निश्चय (niścaya)

Kurt Godel was the first to prove that mathematics is not certain. According to him, there will be statements about numbers which are true but cannot be proved. These statements can be either true or false. In the novel “A Certain Ambiguity”, mathematics had entered a different dimension in the lives of the two main fictional characters: Ravi Kapoor and Vijay Sanhi. Mathematics had caused an exciting twist in their lives which you will find out after reading the content of the book that was written by two Indian mathematicians Gaurav Suri and Hartosh Singh Bal.

Ravi is the grandson of Vijay. When Ravi was just a child, his mathematician grandfather showed him a certain problem in a calculator. Vijay wanted to give his grandson a mystical insight towards mathematics. He wanted to know what approach the child would do to give the problem a solution. Unfortunately, the next day became the end of Vijay’s life, but at least he started the mathematical career of his grandson. He was the one who gave Ravi the exciting   feeling for mathematics and without him, the whole story of Ravi’s life would be boring. The next stage of Ravi’s life happened when he got admitted to Stanford.

Ravi had difficulty in choosing his major but his father pursued him to have economics for it will attract many corporate recruiters. He did not know that it is in Stanford where he could find a shocking truth about his grandfather. Nico Aliprantis is a 62-year-old math teacher who became friends with Ravi. He was the teacher of the class “Thinking About Infinity” which Ravi had entered because of curiosity. Obviously, the topic of the course is infinity and through this, Ravi was exposed to notions concerning Zeno’s paradox, convergence of infinite sums, Cantor’s theory of transfinite cardinals, Zermelo-Fraenkel axioms of set theory and especially the Continuum Hypothesis. It was a coincidence having Nico’s subject of dissertation to be Vijay’s specialization, the algebraic number theory. Nico showed Ravi a book and turned to the page where his grandfather’s name was. Ravi had read a footnote saying that the key ideas contained in the paper were formulated by his grandfather while he was serving a prison sentence in Morisette, New Jersey, in 1919.

Vijay was imprisoned due to a blasphemy law. A sheriff complained over his anti-Christian speech in an event. But despite this accusation, Vijay offered no resistance. He and the judge had a conversation in jail which was mainly focused on the axiomatic method of mathematics, first with the Eucledian Geometry then later with the non-Eucledian geometry. There became a conflict between the judge’s Christianity and Vijay’s view of mathematics as a mathematician. Despite the fact that he’s in jail, Vijay was never afraid to reveal his ideas which he got from Euclid’s “Elements”. He believed that like Euclid, we must use the method of incorporating all known geometrical ideas into a self-consistent system based on a handful of axioms.

The mathematical concepts contained in the book began from Zeno’s paradoxes and infinitude of primes down to Godel’s Incompleteness and Paul Cohen’s Consistency theorems. Ravi and his grandfather both faced dilemmas while understanding the certainty of mathematics. They were stuck in the abyss of beliefs and choices while measuring the real scope of their mortal knowledge.

To react with what I read, I appreciate the main purpose of the novel which is to make mathematics beautiful in the eyes of its users. To really love a thing, you must first have a deep connection to it. Through a fictional (story) approach, the authors were able to connect with its readers. The book was able to give entertainment and at the same time knowledge. It encouraged the readers to discover something and enjoy the feeling of discovering it.  Math is all about discoveries and applications, after all. As a mere student who is still in a long-term brain processing, I accepted the book with gladness, but to those grown-ups whom we call now experts, they might have a very different insight. Everyone will surely have their own opinion with regard to this novel but I’m also pretty sure that after reading the book, you will be able to build another perception towards mathematics aside from misery.

A Certain Ambiguity

I never knew that a novel about Mathematics could be possible. And to top it off, it also has such a great plot. This book also tackles about the philosophy, beauty, and the importance of Mathematics to the humans and the world.

On the first part of the story, Ravi Kapoor, the main character, remembers the time when his grandfather gave him a mathematical problem to solve on his calculator but then the grandfather dies the next day. After he dies, the Ravi becomes indifferent towards mathematics.

Ravi then gets accepted to Stanford and takes up a career in Statistics. While taking the subject “Thinking about Infinity”, he then befriends Professor Nico who also specializes in the same field as his grandfather. Nico finds a paper which was developed by Ravi’s grandfather while in jail. Then the story separates into two narrations, Nico presents math topics where “infinity” is the main topic and Ravi researches on his grandfather and discovers that he produced records of philosophical discussions on the nature of truth, certainty and mathematics while on jail.

This book discussed about Zeno's paradoxes and infinitude of primes through Godel's Incompleteness and Paul Cohen's Consistency theorems. This book also talks about the importance of axiomatic method. Overall it was an informative and interesting read. The book was full of informations about mathematics and even philosophy.

Rock, Paper, Scissors: A Book Review

Rock, Paper, Scissors is a popular science book that connects game theory to day to day situations in life and offers strategies for achieving cooperation.

The first chapter “trapped in a matrix” mainly describes the Prisoner’s dilemma and gives the negative connotation that the Nash equilibrium is a logical trap. The book may start off a little bit slowly, either way; these setbacks did not stop me from reading the rest of the book which offers a wide range of interesting examples and explanations.  The second chapter “I cut and you choose offers a nice introduction to the concepts of fair division. Fisher shares fair division with anecdotes like how he got in trouble as a kid shooting fireworks, and as a consequence had to yield fireworks with his brother. The answer he arrived to as a kid was what he knows realizes was an application of the minimax principle. I was also impressed that Fisher discusses the principle of equal division of the contested sum. The third chapter emphasizes a great summary of such problems as the free rider issue and the game of chicken. It is also about the seven of the most interesting game theory problems, which Fisher calls as “the seven deadly dilemmas.”On the other hand, Chapter four is a humorous one, and is about the game “rock, paper, scissors.” I was also amused at how rock, paper, scissors can be used in conflict resolutions, and how it is also played in different countries and races (which, also, has different names). Although the game is used in different day to day situations, the game itself has no pure strategy that may dominate others. On the other hand, conflicts, situations can be solved by adding strategies and converting them to rock-paper-scissors situation. Chapters five through eight are all about cooperation: how we can achieve trust, bargain effectively, and change the game to avoid the “trap” of the Prisoner’s dilemma and other undesirable outcomes. The main fun points are similar in nature as to the other chapters which are the narratives and interesting examples from science.

Fisher used different strategies in the book, one of which is the game “I Cut and You Choose”, which indicates that: When slicing a cake, a person slices two pieces, and then the other person chooses the pieces he/she wants to and for the other person as well. Assuming that the person slicing the cake likes cakes, he/she has the right to make both pieces equal. Fisher suggests using the same strategy to solve disputes amongst different countries.

In conclusion, this author clearly denotes the importance of day to day strategies and techniques in life to achieve unity and cooperation; may it be it in a team or in a group. This is a must read book for achieving coherence and how we maximize gain in competitive situations. In this book Len Fisher starts by demonstrating the limits of game theory: What’s best for you isn’t always what’s best for everyone else. Fisher uses game theories to show how cooperation can be evolved and how cooperation can be achieved even in the most difficult circumstances.  Even so, Rock, Paper Scissors is a wonderfully entertaining introduction to game theory and the science of cooperation. 

Complicated Math that Made Life Easier

Finally, we are at the last instalment of The Story of Maths but actually this is where math really went nuts. Most of the mathematician had a bad life, but that didn’t keep them from being great, instead they became one of the people who made the world what is it now, easier and secure and pleasurable. We owe mathematics a huge amount of recognition for what great things that math contributed to our world.

                The first mathematician featured in this episode is Georg Kantor. He was the first one to really understand the concept of infinity by comparing numbers 1, 2, 3 and so on to 10, 20, 30 … that it has the same size. So what about the fractions in between the numbers? He created an infinite grid which the first round contain the whole numbers fractions with one on the bottom in the second row. 10 harms fractions with on the bottom and so on. Every fraction appears somewhere in this grid for example 4/5. So 4^th column 5^th row is the location where 4/5 is located. He then snaked the grid and turned it into a straight line which every fraction has a corresponding whole number. So the fraction can go to infinity as well as the whole numbers and this created a new idea, a new hypothesis which is the continuum hypothesis. The next person featured is Henri Poincare. He solved the problem in the solar system which involved the revolving of the sun earth and moon together and how to solve each missing variables like exact coordinates, velocity, and direction. He sorted these out and simplified them by making approximations to the orbits. He was the guy that likes techniques in solving such complicated math expressions. But one person found out that the simplifications of Poincare are wrong because a small difference in values could create a new orbit. This ‘mistake’ of Poincare was an indirect discovery of a theory now known as the Chaos Theory. He was also an author writing about the importance of math but the work that made him to the famous status was topology, enabling us to have a new way of looking at shape. Topology gave Poincare a question he cannot seem to answer, If you've got a flat two-dimensional universe then Poincare worked out all the possible shapes he could wrap that universe up into. It could be a ball or a bagel with one hole, two holes or more holes in. But we live in a three-dimensional universe so what are the possible shapes that our universe can be? That question became known as the Poincare Conjecture. This was solved by Grisha Perelman by linking it to as different mathematical area and his proof was very hard to understand even for mathematicians.

 David Hilbert was a mathematician who studied the number theory and revolutionized the theory of integral equation but the work that made him stand out is his work on equations showed that although there are infinitely many equations, there are ways to divide them up so that they are built out of just a finite set, like a set of building blocks. The interesting part of his proof is that he didn’t construct it but he just proved that they should exist. It is just like telling that there is a way from Los Angles to Bay Area but I just don’t know how to get there. He said that there are no equations that can’t be solved and we should and will know. You know when you are full of hope and your day seems to be good then just one person shatters all those? That’s what Kurt Godel did to Mr. Hilbert, and he also put uncertainty to mathematics. He was a person of overflowing curiosity when growing up. Godel proved that within any logical system for mathematics there will be statements about numbers which are true but which you cannot prove known as the Incompleteness theorem. He started with a statement, “This statement cannot be proved”. If the statement is false, that means the statement could be proved, which means it would be true, and that's a contradiction. So that means, the statement must be true. In other words, here is a mathematical statement that is true but can't be proved.

When David Hilbert died, the mathematical prowess of Europe died too and the baton was passed to the Americans. It was in Princeton, New Jersey that they tried to rebuild the environment of mathematics that was lost. Kurt Godel was one of the European exiles from the Nazi empire and he was neighbours with the great Albert Einstein. The sanguine type personality of Einstein made Godel a much more quiet and pessimistic person that led to his lunacy. The next mathematician that evolved during the hedonistic and fast food life of the Americans was Paul Cohen. He was brilliant winning awards and prizes but he wanted to make a mark in mathematics and what field should he leave a mark. He then began to work with Cantor’s problems where he can’t seem to solve but Cohen found a proof that did prove the Continuum hypothesis. Many doubted the solution of Cohen because it is new but there is one man that everyone looked up to for the solution’s approval, Kurt Godel. He then went to Princeton so his work could be checked by Godel and Godel did give his stamp of approval giving Paul Cohen more recognition and prizes. Then after gaining confidence from solving Hilbert’s 1^st problem, he attempted to solve the most important problem in Cantor’s problem, #8 the Riemann Hypothesis. When he died, he was still trying to solve it but he just couldn’t just like other mathematicians before him. That doesn’t mean that he was forgotten, in fact he was a successful product of the American dream because from being an exile he became a top professor.

Then the next thing just caught my attention. A female mathematician was featured and this is interesting because there are just 3 mathematicians that have been really known. The female American mathematicians is the first ever woman to be the president of the American Mathematical Society, she is Julia Robinson. She had an obsession to solve Hilbert’s 10^th problem and she said that she just wouldn’t want to die without knowing the answer. With the help of her colleagues, she then developed the Robinson Hypothesis She had inspiration from Yuri Matiyasevich from St. Petersburg, Russia. Yuri found the last important piece to solving Hilbert’s 10^th problem. He saw how to capture the Fibonacci sequence in the problem using the equations at the heart of Hilbert’s equations. The next mathematician featured was Evariste Galois, a republican revolutionary in France. He believed that mathematics should not be the study of shape and number but of structure. Galois had developed a technique where certain equations could have solutions or not, and shown how new mathematical structures can be used to reveal the solutions to the equation and this is because of his idea to analyse these equations. This idea made light to Andre Weil, a prisoner of a war but he became a soldier. In his jail time, he first developed algebraic geometry and built on the ideas of Galois, a whole new way of understanding solutions to equations. This led to the connections to theorems connected to geometry, topology, and Algebra, all this led to one of the greatest achievement of modern mathematics. The next man is Alexander Grothendieck who is a pretty interesting person because he is a structuralist, interested in hidden structures underneath mathematics. I like him because we can understand the most complicated things when we go back to the basic then from there we could work our way up. He discovered a new way to look at patterns in mathematics which are being used until today to solve problems and equations. But sadly, he turned his back on mathematics when he entered politics. One thing that still remains in doubt today is the Riemann Hypothesis.

 Maybe one of you reading this article may have that world changing idea, not just math. Sometimes math may intimidate most of us, but it is the people who step out and get that intimidation out of their head who can truly get them to that changing idea. People wanted to do simple things and in order to do simple things, some should do the hard things first. This is what all the mathematicians showed us, erase that uncertainty and doubt so we could have that feeling that we can do things very easy. Math isn’t all about hard equations to solve, it is how we use the basic things to solve the hard ones, like what Grothendieck did, he understood complicated things easily because he used the general and basic terms first before all the hard ones so math is possible for all of us.  I hope this will serve as an inspiration to all of you out there, Peace!

What I learned from Number Systems...

This is not your typical math class where in you do a little solving and not the conventional way of solving. I did not expect to learn 'new numbers' here. I knew how to write numbers in Chinese and Mayan and Binary! It was so awesome.

The numbers we have now is not what it looks like before. The system today, Hindu-Arabic, was developed from different kinds of number systems and it was not an easy process because years and years of developing was needed. I also learned that 1 in Hindu-Arabic is different in Mayan or Binary because they have different bases. I thought that there was nothing very important about the history of numbers but I was mistaken. There is a very utmost importance because these numbers are what masks us in the internet, make it very hard to hack passwords of different accounts online. Although this are only numbers we are dealing with, there is also mathematics here of course because they were decoded through mathematics. So, no matter how much we hate or dread math, it is just everywhere my friend and we just have to deal with it.

EXCEPTIONAL THINKERS, GREAT MATHEMATICIANS

“Many mathematicians derive part of their self-esteem by feeling themselves the proud heirs of a long tradition of rational thinking; I am afraid they idealize their cultural ancestors.” – Edsger Dijkstra 

Behind every mathematician, there is a story of great challenges and a journey to the world of mathematics along with self-discovery. A mathematician started off as an everyday casual person who, by influence or circumstance, finds himself drawn to mathematics and its magic.

In this fictional creation called “A Certain Ambiguity” written by Gaurav Suri and Hartosh Singh Bal, Ravi Kapoor takes us along for the ride of his life. It all started on the day his grandfather gave him a calculator for his 12^th birthday. His grandfather showed him some number magic on three-digit numbers which stoked his interest and left him astonished and dazzled. Mathematics has that effect on most people, how it can lead you to where you go and leave you amazed at how you got there in the first place. It leaves you breathless. Ravi felt this way but he didn’t stop there, he wanted to find out how, so he did long divisions by hand and ended up believing that what he was witnessing was a property of all three-digit numbers. But he wanted to know why this was happening and what was behind this problem. Later, days after his birthday, he had a sudden insight of an idea or that Eureka moment. His realization came from the fact that division was a reverse of multiplication and this unlocked the solution to the problem. And after telling his grandfather, they both started their passion for mathematics.

When his grandfather died, he was devastated. It was only years later that he found out how his grandfather believed from the start that he would be a great mathematician. The calculator his grandfather gave him was to mark this path for him, this path to mathematics. His grandfather used to challenge him with mathematical questions and problems everyday and they would bond over the solutions and insights of the case. His grandfather had a very important role in his mathematics. When his grandfather left him money for college in America, he worked hard and excelled in all his subjects yet he saw beauty in none. Losing his grandfather affected his passion for mathematics. Before his 18^th birthday, he got a letter from Stanford University. It was there that he met Peter Cage whom he became friends with and who ended up being his roommate. They were opposites but it was Peter’s determination that drew Ravi to him. As he went through college life, he had this lack of interest and certainty until Peter introduced him to Nico Aliprantis, a math teacher who taught math as a natural thing instead of a bunch of rigid rules. Nico invited them to join his Math 208 class. In his class, Nico tackled topics such as infinity, geometry, sets and so much more. He talked about different mathematicians like Georg Cantor. As the sessions went on, Ravi found interest in mathematics again. Nico had reintroduced him to the excitement and pleasure that the mathematical hunt or puzzle brings. He then found a friend in Adin Kaminker and Claire Stern.

After meeting with Nico, Ravi found out that his grandfather went to America and was imprisoned in Morisette, New Jersey in the year 1919. His classmate, Claire, and her mother helped him find out what happened and why his grandfather was imprisoned. With the newspapers of the town of Morisette and the help of Claire’s mom, he discovered that his grandfather, Vijay Sahni went to jail on the charges of blasphemy. He found recordings of the conversations between his grandfather and a judge named John Taylor. Though the two men have different philosophical views, they somehow bonded through mathematics. The judge believed that in order to understand why Sahni said what he said about Christianity, he should indulge Sahni through mathematics as how he thinks normally is connected to mathematical thinking. They went to Peter’s house to celebrate him getting a job and they also attended the performance of Adin’s band. Later on, Ravi, Adin, PK, Peter, Claire and Nico went to Redwood Park where they talked about Ravi’s grandfather and the points that he was making. It was then that Adin understood Ravi’s grandfather and how he thinks. He believes Vijay Sahni was driven by a quest for meaning and that he wanted to acquire that through a process he could feel certain about.  After reading the last newspapers on his grandfather’s release, he went to New York for his job interview where he was asked an odd mathematical question. In the end, he was able to crack it and was offered the job. But at the same time, Nico arranged him for a full scholarship to study mathematics at a graduate school. And just as he was having difficulty on making his decision, Claire brought him the journal entries of John Taylor which talked about how he and Sahni bonded over mathematics. After much deliberation, Ravi chose mathematics and ended up publishing papers and being a teacher. He later on lived with Claire.  

           This book was a pleasant read. It brought the classroom to the reader. I learned a lot on different topics on mathematics like infinity and geometry. It was very informative but not in a way that seemed boring. It felt like I was interacting with the book and that I was learning with the characters. This was by far the most entertaining book on mathematics that I have read. It pointed out certain topics on mathematics and proved it using mathematical thinking but unlike the other books, this one incorporated that into a story. It was fun seeing how the characters observe and perform mathematical thinking on a day to day basis. I learned about how mathematics has a different meaning or use to different people. I understood from the book that by stripping mathematics down to its core, you see its true beauty just like a musical piece. Just as a painter creates patterns with shapes and a poet creates patterns with words, a mathematician creates patterns with ideas. The book taught me about the aesthetic sense to use different words or ideas to create a beautiful piece of work and this is the connection between poets and mathematicians. It shows how mathematics provides a different way of comprehending the universe. Reason is its foundation and leads their thinking to certainty; this type of thinking can be problematic for some people because their world view might end up offending others. Mathematicians are exceptional thinkers but just like other geniuses of different fields, they face great challenges and scrutiny. Some even end up in jail for standing by their beliefs but it is their passion and determination to their craft that gives them glory. Mathematics has power over people especially with its promise of certainty.

A Certain Ambiguity: A Mathematical Novel

Where's Infinity?

            This book was a written in a novel style for the readers to understand the flow of the story and to easily get related to the subject being discussed, it tried to explain the philosophical idea behind the concept of “Infinity” by the Indian characters; they are Ravi and his grandfather Bauji. His grandfather was a mathematician who was trying to discover another theory from one of the theories in mathematics. Bauji was trying to let Ravi discover the beauty of mathematical equations and its purposes to the present application.

            The story would somehow connect the principle of Infinity and the Euclidean Geometry. The enthralling relationship of the two would make the world think deeply. This issue was being compelled by the characters. This book was arguing about the Infinity and what does it mean to encounter the limits of it to the human understanding. The book also discussed the comparison between theology and mathematical views that introduced some implications to math. It was also asking about the certainty of it to our lives that revealed the insights of mathematics.

            The value of infinity is condensed to almost certain because of the combinations of different theories, hypotheses and statements of those mathematicians. Some people may think if infinity does exist or has a value, but when we come to think of it, we cannot really have a numerical representation of infinity because it was very far from the truth or certainty of mathematics.

            Finding certainty in mathematics and philosophy took some years for the mathematicians before they found out some techniques but not all of them found solutions there were also some hypotheses that were left hanging because they were not able to come up with solutions and so younger generations were triggered to continue their legacies.

            Mathematicians wanted their offspring to continue expressing the point of views of their mathematics to creatively achieve the development of the world using their new ideas. In this book Ravi’s grandfather wanted him to be involved with mathematics since his grandfather saw something in him that would connect them to mathematics.

            This book does not only wrote about math but also the implications brought by the faith and religion. I learned a lot from this book because the topic was explained very well for the readers to understand them. The different contradictions to this narrative novel were adopted by the mathematical community and was continued to find the solution for the problem of Infinity.

The Story of maths 4: To Infinity and Beyond

For many people, mathematics is just about solving problems, creating an equation and theories, but there are more than what you can think of, if you look into its deeper nature like what mathematicians did. Discovering such mathematical breakthroughs can easily change world. One of the great mathematicians who became the first person to comprehend the notion of infinity was Georg Cantor; he then created a mathematical meaning to this. His theories include infinity of whole numbers and fractions, and the relationship of each other.

Another great mathematician named Henri Poincare formed different kind of mathematical techniques and spend his whole life creating applications of his mathematics. He sorted these equations to lessen the problem by simplifying and approximations but he made tiny changes that may have a great impact in the result; another mathematical story about puzzle that he come up with but it then solved by a great mathematician named Leonhard Euler, who proved that it was not possible to cross seven bridges only once. He made several solutions when suddenly he jump into a concept that problem about these bridges and realized that it was a new type of geometry and was called problem of topology.

Topology is everywhere because we use it every day and Poincare saw this as a new notion of shape but some refer this as bendy geometry. They threw some question about the possible shapes in a three-dimensional universe and were called Poincare Conjecture. Then it was solved by a Russian mathematician named Grisha Perelman. He solved this using a different range of mathematics that was very difficult to comprehend that even other mathematicians can’t. David Hilbert studied the number theory and brought everything together but he left that and developed the theory of integral equation. Hilbert was still attached to theorems that he created; he was creating a new style of mathematics.

Kurt Godel, an Austrian mathematician, doubts the mathematics of Hilbert. He came up with an impression that would change mathematics; he created the Incompleteness Theorem or a statement that can be converted into a mathematical expression and is answerable by true or false. After them a lot of different mathematician succeed in constructing mathematical theories and most of them were men, but there were also female mathematicians who succeeded in math and established their name by achieving some goals. This would tell us that mathematics is for everyone and does not depend on gender, age and ability of a person to create something that would contribute even bigger in the world of mathematics.

            The concept behind the number system, algebra, geometry and topography has a relationship with each other. Patterns, structure and logic can be obtained in these cores of mathematics that explains how the world works. It was said that Riemann hypothesis is a corner-stone of maths and difficult math would have more application in the real world. Mathematics is about proving a statement and removing the doubt in it. The four movies revealed the beauty and nature of math. Unsolved problems pushed the mathematicians to solve them until they would come up with the right answer. Hilbert’s words “We must know, we will know” encourage the world to do mathematics.

A Review on Certain Ambiguity ( by Gaurav Suri)

 A Review on Certain Ambiguity

The book was great.

Twelve-year-olds would normally spend the remaining days of their childhood doing games. Younger Ravi was no exception to that, however his method of playing might have been different from his peers--calculators instead of balls and figuring out math patterns instead of hide-and-seeks. Bauji, his grandfather, ignited Ravi’s mathematical abilities writ on his genes by presenting him the magic of repeating a three digit number twice( it now becomes six-digit) and having the original three-digit number as the final answer after different stages of division. Their relationship is more than that of a grandfather-grandson; in fact, Ravi even said on the middle parts of the book that “ he was him”. They were so close that it really pained Ravi loosing him permanently in his life. After that tragic incident, he decided to obey his grandfather’s will, went into the university at US, studied hard as an economics major and met a couple of people who shared the same lust for math- Nico ( their professor), Peter, Adin and Claire, who helped him find the reason why he's granddad got sued eight decades ago and who  became his wife later on. 

Zeno’s paradoxes. Sets and their cardinalities. How Giordano Bruno argued the inferiority of faith to philosophy, how Galileo awakened people with the possibility of a part to equal the whole in the case of infinite series, how Bauji  used his basket of math concepts and presented the Pythagorean theorem in justifying his blasphemy of Christianity to the judge, how harmonious series works, how Cantor and his Continuum Hypothesis attracted attentions of billions and sprinkled a  light of greater understanding and wider perspective on the parts of mathematicians on treating intangible numbers, what axioms are and how other big names such as Euclid, Gödel, Riemann, Einstein, Hilbert, Lobachevsky , Gauss ,Cohen and Bolyai and their respective geometry, Incompleteness theorem, million-dollar Hypothesis, theory of relativity, number theory, complete theory of parallels, electromagnetism and other important contributions that shaped math through the ages and how theology and math fights, dominates, contradicts or supports each other—these were all talked about in Math 208 classes or mentioned in the conversations of the accused Bauji and his judge, Mr. Taylor. 

Dialogues, Dear-diaries of math aces and the concept of having a 62- year-old sharp-minded professor to teach both fictional characters and all blood and flesh readers were the author's undeniably effective strategies in sharing the knowledge he have of this field of beauty and complexities, certainties and its uncertainties, kingdom of truths and its stairways of proofs, without boring readers and serving us a platter filled with math principles, life-changing ( slightly rare but extreme scenarios of overwhelming discernment) insights and a good story line featuring  alive and relate-able characters. Sungo-tic moments do occur but that is the inevitable reality of reading a mathematical novel-- of life itself. 

Glory to God 

Spur of the Moment

Algebra, the branch of mathematics in which symbols, usually letters of the alphabet, represent unknown numbers and this is also the topic that we chose for our group presentation thinking that it would be a lot easier discuss than those other topics that was presented to us and I guess we did the right choice. We were the second group who reported in our class and sad to say, we weren’t able to prepare that much. Even though we were given a lot of time to prepare, we wasn’t able to do so for our schedules did not match and there was also exams to prepare for. Still, we were able to discuss our topic in a presentable way.

Instead of reporting and having a debate on the first day, we did the games in its place. The first part of the game was done in a Quiz bowl manner and was managed by Kuting and I. The mechanics of the game was clearly stated and also the scoring system. Ate Nicole was the one who stated the objectives of the game and to guard or to lookout for those groups who was able to answer first. After the first part of the game, the Number Systems was announced as the one who got the highest points then we proceeded to the second part of the game which was managed by Queenie and Kim. The mechanics, scoring and objectives of the game were clearly stated by Quuenie. After that, the four remaining groups went outside to continue with the game and after a few minutes, went back inside the room with tired looking faces (guess they weren’t prepared for the second part). The Statistics was then declared the winner for they garnered the highest points and was promised with a reward by Ate Nicole.

On our second day, we finally reported. We did the reporting through an impromptu skit which for me turned out well for the topics under Algebra was clearly stated. The skit was all about a young girl played by Kuting who was having troubles with Algebra. Then we acted as her friends that would comfort her and tell her that Algebra wasn’t that hard. Each and one of us were able to discuss the topics with ease from the history of Algebra to Consumer Mathematics which is also related to Algebra. After the skit, we proceeded with our debate. The debate was about whether the branch of Algebra is essential in our daily lives. I was in the opposition group alongside Ate Nicole and Kim while Queenie, Kuting and Roma were in the affirmative group. As we were in the opposition we stand our ground defending that Algebra isn’t that necessary in our daily lives for it is not a basic necessity. We then expressed our own opinions about Algebra and how it affects our daily lives. In the end of the debate, Ate Nicole concluded that each and every one of us has our own perspectives about Algebra may it be bad or good or helpful or not. It depends on how Algebra was used on the field of study that that certain person is studying.

On the whole, our presentation was good enough. We were able to discuss the topics under Algebra and we were able to have fun. If only we were able to well prepare then I guess, we could have done better. But all is done and it is time for the next time to shine and to give their best shot J

Absolute Certainty

A book review to A Certain Ambiguity, A Mathematical Novel

A Certain Ambiguity, A Mathematical Novel is a brilliant and unusual novel which is a book that subjects mathematics, regarding its philosophy, beauty and about its relevance to the surrounding world.

The story mainly gyrates with Ravi Kapoor, a Stanford student taking a course on "Infinity" and his grandfather who was jailed for blasphemy in constitute to a philosophical investigation of the nature of truth, faith and certainty in mathematics. The narrative started with a flashback of the experience of the main characters, to the time when his mathematician grandfather had given Ravi a mathematical problem which conferred him the appreciation of the magical effect of a solution might have. Ravi taking up a course thinking about infinity, he had a professor who specialized in the field of Ravi’s grandfather. As the story goes on, a lot of mathematics had been discussed. The Zeno's paradox, Convergence of Infinite Sums, Cantor's Theory of Transfinite Cardinals, Zermelo-Fraenkel Axioms of Set Theory and the Independence of the Continuum Hypothesis, also about the axiomatic Euclidian geometry. Those were at the heart of the mathematical discussion. Throughout the book it presented ideas in the form of journal entries from famous historical mathematicians including Euclid, Riemann, Gauss and Cantor. Later he had been married to atalented mathematician named Claire.

This novel illustrated the making of difficult mathematical ideas accessible in which it does not only expose the tremendous difficulty of blending science and logic with the emotion and dramatic tension. It has the equal proportions of certainty, ambiguity, frustration and joy that succeeded both as a compelling novel and in the intellectual tour through some starting of mathematical ideas being discovered.

For me, the greatest positive feature of this book is letting the readers understand the starting point of mathematics as a discipline. We all know about great mathematicians and all the theorems, axioms and principles they had discovered and added to the horrific long list of those, but only in this book I had learned about its basis as a subject.  Also this book had a very unique approach for dealing mathematics with a touch of a real story going on between the characters. It took as to another side which is the emotional mathematics most mathematician feels. It can be construed that mathematics can be an entryway to absolute truth and absolute certainty through pure reasoning. The book did a good job of presenting this viewpoint that had given anyone an appreciation to mathematical logic.

Hilbert's 23 Problems

        The last installment of The Story of Maths introduced us to more mathematicians that have been committed in to different problems, equations and even hypotheses. These people are inspiration for the new generation of mathematicians.
A German mathematician, David Hilbert laid 23 problems for mathematicians to solve. These problems are considered to define the modern age mathematics. The first problem was about the infinity. George Cantor was the first one to understand infinity. He concluded that there are many infinite fractions between to whole numbers. The continuum hypothesis was a problem that Cantor wrestled for the rest of his life.
Henri Poincare was a man who was good in everything; algebra, geometry and analysis. He was the man behind the chaos theory. The question about what are the possible shapes of our universe was known as the Poincare Conjecture. This question was solved by a Russian mathematician, Grisha Perelman. His theory was difficult to understand but he won prizes and offered professorship in various universities.
A mathematician named Kurt Godel tried to solve Hilbert’s second problem but instead of proving it true, he proved the opposite which is called the Incompleteness theorem. In this theorem, there will be statements about numbers that are true but cannot be proven.
Paul Cohen was a young man whose interest was in the field of mathematics. He was not the typical teenager that would indulge in the normal pursuits. He proved that there is and there’s not an infinite set of numbers bigger than the set of all whole numbers but smaller than the set of all decimals.
The first woman ever to become the president of American Mathematical Society was Julia Robinson. She and her colleagues were the people behind the Robinson Hypothesis.  The tenth problem of Hilbert which Robinson tried to solve was answered by a Russian mathematician named Yuri Matiyasevichand.
One man had not given the chance to showcase his brilliance in mathematics; his name was Evariste Galois. He discovered techniques to tell whether an equation could have solutions or not but he was killed at the age of 20.
I have a little hatred to David Hilbert, he started the 23 problems that made mathematicians discover more about mathematics and it became difficult but on the contrary, he was also one of the reasons why great mathematicians came out of their comfort zones and tried something that opened the eye of the humanity towards mathematics.

Read it!

*book review: A certain Ambiguity A Mathematical Novel by Gaurav. Suri and Hartosh Singh Bal

A certain ambiguity is a story about what it means to face the extent- and the-limits- of human knowledge. ~ G. Suri

Yes! Finally!  There’s a math book written in novel, a different way to establish mathematical ideas. Two points of the book were the fascinating story line and the math behind. The book succeeds to establish knowledgeable insights through astonishing mathematical ideas at the same time convincing novel which I appreciate the most. Importantly, engaging about the book is the supposition that elucidate even the relationship of certainty, ambiguity, frustration and pleasure in the evidence of our humanness.

Actually I really don’t have an idea on how to start making a review for this book. Of whether I’ll attack it on its math side or appreciate the author for a novel well done. Well, I ended up applauding both.

It’s surprising, to think that a mathematical novel can be made, well, it is made. A brave act for the authors. Many people considered math as boring and complicated and it’s hard to believe that they were able to blend it with emotion and something drama effect which truly comprises a good literature piece. If you think you’re a math illiterate, then don’t worry too much. I assure you that it won’t be a barrier to truly enjoy this. Fits to any age.

It circles on math and its philosophy, religion, faith, the beauty of math and its significance to human in understanding the world. Math is embedded on every page of the book and is evolving as the story goes on. Starting with the Pythagorean Theorem, it steps through number theory and geometry, Continuum Hypothesis to Cantor's alephs, non-Euclidean geometry and the discoveries of mathematical proofs based on axiomatic theory, Gödel, and even relativity.

The author wants to clear out questions of whether math and the theorems, ideas, objects underline it is independent or dependent of how human think. This book, I must say is made to captivate those who have still undeveloped knowledge in math.

The novel is about a young man’s search for his grandfather’s life and the enthralling mathematics. I’m warning you, once you start reading it, you can’t stop reading ‘til it’s finished. The book started with a flashback of the childhood life of Ravi Kapoor (main character) to the time his grandfather gave him a math problem to solve in a calculator. The next day, the grandfather died but the idea of the memory of the grandfather played in the grandson’s life remained which I think guided Ravi in his math life.

Ravi Kapoor went to USA to further his education. He’s engrossed both by mathematics and philosophy. There he discovered about his grandfather (who had been a mathematician and so called “atheist” being jailed in 1919. It talks about his college life experiences and his search to find out cause for his grandfather's capture.  As he was into his quest, Kapoor was hooked into confronting the same math and philosophical dilemma that his grandfather encountered which is the reason for his grandfather to land in jail (charged under vague blasphemy law). I’ll be not tackling the whole plot, I leave it to you, and I’m not a spoiler. If you’re eager enough, read the book! Swear you’ll really appreciate it.

One thing that disturbs me most about the book, is the question, “Is the existence of God can be prove or disprove mathematically?” Fascinating enough for anyone (I believe) to finish reading the book. If the book answers the question, it really depends on the reader. The book may not show you equations to answer the question but in some philosophical way it answers the question, if you know what I mean. I highly recommend reading this book.

This piece of fiction push me to understand the number theory, continuum hypothesis and especially the different kinds of infinities which I had a hard time dealing with before. I came to comprehend fully things which I considered new and I find it really interesting and thought-provoking.

I believe the authors were successful on their set goals as stated in the author’s note quoted below inspite of some flaws:

"Our principal purpose in writing A Certain Ambiguity is to show that mathematics is beautiful. Furthermore, we seek to show that mathematics has profound things to say about what it means for humans to truly know something. We believe that both these objectives are best achieved in the medium of a novel. After all it is human beings who feel beauty and it is human beings who feel the immediacy of philosophical questions. And the only way to get human beings into the picture is to tell a story."