### Zero

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.

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##### Stuart Manins 19 January 2017, 12:18 PM

I find Bayes' Theorem difficult to explain to others which is frustrating because I think I can follow it from the videos. But I don't feel confident yet with all the steps. Is there any chance of some further explanations, please?

##### David Bell 20 January 2017, 12:45 PM

Like your use of that word chance in this context! I will be working on it over the next few weeks.

##### Stuart Manins 26 January 2017, 7:28 PM

Another surprising video with the Monty Hall problem. I opted for making another choice, though not through predicting a  two out of three chance of success, but because I thought that with only two doors left there was a one in two chance of getting a goat or a car.

##### David Bell 26 January 2017, 9:07 PM [Updated: 26 January 2017, 9:09 PM]

Yes indeed. It is not the expected result. Almost everyone thinks there is 50-50 chance after one door is opened, and so there's no real point in swapping. If you check out some online referencing (wikipedia, for example) it seems that some highly distinguished mathematicians - including Paul Erdös - wasn't initially convinced. It's easy to set up a computer simulation and when he played the game many times it soon came close to a 2/3 chance of success by swapping.

Erdös was not alone  in thinking there was something a bit odd about Bayes' Theorem. Bayesian probability was given a wide birth by the mathematical community for a few hundred years! In recent decades it has become quite central in a number of disciplines but there are different ways of interpreting the results, and deciding the validity of inferences. It requires higher order mathematics to enter this debate.

What I found interesting as we developed the three results was how I had to keep going back to basics to convince myself - I mean I needed the help of diagrams to do it.  Max and Stuart in the forums had little idea how I struggled to express it, i.e. turn the verbal data into visual data.

##### Win Davies 11 February 2017, 11:20 AM

Here is Amanda's paper, well worth looking over.