Mathesophy: Two into One

                What would you do if the person you love suddenly goes away for good? Would you still have the courage to finish what you have started? Or will you start over and change the course of your path?

            When Ravi Kapoor’s grandfather, Bauji, suddenly died without prior notice, Ravi’s world was crumbled into pieces. His love for mathematics that was established by Bauji began to inflict sadness rather than happiness.   Following his father’s intuitions, Ravi took a major course in economics during college. However, things changed when he enrolled extra units in mathematics and met Nico, his professor.

            Nico’s class was about the infinities of mathematics. The first class discussion dwelt on the paradoxes of Zeno. Cantor’s theory on transfinite cardinals, the axioms of set theory, and the Continuum hypothesis were also explored. As the lectures passed by, Ravi found himself falling in love with mathematics the second time around. However, a big revelation was unearthed when he was reading a paper that was given to him by their professor. He discovered that this grandfather, the author of the paper, had written it while he was doing his time in the New Jersey Jail.

Shocked by this discovery, Ravi piece by piece recollected about his grandfather’s past. In the town of Morisette, New Jersey, Ravi’s Bauji was put into prison for violating the Blasphemy Law. During the conversations between the judge and Bauji, matters of faith, truth and certainty were heavily discussed. The concepts on Euclidean and non-Euclidean geometry were tackled, and a heated deliberation on mathematics and religion took place. Reading the chapter three of this book surely made gave me a hard time. Different concepts supporting the non-existence of God were really alarming. I advise those who have fresh and young faith to not yet read this book.

            The Euclidean geometry tackled five postulates which had given rise to the concept of the Elements. The first postulate states that it is always applicable to draw a straight line between any two points.  Secondly, forming a finite straight line from a straight line is also always possible. The third postulate shows that a circle with a center and a radius can always be defined, and that no two right triangles are dissimilar. Lastly, “if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” It is with the fifth postulate that the non-Euclidean geometry was born. Godel’s Incompleteness and Paul Cohen’s Consistency theorem were some of the non-Euclidean geometry concepts that had been reviewed in the book. The book ends with marriage of Ravi to his two trophies: mathematics and Claire. The mysteries of mathematics are what led him to where he is now.

            Upon reading the introductory story of the book, I actually found myself having a genuine interest on it. I like how Ravi first got hooked on mathematics, and the way Bauji had introduced math to him. However, the third chapter of the book which focused on the arguments against God’s existence had really put me down. But as the story progressed, the conversation between Bauji and Judge Taylor in the prison finally became more comprehensible and more meaningful. “We are free to chart our course, free to pursue our passions, and free to create the axioms of our lives. And it is in this glorious freedom that I find grace. This freedom, then, is my proof of His existence.” These words of Bauji really touched my heart and restored my interest on the book.

            Overall, the authors had made a masterpiece. This book had told the story of mathematics in a dramatic ad remarkable way.  The words encrypted within will make the reader fall in love with the mysteries of mathematics. Moreover, as a fan of philosophy, encountering the lines of this story made me realize that mathematics is a great approach in addressing the uncertainties and truths about life. Hence the title Mathesophy: mathematics and philosophy into one. And most importantly, I now finally see that mathematics does not complicate life. In fact, it simplifies and gives more sense to life. 

The Land of the Unexplained

                Philosophy introduces us to infinity: infinity of ideas, infinity of questions, and infinity of answers. It entices us to indulge into the world of the unknown...into the world of infinity. Faith, on the other hand, shows us that God is infinite.  The laws of this world are inapplicable to Him for He’s the only abstract thing that’s tangible, and His love is the only one that lasts forever.  Mathematics, one of the building blocks of human life, also exhibits infinity and explores the land of the unexplained.

                From the frontiers of space, Marc du Sautoy now brings us to the infinity and beyond. The story starts off with the Hilbert’s 23 problems. These 23 problems became the stars that guided the mathematics to the Promise land, and the same problems that had saved the world. Georg Cantor was one of the geniuses that had stepped onto the world of infinity. He discovered the infinity of whole numbers, as well as unending list of fractions. One of his legends was the Continuum Hypothesis, which explores on the possibility of different infinities and their sizes. Henri Poncaire was also puzzled by these infinities. As he was struggling to solve the mystery at hand, Poncaire has unintentionally unearthed another masterpiece: the Chaos theory. Topology was then established by Leonard Euler. Two of the greatest theorems on the logic and philosophy of mathematics, generalized as the Incompleteness theorem,  were discovered by Kurt Godel. The first Incompleteness Theorem tackles that  mathematical concepts are true, but there are also those that cannot be proven despite the assumed validity. The second theorem supports that the consistency of such concepts cannot be established.

                Just like philosophy, mathematics also believes that answers must not silence questions; they must stir up curiosity and hunger for proof. The mathematician that had claimed this was Paul Cohen, who stated that there are different mathematical questions, and a question may be answered a dozen, a thousand, or a million times.  But who would’ve thought that women back then also rocked the world of mathematics? Julia Roberts, the formulator of the Robinsons Hypothesis, did rock mathematics well.  Robinsons solved Hilbert’s 10^th problem. Hilton’s problem teased mathematicians to discover a solution composed of integers that would solve the Diophantine equations.   Julia Robinsons, along with colleagues, sufficed this question, not by providing the integral solution, but by proving that such solution does not exist at all. A group of French mathematicians, called Nicolas Bourbaki, had also contributed to the infinity of mathematics. Another great mathematician discussed in the story had focused on the structures of mathematics, and believed that breaking down a concept to its basic units are essential to understanding the patterns of mathematics.

                In biology, cell is the structure of life. Prime numbers, on the other hand, are the building blocks of mathematics. Riemann’s Hypothesis dwells on the distribution of prime numbers. And it is with this hypothesis that the narrator himself, Marcus du Sautoy, fell deeply in love with mathematics.

                This last installment of the Story of Maths was indeed the best one. Truly, the different facades of mathematics were shown, from it being the universal language up to being the gate to infinity and beyond. With mathematics, confusions are cleared, and the world becomes more concrete. This I guess is the essence of mathematics.

 May the words of Hilbert saying “We must know. We will know.” keep the fire burning. J

A Grandfather’s clockwork

To start off, I am not a fan of novels that long. Books of pages more than 250 don’t attract me that much. An exception is when the plot is most interesting. The book “A Certain Ambiguity” by Indian authors Gaurav Suri and Hartosh Singh Bal was quite a good book. I cannot believe I had spent my whole day reading it. At first I thought it was another head spinning boring Math philosophy book. There I was wrong. Well, of course it has the usual math concepts and a typical merge of philosophy, yet it was made in a different way. It was inspiring. A Math novel. How had I appreciated a “Math” novel?

                It is true that at the first part, you cannot resist to get your calculator out of your bag (Jacinto,2014). Bauji had not only got his Ravi to be amazed by his math magic. I had already tapped a lot of 6 digit numbers in my calculator. All of which has ended to the 3 digit number I had thought of.  And the fact that Ravi deduced the pattern and the solution quickly was not as what I would have done. Oooh these math people. Then the next thing I read, Bauji died. It was quick and dramatic. Well I thought the “Bauji and Ravi” part will end like that. Good it had not. The whole book is about the incorporation of Bauji in Ravi’s life and how he influenced Ravi even though he is dead.

                Ravi was sent to a college in America where he had used Bauji’s given wealth for his schooling. There he met friends and had taken a math class “Thinking about Infinity” with Nico as the professor. He had taken the subject due to his interest and his grandfather. After his death, Ravi was more connected and inspired by his grandfather’s work. His mother had tried to persuade him to move on. It was misunderstood, Ravi thinks himself as his grandfather.

                At his class, with Claire and Adin, I had also learned a lot. The concept of infinity is most emphasized. By definition, infinite is endless, boundless, it is time, it is space and generally, it is God himself. There is no exact or certain fixed complete idea of infinity. It is an idea of ambiguity. Before, I just thought of infinite as the biggest unending trail of numbers that is used to exaggerate numbers a bit. Later on I realized a bigger and more complex definition of it. Unlike finite numbers, infinite plus one or minus something will still be infinite. Whatever number, be it multiplied by itself, will still yield infinite. To say, it is invincible. The only person who completely understands infinity is Georg Cantor. Cantor and his ideas were degraded and ignored by people. They find it absurd and uncertain. And it is very unfortunate that people will only realize the great works of a person after a very long time.  

                Not only did Infinity have my attention, it is the Primes too. Claire’s analogy of primes is odd and amazingly logical. It is like the Euclid’s 5^th postulate where either is acceptable and the other contradicts. I remember the variables used: P as the largest prime number ad N is all primes multiplied. N+1 should be a prime on the other hand, if P is the largest prime, it cannot be a prime. The paradox of primes gives me the goose bumps. It just depends on how you look at it. This kind of analogy is like the belief of creation or evolution. People have different views on things they perceive. It is quite a fight. Mathematics is all about proof of a certain thought as it bears truth. Mathematicians work for proofs of their study and as much as possible they avoid axioms. However, if they come to a point of paradox like what has existed in the analogy of primes, they settle to axioms to avoid it.

                The complexity of Euclid’s 5^th postulate is also a paradox. Contradiction and acceptance both occurs at the same time. It is very exhausting putting an effort on which weighs more to believe in one concept over the other. I can’t say more of this anymore.

                The story also revolved on the law versus the belief. ‘Bauji’ is a prisoner held by a blasphemy case. Him being an Indian, I also thought that he is much of a religious person. Indian people are known for their devotion to their gods. It is unusual that one can become an Atheist. But well, his devotion to knowledge, logic and proofs through mathematics has turned him an Atheist. At his trial, Judge Taylor had a long court conversation with Vijay Sahni (Bauji). The court trial proceeded with violence-free and peaceful talk.  Vijay explained with elegance his proposition of mathematical ideas and had already influenced ad inspired the judge yet it had still ended at his arrest and imprisonment due to state laws.

                Vijay is a mathematician worth praising as we had been inspired by Plato’s teachings. He made mathematics define the object and language of reality. Human beings live to do assumptions, rationalize, analyze and deduce facts gracefully. Thinking logically upon things is what Vijay is trying to get our minds at. I strongly believe in God and I am inspired by Vijay’s logic. It’s not much of a paradox.

Mathematics is beautiful, isn't it?

When I read the first sentence, I was so glad that the book was not like the previous books I’ve read that gave us information about mathematics; what it is, how it is used in biology, what are its deeper meanings, but this time, the book is actually a novel! *fireworks*. It has its own story. I didn’t have a hard time reading it.

The book is entitled “A certain ambiguity”. This book is a novel but still, mathematics is really inserted in every page of the book. The first sentences really amazed me since it was still the first part I was reading back then, not knowing that the next parts would still be as amazing as the first part. It was where Ravi Kapoor was given a calculator in his birthday and told by his grandfather to enter 3 random numbers in the calculator and enter the same numbers again so that he had a total of 6 digits. When his grandfather told him that the number would exactly be divisible by 13, I quickly got my calculator and tried the technique and it really was true! I was really surprised, I actually said “woah” (haha). I read again and knew that he wasn’t finished yet. He said that the answer that Ravi got can still be divided by 11 and then by 7, then he got the same answer to the original 3 numbers he entered in the calculator. I did that also and that was awesome. Because of Ravi’s curiosity, he tried to find out the principle behind those numbers and he succeeded. He found a solution and the pattern to it which made his grandfather very proud of him.

Anyway, that was just a nostalgic flashback of Ravi. He loves his grandfather, whom he calls Bauji, so much. His Bauji made him appreciate math even more. Bauji’s real name was Vijay Sahni. He died the next day after he gave the calculator to Ravi but his Bauji’s life story was still important in the story’s plot. Then, Ravi went to Stanford with his career in economics. He met a guy and became his buddy who named Peter. He also met Nico who was an amazing Math professor. Ravi took one of Nico’s classes which was the “Thinking about Infinity”. Nico also specializes in the field of Ravi’s Bauji. One time, Nico found a paper with a footnote saying that the author of the book, which was Ravi’s grandfather, conceived the main ideas developed in the paper while he was in the jail in New Jersey. Ravi was so curious thus, he researched about the reasons why his grandfather was imprisoned. He found lots of things; philosophical discussions on the nature of the truth, certainty and mathematics that his grandfather made in jail. He was jailed because of a blasphemy law in New Jersey. Ravi found out the secrets of his grandfather when he was still inside the jail. He knew that his Bauji was saying things that were against Christianity.

On the other side of the story, Ravi attends to Nico’s class where he found some friends there and accompanied him all the way. Nico lectures math topics where infinity has a very important role in each topic discussed. Nico was also a very brilliant man who also had the same interest in life with Ravi’s Bauji. Lots of topics were discussed in Nico’s class. These include Zeno’s paradoxes and the infinity of the primes, convergence of infinite sums, Godel’s incompleteness and the consistency theorems of Paul Cohen’s. I have observed that the topics discussed in the book were also discussed in the Story of Maths, particularly in the first video.

The jailhouse conversation of the Judge Taylor and Ravi’s grandfather covered a lot of space in the book which focused about the obvious method of mathematics, especially the Euclidean geometry. However, the emotional side of the mathematics that many mathematicians feel but cannot directly share to those who are non-mathematicians were exposed because of the continual focus on the comparison of the mathematical methods and the religion and, the questions and certainty in our everyday lives. Their conversation was very brilliant. About the Christianity, for me, it is real. Their theorems could not stop me from believing God is real. We just can’t see him, of course. God is in our hearts, right? He is everywhere. But still, every person has different views in life and I can’t blame Ravi’s grandfather for acting that way.

Nico helped Ravi pursue his chosen career which was being a Mathematician and fulfilling his grandfather’s wish for him. He just finally realized his desire to be one. This was also one of his uncertainties and it is clear for him now. Also, the book ended with a little love story in it where Ravi married Claire. Claire was also his friend and classmate in Nico’s “Thinking about Infinity” class.

The book contains a lot of interesting math. It deals with questions that the philosophers have struggled to focus and answer throughout the centuries. The philosophical connections are also well developed and believable. I could not definitely say that I enjoyed both the mathematical and philosophical parts but those were really smart to read. Like what I have said earlier, I was glad that the book was a novel and not a book about mathematics alone. The book is also inspiring. Inspiring in a sense that you can really reach your dreams and gain more knowledge while reaching it. To end this review, I can tell that the author somehow achieved his purpose where he wants to show that mathematics is beautiful. Yeah, it really is.*winks*

Mathematical Fiction: A Review on the book A Certain Ambiguity

Truth. Faith. And Infinity.

These three words resound all throughout the book, A Certain Ambiguity, with the lives of Ravi and Bauji (his grandfather) intertwined along with it. 

Nico Aliprantis, one of the characters in the book said “I think that mathematics is beautiful at its core; it is much more like a musical piece than an accounting formula.” By this I have concluded that the authors primarily wanted to convey to their readers that mathematics is delightful. I wanted to agree; and by seeing and reading many mathematical formulae (which is by the way, too dense for my level) and its philosophy (which for the first time I have encountered with less technicality) in a novel, I agreed that even mathematics is beautiful.

(Well, on an outsiders’ perspective that is. I’m not that into math.)

However, although mathematics is beautiful and interesting and encompasses an extremely wide range of knowledge that most of us can’t dive into, I still believe that mathematics can’t lead us to the ‘truth’ (whatever kind of truth people look for); I still believe that faith is greater than pure reasoning and of human knowledge; And the more I believe that we can’t simply contain the idea of infinity to just some formula or theory or law.

Although some parts of the book disturbed me, I still credit the authors for a good literary piece –gluing math and its concepts with a dramatic plot of a person’s life is hard work.

And even though a lot of passages in the book shook my convictions and angered me, I still found something valuable from this reading, just like what Prof. Aliprantis told Ravi:  “Ravi, in my opinion every person should choose a life path that allows himself to nurture this sense of order and connectedness. That’s what your grandfather did, and that’s what –in his way– Judge Taylor did as well, and that’s what I think you should do.”

And that’s what I’m going to do. 

The Truth of Being Limited

The last episode of the installment The Story of Maths: To Infinity and Beyond, Marcus du Sautoy examined twenty-three unsolved problems of mathematics as posed by David Hilbert at the International Congress of Mathematicians in Paris on year the 1900.

Du Sautoy explored the ideas of many great mathematicians about infinity especially those of Georg Cantor and Henri Poincaré.

According to Cantor, there are different infinities, some bigger than the others. He was also the proponent of the Continuum Hypothesis –which Hilbert listed as the first problem of the 23 great unsolved problems of mathematics.

Henri Poincaré worked on about ‘bendy geometry’ and left a problem he could not solve regarding the shapes that could be made possible for a 3D universe. This was solved in 2002 by Grigori Perelman whose identity is concealed from the public.

Du Sautoy looked at the discoveries of the American mathematician Paul Cohen, the work of André Weil on algebraic geometry (which helped solve Fermat’s Last Theorem) and on the contributions of Alexander Grothendieck. On the last part of the documentary, all the other great unsolved problems of mathematics today.

But what I like best was the part about Kurt Gödel. He showed that mathematics could not prove its own consistency or that the unknown could never be separated from mathematics. This just tells us that even in mathematics there are things that are true but there is incapability of proving it.

I honor this statement because I do believe that there are things which are far from our reach to prove. That even the lack of proof can’t defeat the truth –like the existence of the Higher Being. 

Confusingly understandable

A mathematics lecture book for most students is obviously only essential for educational purposes and not for leisure reading. Compared to other novels, a mathematics lecture book would not stand a chance in overpowering the charismatic beauty of a novel. But imagine mathematics incorporated in a novel, could this be even possible? How complicated and boring it is to read a story about mathematics, who would even bother reading one? The characters of the story would probably be bored people trying to solve the uncertainties and phenomenon of life (though I thank all mathematicians for most innovations we have to day) using mathematical equations. The novel’s story plot would probably be complicated as the subject itself. Now the book would even sound worst if philosophy would be added in it. A philosophically inclined mathematics novel, I’m sure that only a few people would read one.

Those hasty generalizations above was proven wrong by the authors Gaurav Suri and Hartosh Singh Bal who wrote a novel about mathematics and the philosophy behind mathematics. Their book was entitled A Certain Ambiguity. Even the title of the book itself is confusing how much more would be the story.  It is a novel that talks about how mathematics is relevant to human understanding of everything that encompasses out universe. All the pages in the book talks about mathematics, not a single page was left untouched by mathematics. Mathematical philosophy was the backbone of the story and mathematics itself was the heart.

The story started with a sentimental flashback experienced by Ravi Kapoor the main character, back when he was a child, during the time when his mathematician grandfather was still alive. It was the time when his grandfather had given him a “magical” math problem to try on a calculator. The magical math problem has loose its enchantments when Ravi was able to solve the pattern and solution behind the problem. This certain event in Ravi’s life had opened the doors of his heart to the beauty and world behind mathematics. His grandfather died the next day however, this had become his inspiration in pursuing his love for mathematics by continuing the mathematical journey his father had started.

During his adolescence life, he was accepted to Stanford University and was pursuing towards a career in economics. Ravi took a course entitled: “Thinking about inifinity” thought by professor Nico who later on became his friend. Professor Nico was incidentally a person who specializes in the field of Ravi’s Grandfather. Nico once found the paper of Ravi’s grandfather, where there was a footnote that stated that the main ideas of his paper were developed or were analyzed inside New Jersey jail. During this time there were two separate but linking narratives in the story. One of it was Professor Nico’s lectures about different math topics that mainly discuss about the significance and importance of infinity. The other one is Ravis’ research into unveiling the story behind his grandfather’s philosophical discussions on the underlying principles of truth, certainty and mathematics in the jail with a judge assigned to investigate his case during the time he was imprisoned. He found out that his grandfather was jailed due to a blasphemy law.

The main mathematical topics that were discussed in the book was properly written and discussed by the authors as expected because they are both wise mathematicians. However due to lack of mathematical background about some of those principles mentioned it was hard for me to understand most of it. For readers who did not have that enough background in different philosophical mathematics it would be hard for them to grasp the concepts that were thought or mentioned in the book that easily. But most of the math topics discussed were Zeno’s paradoxes, Godel’s incompleteness theory and Paul Cohen’s consistency theory. 

During Ravi’s grandfather’s imprisonment the grandfather and the judge discussed of many clashing ideas about certainty and the axiomatic method. There were many questions and ideas that were raised about the uncertainties of life. Euclid’s axioms were mentioned about how he had expressed the geometrical structure of the world which also had led to a question if his axioms were just a product of faith that was passed on from generation to generation. If ever this was true why do the non-Euclidean geometries exist? There were many clashing points that were discussed but were indeed fascinating.

At the end of the story the authors had even added a little love story in the book where Ravi married Claire one of professor Nico’s math student, Ravi’s classmate in his infinity course. It was also mentioned that Ravi preferred a mathematical career instead of an economics career where he would probably be wealthier. The book as a whole was never boring as expected of a mathematically philosophical book.

Though the story has a rough outline, a hard one to discuss and be incorporated in a novel but the author had pulled it off and had created an outstanding outline. The authors of the story had managed to create a plot that was relevant to present day viewpoints. The book was field with learning where each page a reader’s mind would keep on working trying to understand mathematical philosophy. This book would exercise the brain to think deeply and think beyond what we are capable off. The authors were able to present and convince me as reader that math is an essential part of mathematics of course, scientists, each human life and all that encompasses the universe. Though this book was a fiction the author was able to instill in our mind that mathematics is not a fiction and it is indeed part of our reality.