The Story of Maths 3: The Frontiers of Space

By 17^th century, Europe has begun to replace Middle East as the world’s powerhouse for mathematical ideas. In France, Germany, Holland and Britain the race was about understanding the mathematics of objects in motion.  It was in a village at the heart of France where the pursuit of new mathematics has started. The village was a home to Descartes and was named after him as well.  For Descaartes, removing all distractions is a requirement before starting with mathematical works.  He even thought that the bed was the best place to achieve such meditative state. I was quite wondering whether or not Descartes could sleep whilst he’s on his bed.

Descartes found a home in Holland. He’d been one of the champions of a new scientific revolution which rejected the dominant feud that the sun went round the earth. Descartes may not have been the most sociable person but there is no doubt that his intelligence, his contribution and insights on the connection of algebra and geometry transformed mathematics forever.

Another first class mathematician, Marin Mersenne, went to the same school as Descartes. He saw Math and Science as evidence of the existence of God. One of the discoveries of prime numbers is still named after him. He published new findings on the properties of numbers by an unknown amateur who end up rivaling Descartes, Pierre de Fermat. He loved looking for patterns in number and proved that these patterns will be there forever. One of his theorems is the basis of the codes that protects our credit cards. But the usefulness of Fermat’s mathematics is nothing compared to Isaac Newton’s.

 Isaac Newton developed a new theory of light, discovered gravitation, scribbled out a revolutionary approach to math, the calculus. Calculus enables us to understand the changing world, the motions of fluid, and the orbits of the planet. Newton decided not to publish his work on Calculus.

However, a rival, Gottfried Leibniz, came up with the same idea as him.  Although he did not get as famous as Isaac Newton, he continued his work on mathematics. He was also one the few people who invented a calculating machine that worked on a binary system. The similarity between his work and Newton’s gained controversies and debates. Isaac Newton was credited the work while Leibniz was credited for the first publication. But, as it was mentioned, it was the Bernoulli family who helped developed Calculus.

In the last part of the documentary, it was said that Riemann’s mathematics changed how we see the world. The potential was there in Descartes’ ideas but it was Riemann’s imagination that made it happen. It was with Riemann’s that we finally have the mathematical glasses to be able to explore the worlds of the art.

There was a spirit of mathematical communication in the 17^th century. We could clearly see it in the documentary. All events and things during that time have a connection to Mathematics. In France, for example, the revolution emphasized the importance of mathematics, especially when it comes to artilleries. The buildings, the architecture which was shown in the video, all had a link to math. 

The Different Truth Math Could Offer

(A Book Review on A Certain Ambiguity: A Mathematical Novel by Guarav Suri and Hartosh Singh Bal)

          I guess Gaurav Suri and Hartosh Singh Bal, the authors of the book are worthy of praise. And as much as I want to elaborate all my points for this book, I’d want to keep this review plain and simple.

          First point, the book was an excellent story teller. It is one thing to have made a storyline out of the numerous journals and personal entries of past mathematicians but it is really another thing when you are able to create an effective story telling mood from the storyline and to actually talk to the readers. This book is one of the few academic related books that I think you could vividly imagine the scenarios. The authors might not have the writing caliber of renowned authors but still, their ability to have connected with their prospect audience give this book and edge.

          Second point, it is not an easy job to review and aggregate a lot of journal entries into one and the book did good justice on the specifics. This part I guess was the hardest for the authors since they’d have to create a self-sufficient story out of the tiny scraps they could extract from the different entries. Now that is hardcore recycling.

          Third point, the fiction, the make-up story and the fictitious characters are effective enough to have proven both the author’s point. The book managed to have slipped off the boring parts of the entries and instead created a comfortable environment for the readers. It is admirable how the main character Ravi was able to revolve around the concepts of mathematics while in a state of being normal for a fictitious character. I mean, usually, being super and awkwardly talented in a certain thing means that character would usually be boring. But in the novel, not only did he entertains the readers with a certain kind of eagerness to retrace and learn the steps his grandfather did but he also gave the readers a sense of responsibility to have to explain everything for the sake of being closer to the certainty of math that he was after to.

          Fourth point, the book I might say is a “mid-brainer” and that is its best asset. No-brainer books would go off the charts usually and eccentrically made books would usually have plots which are - uhm, eccentric? But this book was not a trying-hard one. It is not trying-hard to be something; it is trying just enough to be at the middle between nothingness and fullness. It was jam-packed with theories and insights like Zeno’s paradox and Godel’s Theorem but it never gets to the point where it masks the story and somehow shifts the book from a novel to a mathematics book. As to shorten, I’d say the book is just the right flavor and is palatable (in a metaphoric sense that is).

          On to my last point, after all the praises I’ve been giving, it would be totally unfair to have not noticed anything offensive about the book. I am a believer, I rely on faith and the book just ain’t recognizing people who have religion as a pillar. They believe that whoever relies on religion as a basis for things occurring in the world already gave up on the concept of reasoning. If I take myself for example, religion is another thing and so is rationality. Reasoning is a concept and must not be intertwined with religion because as much as people wanted to believe in the Creator, they’d also want to recognize the reasons why everything were made. We cannot generalize that since these people believes only in one reason to how everything works that they have already grown tired of looking for the reasons. That is why people have ears and brains. People are always ready to listen to the newest concepts every scientist, mathematician, physicist or the likes could create or prove. Then, the brain works on rationalizing these things. Humans are sponges for information and they tend to rationalize whichever would make greater sense. And the book is questioning the ability of every person to absorb things naturally and that would be really offensive.

          I know this book has been very awesome and Suri & Bal really did a great job on creating a masterpiece novel about mathematics and the truth it can and cannot offer. And as expected from great masterpieces, its either they’ve got a simple flaw or a certain selfish concept and that is religion out of the picture from the book. Even though the book tied philosophy with it, the messy parts would have just become messier. However, the book is a magnificent and splendid work. Not once would have I known for a certain book to have existed that could really open math in a unique way. 

Ambiguous Certainty

So, uhm. Let my situation while writing this book review be describe. I am sitting alone on the beach side while the tide’s low and chains of city lights glow in the distant shores. I’m 72 minutes away from the deadline and yet I don’t mind spending with the unusual and contradicting coldness and party music in the air around me. Tonight I’m entering a contemplation I’m not sure I can express well in this review, but I’ll do my best.

I introduced you Ravi. For all that I have read, he was the promising protagonist. And frankly, I thought the author would start with a low note on mathematics. Instead, he started fast with infinity. Gaaah. Infinite sets, in particular, were discussed earlier in the book. Relate The Story of Maths Part 4 here and you’d be sure to understand all the points this book is trying to portray.

What amused me are the conversations between Vijay Sahni and Judge Taylor. Throughout the book, these exchange of views and criticisms were what made this book an essential read when learning complicated complexities between science and faith. Or rather, mathematics and faith. ;)

‘Medieval mathematicians saw infinity as an awe-inspiring and sometimes a fear-inspiring idea. ‘Only God is infinite’ was their conclusion; everything else is limited. An Italian thinker, Giordano Bruno, was tortured for nine years in part because he refused to retract his idea that the universe was infinite and extended forever. Bruno believed that reason and philosophy are superior to faith, and to knowledge founded on faith. He refused to accept the finiteness of the universe merely because the Church decreed that only God could be truly infinite. At his trial, which ended in 1600, he was as defiant as ever. Upon hearing his death sentence, he responded, ‘Perhaps your fear in passing judgment on me is greater than mine in receiving it.’

Personally, I’m open to such views but it disheartens me to know some people forget their faith in God when it comes to explaining both the natural and unnatural phenomena around them. There’s even Galileo’s insight about realizing that you cannot apply the laws of finite mathematics to infinite sets. How would you ever make sense of such statements? Sure, it opens up new and definitely bright ideas. But what do you make of that? There is a statement said by Vijay Sahni.

‘There is no democracy in the realm of ideas. Just because so many people believe something or live in a certain manner is not reason enough for me to concede they are right. And your own laws allegedly give me the freedom to speak about such matters without fear of reprisal or arrest. My confinement violates some of the most basic principles of your constitution.’

What do you make of that? Suppose we debate on math and religion. Isn’t that debating also logic and reason against faith and God Himself? I’m on the latter side always, and truthfully speaking, this book may have hurt me for real but actually, I’m enlightened to the fact that not everyone would see both sides and I guess I’m privileged enough to see the contradicting points pointed out in this book. Math relies on reason, not mysticism. The book offers more than just arguments. It took me to new places, sure, but I'm sticking with my faith.

It comprehensively discusses Pythagorean Theorem, Georg Cantor’s Continuum Hypothesis, Euclid’s geometry and postulates in The Elements, Einstein Theory and David Hilbert. Also, I’m amazed to find out that Descartes is a devout Christian. Sighs.

Euclid’s fifth postulate and the Continuum Hypothesis are concrete examples of important, interesting questions that were shown to be undecidable in the context of their respective axiom schemes. Gödel provides a measure of finality to the conclusions we came to from the specific instances of geometry and set theory. No matter what the axiom system, truth will outrun proof.

Throughout the talks of JT and VS, I noticed one thing, though. I know, JT was trying to question VS on his stand and most of it talks strength in VS’s views. It would have been more entertaining and more pleasing if JT also voiced out more significant discussions on his side. Anyway, they ended well after 5 sessions of listening mostly to VS’s deductions and explanations of some traditional mathematics. It continues to test what we already knew and discover things we never knew.

Like the party music, sometimes we get distracted to what we should do or focused on. Our beliefs and principles, the very essence of our faith, would always be questioned with things very different from the rhythm we’re used to.  Like the faint little lights across the dark horizon, once in a while we glimpsed into the reality that when a greater light comes to eat all the darkness, they would be turned off to make way for a new truth we embraced. Laws and axioms and lists of standards and bases will be revised over time. Like the passing ships swimming through the liquid glass, mankind would be always willing to wander through known and unknown, through certainties and uncertainties alike, challenging themselves. But like the tides of the sea, it would exposed now and then every hope that once in a while, the truths we needed are nearer than we thought they’d be. It only matters how high or low our views amongst such things. I think mathematics already questioned faith way before numbers are presented as unknown quantities. Like they said, people change, hearts move on. But never can a heart change if people do not move on. So forward is all the direction we needed to continue understanding all we need to know and explain. We needed the brain and sadly, the heart was left to accept the rest.

It’s hard saying what this book is all about. But one thing’s for sure. Ravi became successful in his career. He was molded through thick and thin with mathematics as his passion and foundation. And true enough, a calculator makes you pass the test. Cheating is strictly discouraged. Long live book review! And one more thing. Read it when you're not in a hurry. And not on evenings near the beach with party music in the cold air. ;)

Math Was Not As Absolute As You Thought

The mathematical novel: "A Certain Ambiguity" tells us the story of Ravi Kapoor as he reminisces about his past quite thoroughly. It starts with a simple gift that first stimulated his mathematical interest- a calculator. It was given by his grandfather who happens to be a brilliant mathematician. They work up with some tricks regarding prime numbers and find beauty in it. Unfortunately this was Ravi’s last memories with his Bauji, Vijay Sahni. All he had left was money to ensure his future in a good university and his grandfather’s legacy.

Next, Ravi gets to Stanford University not exactly sure of what to make of his life in the future. Until a professor, Nico, came along and sparked his interest in math much like how his Bauji did. So he joins this class: “Thinking about Infinity”. From here on we sit in together with the other characters in various discussions about infinity. There’s Peter, Ravi’s roommate, straight-A student ; Claire, a gardener who happens to be inclined with math and is Ravi’s love interest; Adin, a philosophy student who seeks for something absolute. They started with Zeno’s paradox about the running man who runs on an infinite distance. I can’t really elaborate on it but I must say, the book did a good job of doing that. Shortly after this, Ravi finds out that his grandfather had been imprisoned without his family’s knowledge. And he is eager to find out why.

Ravi unravels the mystery of his Bauji’s imprisonment that it was because of blasphemy. All of this while his infinity class continues. They discuss further about Cantor's theory of transfinite cardinals, Zermelo-Fraenkel axioms of set theory and the independence of the Continuum Hypothesis. We also eavesdrop on the axiomatic method of mathematics and Euclidean geometry through Mr. Sahni and the Judge’s conversations.

This book is highly mathematical but interestingly written in bits that make sense in its own right. It is closely-knit to philosophy and religion, which bothered me somehow. Especially with the line: “Christians are people who gave up on reason”. Quite heartbreaking, really. But moving on, the book makes its reader realize that math isn’t a doorway to absolute truth. Gödel's Theorem proves that there will always be statements which are impossible to prove or disprove.

This is all a rough plot of the journey but in the end we find out that Ravi pursued a career in mathematics where he can “nurture this sense of order and connectedness” as his favourite professor, Nico, advised him. We are also given the understanding that he married Claire.

I think this book has achieved what it wanted: to give its readers an investigation on the nature of truth, faith and certainty in math, religion and life. It is also perhaps the first time that I actually came to understand, in the minutest way, the idea of infinity. This book gives fascinating arguments with mathematical proofs which are surprisingly made easy. I would have to give it to the authors that this is a job well done. Because sewing mathematics, philosophy, religion, and a bit of romance in a novel probably isn’t the easiest thing to do. And you come up with the thought that, mathematics can be a certain ambiguity.

Should you be certain of anything?

This is an ambitious attempt of reviewing the book entitled A Certain Ambiguity by Gaurav Suri and Hartosh Singh Bal.

            The book is a mathematical novel about a boy (eventually became a man) named Ravi and his journey in life. Ravi was greatly influenced by his grandfather whom he calls Bauji (which means grandfather in India). Ravi received a calculator from Bauji for his 12th birthday. (I also had my first personal calculator at that age but I was not as enthusiastic about it as compared to Ravi.) He loved the mathematical problems presented to him by his grandfather and his ‘aha’ moments were very remarkable for me. He really loves his grandfather as well as his mathematics. When Bauji died, it was no shock that Ravi was devastated. (You would be if you are that close to your grandfather.) Ravi wanted to become like his Bauji and Bauji longs for Ravi to be a mathematician because he saw that it was innate in him.

           It was easy for Ravi to get A’s in his classes during high school and he passed at Stanford University for college. His mother was worried about the dues they need to pay for Ravi to continue his studies but fortunately Bauji left money that was intended for Ravi to be able to study college in a good school in America. When Ravi flew to California, the first person he met was Peter Cage. In fact, they were roommates. He was constantly reminded by his mother to just finish his course because he needs to be earning due to financial difficulties. It wasn’t hard for Ravi to achieve high grades in his classes but no certain study yet has caught his attention. Not until Peter introduced a math class he was going to be enrolled to under a brilliant professor by the name of Dr. Nico Aliprantis. They had met Nico in an event where he had played the saxophone. Ravi was also in love in music just as his grandfather. However, Ravi was still doubtful if he is to enroll in that class. He needs to finish up his majors and they lack financially. Fortunately, he decided to take the class because it is the first time he found an interest in and Peter offered him a loan. He knew that if he told his mother about his plans, she would be mad and he was right. Nevertheless, he pursued enrolling. This was the start of his ‘journey’ in mathematics.

            He found his circle of friends (PK, Adin, and Claire) in this math class. The class discussed about the certainties of theorems by mathematicians. Nico left a problem during their first meeting. After the class, Nico spotted Claire (whom turned out to be Ravi’s destiny) and Ravi and called them to step into his office. It seemed that only the two of them got the correct answer though their approaches were different. Then the conversation went on until Ravi has mentioned his grandfather, Vijay Sahni. It appeared that Nico had read a book written by Sahni and quickly looked for it. Claire was keen enough to observe a footnote saying that the author (whom is Vijay Sahni) wrote the book when he was imprisoned. This was a fact unbelievable for Ravi. He had never knew of this, so he quickly called for his mom who also new nothing. Wanting to know the truth behind, they went to Claire’s mother whom happened to be a librarian to seek for information about the case. Fortunately, Claire’s mom was happy to help and was very helpful. Ravi had accessed the newspapers in Morisette, New Jersey, the small town where his grandfather was imprisoned.

            As he read the newspapers, he found out that Bauji was imprisoned due to his beliefs that God does not exist. He questioned the certainty of Christians about their God. The town of Morisette being full of people believing in Christ, they were maddened about this visiting Hindu. He was then accused of blasphemy. Due to the disputes this was creating, the town sheriff was forced to arrest Vijay Sahni. However, the governor was not that sure if the case needs to proceed to trial or if Vijay Sahni needs to be left alone. With this, the governor assigned Judge John Taylor to decide with proceedings. He talked to with Vijay Sahni and these conversations aimed to determine whether Sahni was just driven by his emotions at the time he said what he said, or he did blaspheme the name of the Lord.

            Ravi got hold of the transcripts and he waited patiently week after week because he was only allowed to read a transcript a week since these papers are very old and antique. Within the conversations, both the judge and Sahni were reviewing the concepts and theorems of mathematicians such as Euclid and Pythagoras. These theorems were trying to determine the certainty in mathematics and relating it to reality. After the judge had numerous conversations with Sahni, he sent a letter to Sahni saying that his decision was to proceed with the trial but there was a twist after that because when they had their last conversation, Sahni realized in their discussion of Euclid’s postulates that there could really be no certainties.

            Ravi had an interview with a well-known bank but he was quite doubtful about taking the job. His mom was very happy because he got the job offer but he wasn’t that interested. Also, Nico thinks that Ravi has a blood of a mathematician rushing through his veins and offered him a scholarship on doctoral studies. Ravi was uncertain but followed his heart and interest; of course those were Mathematics and Claire.

            This novel makes me realize that you can never be a hundred percent sure of something. I suggest that before you read the book; make sure that your foundation is solid rock in what you believe in because I am quite sure that it will really try to shake your beliefs. I never realized that you could write a novel about math or with math in it. Though I found myself lost in some mathematical problems introduced in the story, it was a pretty good decision finishing this book; KUDOS TO THE AUTHOR.

Another Ambiguous Math Book

Another Ambiguous Math Book

Based on a simple google search, ambiguous means that something can be interpreted in more than one way, it has a double-, triple-, …, n-tuple meaning. Trust me, this book had a LOT of interpreatations and meanings. It was basically a novel plus some math stuff (weird combination, I know). The story revolves around Ravi Kapoor and his path towards a life of mathematics.

It all starts with a scene where Ravi’s grandfather, Vijay Sahi, gives him a calculator and a math problem for his birthday. Who does that to a child? I thought it was sad, I mean, what would you think if you were Ravi that night? “Oh, it’s my BIRTHDAY! Yey! A gift from gramps! It’s a…It’s a… CALCULATOR, BOOM”. Fortunately, Ravi was also a nerd so it actually worked out. That event motivated Ravi to explore the world of mathematics, until, like in all novels, someone dies, in this case his nerd idol the grandfather. And like all protagonists, he goes into a state of insecurity and indecision which leads him to a course in economics at Stanford. I thought he was running away from math, but maybe he still had some attachments (economics is somewhat like math with money and social science, right?). Here he meets another old dude, kinda like his new mentor-ish. And here starts the complex and deep parts, mainly about his grandfather being IMPRISONED FOR INFINITY (funny pun, hahaha). I think this is where the “ambiguity” lies. The novel elaborately displays the many perspectives between math, philosophy and religion. I didn’t fully understand, but Ravi’s grandfather was imprisoned because his mathematical ideas (some concerning infinity) were considered as blasphemy. At first I thought it was weird, science yeah, but math? How can that be blasphemous? The first thing that came to my mind was: “How can something that came from simply counting stuff, be a threat to religion?”. Eventually it became understandable… sometimes old laws were inevitably stupid and unreasonable.

The novel then tackles some not so memorable and boring events on how Ravi “partially” follows in his grandfathers footsteps. The ambiguity in this part is how society sees the concepts of infinity and mathematical philosophy or thought. During his grandfathers time it was considered blasphemous, useless and impractical but the same ideas were somewhat welcomed by people of Ravi’s time. I think it was mainly about certainty and uncertainty, with a little bit of infinity concepts. Another reason why I admired this book (but not enjoyed) was how the authors smoothly added the math mumbojumbo into the story. There were so many concepts and background information but it wasn’t as overwhelming as I thought it would. I felt proud whenever the novel mentioned mathematicians who I actually recognized from the Story of Math like Reiman, Cantor and Pythagoras. New names also popped up, like Zeno who did some notable stuff on axiomatic something.

There wasn’t anything that I thought was noteworthy at the last parts, maybe because I was frantically trying to read as much as possible. Almost forgot to mention the “bad guy”, Judge Taylor, who wasn’t actually that bad. I agree he was stubborn but his actions weren’t completely unjustified. Another ambiguity for me since, belief in itself is relative and subjective which is the very essence of ambiguity.

The ending was somewhat boring since it was a “realistic” story. So obviously it included the decision to continue economics or pursue mathematics. The answer is obvious, it is a MATH book after all. As all stories go, he lived happily ever after. And of course at the end there is always the “moral of the story”. in this case, I think there was a message to pursue your goals while hoping that everyone else will keep an open mind.  There was also the subtle believe in yourself, trust people, don’t judge, and many other clichéd ideas. Now that I think about it… I can’t, too much information math and literature should really limit their relationship.

Counting Infinity

              The last episode of the Story of Maths dealt with David Hilbert’s 23 most important problems math have answered which he believed have greatly influenced the mathematics of today. Hilbert’s first problem was the concept of infinity. A man named Georg Cantor was the one who really understand infinity. He also said that there is not only one infinity but a lot of infinities and that these infinities have the same sizes. For example, he paired whole numbers to fractions and concluded they have the same size. Then, he considered the set of all infinite set of decimal numbers in which he proved that this set has a bigger infinity. Yet, Cantor was unable to solve one problem- if there is an existence of infinity in between smaller and larger infinities. He was so bothered with this continuum hypothesis and he believes that this kind of set does not at all exist.

            Another problem in Hilbert’s list was the Poincaré conjecture. Henri Poincaré worked on ‘Bendy Geometry’ wherein he said that a shape can be molded to look like another and vice versa. However, a problem arose when he found out a two-dimensional figure in which he could not solve. Thus, it was called Poincaré conjecture. On the other hand, he had developed mathematical techniques to cosmological studies. In 2002, man named Grigori Perelman found an answer to this conjecture. He stated that a three-dimensional figure could be wrapped up figure with a higher dimension.

            Hilbert, despite being a brilliant mathematician, could not construct a finite set but believes it to exist. His famous line was “We must know, we will know.” However, Kurt Gödel destroyed this belief of Hilbert. As a kid, Gödel was described as ‘Mr. Why’ since he couldn’t stop asking questions. He proposed the incompleteness theorem. You could not say a statement is true but not prove. If a statement is false, it could be proved then it is true and that contradicts your assumption that it is false because it should be true. The problem here is, it should be true but could not be proved.

            During the Nazi’s regime, most of the mathematicians flew to America including Kurt Gödel. Personally, I was shocked to see that Gödel was actually Einstein’s friend more specifically to the fact that Einstein has friends (I always thought he was introvert, proved me wrong). In 1950’s, Paul Cohen worked on mathematical universe. He also tried to solve the Reimann hypothesis, Hilbert’s eighth problem on the list. Hilbert’s tenth problem is if there is a sort of universal method which could tell if any equations have whole number solutions or not. This is when a woman enters a man’s world, Julia Robinson; the first woman elected President of the Mathematical Society (I am so a fan). In 1952, Julia was married to Rafael Robinson and the couple answered the tenth problem on Hilbert’s list, the Robinson hypothesis. This hypothesis stated that in able for you to show there is no method, you choose an equation which solutions is a very specific set of numbers, and that this set should exponentially grow but still be in the premise of the equation. The problem is that Robinson was not able to find this set but fortunately, Yuri Matiyasevich was up for the job. He was able to solve Hilbert’s 10^th problem using the Fibonacci sequence when he was just 22 (I wonder what I would be at my 22^nd year on this planet).

            As the episode approaches its end, it discussed briefly about algebraic geometry. Evariste Galois believed that mathematics should not be a study of shapes and numbers but of structure. He had answered the question whether or not an equation has a solution or not (the Riemann Hypothesis). However, he had a sad fate and ended his life when he was 20. Another mathematician, Andre Weil, was the one who structured algebraic geometry through the works of Galois. He was the leader in creating a fictional mathematician by the name of Nicholas Bourbaki alongside with another mathematician, Alexander Govendick. Bourbaki was made because the above mentioned mathematicians do not desire personal glory. I personally admire these two mathematicians. Then again, this is the last episode of The Story of Maths where Du Sautoy broadened our view and history of mathematics. KUDOS for Marcus du Sautoy.

23 Problems That Changed the World Forever

Most summers are spent enjoying the beach, doing movie marathons or simply enjoying the free time. But one particular summer was the mark of the start of modern mathematics era and it happened in the city of love, Paris. David Hilbert gave a lecture on the International Congress of Mathematicians and pointed out 23 problems that he sketched as the most important ones that mathematicians should answer for it will define 20^th century mathematics.  Now one might ask, just because one guy pointed these out, it’s definitely worthwhile? Apparently, Hilbert wasn’t just any guy. He is the guy whose works are still widely talked about today and whose name is attached to various mathematical terms. These problems said to have changed the world. Imagine what it did to the lives of those who tried answering it. That is where the movie revolves while delving on the mathematical concepts involved.

Some won prizes for their brilliance. When king Oscar of Sweden wanted to know if the orbits in the solar system was stable or not, the mathematician Poincare thought of the answer while riding a bus. Yes, while riding a bus. Who knows, you might also get one while brushing your teeth or something. He won the prize for this answer because of his successive estimation of the orbits. However, he realized that he had made a wrong answer and kept the book from publishing. But he did came up with the famous Chaos Theory which deals on the random lack of order in a system that nevertheless obeys particular laws or rules. Like how a butterfly wing can produce a tornado in the other side of the world. Pretty weird, but the movie didn’t really expound on this idea. You can always refer to Mr. Google of course. Unlike Poincare, one mathematician chose not to claim his prize. Im talking about Grigori Yakovlevich Perelman, the one who answered Poincare’s conjecture. Despite his achievement which was accompanied by fame and glory in the mathematical world, he chose to shut the noise and live a simple life with his mother.

Others withdrew to their own interior world... The first problem was answered by George Cantor. He was the first to finally have a grasp on the concept on infinity. Unfortunately, he spent a lot of time in a sanitarium because of his manic depression and paranoia. It may have been because of what he unravelled or what he can’t—The Continuum Theorem. Another beautiful mind shared Cantor’s fate. Kurt  Gödel uncovered the Incompleteness Theorem in the attempt to answer one of the 23 problems. He demonstrated that it is impossible to prove everything in mathematics. As a kid he was called Mr. why because of his curiosity about his world but growing up in World War II must’ve messed with his mind. This caused him to be severely pessimistic which later lead to paranoia.

Some found a friend..  Julia Robinson was probably one of the most passionate mathematician of all time. She had to fight an awful lot of adversities to reach her dream- to be a part of the world of maths. And so she did. Although she just couldn’t find the answer to Hilbert’s 10^th problem but she did came up with a hypothesis and named it after herself (Robinson’s hypothesis). Until a young man from New York, Martin Davis, came along with the last puzzle piece. When he found out he had the answer to Hilbert’s 10^th problem he immediately made friends with the person who practically have been his mentor through her works- Julia Robinson.

This isn’t nearly half the story. There were several other mathematicians who contributed so much to the 20^th century mathematics by providing thoughts on Hilbert’s 23 problems. The mathematical world is indeed majestic. It creates developments that can change the world and yet stay modest in its very core. When Hilbert sketched out the 23 problems, no prize was offered beyond the admiration of other mathematicians.

I think a mathematician’s endeavour is very noble. He spends his life working on a problem, uncertain if he could actually answer it, but very determined. Not expecting a literal pot of gold at the end of the rainbow; but works for the ultimate fulfilment of finding the answer to a very important question. And because math is closely knit with pretty much everything, we have quite a lot to thank them for. These next words I used to say sarcastically, but not anymore: “Thanks, math”.

Mathematical Journey of Ravi

                   A Certain Ambiguity, a mathematical novel, a book written with the plot of following the steps of Ravi Kapoor in realizing his passion and interest in the course of mathematics. It was able to cover the topics of infinity, certainty and faith.

                The plot started with the flashback of Ravi to the time where his Bauji (Hindi for grandfather), gave him a calculator as a gift on his 12^th birthday. From this day on, the calculator introduced him to the world of numbers, shapes and patterns—mathematics. His grandfather showed him a trick involving three-digit numbers. With his fascination, the number that he typed on his calculator, even after the division of 11, 13 and 7, was retrieved at the end of the process.  The next morning, he found out that his bauji died without fulfilling the promise of giving him another puzzle to be solved. With his calculator in hand, he found his mother crying over the lifeless body of his grandfather. From then on, the story showed how the death of his father, Vijay Sahni, affected the life of Ravi and his interest in mathematics.

                With his grandfather’s money left for his education, Ravi enrolled in Stanford and pursued a degree in economics. He soon gets acquainted with a professor in mathematics, Nico Aliprantis, who, later on, had a great influence in Ravi’s rediscovering his passion in math. Ravi, rather than signing up for his economic subjects, chose to attend Nico’s subject, “Thinking about Infinity”. There, he found a friend other than Peter, in the form of Adin and his later love interest Claire. During the discussions in the class, several topics such as Zeno’s argument, Cantor’s theory and Continuum Hypothesis were covered. The good thing about this book is that through every situation, mathematics is always mentioned. Also, the way of explaining such complex theorems and concepts were surprisingly understandable and can be easily comprehended. This may be due to the delivery of the characters and how they can connect it with several examples that can be easily related to.

                 At one point in the story, Ravi discovered that his late bauji was imprisoned for some reason in New Jersey, USA in 1919. With his great curiosity and disbelief that his great grandfather can be involved in such event, led him to investigate the reasons behind how Sahni was in that circumstance. He sooner found out, with the help of Claire’s mother, that it was because of Sahni’s view over the religious belief of Christianity that, according to the people of New Jersey, was an act against the Blasphemy Law of the state. He pointed out, as what Ravi read in some documents recording the conversation of Judge Taylor and his grandfather, that religion is an illogical concept of belief since there is no solid proof of what the people believe in. He related this to the concept of mathematics, because of his understanding in numbers and the patterns behind, lead him to realize such view about religion. This part of the story can be a very confusing one. It is because your personal faith and beliefs are challenge by the arguments and explanations of Sahni. He, according to Ravi, was a very open-minded person and never criticized the beliefs of others, especially religion, when he was still alive. This discovery has left Ravi confused to the true personality of his beloved bauji. From this on, the story evolved until such time that Ravi found his passion again and chose to be a mathematician rather than a successful economist and also found his love in the persona of Claire.

                Also, through the perspective of Adin, a person who is interested in finding out the true reason of life, mathematics was characterized as a doorway into answering the questions behind the obvious things in nature where a certain proof and solid reasoning is offered. In this view, the philosophy, another complex field of study, can be understandably connected with mathematics. Philosophy is a study of knowledge, of how things are considered certain or how such idea can be accepted as true. It was clearly pictured out by the authors, that through mathematics, these questions can be answered.

                I was fascinated with the concept of infinity. Because of this book, I was able to discover that mathematics can never be contained into just equations and shapes. There is more coming and more discoveries are still left unveiled, left for the next generation to find out and conceptualized. This concept was very new to me and can be considered as a new knowledge.

                Generally, the book was well constructed by the authors. They were able to relay some of the complex concepts into the readers in such a fashionable and understandable way. it was surprising at the end of the story that you were able to understand what the characters were talking about and can even feel their passion for mathematics. I give my appreciation to the writing skills of the authors.

                Halfway through the story, you can feel the developing love between the Ravi and Claire. But expecting a more focus on this aspect, left me a bit disappointed and hanging. The authors should have expanded their story and showed how this romance was able to inspire and influence the life of Ravi. Overall, the book was very informative and can be helpful in realizing the importance and concepts behind mathematics. A course that can be reflected in a very wide range of other studies and a very wide range of people’s understanding of the world and the numbers behind it.

               

               

               

Complexity of Infinity and Numbers

              Mathematics is all about solving equations which consist of different variables and numbers arranged in patterns. The solved equations are now the basis of the mathematics we are studying and those that are unsolved are the ones that keep math alive and growing. In the last episode of the Story of Maths, the topic was now more complex as the evolution of mathematics takes place in the 18^th and 19^th century. The episode focuses on the unsolved problems proposed by David Hilbert in 1900 during a proceeding of the International Congress of Mathematics in Sorbonne, Paris. These 23 problems were considered as the most important problems that mathematicians should solve. Some of these problems, especially those that were solved, were discussed. The greatest minds that were involved were also elaborated such as Georg Cantor, Henri Poincare, Kurt Godel, Paul Cohen, Julia Robinson, Yuri Matiyasevich, Andre Weil and Alexander Grothendieck.

                Among these mathematicians, I was pretty amazed by Julia Robinson and Yuri Matiyasevich, a woman who competed with the other dominant male population of mathematicians and the 22 year old man who was able to solve the 8^th problem of Hilbert. I did not expect how the journey of Kurt Godel in life ended and how he suffered from several mental lapses all throughout his life. Yes, he is eccentric but he had a very great mind. Also, the meeting with Yuri Matiyasevich was really inspiring and such an amazing thought that one of the greatest alive mathematicians was featured in the episode.

                Although the episode was informative and personally, I considered as the most interesting one, there is some parts that I was not able to comprehend due to the complexity and technicality of the topic. But because of this episode, I was able to realize that there is a broader and a much wider future for mathematics. It was able to open my mind that mathematics is not as simple as one may think. It has a deeper purpose and very complex derivatives which are from the simplest forms of patterns developed by the early era of numbers.

                Marcus Du Sautoy has been a very good host for the show. His actions speak of his love and passion for mathematics. It was never just a job for him, but an exploration of the numbers and patterns of his own personal interest. The way he explained and relay the information to the viewers can give the impression that mathematics is a very interesting yet complex field of study. It can never end as long as there are equations and patterns that are unresolved and unanswered. Just like what David Hilbert said, “We must know, we will know.” the answers behind these mysteries that will keep the subject alive and continually growing.

               

END OF INFINITY

END OF INFINITY

                Honestly speaking, the first three episodes of “The Story of Maths” seemed distant and somewhat irrelevant since they were either ancient history or futuristic mumbojumbo concepts. Almost everything from those three episodes was foreign to me, I mean, they happened so long ago and not anywhere near here. But this fourth episode was different somehow, I can’t explain it but it felt “real”.

                Weird right? The episode that dealt with “infinity and beyond”, intangible concepts, actually felt more “real” than those about practical mathematics. Why? I don’t know… maybe because it was something I wanted to know? Maybe because it actually happened within this century? Maybe because the documentary actually aimed to brainwash students like us to pursue “INFINITY AND BEYOND”? most likely, all of the above… and other still unexplainable reasons.

                This episode was an improvement from the third, I wasn’t sleepy most of the time and it actually got my attention on many scenes. Like with Cantor, “Mr. Infinity”, who else would think to make math harder by adding unattainable numbers? And it wasn’t even just one, but INFINITE INFINITIES. That may have explained why he was in a looney bin (no offense). Or was it Hilbert? Anyway both of them added infinities to math, so we should blame them both. There was also that Russian mathematician who became a weird recluse, I could barely believe that he was still alive (mainly because the documentary seemed to tackle old stuff). Ah, that statement about there being no unsolvable puzzles, and that only God can comprehend everything (or something similar), was my favorite from all the episodes. It felt meaningful to me.

                There wasn’t much to remember in that episode, mostly about fleeing form nazis and war time maths. I don’t remember the ending too much… I think it summarized everything, but I don’t remember how or in what way. The four films were fun to watch but I honestly think that they were for one-time viewing only (unless you actually want to feel bored). To sum up my report, I would like to quote a well known figure from my childhood:

TO INFINITY AND BEYOND. – BUZZ LIGHTYEAR

From here to there, to there to there, to here!

4th Installment- Story of Maths "to Infinity and Beyond"

True mathematicians have always sought for adventure more than anything else. It might not be in fields, the wilderness or basically outdoors but these extraordinary adventures occur in the mind but does not limit there. Like astronauts kept on seeking for excitement in space, they discover the math behind it. Indeed math is the universal language.

Sad to say but this is the last exposure of Marcus du Sautoy talking in our mathematics 1 class. But this doesn’t make the fun in math any lesser. Today is the twenty-first century and a century before left amazing problems that’s still unsolved. Georg Cantor a brilliant mind dared to study and understand what lies beneath an amazing mathematical phenomenon; infinity. He began to measure infinity which is impossible. Measure in a way that it is not literally bounded but he found out that there were kinds of infinity not only one and that there are smaller or larger than the others.

Who would have thought that the seven bridges of Königsberg are responsible for the graph theory thanks to Euler! Who is now my second favorite mathematician since he was curious enough and that he created a pathway to the breakthrough of topology. Most of math discoveries and theories are accidental. Like Henri Poincaré who was just trying to solve a mathematical problem by chance stumbled on chaos theory. This breakthrough was the beginning of a new revolution in technology which they call “smart”. It started new sets of machines that we people now benefit from; like ones that control regularity of heartbeats. This modern math was phenomenal until it became a chaos in the mid-20^th century. Even if math is diverse and beneficial its complexity also leads to inconsistency. That is why continuous validations are done in the theories that are already proven. We need to face the truth that our limited minds cannot overlook on the unknowable.

Today new discoveries are made, still there’s these unsolved problems waiting to be understood. The prime numbers’ complexity is continuously studied. There’s even grand cash waiting to the one that can prove Riemann’s theorem and of course patent and appearance in books. Mathematics may be formed either by an accident, coincidence or maybe just plain intelligence. Top of it all curiosity make our minds work infinitely ∞

The Complexity of an Unending Cycle

 “Anyone who wastes his time writing a review of a book that he or she dislikes, is a frustrated mathematician, who has an axe to grind, and just enjoys being mean” quote by Doron Zeilberger. I always make sure that in every side of a story I see its beauty. The word ambiguity reminds me of a Taiwanese song “Ai Mei” (ambiguity). The word is the best adjective in venturing the world of the infinite. Can we define infinity? The novel is not just a story involving people but mathematics itself that is found in each and every one’s life.

Ravi Kapoor is the main character in the novel, he’s grandfather is a mathematician who gave him a math problem to try on a calculator. It was actually a fun one wherein Ravi and I appreciated the amazing effect of his grandpa’s solution. The next day his grandpa died but then realized the importance of the memory that his grandfather left him. This had an impact in the life of Ravi who is wise like his grandpa. The next chapter in Ravi’s journey was his education. He passed and entered Stanford establishing a career in economics. There was a point Ravi took a course "Thinking about Infinity" wherein he met Nico who handled the course. Coincidently like his granpa; his prof specializes in the same field. Nico's lectured on the current math topics where infinity is the star. Ravi discovered he’s confronting the same dilemmas on math and philosophy that his grandpa also faced years ago putting him into jail. Ravi has researched into his grandpa's imprisonment. He saw discussions on the philosophy of the truth’s nature, certainty as well as uncertainty and math that his grandpa conducted in jail due to a distrustful judge who challenged him to defend his principle that the certainty of mathematics is possibly extended to every bit of human knowledge.

Genes are interconnected. One can inherit the intelligence that runs in the blood from generation to generation. Indeed Ravi’s grandpa has inspired him to beat the existing beliefs that are needed to be modified. Like Ravi, I am inspired by my grandfather; who is a forester and an advocate for the environment. Sadly I didn’t have a chance to meet him, to ask him a lot of questions and specially to learn so much from him. That is why today similarly I am studying biology with the goal of protecting life in every bits of its kind. So much for that nostalgic story, let’s go back to Suri and Singh’s masterpiece. Math uses logic, the authors have pointed out philosophy for putting up arguments against certainty and uncertainties. As a philosophy 1 student for this semester in my opinion this book is open ended and that for me mathematics is uncertain till the end of time because time has always been a factor.

The book has discussed math ranging from the paradoxes of Zeno and infinitude of primes through Godel's Incompleteness which is still a challenge to mathematicians until now and Paul Cohen's theorems on consistency. I can say that the novel seizes complexity and is not afraid in unravelling the beliefs of the authors’ perspectives but there will always be downfalls. I don’t see the point of connecting mathematics and religion. The human knowledge is complex as ever but even if it doesn’t limit in imagination we again go back to time that bounds it; the time of its existence in this world.

Math dwells in the ideality but to unravel it we must first seek the pieces to form a concrete concept ∞

From Infinity to Forever and Ever

Mathematics is the Empress of the Sciences. Without her, there would be no physics, nor chemistry, nor cosmology. Any field of study depending on statistics, geometry, or any kind of calculation would simply cease to be. And then, there are the practical applications: without maths there’s no architecture, no commerce, no accurate maps, or time-keeping: therefore no navigation, nor aviation, nor astronomy. She is all-powerful: and she rules ruthlessly. Imperious and unyielding, mathematics brooks no dissent and tolerates no error. In an age of uncertainty, mathematics is the only discipline that generates knowledge that?s immutably, incontestably, and eternally true.

 In a journey that takes him through the ages and around the world, he examines the development of key mathematical ideas and shows how, in a multitude of surprising ways, mathematical ideas underpin the science, technology, and culture that shape our world. As Marcus shows, mathematics was part of the bedrock of intellectual life in the world?s great civilisations. It was central to the survival of some of the world?s most powerful empires. And even today, mathematical knowledge remains the motor-force that drives the modern world. The films in this ambitious series offer clear, accessible explanations of important mathematical ideas but are also packed with engaging anecdotes, fascinating biographical details, and pivotal episodes in the lives of the great mathematicians.

In this program, Marcus looks at the startling discoveries of the American mathematician Paul Cohen, who established that there were several different sorts of mathematics in which conflicting answers to the same question were possible. He also examines the work of André Weil and his colleagues, who developed algebraic geometry, a field of study which helped to solve many of mathematics' toughest equations, including Fermat’s Last Theorem.

He also reflects on the contributions of Alexander Grothendieck, whose ideas have had a major influence on current mathematical thinking about the hidden structures behind all mathematics. Marcus concludes his journey by considering the great unsolved problems of mathematics today, including the Riemann Hypothesis - a conjecture about the distribution of prime numbers – which are the atoms of the mathematical universe. There is now $1 million prize and a place in the history books for anyone who can prove Riemann’s theorem.

Science and Faith: Where you can break everything down to chemicals

What is faith? why do u believe in something- what is true, what can be true, is truth relative?
No, this book does not answer any of these questions, but it does take u on a journey- a journey which is about revelations;and this journey is made more beautiful by using mathematics as a tool.
The author has questioned certain paradigms believed to be true by using the simplicity of Eucladian Mathematics.He tries to establish a line of thought which says that just like mathematical theorems are based on axioms which are inevitably true, religion and faith in god, infact faith in anything is also based on the simple axiom that everything was created by someone!

Though astounded one might be by this reasoning, the beauty lies in the simplicity of it - that everything ultimately is based on  one starting point, and that starting point can be different for different people.
The most delightful aspect about this book was that how intricately the authors have woven a piece of fiction, around non-fiction. The book takes you back in time - when mathematics was evolving; (not that it isnt now) but the likes of Einstein and Ramanjunam were still formutaling theories and refuting various hypotheses, but even back to the time of the Greek mathematicians and philosophers. And amidst all this, the authors have also highlighted the dilemma faced by Gen Y today(passion or money), and have also been successful in bringing out the contrast between the clear-headed and the confused.

The book follows the life of an Indian mathematician and tries to draw parallels between maths and faith and life, in general. While the story and the characters are fictional, the intriguing mathematical proofs are as real as you and I are. The famous Pythagorean theorem, the Euclidean geometry find a place in this book. What occupies the most part of the book is infinity. I had never thought of infinity in terms of an infinity being bigger or smaller than another infinity. The book taught me a lot about infinity and maths in general, which I didn’t know. Was I sleeping through my math class when my teacher covered these?

When I picked up this book, I never expected it to be a page-tuner. For the readers who have forgotten basic math, the authors have thankfully included an introduction to all basic concepts wherever necessary. Anybody having any inclination towards mathematics will love this book. The proofs are beautiful. Even for those who hated maths in school, that non-mathematical part of the book will be interesting. Characters, story, style of writing don’t matter, atleast didn’t matter to me. The very question of ‘Can you prove or disprove mathematically that God exists?’ is intriguing enough for anybody to pick up the book, what say?

Does the book answer that question? Well, that is for you to find out. It might not give you a page of equations which leads to the proof, but it does answer in some way. Only if you think so. You will know what I mean when you read the book.

This is a charming way to become introduced to formalism in mathematics, as the authors have very cleverly inserted a lot of math theory into a light narrative around a Stanford University student's life.

Past some simple and pleasing proofs (pythagoras, no-greatest-prime-number, etc.) The book dives into Georg Cantor's work in set theory and his continuum hypothesis, the discovery of non-Euclidean geometry, and how these discoveries challenged our understanding of mathematical proofs based on axiomatic theory.

Things never get too complicated, so if you're interested in getting a taste for these mathematical ideas, I think this is probably the best place to start.

I was disappointed that Kurt Godel wasn't mentioned until the final 2 pages of the book. He probably made the most significant contributions in this field, and he was certainly relevant, and perhaps entirely conclusive, this narrative. and I think the authors probably had to cut him out because they couldn't find enough time to fit him into the story.

The best parts of the novel focus on Bauji and the reasons for his incarceration and on what Ravi learns about infinity in his math course. I really enjoyed this novel because of what I, as a liberal arts person, was able to learn about mathematics. Prior to reading this novel, I had read about Georg Cantor and his continuum hypothesis, but I never really understood the importance of Cantor's contributions to mathematics despite spending some time researching the issue on line. Amazingly, this work of fiction brought to life and made understandable to me, for the first time, Cantor's Aleph numbers in his set theory as well as his continuum hypothesis.

This novel was a true delight not because of its narrative quality but because I came to understand fully something truly new and interesting. I now see why and how there are different types of infinities and why and how some infinities are actually larger than others. And I came to appreciate what a true genius Cantor really was.

Do You Believe In Forever?

Infinity is such an unimaginable abstract that enables human mind to think and learn beyond barriers. The irony is that it is just inconceivable. The last episode of the series “The Story of Maths” entitled “To Infinity and Beyond” was the one that has greatly boggled my mind relative to the other episodes. It has efficiently shown to the viewers that the patterns that mathematics deals with are indescribable and unquantifiable. Georg Cantor had given points on ‘infinities’ in numbers. He affirmed that the infinity of fraction is relatively bigger than that of whole numbers. It’s confusing to think that the size of infinities can be relatively compared though they are technically infinite.

            In relation to infinity, another problem raised in the episode was that “What are the possible shaped for a three-dimensional universe?” I, myself haven’t asked myself this question. Shapes impart different patterns. As illustrated in the episode, when to geometric shapes are merged or molded, they create a common topology. Now, I have then wondered the number of possible combinations of several shapes that can be made from morphing them. This part of the episode also caught my attention as it included interesting visuals. Indeed, I was overwhelmed by some interesting illustrations from simple concepts that I haven’t even encountered before.

            The next part of the episode was quite technical for me as it tackled some of the mathematical problems raised and formulated by David Hilbert. These questions included infinities of decimal and whole numbers, mathematical equations and others. The fact that I honestly did not understand this part of the episode implies that indeed, mathematics is a complex science that covers variety of questions and some of these questions couldn’t even be answered by logic. Another implication is that mathematics is also interconnected not only to the natural science but by the concepts of philosophy and logic as well as to analyzing answers and existence of such entities.

            The last part wrapped it all up as it established the interconnection of all problems previously discuss. It introduced algebraic geometry which is said to be a new language in mathematics as it studies and analyzes structures. The discovery of algebraic geometry led to the solution of certain equations. I wish to see a second season of the series since it is indeed informative and an effective learning material for student who are not into mathematics as well as those who are in the mathematical field.

Infinity: The End Is Not What It Seems

            The last episode of the Story of Maths entitled “To Infinity and Beyond” showcased the continuous development of Mathematics at the modern era. Du Sautoy took us to another journey into the other realm of infinity mathematics.

            The episode started with the introduction of Hilbert’s twenty-three (23) unsolved problems in mathematics discussed at the International Congress of Mathematics in Paris on August 1900. The concept of infinity was championed by George Cantor in 1878. Cantor relates similarities between two infinite sets of numbers. While constructing a list of numbers, he found that bigger infinity was produced. This led to another problem where Cantor deduced the Continuum Hypothesis where it describes that “there is no set whose cardinality is strictly between that of the integers and the real numbers.” Grasping the concept of infinity is ambiguous, leading me to confusion.

            Another prominent figure who rose during the early 20^th century was Kurt Godel, famous for the line “This statement cannot be proved.”  He ignited thinking that proving mathematics and the concept of uncertain is considered an important concept in mathematics. He also examined the works of the American mathematician Paul Cohen. Among Cohen’s contributions to mathematics is finding the answer to the Continuum Hypothesis questioning its validity. Algebraic geometry also came into existence. Algebraic geometry was able to decode mathematics’ hardest equations such as Fermat’s Last Theorem. Julia Robinson and Sophia Covaleskia were among the female mathematicians that I have known from this episode and women in the realm of mathematics were rare.

            Hilbert’s 23 mathematical problems have become the foundation for the inquisitive minds of modern mathematicians. Other problems have led to discovery of answers while some problems have left to uncertainty such as the Riemann’s hypothesis where no one has ever provided an answer to it. The Clay Mathematics Institute would give a stunning US $1,000,000 to anyone who can provide a solution to the Riemann hypothesis. (So better get your pen and tons of papers and try to find the solution. :D)

            Among the four episodes, this was the most difficult for me to comprehend. The concept of infinity has led me to the infinity of uncertainty. Yet, I appreciate Marcus du Sautoy on his work to reach the people and see into another dimension of mathematics. Sadly, I am going to miss this series starting from the Egyptians up to the modern 21^st century mathematics. We owe too much to these people who devoted their time to make our lives easier. 

Ambiguous: A Mathematical Dream Turned To Reality

(Book Review of A Certain Ambiguity)

            Ambiguity, as we all know, means uncertainty or undefined. It is a state of vagueness from the truth. This was the main highlight of next book review in this course. I have read different kinds of novels and this next reading material would be very interesting.

          “Wow, a novel with a calculator on a cover!” These were my first words when I saw the front cover of the book entitled “A Certain Ambiguity” by Gaurav Suri and Hartosh Singh Bal. I expect the book to be full with mathematical stuff as the front cover shows a scientific calculator (Gasp). This kind of genre is a mathematical novel, and I got to see it rarely. The novel revolves on mathematics and philosophy and how these two elements lead to the human understanding of mundane things. Mathematics was used as an inspiration, a tool for achieving one’s dream.

           First words of the novel revolve on the young Ravi Kapoor, the main character. It was a throwback scene where he remembers his mathematician grandfather, Vijay Sahni. Ravi received a calculator as a birthday gift partnered with a math problem which he solved on the calculator. He was distracted with this that he almost skipped his dinner. Ravi was not allowed by his mother to leave the table but his grandfather’s encouragements have won. The warm relationship of Ravi with his grandfather was well expression. Also, Ravi’s intuitive skills and love for learning had sparked motivation for him to pursue a job in the mathematical realm. While Ravi is experiencing the ecstatic events in his younger years, his grandfather succumbed to death.

            Ravi was then getting ready for college. He was accepted to Stanford University and inclined on a career in economics, leaving his dreams to be in the mathematical career. He then met Professor Nico and shares the same specialization with Ravi’s grandfather. The novel then continues with the intertwining threads of philosophy and mathematics. Ravi’s research on his grandfather’s imprisonment focused on philosophy the transcripts which cost him to imprisonment. Also, among the topics focused by Ravi’s grandfather is on the Euclidean and Non-Euclidian Geometry. The imprisonment was due to the violation of the blasphemy law where said against Christianity in New Jersey. Professor Nico’s lecture, on the other hand, discusses on the importance of infinity and related it to the Continuum Hypothesis.

        Mathematical topics got also a share in the book. It is impressive how the authors were able to incorporate the works of famous mathematicians like Zeno, Galileo, Bhaskara, Pythagoras, Cantor (which he described having Sherlock Holmes’ nose), Baruch, Euclid and Riemann. Among the topics discussed on the book ranged from Zeno’s paradoxes and infinitude of primers through Godel’s Incompleteness and Paul Cohen’s Cosistency theorems. Zeno and Cohen are known to have master’s degree in mathematics and have significant contributions to the field. They heavily rely on philosophical discussions in studying the concepts and recognized the significance of the axiomatic method. Professor Nico’s lectures emphasized the set theory concept and the possible relationship of it with the Continuum Hypothesis. Again, the works of Ravi’s grandfather on the Euclidian and Non-Euclidian Geometry was cited between the conversations of the former with Judge Taylor. By looking at the philosophical aspects of the judge, he cannot connect to the modern monotheistic religions because it breaks down immediately as soon as the first deductions are made from it about actual human lives.

           I think the title itself was connoted by the ambiguous principles and beliefs of Vijay but in actuality, this is absolutely concrete. The conversation heated up when Vijay talked with Judge Taylor, who just wants to stick on his decision. Taylor would not bargain his belief as a Christian and how philosophy mended mathematics and existence reasoning. Ravi ended contemplating on everything what was written.

           In the last chapters, Ravi was torn between choosing his desire to be an economist or pursue a career in the academe as a mathematician. Weighing both possibilities, the reading materials – newspapers and court notes opened his mind and recognize his father commitment to motivate him and pursue mathematics. Eventually, he was aided by Professor Nico on pursing his dream to be a mathematician and soon his grandfather’s dreams would be realized. We learn Ravi chose to pursue career on mathematics over economics. The death of Ravi’s grandfather shows that death comes to everyone. The only thing that will bug us is we don’t know where and when. We should just prepare for it and leave a legacy that everyone will able to cherish and reflect.

        Oh, I almost forgot to mention, Ravi was also married to Claire where she met at the infinity course. I suggest that the authors should expand on this part. Truth, faith and religion, and mathematics and life added spice to the book. The fusion of philosophy and mathematics were unbelievable. The book is non-fiction work, as it shares some present issues in the society (e.g. religious intolerance of laws). The theme on the life’s perspective may lead you to think critically as it clashes with your beliefs. The outline of the story was well laid out. This masterpiece is very spectacular as the authors were able to present mathematics as a career. The book is mainly a work of fiction, but it succeeds infallibly in meeting mathematics with a human face. Overall, this book must be cited with recognition. This is another book capable of combining philosophy and mathematics.