Proto Calculus

In the first lecture we looked at Proto calculus. It is a calculus which predates even Archimedes. The derivative of f is defined there as Df(x) = f(x+1)-f(x) and the integral is defined as Sf(x) = f(0)+f(1) + ... + f(x-1). We have seen in class that DS f(x) = f(x) and S D f(x) = f(x)-f(0). This is the fundamental theorem of discrete calculus. It is completely analogue to the fundamental theorem of calculus ∫0x (d/dt) f(t) dt =f(x)-f(0) and (d/dx) ∫0x f(t) dt = f(x). You see the 20 second proofs here:
Mathematicians have used tally sticks or pebbles to compute at first. An other name for proto calculus is business calculus or quantum calculus. There are no limits taken yet. You see something about Pebble calculus the movie "The Clan of the Cave Bear". You can find something about the development of arithmetic here:
The concept of limits took a long time to develop. Zeno, the Greek philosopher has thought a lot about it. He had asked himself questions like: if an arrow is at a fixed position at any time, how can it move? A bit more about the history of calculus is on these slides: