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I wonder What were the obstacles that made the discovery of calculus very late ?

Why the discovery of calculus took so long? I know that some of the ideas and techniques of calculus appeared in ancient Greece, but they were not developed into a systematic theory until the $17$th century by Newton and Leibniz. What were the main obstacles or challenges that prevented the earlier mathematicians from discovering calculus?

One of the factors that I think might have played a role is philosophy. I have heard that some of the ancient and medieval mathematicians in Islamic world were influenced by Aristotle’s philosophy, which had a negative view of infinity and rejected the concept of limit and convergence. Aristotle also preferred geometric methods over algebraic methods, which might have limited the scope and applicability of calculus. Is this true? How did philosophy affect the development of calculus? And is philosophy to blame for the discovery of calculus taking more than 1000 years?

Another factor that I think might have motivated the discovery of calculus is the problem of finding the area of a curve and the tangent line to a curve. These are two important problems in geometry and they require the concepts of derivative and integral. I know that Archimedes and other ancient mathematicians used the method of exhaustion to approximate the area of a curve, and that Fermat and other 17th century mathematicians used the method of adequality to find the tangent line to a curve. But why did it take so long to generalise and formalise these methods into calculus?

I know this question might have a long and complex answer, so I will ask for a book or reference if the answers are too long or complex.

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    $\begingroup$ The modern version of the calculus (Newton-Leibniz) started with the development of a specific symbolysm that is "algebraic-like" and this followed the development of modern algebra during the Renaissance. See here for a discussion of pre-modern algebra: how difficult was to "describe" a simple (in modern terms) equation. $\endgroup$ Jan 18 at 15:33
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    $\begingroup$ "Aristotle’s philosophy... had a negative view of infinity and rejected the concept of limit and convergence." — Really? $\endgroup$
    – Michael E2
    Jan 20 at 19:18
  • $\begingroup$ @MichaelE2 well if that wasn't true then what philosophy that had a negative view of infinity $\endgroup$
    – pie
    Jan 20 at 20:08
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    $\begingroup$ I wasn't questioning the attribution. I was questioning the three assertions as not completely accurate characterizations of Aristotle's philosophy. $\endgroup$
    – Michael E2
    Jan 20 at 20:25
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    $\begingroup$ @MikhailKatz you know what? you are right. The authorities can be very drastic when you go against the paradigm, as with Copernicus. Very probably Aristotle was the main science paradigm for more than 1000 years. What I would like is that we also recognize is that progress in math does not happen in isolation. Despite being men of the church, Cavalieri and Copernicus challenged the paradigm successfully. Shattering a paradigm results from accumulated evidence and progress in those 1000 of years. $\endgroup$
    – Mauricio
    Jan 22 at 18:47

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I would like to make several points with regard to this interesting question.

  1. The discovery of some Taylor series of trig functions by the Kerala school is a very impressive early breakthrough. However, I believe the consensus of historians is that this does not in any way constitute significant progress toward the invention of something on the scale of the calculus. Similar remarks apply to many other multicultural achievements touted at answers to some parallel questions.

  2. Note that by the time of James Gregory, for example, many power series were already known, before the invention of the calculus. Today we tend to think of power series as part of the calculus, but historically they were available well before, including (as I mentioned) in Kerala early on.

  3. With regard to Aristotle: The issue is not so much Aristotle himself but rather some Aristotelian doctrines as developed by medieval catholic theologians, including Aquinas. One of such doctrines was the doctrine of hylomorphism which, so they claimed, stemmed from Aristotle. Hylomorphism was viewed as the rival doctrine of atomism/indivisibles.

  4. Hylomorphism since at least Aquinas was viewed as a theoretical background necessary for the catholic interpretation of the eucharist. Therefore indivisibles were viewed with great suspicion by some catholic theologians, including jesuit mathematicians such as Guldin, Tacquet, and others.

  5. In a forthcoming article on Cavalieri, we show how religious resistance to his method of indivisibles (a precursor of integration) was the source of much of the opposition to his work and the work of his disciple Stefano degli Angeli. Both Cavalieri and degli Angeli were jesuats. The order was suppressed by papal bull in 1668.

  6. In this sense, it can be said that philosophical and theological doctrines were an impediment to the development of the calculus, where ideas related to indivisibles were needed for the eventual breakthrough by Leibniz and Newton. Such doctrines may have contributed to the delay that you mentioned.

  7. You ask also for a book dealing with these issues. While we don't have a book, there is a number of articles you can consult here.

  8. You ask also about Aristotle and infinity. Medieval scholastics interpreted Aristotle as introducing a distinction between potential infinity and actual infinity, and rejecting the actual sort. This dogma was similarly used against scholars attempting to explore infinitesimal mathematics. Thus, Paul Guldin wrote in his book that Cavalieri's indivisibles involve viewing a plane region as consisting of an actual infinity of parallel lines (Cavalieri himself denied this), and therefore meaningless, and therefore any results obtained by means of indivisibles are false. Cavalieri and degli Angeli were never able to establish a school practicing these methods. James Gregory, who visited degli Angeli in the 1660s, saw his books suppressed in Italy.

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    $\begingroup$ The problem about Kerala School is that it never made it outside Kerala... However other "multicultural achievements" did made it they were just not calculus. $\endgroup$
    – Mauricio
    Jan 18 at 16:40
  • $\begingroup$ Another impediment I see is the geometric-based math, it was not until the development of algebra and algebraic geometry that calculus was able to appear. Abstract analysis would appear later. Reading Newton you can see that calculus was mostly invented based on geometrical grounds (that required as you guess, the Middle Ages developments). $\endgroup$
    – Mauricio
    Jan 18 at 16:43
  • $\begingroup$ @Mauricio, I made a related point about Vieta at a parallel question. However, I wonder how this could be squared with Archimedes' tremendous achievements in area calculations. Apparently he managed to do it without "proper" algebra :-) $\endgroup$ Jan 18 at 16:47
  • $\begingroup$ If Archimedes work on getting areas and approximations of $\pi$ are considered near-calculus then this never stopped, al-Kashi and al-Khwarizmi also continued to get better and better approximations by slicing things into polygons. $\endgroup$
    – Mauricio
    Jan 18 at 17:06
  • $\begingroup$ @Mauricio, I didn't say that Archimedes' work on areas is near-calculus. Note by the way that Cavalieri did not use the Vieta formalism (unlike Fermat and others). It is undeniable that Cavalieri's indivisibles were a major step toward integral calculus. $\endgroup$ Jan 18 at 17:45
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My opinion is decidedly against the current, and I will also write quickly from memory without looking for precise references. So, if the quality of this answer is considered low, I can accept it. Let's start immediately with an important question: who invented infinitesimal calculus? Usually, the common response is: Newton and Leibniz. This would suggest that it was an invention in the air if two scientists (albeit mythical) had arrived at it together. Instead, if the author was only one, one could think of a stroke of genius, a sudden leap into the future. Now, in my opinion, the author is clearly only one: Leibniz. I won't go into the "procedural" details, but the point is that it is too particular, too unstable, too contradictory (yes, contradictory) to be independently produced by two different people; in fact, it took about two centuries to give infinitesimal calculus a satisfactory foundation (first by replacing infinitesimals and infinites with limits, then finding an acceptable definition of a limit). The second important question is: what is infinitesimal calculus? Leibniz explains it in "Historia et origo calculi differentialis": it is a way to represent algebraically the curves that Descartes could not represent with his system. So, the heart of the new system is given by the differential equations that are tasked with describing the curves that we now call transcendental. Finally, we come to the initial question: why did it take so long? In my opinion, it's the opposite: between Descartes and Leibniz only a few years pass (1684 - 1637 = 47), which is a miracle. If Leibniz had never existed, I don't know how mathematics would have developed; perhaps it would have focused on infinite series.

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  • $\begingroup$ This question is relayed to this hsm.stackexchange.com/questions/16120/…. I think the development of mathematics kinda stopped between ancient Greece and desecrates, there was not any major breakthrough that I know in this period. $\endgroup$
    – pie
    Mar 28 at 0:40

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