This is some latest results from ALMA a hugh like hundreds of micrometer wavelength interferometer in South America which should get exciting scientifically.
"Philosophy is harder than mathematics." - Plato
Calculus courses usually start with, more or less, Fermat's tangeant method and the algebraic equivalent. They then work out various general derivative formulas. The Integral is introduced as a sum first and then they point out there's these anti-derivatives. It's a mystery and everyone just says Newton was a genius for noticing and making the connection between the derivative and the integral in the "Fundamental theory of the Calculus." But, John Stillwell, in his "Mathematics and It's History" does some history detective work to show how Isaac Newton knew about this connection.
The Arabs brought the indian idea of zero and place value numerology to recast much Greek mathematics in a simpler form. This new expression of mathematics was more generalizable as well to higher dimensions; but, that is getting beyond even them. Did the Hindu/Arab numerals really make Greek mathematics obsolete? Not really, not until Descartes coordinate geometry. But, as it turns out Descartes had a precursor as well.
Oresme around 1300 introduced coordinates and noticed that distance is the area of velocity. That mere observation is as far as he could take it and then the European 'Black plague' happened. How this bit of information survived and got into Galileo's hands nobody knows(as far as I know!).
The idea of speed as the slope of distance appears to be due to Torricelli(remember him as the guy who solved the water in the mines in James Burke's 'Connections: episode 3'? And, he got the bit about distance as the area of velocity from Galileo. How this bit got to Isaac Newton, I havn't heard(apparently neither has John Stillwell). But it did.
In fact, Isaac Newton got his news from an Isaac Barrow. His claim to fame is that he knew of this and passed it on to Isaac. So that from Italy all the up to England pretty quickly. And so we see how people are finding and combining ideas from previous people. This is no flash in the pan insights. Isaac Newton got his idea for actually computing derivatives from Fermat of course.
I bring this up cause it shows how ideas are spread and combined in weird ways(even if we still can't quite see how ideas are preserved despite plagues and distance . . . including cross culture idea transfers), and to point out something about those initial observations later to be combined and then generalized by Fermat's tangent method. The observations of distance is the area of velocity and velocity is the slope of distance are like Jacob Bronowski's 'inferred units' which I explain in my "Origins of Mathematical knowledge"(third post of this blog). I'll just review the point about sentences; how in a sentence, certain noun words make sense with certain verbs. Here we have certain concepts generated by others and put in sentence form. What we have further is the combining of the two. By seeking to define this creative combination, mathematicians opened up a river of knowledge - differential equations, calculus of variations, complex analyses, and real analyses, integral equations. It all comes out filling in the gap between the creative combining of the two observations.
----------------------science/technology extra
A year or so ago, there was a report that Europe made making memrister a.i. happen as a national priority. Here, we see just one recent news about making memrister chips, Memrister chips within weeks . This is a jump in computing ability that's pretty big; it should be interesting just in the next year to see what affect it has on all of humanity really.
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