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I asked this question on MSE here


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.

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    $\begingroup$ Did you miss all contribution from Islamic Middle Ages and India? Algebra, polynomial equations, cubic roots, modern numerals, negative numbers, trigonometric functions and much more came from that period $\endgroup$
    – Mauricio
    Jan 15 at 9:50
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    $\begingroup$ I think you are just dismissing it because they are sometimes not correctly attributed in Western learning. But the history of mathematics shows a continuous flow from Greek sources to Medieval Islam and back to Europe. Are you telling me that Al-Khwarizmi solution for quadratic equations was not consequential? Algebra is a major topic in math from that era. $\endgroup$
    – Mauricio
    Jan 15 at 10:00
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    $\begingroup$ When your say that mathematics "is not like physics that needs experiments" and it "only relies on logic", you seem to underestimate its connections with other fields and with society (or the reasons we do math in the first place). A lot of development in math from 1600s onwards can be attributed to developments in physics, engineering, astronomy, etc. $\endgroup$ Jan 15 at 10:12
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    $\begingroup$ We have to consider the complexity of historical events... Today we are accustomed to think in term of progress: an ever growing progress that accumulates knowledge. But "modern progress" lasted since Renaissance: about 500 years. Ancient Greece culture lasted from 500 BC to 300 AC (around 800 years). The "Dark Ages" covered in Europe several centuries, but in the meantime there were mathematics living in Islam, including some part of Europe, Baghdad, etc. And there is India, China. Historical facts are consistent with the possibility that there are "waves" of growing and decay... $\endgroup$ Jan 15 at 15:26
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    $\begingroup$ @pie A fascinating book I recently found is Series and Products in the Development of Mathematics by Ranjan Roy. Many formulas for pi were discovered from antiquity to modern math. $\endgroup$
    – qwr
    Jan 16 at 15:32

5 Answers 5

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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.

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    $\begingroup$ The invention of the printing press in the mid 1400s played a crucial role in the acceleration of scholarly work. For what it's worth, I think there are a lot of parallels with the rise of the internet beginning in the mid 1990s. Both brought about faster and wider dissemination of information, more outlets for scholarly writing, more lower quality stuff getting through $\ldots$ $\endgroup$ Jan 15 at 14:44
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    $\begingroup$ @DaveLRenfro certainly the speed of progress is changing and accelerating. Much more math is produced now than a century ago, more math was produced in the 20th century compared to the 19th century and so on. But I think that is the way no matter the century, I do not see what's special about the Middle Ages in terms of acceleration. It never "stopped" as OP wrote originally in his question. $\endgroup$
    – Mauricio
    Jan 15 at 14:51
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    $\begingroup$ @gidds the Dark Ages is the idea that due to the decline of Rome, the Middle Ages had little to no advancement in knowledge and science. Many authors have written against this idea. The Middle Ages saw a lot of progress (mainly outside Europe but in Europe too!). $\endgroup$
    – Mauricio
    Jan 16 at 1:01
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    $\begingroup$ @gidds: The notion that we don't have much written evidence from the so-called Dark Ages is also debatable at best. The medievalists Matthew Gabriele and David M. Perry have recently written a popular book, The Bright Ages: A New History of Medieval Europe, that you may be interested in. (The "bright" of the title means "well-lit" more than "wonderful", though the authors do also emphasize that the Middle Ages were much better than most people assume today.) $\endgroup$
    – ruakh
    Jan 16 at 8:51
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    $\begingroup$ @pie according to legend (written some centuries after), "so many manuscripts were tossed into the River Tigris that its waters turned black from ink". We can only wonder. $\endgroup$
    – Mauricio
    Jan 16 at 13:06
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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.

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    $\begingroup$ After reading through about 13 comments, I got to your answer and thought "Finally, someone mentions historical aspects of the middle ages!" (often called the Dark Ages), which at least for me was one of the 3 or 4 main topics in my high school world history class. Regarding the OP's question, to me the amount of mathematical pursuits done during this time is actually rather amazing (most of it in the middle east regions) given the harsh conditions and lack of resources for scholarly pursuits (time, travel, communication, etc.). $\endgroup$ Jan 15 at 14:29
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    $\begingroup$ Note that these sources are about the decline of specific civilizations, but if the discussion is about mathematics as a whole it is hard to say that there was a global decline. Maybe OP should specify if this is about European mathematics. $\endgroup$
    – Mauricio
    Jan 15 at 16:34
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    $\begingroup$ @AlexandreEremenko, you can reasonably claim that Indian and Chinese mathematics didn't have a major influence, but you can't ignore the contributions of the Islamic mathematicians. $\endgroup$
    – Mark
    Jan 16 at 4:15
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    $\begingroup$ @DaveLRenfro While I agree with you in principle, I feel that there is a risk that your education, like mine, brushed everything from the Fall of Rome to the emerging struggle between Rome and Canterbury under the carpet. No Eastern Roman Empire, no Byzantium, no Islamic culture and /definitely/ no Avignon Papacy to challenge the narrative. $\endgroup$ Jan 16 at 7:38
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    $\begingroup$ @AlexandreEremenko slow with respect to what? Is this statement focused in Europe? $\endgroup$
    – Mauricio
    Jan 16 at 17:27
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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.

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    $\begingroup$ I would love to see a metric on this "acceleration", do you think that the acceleration was different during the Middle Ages under which criteria? $\endgroup$
    – Mauricio
    Jan 17 at 15:22
  • $\begingroup$ @Mauricio That's a good question and we have to be careful how we quantify terms like acceleration as well as time period and location (I know "Western Europe" is vague). The simplest metric would be counting the number of mathematicians who made a significant contribution. I think it's safe to say something special was happening at certain times and places in history, and at other times there was a steady trickle of new thinkers and innovations. $\endgroup$
    – RobertF
    Jan 17 at 22:54
  • $\begingroup$ But that's where I do not get it. Do people really see a decline in the Middle Ages in the number of mathematicians and contributions? This period was as huge as in any other times I do not even see how would somebody would separate this age from the rest. $\endgroup$
    – Mauricio
    Jan 18 at 8:32
  • $\begingroup$ There were a few bright lights of learning during the Dark Ages in W. Europe, for example under Charlemagne's reign. But nothing like the Library of Alexandria existed until the first universities were established. You're correct there was activity in other cultures during the 5th to 16th centuries, such as the Islamic world and Chinese Empire. But there was no Islamic or Chinese Scientific Revolution. I'm not belittling the contributions; they were invaluable. There was a missing ingredient that allowed science to flourish in Europe after a long dry spell. $\endgroup$
    – RobertF
    Jan 18 at 15:46
  • $\begingroup$ I agree, as long as we focus on Europe. But to put a "slow" label on the Middle Ages just because there was no revolution is to me is like saying that in between the Scientific Revolution (from Copernicus to Newton) and the 20th century nothing happened or progress slow down because there was no revolution. $\endgroup$
    – Mauricio
    Jan 18 at 15:59
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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.

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  • $\begingroup$ I didn't mean Descartes specifically but the era that he lived in. Cavalieri, Fermat and Kepler were from the same era i.e $1550 -1650$ $\endgroup$
    – pie
    Jan 15 at 11:58
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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.

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