Since my college days I hear about an on-again off-again debate on whether mathematics is developed (invented) or discovered.

Did Newton build his theories off of the research of others (discovery)?
Or did he create a whole new way to apply what was already known (developed)?

Much like the physical sciences, it's always been there since the start of the universe, but it takes humans a while to discover it (like graphite). Once discovered, it takes a different kind of mind to do something with it (like graphene).

--Scott
www.aerornd.com

Quote (Isaac Newton)

If I have seen further it is by standing on ye sholders of Giants

So... What's there to debate? Newton said it himself.

Quote:

(Isaac Newton)

If I have seen further it is by standing on ye sholders of Giants

I guess my question is: who are the "giants"? (And how big of a role did they play?)

I can name Kepler and Galileo, and their role was... gigantic

Archimedes might have realized the calculus if he’d had access to a number system that included zero. When/where/how depends on lots of things. It takes being the right person at the right place at the right time. If the plague hadn’t sent Newton back to country side at just the right time would he have produced the same work at a different time? Or was the plague an essential component?

Mathematics, I think, is a combination. It's not like you're going to find the rules of differentiation and integration cast in stone somewhere. Someone had to work out the rules, like the basic finite difference equation (f(x+dx)-(f(x))/dx, the limit of which is df(x)/dx.

But, all of that requires advanced algebra, and calculus is incomplete without trigonometry.

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Quote (DavidBeach)

Archimedes might have realized the calculus if he’d had access to a number system that included zero.

He may have been able to do some parts of calculus without having a zero... They had a pretty good understanding of triangles, similar triangles, slopes and ratios.

Dik

I don't think Archimedes had sufficient algebra to derive the basic derivative.

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Quote (IRS)

I don't think Archimedes had sufficient algebra to derive the basic derivative.

Agreed... but he was involved with proportions and slopes... the latter being fundamental derivative...

Dik

The word "algebra" is from Arabic. We also use Arabic numerals. Many mathematic and astronomical terms have Arabic roots.

Note that while the Greeks didn't have a symbol for zero, they did understand the concept of infinitesimals, which is part and parcel to doing a derivative. Zeno's Paradox is part of the reasoning of what infinitesimals do. But, clearly, they weren't at the point of connecting geometric slopes to abstract mathematical slopes, again, since they didn't have anything close enough to algebra to do a "find x" problem.

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From the wikipedia article on Rene Descartes,

"Current opinion is that Descartes had the most influence of anyone on the young Newton, and this is arguably one of Descartes' most important contributions. Newton continued Descartes' work on cubic equations, which freed the subject from the fetters of the Greek perspectives. The most important concept was his very modern treatment of independent variables."

The cartesian coordinate system (or orthogonal coordinates if you like) is a pretty useful idea underpinning calculus.

You don't think that would stop them... they likely had a workaround... anyone that could measure the circumference of the earth...

"anyone that could measure the circumference of the earth"

That was a straight proportions problem, neither trig nor algebra required.

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I remember in pre calculus being told to calculate the volume of a sphere by slicing .

B.E.

You are judged not by what you know, but by what you can do.

My Xmas reading was this excellent book. It covers the history of calculus too. A good read.

e: The Story of a Number
Eli Maor

An epic voyage of discovery, spanning the centuries and still very relevant today. Starring all the greats, including: Napier, Archimedes, Fermat, Galileo, Kepler, Laplace. The battle between Newton and Leibniz. And a late appearance from perhaps the greatest player of all: Euler.
(Not about MDMA)

Steve

IRS: the reasoning was there...

Steve "e: The Story of a Number, by Eli Maor"

That book has been on my Amazon 'Math' wish list for about a year and a half, with me impatiently waiting for the 'hardcover price and condition gods' to smile upon me. Your post prompted me to check the availability again, and now a cheap and 'very good' copy is on the way. Well timed reminder.