Why the grass is always greener on the other side

Schrodinger, when not experimenting with cats, entertained himself by experimenting on rational men:

Choo and Png met and decided to compare the size of their wallets. Neither man knows how rich the other is. Schrodinger suggests that they play a simple game --- the poorer person gets to keep the contents of the richer person's wallet (along with his own money). Choo, ever so eager to assert himself as his own man, reasons that by playing the game, he stands to lose only his own money, but stands to win more than what he already has. Png, latching on to the same line of reasoning, exclaims "huat ah!" and joins in the fun.

The game should be fair by its symmetry, yet both men decided that it was to their advantage to play it. Who is right? I suppose we should take a vote?

Incestuous Ancestors

Each human being has two biological parents. Consequently, he has four grandparents, eight great-grandparents, sixteen great-great grandparents etc.

Tracing back for 30 generations, one has 2^30, or slightly more than one billion ancestors (at that generational level). The average generational gap is something like twenty years, so those ancestors lived about 600 years ago, in the Middle Ages perhaps. The population of the entire world breached one billion only in the 19th century. This suggests that some inbreeding had gone on, probably unwittingly. To avoid this controversy, one could argue that the one billion ancestors were not all alive simultaneously. But like compound interest, we can see that this exponential growth quickly takes us into perilous territory.

A conservative estimate for the total number of humans who have ever lived is about 100 billion. Over 37 generations, one would have had more ancestors than the number of humans who had ever been alive! But perhaps the models by the historians and anthropologists are unreliable, so let's jack it up to 50 generations, so that each person has more than one quadrillion ancestors. Hardly enough room (or carbon) on earth for that many people! All this without even breaking into the BC era.

Whether or not we evolved from more primitive lifeforms, we can make some conclusions from these considerations. To curb this exponential growth, it seems like we have to either truncate the backward projection (i.e. invoke some sort of "creation"), or accept that the calculated number of ancestors does not represent unique individuals. Unwitting incest becomes exponentially (I think) more likely as we go back in time, so I suppose it acts as a nice counterbalance.

Let's consider a bolder claim: Not too long ago, a woman walked the face of the earth, whose genetic content is in every single person alive today.

I did not make this up. It's actually a widely-accepted claim, and that woman even has the pompous name of Mitochondrial Eve. She was supposed to have lived about 200,000 years ago. Naturally, creationists would be tempted to see this as scientific vindication. The idea of a common ancestor to all living humans intrigued me enough, that I paused to think why this had to be the case. In the end, I was only able to conclude that it could have been the case, so I looked to the literature for a convincing argument.

First of all, we should remind ourselves why this is of any scientific interest. Apparently, each of us has mitochondrial DNA (mtDNA) which is inherited only from the mother. DNA carries our genetic information, which is supposed to give us our characteristics. Mitochondrial Eve, if she had existed, would thus be the Great Grand-mother of humanity today --- her extremely diluted blood would literally be coursing through every human vein today. (But see the wikipedia article for common misconceptions. I'm also not sure whether blood and mtDNA are related.)

How do we know she existed? According to my quick and probably careless research, it is not merely a theory dreamt up by geneticists, but even a simple mathematical fact. This got alarm bells in my head ringing, so I proceeded to look for a proof. It is stunningly elegant. The first thing to note is that each person has only one mother. Consider the set of mothers of all the people living today. This set is smaller than the number of people alive today. Now consider the set of mothers of mothers. This is a smaller set than the previous one. Then consider their mothers, and their mothers' mothers. We arrive at a sequence of sets of decreasing size, which eventually becomes a set with one member --- Mitochondrial Eve herself!

I found a number of problems with this supposed "proof". First of all, the sequence of sets is not required to decrease strictly. It only has to be non-increasing, in which case, we cannot conclude that it will eventually decrease to a singleton set. I've seen a slightly weakened claim in response to this: With near certainty, Mitochondrial Eve did exist. The "proof" of this assumes that there is a non-negligible probability that a mother has more than one daughter (which is probably true). In this case, the sequence of sets has to decrease in size eventually. (But why must it decrease to one? Why not zero???? Or two? Or 1 million???)

Of course, this line of reasoning doesn't explain where Mitochondrial Eve came from. We have nothing to worry about if we believe that she was simply put there by a deity of some sort. Otherwise, we would have to continue this ancestry chasing indefinitely. Dangerous territory. Also, note that there were other women alive alongside Mitochondrial Eve, but their line of daughters didn't manage to persist to the present day. Which prompts two questions, (i) Who was their Mitochondrial Eve? (ii) How likely is it that all of Mitochondrial Eve's contemporaries failed to leave behind a line of daughters over 200,000 years? Despite the fact that Mitochondrial Eve did? (Warning: anthropic reasoning imminent...)

Actually, I discovered that the proof was wrong. It leads to an absurd conclusion. Suppose we arrived at Mitochondrial Eve at the N-th step. That would mean that she is precisely the N-th maternal ancestor of all people alive today. Yes, that means you, your children, and your great-grandparents, (if they are still alive), have the same N-th maternal ancestor.

We abuse the the word "proof" in everyday language to prove all manner of things, which is fine. But the claimant of a mathematical proof ought to bear the full burden of proof.

Mathematics and Power

It's strange... I've only just realized that the two most powerful men in Singapore were once mathematicians. I'd always known this fact for the two men separately, but failed to make the most obvious of connections. Beware of the mathematician... he is not simple!

Panasonic --- now you can't

I don't have many possessions, maybe due to a fear of attachment to worldly stuff. Little did I know that I had been carrying something very personal with me for about ten years --- yes, since my centre-parting JC days.

From the beginning, it had always been there when I needed it, and I reciprocated by lending a listening ear whenever it had something to communicate to me. It accompanied me on long, lonely trips, where it would find a comfortable resting spot right beside my ears, always fitting into a familiar position. When it is with me, all the surrounding buzz and chatter would be drowned out, and all sound turned into music in its presence. There and then, nothing else would matter.

Today, it fell silent. Like all things I've grown to take for granted, I never appreciated it until it was no longer there for me. Few things these days stick around for ten years, not clothes, not cars, not even wives. I'm not sure if I'll ever find anything quite like it again. A quick search suggests that I'd be lucky to find another one that sticks around for even 6 months. A big lump formed in my throat as the memories of our time together came flooding back to me. A wonderful time of Discmans, non-air-conditioned-double-decker buses, and dumb excuses for taking long, windy, and windy bus rides.

Thanks, and sorry for the neglect.

A foray into the real world

When the company of "abstract nonsense" becomes too bewildering, it seems appropriate to return to reality and consider some real-life problems. Well, probably...

So suppose I have two children, and I tell you that the older one is a boy. What is the probability that I have two boys?

1/2 of course. Probability is easy.


Suppose, instead, that I tell you that at least one of my children is a boy. What is the probability that I have two boys?

1/3. Probability is tricky, but still easy.



Just for fun, suppose I told you that my son, Mark, is celebrating his birthday next Tuesday. What is the probability that I have two boys?

1/2. Tuesday is a red herring, surely.



Just for the hell of it, suppose that I tell you that at least one of my children is a boy, and that his birthday is on a Tuesday. What is the probability that I have two boys?

Slightly less than 1/2. Actually, it's 13/27, if I didn't mess up.


Suppose I mumbled, "At least one of my children is a boy, and his birthday is on a @?£$day (...inaudible...)." What is the probability that I have two boys?

Still 13/27.


Finally, suppose I mumbled even more. "At least one of my children is a boy, and he ... (inaudible)..." What is the probability that I have two boys?

Infinitesimally greater than 1/3.


At this point, the classical world just seems plain weird to me. I'm going back to the quantum one.

Slovenian madman

I recently discovered Slovaj Zizek. He's amazing in his biting critique of culture, society and politics, not to mention controversial, but at least, never dumbed down. Certainly you'll have to entertain dangerous thoughts, in order not to see him as a complete basket case.

Among other things, Neils Bohr was a self-styled philosopher, and Zizek reminded me of an interesting anecdote I'd read somewhere sometime ago. A visitor of Bohr was puzzled to see a horseshoe hung in Bohr's home, and asked him why he had displayed an object of superstition in his home. Bohr explained that he was not an idiot, but that he had been told that the horseshoe works... even if you don't believe in it.

Zizek also had a gentle remark about an old movie I'd watched about ten years ago. It's an Italian movie called "La vita e bella" --- Life is beautiful. If you had watched the movie, you'll remember the heart-wrenching scene at the end when Giosue was greeted by the tank that his father Guido had promised him. Guido had created an elaborate game for his son in order to shield him from the realities of the concentration camp they were living in, and ultimately sacrificed his own life in doing so.

Zizek suggested an alternative, and devastatingly sombre double-bluff ending. That Giosue had known all along what Guido was trying to do... and that he was merely playing along so as not to disappoint his father. 

What is the smallest uninteresting number?

You can try to find the answer on Wikipedia. The smallest number without its own wikipedia entry is 224 (as of today). Despite it being the sum of four consecutive cubes (224 = 8+27+64+125), it only manages to share space with the article on the number 220, appearing in the "221-229" subsection.

Surely that's a lousy criterion! Sooner or later, some bored guy behind a computer is going to write up an article on the number 224, following which, 225 would become the smallest uninteresting number. That's despite it being more interesting than 224... after all it is the sum of five consecutive cubes.

It might be better to ask an actual mathematician. G. H. Hardy, the British mathematician who wrote "A Mathematician's Apology", was visiting his advisee, the great Indian mathematician Ramanujan, at a hospital. Being the typical awkward conversationalist, Hardy began by remarking on the dullness of the license plate number of the taxi he was in -- 1729 --- and was hoping that it would not turn out to be an unfavourable omen. To which Ramanujan replied, "No, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways".

I learnt of this lovely anecdote while watching QI, which is short for "Quite Interesting", which is a most interesting show on the BBC where Stephen Fry and four blokes would converse about obscure trivia --- fascinating, funny and most definitely repulsive to the pragmatist. 

According to QI, the "correct answer" is 12407. This sleep-inducer holds the distinction of being the smallest number absent from all 200000 sequences in the Online Encyclopaedia of Integer Sequences. That is to say, it's not interesting enough to appear in any number sequence that mathematicians might be interested in.

But surely, that is interesting? To be the smallest uninteresting number is probably one of the most interesting attribute any number can have! And so we have a paradox. If there were a smallest uninteresting number, then it would in fact be interesting by the mere fact that it is so small and so uninteresting. Therefore, there can't be a smallest uninteresting number.

And so what have we gotten ourselves?

For all the math-haters out there, we have a proof that all numbers are interesting.

(I am of course, referring to natural numbers 1,2,3,... As always, the real numbers are a different beast altogether, and one might be able to argue that there are uninteresting real numbers.)