists sacrificing pleasures of life and working real hard to contribute to Science. Such tales are written by historians and parents who want to “inspire” thier kids. These are all false stories. A Scientist loves science. They appreciate the beauty of the laws of nature. They want to spend their lives surrounded by this beauty. Science makes them happy.They do not sacrifice the pleasures of life. Science is their pleasure of life. So the true story would be ” Marie Curie enjoyed herself for 10 years finding why radioactivity happens. Everyday
was a new adventure. In the end she also got a lil medal called the Nobel Prize.”

       Today in VERITAS we will remember the story of Sophie Germain. You might think that it is a story of struggle and sacrifice. Try and read it from the correct angle: it is a story of love.

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       Sophie Germain was born in 1776 in Paris.  When she was 13, Sophie read the story of Archimedes. She was surprised to read that Archimedes was so engrossed in doing mathematics that he ignored the soldier who threatened to kill him(and killed him). She thought that this subject must be really interesting. She started studying mathematics on her own and she loved it.

       Now in those days women were not supposed to study Science and Mathematics. Her parents discouraged her. They even went to the extent of taking away her clothes at night and depriving her of heat and light so that she would only sleep and not get up to do mathematics. She
could not study in the daytime because they would not let her. But she loved the subject and eventually her parents realized the futility of keeping her away from it. They let he do whatever she wanted.

       Women were not allowed to study in universities so she studied on her own. She borrowed lecture notes from Lagrange’s students to study. She assumed the male name of M. LeBlanc and even submitted a paper to Lagrange. Lagrange was impressed by the work and wanted to meet
his student. He was surprized when he realized that his student was actually a female and that she had been studying on her own. He advised
her on her work and encouraged her to do more.

       Sophie became interested in number thery and started corresponding with Gauss. She did not reveal her real name and corresponded as LeBlanc. They discussed mathematics for three years(1804 to 1807 ) before Sophie revealed her true identity. Gauss was thrilled to find that his
“pen-friend” was a very talented woman. She even saved Gauss’ life. In 1806 when Napoleon invaded Germany she contacted a french commander, who was a friend of her family and told him that Gauss should not be harmed. She remembered Arichmedes and did not want Gauss to have the same fate.

Sophie proved that if x,y and z are integers then if x^5+y^5=z^5 then x,y or z must be divisible by 5. Sophie’s proof was a major step in proving Fermat’s last theorem.

In 1808 the physicist Chladini conducted experiments on vibrating plates which exibited some strange figures called  Chaldini figures. The
Institut de France set a prize for anyone who could mathematically explain these figures.

       Sophie Germain started thinking about this problem. In 1811 her’s was the only entry trying to explain the figures. But her analysis was wrong( since she was not formally educated and did not have access to modern journals she had holes in her knowledge). She did more research. In 1813 she came up with a better explanation of the figures. This work was rejected too. In 1815 however her third work on the subject got her the prize.

       Most Scientists and Mathematicians however chose to ignore her and her work.  Sophie was not discouraged by this and kept working on mathematics till her death in 1831. Her death certificate did not list her as a Scientist or a Mathematician, but a property holder.

      Sophie did not have any formal education.  Despite that she made great contributions to number theory, theory of elasticity, theory of vibrations etc.

       Most of us lay too much stress on degrees acquired in colleges and universities. Are these degrees really important? Isnt having a “fire within” more important. Isnt love the supreme conquerer?

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kanwar

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 Go, wondrous creature! mount where Science guides:
 Go, measure earth, weigh air, and state the tides:
 Instruct the planets in what orbs to run,
 Correct old time and regulate the Sun;
|======================================================|

Creative Commons License
Veritas by Kanwarpreet Grewal is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License.

The Brachistochrone Problem

 

      You should be very careful when you talk about a scientist named Bernoulli. Because  Bernoulli was not one famous scientist. The Bernoullis  were a family of mathematicians and  scientists.

In three generations the Bernoulli family produced 8 eminent mathematicians. Most of them did not choose mathematics as their profession but “drifted into it inspite of themselves as a dipsomaniac returns to alcohol”(E.T Bell). They fought with each other over mathematics. Once Johannes Bernoulli threw his son out of the house because the son had won a mathematics prize for which the father had also competed! “After all, if rational human beings get excited over a game of cards, why should they not blow up over mathematics which is infinitely more exciting?”(E.T Bell) 🙂

        In the year 1696 Johann Bernoulli challenged the mathematicians of Europe with a problem:”If two points are fixed at random(one at a lower height than the other but not vertically below) then what is the shape of the incline along which a body should slide so that it reaches the other point in the least possible time?” The answer, VERITAS readers is not a straight line! The mathematicians spent six months trying to solve this problem but they all failed. One evening this problem reached the great Newton. Newton had just returned from his duties as the warden of the royal mint. He was tired but not tired enough for mathematics. The problem was solved after dinner and the correct answer was despached anonymously to Bernoulli.

Bernoulli on reading the solution immediately recognised that it was Newton who had solved the problem. He said “one can recognise the lion by the impression of his paws”!

     The problem is known as the brachistochrone(“shortest time”) problem . And the correct answer to the problem is a cycloid. So if the shape of the curve between the two points is a cycloid then the body will take the least time to travel between the two points. See attached file to find what a cycloid looks like.  The cycloid and similar curves were studied extensively by the Bernoillis. They also found that the cycloid is also a tautochrone i.e a body placed at any height on a cycloid would take the same time to come down! See attached figure:

if a body is placed at the highest point A or a point lower than A (near B) the time taken to come to the bottom of the curve under gravity would be the same!

      The branch of mathematics that deals with finding the path of least/most length or least/most time between two points under various conditions is called the calculus of variations. The Bernoullis laid the foundations of this branch of mathematics.

      Now I shall attempt to shock you, VERITAS readers.  There is a principle in Physics called “Fermat’s principle of least time”. According to this principle light travels along a path between two points which takes it least time. Note that this is true even if light has to bend or pass throught objects of different kinds. Light chooses the path that will take the least time! This principle is so deep and fundamental that you can derive the laws of reflection and refraction of light using this principle alone! How much does light bend when it goes into a material such as water. It will take the path that will minimise time and you will know how much it will bend! How does light decide which path to take? Surely light cannot select paths! Is there intention in nature? Does light make decisions? The answer to this question lies deep inside the laws of Quantum Mechanics. Such minimum principles are fundamental in nature and surprise everyone including most Physicists. 

                  Give me to learn each secret cause;

                  Let number’s, figure’s, motion’s laws

                  Reveal’d before me stand;

                  These to great Nature’s scenes apply,

                  And round the globe, and thro’ the sky,

                  Disclose her working hand

                              –Mark Akenside

(Most of the material for this VERITAS has been taken from the book “Men of Mathematics” by ET Bell and the book “Classical Mechanics” by Herbert Goldstein)

 

Kanwar

 

|======================================================|

  Go, wondrous creature! mount where Science guides:                 

  Go, measure earth, weigh air, and state the tides:                 

  Instruct the planets in what orbs to run,                          

  Correct old time and regulate the Sun;                              

|======================================================|

Diophantine Equations

Diophane was a Greek Mathematician in whose honour a whole field of mathematics is named . The fied is Diophantine equations and is a part of the number theory . ( Number theory is a field that deals with numbers and their properties 🙂 .. sometimes I make such informative statements )

 

      Okay . Now the difference between Diophantine Equations and other kinds of equations is that in Diophantine Equations we insist on integral solutions . That means that we dont want decimal solutions .

For example one Diophantine Equation can be :

 

      1027x + 712y = 1

 

Note that we have one equation in 2 variables and we are supposed to find integral solutions to the problem . One solution is x= -165 and y = 238 .

Of course there may be others.

 

      One famous incident related to Diophantine equations : One day

Hardy( Ramanujan’s mentor ) went to visit Ramanujan in the hospital

(Ramanujan was suffering from tuberculosis ) . Hardy says that the cab

he took had a very uninteresting number 1729. Ramanujan smiled and said :

1729 is the smallest number that can be expressed as the cube of two

numbers in two different ways . What Ramanujan had solved was the Diophantine

Equation :    a^3 + b^3 = c^3 + d^3

 

 Fermat’s last thoerem is also an example of Diophantine Equation .

He said that there cannot be a solution to the equation :

 

x^3 + y^3 = z^3 .

 

Note that u are allowed only whole numbers in place of x,y and z .

 

      There is no general way to solve problems of this kind .

Hilbert the famous German Methematician challenged mathematicians

to find a algorithmic approach to solve problems of this kind .

But it doesnt work ! In fact there are theorems which say that

given a Diophantine Equation there is no way to determine if it

has a solution or not . 

Leonhard Euler: Analysis Incarnate

Leonhard Euler was one of the greatest mathematicians of all time .

He made a huge number of contributions in almost all fields of mathematics.

He was the most prolific mathematical writer of all times. It is said that “Euler calculated without effort , as men breathe … ” .

 

      Euler was born in Basel,Switzerland on April 15 1707 . His father wanted him to become a clergyman and send him to University of Basel . Euler graduated from the univ in 1724 where he had studied theology and Hebrew. But he came under the influence of a famous mathematician Johann Bernoulli and Bernaulli was impressed with the mathematical abilities of Euler and convinced his dad to let him become a methematician .

 

      In 1727 he joined the math dept of Academy of St.Petersnerg , Russia.

He married and had kids . But the amount of math he did was just too much !

While in Russia he became blind in one eye , because he worked on a problem for for 3 days and nights without rest !

 

      In 1741 he moved to Berlin and took up a position there .

In 1766 he went back to Petersburg to become the director of the academy there . There is an interesting story here : The queen of Russia Catherine was disgusted by a philosopher Diderot who would find new arguments to prove that GOd does not exist . Catherine invited Euler to shut up Diderot . Someone told Diderot that Euler has a mathematical proof to prove the existance of God . Diderot asked to hear it .

Euler said “Sir  (a + b^n)/n = x , hence God exists . Reply .”

Diderot had no idea of what hit him and soon returned to france .

 

      something about his contributions : He combined Leibniz’s Diffrenential calculus and Newton’s Method of fluxions to form what we know as calculus.

He invented the theory of functions . He invented Calculus of Variations .

He made NUmber theory into a science . He made significant contributions to the mechanics of rigid bodies . He was responsible for many commonly used mathematical notation like e , pi , the sigma symbol . He also did a lot of work in goemetry .

 

      He produced more mathematical work that any other dude . His combined works will fill 70 large volumes . It is said that he could produce a Mathematics paper in 30 minutes . And he had a huge pile lying on his desk . Whenever some Mathematics journal ran short of papers they would just pick up a few from the pile and publish them without review !! And Euler could work anywhere . He didnt need peace or quiet ! He would have a child in his lap and be thinking of cutting edge mathematics .

 

      In 1766 he became completely blind . But that didnt slow his mathematicsl pace . He had a wonderful memory and could produce mathematics even without writing . He would think and talk and his son would write !

 

      He was just too good . Prople called him “Analysis Incarnate ” .

He died in 1783 .

 

Godel’s Theorem

First of all : thanks to Sandeep Seth for lending me a wonderful book that introduced me to Godel’s theorem ( and Escher’s Paintings ) The name of the book is “Godel,Escher,Bach The Eternal Golden Braid”.

A wow book !

      After the theory of sets was invented Mathematicians started facing

a new kind of problem . Problems related to statements that were neither

true or false . Statements whose truth was indeterminate . FOr example take

the following statement :

   A set G that contains all sets that dont contain themselves .

         Now does G contain itself ?

Lets suppose it does . But then there is a problem because then the

definition of G would be wrong .

Now lets suppose that it does not . But then the set G would not contain a

set G ( which does not contain itself ) . And that is incomplete .

 

Paradox ! Paradoxes of this kind are called Russel’s Paradox .

 

Mathematicians didnt like such inconsistencies in their Math . So

Bertrand Russel and Whitehead tried to make a new system that has

no inconsistiencies. They called it Principia Mathematica .

 

Another Great Mathematician Hilbert had bigger plans . He challenged

Mathematicians to make a formal system that is complete and consistent

and we could base all mathematics on the top of that .

 

What is a formal system :

 

Any system in mathematics contains some axioms . These axioms are

things you assume to find theorems . For example lets invent a new

system . We have a string : KO . The rules of the system are :

 

a) if we have O at the end then we can add a T at the end of the

   string .

 

b) if we have OT ate the end of the string we can add another OT at the

   end .

c) If we have 2 sets of OT at the end then we can remove them .

 

so our three rules and our strings are axioms . Stuff we start off with .

 

And a new string that you may invent is a theorem .

 

Is the string K a theorem in our system ? lets check :

 

start off with KO and apply rule a to it . Our string becomes KOT .

Now apply rule b we get KOTOT . NOw apply rule c we get K .

 

So K is a theorem in our system . Any mathematical system proceeds

in a similar way . Theorems and axioms . SO either a statement can be

a theorem( it is true ) or it is false ( its opposite is a theorem ) .

 

How Hilbert’s program was to reduce mathematics to string analysis .

So that all you need to know is the axioms and you can prove anything .

Just keep applying axiom after axiom and u get a new string which you

call a theorem . You can express any theorem in math as a string :

 

for example :

there exists no x,y,z : x^3 + y^3 = z ^3

 

is a statement of fermat’s last theorem

 

So Hilbert’s program tries to reduce Mathematics to string analysis .

And he propopsed to start off with Principia Mathematica .

 

But Kurt Godel published a theorem in 1931 that destroyed such hopes .

He said 2 important things :

 

1) YOU CAN REDUCE ANY FORMAL SYSTEM TO A SYSTEM INVOLVING NATURAL NUMBERS.

   for example in our system we can give K a no 10 . O is 2 and T is 3 .

   so KO is 1023 and rule c) is that if you have anything followd by 2323 then

   you can remove that . SO K is a theorem means that 10 is a valid

   no in our system . Similarly all formal systems can be expressed in

   terms of natural numbers and then in terms of each other .

 

 

b) YOU CAN ALWAYS CREATE A NEW THEOREM THAT CANNOT BE PROVEN ( AND ITS

   NEGATION CANNOT BE PROVEN TOO . Ie ANY COMPLETE FORMAL SYSTEM CONTAINS

   UNDECIDABLE THEOREMS ) . THis means that there will always be theorems

   that will escape the net of our formal methods . These are kind of

   theorems that reference themselves . Godel showed this for Principia

   Mathematica and then since all formal systems are equiivalent thus

   it applies to all branches of Mathematics . Thus provability

   is weaker than a formal system .

 

So what this showed was that Mathematics cannot be made into a formal

system and that the role of intution is still very important .

THis was a great blow to Hilbert’s program . Now no one tries to make

systems that dont have inconsistencies . Because Godel showed that

you cannot do that . If a system is strong enough then there will

always have inconsistencies .