Zeno and more
Bayes Theorem For Life Decisions
A Second Scenario
Win a Car | A Third and Final Scenario
Music, Number, Mysticism = Pythagoras
At the Intersection of Faith & Mathematics
We are so used to the idea of zero we regard it as a number like any other. But it is not really a natural number. The natural numbers are 1, 2, 3, 4, 5, ... Let's call them the set of natural numbers N
Charles Seife, Zero, the Biography of a Dangerous Idea, cites the great philosopher and mathematical logician A N Whitehead, "The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish."
Dutch mathematicians (and some others) define natural numbers as including zero, but that isn't a widespread practice precisely because 0 isn't like the natural numbers. Zero is much more extraordinary.
Perhaps it's easier and best to call 0, 1, 2, 3, 4, 5, ... the set of whole numbers, in short, W.
And 0, ±1, ±2, ±3, ±4, ±5, ... the integers, I.
Pythagoras and the Greeks loved the Natural numbers, N. They could see relationshis between various numbers and they greatly enjoyed making ratios with the natural numbers. Today we would call those ratios fractions, like ¼ or ½, but in the time of Pythagoras the Greeks didn't have numbers as such. Instead, as we've already mentioned, they had to make do with using Greek letters for numbers. So addition, subtraction, multiplication were very complex to do. Division was truly horrendous. Instead of fractions they used an equivalent concept. This was Ratio.
Ratio was absolutely, truly a fundamental concept. It was a matter of belief that ultimately every single thing in the universe could be measured as a ratio of natural numbers. Then Zeno came along, and undid it all.
Zeno's Paradox: Achilles and the Tortoise
Zeno argued that motion in the universe is an illusion. Everything, he thought, was at rest. He produced a number of paradoxes, some of which have survived, including Achilles and the Tortoise. He introduced the logical method of reductio ad absurdum to generate the paradoxes, but as the video shows, the implicit assumptions are fallacious. In the instance cited, the mathematics of the situation - it is possible to have an infinite series sum to a finite number - is the foundation of calculus. But that is a number story for later.