More quotes from Randomness

I have read quite a lot of Randomness, but haven't had time to blog. Instead, I noted the pages of the quotes that I have come across and waited until I had time to blog (now) and type up all of them.

"The first atomist, Leucippus (circa 450 B.C.), said, 'Nothing happens at random; everything happens out of reason and by necessity'. The atomic school contended that chance could not mean uncaused, since everything is caused. Chance must instead mean hidden cause."

"[...] Newtonian physics - a system of thought which represented the full bloom of the Scientific Revolution in the late seventeenth century. [...] a belief developed among scientists that everything about the natural world was knowable through mathematics. And if everything conformed to mathematics, then a Grand Designer must exist. Pure chance or randomness had no place in this philosophy."

The above quotes are reasons to way random cannot exist. When I read these passages, I thought of the domino effect and how every event has a cause, and that cause has its own cause and so on. It doesn't seem correct to say that an event has no cause, and there is no way to understand how it came to be.

Robert R. Coveyou (american mathematician) - "The generation of random numbers is too important to be left to chance."

"Within any sequence generated by the computer through a programmed algorithm or formula, the next digit is a completely deterministic choice, not random in the sense that a dice throw, a spinning disc, an electronic pulse or even the infinite digits of the mysterious pi are random. The very notion that a deterministic formula could generate a random sequence seemed like a contradiction".

This is what I have breifly mentioned before. Surely pure random doesn't truly exist if it is possible to produce a sequence that is identical to a random one.

"Today, primarily three types of generators are in use: (1) congruential generators, which are based on modular arithmetic, or remainders after division,(2) generators which use the binary (bit) structure of computer-stored information, and (3) generators based on number theory."

(1)"Congruential generators use modular arithmetic, or the remainder after division, as the next digit in the sequence. For example, a number mod 7 is replace by the number's remiander after dividing the number by 7."

I have come across modular arithmetic very breifly when I attended one of the mathematics taster days at Queen Mary's University.

More research on congruential generators:
"An LCG generates pseudorandom numbers by starting with a value called the seed, and repeatedly applying a given recurrence relation to it to create a sequence of such numbers. At a glance, the graphs will always look random (except in trivial cases, such as when the modulus is a multiple of the multiplier), but there is actually a sophisticated study of how closely pseudorandom number generators approximate processes that are truly random." - http://demonstrations.wolfram.com/LinearCongruentialGenerators/

"The linear congruential generator (LCG) was proposed by Lehmer in 1949" - http://random.mat.sbg.ac.at/results/karl/server/node3.html

(2) "A new class of number theoretic generators has recently been developed by George Marsaglia and Arif Zaman. [...] Called add-with-carry and subtract-with-borrow generators, their technique relies on the Fibonacci sequence and the so-called lagged-Fibonacci sequence [...]"

I have come across the Fibonacci sequence many times before. The sequence begins with 0,1, and the next number in the sequence is found by adding together the two previous numbers. So, the sequence starts off like so: 0,1,1,2,3,5,8 etc.

"A logged-Fibonacci sequence begins with two enormously large starting numbers, or seeds, instead of 0 and 1.[...] As in the Fibonacci sequence, in the add-with-carry method each new number will be obtained by summing up the two digits previous to it. If the sum is 10 or more, we use the right-most digit only and carry the 1 (to be used in obtaining the next digit.) [...] For instance, beginning with the two initial seeds 0 and 1, we obtain the same beginning of the Fibonacci sequence 0,1,1,2,3,5,8 until we reach 13. Here, the 3 is used and the 1 is carried. The next number in the sequence is obtained by summing the previous two, 8 and 3, and the 1 that was carried, 8+3+1 is 12, so the 2 is used and the 1 is carried."

(3) Number Theory. Deborah J. Bennett does not discuss number theory in the book but I did a bit of research: There are many branches of number theory but I think that the most relevant one to my project is probabilistic number theory:

"In probabilistic number theory statistical limit theorems are established in problems involving "almost independent" random variables. Methods used include a combination of probabilistic, elementary and analytic ideas.
One of the first achievements in this area was the Erdos-Kac theorem, which asserts that properly normalized values of a rather general additive arithmetical function have a Gaussian limit distribution. The determination of necessary and sufficient conditions for such functions to have a limit distribution is an outstanding problem."

I'm not going to do any deeper research into the number theory because the topic has many different branches and I don't think that it will be extremely worth it.


All the above research will help me for when I write my "random number generator" part of my dissertation, because I now understand how these generators work. I can see the advantages and disadvantages of them. I feel as though I am almost ready to write the chapter, which is good because I have been meaning to write it for the past week.

I found a passage about Chaos in the book:
"Chaos theory, the science which predicts that the future state of most systems is unpredictable due to even small initial uncertainties, holds new meaning for the notion of randomness, and simulating these systems requires huge numbers of random digits. It has been shown that with even small deterministic systems, initial observational error and tiny disturbances grown exponentially and create enormous problems with predictability in the long run".

I think this quote summarises chaos theory really well and will be useful to me when I write my chaos chapter of my dissertation.