The Wild Ride of Math: From Goats to Rockets

Gabrielle Birchak/ October 14, 2025/ Ancient History, Archive, Enlightenment, Modern History, Post Classical

(PODCAST TRANSCRIPTS, GOATS INCLUDED)

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 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? 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. 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.

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. 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. 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 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 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 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. 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 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.

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.

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. 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 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.

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, 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 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, 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 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 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 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.

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. 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 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, 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 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, 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 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 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.

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 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 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 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 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 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 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 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.

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 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 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 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 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. 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.

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 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 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.

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 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. 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.

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 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 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, 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 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. 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 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.

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 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.

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 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.

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.

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 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 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 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. 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 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.

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. 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 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. 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-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 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.

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 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 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 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 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 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. 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 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 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 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 Noe­ther, who for­mal­ized abstract alge­bra and tied it to physics.

Her the­o­rems still dri­ve mod­ern sci­ence. If alge­bra were a messy teenag­er, Noe­ther 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 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 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 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. 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. 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 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 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. 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. 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 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. 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.

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. 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. Again, thank you for tun­ing in.

And until next time, carpe diem.