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.

A useful website: http://www.fortunecity.com/emachines/e11/86/random.html

I have written roughly 240 words of my introduction so far and it should be finished by tomorrow.

Randomness - Deborah J. Bennett arrived in the post today so I will begin reading that once I have finished reading Chaos.

One more book to look at.

While browsing the internet for sources that may be useful to my project, I came across this book:
Does God Play Dice? - Ian Stewart

This may be a very helpful book to read as it may take into account religious arguments about random which I have not researched yet.

Reading through Chaos...

The book begins by talking about a meteorologist called Edward Lorenz. He created a "toy weather". "He appreciated the patterns that come and go in the atmosphere, families of eddied and cyclones, always obeying mathematical rules, yet never repeating themselves."

"To make the patterns plain to see, Lorenz created a primitive kind of graphics. Instead of just printing out the usual lines of digits, he would have the machine print a certain number of blank spaces followed by the letter a. He would pick one variable - perhaps the direction of airstream. Gradually the a's marched down the roll of paper, swinging back and forth in a wavy line, making a long series of hills and valleys that represented the way the west wind would swing north and south across that continent, The orderliness of it, the recognizable cycles coming around again and again but never twice the same way, had a hypnotic fascination."

"A partiular kind of fluid motion inspired Lorenz's three equations: the rising of hot gas and liquid, known as convection. In the atmoshpere, convectiong stirs air heated by the sun-baked earth, and shimmering convective waves rise ghost-like above hot tar and radiators. Lorenz was just as happy talking about convection in a cup of coffee. [...] How can we calculate how quickly a cup of coffee will cool? [...] Convection in coffee becomes plainly visible when a little cream is dribbled into the cup. The swirls can be complicated. But the long-term destiny of such a system is obvious. [...] the motion must come to an inevitable stop."



"When a liquid or gas is heated from below, the fluid tends to organize itself into cylindrical rolls (left). Hot fluid rises on one seid, loses heat, and descends on the other side - the process of convection. When heat is turned up further (right), an instability sets in, and the rolls develop a wobble that moves back and forth along the length of the cylinders. At even higher temperatures, the flow becomes wilds and turbulent."

I thought the image of the fluid would be useful and so I took a picture of it with my phone. I'd like to find a better picture on the internet.

These quotes are helpful to my project because it is the first evidence I have to argue that pure randomness does exist. Also, fluid dynamics is mentioned so a different side of mathematics is involved, rather than probability which I have mainly come across. I feel as though I am making a lot of progress in terms of collecting differenct sources for my dissertation.

What have I done this week?

Firstly, I researched random number generators to try and understand the mathematics behind them. I came across the website RANDOM.ORG which claimed to be a "True Random Number Service".

"Perhaps you have wondered how predictable machines like computers can generate randomness. In reality, most random numbers used in computer programs are pseudo-random, which means they are a generated in a predictable fashion using a mathematical formula. This is fine for many purposes, but it may not be random in the way you expect if you're used to dice rolls and lottery drawings.

RANDOM.ORG offers true random numbers to anyone on the Internet. The randomness comes from atmospheric noise, which for many purposes is better than the pseudo-random number algorithms typically used in computer programs. People use RANDOM.ORG for holding drawings, lotteries and sweepstakes, to drive games and gambling sites, for scientific applications and for art and music. The service has existed since 1998 and was built and is being operated by Mads Haahr of the School of Computer Science and Statistics at Trinity College, Dublin in Ireland."

So, this shows that when computers generate random number, it is infact not random, as there is an algorithm involved.

I then found out more about pseudo-random:
"A pseudorandom number generator is an algorithm for generating a sequence of numbers that approximates the properties of random numbers. The sequence is not truly random in that it is completely determined by a relatively small set of initial values, called the PRNG's state. [...] Careful mathematical analysis is required to have any confidence a PRNG generates numbers that are sufficiently "random" to suit the intended use."
http://en.wikipedia.org/wiki/Pseudorandom_number_generator

The fact that the algorithm can produce numbers that seem random to anyone that uses it implies that which more research, a random number generator may be created which will produce truly random numbers.

I had a quick read through this website and would like to note it here for futurereference. http://www.scientificamerican.com/article.cfm?id=how-randomness-rules-our-world

A book I came across while browsing on Google: Randomness by Deborah J. Bennett.


I now have a rough idea of the structure of my dissertation.
Introduction: Includes definitions of random and examples of random in everyday life. (approximately 500 words)

Probability: Probability explained and why it implies that random does not exist, and how random events are not as unpredictable as people think. Possible topics to include : reference to Chance by Amir Aczel, gambling, dice throws. (approximately 1000 words)

Short Introduction to Chaos and Quantum Mechanics: Proof to why pure randomness does exist. May also include other topics that I way discover. (approximately 1000 words)

How do some random mechanisms work?: Random number generators and other objects that have been programmed to be "random". (1000 words)

Uncertainty and Unpredictability. How should we cope with random events? How should one go about handling the subject of random? (1000 words)

Conclusion: Do I believe that random really exists? What have I learnt about random by doing this investigation? (500 words)

Hopefully this structure will give me a clearer idea of what I need to look for when I am looking at sources.
This week I will try to finish reading Reckoning with Risk (which is not seeming very relevant to my project at the moment) and maybe try to get primary opinions from mathematicians. I will also find more internet sources.

Here is my updated GANTT chart:


Here are a list of things for me to do for the upcoming week:
Collect quotes from Chaos
Research more books and internet quotes from the internet.
Begin writing the introduction (a very rough draft)

This week I have been working as an intern in an investments firm in London Bridge, so haven't managed to get as much done as I wanted. The only reason I've actually been able to not completely lag behind on my GANTT chart is because the journey to the firm is quite long and so I have time to read Reckoning with Risk. At the end of this week I will evaluate the GANTT chart and set some targets for the upcoming weeks. I feel as though I am doing a fair amount of research but I really want to knuckle down on structuring, planning and writing the first draft of my dissertation.

Introduction?

I have been brainstorming ideas of the introduction of my dissertation. I was to begin with something relatable and simple in order to draw in the reader. Ideas of things I may mention (that I need to research):
How a calculator/computer generates random numbers (possibly there is a complicated algorithm involved, but an algorithm nonetheless)
Children's games involving a random aspect, for example, Buckaroo! and Pop-up pirate.
Picking a coloured ball from a bag of balls of different colours.

I have also realised that the majority of my research is in some way related to probability. Depending on how things turn out, I may have to change the title of my project, perhaps to something along the lines of "Has probability done enough to eliminate the idea of pure randomness?"

Reading through Reckoning with Risk...

Quoted from the blurb:
"However much we want certainty in our lives, it feels as if we live in an uncertain and dangerous world. But are we guilty of wildly exaggerating the chances of some unwanted event happening to us? Are we misled by our ignorance of the reality of risk?"

So far, the book mainly talks about innumeracy, which is the misunderstanding of mathematical concepts. The book is more about misinterpreted probabilities and uncertainty rather than the existance of random, but it is very informative and may become useful to me in the future.

"The following information is available about asymptomatic women aged 40 to 50 in such a region who participate in mammography screening:

The probability that one of these womem has breast cancer in 0.8 percent. If a woman has breast caner, the probability is 90 percent that she will have a positive mammogram. If a woman does NOT have breast cancer, the probability is 7 percent that she will have a positive mammogram. Imagine a woman who has a positive mammogram. What is the probability that she actually has cancer?

Let us try to turn [...] innumeracy into insight by communicating in natural frequencies rather than probabilities:

Eight out of every 1,000 women have breast cancer. Of these 8 women with breast cancer, 7 will have a positive mammogram. Of the remaining 992 women who don't have breast cancer, some 70 will still have a positive mammogram. Imagine a sample of women who have positive mammograms in screening. How many of these women actually have breast cancer?

The information is the same as before (with rounding) and it leads to the same answer. But now it is much easier to see what that answer is. Only 7 of the 77 women who test positive (70+7) actually have breast cancer, which is 1 in 11."

This may be an interesting topic to use in my presentation. It shows that before we can even begin to question randomness, we must be sure that we are certain about probabilities and possible misinterpretation of them. Reading this book has shown me that there is so much more to randomness than just trying to predict what will happen. Many probabilities must be proven and understood, and numerous tests/experiments need to be carried out.

So what next?

I have ordered and received the book "Reckoning with Risk" - Gerd Gigerenzer. I was originally going to read Chaos after finishing Chance. I began the first chapter but for me, it is extremely hard to get my head around. I flicked through Reckoning with Risk and it seemed a bit easier to read, so I am now reading that. I don't want to get stuck in Chaos (and I dont mean 'stuck' in a good way) and end up not being able to find relevant quote to use, as I am too occupied in understanding it. Perhaps after reading a few more books and researching the internet, I will attempt to read Chaos again and it may seem a bit more understandable.

Finished reading Chance

I haven't blogged in a while, so here we go.
I have updated the GANTT chart:


The last few chapters of Chance talk about a variety of things, such as The Birthday Problem, which I meantioned in an earlier blog (about the chances of two people having the same birthday). Amir Aczel also talk about a similar situation - the probability that 2 balls land in the same box, if a number of balls are dropped into a number of boxes at random. There are 2 equations that are stated which I found quite interesting:

"1.2 times the square root of the number of categories gives the number of 'balls' required for even odds that at least two share some characteristic.
And:
1.6 times the square root of the number of categories gives the number of 'balls' required for ninety-five percent probability that at least two share some characteristic."

So, if we go back to the Birthday Problem that I came across at the maths taster day, I can now work estimate:
1) 1.2 times the square root of 365 = 22.926, so 23 people have a 50 percent probability of at least one shared birthday.

2) 1.6 times the square root of 365 = 30.57, so 31 people have a 95 percent probabily of at least one shared birthday.

Now it is easy to see why, in my group of 50 people, there was an extremely high probability (97 percent) that we would find a shared birthday.

The book then goes on to talk about the normal curve:
"De Moivre discovered the "normal law of errors." He found out that when many random factors accumulate, they form a bell-shaped curves, with lesscommon valuse tailing off on either end, and the more average valuse grouping in the middle."
A few probabilities are stated, involving standard deviation, for example, "1. The probabilist that a normally distributed random quantitt will fall within one standard deviation of the mean is about sixty-eight percent."

Here is a picture of the normal distribution curve that I found on google.

And here is the website that the picture if from: http://www.igs.net/~cmorris/index_subject.htm

I already know these probabilities at they are used in the Statistics 1 module of the maths A-level.

The final chapter concludes the book:
"Probability is humanity's attempt to use pure mathematics to understand the un-understandable. It is our way of trying to learn something about the workings of chance. Chance remains forever untamable, for fate does what it wants to us and to the world around us."

This passage shows that Amir Aczel believes that pure randomness does exist, and probability is a way of helping us understand it. Nothing in the book proves or suggests that randomness does not exist, however Amir Aczel indicates that with enough maths, it is relatively easy to be able to predict random events, which therefore make them less random, and more of a pattern.