Why was the development of mathematics very slow between Ancient Greece and Descartes?

Question

I asked this question on MSE here

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In my studies of mathematics (I am not very good at mathematics, I only studied real analysis, some linear algebra, geometry and calculus ), I noticed that mathematics can be divided into two major parts. One part is Ancient Greek mathematics . This part has achievements in geometry and arithmetic and their connections. The other part is modern mathematics from Descartes to today. This part has discoveries in calculus, number theory, topology, and many other fields (areas). But what happened between these two eras? In my studies, I did not find many theorems from this period or any great breakthrough at all.

It seems like mathematics stopped there and only resumed in the 1500s and 1600s here I am not talking about European mathematics but mathematics in general (Middle East, China, India, etc.). But why is that? Mathematics is not an empirical science like physics that needs experiments to test theories and wait for centuries such that their tools become good enough. Mathematics only relies on logic, and for some reason Mathematics was just like other empirical sciences -- it began really to develop after the 1600s.

As far as I know, mathematics can be divided into fields (areas) like topology, analysis, number theory, algebra, geometry, combinatorics, etc. Most of these fields or areas took more than a millennium to appear. The first major one was geometry, and it appeared in Ancient Greece. All the others did not exist until after the 1600s, like topology or analysis, or they were too small and narrow to be considered as separate fields before. So why is that? Even if calculus took 1000 years to be discovered (Which I don't know why! see This question ) I heard that many mathematical fields don't rely on analysis very much, like abstract algebra, graph theory, etc.

I heard that philosophy held back mathematics for millennia, Archimedes could have discovered calculus if he did not think too much about philosophy and infinity. Is this the reason?

So why was the development of mathematics slow for a millennium and why did many mathematical concepts take millennia to develop? or in other words why the development of mathematics was very slow between Ancient Greece and Descartes (I don't mean Descartes specifically but the era that he lived in. Cavalieri, Fermat, Kepler, etc. were from the same era, i.e., 1550−1650 .)

I think this question might have too long answers as a user point out before in one of my old questions: "You must realize that the history of mathematics is not going to be broken down into a couple of paragraphs," and so I will ask for books or references that answer this question if the answer is too long.

Answer 1

Making my comment into an answer:

Was there a gap of knowledge or slow-down of progress in math as a whole between Ancient Greeks and the 17th century?

The answer is probably no.

Islamic Medieval World and Medieval India contributed greatly to math during the Middle Ages. The fields of algebra, polynomial equations, cubic roots, modern numerals, logic, negative numbers, trigonometric functions and much more came from that period. These contributions are not minor and contributed continuously to the development of math. Islamic sources in particular translated and preserved a lot of Ancient Greek and Indian sources, and Islamic sources were later used by Renaissance Europeans like Fibonacci.

Wikipedia's history of mathematics also cites:

Other achievements of Muslim mathematicians during this period include the addition of the decimal point notation to the Arabic numerals, the discovery of all the modern trigonometric functions besides the sine, al-Kindi's introduction of cryptanalysis and frequency analysis, the development of analytic geometry by Ibn al-Haytham, the beginning of algebraic geometry by Omar Khayyam and the development of an algebraic notation by al-Qalasādī.

As for well known theorems of this time you have: the law of cosines (a.k.a. Al-Kashi's theorem), Chinese remainder theorem, Merton mean speed theorem, Wilson's theorem, Thabit's rule, divergence of the harmonic series, and the binomial theorem.

But even if we forget the Middle Ages as some "Dark Ages" (a term now considered as misnomer) and focus on Europe, we still have the whole Renaissance. People like Tartaglia and Cardano helped develop the solution to the quartic equations. Bombelli developed the algebra of imaginary numbers.

Many theses could be written on why the developments between Ancient Greeks and 17th century math seem "slow" or minor. I propose three conjectures:

1. Lack of proper sources and attribution: sources foreign to Europe were not translated or properly attributed. This makes a lot of contributions that have no authors attached to them (even if we have words like algebra that come from Arabic or algorithm which comes from Al-Khwarizmi).
2. Utility: Some mathematical content is preserved and transferred to the next generation better than others. Most of the time if there is some application (engineering, physics or astronomy) there was a higher chance for the content to survive. Kerala School might have invented calculus, but clearly their results were never largely popularized (maybe even never left Kerala).
3. Transition of power: As Europe declined in the Middle Ages and the Medieval Islamic World became an important center of knowledge and power, clearly there was a lot of friction. Text had to be translated from Greek and Latin (and Sanskrit from India) to Arabic - and when Europe "reemerged", back from Arabic to Latin. Religious conflicts and other power struggles also did not help. Imagine how hard it was to introduce Hindu decimal numerals from the Indian subcontinent into West Asia and into Europe without the Internet, printing, long distance sailing and so on. Nevertheless, it made the trip because it was an important breakthrough.

However for every reason that I can find for this possible setback in progress in math, I can find a ton of contributions from those times that make the setback negligible. How much importance we give to these developments over some more recent ones is more a matter of opinion or perspective - though this is not the forum for opinion-based discussions.

The Dark Ages is a myth. For a light read, check for example The Light Ages: The Surprising Story of Medieval Science by Seb Falk (Time excerpt), and The West: A New History in Fourteen Lives by Naoíse Mac Sweeney (Smithsonian excerpt).

MathSE recommends A Short Account of the History of Mathematics by W. W. Rouse Ball for the history of math in general. It divides all math in three great periods: Ancient Greeks, Middle Ages and Renaissance, and Modern (beginning with Descartes). Chapters cover math from the last period of the Roman Empire, Middle Ages math (both in Islamic and Europe) and Renaissance math.

I end by adding this good quote I found in MacTutor History of Mathematics Archive article on "Arabic mathematics":

There is a widely held view that, after a brilliant period for mathematics when the Greeks laid the foundations for modern mathematics, there was a period of stagnation before the Europeans took over where the Greeks left off at the beginning of the sixteenth century. The common perception of the period of 1000 years or so between the ancient Greeks and the European Renaissance is that little happened in the world of mathematics except that some Arabic translations of Greek texts were made which preserved the Greek learning so that it was available to the Europeans at the beginning of the sixteenth century.

That such views should be generally held is of no surprise. Many leading historians of mathematics have contributed to the perception by either omitting any mention of Arabic/Islamic mathematics in the historical development of the subject or with statements such as that made by Duhem in 3:-

... Arabic science only reproduced the teachings received from Greek science.

Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the sixteenth, seventeenth and eighteenth centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier. In many respects the mathematics studied today is far closer in style to that of the Arabic/Islamic contribution than to that of the Greeks.

Answer 2

Mathematics is a tiny part of human activities, and its historical development cannot be considered in isolation. Since the 2nd century AD, there was a decline in all intellectual activities. Eventually civilization itself collapsed, and its slow revival started only in 16th century. In this period, people mostly lived in the villages, rather than in large cities, most of the economy was on subsistence level, most people were illiterate, technology was lost, and the main intellectual activity was concern about afterlife.

This was not the only such collapse of which we have historical evidence: another one on a similar scale happened around 1200 BC. Historians discuss a lot the reasons of these collapses, the literature on the topic is enormous, and there is no common opinion.

Let me mention just a few books, and one modern paper.

• Edward Gibbon, The decline and fall of the Roman empire.

• Peter Heather, The Fall of the Roman Empire: A New History of Rome and the Barbarians, Oxford, 2006.

• Brian Ward-Perkins, The fall of Rome and the end of civilization, Oxford UP, 2005.

• Adrian Goldsworthy, Fall of the West: the death of the Roman superpower, Phoenix, 2010.

• W. M. Jongman, Gibbon was right: the decline and fall of Roman economy, in the book: Crises and the Roman Empire. Proceedings of the Seventh Workshop of the International Network Impact of Empire (Nijmegen, June 20-24, 2006), Freely available on Internet.

  (This paper checks Gibbon's insight with modern hard evidence).

To be sure, decline of mathematics, and of science in general, began several centuries before the decline of the Roman empire, see on this:

• L. Russo, The forgotten revolution. How science was born in 300 BC and why it had to be reborn, Springer, 2004.

So decline in science was followed by a collapse of civilization.

Answer 3

Perhaps the question ought to be turned on its head: why did mathematical learning hit the accelerator in Ancient Greece and during the modern era in Western Europe (17th century to present)?

Some historians (e.g., Walter Scheidel) have reasoned there were unique conditions in both eras and locations that helped foster the growth of mathematical and scientific knowledge: a fractured, competitive political geography that prevented a single authority controlling or stifling higher learning, and the presence of an economically prosperous, literate and educated class.

Humans are naturally curious and certainly human knowledge has expanded in other times and places (the Roman and Chinese empires, ancient Egyptian, Mesopotamian, Indian civilizations) but at a slower pace.

Answer 4

There reason for the speed-up in the 1600s was Vieta's development of an algebraic formalism that allowed mathematicians for the first time to write down mathematical formulas rather than merely abbreviated phrases in natural language. In your title you mention Descartes specifically, but several dramatic advances occurred before Descartes. The names that come to mind are in alphabetical order Cavalieri, Fermat, Kepler, and others.

Answer 5

An aspect that has not been mentioned and is at least a partial answer is that progress must strongly depend on the number of people working in the field, which is the product of the population and the fraction of them working in the field. I'll focus on Europe here.

The population in Europe doubled between 1500 and 1750; consequently, ceteris paribus, the number of people working in the field (I avoid the term "mathematician" because many scholars at the time were polymaths) should have doubled. Without any reference I can imagine that there was only a handful of people of the kind who would advance mathematics (or any other field) until maybe the early 1600s, in the entirety of Europe.

And the fraction of the population working in the field should also have increased. Life in Europe until the late middle-ages intermittently devolved into a fight for survival, often unsuccessful. Europe-scale famines and pandemics ravaged the continent in the middle-ages. While both didn't disappear later, they became less frequent.¹.

With greater wealth and a growing surplus in food came the opportunity for a division of labor; professionals who did not directly contribute to food production could be sustained, like musicians, artists and scholars. At the same time, trade and crafts made education more important; in the cities, an educated citizenship emerged, among them teachers. Courts were able to host scholars like Leibniz and Voltaire.

That alone must be one reason the speed of mathematical development, like everything else, picked up after 1500. Working hypothesis: That other regions, like the Arabian world, were more productive may be an indicator for greater prosperity.