The desire to be puzzled

Here is a very interesting post on possible alternatives to Cantor’s Transfinite Numbers Theory! Was Cantor’s notion of infinite really “unavoidable”?
The topic is particularly interesting when considered within the general framework of mathematical development. How does mathematics evolve? What “forces” mathematics to take some roads instead of other roads? And how do these choices influence the way in which we model and understand the natural world?

Logitations

A prominent professor in the philosophy of mathematics once told me that the key to writing an attractive philosophy paper is to present the reader with a puzzle. “Give me a puzzle, and I’ll be interested”, he said. As I was surrounded by mathematicians and philosophers of mathematics which were steadily exchanging puzzles, I had no doubt that he was right: mathematicians and philosophers of mathematics like puzzles. But then, mightn’t it be the case that this fondness of puzzles influences much more than just our judgment of a philosophy paper (and our conversations over dinner)? Here’s a crazy idea – or maybe not so crazy – does our desire to be puzzled affect our judgement of a certain foundational mathematical theory?

The foundational mathematical theory which I have in mind is, of course, Cantor’s transfinite set theory. Given its general acceptance nowadays, it is easy to forget that in order to generalize arithmetic from the finite to…

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Yitang Zahng and prime numbers

On the New Yorker, an interesting article on Yitang Zhang, the mathematician who in 2013 proved a theorem in number theory according to which there are infinitely many pairs of prime numbers that differ by 70 million or less. The theorem may not sound really interesting, but actually it is the first proof to establish the existence of a finite bound for prime gaps, resolving a weak form of the twin prime conjecture.

Here you can find the article. Good reading!

The coconut puzzle

Some time ago, I chanced upon the following puzzle:

Five men find themselves shipwrecked on an island, with nothing edible in sight but coconuts, plenty of these, and a monkey. They agree to split the coconuts into five equal integer lots, any remainder going to the monkey.
Man 1 suddenly feels hungry in the middle of the night, and decides to take his share of coconuts at that very moment. He finds the remainder to be one after division by five, so he gives this remaining coconut to the monkey and takes his fifth of the rest, lumping the coconuts that remain back together. A while later, also Man 2 wakes up hungry, and does exactly the same thing: takes a fifth of the coconuts, gives the monkey the remainder, which is again one, and leaves the rest behind. So do men 3, 4, and 5. In the morning they all get up, and no one mentions anything about his coconut-affair on the previous night. So they share out the remaining lot in five equal parts finding, once again, a remainder of one left for the monkey. Find the initial number of coconuts.1

I won’t give you the answer, so you can enjoy finding it by yourself. The solution is not trivial, for you have to solve a Diophantine equation, but in the end it is not too much complicated. Obviously, there is an infinite number of possible solutions, but the problem implicitly asks for the smallest one.

The puzzle, per se, is not very intriguing, but what makes it interesting is a story that usually goes with it. Continue reading