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The history of Mathematics shows a peculiar pattern of progress. The development was quite steady but one-sided during the Greek times and then was almost very slow during the Roman and Byzantine times. But, the growth plummeted from the Renaissance and post-Renaissance era in a remarkable manner and currently is growing at an exponential pace.

What could be the philosophical and psychological reasons for such a development? Is it just the ease of access to information that is behind all this? If so, then how do we explain the progress from the post-renaissance to pre-20th century development in Mathematics? Is there a particular method of thought which is the cause?

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    Just a suggestion: the so called "Greek miracle" owes much to the invention of phonetic writing (see e.g. E. Havelock); mathematics explosed once a good symbolic notation developed, that is roughly from Viete to Leibniz with Descartes towering centerstage - geometrical (quasi)objects became secondary to functions.
    – sand1
    Aug 23, 2020 at 21:15
  • @sand1 also useful was the indian or arabic numerals for the help
    – vidyarthi
    Aug 23, 2020 at 22:14

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It's an interesting question. Others will be better able to answer, but I would note that the proper evolutionary model here may be punctuated equilibrium or, more pertinently, Kuhn's "Structure of Scientific Revolutions," with its near-random paradigm shifts.

A truly authoritative work, such as Aristotle's logic or Euclid's geometry will hold sway for centuries, until it accumulates too many failures or complications, in the manner of Ptolemy's astronomy. Then some opening will bring cascading change.

Because of its very formalism, I suspect, mathematics may be unusually dependent on taboos. The Pythagoreans felt the the irrationals menaced the cosmic order, or so the story goes. The Greek mathematicians were constrained by their superstitious rejection of "nothing" or "zero" or negative numbers.

Which is not so unreasonable. Imaginary numbers were resisted for similar reasons and many mathematicians, notably Kronecker, had an almost visceral dread of Cantor's set theory, which seemed to be turning math in pure fantasy. The boundary line between "proper math" and wild conjuring is viewed by many as a slippery slope with almost cosmic implications. And concepts like infinity always raised theological shudders.

So, as I say, strong taboos with punctuated equilibria. Someone dares an idea at the very moment some controversy allows an opening and the field can no longer shut that barn door. Sometimes it seems overdue. It is hard to see why Euclid's fifth postulate lasted so long on a spherical planet. Similarly, Descartes' merging of algebra and the grid seems like it could have come at any time, with practical applications.

After Newton, of course, mathematics was developing alongside physics, so practical possibilities drove things on. My favorite "math taboo" was Berkeley's complaint about the infinitesimals or limit in Newton's calculus, which he described as "the ghosts of departed quantities." In this case, the calculus was just too useful to be derailed by metaphysical niceties.

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    Gillies D., Revolutions in Mathematics, Oxf.UP 1992 is a book of essays which more or less consensually reject the application of Kuhn's model in mathematics. I was tempted to write something about what was called by Koyre and al. "The scientific revolution" but refrained because of the confusion(s) created by Kuhn.
    – sand1
    Aug 24, 2020 at 8:56
  • Thank, I'll look that up, sounds interesting. Frankly, I never really thought about it until considering the question. I'm also a big Havelock fan, so would love to see something similar on math notation. Aug 24, 2020 at 14:03

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