Alternative Histories for the Emergence of Science

quote
Before embarking upon the answer, I explain how I characterize science for purposes of
the following discussion. I take science to be an enterprise, or more accurately, a group of
mutually supporting like-minded enterprises, that originated in western Europe several hundred
years ago. Both empirical research and mathematical formulation, in varying combinations, are
prominent. Galileo (1564-1642) and Newton (1642-1727) are two recognized landmarks. It is
important that science has turned out to be systematic, comprehensive, cumulative, selfcorrecting, and self-sustaining. At the outset these traits were perhaps largely implicit, but over
the years they have become explicit aims. A key feature is that science continues unbroken
through time. It is not an activity that stops and then has to restart, at least not so far. Since its
beginning science has spread across the earth, and it is now practiced in every country, or nearly
every country, by people from many different backgrounds, traditions, and cultures.
To characterize science this way is not meant to deny the fact of important inquiries
undertaken in many parts of the globe before the commencement of science. Still less is this
characterization intended to demean those researches. In fact, science could not have begun how
it did without many contributions from elsewhere and earlier times. Most likely, without those
contributions science could not have begun at all. However, for purposes of this study, we focus
on the internally connected and cumulative social practice that began in early modern Europe.
end of quote
Where do I begin? OK, here is a big one
quoteOccam's razor is a principle often attributed to … 14th century friar William of Ockham that says that if you have two competing ideas to explain the same phenomenon, you should prefer the simpler one.
end of quote
I.e. the virtue is in the pursuit of simplicity and this has connotations world wide. And I mean WORLD WIDE
Next, was Calculus primarily a EUROPEAN invention? er, not exactly and also we have this one as to how to reconcile what we can do with our MATH.
quote
What does Gödel's incompleteness theorem say?
Can you solve it? Gödel's incompleteness theorem ...
In 1931, the Austrian logician Kurt Gödel published his incompleteness theorem, a result widely considered one of the greatest intellectual achievements of modern times. The theorem states that in any reasonable mathematical system there will always be true statements that cannot be proved.
end of quote
These two oldies but goodies are literally ALL over the globe and they are not exactly European.
Now how does the following "grab you" ?
quote
Between 1600 and 1868, when the nation was largely isolated from outside influences, Japan developed a highly elaborate indigenous mathematics known as wasan. This book is an unaltered reprint of Smith and Mikami's classic account of wasan, first published in 1914. We owe Dover yet another vote of thanks for bringing an important book back into print at a very reasonable price.

The book includes a history and discussion of arithmetic algorithms for the soroban, the abacus which served Japanese society as a powerful calculator, and a number of impressive magic squares and magic circles (think of magic squares in polar coordinates).

However, the focus is on the two great achievements of tenzan algebra and the yenri ("circle principle"), both of which grew out of the work of Seki Kowa and his student Takebe Kenko in the second half of the 17th century. Japan inherited from Chinese algebraists of the 13th century some polynomial techniques and the method for approximating roots of polynomial equations known in the west as "Horner's method." Seki and Takebe adapted these into the powerful tenzan technique for setting up and solving polynomial equations arising from geometric problems. As a famous example, Seki solved the problem of finding the lengths of the sides and diagonals of a quadrilateral given the differences of the cubes of those six lengths, by setting up and solving a polynomial equation of degree 1458.

The yenri was an ingenious kind of proto-integral-calculus, which originally gave infinite series for, for example, the length of the arc of a unit circle cut off by a segment of height h. Later mathematicians like Ajima Chokuyen in the 18th century and Wada Nei in the early 19th century used it to solve problems like finding the arclength of an ellipse or the volume of various intersections of spheres, cylinders and cones.

Smith and Mikami assess wasan as follows: "The work was exquisite in a way wholly unknown in the West. For patience, for the everlasting taking of pains, for ingenuity in untangling minute knots and thousands of them, the problem-solving of the Japanese and the working out of some of the series in the yenri have never been equaled."

Phil Straffin (straffin@beloit.edu) is Thomas White Professor of Mathematics at Beloit College, where he regularly teaches a course on mathematics in other cultures. His survey of Chinese mathematics from the third to the fifth century C.E. appeared as "Liu Hui and the First Golden Age of Chinese Mathematics," in Mathematics Magazine (volume 71 (1998), pages 163-181).

Tags:
History of Mathematics
Non-Western Cultures
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end of quote
Necessity is the mother of invention and one can say that the Japanese had a Proto calculus, as outlined above.
Next, would this mean Japanese society was similar to European ?
OMG are you kidding me?
NOPE
Further investigations reveal that the Babylonians for work on the star maps intuited the idea of a DERIVATIVE and going back to Egypt, the Egyptians prior to 1000 BC had at least the first 10 digits of Pi, worked out

Intuitive common sense would mean that some variant of Occam's razor was world wide, although not necessarily formulated as given above, and that Godel's incompleteness theorem is used, REGARDLESS of if the society involved is European based.

I.e. clever essay but in some ways very Eurocentric

Hi TangerineMandrill

I see you have 6 ratings and so need another 4 to qualify for the next stage of the contest. As do I. Would you like to help each other get across the line by reading and rating each others essays over the weekend?

Cheers
Swan