October 14, 2025 • 36 mins
From tallying goats to launching rockets, this episode takes you on a fast-paced, lighthearted journey through the complete history of mathematics. Inspired by D.E. Smith’s The History of Mathematics, this is the espresso version — goats, Greeks, calculus cage matches, infinity, and beyond.
3 Things Listeners Will Learn:
- How early humans counted goats, stars, and time with sticks, pebbles, and sky-watching.
- Why Newton and Leibniz’s feud over calculus was the most brutal “cage match” in math history.
- How modern math evolved into the language of infinity, chaos, quantum physics, and AI.
Resources Mentioned:
Explore more on our website: mathsciencehistory.com
To buy my book Hypatia: The Sum of Her Life on Amazon, visit https://a.co/d/g3OuP9h
🌍 Let’s Connect!
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Threads: https://www.threads.com/@math.science.history
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Episode Transcript
Available transcripts are automatically generated. Complete accuracy is not guaranteed.
(00:01):
One of my favorite books is The History
of Mathematics by D.E. Smith.
It's over a thousand pages and there are
two volumes and in my opinion it is
a masterpiece of scholarship.
I highly recommend it.
But come on, let's be honest, you don't
have a thousand pages worth of patience today.
You're here for the synopsis, right?
So this podcast, spoken off the cuff, is
(00:24):
the espresso version.
Here we go.
Nope, sorry, wait.
First, a word from my advertisers.
Before there were calculators, pencils, or even pockets
to lose pencils in, there were humans.
They had two urgent questions.
One, is that berry poisonous?
(00:45):
And two, how many goats do I have
because the goats keep leaving?
Now, we don't know if early humans invented
numbers to count goats or to count the
times the goats escaped.
Either way, they needed a system and by
system I mean a stick.
They carved little notches in bones and sticks.
Congratulations, you are listening to the birth of
accounting.
(01:06):
The first ledger is a bone.
No monthly subscription required.
If you've ever tried to count sheep, you
know the problem.
Sheep move.
They drift in and out of the frame
like fuzzy clouds with opinions.
Early humans discovered a genius hack called one
-to-one matching.
Touch a pebble for each sheep.
(01:27):
If you run out of pebbles, you have
more sheep than pebbles.
And if you run out of sheep, you
have more pebbles.
It's elegant and dusty.
But soon a pattern appears.
Ten keeps happening.
Fingers are extremely persuasive.
Ten is handy because hands are handy.
(01:50):
But our ancestors were creative.
So sometimes they counted to 20 with toes.
That gives us base 20, which is a
vibe in parts of the world.
If you have ever said score to mean
20, you are doing toe math in a
tuxedo.
Other groups like even bigger bundles.
In Mesopotamia, traders and scribes worked with 60s.
(02:13):
Why 60?
Well, 60 is kind of polite.
It divides many numbers, which makes fractions friendlier.
Also, nobody wants to slice a pizza into
seven equal parts.
60 says, I got you.
It is why your minute has 60 seconds,
not 100.
Thank you, Mesopotamia.
Our watches are weird because your fractions were
(02:35):
excellent.
Now, let's talk about this whole base 60
thing.
How on earth did Mesopotamians decide to count
all the way to 60 when the rest
of us are content with fingers and toes?
Well, here's the trick.
They didn't just count fingers.
They counted phalanges.
That's the fancy word for finger bone.
Each finger, not the thumb, has three little
(02:56):
sections.
Use your thumb as a pointer and suddenly
each hand gives you 12 spots to tally.
Multiply by five fingers on the other hand
and boom, you've got 60.
That's clever.
It's practical.
And it's probably the only time your thumb
got promoted to human abacus.
While numbers are learning to walk, language is
(03:17):
teaching numbers how to talk.
Words like one and two hitch a ride
on breath and memory.
Some languages even have special counting words depending
on what you are counting.
Fish get one set of words.
Long objects get another.
Goats probably got their own because goats insist
on special treatment always.
(03:39):
Now, while the math of how many is
getting sorted, something else is happening at the
campfire.
People were looking up because the ground kept
changing.
Rivers flooded.
Herds migrated.
Seasons refused to send email reminders.
But the sky is gloriously punctual.
The sun rises where it should, sets where
(04:01):
it must, and the moon cycles like a
gentle metronome.
If you want to know when to plant
barley or when the fish return or when
your in-laws traditionally visit, the heavens are
a calendar that doesn't forget its password.
Enter astronomy.
First, it is just noticing.
A bright thing rises here in the winter.
(04:21):
Another bright thing rises there in the summer.
We should mark that rock.
Rock gets marked.
Then someone says, Hey, if we put two
stones here and one over there, the sunrise
on a certain day will peak right through.
Now you have an observatory and a tourist
attraction.
Please bring snacks.
(04:42):
From 5000 BCE in Mesopotamia, we get records
on clay tablets that tie sky events to
earthly life, like eclipses and the motions of
planets.
Over centuries, they turned notes into tables, which
were the ancestor of spreadsheets, but with more
cuneiform etchings and fewer macros.
(05:05):
From Egypt, the Nile's rhythm and the appearance
of Sirius helped anchor calendars that could reference
floods, planting and harvesting.
The stars were not decoration.
Priests tracked rising stars called deacons to tell
time at night.
From China, they saw long traditions of sky
watching.
Court astronomers kept careful logs because emperors cared
(05:27):
about omens and calendars.
And frankly, they also cared about punctuality.
So if the calendar slipped, tax day showed
up on the wrong Tuesday, and nobody wants
that.
So they audited the sky.
Do you notice the feedback loop?
The more you watch the sky, the more
you need numbers.
The more you use numbers, the better you
get at predicting the sky.
(05:48):
They invented better counting.
They invented better symbols.
Eventually, they invented the recipe book of the
universe, also known as mathematics.
And they passed it on with clay, papyrus,
bamboo slips, and whatever your local stationery store
carried in 2000 BCE.
So back by the goats, our hero, math,
(06:08):
was upgrading.
The tally stick got columns for different herds,
maybe small notches for kids, larger notches for
adult goats.
It was a positional idea in the wild.
And so in time, shepherd math met market
math, and they had to figure out how
do they divide these.
And so fractions appeared.
Everyone sighed, and the Mesopotamians brought 60 to
(06:31):
the party.
So the slicing was nicer.
Scales of 10, scales above 10, scales of
20, you could pick your favorite.
It was kind of like an a la
carte for math.
And by then, humans had two superpowers that
loved each other, pattern hunting and storytelling.
Pattern hunting said, the moon grows and shrinks,
(06:52):
there must be a rhythm.
Storytelling said, we should name that rhythm and
sing about it.
Soon, mathematics was in myths, rituals, architectures, it
had become a way of life.
It was baked into when you planted, how
you traded, and where you built the doorway,
and which night the whole town met to
(07:14):
watch the sky do a trick.
So prehistoric mathematics was not homework.
It was a lifestyle.
It was goats and grain, stars and seasons,
pots and patterns.
It was fingers, toes and a very charismatic
number 60.
It was the moment we realized that the
world liked to repeat itself if you paid
(07:34):
attention.
And once you noticed repetition, you could predict
things.
Once you could predict things, you could relax
until the goats escaped again.
Coming up next, we trade bone notches for
clay tablets and step into the river valleys
where writing, measurement and math started wearing real
(07:55):
shoes.
Bring your stylus and maybe a spare pebble
for old time's sake.
We'll be right back after a quick word
from my advertisers.
After our goat counting, we stepped into the
historic period down to 1000 BCE.
And by now humans were doing something revolutionary,
not just noticing patterns, but writing them down.
(08:17):
In the Stone Age and Paleolithic Age, mathematics
was simple.
They would say, that's a lot of bison,
or that's not enough berries.
But eventually we graduated from scratches on bones
to scratches with meaning.
And by around 4000 BCE, the advent of
writing turned math from finger wiggling into something
permanent.
(08:37):
Clay tablets and inked bamboo slips meant you
could keep score without actually having to be
there.
The world's first group project in history had
begun.
And unlike modern group projects, at least this
one got finished.
Meanwhile, in China, mathematics was already flexing.
Certain historians believe that some of the first
descriptions of astronomy go as far back as
(09:00):
3000 BCE.
Early Chinese thinkers were using numbers in governance
and divination.
The legendary I Ching, also known as the
Book of Changes, was a mashup of philosophy,
mysticism and binary math before binary math was
cool.
By the 8th century BCE, under kings like
Wu Wang, often translated as Wu of Zhu,
(09:23):
China was formalizing rituals, governance and record keeping.
Numbers weren't just handy, they were political tools.
Then we come to the legendary Huang Ti,
the Yellow Emperor.
Chinese tradition credits him with inventing all sorts
of useful things like medicine, music, even government
structure.
And woven into these myths are the seeds
(09:44):
of mathematics as canonized knowledge.
Eventually, Chinese learning was framed around the five
classics, with the I Ching among them.
It was proof that math was not just
for merchants and scribes, but part of the
state's intellectual DNA.
Meanwhile, in India, Hindu mathematics was stirring.
From the early Vedas to the later treatises,
(10:05):
numbers shaped rituals, altars and calendars.
Geometry told you where to place the sacred
fire, astronomy told you when to light it,
and arithmetic told you how much food you'd
need to keep the priests happy.
One of the most famous astronomical texts, called
the Surya Siddhanta, would eventually appear in its
first form, offering trigonometry, planetary motion and calendar
(10:29):
rules.
Not bad for a handbook with a title
that translates roughly to knowledge of the sun.
Across Persia, India and their neighbors, knowledge cross
pollinated.
When your trade routes included camels, spice caravans
and the occasional invading army, ideas tend to
travel.
Math was portable, kind of like a good
(10:50):
joke.
All right, let's march back to the fertile
valleys of Mesopotamia and Babylonia.
Around the Tigris and Euphrates rivers, scribes pressed
their first cuneiform tablets with columns of numbers.
The Chaldeans, keen sky watchers, combined mathematics with
astronomy.
They were tracking eclipses and predicting omens.
(11:10):
So if you've ever opened an Excel file
called budget underscore final underscore final in all
caps underscore, this one is the final final,
I think you understand their energy.
Clay tablets filled the shelves, recording everything from
goat counts to planetary movements.
And by the way, some of those earliest
(11:31):
math tablets are 4000 years old, and they
are still readable.
Meanwhile, I still can't open a Word document
from 2003.
Then there's Egypt.
By the third millennium BCE, Egyptian engineers were
doing geometry with precision that would impress your
high school math teacher.
They designed pyramids, surveyed farmland, and used sundials
(11:55):
to keep time.
Forget daylight savings, Egyptians had sunlight accuracy.
The great pharaoh Ramses II even divided land
among his people, which meant mathematics wasn't just
for pyramids, it was also for property disputes.
Picture the court.
Yes, yes, Your Majesty, we measured your cousin's
(12:16):
field three times, and no, he cannot count
the neighbor's goats as his own.
Yeah, I could hear it.
From Stone Age scratches to Egyptian sundials, mathematics
had gone from survival skill to statecraft.
The world was realizing that math wasn't just
about tallying sheep, it was about running civilizations.
And that realization leads us westward into the
(12:39):
world of the Occident, where philosophers, merchants, and
would be mathematicians were about to take center
stage.
Occident simply means the West and the place
where the sun goes down.
Its opposite is the Orient, the East, where
the sun comes up.
Basically, a poetic way to say sunrise over
(13:01):
here, sunset over there.
In the Occident, math got philosophical.
The Greeks weren't satisfied just counting sheep.
They wanted proofs.
They wanted reasons.
They wanted to argue about numbers until everyone
else at the symposium pretended to fall asleep.
And then came Pythagoras, the guy everyone credits
(13:21):
with discovering that right triangles have a thing
for squares, except he did not discover it.
Babylon in India had already been doing that
math for about 2000 years.
So what happened?
Well, Pythagoras just slapped his name on it.
So technically, math is like comedy and the
Pythagorean theorem is the oldest borrowed joke in
(13:41):
the book.
So while Pythagoras was busy with triangles, the
Chinese already had rods, shadows, and proof sketches.
Think of them as the silent backup guitarists
in the global math band.
In India, they were messing with string, bricks,
and fire altars, making precise right angles thousands
of years before Pythagoras could trademark the idea.
(14:04):
So Pythagoras is kind of like the spinal
tap of math, always leaning into the common
cliches.
I could see that.
Okay, so then there's Plato, who said that
math was the language of perfect forms, triangles,
circles, cosmic ideals, lounging in some celestial VIP
room.
And Aristotle, who wrote down rules for logic
(14:25):
and classification, basically invented the world's first math
adjacent buzzfeed listicles.
So that's the Occident, proofs, philosophy, and the
first math club with actual membership rules.
But the sun doesn't just set, it also
rises.
While the Greeks were busy arguing about triangles,
the Orient was quietly doing math of its
(14:47):
own, often with a lot less drama, and
a lot more practical results.
In China, the Zhoubi Zhuangjing gave us the
Gaozhu theorem, which, yes, is basically the Pythagorean
theorem, except a few centuries earlier.
They also had counting rods, little sticks arranged
on a board that let you do place
(15:08):
value arithmetic, negatives, and even fractions.
Basically, the world's first calculator app, but in
wood.
Over in India, the Shulbasutras showed priests laying
out fire altars with ropes and pegs.
And wouldn't you know it, right triangles kept
showing up.
They knew there are three, four or five
(15:29):
triangles long before Greece tried to trademark them.
So while Western history textbooks sometimes made it
sound like Pythagoras invented everything, Pythagoras, Pythagoras, Pythagoras,
Marsha, Marsha, Marsha, the truth is the Orient
was already playing the same game.
(15:49):
They just weren't as noisy about taking credit.
They were quietly and diligently doing their homework.
So after the Greeks had their fun proving
that triangles weren't lying to us, mathematics packed
its bags and moved to Alexandria.
Picture a university slash library slash nerd paradise.
(16:12):
I can.
Shelves of scrolls, scholars scribbling everywhere, and enough
papyrus dust to choke a camel.
It was the beginning of paperwork.
Yay.
At the front of the class, Euclid.
He wrote the Elements, a math textbook so
good it outsold the Bible for centuries.
No joke.
Every proof, every theorem laid out like IKEA
(16:34):
instructions for the universe, except his triangles actually
fit together at the end.
Then there was Archimedes, the inventor, the mathematician,
the engineer, and basically an all around show
off.
You know, the overachiever who tries so hard
for the sake of trying, who just wants
to do better than everybody else, but really
(16:55):
feels isolated inside.
Okay, he calculated pi, he built war machines,
and he supposedly ran naked through the streets
shouting Eureka after figuring out how to measure
volume in a bathtub.
That's right.
The greatest breakthrough in fluid mechanics began with
public nudity.
And we have Pythagoras.
Nope, we have Archimedes to thank for that.
(17:20):
Pythagoras, Pythagoras, Pythagoras.
Speaking of astronomy, Ptolemy, his Almagest gave us
a geocentric model of the universe with the
earth at the center.
It was complicated, elegant and totally wrong.
But hey, it lasted for 1400 years, which
(17:40):
means either he was convincing, or nobody wanted
to redo the math.
And I'm going to lean into the latter
because I actually did a podcast about that
way back in 2020, I think.
Then comes Diophantus, the father of algebra.
He loved equations with whole number solutions, what
we now call Diophantine equations.
(18:02):
It was the kind of math that looks
innocent, but can actually ruin an entire Saturday.
And centuries later, I kid you not, Fermat
would scribble in the margin of his book,
something along the lines of I've got a
proof for this, but the margins too small,
which is basically the mathematical equivalent of a
mic drop.
(18:22):
And then my personal favorite, Hypatia of Alexandria.
She was a brilliant teacher, a philosopher, a
mathematician.
She taught geometry and astronomy.
She edited her father's works and wrote commentaries
that preserved Greek math for future generations.
She was admired by students respected across the
(18:43):
ancient world, and tragically was murdered by a
mob of monks in 415 CE.
Her story is why I wrote a book
about her because history should remember not just
her death, but the sum of her life
and legacy.
And that actually happens to be the title
of my book called Hypatia, the sum of
(19:03):
her life.
And you can find it on Amazon and
the link will be in my show notes.
We'll be right back after a quick word
from my advertisers.
Here's what's interesting.
After Hypatia was murdered, things got weird.
And about 100 years later, the West hit
pause.
Europe slipped into the so called Middle Ages,
(19:24):
where math mostly huddled in monasteries, trying not
to be forgotten.
It wasn't a total blackout, just a very
long math nap.
Luckily, while Europe was dozing, the Islamic Golden
Age was wide awake.
Scholars across Baghdad, Damascus and Cordoba translated Greek
works and added their own discoveries and gave
(19:47):
us new tools.
There are even tracings of Hypatia's work in
some of the translations from the Islamic Golden
Age.
Algebra got its very name from Al-Khwarizmi.
Trigonometry blossomed, algorithms were born.
And without them, Europe would not have had
anything to wake up to during the Renaissance.
(20:09):
In short, while Europe was snoring, the Islamic
world was basically babysitting math, feeding it, clothing
it and teaching it how to walk.
So when the Renaissance finally rolled around, Europe
didn't invent math all over again.
It snuck out the window with dad's car
keys and went for a joyride taking credit
(20:30):
for all the fun.
By the 1300s, the great centers of the
Islamic Golden Age had dimmed.
The Mongols sacked Baghdad in 1258.
And let's just say that the House of
Wisdom didn't exactly get a renovation budget afterwards.
So without strong rulers funding scholars, math and
science lost some of their shine.
(20:50):
Theology and law were safer bets than astronomy
or algebra.
And while the Islamic world was arguing over
which star catalog to keep, Europe was busy
stealing the whole frickin library.
Spain, Sicily and Crusaders carried translations into Latin.
And suddenly the West was cramming algebra like
(21:11):
a college kid before finals.
And that doesn't mean math disappeared in the
East.
Far from it.
Ulaanbaatar and the Samarkand built an observatory so
massive, it made Stonehenge look like a backyard
lawn decor.
But the Golden Age of invention had slowed.
The math car keys were now jingling in
Europe's pocket.
(21:31):
Meanwhile, in China, scholars like Cheng Dewei printed
arithmetic manuals that spread like wildfire.
Counting rods, abacuses and math tables went from
scholars' desks to merchants' toolkits.
Math was basically hitting the shelves like the
latest bestseller.
Trade routes in the 13th century carried more
(21:52):
than silk and spices.
In India, Siddhara and Bhaskara II wrote on
astronomy, arithmetic and algebra, predicting eclipses while Europe
was still suspicious of zero.
Meanwhile, in Persia, Al-Qaraji pushed algebra into
new territory with powers and roots, laying groundwork
for future algebraic thinking.
(22:14):
And Chinese astronomical methods even traveled into Japan,
shaping calendars and sky charts.
So by the 1500s, the Orient had printing,
algebra, astronomy and trade all fueling with mathematics.
Which meant that when the Renaissance revved its
engines, Europe wasn't inventing math from scratch.
(22:34):
It was merging into an already busy highway.
So Europe sneaks out with dad's car keys,
takes math for a joyride, and suddenly it's
the Renaissance.
The printing press is the turbo boost.
Math isn't just locked in libraries anymore.
It's mass produced.
Textbooks, tables, diagrams, all spreading faster than gossip
(22:57):
in a small town.
Math memes, but in ink.
Enter François Viet, whom I've also done a
podcast on.
Did that about a year ago.
Please go visit mathsciencehistory.com and dig around.
There's some good stuff there.
Anyhow, François Viet, he was the guy who
gave algebra its makeover.
Before him, equations were basically long winded novels
(23:19):
like The unknown multiplied by itself added to
four times the unknown is equal to 21.
Viet said, Nope, let's just use letters.
Suddenly, algebra looked like Twitter.
Short, sharp, and easy to argue about.
Meanwhile, trigonometry was getting a glow up.
Navigation needed it.
And when your ships are crossing oceans, close
(23:42):
enough, right, is not good enough.
Tables of sines and cosines went into print,
and sailors could finally aim for the new
world without relying entirely on guesswork and rum.
Though to be fair, rum was still heavily
involved.
The Renaissance also saw a mashup of art
and math.
Perspective drawing turned flat canvases into 3D illusions.
(24:05):
Architects rediscovered geometry, making domes and cathedrals that
look like math problems you'd actually want to
live in.
So the 16th century gave us algebra with
style, trig for sailors, geometry for artists, and
math books going viral thanks to Gutenberg's machine.
It was less about inventing new math and
(24:25):
more about broadcasting it to the masses.
I love it.
And next, well, things really speed up.
The 17th century, where Descartes brings us analytic
geometry, Fermat and Pascal invent probability, and Newton
and Leibniz get into the world's most famous
calculus cage match.
It's good.
(24:46):
The 17th century was Europe's math firework show.
France, Britain, Germany, the Netherlands, every country wanted
a math genius to brag about.
In France, Rene Descartes invented analytic geometry, turning
curves into equations.
Suddenly, algebra and geometry weren't two subjects that
hated each other in high school.
(25:08):
They were dating.
In the Netherlands, Christian Huygens studied pendulums and
probability, basically building the first math-powered clock
and helping gamblers figure out their odds.
You gotta know when to hold them and
know when to fold them.
In Britain, a certain Isaac Newton came along,
invented calculus, and then decided gravity should be
(25:29):
his side hustle.
Meanwhile, in Germany, Gottfried Wilhelm Leibniz also invented
calculus.
Cue the most brutal math feud in history,
the Newton-Leibniz cage match.
In the red corner, from Woolstorp, England, the
apple dropper, the gravity guy, the calculus crusader,
Isaac, don't steal my derivatives, Newton.
(25:51):
And in the blue corner, from Leipzig, Germany,
the philosopher, the polymath, the notation ninja, Gottfried
Wilhelm integrals forever Leibniz.
Yep, Newton claimed he invented calculus first.
Leibniz said his notation was better.
Their fans went to war, pamphlets, accusations, full
(26:12):
-on character assassinations.
Newton, who just happened to run the Royal
Society, stacked the jury against Leibniz.
That's like being both the boxer and the
referee.
Brutal.
Leibniz died with his reputation in tatters.
Newton, meanwhile, lived long enough to look smug
(26:34):
about it.
And yet, both notations survived.
Today, we use Leibniz's slick symbols and Newton's
methods.
Back in France, the 18th century saw the
rise of giants like Euler and Lagrange, turning
out equations like a bakery turns out baguettes.
Theoretical math was the center stage.
(26:56):
And this is a really fun era to
study.
And remember Euclid?
His Elements, written way back in Alexandria, was
first printed in 1482.
So by the 17th and 18th centuries, it
was still the gold standard.
Imagine writing a textbook that stays in print
(27:17):
longer than Shakespeare.
Meanwhile, European math didn't stay in Europe.
By the 16th and 17th centuries, translations carried
these works eastward.
Chinese scholars began building essential treatises based on
European models.
And through Dutch traders at Nagasaki, math also
slipped into Japan, where scholars like Nakane Genkai
(27:38):
adapted Western astronomy and algebra into Japanese frameworks.
So the 17th and 18th centuries gave us
analytic geometry, probability, calculus, a feud worthy of
pay-per-view, and the first global math
exchange program.
Not bad for a couple of centuries.
So up to now, math had been busy
(28:00):
building its toolkit.
Geometry, algebra, trigonometry, probability, calculus.
Each century added another gadget to the box.
By the 19th century, the box was overflowing.
And that's when mathematicians decided, hey, let's see
how far this thing can go.
So remember Descartes' analytic geometry, that marriage of
(28:20):
algebra and geometry?
By now, it was producing children, whole families
of curves, surfaces, and equations.
Calculus, invented in the 17th century for physics
and astronomy, had grown into a universal tool.
It could describe how planets move, how heat
spreads, how a pendulum swings.
And trigonometry, once about triangles and shadows, was
(28:44):
now deeply woven into calculus, making it possible
to tackle wave motion, oscillations, and even music.
The triangle wasn't just hanging out on chalkboards
anymore.
It was solving differential equations.
The glow-up was complete.
Enter the 19th century heavyweights.
(29:06):
In Germany, Gauss ate theorems before breakfast.
Actually, he didn't eat them.
He just solved them.
He dabbled in everything from number theory to
magnetism and made everyone else feel underachieving.
I've been around people like that.
It's no fun.
His student, Riemann, bent geometry itself, laying the
groundwork for curved spaces, a concept that would
(29:29):
later fuel Einstein's relativity.
Meanwhile, a fiery young genius named Evariste Galois
practically invented abstract algebra before dying in a
duel at the age of 20.
And if you want to hear a podcast
about that, I also do one about that
as well, which was done about, oh, 15
months ago.
So again, please visit MathScienceHistory.com and dig
(29:52):
around in the archives and you'll see some
great podcasts.
And while you're there, click on that coffee
button because every donation you make keeps the
podcast up and running.
Anyhow, back to Evariste Galois.
Think of what he did.
He created group theory, field theory.
It was all sketched out, literally, in a
few frantic manuscripts the night before his last
(30:13):
pistol match.
Mind-blowing.
This kid was brilliant and he was only
20.
So yeah, math can be dangerous.
Don't get involved in duels.
Then comes Cantor, who looked at infinity and
said, one size doesn't fit all.
He showed that some infinities are bigger than
others.
And then, finally, Emmy Noetcher, who formalized abstract
(30:35):
algebra and tied it to physics.
Her theorems still drive modern science.
If algebra were a messy teenager, Noetcher was
one who said, clean your room, organize your
variables, and respect symmetry.
Like a true boss.
So, by the 19th century, math had evolved
from counting dots and proving triangles to taming
(30:57):
infinities, curving space, and inventing algebra so abstract
it made even mathematicians nervous.
And in my case, when I was studying
math, it made me cry.
I'm going to admit it.
It made me cry.
It's hard.
And just when math seemed to have reached
its limits, the 20th century cracked the door
wide open, bringing machines that could calculate faster
(31:20):
than any human, physics that bent reality itself,
and logic that questioned what we even mean
by truth.
Math wasn't just sneaking out with the car
keys anymore.
It was hot wiring the whole garage.
Buckle up.
Things are gonna get wild.
By the dawn of the 20th century, math
(31:40):
wasn't just sneaking out with the car keys
anymore.
It was stealing the whole car, peeling out
of the driveway, and never looking back.
First up, David Hilbert, the grandmaster of rigor.
He laid out 23 problems in 1900, like
the ultimate math scavenger hunt, challenging mathematicians to
solve them over the next century.
(32:01):
Some got solved.
Some are still haunting blackboards like math's version
of the unsolved true crime.
Then came Kurt Gödel.
He dropped his incompleteness theorem in 1931 and
proved that math itself has limits.
There will always be true statements that can't
be proven.
Mathematicians cried.
(32:21):
Philosophers panicked.
Logic had just punched certainty in the face.
Meanwhile, Alan Turing was teaching machines how to
think.
His work on computation and algorithms birthed the
modern computer.
Suddenly, math wasn't just in books.
It was humming inside machines, calculating, encrypting, breaking
(32:42):
codes, and eventually playing chess better than humans.
And over in physics, math was bending reality
itself.
Einstein's relativity used Riemann's curved geometry to show
space and time aren't fixed.
Then, quantum mechanics arrived with wave functions, probability
amplitudes, and equations that said particles could be
(33:04):
cats, both alive and dead, until you check.
So math had gone from let's count goats
to let's describe parallel universes.
That escalated quickly.
Then there was chaos theory.
(33:26):
Suddenly, simple equations produced wildly unpredictable outcomes.
The butterfly effect became a household phrase, though
mathematicians never did prove whether butterflies really control
the weather or just have really good PR.
As the century rolled into the 21st, math
embedded itself everywhere.
(33:47):
In cryptography, securing your credit cards, in data
science, running everything from Netflix recommendations to climate
models, and in AI.
So, math has officially taken the wheel.
And no, it's not bringing the car back.
So here we are, from tallying sheep bones
to infinity, from sundials to supercomputers, math has
(34:09):
been our universal language, our secret decoder ring
for the cosmos, and occasionally, our favorite excuse
for why we are bad at taxes.
And the ride isn't over.
Math began as a baby.
It was coddled by the Islamic golden age.
And then it became a car sneaking out
of the driveway, then tearing down the highway,
inventing new lanes as it went.
(34:30):
But here's the thing.
Highways only take you so far.
By the 20th and 21st century, math wasn't
satisfied with asphalt.
It strapped on boosters, pointed upward, and launched
from quantum mechanics to AI, from curved space
to infinity itself.
Math has become our rocket.
(34:50):
It's our ticket into the vast unknown.
Every equation is a launchpad.
Every proof, a countdown.
Every theorem, a trajectory.
And where does it go?
It goes into a universe that has no
end, no edges, just limitless space for discovery.
So math isn't just driving anymore.
It's flying.
(35:11):
It's carrying us past what we can see,
past what we can touch, into the infinite.
And the ride?
Well, the ride has only just begun.
Thank you for listening to Math Science History.
And until next time, carpe diem.
(35:53):
Again, thank you for tuning in.
And until next time, carpe diem.
One of my favorite books is The History
of Mathematics by D.E. Smith.
It's over a thousand pages and there are
two volumes and in my opinion it is
a masterpiece of scholarship.
I highly recommend it.
But come on, let's be honest, you don't
have a thousand pages worth of patience today.
You're here for the synopsis, right?
So this podcast, spoken off the cuff, is
(00:24):
the espresso version.
Here we go.
Nope, sorry, wait.
First, a word from my advertisers.
Before there were calculators, pencils, or even pockets
to lose pencils in, there were humans.
They had two urgent questions.
One, is that berry poisonous?
(00:45):
And two, how many goats do I have
because the goats keep leaving?
Now, we don't know if early humans invented
numbers to count goats or to count the
times the goats escaped.
Either way, they needed a system and by
system I mean a stick.
They carved little notches in bones and sticks.
Congratulations, you are listening to the birth of
accounting.
(01:06):
The first ledger is a bone.
No monthly subscription required.
If you've ever tried to count sheep, you
know the problem.
Sheep move.
They drift in and out of the frame
like fuzzy clouds with opinions.
Early humans discovered a genius hack called one
-to-one matching.
Touch a pebble for each sheep.
(01:27):
If you run out of pebbles, you have
more sheep than pebbles.
And if you run out of sheep, you
have more pebbles.
It's elegant and dusty.
But soon a pattern appears.
Ten keeps happening.
Fingers are extremely persuasive.
Ten is handy because hands are handy.
(01:50):
But our ancestors were creative.
So sometimes they counted to 20 with toes.
That gives us base 20, which is a
vibe in parts of the world.
If you have ever said score to mean
20, you are doing toe math in a
tuxedo.
Other groups like even bigger bundles.
In Mesopotamia, traders and scribes worked with 60s.
(02:13):
Why 60?
Well, 60 is kind of polite.
It divides many numbers, which makes fractions friendlier.
Also, nobody wants to slice a pizza into
seven equal parts.
60 says, I got you.
It is why your minute has 60 seconds,
not 100.
Thank you, Mesopotamia.
Our watches are weird because your fractions were
(02:35):
excellent.
Now, let's talk about this whole base 60
thing.
How on earth did Mesopotamians decide to count
all the way to 60 when the rest
of us are content with fingers and toes?
Well, here's the trick.
They didn't just count fingers.
They counted phalanges.
That's the fancy word for finger bone.
Each finger, not the thumb, has three little
(02:56):
sections.
Use your thumb as a pointer and suddenly
each hand gives you 12 spots to tally.
Multiply by five fingers on the other hand
and boom, you've got 60.
That's clever.
It's practical.
And it's probably the only time your thumb
got promoted to human abacus.
While numbers are learning to walk, language is
(03:17):
teaching numbers how to talk.
Words like one and two hitch a ride
on breath and memory.
Some languages even have special counting words depending
on what you are counting.
Fish get one set of words.
Long objects get another.
Goats probably got their own because goats insist
on special treatment always.
(03:39):
Now, while the math of how many is
getting sorted, something else is happening at the
campfire.
People were looking up because the ground kept
changing.
Rivers flooded.
Herds migrated.
Seasons refused to send email reminders.
But the sky is gloriously punctual.
The sun rises where it should, sets where
(04:01):
it must, and the moon cycles like a
gentle metronome.
If you want to know when to plant
barley or when the fish return or when
your in-laws traditionally visit, the heavens are
a calendar that doesn't forget its password.
Enter astronomy.
First, it is just noticing.
A bright thing rises here in the winter.
(04:21):
Another bright thing rises there in the summer.
We should mark that rock.
Rock gets marked.
Then someone says, Hey, if we put two
stones here and one over there, the sunrise
on a certain day will peak right through.
Now you have an observatory and a tourist
attraction.
Please bring snacks.
(04:42):
From 5000 BCE in Mesopotamia, we get records
on clay tablets that tie sky events to
earthly life, like eclipses and the motions of
planets.
Over centuries, they turned notes into tables, which
were the ancestor of spreadsheets, but with more
cuneiform etchings and fewer macros.
(05:05):
From Egypt, the Nile's rhythm and the appearance
of Sirius helped anchor calendars that could reference
floods, planting and harvesting.
The stars were not decoration.
Priests tracked rising stars called deacons to tell
time at night.
From China, they saw long traditions of sky
watching.
Court astronomers kept careful logs because emperors cared
(05:27):
about omens and calendars.
And frankly, they also cared about punctuality.
So if the calendar slipped, tax day showed
up on the wrong Tuesday, and nobody wants
that.
So they audited the sky.
Do you notice the feedback loop?
The more you watch the sky, the more
you need numbers.
The more you use numbers, the better you
get at predicting the sky.
(05:48):
They invented better counting.
They invented better symbols.
Eventually, they invented the recipe book of the
universe, also known as mathematics.
And they passed it on with clay, papyrus,
bamboo slips, and whatever your local stationery store
carried in 2000 BCE.
So back by the goats, our hero, math,
(06:08):
was upgrading.
The tally stick got columns for different herds,
maybe small notches for kids, larger notches for
adult goats.
It was a positional idea in the wild.
And so in time, shepherd math met market
math, and they had to figure out how
do they divide these.
And so fractions appeared.
Everyone sighed, and the Mesopotamians brought 60 to
(06:31):
the party.
So the slicing was nicer.
Scales of 10, scales above 10, scales of
20, you could pick your favorite.
It was kind of like an a la
carte for math.
And by then, humans had two superpowers that
loved each other, pattern hunting and storytelling.
Pattern hunting said, the moon grows and shrinks,
(06:52):
there must be a rhythm.
Storytelling said, we should name that rhythm and
sing about it.
Soon, mathematics was in myths, rituals, architectures, it
had become a way of life.
It was baked into when you planted, how
you traded, and where you built the doorway,
and which night the whole town met to
(07:14):
watch the sky do a trick.
So prehistoric mathematics was not homework.
It was a lifestyle.
It was goats and grain, stars and seasons,
pots and patterns.
It was fingers, toes and a very charismatic
number 60.
It was the moment we realized that the
world liked to repeat itself if you paid
(07:34):
attention.
And once you noticed repetition, you could predict
things.
Once you could predict things, you could relax
until the goats escaped again.
Coming up next, we trade bone notches for
clay tablets and step into the river valleys
where writing, measurement and math started wearing real
(07:55):
shoes.
Bring your stylus and maybe a spare pebble
for old time's sake.
We'll be right back after a quick word
from my advertisers.
After our goat counting, we stepped into the
historic period down to 1000 BCE.
And by now humans were doing something revolutionary,
not just noticing patterns, but writing them down.
(08:17):
In the Stone Age and Paleolithic Age, mathematics
was simple.
They would say, that's a lot of bison,
or that's not enough berries.
But eventually we graduated from scratches on bones
to scratches with meaning.
And by around 4000 BCE, the advent of
writing turned math from finger wiggling into something
permanent.
(08:37):
Clay tablets and inked bamboo slips meant you
could keep score without actually having to be
there.
The world's first group project in history had
begun.
And unlike modern group projects, at least this
one got finished.
Meanwhile, in China, mathematics was already flexing.
Certain historians believe that some of the first
descriptions of astronomy go as far back as
(09:00):
3000 BCE.
Early Chinese thinkers were using numbers in governance
and divination.
The legendary I Ching, also known as the
Book of Changes, was a mashup of philosophy,
mysticism and binary math before binary math was
cool.
By the 8th century BCE, under kings like
Wu Wang, often translated as Wu of Zhu,
(09:23):
China was formalizing rituals, governance and record keeping.
Numbers weren't just handy, they were political tools.
Then we come to the legendary Huang Ti,
the Yellow Emperor.
Chinese tradition credits him with inventing all sorts
of useful things like medicine, music, even government
structure.
And woven into these myths are the seeds
(09:44):
of mathematics as canonized knowledge.
Eventually, Chinese learning was framed around the five
classics, with the I Ching among them.
It was proof that math was not just
for merchants and scribes, but part of the
state's intellectual DNA.
Meanwhile, in India, Hindu mathematics was stirring.
From the early Vedas to the later treatises,
(10:05):
numbers shaped rituals, altars and calendars.
Geometry told you where to place the sacred
fire, astronomy told you when to light it,
and arithmetic told you how much food you'd
need to keep the priests happy.
One of the most famous astronomical texts, called
the Surya Siddhanta, would eventually appear in its
first form, offering trigonometry, planetary motion and calendar
(10:29):
rules.
Not bad for a handbook with a title
that translates roughly to knowledge of the sun.
Across Persia, India and their neighbors, knowledge cross
pollinated.
When your trade routes included camels, spice caravans
and the occasional invading army, ideas tend to
travel.
Math was portable, kind of like a good
(10:50):
joke.
All right, let's march back to the fertile
valleys of Mesopotamia and Babylonia.
Around the Tigris and Euphrates rivers, scribes pressed
their first cuneiform tablets with columns of numbers.
The Chaldeans, keen sky watchers, combined mathematics with
astronomy.
They were tracking eclipses and predicting omens.
(11:10):
So if you've ever opened an Excel file
called budget underscore final underscore final in all
caps underscore, this one is the final final,
I think you understand their energy.
Clay tablets filled the shelves, recording everything from
goat counts to planetary movements.
And by the way, some of those earliest
(11:31):
math tablets are 4000 years old, and they
are still readable.
Meanwhile, I still can't open a Word document
from 2003.
Then there's Egypt.
By the third millennium BCE, Egyptian engineers were
doing geometry with precision that would impress your
high school math teacher.
They designed pyramids, surveyed farmland, and used sundials
(11:55):
to keep time.
Forget daylight savings, Egyptians had sunlight accuracy.
The great pharaoh Ramses II even divided land
among his people, which meant mathematics wasn't just
for pyramids, it was also for property disputes.
Picture the court.
Yes, yes, Your Majesty, we measured your cousin's
(12:16):
field three times, and no, he cannot count
the neighbor's goats as his own.
Yeah, I could hear it.
From Stone Age scratches to Egyptian sundials, mathematics
had gone from survival skill to statecraft.
The world was realizing that math wasn't just
about tallying sheep, it was about running civilizations.
And that realization leads us westward into the
(12:39):
world of the Occident, where philosophers, merchants, and
would be mathematicians were about to take center
stage.
Occident simply means the West and the place
where the sun goes down.
Its opposite is the Orient, the East, where
the sun comes up.
Basically, a poetic way to say sunrise over
(13:01):
here, sunset over there.
In the Occident, math got philosophical.
The Greeks weren't satisfied just counting sheep.
They wanted proofs.
They wanted reasons.
They wanted to argue about numbers until everyone
else at the symposium pretended to fall asleep.
And then came Pythagoras, the guy everyone credits
(13:21):
with discovering that right triangles have a thing
for squares, except he did not discover it.
Babylon in India had already been doing that
math for about 2000 years.
So what happened?
Well, Pythagoras just slapped his name on it.
So technically, math is like comedy and the
Pythagorean theorem is the oldest borrowed joke in
(13:41):
the book.
So while Pythagoras was busy with triangles, the
Chinese already had rods, shadows, and proof sketches.
Think of them as the silent backup guitarists
in the global math band.
In India, they were messing with string, bricks,
and fire altars, making precise right angles thousands
of years before Pythagoras could trademark the idea.
(14:04):
So Pythagoras is kind of like the spinal
tap of math, always leaning into the common
cliches.
I could see that.
Okay, so then there's Plato, who said that
math was the language of perfect forms, triangles,
circles, cosmic ideals, lounging in some celestial VIP
room.
And Aristotle, who wrote down rules for logic
(14:25):
and classification, basically invented the world's first math
adjacent buzzfeed listicles.
So that's the Occident, proofs, philosophy, and the
first math club with actual membership rules.
But the sun doesn't just set, it also
rises.
While the Greeks were busy arguing about triangles,
the Orient was quietly doing math of its
(14:47):
own, often with a lot less drama, and
a lot more practical results.
In China, the Zhoubi Zhuangjing gave us the
Gaozhu theorem, which, yes, is basically the Pythagorean
theorem, except a few centuries earlier.
They also had counting rods, little sticks arranged
on a board that let you do place
(15:08):
value arithmetic, negatives, and even fractions.
Basically, the world's first calculator app, but in
wood.
Over in India, the Shulbasutras showed priests laying
out fire altars with ropes and pegs.
And wouldn't you know it, right triangles kept
showing up.
They knew there are three, four or five
(15:29):
triangles long before Greece tried to trademark them.
So while Western history textbooks sometimes made it
sound like Pythagoras invented everything, Pythagoras, Pythagoras, Pythagoras,
Marsha, Marsha, Marsha, the truth is the Orient
was already playing the same game.
(15:49):
They just weren't as noisy about taking credit.
They were quietly and diligently doing their homework.
So after the Greeks had their fun proving
that triangles weren't lying to us, mathematics packed
its bags and moved to Alexandria.
Picture a university slash library slash nerd paradise.
(16:12):
I can.
Shelves of scrolls, scholars scribbling everywhere, and enough
papyrus dust to choke a camel.
It was the beginning of paperwork.
Yay.
At the front of the class, Euclid.
He wrote the Elements, a math textbook so
good it outsold the Bible for centuries.
No joke.
Every proof, every theorem laid out like IKEA
(16:34):
instructions for the universe, except his triangles actually
fit together at the end.
Then there was Archimedes, the inventor, the mathematician,
the engineer, and basically an all around show
off.
You know, the overachiever who tries so hard
for the sake of trying, who just wants
to do better than everybody else, but really
(16:55):
feels isolated inside.
Okay, he calculated pi, he built war machines,
and he supposedly ran naked through the streets
shouting Eureka after figuring out how to measure
volume in a bathtub.
That's right.
The greatest breakthrough in fluid mechanics began with
public nudity.
And we have Pythagoras.
Nope, we have Archimedes to thank for that.
(17:20):
Pythagoras, Pythagoras, Pythagoras.
Speaking of astronomy, Ptolemy, his Almagest gave us
a geocentric model of the universe with the
earth at the center.
It was complicated, elegant and totally wrong.
But hey, it lasted for 1400 years, which
(17:40):
means either he was convincing, or nobody wanted
to redo the math.
And I'm going to lean into the latter
because I actually did a podcast about that
way back in 2020, I think.
Then comes Diophantus, the father of algebra.
He loved equations with whole number solutions, what
we now call Diophantine equations.
(18:02):
It was the kind of math that looks
innocent, but can actually ruin an entire Saturday.
And centuries later, I kid you not, Fermat
would scribble in the margin of his book,
something along the lines of I've got a
proof for this, but the margins too small,
which is basically the mathematical equivalent of a
mic drop.
(18:22):
And then my personal favorite, Hypatia of Alexandria.
She was a brilliant teacher, a philosopher, a
mathematician.
She taught geometry and astronomy.
She edited her father's works and wrote commentaries
that preserved Greek math for future generations.
She was admired by students respected across the
(18:43):
ancient world, and tragically was murdered by a
mob of monks in 415 CE.
Her story is why I wrote a book
about her because history should remember not just
her death, but the sum of her life
and legacy.
And that actually happens to be the title
of my book called Hypatia, the sum of
(19:03):
her life.
And you can find it on Amazon and
the link will be in my show notes.
We'll be right back after a quick word
from my advertisers.
Here's what's interesting.
After Hypatia was murdered, things got weird.
And about 100 years later, the West hit
pause.
Europe slipped into the so called Middle Ages,
(19:24):
where math mostly huddled in monasteries, trying not
to be forgotten.
It wasn't a total blackout, just a very
long math nap.
Luckily, while Europe was dozing, the Islamic Golden
Age was wide awake.
Scholars across Baghdad, Damascus and Cordoba translated Greek
works and added their own discoveries and gave
(19:47):
us new tools.
There are even tracings of Hypatia's work in
some of the translations from the Islamic Golden
Age.
Algebra got its very name from Al-Khwarizmi.
Trigonometry blossomed, algorithms were born.
And without them, Europe would not have had
anything to wake up to during the Renaissance.
(20:09):
In short, while Europe was snoring, the Islamic
world was basically babysitting math, feeding it, clothing
it and teaching it how to walk.
So when the Renaissance finally rolled around, Europe
didn't invent math all over again.
It snuck out the window with dad's car
keys and went for a joyride taking credit
(20:30):
for all the fun.
By the 1300s, the great centers of the
Islamic Golden Age had dimmed.
The Mongols sacked Baghdad in 1258.
And let's just say that the House of
Wisdom didn't exactly get a renovation budget afterwards.
So without strong rulers funding scholars, math and
science lost some of their shine.
(20:50):
Theology and law were safer bets than astronomy
or algebra.
And while the Islamic world was arguing over
which star catalog to keep, Europe was busy
stealing the whole frickin library.
Spain, Sicily and Crusaders carried translations into Latin.
And suddenly the West was cramming algebra like
(21:11):
a college kid before finals.
And that doesn't mean math disappeared in the
East.
Far from it.
Ulaanbaatar and the Samarkand built an observatory so
massive, it made Stonehenge look like a backyard
lawn decor.
But the Golden Age of invention had slowed.
The math car keys were now jingling in
Europe's pocket.
(21:31):
Meanwhile, in China, scholars like Cheng Dewei printed
arithmetic manuals that spread like wildfire.
Counting rods, abacuses and math tables went from
scholars' desks to merchants' toolkits.
Math was basically hitting the shelves like the
latest bestseller.
Trade routes in the 13th century carried more
(21:52):
than silk and spices.
In India, Siddhara and Bhaskara II wrote on
astronomy, arithmetic and algebra, predicting eclipses while Europe
was still suspicious of zero.
Meanwhile, in Persia, Al-Qaraji pushed algebra into
new territory with powers and roots, laying groundwork
for future algebraic thinking.
(22:14):
And Chinese astronomical methods even traveled into Japan,
shaping calendars and sky charts.
So by the 1500s, the Orient had printing,
algebra, astronomy and trade all fueling with mathematics.
Which meant that when the Renaissance revved its
engines, Europe wasn't inventing math from scratch.
(22:34):
It was merging into an already busy highway.
So Europe sneaks out with dad's car keys,
takes math for a joyride, and suddenly it's
the Renaissance.
The printing press is the turbo boost.
Math isn't just locked in libraries anymore.
It's mass produced.
Textbooks, tables, diagrams, all spreading faster than gossip
(22:57):
in a small town.
Math memes, but in ink.
Enter François Viet, whom I've also done a
podcast on.
Did that about a year ago.
Please go visit mathsciencehistory.com and dig around.
There's some good stuff there.
Anyhow, François Viet, he was the guy who
gave algebra its makeover.
Before him, equations were basically long winded novels
(23:19):
like The unknown multiplied by itself added to
four times the unknown is equal to 21.
Viet said, Nope, let's just use letters.
Suddenly, algebra looked like Twitter.
Short, sharp, and easy to argue about.
Meanwhile, trigonometry was getting a glow up.
Navigation needed it.
And when your ships are crossing oceans, close
(23:42):
enough, right, is not good enough.
Tables of sines and cosines went into print,
and sailors could finally aim for the new
world without relying entirely on guesswork and rum.
Though to be fair, rum was still heavily
involved.
The Renaissance also saw a mashup of art
and math.
Perspective drawing turned flat canvases into 3D illusions.
(24:05):
Architects rediscovered geometry, making domes and cathedrals that
look like math problems you'd actually want to
live in.
So the 16th century gave us algebra with
style, trig for sailors, geometry for artists, and
math books going viral thanks to Gutenberg's machine.
It was less about inventing new math and
(24:25):
more about broadcasting it to the masses.
I love it.
And next, well, things really speed up.
The 17th century, where Descartes brings us analytic
geometry, Fermat and Pascal invent probability, and Newton
and Leibniz get into the world's most famous
calculus cage match.
It's good.
(24:46):
The 17th century was Europe's math firework show.
France, Britain, Germany, the Netherlands, every country wanted
a math genius to brag about.
In France, Rene Descartes invented analytic geometry, turning
curves into equations.
Suddenly, algebra and geometry weren't two subjects that
hated each other in high school.
(25:08):
They were dating.
In the Netherlands, Christian Huygens studied pendulums and
probability, basically building the first math-powered clock
and helping gamblers figure out their odds.
You gotta know when to hold them and
know when to fold them.
In Britain, a certain Isaac Newton came along,
invented calculus, and then decided gravity should be
(25:29):
his side hustle.
Meanwhile, in Germany, Gottfried Wilhelm Leibniz also invented
calculus.
Cue the most brutal math feud in history,
the Newton-Leibniz cage match.
In the red corner, from Woolstorp, England, the
apple dropper, the gravity guy, the calculus crusader,
Isaac, don't steal my derivatives, Newton.
(25:51):
And in the blue corner, from Leipzig, Germany,
the philosopher, the polymath, the notation ninja, Gottfried
Wilhelm integrals forever Leibniz.
Yep, Newton claimed he invented calculus first.
Leibniz said his notation was better.
Their fans went to war, pamphlets, accusations, full
(26:12):
-on character assassinations.
Newton, who just happened to run the Royal
Society, stacked the jury against Leibniz.
That's like being both the boxer and the
referee.
Brutal.
Leibniz died with his reputation in tatters.
Newton, meanwhile, lived long enough to look smug
(26:34):
about it.
And yet, both notations survived.
Today, we use Leibniz's slick symbols and Newton's
methods.
Back in France, the 18th century saw the
rise of giants like Euler and Lagrange, turning
out equations like a bakery turns out baguettes.
Theoretical math was the center stage.
(26:56):
And this is a really fun era to
study.
And remember Euclid?
His Elements, written way back in Alexandria, was
first printed in 1482.
So by the 17th and 18th centuries, it
was still the gold standard.
Imagine writing a textbook that stays in print
(27:17):
longer than Shakespeare.
Meanwhile, European math didn't stay in Europe.
By the 16th and 17th centuries, translations carried
these works eastward.
Chinese scholars began building essential treatises based on
European models.
And through Dutch traders at Nagasaki, math also
slipped into Japan, where scholars like Nakane Genkai
(27:38):
adapted Western astronomy and algebra into Japanese frameworks.
So the 17th and 18th centuries gave us
analytic geometry, probability, calculus, a feud worthy of
pay-per-view, and the first global math
exchange program.
Not bad for a couple of centuries.
So up to now, math had been busy
(28:00):
building its toolkit.
Geometry, algebra, trigonometry, probability, calculus.
Each century added another gadget to the box.
By the 19th century, the box was overflowing.
And that's when mathematicians decided, hey, let's see
how far this thing can go.
So remember Descartes' analytic geometry, that marriage of
(28:20):
algebra and geometry?
By now, it was producing children, whole families
of curves, surfaces, and equations.
Calculus, invented in the 17th century for physics
and astronomy, had grown into a universal tool.
It could describe how planets move, how heat
spreads, how a pendulum swings.
And trigonometry, once about triangles and shadows, was
(28:44):
now deeply woven into calculus, making it possible
to tackle wave motion, oscillations, and even music.
The triangle wasn't just hanging out on chalkboards
anymore.
It was solving differential equations.
The glow-up was complete.
Enter the 19th century heavyweights.
(29:06):
In Germany, Gauss ate theorems before breakfast.
Actually, he didn't eat them.
He just solved them.
He dabbled in everything from number theory to
magnetism and made everyone else feel underachieving.
I've been around people like that.
It's no fun.
His student, Riemann, bent geometry itself, laying the
groundwork for curved spaces, a concept that would
(29:29):
later fuel Einstein's relativity.
Meanwhile, a fiery young genius named Evariste Galois
practically invented abstract algebra before dying in a
duel at the age of 20.
And if you want to hear a podcast
about that, I also do one about that
as well, which was done about, oh, 15
months ago.
So again, please visit MathScienceHistory.com and dig
(29:52):
around in the archives and you'll see some
great podcasts.
And while you're there, click on that coffee
button because every donation you make keeps the
podcast up and running.
Anyhow, back to Evariste Galois.
Think of what he did.
He created group theory, field theory.
It was all sketched out, literally, in a
few frantic manuscripts the night before his last
(30:13):
pistol match.
Mind-blowing.
This kid was brilliant and he was only
20.
So yeah, math can be dangerous.
Don't get involved in duels.
Then comes Cantor, who looked at infinity and
said, one size doesn't fit all.
He showed that some infinities are bigger than
others.
And then, finally, Emmy Noetcher, who formalized abstract
(30:35):
algebra and tied it to physics.
Her theorems still drive modern science.
If algebra were a messy teenager, Noetcher was
one who said, clean your room, organize your
variables, and respect symmetry.
Like a true boss.
So, by the 19th century, math had evolved
from counting dots and proving triangles to taming
(30:57):
infinities, curving space, and inventing algebra so abstract
it made even mathematicians nervous.
And in my case, when I was studying
math, it made me cry.
I'm going to admit it.
It made me cry.
It's hard.
And just when math seemed to have reached
its limits, the 20th century cracked the door
wide open, bringing machines that could calculate faster
(31:20):
than any human, physics that bent reality itself,
and logic that questioned what we even mean
by truth.
Math wasn't just sneaking out with the car
keys anymore.
It was hot wiring the whole garage.
Buckle up.
Things are gonna get wild.
By the dawn of the 20th century, math
(31:40):
wasn't just sneaking out with the car keys
anymore.
It was stealing the whole car, peeling out
of the driveway, and never looking back.
First up, David Hilbert, the grandmaster of rigor.
He laid out 23 problems in 1900, like
the ultimate math scavenger hunt, challenging mathematicians to
solve them over the next century.
(32:01):
Some got solved.
Some are still haunting blackboards like math's version
of the unsolved true crime.
Then came Kurt Gödel.
He dropped his incompleteness theorem in 1931 and
proved that math itself has limits.
There will always be true statements that can't
be proven.
Mathematicians cried.
(32:21):
Philosophers panicked.
Logic had just punched certainty in the face.
Meanwhile, Alan Turing was teaching machines how to
think.
His work on computation and algorithms birthed the
modern computer.
Suddenly, math wasn't just in books.
It was humming inside machines, calculating, encrypting, breaking
(32:42):
codes, and eventually playing chess better than humans.
And over in physics, math was bending reality
itself.
Einstein's relativity used Riemann's curved geometry to show
space and time aren't fixed.
Then, quantum mechanics arrived with wave functions, probability
amplitudes, and equations that said particles could be
(33:04):
cats, both alive and dead, until you check.
So math had gone from let's count goats
to let's describe parallel universes.
That escalated quickly.
Then there was chaos theory.
(33:26):
Suddenly, simple equations produced wildly unpredictable outcomes.
The butterfly effect became a household phrase, though
mathematicians never did prove whether butterflies really control
the weather or just have really good PR.
As the century rolled into the 21st, math
embedded itself everywhere.
(33:47):
In cryptography, securing your credit cards, in data
science, running everything from Netflix recommendations to climate
models, and in AI.
So, math has officially taken the wheel.
And no, it's not bringing the car back.
So here we are, from tallying sheep bones
to infinity, from sundials to supercomputers, math has
(34:09):
been our universal language, our secret decoder ring
for the cosmos, and occasionally, our favorite excuse
for why we are bad at taxes.
And the ride isn't over.
Math began as a baby.
It was coddled by the Islamic golden age.
And then it became a car sneaking out
of the driveway, then tearing down the highway,
inventing new lanes as it went.
(34:30):
But here's the thing.
Highways only take you so far.
By the 20th and 21st century, math wasn't
satisfied with asphalt.
It strapped on boosters, pointed upward, and launched
from quantum mechanics to AI, from curved space
to infinity itself.
Math has become our rocket.
(34:50):
It's our ticket into the vast unknown.
Every equation is a launchpad.
Every proof, a countdown.
Every theorem, a trajectory.
And where does it go?
It goes into a universe that has no
end, no edges, just limitless space for discovery.
So math isn't just driving anymore.
It's flying.
(35:11):
It's carrying us past what we can see,
past what we can touch, into the infinite.
And the ride?
Well, the ride has only just begun.
Thank you for listening to Math Science History.
And until next time, carpe diem.
(35:53):
Again, thank you for tuning in.
And until next time, carpe diem.
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