Imagine for just a second that you're trying
to build a functioning spacecraft. Okay a spacecraft.
Yeah, but here's the catch you Have never seen
a blueprint You haven't like been taught the
laws of physics or anything and you're working
entirely alone in a cave, right? So impossible,
basically. Exactly. But against all conventional
logic, the ship you build doesn't just fly. It
travels faster and further than anything the
top engineers in the world have ever even conceived
of. Wow. I mean, it sounds like the origin story
of like a comic book character or something,
just a mind operating entirely outside the boundaries
of human knowledge. It really does. But today,
we're actually exploring a reality that mirrors
that exact scenario. Welcome to today's Deep
Dive. Glad to be here. So we have a massive stack
of sources today. We're looking at a really comprehensive
biographical and mathematical overview of Srinivasa
Ramanujan. Ah yes, a legendary figure. Our mission
today for you listening is to basically extract
the ultimate aha moments from a man who, well,
he essentially pulled advanced mathematics out
of thin air. He really did. We want to figure
out what his story reveals about the true nature
of knowledge and, frankly, genius. Like, how
does an untrained mind perceive these universal
truths that experts just, they can't see them?
It's the ultimate question. Okay, let's unpack
this. Because to understand how his mind operated,
we have to look at the environment that forged
it, right? Definitely. Context is everything
here. So we are starting in 1887. where in British
-ruled India, Ramanujan is born into a Tamil
Brahmin family, and they're living in this small,
traditional home in Kumbakonam, and they are
living in deep, deep poverty. Right. I mean,
the environment was incredibly resource -scarce,
but the mathematical anomaly that he presented,
it was just, it was apparent almost immediately.
Even as a kid, right? Yeah. By the age of 11,
he had completely exhausted the mathematical
knowledge of two college students who were renting
rooms in his family's home. Wait, 11 years old?
11. And by 13, he had mastered advanced trigonometry
totally independently. He was already discovering
complex theorems on his own. Just completely
disconnected from the global math community.
Exactly. Entirely isolated. But the crucial spark,
the thing that really sets him off, happens when
he's 16. Yes, the book. Right. He gets his hands
on this library copy of a book by G .S. Carr.
It's called A Synopsis of Elementary Results
in Pure and Applied Mathematics. And the structure
of that specific book. That is the key that unlocked
Ramanujan's unorthodox brain. How so? What was
different about it? Well, it wasn't a standard
textbook. It didn't walk the reader through the
logic of how to solve a problem step by step.
It was essentially just a compilation of about
5 ,000 theorems with almost no proofs provided.
Just the answers? Just the final equations, literally.
Let's cause on that, actually. Because the concept
of a proof is super foundational here. Oh, absolutely.
In math, a proof is like the logical scaffolding,
right? It's the step -by -step argument that
takes you from a basic assumption to a final,
undeniable conclusion. Precisely. A proof is
how a mathematician actually demonstrates that
their answer is correct. It's the show your work
part. Exactly. It's show your work scaled up
to a professional academic level. And Carr's
book basically stripped all of that away. It
was just a catalog of final answers. It's like,
OK, it's like someone trying to become a master
chef, right? But they're only reading a list
of ingredients. They never see the recipe steps.
They never learn what a stole even is. That's
a great way to put it. You know what the final
dish is supposed to look and taste like, but
you have to invent the underlying physics of
cooking entirely on your own just to recreate
it. And because he lacked those formal recipes.
You know the proofs. Ramanujan had to forge his
own neural pathways to reach those mathematical
conclusions. He had to rewire his own brain.
Exactly. He rewired his brain to leap straight
from the problem to the answer, totally bypassing
the standard logical stepping stones that every
formally trained mathematician relies on. Which
is just wild to think about. And stating incredibly
complex mathematical results without ever showing
the derivations, that just became his lifelong
habit. Which is fascinating, but I mean, it completely
torpedoed his conventional life. Oh, yeah. It
was a disaster for his schooling. He becomes
so intensely focused on this bizarre method of
independent research that he literally fails
his college exams. In English, in Sanskrit, in
physiology. He loses his scholarship. Yeah, he
actually runs away from home at one point, and
he's living on the absolute brink of starvation.
His isolation was this profound double -edged
sword. I mean, on one hand, it caused him immense
academic failure and physical suffering. Real
suffering. Yeah. He had no degree, no money,
no traditional prospects at all. But intellectually,
his mind just wasn't boxed in. Because he hadn't
been taught what the academic establishment considered,
you know, impossible, he wasn't constrained by
traditional mathematical paradigms. He didn't
know the rules, so he didn't know he was breaking
them. Exactly. He was free to explore dimensions
of numbers that formally trained mathematicians
wouldn't even think to look at. Which sets up
this massive collision course because you have
this isolated college dropout, right? He's working
as a clerk at the Madras Port Trust for like
30 rupees a month. doing math on a literal slate
chalkboard. Right. And he knows he's seeing things
no one else is seeing. And he desperately needs
a mathematical peer who can actually read his
notebooks. So he starts writing letters to British
academics. And those initial attempts went exactly
how you might expect. Not well. Not well at all.
Yeah. He sent his papers to two prominent professors,
H .F. Baker and E .W. Hobson. and they returned
his work without a single comment. Just completely
dismissed him. Totally ignored him. But then
in January of 1913 he sends a nine -page letter
packed with unproven theorems to G .H. Hardy
at Cambridge University. Yes. a major turning
point. And Hardy is like one of the most prominent
rigorous mathematicians in the world at this
time, right? Oh, absolutely. Top of his field.
So Hardy opens this letter from this unknown
clerk in India and just sees page after page
of alien looking formulas, just bizarre assertions
about infinite series and continued fractions.
And just to give you a sense of what Hardy was
actually looking at. A continued fraction is
a mathematical expression where the denominator
of a fraction contains another fraction. Okay,
so a fraction inside a fraction. Right, which
contains another fraction. potentially go on
infinitely. That sounds like a nightmare to calculate.
It is complex. It's a way to represent irrational
numbers. Numbers that can't be expressed as simple
fractions like pi with incredible precision.
Gotcha. But Ramanujan's continued fractions were
wildly complex. They connected seemingly unrelated
mathematical concepts in ways that looked almost
nonsensical to a traditionally trained eye. Okay,
here's where it gets really interesting though.
I have to push back on this narrative a bit.
Because something doesn't add up for me. Sure,
what is it? If this letter was just nine pages
of alien equations with zero proofs, why didn't
Hardy just throw it in the trash like Baker and
Hobson did? That's a fair point. Right! Like,
in a pre -internet era, how does a Cambridge
professor distinguish a once -in -a -century
genius from a random person writing literal gibberish?
That is actually the pivotal question of this
entire story. Because it just seems so unlikely
that he would even read it. Well, Hardy didn't
just blindly trust the alien equations. He needed,
um... A Rosetta Stone, basically. An anchor of
credibility. Oh, okay. So there was something
familiar in there. Exactly. Amidst all the wildly
unfamiliar theorems, Hardy recognized a few formulas.
Like what? Specifically, he saw a result concerning
the convergence of a certain series that had
already been determined by a mathematician named
G. Bauer back in 1859. And Ramoujian had just
independently rediscovered it. Yes. He rediscovered
it entirely on his own. So because Ramanujan
got that one obscure known theorem correct, it
validated all the rest of the crazy stuff. It
provided the context Hardy needed to trust the
unknown. You know, Hardy realized that no one
could accidentally stumble onto Bauer's theorem
while just writing gibberish. That makes a lot
of sense. As Hardy later concluded regarding
those wildly unfamiliar continued fractions,
he said they, quote, must be true because if
they were not true, no one would have the imagination
to invent them. Wow. If they weren't true, no
one would have the imagination to invent them.
That's a brilliant way to look at it. It really
is. So Hardy is convinced. He knows he's holding
lightning in a bottle here. But to do anything
with it, Ramanujan has to cross the globe to
Cambridge. Which was a massive monumental hurdle.
Right. Because of his background. Ramanujan's
orthodox Brahmin upbringing strictly forbade
overseas travel. I mean, crossing the ocean meant
losing his caste. It wasn't just a longboat ride.
It was a huge cultural taboo. He only agreed
to leave for England after his mother had this
vivid dream, right? Where the family goddess
Namajiri commanded her not to stand in her son's
way. That's the story. And when he finally arrives
in Cambridge, we see this incredible collision
of two Brilliant, but fundamentally opposite
minds. I can only imagine. Ramanujan was deeply
religious. He relied almost exclusively on intuition.
He famously stated, an equation for me has no
meaning unless it expresses a thought of God.
A thought of God, wow. Yeah. He claimed to have
literal visions of blood drops, and he'd see
the goddess Namajiri unfolding scrolls of complex
mathematical content right before his eyes. And
then sitting across the desk from him is G. H.
Hardy. The exact opposite. Literally. A staunch
atheist and what his peers called an apostle
of proof. Yes. For Hardy, mathematical rigor
was the only religion. If you couldn't prove
it step by step, it wasn't mathematics. So the
cultural and philosophical friction there was
just intense. I mean, Hardy was constantly trying
to teach Ramanujan the formal math he had missed,
like how to actually write a proof. Without destroying
his natural intuition, I assume. Exactly. That
was the delicate balance. It's like, OK, it's
like taking a completely self -taught. wildly
brilliant jazz improviser, someone who plays
entirely by ear, and forcing them to jam with
a strict sheet music only classical conductor.
That's a perfect analogy. The jazz player is
just feeling the music, jumping straight to the
climax of the song, and the classical conductor
is shouting, wait, what time signature is this?
You have to write down the notes so the rest
of the orchestra can play along. If we connect
this to the bigger picture. Progress often requires
this exact kind of friction. I know you mean.
Well intuition needs rigor to become universally
applicable. You need both. Yeah. And their collaboration
on the partition function is the absolute perfect
example of this. Okay, let's break that down
for the listener because the partition function
sounds super intimidating. It does sound technical.
But if I'm reading the sources correctly, a partition
is just figuring out how many ways you can add
up whole numbers to get a specific target number.
That's it. Exactly. It's actually very simple
at its core. Take the number four. Okay, four.
You can make four by just having four. Right.
Or three plus one. Right. Or two plus two. Exactly.
Or two plus one plus one. Or one plus one plus
one plus one. So the number of partitions for
four is five. Yep. That seems really simple.
It is simple until the numbers get bigger. By
the time you reach the number 200, the number
of partitions explodes to nearly four. trillion
ways. Four trillion just for the number 200.
Almost four trillion, yeah. And mathematicians
had absolutely no idea how to calculate these
partitions for large numbers without just counting
them one by one. Which is impossible. Totally
impossible. So Ramanujan and Hardy set out to
create a formula that could predict the exact
number of partitions for any integer to infinity.
And to solve it, they created this thing called
the circle method. Yes, the circle method. I
read through this part of the source material
a few times, and I'll admit the mechanics of
it are really hard to visualize. Like, how does
a circle help you count whole numbers? It's tricky,
but imagine you have a massive complex landscape
of data. but you only care about specific isolated
peaks within that landscape. Okay, peaks in a
landscape. The circle method uses the boundary
of a circle. in the complex plane, which is a
continuous, smooth curve, to enclose and analyze
the behavior of functions near the edges of that
circle. So you're drawing a boundary. Right.
By using calculus to integrate around this path,
they could essentially sift out the exact discrete
whole numbers they were looking for. Oh, wow.
They used continuous flowing mathematics to perfectly
cache discrete, jagged numbers. That is so cool.
So they basically built a mathematical net to
catch the trillion different ways to add up to
200. exactly what they did. And Rimojin's intuition
saw where the net needed to be cast, while Hardy's
rigorous calculus built the actual mesh of the
net so it wouldn't break. Without Hardy, Rimojin's
insights would have just remained incomprehensible
visions on a slate. Right. But without Rimojin,
Hardy would never have had the raw, otherworldly
material to build those mathematical roads in
the first place. It's the ultimate partnership.
But this incredible output came at a massive
physical cost. It really did. The sources paint
a pretty grim picture of Ramanujan's time in
England. I mean, it's World War I. Rationing
is in full effect. Food was scarce for everyone.
And Ramanujan is struggling immensely to maintain
his strict vegetarian diet in a country that
barely has enough food to begin with. Plus, he's
working obsessively. He would sometimes forget
to eat entirely. And his health just deteriorates
rapidly. He ends up confined to a sanatorium.
And it's during this really dark period that
we get one of the most famous anecdotes in all
of mathematics. Oh, the 1729 story. Yes. It perfectly
encapsulates how Roman Nugent viewed the world.
So Hardy goes to visit Roman Nugent when he is
sick in the hospital. Okay. Hardy walks in feeling
kind of awkward, you know, and he bluntly complains
that the taxicab he rode in had the number 1729.
Classic Hardy. Just awkward small talk. Right.
And Hardy remarks that it seemed like rather
a dull one, and he hoped it wasn't a bad omen.
And Ramanujan is lying there, right, severely
ill. And he instantly replies, no, it is a very
interesting number. Yes. It is the smallest number
expressible as the sum of two cubes in two different
ways. Just think about how fast his brain had
to process that to do it in a split second. It's
insane. He instantly knew that 10 cubed is 1
,000, and 9 cubed is 729. Add them together,
you get 1729. He also instantly knew that 12
cubed is 1728 and one cubed is one. Add them
together, you also get 1729. And no other number
smaller than 1729 has that property. Exactly.
Most people would need a calculator and like
an hour to figure that out. And his brain just
produced it as a reflex. J .E. Littlewood, who
was another brilliant mathematician who worked
with him, he later said, Every positive integer
was one of Ramanujan's personal friends. That's
a beautiful way to phrase it. He didn't view
numbers as abstract quantities, you know. He
recognized them as distinct entities with unique
characteristics and relationships. But sadly,
that genius couldn't save his body. No, he couldn't.
The mental and physical toll combined, and it
led to a suicide attempt where he actually threw
himself in front of a train on the London Underground.
Just incredibly tragic. He survived and Hardy
actually intervened with Scotland Yard to keep
him out of prison. But finally, in 1919, his
health is so completely shattered that he returns
to India. And he dies there the following year.
at just 32 years old. 32! It's just devastating.
And the tragedy of his early death is really
compounded by a medical analysis that was done
decades later in 1994. Right, because at the
time of his death he was diagnosed with tuberculosis.
Exactly. But the 1994 analysis of his medical
records, looking at his specific symptoms, it
strongly suggests he actually had hepatic amoebiasis.
Which was what exactly? It was a complication
from episodes of dysentery that he had suffered
years earlier back in India. And the truly heartbreaking
part of that finding is that if he had been properly
diagnosed, hepatogamy biases was a completely
treatable and curable condition at that time.
A simple misdiagnosis robbed the world of decades
of his mind. Which brings us to a major question
for today. When he died at 32, did mathematics
stall? Or are we still playing catch -up to the
equations he left behind? Oh, we are absolutely
still playing catch -up. The sheer density of
what he produced in such a short time is staggering.
Really? Even today? Yes. In 1976, a mathematician
named George Andrews found what is now known
as Ramanujan's lost notebook in the library at
Trinity College. The lost note? Yeah. It contained
87 pages of unorganized, incredibly complex material
from the last year of his life in India. Mathematicians
have spent the last 50 years just deciphering
those final scribbles. Give us an example of
how those abstract scribbles actually impact
the world today. We aren't just talking about
academic parlor tricks, right? Far from it. Consider
his work on the tau function. Okay, the tau function.
Without getting too lost in the weeds, the tau
function deals with deep symmetries in something
called modular forms. All right. Ramanujan proposed
a conjecture about these symmetries based purely
on his intuition. And decades later, Proving
the Ramanujan conjecture laid the groundwork
for a massive theory involving Gullo representations.
Which do what? They essentially map the hidden
symmetries of algebraic equations. Wait, symmetries
of algebraic equations? Isn't that the exact
framework Andrew Wiles used in the 1990s to finally
prove Fermat's last theorem? Exactly. A problem
that had stumped the brightest minds on Earth
for 350 years was finally solved, in part because
the mathematical architecture Ramanujan envisioned
decades earlier. That is mind blowing. His ideas
permeate modern physics, the theory of black
holes, and even the highly complex algorithms
used by computer scientists to calculate pi at
unprecedented speeds today. It really puts into
perspective how uniquely powerful his intuition
was, especially compared to the established academic
elite at the time. It does. The mathematician
Paul Erdos once passed on a personal rating system
that Hardy used to rank mathematical talent.
Oh, I love this part. It was on a scale from
0 to 100. And Hardy... who was considered a titan
in the field, gave himself a 25. A 25 out of
100. Right. He gave his brilliant colleague Littlewood
a 30. He gave the great German mathematician
David Hilbert an 80. And Ramanujan. Ramanujan
received a 100. Wow. Hardy recognized that Ramanujan
wasn't just better at mathematics. He was fundamentally
playing a different game. Which brings us all
back to you listening right now. Ramanujan's
story is the ultimate reminder that you don't
always need the traditional pedigree, you know.
Absolutely not. You don't need the pristine resume
or the proper educational background to uncover
profound truths. Yes, structure and rigor are
important to translate those truths to the rest
of the world. We saw that with Hardy providing
the mathematical scaffolding. Right. But viewing
a problem from a completely isolated unorthodox
angle isn't inherently a weakness. Sometimes
it is your greatest superpower. Ramanujan didn't
know the rules, which meant he didn't know the
boundaries of what was supposedly impossible.
There is a really compelling thought from the
theoretical physicist Freeman Dyson that captures
the enduring lesson of Ramanujan's life perfectly.
What did he say? Dyson once noted that whenever
he receives letters written in an illegible scroll
from obscure places, you know, from people entirely
outside the scientific establishment, he always
makes a point to read them. Just in case. Exactly.
He reads them because he hoax against all odds
that he might be opening a letter from another
Ramanujan. And in today's hyper -credentialed
world, a world of absolute information overload
where algorithms and gatekeepers instantly filter
out anything that doesn't look professional or
formatted correctly, it leaves you wondering.
It really does. How many ramanujans are alive
right now? How many brilliant, world -changing
insights are we actively ignoring simply because
they don't arrive with the proper pedigree? That's
a profound question. Keep your eyes open for
the scroll!