Srinivasa Ramanujan and Lessons for Today's India

Today, April 26, 2020, marks the centennial death anniversary of Srinivasa Ramanujan, the self-taught math genius born into a poor family in Southern India in 1887 under British colonial rule, who conjured theorems and formulas in his short 32 years of life that continue to intrigue and challenge mathematicians to this day.

Srinivasa Ramanujan

The anniversary comes at a tragic time for India. Sectarian and police violence have claimed many lives, mostly Muslim, following the passing of a discriminatory Citizenship Amendment Act in 2019 aggressively promoted by Prime Minister Narendra Modi and his Hindu-Nationalist Bharatiya Janata Party (BJP). Indians of all faiths have been demonstrating against the law, although the coronavirus contagion has halted it for now. (But the pandemic has added its own macabre twist to the story. Reports are now emerging that right-wing fanatics are attacking and killing Indian Muslims for spreading the virus!) The original Citizenship Act of 1955 guaranteed citizenship to every Indian irrespective of religion, reflecting the aspirations of a secular, democratic nation. The amended bill favors all major religions of South Asia except Islam, a blatant discrimination against India’s 200-million Muslim community.

Remembering Ramanujan against today’s India offers lessons about the insidious ways by which bigotry and injustice can erode the soul of a nation and stifle the flowering of its geniuses.

Ramanujan would have withered away as a shipping clerk in Madras (now Chennai) had it not been for the patronage of an Englishman. In 1913, the same year that Rabindranath Tagore, another son of India, won the Nobel Prize for Literature, Ramanujan wrote a letter to the Cambridge mathematician G. H. Hardy to rescue him from oblivion. Trying to digest the 50 or so theorems included in the letter, Hardy realized he was dealing with a rare phenomenon. The mathematics seemed so sublime that Hardy was convinced “the results must be true, because, if they were not true, no one would have the imagination to invent them.”

Ramanujan's Letter to Hardy

G.H. Hardy

Hardy managed to bring Ramanujan to England through a grant, giving him an intellectual home at Cambridge where he flourished for five years, producing pioneering work in such branches as number theory and partition function that secured his reputation as among the greats of mathematics. The serendipitous discovery of Ramanujan’s “Lost Notebook” by Wisconsin University Mathematics professor George Andrew in 1976 that included calculations involving “mock theta functions,” 56 years after he worked them out in India, has added to the enigma and the otherworldly genius of Ramanujan. The late physicist Freeman Dyson (1923-2020) aptly summed up Ramanujan’s achievement: “That was the wonderful thing about Ramanujan. He discovered so much, and yet he left so much more in his garden for other people to discover … I have intermittently been coming back to Ramanujan’s garden. Every time I come back, I find fresh flowers blooming.”

From Ramanujan's Notebooks

Ramanujan’s theorems are now applied in fields ranging from particle physics and statistics to cryptology and computer science. His approximation for pi, the ratio of a circle’s circumference to its diameter, finds use in today’s fastest algorithms in computing the irrational number to trillion decimal places.

Ramanujan was lucky. (So was Tagore. Had not the Irish poet Yeats championed his work in the West, the Bengali poet would unlikely to have won the Nobel.) But for every Ramanujan who succeeded, how many Ramanujans must have died on the vine when the British ruled the Subcontinent for over a century? Colonial powers by their nature subjugate, exploit and massacre. But when the oppressors are one’s own leaders, what’s the excuse? Modi and his BJP passed the Citizenship Act to marginalize Muslims or drive them out of India. Such overt discrimination affects not only the target population but the entire nation, preventing potential Tagores and Ramanujans from ever seeing the light of day.

The human potential of India is incalculable. We get a sense of it by the disproportionate number of Indian American engineers and CEOs keeping hi-tech companies humming right here in Silicon Valley. Brain drain from India and other countries will continue so long as the West offers better opportunities to immigrants. While India cannot provide such opportunities for all, passing discriminatory laws against a segment of its population diminishes the nation and shackles its geniuses, be they Hindu, Sikh, Christian or Muslim. When a government becomes tyrannical, the creative vitality of its people seeps away as from cut flowers in a vase.

Ramadan in the Time of Pandemic

You can read a slightly different version of this in the San Jose Mercury News

This year’s Ramadan (April 24 to May 23) comes at perhaps the most challenging time of our lives. A killer virus is advancing like wildfire across the globe, leaving heartbreak and death in its wake. Confined to our homes, we are stressed and uncertain of the present and the future.

But hope is native to our nature. The Quran, the Islamic Book of Divine Guidance, instructs us never to give in to despair. Certainly no one despairs of God’s Mercy … (12:87)

When we become complacent and think that we can dictate our destiny, God warns us: And We will surely test you with something of fear and hunger and a loss of wealth and lives and fruits, but give good tidings to the patient, who, when disaster strikes, say, “Indeed we belong to God, and indeed to Him we will return.” (2:155-156)

But God also assures us that Surely with hardship comes ease. Indeed, with hardship comes ease. (94:5-6) 

In this trying time, Ramadan reminds us to slow down, to pause and reflect on life’s big questions of meaning and purpose. So, where are you going? (81:26) asks the Quran. We must ask ourselves: Where, indeed, am I going with my life? Is it aligned with God’s expectations of me as defined by God, or am I going astray from a combination of ego, ingratitude, impatience, forgetfulness and other assorted flaws?

As we Muslims welcome Ramadan into our lives, we must remember the many in our community who have lost their jobs - cabdrivers, office assistants, caregivers, domestic help, daily wage earners, and others. They are among the more than 22 million of our jobless fellow-Americans facing a frightening future. We are traditionally most generous during Ramadan, and we pray that we can be even more generous during this Ramadan. Charity, the Quran tells us, is for the neediest, regardless of religion, particularly those who do not ask for help due to modesty or embarrassment. We must find and help them without in any way hurting their sense of dignity and self-respect

The doors of our mosques are closed, as are the doors of churches, synagogues, and temples. But thanking God for His blessings, mercy and forgiveness in the sacred days and nights of Ramadan can open windows in our hearts. Never for a moment should we forget that repentance can lead to redemption and remembrance of God to renewal. The dark shadow of life’s pain will, God Willing, be dispelled soon by the bright sunshine of life’s promises.

As it happens, today, Wednesday, April 22, 2020, is Earth Day. Fifty years ago, it dawned on us that there was no Planet B, that the Earth was the only home we had, and unless we cared for her, we were doomed. The forced isolation of the coronavirus has done one good thing: with little pollution defiling Mother Earth, the sky looks clearer and the vegetation pristine. Covid-19 pandemic has taught that we are not really the masters of the universe that we think we are. We know very little and our capacity is extremely limited. We have been committing crimes against nature, against animals, against all the sentient beings that we share the earth with. A single novel virus has collectively brought us to our knees. To survive, we must replace our hubris with humility. Repentance can lead to redemption and responsibility to renewal. That is also the lesson of Ramadan.

Uncertainty of the Coronavirus Pandemic: Humility Before Nature

William Goldman, the talented Hollywood screenwriter (1931-2018), summed up the impossibility of predicting which movie will be a hit and which not, with this pithy perspective: “Nobody knows anything.” (He qualified his statement with “Not one person in the entire motion picture field knows for a certainty what's going to work. Every time out it's a guess and, if you're lucky, an educated one.” But no one quotes the long qualifier when three words do a far better job than thirty.)

William Goldman - "Nobody knows anything"

Goldman’s 3-word phrase springs to mind when we hear all the punditry and stupidity (the latter emanating mostly from the clueless, ignorant liar-in-chief narcissist in the White House) surrounding the coronavirus pandemic. Nobody knows anything. Well, some experts in the field, mostly researchers who have studied pandemics all their lives in the field or in the lab, probably know something but what they do not know - and they will be the first one to tell you - is so much more than what they do that, even for them, the ratio of the known to the unknown is practically zero.

What is worth reflecting on is the rate at which the killer virus is leaving death in its wake. One word that sums it up is “exponential,” a mathematical concept many of us are familiar with from such media phrases as “CEO compensations are increasing exponentially” or “Many species of the world are decaying exponentially.”

If the coronavirus infects 1 person today, who in turn infects 2 tomorrow, each of whom infects 2 the following day to make it 4, and so on, it is an exponentially increasing (doubling) function. It is base 2 raised to d, the number of days since the 1^st infection, that is, 2^d. What if the 1^st person infects 3 rather than 2, and those 3 each infect 3 the following day, and so on. In that case, it is 3^d, the base being 3 (tripling) rather than 2.

You can see what we need to do to make the infection rate decay: Make the base less than 1, until the number of infections “asymptotically” (that is, very close to but not exactly equal to) approach 0.

But that decay rate is nowhere in sight.

For example, on Tuesday, March 17, it was reported that there were just more than 5,700 confirmed coronavirus cases in the United States. That number climbed above 11,500 on Thursday, March 19, and officials indicated the number will continue to rise sharply as more test results become available.

The number doubled in 2 days. The exponentially increasing function is now (5700)2^d/2, where 5,700 is the initial count and d is the number of days since the initial count.

If somehow we could replace that base 2 (doubling) with say, 0.7, then (5700)(0.7)^d/2 will rapidly “approach” 0.

One way to understand the numbers is to take this paragraph from a recent opinion piece by the NY Times columnist Tom Friedman:

“One of the hardest things for the human mind to grasp is the power of an exponential, something that just keeps relentlessly doubling and doubling, like a pandemic. The brain just can’t appreciate how quickly 5,000 cases of confirmed coronavirus infection in America can explode into one million if we don’t lock down now. Here’s a simple way to explain the exponential threat we face, in a way an oft-bankrupt real estate developer like Donald Trump might understand. It was also offered by Bill Joy: ‘The virus is like a loan shark who charges 25 percent a day interest. We borrowed $1 (the first coronavirus to appear here). We then fiddled for 40 days. Now we owe $7,500. If we wait three more weeks to pay, we’ll owe almost $1 million.’”

Let’s do the math.

A loan shark charges me $0.25 per day compounded on a $1 I borrowed from him (I am not aware that there has ever been a woman loan shark. It's always men!) So, at the end of the first day, I, the borrower, owe the shark $1.25. At the end of the 2nd day, I owe the shark interest on interest, which amounts to (1.25)² (1.25 raised to the power of 2.) So, at the end of 40 days, I will owe him (1.25)⁴⁰ = $7,523.16, which rounds off to $7,500. After 3 more weeks, that is, 61 days later, I owe the shark (1.25)⁶¹ = 815,663.05, which is close to 1 Million Dollars!

The way to win this battle, and also the war, is to make that 1.25 LESS than 1, say, 0.8, so we experience exponential decay. In that case, after 40 days of decay, the number reduces to (0.8)⁴⁰ = 0.0001329, which is one-hundredth of 1%! It's in the math. We need to make that number in the parenthesis less than 1 before we can resume normal life.

So how do we make the base of the exponential function less than 1? That’s the trillion-dollar question facing us today. Social distancing, better-equipped health workers and hospitals, proper hygiene, and a combination of many more yet unknown factors can perhaps force the killer virus to be on the decaying curve. But again, we just don’t know! No matter how much fancy statistics and solutions we weave and spin, we just don’t know!

While the mathematics of increasing and decaying exponential models are known with certainty, the nature and the destructive power of the coronoavirus pathogen is unknown. We only know that unchecked, it grows exponentially, but if somehow checked, will decay and eventually, for all practical purposes, disappear.

Consider Hemingway’s insight in this dialogue from his 1926 novel “The Sun Also Rises”: “How did you go bankrupt?" Bill asked. “Two ways,” Mike said. “Gradually and then suddenly.”

The initial increase from 1 to 2 to 4 seems gradual when suddenly, before we know it, the virus has infected over 65,000 and then more, and yet more, with its limitless, voracious appetite.

People who spout certainty when life is fundamentally uncertain are the real ignoramuses, no matter how powerful or wealthy or apparently smart they may appear be. In contrast, those with real knowledge are humble when confronted with the unknown. They know that nature’s imagination is superior to human imagination. To glimpse into the inner workings of nature, whether of physical laws or of pathogens, they know that first they must acknowledge, “I don’t know.” This is also true from a theodicy perspective. Priests, Rabbis and Imams afflicted with religious chauvinism who think they know how God’s mind works with certainty are as guilty as arrogant scientists and political leaders in misleading us.

“Doubt is an uncomfortable condition,” said Voltaire, “but certainty is a ridiculous one.”

In the next year or two, brilliant but humble researchers will undoubtedly have made slow but significant progress in deciphering the working of the killer coronavirus and ways of neutralizing it. The rate of their progress will be anything but exponential. Yet they offer us hope that one day this virus will have met its match in their ingenuity. If you were to ask one of them what the catalyst was for her breakthrough, she will unhesitatingly tell you, “I began by acknowledging that I didn’t know.”

Leaplings of the World, Unite!

Today’s “Word of the Day” is “intercalary.” It refers to a day inserted in the calendar to bring it into sync with the solar year. That day is today, the 29^th of February, the Leap Day of the Leap Year 2020.

People born on this day, the Leaplings, are special. They get to celebrate their birthday every four years (although some cheat by celebrating three years on the 1^st of March and one on the 29^th of February) and so make it more special than the yearly birthdays we normal folks celebrate.

Everything comes down to counting. We count a year to consist of 365 days, but the Earth takes a little longer than that to go around the sun in an elliptical orbit, although, since the eccentricity is only 0.0167, or almost zero, the orbit is almost circular. The actual number of days is not 365 but 365.2422 days. Think of that rather annoying additional decimal day. Why couldn’t the number of days be an integer instead of a decimal? Because celestial phenomena do not follow the neat geometry of humans. They follow what they must, according to the laws of physics.

Still, let’s approximate and assume that the number of days in a year is 365.25. A day of 0.25 means quarter of a day, which is 6 hours. So, for three years, we fall short in our counting by a total of about 18 hours. The solution? Make it 24 hours by the 4^th year and add that extra day to the fourth, that is, the Leap Year, and we are in sync with the true celestial year, on the average. Except, we are overcounting the solar year by about 12 minutes on the average per year. The solution? Normalize these extra minutes by skipping “intercalary” for some of the Leap Years. Which ones? Those year that are not exactly divisible by 400. That means the years 1700, 1800 and 1900, although divisible by 4 but not 400, were not Leap Years. The year 2000 was but not 2010. This year 2020 is but 2030 will not be, and so on.

Except that this calculation is still not completely right. 365.2422 of a true solar year means 365 days, 5 hours, 48 minutes and 46 seconds. So, when we add a day every Leap Year, we are overcounting the length of the year by (6 hours - 5 hours 48 minutes 46 seconds), that is, 11 minutes and 14 seconds, not 12 minutes. In other words, the Gregorian Calendar we currently use and adjust with a Leap Day overestimates the average length of a year by 11 minutes and 14 seconds, not 12 minutes. The correction necessary is 365.2425 – 365.2422 = 0.0003 days per year, which means another correction will be necessary in 3,030 years. It is safe to say that most of us will NOT be around to experience THAT correction!

Transcending these 4^th place of decimal correction is, however, this question: What is the probability that a person selected at random from the population will have a 29^th of February birthday?

Leapling, anyone?

Well, in 4 years, there are 4 x 365 days = 1460 days + 1 Leap Day = 1461 days. So the probability that your spouse, niece or cousin, or anyone, was born on the 29^th of February = 1/1461, which is approximately 0.000684, or about seven hundredths of 1 percent. So how many people in the world can we expect to have their birthdays fall on the 29^th of February? This expectation value (not the actual number but close, according to laws of probability), assuming that the population of the world in 2020 is 8 billion, is 1/1461 x 8,000,000,000, or about 5.8 million. In other words, we can expect at least 5 million Leaplings in the world today.

If you are looking for an ice-breaker at a party, how about this as you hone in on a potential friend: “Are you a Leapling?” When the response is a puzzled, “Say what?”, you launch into an explanation of Leap Years, why and how they came about, and how, using probability, you can calculate that there are over 5 million Leaplings in the world now.

As you look up with a triumphant smile after your brilliant monologue, you realize with a shock that your "target" has vanished!

Women's March 2020, San Jose, CA: Dump Trump

About 8000 of us gathered today (Saturday, January 18, 2020) in San Jose, California - 75% women and 25% men from what I could gather - at the Arena Green in downtown San Jose, California, to participate in Women’s March 2020. Similar marches - estimated at 180 - are taking place in small and big cities throughout America today. The march has three objectives: Dump Trump in the November 3^rd presidential election, empower women and enshrine their rights in laws and in practice, and save Planet Earth from the catastrophic effects of Climate Change.

I listened to Karrington Kenney, the 16-year-old junior and VP of the Black Student Union at Milpitas High School. “Make sure to vote in December,” said the poised and persuasive speaker. Under Trump, we are witnessing unabashed racism. We must do our part in defeating the haters and the White Supremacists. Kenney was instrumental in outing a history teacher in her school who wore blackface last year at Halloween. Although the teacher apologized, Karrington reminded us to be vigilant and proactive in calling out racism wherever and whenever we see it. She quoted the names of the African-American women who, as NASA engineers and mathematicians, made incalculable contributions to America’s space program. That they did it in Jim Crow South made their contributions that much more remarkable.

Karrington Kenney, Milpitas High School Student, Addressing
Women's March in San Jose, January 18, 2020.

Sabrina, a third-generation Jewish American, was holding up signs with her teenage daughter that read, “You Can’t Complain If You Don’t Vote!” Her grandmother was born in Stockton, CA. When she was 3, she was sent to Germany to be looked after by relatives. Then the Nazis took over. Her grandmother and family narrowly escaped being Holocaust victims: A church “certified” that they were Catholics. When Sabrina’s mother was 3 years old, her grandmother migrated to the U.S. with her daughter for good in 1948. “And so here I am,” said Sabrina. “I want everyone to understand how lucky we are that we can assemble like this and vote. And unless we vote, we deserve our leaders. That’s how we got Trump. So make sure you vote on 3^rd November!”

Sabrina tells San Jose Women's March participants that
unless they vote, they have no right to complain.

Can we restore the rule of law, decency, justice and democracy in the United States, come November 2020? Only if we do our part and vote our conscience. Words are meaningless unless backed by action. "Vote blue no matter who," read an impassioned poster. Evidence of Women Power was everywhere. The nation is counting on America's women to rise and lead the nation to justice, decency and rule of law by decisively defeating Trump.

Here are some more photographs to give you an idea of the energy, passion and activism that are beginning to animate Americans in this critical election year.