Conclusion of Reckoning with Risk

While surfing Google I came across this website which may help me later on, when I am writing my section about quantum mechanics:
http://www.faqs.org/docs/qp/chap01.html

Here is an extract of a Reckoning with Risk review, written by Helen Joyce:

"Gerd Gigerenzer is not a mathematician or statistician per se, but primarily a psychologist, working across disciplines to understand how human beings make decisions in the face of uncertainty. What he offers here is nothing less than a prescription for how to think, how to choose, and how to live, when the information on which we base our decisions is necessarily incomplete and flawed. For example - how worried should you be if you have a positive mammogram as part of a screening programme for breast cancer, or a positive HIV test despite the fact that you are in a low-risk group? You may be surprised to learn that the answer may well be "not too worried" - what should really worry you is that not many medical personnel know this!

The book also looks at the way courts deal with uncertainty, and offers some suggestions for improving the handling of statistical evidence such as DNA testimony"

The fact that Gerd Gigerenzer is not a mathematician or statistician may make the book less reliable. The book in general does not seem to talk about randomness, but more about uncertainty. There is a clear difference between the two because something that is uncertain may not be random. However, I did find a lot of information that will help me with my probability section of my dissertation. For example, at the end of the book, there is a chapter called "Fun Problems". The Monty Hall Problem is included in the chapter, along with many problems that I have never come across:
"The First Night in Paradise.
It is the night after Adam and Eve's first day in paradise. Together,they watched the sun rise and illuminate the marvelous trees, flowers, and birds. At some point the air got cooler, and the sun sank below the horizon. Will it stay dark forever? Adam and Eve wonder, What is the probability that the sun will rise again tomorrow?
[...] If Adam and Eve had never seen the sun rising, they would assign equal probabilities to both possible outcomes. Adam and Eve represent this initial belief y placing one white marble (a sun that rises) and one black marble (a sun that does not rise) into a bag. Because they have seen the sun rise once, they put another white marble in the bag. [...] Their degree of belief that the sun will rise tomorrow has increased from 1/2 to 2/3. [...] According to the rule of succession, introduces by the French mathematician Pierre-Simon Laplace in 1812, your degree of belief that the sun will rise again after you have seen the sun rise n times should be (n+1)/(n+2)."

This quote shows a different, unusual way to use probability which may be very useful to my dissertation.

Reckoning with Risk was a very interesting book but I don't think that it really gave me a lot of information that helped me progress in my project. I guess it showed me a more psychological side to mathematics but I do not know how relevant this actually is to randomness. In general, there was barely any information about the concept of randomness but it has shown me that randomness and uncertainty are two very different things.

Reading through Randomness...

The book begins by discussing the misconceptions that occur when probability is used. The cases are similar to that of the ones used in Reckoning with Risk i.e. false positives and negatives, and so I won't use them.

She then talks about the use of randomness in decision making:
"Chance is a fair way to determine moves in some games and in certain real-life situations; the random element allows each participant to believe, 'I have an oppurtunity equal to that of my opponent.'"

I haven't really thought about the role of randomness is something like decision making. The quote suggests that randomness may be the only unbiased way of making decisions in certain situations.

Bennett then moves onto the randomizers that were used in ancient history.
"Though not always recognized or acknowledged as such, chance mechanisms have been used since antiquity: to divide property, delegate civic responsibilities or privileges, settle disputes among neighbors, choose which strategy to follow in the course of battle, and drive the play in games of chance."

It seems as though randomness has always been around, and people took advantage of its unpredictability to make quick decisions about certain events.
Deborah J. Bennett then talks through the various randomizers that were used in ancient times, many resembling the dice that is used in our time. Chapter three discusses religion and the role of God in the concept of random:

"The purpose of randomizers such as lots or dice was to eliminate the possibility of human manipulation and thereby to give the gods a clear channel through which to express their divine will. Even today, some people see a chance outcome as fate or destiny, that which was 'meant to be'"

This quote coincides with one I found in March: "God does not play dice" - Einstein

Both quotes seem to indicate that random does exist but it is an act of God. Randomness, like many other things in the world that humans do not fully understand, is a concept that we must accept and put faith in because it is a part of God.

New GANNT Chart

Since my last post, I haven't managed to get any work done. I feel as though I worked really hard on this project at the beginning of the summer and I'm no longer as motivated as I was before. And the fact that I'm still behind on my GANTT chart is not helping at all:



Here are the problems that I have come across this week (which are the reasons why I have barely made any progress):

I don't know why it is so difficult for me to make progress on reading Chaos. It just seems like such long book with so much context. When I read it, I understand what is being said, however I can't really understand the point that James Gleick is trying to get across. I am only a quarter of the way through the book and this is affecting the rest of my research because I have other books to read. Also, I thought that I would be able to finish reading this book quite quickly but I can't.

I have a list of books that I have not bought yet:

Introduction to random time and quantum randomness - Kai Lai Chung

Quantum Theory: A very Short Introduction - John Polkinghorne

Does God Play Dice? - Ian Stewart

I've realised that I will have to ask for a grant when I school begins. Luckily, the books that I have already managed to buy were no more than £3 each but the cost of the three above books will add up to roughly £35. This will affect my current GANTT chart as I will have to wait until school starts to order the books, and then I will have to wait for the books to actually be delivered to my house. To avoid this, I will search for these books at my local library.

I have not managed to get any opinions yet. I guess I didn't really know where to look; I doubt that important mathematicians will have time to respond to my enquiries. I have realised that the best way to get opinions will be to research different university lecturers (physicians, mathematicians) and email them about my project so I will try to do so this week. I believe that they will be most willing to help me.

I need to begin gathering a large amount of internet sources because at the moment, all my sources are books and although it is good that I have so many books that I will be able to refer to, my research needs to be more varied. So far, my approach towards the internet has been quite half-hearted, so I need to focus on searching and evaluating as many sources as possible.

All in all, I've had an extremely unproductive week. I have decided to update my GANTT chart again. It starts from this week and ends in mid-November. It is much more detailed than the previous GANTT chart and I hope this will allow me to really focus on what needs to be done and not lag behind:


I am going to begin reading Randomness and will come back to Chaos because it is really stopping me from making progress. Hopefully I can get this project back on track now!

My conclusion of Chance by Amir Aczel.

The book Chance is accurate and reliable because Amir Aczel is a lecturer in mathematics and the history of mathematics and science. The book has been very useful to my research into whether randomness exists because it is the first book I have read which clearly explains the role of probability in the subject of randomness. It shows that randomness does exist, but the concept of random itself has been made out to be much more difficult that what it actually is. We can predict the frequency of an event occuring thanks to probabilty.

"Chance and probability play a part in many aspects of our everyday lives. Simple examples include playing the lottery to win millions of dollars and the chance that you will have inclement weather tomorrow. However, Amir D. Aczel, author of Chance: A Guide to Gambling, Love, the Stock Market, and Just About Everything Else shows that chance and probability can be taken to a whole different, much more interesting level.

Aczel's book is only 160 pages long, making it an easy read without much numerical theory and complicated mathematical equations."

This is an extract from a review of Chance that I found at: http://ezinearticles.com/?Book-Review---Chance,-by-Amir-D-Aczel&id=3507603
The review is written by Daniel Breedlove - business owner and engineer.

Another review:
"In Chance, celebrated mathematician Amir D. Aczel turns his sights on probability theory - the branch of mathematics that measures the likelihood of a random event. He explains probability in clear, layperson's terms, and shows its practical applications." What is commonly called "luck" has mathematical roots - and in Chance, you'll learn to increase your odds of success in everything from true love to the stock market."
http://www.goodreads.com/book/show/441215.Chance

This review implies that randomness is can be mathematically understood quite easily. The idea of a random event isn't as mind-blowing as people think it is - the problem is that people are not educated enough about probability.

Chaos

The Lorenzian Waterwheel:

Image from Google - http://t1.gstatic.com/images?q=tbn:ANd9GcQk5Bw3BdGRlboekIxXWw8YvWhsHzVQFK8tM8s7vSRCWEUaOsE&t=1&usg=__CxFNmGp1uPBw86HSTTmeF9oBhEw=
This image is identical to the one in Chaos.

"The rotation of the waterwheel shares some of the properties of the rotating cylinders of the fluid in the process of convection. [...] Water pours in from the top at a steady rate. If the flow of the water in the waterwheel is slow, the top bucket never fills up enough to overcome friction, and the wheel never starts turning. [...]
If the flow is faster, the weight of the top bucket sets the wheel in motion (left). The waterwheel can settle into a rotation that continues at a steady rate (center).
But if the flow is faster still (right), the spin can become chaotic, because of the nonlinear effects built into the system. As buckets pass under the flowing water, how much they fill depends on the speed of the spin. If the wheel is spinning rapidly, the buckets have little time to fill up. [...] Also, if the wheel is spinning rapidly, buckets can start up the other side before they have time to empty. As a result, heavy buckets on the side moving upward can cause the spin to slow down and then reverse."

It is impossible to predict the actions of the waterwheel so in this case, randomness does exist. There are many examples similar to the rotating cylinders and the waterwheel in the book such as pendulums and oscillations. They are all examples of chaos and so I will not find quotes for all of them as they are all talking about the same things.

Updated GANTT chart

It is the 21st of August and it is obvious to see that although I have begun writing my dissertation earlier than planned, I am behind on my GANTT chant. Here are a list of tasks that I will prioritise for the upcoming week:
Finish reading Chaos - I am about a sixth of the way through the book so it is vital that I finish this as soon as possible as I have otherbooks to read.
Collect opinions - I could try to email mathematicians and mathematics teachers/lecturers in order to obtain quotes from primary sources.
Conclude the books that I have already finished reading - Talk about what I have learnt and how this has helped my project.

Introduction.

Although randomness is very common, it is a topic with a lot of loose ends. Whether it’s something as small as the rolling of a die or something a bit more extreme, like lightning during a thunderstorm, random events are all around us. However, the reason to why such things happen in unknown. Is it really possible to not be able to predict an event at all? Can an event really have no pattern, or is the pattern just too complex for the human mind? Perhaps random is just an adjective that we use to describe things that we can’t understand.
The concept of randomness can sometimes be very hard to grasp. It is important to firstly ensure that we understand what randomness actually is. The oxford English dictionary defines random thus:
"having no definite aim or purpose; not sent or guided in a particular direction; mad, done, occurring etc. without method or conscious choice; haphazard". This definition is easy to understand however, it indicates that randomness is subjective. What may seem like a totally random, unpredictable event to one person may be seen as a clear pattern to another. Wikipedia discussed this issue: “Randomness, as opposed to unpredictability, is held to be an objective property -determinists believe it is an objective fact that randomness does not in fact exist. Also, what appears random to one observer may not appear random to another.” This statement shows that my suggestion may be correct; events that seem random may just be patterns that are too difficult for people to understand. It may be possible that in the future, with enough extensive mathematical research, the concept of random can be eliminated but this is highly unlikely.
Probability gives us a way of understanding more about random. It is now easy to make rough predictions. For example, if a die was rolled 300 times, each number should show up roughly 50 times because the probability of a number showing is 1 in 6. However, if one was asked to predict the next number that will come up when rolling the die, it is no longer a simple process. This quote from Wikipedia explains this point more effectively: "Closely connected, therefore, with the concepts of chance, probability, and information entropy, randomness implies a lack of predictability. More formally, in statistics, a random process is a repeating process whose outcomes follow no describable deterministic pattern, but follow a probability distribution, such that the relative probability of the occurrence of each outcome can be approximated or calculated. For example, the rolling of a fair six-sided die in neutral conditions may be said to produce random results, because one cannot compute, before a roll, what number will show up. However, the probability of rolling any one of the six rollable numbers can be calculated, assuming that each is equally likely."
So, random is a word used to describe a series of events that never repeat themselves and cannot be accurately predicted. This dissertation is an investigation into whether pure randomness exists. I will collect evidence from a variety of sources supporting both sides of the argument in order to allow myself to make my own conclusion about whether randomness really exists. This conclusion will be entirely my own opinion, based on my research; I am not trying to prove whether pure randomness exists, I am just hoping to learn enough to allow myself to understand the arguments around it and therefore have my own thoughts about randomness.