Imagine locking yourself in an attic for six
years. Right. Just keeping a massive career -defining
secret from your colleagues, your friends, I
mean, the entire world. Which is incredibly hard
to do in academia. Exactly. And you're doing...
all of this to solve a 300 year old riddle that
the absolute greatest minds in history have already
deemed basically physically impossible. Yeah,
a true lost cause. So today we are taking a deep
dive into a stack of Wikipedia excerpts about
Sir Andrew Wiles. Now you hear British mathematician
and you might immediately brace yourself for
a dusty biography. Or an hour of chalk squeaking
on a blackboard. Right, exactly. But this is
not that kind of story. This is a story of childhood
obsession, a devastating, highly public failure,
and a triumphant comeback that literally unified
entirely different branches of human knowledge.
It really is a massive saga. It is. Our mission
today is to figure out how one man cracked the
most stubborn problem in mathematics, Fermat's
Last Theorem. OK, let's unpack this. Whoa. To
really understand Wiles, we have to start with
the detail from his early life that kind of sets
the stage for his entire personality. Okay, where
are we starting? He was born in Cambridge in
1953, but he spent a significant chunk of his
early childhood living in Nigeria with his parents.
Right. And when it came time for him to start
formal schooling there, according to his parents'
letters, he just flat out refused to go. Wait,
really? He just refused? For months. Just completely
refused. That is incredibly ironic. I mean, the
man who goes on to become one of the greatest
academic minds of our century basically started
out staging a toddler protest against the very
concept of school. Exactly. He hated the classroom.
But the sources make a point to highlight. that
he already loved solving math problems on his
own terms. So it wasn't the learning, it was
the environment. That distinction is everything.
He was avoiding intellectual work, he was avoiding
institutional structure, he was drawn to the
purity of the puzzles themselves, where he could
just sit and think. Which perfectly brings us
to a fateful walk home from school when he's
10 years old back living in Cambridge. Yeah,
this is the inciting incident of his whole life.
He stops at his local library and he picks up
a book by Eric Temple Bell called The Last Problem.
And inside he discovers Fermat's Last Theorem.
The ultimate mathematical white whale. Truly.
Now, for anyone a little rusty on their 17th
century math puzzles, the theorem is deceptively
simple. It really is. It states that no three
positive integers a, b, and c can satisfy the
equation a to the power of n plus b to the power
of n equals c to the power of n. Right, for any
integer value of n that is greater than 2. Exactly.
And if you think back to basic geometry, if n
is 2, that's just the Pythagorean theorem. A
squared plus b squared equals c squared. Which
we all learned in school. Right, and there are
endless combinations of whole numbers that make
that work. 3, 4, and 5, for example. Sure. But
in the 1630s, period of Fermat claimed that if
you raise those numbers to the power of 3 or
4 or 5 or anything higher, the equation just
breaks. You can never find whole numbers to make
it balance perfectly. Exactly. And the crazy
part is, Fermat wrote this claim in the margin
of a book, adding that he had a truly marvelous
proof for it, but the margin was too small to
contain it. The augacity of that guy. I know.
It's basically the ultimate mathematical mic
drop. He just leaves it there for history. to
deal with, but here's what really blows my mind.
What's up? Wiles is 10 years old. He's looking
at a problem that has baffled humanity for over
three centuries. It's like finding a puzzle box
with instructions so simple a child can read
them, but the box itself is made of impenetrable
titanium. That's a great way to put it. Why would
a 10 -year -old look at that centuries -old titanium
box and confidently think, yeah, I'm going to
be the first person to crack this? What's fascinating
here is the psychological appeal of accessible
mysteries. OK, tell me more about that. For a
mathematically inclined kid, The fact that the
problem is so easy to state is exactly the trap.
It doesn't require advanced calculus to understand
the premise. It just requires basic arithmetic.
Right, so it feels solvable. Exactly. Wiles saw
a universal truth sitting right there, unproven,
just mocking everyone. But as the sources note,
reality eventually set in. Because he was still
just a kid. Right. He realized that his childhood
toolbox was just too limited. Passion without
the proper tools simply isn't enough. That makes
sense. He had to put the puzzle box back on the
shelf and wait for the rest of mathematics to
catch up. And when he did, I mean, he gets his
bachelor's degree at Oxford, gets his PhD at
Cambridge, working with his advisor John Coats,
becomes a professor at Princeton. He's building
this massive, highly successful career in number
theory. But the dream is still there, just on
the shelf until 1986. Yes. Decades later, a specific
mathematical breakthrough drags him right back
in. Wiles is 33 at this point. Two mathematicians,
Yourhard Frey and Ken Rabette, managed to build
a theoretical bridge between Furman's ancient
equation and a completely different, highly modern
area of mathematics. Rabette proved something
called the epsilon conjecture, which cemented
a link to the Taniyama -Shimura -Wile conjecture.
OK, let's not just drop those massive terms without
breaking them down a little bit for everyone
listening. Sure enough. We've got this 300 -year
-old equation on one side and this new Taniyama
-Shimura -Wild conjecture on the other. What
is the actual bridge connecting them? Think of
the Taniyama -Shimura -Wild conjecture as a sort
of mathematical Rosetta Stone. A Rosetta Stone,
OK. It proposed that every single elliptic curve
which our sources describe as equations that
look like the surface of a torus, or basically
a donut shape. Oh, I like a donut analogy. Right.
It proposed that these curves could be perfectly
translated into something called the modular
form, which is a highly symmetrical object in
a totally different branch of math. Okay, so
it translates donut math into symmetrical math.
Basically, yes. Fray and Rabette proved that
if Fermat's equation did have a solution, meaning
if those whole numbers for powers greater than
two actually existed, you could use those numbers
to create a very peculiar elliptic curve. A curve
that shouldn't exist. A mathematical monster.
It would be an elliptic curve so bizarre that
it fundamentally could not be translated into
a modular form. It would break the Rosetta Stone
entirely. Ah, I see the logic. So if you can
prove the Rosetta Stone is real, that the Taniyama
-Shimura -Weil conjecture is true, and all curves
must translate by strict rules, then you prove
the monster curve cannot exist. Precisely. And
if the monster can't exist, Fermat's equation
has no solutions. You prove Fermat by proving
the impossibility of the alternative. Exactly.
Proof by contradiction. You don't actually attack
Fermat directly at all. You attack the Taniyama
-Shimura -Wile conjecture. Here's where it gets
really interesting. Wiles sees this connection,
and he decides he is going to be the one to prove
the Taniyama -Shimura -Wile conjecture for these
specific curves. Which is a monumental task.
Huge. But he doesn't start publishing papers,
and he doesn't collaborate with his brilliant
colleagues at Princeton. No, he doesn't. He vanishes
into his attic. He dedicates all of his research
time to this one problem for over six years in
near total secrecy. Six. He only tells his wife.
He even covers his tracks by releasing older,
unrelated work in small segments just to throw
people off the set. It's extreme. I really have
to push back on this approach. was this pure
hubris. It's like a chef secretly trying to invent
a completely new flavor of food for six years,
occasionally tossing out fake recipes just to
confuse rival chefs. That's a funny way to look
at it. Why the intense secrecy? Was it arrogance
or absolute necessity? It's a really vital question.
To understand it, you have to look at the reality
of the mathematical community at that time. Okay.
The Taniyama -Shimura -Wile conjecture was viewed
by contemporaries as extraordinarily difficult,
perhaps completely impossible to prove. Like
a career killer. Yeah. John Coates, his former
supervisor, stated flatly that it seemed impossible.
Ken Rabette considered himself one of the vast
majority who believed it was completely inaccessible.
Wow. Rabette noted that Wiles was one of the
few people on earth who had the audacity to even
dream of trying. Audacity being the polite academic
way of saying he was crazy for trying. Precisely.
If Wiles had announced his intentions, every
mathematician, journalist, and, you know, crank
with a calculator would have been calling his
office. Oh, the pressure would be insane. Pointing
out potential pitfalls, demanding updates, the
sheer noise would have been deafening. Working
in a silo allowed him unbroken, obsessive concentration.
He needed the quiet. He needed to hold a massive
architecture of abstract concepts in his mind
without a single interruption. But working in
a silo comes with a terrifying risk. It does.
When you don't have anyone checking your math
for six years, you are basically walking a tightrope
without a net. And it almost ended in disaster.
Which brings us to June of 1993. Wiles finally
steps into the light. He's at a conference in
his hometown of Cambridge giving a three -day
lecture series. Right. And Gina Collada, reporting
for the New York Times, noted that he titled
his lectures, Modular Forms, Elliptic Curves,
and Gell -Oll Representations. Notice what is
conspicuously missing from that title. The word
firmat. There was absolutely no hint he was going
to talk about the most famous puzzle in history.
None. He just grinds through his incredibly dense
proof for three days. And then at the very end
of the third lecture, he casually concludes that
he has proved a general case of the conjecture.
Almost as an afterthought. Yes. He notes that
this means Fermat's last theorem is true. He
writes, QED. Thus, it has been demonstrated on
the board. And just stop speaking. The 300 year
mystery is supposedly solved. And the mathematical
community was electrified. It was front -page
news around the entire world. But the triumph
was shockingly short -lived. Very short -lived.
Just two months later, in August of 1993, while
the proof is undergoing rigorous peer review,
a critical flaw is discovered. We're talking
about a proof that was hundreds of pages long,
and the reviewers find an issue. The sources
say the problem involved the properties of something
called the Selmer Group and Wiles's use of a
tool called an Euler system. That's right. Wait,
before we talk about the psychological fallout
of this, What actually broke? What do those terms
mean in the context of our architectural metaphor?
Okay, let's imagine Wiles' proof as a magnificent
100 -story skyscraper. Okay, I can picture that.
To prove the whole structure holds, Wiles had
to prove that a specific foundational pillar,
which we'll call the Selmer group, had a precise
mathematical limit for how much stress it could
take. Because if the stress goes over that limit...
The whole building falls. The Euler system was
essentially the highly specialized theoretical
measuring tape Wiles built to calculate that
exact stress limit. Okay, so he's measuring the
load bearing capacity of the pillar. Right. But
during peer review, they realized his measuring
tape was flawed under one specific set of extreme
mathematical conditions. The tape effectively
stretched. Oh no. Yeah, I couldn't guarantee
the pillar would hold the weight of the logic
above it. Imagine the sheer stomach -dropping
horror of this. You've just told the world you
solved the unsolvable. The confetti has fallen.
Right. And then an inspector whispers in your
ear that there's a microscopic, fatal crack in
the foundation. The 100 -story building is swaying
in the wind, and the entire academic world is
watching you. The psychological resilience required
to survive that moment is honestly hard to fathom.
Wiles tried to repair that broken oiler system
for over a year. A whole year of just staring
at the crack in the foundation. And he failed
over and over. According to the sources, He was
on the verge of giving up completely. Wait, so
after six years of guarding this secret with
his life, being utterly paranoid about someone
stealing his thunder, he just throws in the towel
and decides to ask for help. What wasn't throwing
in the towel, it was accepting reality. He realized
he couldn't fix the crack in the foundation entirely
isolated in his own mind. He had to let someone
in. He had to break his six -year habit of secrecy.
He brought in his former doctoral student Richard
Taylor to help him interrogate the flaw. And
together they just patch up the measuring tape.
That's the most fascinating mechanical shift
of the entire story. They didn't patch it? They
didn't. No. On September 19th, 1994, Wiles had
a sudden insight. He realized that trying to
fix the broken Euler system was the wrong approach
entirely. He could circumvent the flawed area
completely by switching out the core mathematical
framework. The sources mention he used Galois
representations to replace the elliptic curves
in that part of the proof. How did that fix the
problem with the pillar? It didn't fix the pillar.
It bypassed it. Oh. They used Galois representations,
combining them with ideas from a mathematician
named Victor Kalavajian, to salvage a totally
different approach that was originally developed
by Mathias Flack. That sounds incredibly complex.
Basically, instead of trying to repair the flawed
measuring tape, they build a massive steel bypass
around the broken pillar using a completely completely
different material. It was a brilliant synthesis
that reduced the problem to a class number formula,
sidestepping the flaw entirely. And it worked.
It did. Wiles and Taylor published a second paper
containing the circumvention alongside the main
proof. Both are published in May 1995 in a dedicated
issue of the Annals of Mathematics. And with
that, the 300 -year -old mystery was unequivocally
put to rest. The proof held up. It did indeed.
But that brings us to the ultimate question of
value. Exactly. So... What does this all mean?
For anyone listening to this who isn't a number
theorist, why should you care that a guy proved
a 300 -year -old equation about numbers raised
to a power? Fair question. I think about it like
this. It's like someone spending six years inventing
a revolutionary new type of combustion engine
just so they can win a single, very specific,
historic street race. They win the race, which
is great for them, but then the rest of the world
looks at the engine and realizes... Wait a minute,
this technology is the key to commercial space
travel. If we connect this to the bigger picture,
your engine analogy is absolutely spot on. The
real treasure of Wiles' proof wasn't the solution
to Fermat's equation itself. It was. No, I mean,
Fermat's last theorem, in isolation, doesn't
actually do much for the modern world. It's just
a rigid statement about numbers. The profound
value was in the entirely new mathematical tools
Wiles had to forge in the fires of his obsession
to get there. The bridge between the islands.
Exactly. The methods he developed began unifying
completely disparate ideas and theorems, and
the impact was immediate. How immediate. By 1999,
just a few years later, his former students Richard
Taylor, Brian Conrad, and Fred Diamond working
with Christophe Bruhl took the tools Wiles had
built and used them to prove the full modularity
theorem. Wow. Wiles had only proven the limited
case needed for Fermat. Right. But his students
took his engine and drove it across the entire
mathematical landscape. And the mathematical
world recognized this massive paradigm shift.
The list of accolades Wiles received is just
staggering. It really is. Just in 1995 and 1996,
he wins the Shock Prize, the Fermat Prize, the
Wolf Prize, the Royal Medal, and the NASA Award
in Mathematics. And it kept going. He gets a
MacArthur Fellowship, you know, the genius grant
in 1997. He even gets a silver plaque from the
International Mathematical Union because he was
technically 41 when he completed the proof. Making
him just one year too old to receive the famous
Fields Medal, which has a strict cutoff at age
40. Which is an amusingly rigid quirk of academic
bureaucracy, honestly. Very much so. You solved
the biggest problem in history, but you're 41.
So here's a lovely silver plaque instead. Right.
But the honors didn't stop there. No, they didn't.
He was knighted in 2000. He was appointed the
first Regis Professor of Mathematics at Oxford
in 2018. There is literally an asteroid flying
around out there in space named 900 -999 Wiles.
It is a testament to the magnitude of his achievement.
But what I find most compelling is how Wiles
himself views his legacy. When he won the incredibly
prestigious ABLE Prize in 2016, he said something
really revealing. He noted that the methods that
solved Fermat opened up a new way to attack one
of the big webs of conjectures in contemporary
math called the Langlands program. Our sources
mentioned the Langlands program is a grand vision
that aims to unify different branches of mathematics.
It is considered by many to be the grand unified
theory of mathematics. Wiles said his work gave
mathematicians a new way to look at that unification.
So it's all about the process. Yes. This journey
challenges us to recognize that the process of
wrestling with a stubborn problem The dead ends,
the tools you have to invent when the old ones
break, often yields innovations far more valuable
than the solution you originally set out to find.
The struggle wasn't just the price he paid to
solve the puzzle. The struggle was the prize.
It generated the new math. Perfectly said. Well,
we've covered a lot of ground today, from a little
boy in Nigeria refusing to go to school, to a
10 -year -old staring down a 300 -year -old mystery
in a Cambridge library, through six years of
silent obsession, a heart -stopping public failure,
and finally, a collaborative triumph that reshaped
modern mathematics. Thank you for joining us
on this deep dive into the extraordinary, obsessive,
and ultimately unifying life of Sir Andrew Wiles.
But before we let you go, there is one final
detail from this story to mull over. Oh, there
is. We talked about how Wiles' crowning achievement
almost collapsed under the weight of that single
flaw in 1993, an error that existed, at least
in part, because he had spent over six years
working in a near total vacuum. Right, completely
disconnected from the rigorous peer review of
his colleagues. Exactly. It took swallowing his
pride and bringing in Richard Taylor to finally
discover the circumvention that crossed the finish
line, which leaves you with a lingering question.
I think I know where you're going with this.
Think back to that titanium puzzle box. Is the
extreme, isolated obsession that Wiles utilized
the only way the human mind can hold enough complex,
disparate pieces together to solve a seemingly
impossible problem? Or did his intense secrecy
actually needlessly delay the greatest mathematical
breakthrough of the 20th century? It's something
to think about the next time you decide to tackle
a massive project entirely on your own.