I’ve just deleted about 200-300 spam comments that have appeared over the last day or two.  I don’t think I accidentially deleted any actual comments, but if I did, please post again.

Trackbacks (and pingbacks) have now been disabled so there shouldn’t be any more spam and the comment RSS feeds should (hopefully) be safe to subscribe to again.  I’d like to re-enable trackbacks again at some stage, but probably not until I’ve figured out how to block this bulk spam.

Is Math Getting Too Hard?

The Edge Foundation has collected over a hundred essays in response to the question “What is your dangerous idea?“.

I haven’t yet read all of them (75,000 words!), but I thought that Steven Strogatz’s idea was worth mentioning. With reference to the four-colour theorem, classification of simple groups, and sphere packing, he worries that mathematics might be getting too hard, that the use of computer programs in mathematical proofs leaves mathematicians with the ability to show something is true without understanding why.

Obviously the use of computers as an aid in proofs is relatively new. But, is it new that there are results where we dont really understand why they are true? I’ve always thought that on the frontiers things are usually not well understood; but, as the body of knowledge grows, new tools are developed and new insights achieved, and what was hard becomes easier. Computer proofs may have skewed this progression somewhat, but do they signal a more fundamental change? Is it worth speculating whether or not, without computers, mathematicians might have continued working on the four-colour theorem and we might have a “real proof” by now?

I don’t think I’m quite ready to accept the idea that we are now reaching the limits of the human brain.

Aside: Professor Strogatz mentions a recent article by Brian Davies, Whither Mathematics, which talks about similar issues. It also talks about using formal verification of computer programs when they are included in a mathematical proof. Until now I’ve not paid much attention to such things, but I guess that if mathematical proofs are requiring computer programs then we’ll need techniques to verify their correctness so they can be verified like more traditional proofs.

Another aside: Not Even Wrong and Cosmic Variance have some comments about a few of the physics related Edge essays.

Ponder This

It’s the beginning of the month and the solution to last month’s Ponder This challenge is up, as well as the puzzle for August:

For K as large as possible, produce a K-digit integer M such that for each N=1,2,…,K, the integer given by the first N digits of M is divisible by N.
An example is K=4, M=7084, because 7 is divisible by 1; 70 is divisible by 2; 708 is divisible by 3; and 7084 is divisible by 4.

I guess that the largest K is around 28.

Ponder This

IBM Research has put up the July challenge on their Ponder This site.

Update: Here is the problem:

Upon a rectangular table of finite dimensions L by W, we place n identical, circular coins; some of the coins may be not entirely on the table, and some may overlap. The placement is such that no new coin can be added (with its center on the table) without overlapping one of the old coins. Prove that the entire surface of the table can be covered completely by 4n coins.

Symbolic Dynamics

Here is an introduction to Symbolic Dynamics or, if you like books, there is Symbolic Dynamics and Coding.

One of the things that makes symbolic dynamics useful/interesting are Markov Partitions. A Markov Partition is a finite partition of the state space for a dynamical system. For a point x in the state space, look at it’s orbit and write down which element in the partition each point of the orbit belongs to. This infinite sequence of partition elements uniquely defines x. Doing this for all points in the state space yields a mapping from the dynamical system to a shift space. For more details, see the paper Symbolic Dynamics and Markov Partitions.

Infinitude of Primes

For an alternative to Euclid’s proof, there is Furstenberg’s topological proof of the infinitude of primes. The MAA‘s Mathematics Magazine published a variation of Furstenberg’s proof which does not use topology:

For each prime p, let Sp = pZ. Each Sp is periodic (it’s characteristic function is periodic). Let S be the union of all Sp. If S is the union of finitely many periodic sets, then S is also periodic. However, the complement of S is {-1, 1}, so S is not periodic. Hence, there must be infinitely many sets Sp, and infinitely many primes.

Tippe Top

Tippe Top The mathematical model for a tippe top is a sphere with an uneven concentration of mass along the vertical axis, making the lower half heavier than the upper half; a physical model is more practical if you cut the top off the sphere and replace it with a stem. In each case, the key feature is that the centre of gravity is lower than the centre of the sphere. When the top spins, with the help of friction, it slowly tips over — raising the centre of gravity — until it is upside down.

For a history of the tippe top and an overview of how it works see this page. You can also watch an animation of the tippe top in action.

If you want to really understand why it inverts, Richard Cohen was the first to provide a rigorous explanation in The Tippe Top Revisited Am. J. Phys. 45, 12 (1977). Or, if it’s important that you also know why it falls back down as it runs out of spin, see this paper.