The Origins of Greek Mathematics
Though the Greeks certainly borrowed from other civilizations, they built a culture and civilization on their own which is
Basic facts about the origin of Greek civilization and its mathematics.
The Sources of Greek Mathematics
In actual fact, our direct knowledge of Greek mathematics is less reliable than that of the older Egyptian and Babylonian mathematics, because none of the original manuscripts are extant.
There are two sources:
The Greeks wrote histories of Mathematics:
The Major Schools of Greek Mathematics
Thales
The Major Schools of Greek Mathematics
Zeno's Paradoxes Zeno constructed his paradoxes to illustrate that current notions of motion are unclear, that whether one viewed time or space as continuous or discrete, there are contradictions. They are
Dichotomy. To get to a fixed point one must cover the halfway mark, and then the halfway mark of what remains, etc.
Achilles. Essentially the same for a moving point.
Arrow. An object in flight occupies a space equal to itself but that which occupies a space equal to itself is not in motion.
Stade. Suppose there is a smallest instant of time. Then time must be further divisible!
Now, the idea is this: if there is a smallest instant of time and if the farthest that a block can move in that instant is the length of one block, then if we move the set B to the right that length in the smallest instant and the set C to the left in that instant, then the net shift of the sets B and C is two blocks. Thus there must be a smaller instant of time when the relative shift is just one block.
The Eleatic School
How to draw a trisectrix: Imagine a radial arm (like a minute hand of a clock) rotating at uniform speed about the origin from the vertical position to the horizontal position in some fixed period of time. (That is from 12 O'clock to 3 O'clock.) The tip of the arm makes a quarter circle as shown in red in the picture. Now imagine a horizontal (parallel to the x-axis) arm falling at uniform speed from the top of arm to the origin in exactly the same time. The trisectrix is the intersection of the two arms. The curve traced in black is the trisectrix. As you can see, the trisectrix is a dynamically generated curve. The Platonic School and those subsequent did not accept such curves as sufficiently ``pure" for the purposes of geometric constructions.
Pythagorean forerunners of the school, Theodorus of Cyrene and Archytas of Tarentum, through their teachings, produced a strong Pythagorean influence in the entire Platonic school.
``O King, for traveling over the country there are royal roads and roads for common citizens; but in geometry these is one road for all."
The School of Aristotle
Aristotle set the philosophy of physics, mathematics, and reality on a foundations that would carry it to modern times.
He viewed the sciences as being of three types -- theoretical (math physics, logic and metaphysics), productive (the arts), and the practical (ethics, politics).
He contributed little to mathematics however,
...his views on the nature of mathematics and its relations to the physical world were highly influential. Whereas Plato believed that there was an independent, eternally existing world of ideas which constituted the reality of the universe and that mathematical concepts were part of this world, Aristotle favored concrete matter or substance.
Aristotle regards the notion of definition as a significant aspect of argument. He required that definitions reference to prior objects. The definition, 'A point is that which has no part', would be unacceptable.
Aristotle also treats the basic principles of mathematics, distinguishing between axioms and postulates.
Aristotle explored the relation of the point to the line -- again the problem of the indecomposable and decomposable.
Aristotle makes the distinction between potential infinity and actual infinity. He states only the former actually exists, in all regards.
Aristotle is credited with the invention of logic, through the syllogism.
Aristotle 's influence has been immeasurably vast.