As part of a larger project, I needed a simple timeline for the history of mathematics. I kicked around a bunch of candidates but settled on the following:
Old Tally — before 4000 BCE
Bronze River — 4000 BCE to 1001 BCE
Early Philosophic — 1000 BCE to 301 BCE
Euclidean — 300 BCE to 599 CE
Medieval Zero — 600 CE to 1499 CE
Gutenberg — 1500 CE to 1799 CE
Protoformal — 1800 to 1949
Turing — 1950 to present
AIM — future, when AIs eclipse human mathematicians.
Like all timelines, this one is entirely arbitrary. Its primary function is to impose document structure and provide handy bins for sorting material. While this timeline is arbitrary, it is not without justification. Here are my admittedly questionable reasons for each period.
Old Tally — before 4000 BCE
Only a handful of surviving authentic mathematical artifacts can be dated to periods before 4000 BCE. The most famous are notched bones like the Ishango and Lebombo bones. In addition to bones, we also have artworks and tools showing people in the Old Tally period share our affinities for symmetry and balance. We also find many earthworks and impressive monuments that reflect a basic grasp of geometric principles. You cannot infer much about Old Tally mathematics from such artifacts. Still, Old Tally people matched, counted, estimated distances, compared quantities, and appreciated symmetries long before anything resembling “civilization” appeared. Tally is another word for counting. There has never been a human culture without some form of counting, so it’s safe to assume our long-dead ancestors counted.
Bronze River — 4000 BCE to 1001 BCE
The Bronze River Era gets its name from the many prominent Bronze Age civilizations that sprung up near major historic rivers, with the most famous being the Nile, Tigris, Euphrates, Indus, Yellow, and Yangtze Rivers. During the Bronze Age, Egyptian, Mesopotamian, Indic, and Chinese cultures developed sophisticated, practical problem-solving mathematics. Bronze River mathematicians dealt with many accounting, surveying, and astronomical calculations. They tackled area and volume calculations, solved some linear and quadratic equations, extracted roots, handled fractions, and computed loan interest. The largest number of surviving Bronze River mathematical artifacts are Mesopotamian. Mesopotamians wrote on durable cuneiform tablets, and many thousands of them have survived. Hence, we know more about their mathematics than other Bronze Age civilizations, but what survives from the others indicates comparable levels of sophistication. I suspect there was more contact between Bronze Age people than previously supposed. Humans are highly mobile animals that cannot resist the urge to see what’s over the horizon.
Early Philosophic — 1000 BCE to 301 BCE
The seven centuries from the end of the Bronze River to the Euclidean era saw something new develop in mathematics: an emphasis on philosophical proof. Proof, in an almost modern programming language debugging sense, had long existed in mathematics. Mathematicians of earlier eras constantly looked for ways to verify their work. They checked and double-checked their calculations. They came up with general schema to justify their complex methods and, in some cases, produced convincing proofs of various assertions. The notion that proof was a unique invention of the ancient Greeks is misleading. Their unique innovation was more subtle. Debugging proof focuses on exposing local errors and not constructing coherent, logical wholes. The yearning for logical wholes is a philosophical tendency, not a practical one. You do not need logical wholes to solve problems.
The poster boys for insisting on logical wholes were the Pythagoreans and their titular sage, Pythagoras. Recently, Pythagoras has been denounced by ignorant harpies and diverse idiots as an uber-dead white male who stole the work of the BIPOC’ky brown and passed it off as his own. This, in Pauli’s immortal words, “Is not even wrong!” We know very little about the alleged person Pythagoras. We don’t know his skin color, his sexual orientation, his penis length, or perhaps her vagina depth. If you’re going to slander someone, do your homework. But all of this woke whining doesn’t matter. Pythagoras may or may not have existed, but the philosophical school taking his name did! The Romans were bitching about the Pythagoreans centuries after Pythagoras’s alleged lifetime, so the school’s attitudes and outlook certainly had an impact. We know this because the very word “mathematics” derives from one of the Pythagorean schools. The Pythagoreans and others persuasively argued that it wasn’t enough to check your homework; a deeper and more comprehensive understanding was required. You needed coherent, logical wholes. This change in outlook elevated the status of mathematical proof. It was no longer a check or test of useful methods; it became the rarest of things: a philosophical necessity1.
Euclidean — 300 BCE to 599 CE
Once the philosophical necessity of proof was established it was probably only a matter of time until someone constructed a coherent logical whole. Euclid turned out to be that person, and his Elements was the logical whole. While there are arguments about how much material in the Elements dates to 300 BCE and how much can be accurately attributed to its author, there are no arguments about the importance of this singular work. The Elements is easily the most influential scholarly work in history. Its impact is only exceeded by sky fairy manuals like the Bible, Koran, and Mahabharata. Euclid was the first, as far as we know, to organize a large body of results into a systematic, logical, deductive structure. You can quibble about the Element’s defects and omissions (and many have), but the Elements created the theorem-proof world that mathematicians still live in. The book’s impact was so overwhelming that its style and conventions set the tone for centuries.
Medieval Zero — 600 CE to 1499 CE
The Medieval Zero period was characterized by the rise of Algebra by figures like al-Khwarizmi, the global spread of Indian place value numbers, and the emergence of Zero. In the modern world, two numbers, One and Zero, reign supreme. We could not do without them. It’s ironic that perhaps the most important of the two, Zero, took so long to be appreciated. But, better late than never.
Gutenberg — 1500 CE to 1799 CE
It may seem strange to name a mathematical era filled with giants like Descartes, Fermat, Newton, Leibniz, Laplace, Euler, Bernoulli, and many more after a printer, but the giants would not have been so gigantic without lowly movable print. You can see the impact of printing in the enormous increase in mathematical documents after 1500. This is evident in The Mathematical Treasures list maintained by The Mathematical Association of America. The bulk of the entries appeared after 1500.
Protoformal — 1800 to 1949
It took me some time to come up with a polite name for this period. I wanted to call it the age of bogus rigor, but that’s a tad mean. From 1800 until well into the 20th century, mathematicians were anal about rigor. The advent of epsilontics and the driving out of the infinitesimals had to happen. You cannot live in a theorem-proof world without solid proofs, but in their zeal to banish all sloppy thinking, they discovered that perfect formal proofs are horrendously long and yet hopelessly incomplete. The idea that all mathematics could be consistently constructed from a small set of axioms and a sufficiently powerful set of deduction rules was shown to be impossible. Some philosophical necessities cannot be realized.
Turing — 1950 to present
Naming our current era after Alan Turing was a no-brainer. Computers have utterly transformed our world. Many mathematicians, being elite establishmentarians, were slow to adjust. Some were on board almost as soon as computers became available, but most held back. This is changing. While it’s not feasible to formalize vast swathes of mathematics by hand, it’s entirely possible with computers. We’re currently witnessing another shift in attitude about what’s an acceptable proof. In a few more years, mathematicians will be expected to formalize new theorems. It will be seen as a tightening up of the peer review process. It might even become a requirement for publishing in major journals. It will be a brave new world.
AIM — future, when AIs eclipse human mathematicians
If you’ve been paying attention, you’ve probably noticed machines
slowly chipping away at human intellectual benchmarks. I’m old enough to
remember people saying machines would never beat the best Chess players.
Wrong. Then it was Go. Wrong. Then Jeopardy. Wrong. Similarly,
they would never be able to converse fluently. Wrong. With each advance,
the goalposts are moved. Now, machines must demonstrate consciousness
(we can’t do this for ourselves), speak a few dozen languages, play many musical instruments, write novels, and prove deep theorems before we call them intelligent. We are all in a state of AI denial, and mathematicians, despite “being smarter than average bears,” are not exempt. I don’t know when AIs will eclipse human beings at finding and proving theorems, but I suspect it’s only a matter of time. It may be centuries (unlikely), but it will happen2, and when it does, we enter the AI Mathematician or AIM age.
Analyze the Data not the Drivel 
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