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There was this recent discussion, in which I inadvertently got into, which mixed up contexts and concepts such as irrationality,distribution properties, randomness, and practical randomness.
The other players in the discussion did a good job but sounded somewhat ineffective (?) and theoretical. I thought the person who started the discussion really wanted to have an intellectual one. Not to be. It turned out to be an overwhelmingly abusive experience for reasons beyond my understanding. Overall, quite an unexpected turn of events. And after I posted a first version of this blog, I saw another board
where a few days back someone has asked more or less the same question. This seems to be open season for irrational exuberance.

I got into the discussion hoping that all the people involved will come up with some interesting contexts or references which I may not know. That has not happened either. Anyway, the point is that people seem to confuse distribution properties which are usually proved non constructively with (practical) randomness. Measure theoretic proofs are a darling, but when you cross the street, you would prefer to have Brouwer on your side, not Hilbert or Hubris. We all know what happened to Brouwer due to Hilbert's Hubris. And how Einstein remained a silent spectator in the war of frogs.

Normality as it is technically known is difficult to prove for a given irrational number, a fact acknowledged by Hardy, Kac among others. For example, we have a closed form solution and know the transcendence of all the even values of the Zeta function. A result due to Euler. But it was only in 1970's that Apery proved the irrationality of Zeta(3). We do not know whether it is transcendental or not. We do not know the distribution properties of this number either, because as you know, irrational numbers by definition can be approximated. We have many fast converging continued fraction expansions of Zeta(3). Just because you have a notation for Pi, it does not mean that it sits there grinning on the real number line, well understood and smug in its irrationality and randomness. As many including Marsaglia have pointed out, if we set (for a context) certain qualitative criteria for 'randomness', then the expansions of many rational numbers (in different bases) exhibit far better qualitative properties. Poor, well understood rational numbers. And they are all over the place. Counted in all sorts of ways, under appreciated, over utilized, secretly courted, mostly ignored and thought to be plain and marriageable for most applications. We also know due to Tanguy Rivoal, a few years back proved that there are infinitely many odd values of the Zeta function which are irrational. Period. Normality, randomness anyone?

The way the academic world is operating now, I only see NSF and tree cutting in its wake. And possibly more abuse. Possibly Apery or Rivoal types from the land of Grothendieck or Ben Green types from the land of Hardy or Aggarwal types from the land of Ramanujan, Borwein types from the land of Tutte will have something meaningful to say. Apart from cutting more trees in the process.

This season has been quite churlish to me. I wish I can slunk away in ignorance. Recently I received a query on another forum from a stranger on 'real life applications of Combinatorics'. A well meaning and genuine question, but I felt like ranting and raving. Thankfully, I was reciprocated with silence to my rants. Mercifully, it was not misunderstood and there was no torrent of abuse either. Perhaps, I should post my reply to that question too.

This context brings to my mind one of Rota's reviews in Advances in Mathematics about a book. He wrote ' It must be late in the time of the day if pygmies cast long shadows'. A polemic to answer another polemical book, which he disapproved ( I think). He must have got it from somewhere. But I do feel like a pygmy, and the shadows getting longer with no respite.

Gurumurthi V. Ramanan

Views: 278

Tags: Decimal-expansions, Distribution, Irrationality, Life, Normality, Randomness, numbers, polemics


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