Mast Kalandar

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Wed, 14 May 2008

Understanding Large Numbers

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It is very difficult for us (humans) to understand really large numbers. Here are some examples.

Time and space

There are those who say that the complex ecology of the earth could not have "just evolved" and must be the result of "intelligent design". Clearly, these people do not understand the ya...wning gap of time over which the process of evolution has taken place or how even random walks can end up far from the mean over such periods. Of course, there is also the question of whether what has been "designed" is "intelligent" at all!

Similarly, it is difficult to understand "how mind-bogglingly large the universe really is".1 If these people were to understand the low density of the universe and the weakness of the force of gravity, they would be attributing the creation of planetary systems to "intelligent design" or a "supreme being".

When you realise the above two things it becomes clear that the "human condition" is about the here and now. It is about solving problems faced by us humans on this planet. It is not a pholosophical debate about the aeons or the universe or the "supreme being".

Large populations

Even thinking of the population of the earth as "us humans" is really difficult.

India (and China) have really large numbers of people --- most of whom are invisible to those in the west. The people who have become well-off in the post-liberalisation era in India do represent the same size as a large European country; hence their relatively recent well-being is not to be sneezed at. However, an order of magnitude larger number of people remains.

Unlike some others, I am not sure that the better-off Indians have become so at the expense of their counterparts in the west --- it is equally likely that they have expanded (literally!) at the expense of their fellow citizens and rural neighbours.

The Laws of Physics

As Feynman said in his essays on Physical Laws, we are relatively lucky to live in a "temperate zone" of physics. This allowed us to discover the concept of a "physical law". In other zones chatoic, relativistic or quantum-mechanical effects would have prevented us from seeing any order in our surrounding universe. Conversely, living in this "temperate zone" makes it difficult for us to imagine the really large, really small or really unstable.

The discovery of mathematics too depends on the evidence of order and symmetry in and around us. So Feynman's "temperate zone" is important for mathematicians as well.

"Weak" Security

Then there are those who read that RSA is 1-bit insecure and go "I'm switching to something else tomorrow". A back-of-the-envelope calculation shows that if each atom in the universe were a CPU more powerful than the current processors and if all these computers have been trying (since the big bang) to enumerate all numbers from 0 to 1023-bits, they wouldn't have got there yet.

Bruce Schneier once said that if the weakest aspect of your computer's security is the weakness of RSA/MD5/Whatever2 then your system is really quite secure.

Understanding largeness

So this is the nature of mathematics (and most of our knowledge gathering). Take the really large (read "infinite") realm of all that is "out there" and re-formulate it in terms of nuggets that can be digested in terms of our capacity for creating and accessing symbols.

There is a sequence of numbers which grows so fast that it cannot be bounded by any sequence "constructed" inductively! In fact, the only proof that there is such a sequence involves transfinite induction. This sequence places limits on how much of the infinite we can really capture.

  1. A quote from The Hitchhiker's Guide to the Galaxy by Douglas Adams.

  2. Place your latest theoretically attacked cryptosystem component here.

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