The Beautiful Mathematics- Fermat's Last Theorem and Philosophy of History

I was watching one of the episodes of BBC Horizon series. The first one I intended to watch was "Did Darwin Get it Wrong?'. Unfortunately the codec didn't work and the video crashed. I went back to Swaroop's hard disk, from which I had sourced the series. The video is crashed in there as well. I panicked a bit. What if all videos are malfunctioning ones? It took me more than an hour yesterday night to transfer the series to my laptop. It costed me the entire seasons of Psych, Breaking Bad, Two and a Half Men and How I Met your Mother. The latter two are at least saved in Swaroop's hard disk. Even his one terabyte capable portable was running low on space. Thus sacrificing some avoidable stuff became unavoidable.

Pierre de Fermat

Coming back to the series, I settled down to watch the episode on Fermat's Last Theorem. Pierre de Fermat, the 17th century French mathematician, had proved in his theorem that the Pythagorean Theorem wouldn't work for any other whole number exponent but the number two. The Pythagorean Theorem being a^2+b^2= c^2. Or in words, in a right angled triangle, the area of the square of the hypotenuse is equal to to the sum of the square of area of other two sides. What Fermat dared in 17th century was to prove that no other whole number exponent can give you the same results. For instance, a^3+b^3<> c^3 and so on. Now the problem was that over time Fermat's Last Theorem was lost and subsequent mathematicians couldn't reproduce or reprove the conjecture, not even Carl.F.Gauss! The problem continued to the 20th century. Even with the help of modern super computers, the conjecture couldn't be proved. Why? Because simply there are infinite number of numbers which exist. Thus even if we had computed the formula a billion times using a billion number of different whole number exponents, the theoretical possibility of that one number which can topple the formula existed. The mathematicians needed proof, a well thought-out, detailed mathematical theorem which proves the conjecture. In 1993, the British mathematician Andrew Wiles came out with the theorem which proved Fermat's last Theorem after a long and arduous research spanning over seven years. The entire mathematical community was ecstatic. One of the greatest puzzles in the world of numbers finally succumbed to human genius. Or did it? Soon Mr. Wiles's friend who had been given the task of checking the theorem for consistency, emailed Mr. Wiles of the glaring gap he found in the theorem. Andrew Wiles saw for the first time the mistake he had mistakenly overlooked. The error could well hit the death-nail upon his seven-year long work. The fault had put the entire proof at the risk of being unproven. As they say, fortune favours the brave. In 1994, Mr. Wiles found out that the very mistake he had encountered could unlock the possibility of fixing the proof. Finally on a fine day, Andrew Wiles, the man who dedicated seven years of his life to the theorem and worked so single minded on the pursuit of truth, found it. Awesome, isn't it? As he puts it, "I had this revelation...I found it." Thus was put to rest the Fermat's Last Theorem, one of the greatest intellectual challenges dawned upon humanity.

Prof. Andrew Wiles

Here, well. starts my story. I had always been a poor student of Mathematics. There was a period in my life when I hated the subject. I loathed the discipline as my maths teachers ran out patience with this stupid, good-for-nothing fellow. Very late or not until recently have I started admiring the subject. My admiration to mathematics stems from my interest in physics or more precisely, particle physics. Along with fellow physics enthusiasts or fans, I am too waiting eagerly for that day when a Unified Field Theorem would be constructed. The day when the string theorists are able to reconcile quantum mechanics with gravity. To put it poetically, the day in heaven when Albert Einstein shall shake hands with Max Planck. Owing to my personal mathematical ignorance, string theory or M-Theory principles have appealed more to the subjective and philosophical side of my personality than the objective and scientific rationale.

Prof. Max Tegmark

One of the interesting philosophical principles deductible from string theory is, as Prof. Max Tegmark from MIT enlightened the audience, at sub-atomic quantum levels, mathematics becomes the language of nature! Think about it, the universe itself could be nothing but a mathematical super-structure. If it could be true, the nature is not simply speaking to us in mathematics, the nature itself is Mathematics. At quantum levels the particles cannot be seen. The existence of such tiny particles are deduced from experimental observation which shows mathematical consistency. From the tiniest of the proposed strings and quantum particles to super black holes and gamma ray bursts, for some or other reason must exhibit mathematical regularity. It simply shows, as Prof. Max says, Mathematics is not a subject invented by humans but the very building block of the universe. As a student of History we are told to keep an open-mind. We should be open to any idea that comes in our way, however blasphemous it sounds. At the end of the day what we strive to achieve is the greater understanding of the society we live in. Thus the search for objectivity is not a historian's cup of tea, I feel. This constitutes the greatest philosophical division among Science and Social Studies. The search for the Truth, existing and verifiable, by Science is unknown to modern Social Studies. I don't believe the realm of Social Studies have to go after such Truth or such a Truth even exists. But such a rule applies only for Social Studies not for Science. The brain child of Enlightened idea of liberty settles for nothing but the Truth. So be it. While as the Science searches for objective Truth, we Historians and allied Social students shall further our understanding of the nature of such a verifiable and objective Truth, however subjective our interpretations may turn out to be. We are two sides of same coin. The question is but what is the coin?

The mathematics, thus holds the key for the greater understanding of the state of nature, whatever it is. The implications of this idea are far-reaching, even for Social students. The beautiful subject of mathematics can do wonder in our quest for greater excellence in all fields. Maybe we will construct languages of better mathematical consistency which will allow us to communicate our thoughts better. The closed-semantics of many a language including English at times inhibit dissemination of thoughts in their entirety. Unless you are so proficient in the language, mathematics-isation of languages would offer democratization of linguistics on an unprecedented scale. Those who had read Orwell's 1984 and familiar with Newspeak would understand the need for open-semantics in languages better. The computer codes, binary digits and quantum computing are all nothing but significant steps towards such a direction. Such a mathematics-oriented efforts can also do wonders in post-human philosophy, trying to create genetically engineered, superior human beings in a phenomenon that could rightly be labelled as evolution accelerated. The beauty of the subject is that the universe converses in mathematics. But is human-mathematics enough to capture the mystery of the universe in it's full measure? The numbers and characters, which visualizes mathematics to intelligent life, are representations of the principle which has gone in to the making of the universe. If we could push the horizons of human imaginations further, we may be able to come out with a form of 'Higher Mathematics', an improved version of mathematics which is better suited to understand the principle of universe and make our computations easier. It requires help from bio-technology, synthesizing intelligence of greater appeal. Everything demands a multi-disciplinary approach. This is just but one step. The evolution of both mathematics and human imagination have to go miles in evolution before we can feel safe about unlocking nature's secret. In Is Universe a Virtual Simulation? Really?, I had discussed about the research going into unveil whether the entire universe in which we belong is nothing but a PC simulation, run by human beings from future. Sounds like sci-fi, eh? Understanding Mathematics from both a conceptual and practical level again can help us decode our genetic programming, however virtual we are. I have also started encouraging in my mind this weird idea of a possible integration of some sections of Mathematics such as topology with Historiography so as to graph and visualise our world, scanning it across multi-dimensional time. Who knows, such an approach could help us discover attributes and properties of different systems we most commonly bypass in our analysis. And, we can stumble upon that coin, which seems to hold every sides. Thus, Mathematics is that infinitely beautiful subject which offers countless possibilities of reality, stretching our imaginations along with it to unprecedented scale. And that subject whose problems I can't solve for life, in this lifetime.

On a personal note, the story of Prof. Andrew Wiles is encouraging. A reminder of more intellectual nature yet motivating in it's appeal on par with the best of 'follow-your-dreams' kind of stories. In future, I too hope to answer some of the most nagging doubts constantly pricking my understanding about History, it's philosophy and it's motive force, if any. Meantime every generation must also pose problems of unsolvable nature to their successors to keep them busy in pursuit of knowledge. I, too, shall question.

What do picnics mean?

So, what does the three-day trip to Jim Corbett mean to me? A much needed break from Delhi it was. Three days and nights of friends and fun tend to have it's implications on your philosophy as well. Basically it proves the real life is sad and evil ridden at it's extreme. Thus, these kind of picnics and holidays prove to be an escape. Or in other words living a fiction. Novels, films etc. are forms of intellectual escapes from the miseries of real world into a make-believe world, aren't they? Vacations let you indulge not only your intellect but also your body in fantasy. The veil of reality is too tempting to overcome. Yet, some does. Had it not been for such breaks, for sure, humanity would have turned complete insane long back! For those who overcome the veil, it is a bitter acceptance of state of nature. Then, there are others whose hope is rekindled. And maybe there are zillion other categories or none. In what do I belong?

Jim Corbett Diaries- II

And thus started our second day at Ramnagar. The itinerary outlined visits to a local waterfall and a temple. As usual day the day took off with bread, egg, TT and badminton. My TT skills were definitely improving. Add to that Vishrut's, a fresher, tutoring on the game. He could spin the ball and make it land where he intend to on the table. The table was damaged and thus not suited to play proper TT, he opined. He was right. Even a dinner table would have made a better playing surface than the table on which we were playing. I started off with playing ping-pong sort of TT. As we played further, our game skills were progressively improving. Or we could hit the ball fast and yet make it land it inside the table. Brownie points! The badminton court was a different story altogether. To my surprise, I was good at it! I hadn't played the game for a log while. I expected to fare badly. It was not to be. Actually my badminton skills had improved from the last time. Kartik and I beat almost every other team. I must say, Kartik is the better player. The most pathetic one out there was Uday. Despite his height of 6'4" and lanky frame, he couldn't touch the shuttlecock. His above-ordinary height has bestowed upon him the luck to make contact with these high flying projectiles without much ado. Yet, he won't or he can't! Anyway, by 10 o'clock we left for the waterfalls in a bus.

At the waterfalls

The waterfalls was big time fun. Although we had to walk inside the jungle for quite a while to find it. Couple of days of trekking and constant playing had started to show on my calf muscles. Yet nothing could stop me. Simply none of us had the time to reflect on any tiredness. Everyone is moving and there is no time to lag behind. Thankfully our timing was near-perfect. There weren't any outsiders. Thus we owned the waterfalls, nearby woods and fun for the next one hour. Photographers, as usual, were busy capturing waterfalls, greenery and poses. I started stone-skipping, my favourite water sports! I am not bad at it, you know. Later took a small shower in the falls before settling down on a nearby rock. My foot was aching, thanks to walking on rocks barefooted. Nikita and Uday joined me and soon we were talking about books, I guess. After a while, others too had grown tired and came together for another group photo. Other tourists had started coming, mostly college students. The place was getting increasingly crowded. In another ten minutes or so we had started making our walk back to the bus. The resort was another thirty minutes away.

(from right) Uday, Nikita and I- from the waterfalls

Post lunch, boys were back on the field playing badminton and cricket. It was fun. And I can't bat for my life! Aroon, on the other hand is a good batsman and bowler. Kartik's technique is also admirable. Anil is good in Carrom. As I said before, we had another event featuring in the itinerary. We had to visit Garija temple, situated atop mountains. Thanks to the general irreligion prevailing among us, mostly none of us wanted to go. Pratyush&co. bunked it feigning ill-health. I decided to give it a shot. After all, a trip which costed me Rs. 4000/- had to be taken maximum advantage of. The bus trip lasted almost for an hour. I didn't talk much during the ride. I was enjoying the scenic beauty of Ramnagar, as we traveled further inside the village and upwards towards the temple. The roadside view was kind of nostalgic. It reminded me of sights I had come across during my trips to Kanyakumari, my father's place. Finally we reached the temple.

Rakesh, Kartik, I and Uday- on the banks of R. Kosi

Kartik, Uday and I decided to spend our time at River Kosi, which was flowing at a considerable might. Rakesh and Anil joined us in a while. Although I had a digital camera, I didn't make use of it. The entire atmosphere during the trip was so ecstatic, I felt like living the moment than capturing it. At Kosi, Kartik's and Rakesh's phones did the trick for me, though. We played in the river, skipped stones and even tried to determine who can throw the stone the farthest. Uday won, beating Anil, finally! Uday and I ventured quite further into the river. At every step we were careful not to slip on the slippery rocks and stones. The flow of river was quite strong and violent. Had we slipped and fallen, chances of rescue would be dim. Anil had the prasad or temple offerings. He kept it inside my bag. We had spent around an hour and half at the temple before we left. We bought pakoras and snacks on our way back. Vikas bought pizza from a nearby pizza point and Aastha was trying to find a workable ATM to withdraw money to meet the trip's expenses. The entire bus was dancing to Yo Yo Honey Singh's latest number, Break Up Party. It is Uday's birthday by the way. Celebrations were  just starting. Wait till the nightfall.

Uday and I- revel it the way you want!

At night, bonfire was made. It required a lot of effort keep fire burning. We gathered woods and kept feeding the bonfire. I must admit Vikas has a special skill in blowing the ember into life. Let's admit, people were high. That added to the spirit! Uday's birthday cake was cut and everybody sang 'Happy Birthday.' Party songs enthralled the mood and everyone danced. In no time, dancing turned into rain dance. My name was conspicuously missing, though. Simply because I am paranoid about getting sick and I didn't want to invite pneumonia needlessly. Chirag made the plunge into the pool and buddies followed. Again I remained dry, literally. I guess the fun went a bit overboard. Otherwise it wouldn't have woken up Jaya ma'm from sleep. The earlier false alarm just went wrong, ironically. She caught us all red handed. It wasn't scary, but disturbing to upset her that way. She told others to come out of the pool and instructed us to go to bed immediately. We obeyed. The time was around 12 o'clock at night. The night didn't end there, though. We all gathered around in rooms and had fun, cracking jokes and gossiping. Finally we called it a day at 3 o'clock in the morning. Good night.

The third and final day at Ramagar was starting. We were supposed to leave for railway station at night. Our checkout time was around 12 o'clock at noon. We gave back all but three rooms, where we kept our luggage ready. Nothing was on the itinerary for the final day as we had exhausted ourselves of all activities on the first couple of days itself. A quiet and lazy day, it was. We all met at the pool. This time, I decided to take the plunge with everybody. In the pool, the water was biting. We had to acclimatize ourselves to the temperature before any fun could start. Initially we played catch and catch with balls. It is impossible to make diving catches in pools, I learned. It is even harder if you can't swim! Later the game evolved into human tug of war, God knows how! Still, it too was fun, pulling the unsuspecting from both sides. Vatsala captained our side. We were quite good too. We pulled it off! After what looked like an eternity of fun inside the pool, we made ground. I ran for a quick shower and changed dress. 

Tug of War- Anil caught in the middle

Tug of War- Vatsala leading the charge

Kartik, Uday, Vatsala and Aastha opted for sun-bathing, drying up themselves in the process. I joined them shortly. Aastha and Vatsala were talking about the fun we were going to miss once back in Delhi. Aastha complained about withdrawal symptoms. Actually she was right. Too much fun to be left behind. We played cricket and badminton for a while before the lunch. By afternoon we were all tired and could feel the three-day long fun trip coming to it's inevitable end. I went to Pratyush&co.'s place for a nap and Kartik, Uday and Vatsala joined us. Some TV, chit-chats about films and actors and a lot of chips summarize the evening. At night around 8:30, we left the resort for the railway station. Our train was scheduled to leave for Delhi at 9:50 p.m. We comfortably made the distance, waiting for almost forty-five minutes for the train. The train arrived on time and we boarded. The last leg of our trip had begun. I was already missing the fun but wasn't ready to waste the night-time train travel with friends sulking. For some weird reason, what Abhinav to Prtayush to Chirag to Vatsala could talk about was ghosts. Pratyush went on and on about the supernatural experiences he encountered during his school days. Abhinav, Chirag and Aastha also chipped in with their bit. I am very neutral on such matters. they may or may not exist. I don't care. By midnight we were all sleepy and went to our berths. At 4 o'clock in the morning the train reached Delhi and we all deboarded. My father was waiting for me outside the Old Delhi railway station. I said goodbyes to friends and an awesome trip!

Rakesh, Uday, Kartik and I- the fun never ends

Jim Corbett Diaries- I

I had just reached Jim Corbett or Ramnagar railway station, to be exact. We, the touring History association of Sri Venkateswara College, were made to wait for a good twenty minutes before our resort's transport service picked us up. The time was around 4:30 a.m. in the morning. The journey from the station to resort took another thirty minutes. I enjoyed the ride, partly cutting through the jungle as tigers slept, so did the twenty odd friends of mine. Once at the resort, we were all busy unloading. The resort was pleasing in it's appearance. While checking out the surroundings, it happened to look up at the night sky. Uday followed my gaze. The sky was illuminated with infinite stars gleaming the cosmic radiance at us, the lowly mortals, I felt. There is nothing more welcoming than a night sky full of shining stars, I shared my thought with Uday. He nodded in assent. Kartik, Uday and I put up in a single room. Sleep had dawned on us, the dusk had set. More than six hours of train journey at night with friends practically mean little sleep. We said our good nights.

(from right) Rakesh, Abhinav, Kartik, I, Vatsala and Aastha- from the train ride

Room delivery served us tea around nine o'clock. Boys had slept barely for three hours and the day was already starting. There was no stopping Aastha and Vatsala. Aastha was like "Guys, we have only three days in front of us. Do you intend to sleep it off?" The kind of child-like yet contagious enthusiasm was the hallmark of the entire trip. It kept the fun alive.

The entire History tour group

The breakfast was good. A good day to begin. The first on the list was trekking. Sounds fun, eh? We checked with the guide regarding the possible duration of the trekking. He gave us an estimation of thirty minutes. As it turned out to be the longest thirty minutes of my life. First of all, it didn't look like trekking at all. As I climbed the treacherous steps, I kept looking back. The image that came to me was that of a fleeing army, lost and stripped of honour, seeking refuge in some mountain hideout. It is just me. As we trekked our own Mt. Doom, the climb became steeper and murmurs of discontent had risen. Mutiny, my imagination called aloud. Everyone was already feeling tired. Understandable for the urban-pampered cry-babies. Still, the fun was too much to leave it midway. Without turning back, we marched ahead to the tunes of chit-chats. We rested at many spots, photographers didn't seem to have missed a single frame, as the number of Facebook pictures stand testimony to my claim. Some of us lagged behind, playfully enjoying streams and even termite and ant hills. Whatever said and done, once at the pinnacle, the view was breathtakingly awesome. The quaint village of Ramnagar laid in front of us in a panoramic view. Green and lush, wow! The climb was worth it and no one could agree more. As a rule of margin, all things go downhill. The trek is no exception. Our descend was cautious since the step were steep. Yet, thank goodness, we reached safely back. Although Jaya Ma'm vented her fury at the guide for having taken an arduous path, Uday and I later thanked the guide-bhayya for the challenging trek. As Uday says at times, "It's not the destiny; it's all about the journey."

Vatsala and Aastha; the trail follows- during the trekking

We had a great lunch, as the food was generally good throughout the trip. Shortly after the lunch, I stumbled upon the Table Tennis table at the reception. Something of a time-filler for the next three days, the discovery turned out to be. In no time, Kartik was standing at the side, waiting for his turn, as I played against Uday. Such is life, as we played our hearts out, the call for afternoon jungle safari came running in. Earlier we had tried to postpone the safari as we all were kind of tired from trekking that morning. Alas, the jeeps had already come and had to be paid at cancellation too. Thus we took the bet. Our rides were in five open roof Gypsy vans. We were cruising along the roads of Ramnagar. The feel of wind blowing at you, as the Gypsy sped at a speed close to 80 kmph, is wonderful. As a matter of fact Uday feels it should be on everybody's bucket of 'to-do' list before you die. Soon we reached Jim Corbett National Park, in heightened expectations of spotting a tiger. Indeed we spotted, a dozen of monkeys and countless dogs and a handful of sambhals. To our disappointment, we learned the hard way that no one expects you to spot tigers at the fringes of the jungle. Anyway, the ride through the forest was awesome to say the least. One freakish thing that happened during our safari was the breakdown of a Gypsy in which the first years were travelling and everyone stopped. Isn't it sort of scary to be stuck inside a jungle, famed for it's man-eater predators? Well, Rakesh and I took a stroll down the woods, not long before he directed my attention towards a footprint. Apparently a pug mark, he claimed. So much hullabaloo for a pug mark. We all left the scene a little later, halting at a Maggi point inside the jungle before exiting the national park. The Hindi party songs gave us the verve to groove on our way back to resort. At resort, shortly after our dinner, we celebrated Uday's birthday. The boy turned 20 as the clock struck twelve o'clock. All credit goes to Aastha for pulling off a memorable birthday eve. And guys too were ecstatic too. Salute the spirit, literally!

Aastha, Uday, Kartik and I- from the Gypsy ride

Going to Jim Corbett tonight

Yeah, yeah, yeah. Leaving for Jim Corbett National Park tonight. Still gotta take a shower and thank God the packing is done, almost. The train leaves at 10:45 p.m. Aastha has been looking forward to the trip for a pretty long time. And after passing much hurdles, which are comparable to agnipariksha, the platform lies waiting. More than twenty people from the History Association are going including Uday, Kartik and Vatsala. I am hoping for a worthy break from home. A good four days with friends and tigers in the middle of a jungle. Well, what else to say but bon voyage, to ourselves. Stay safe, mate.