Almost the end

I've nearly finished writing my dissertation. Today I looked at the feedback sheets that the audience gave me after my presentation and discussed the results. I have tried to evaluate my sources throughout the project but I need to conclude all my reseach (this will probably be a part of my evaluation instead of a separate section).
TOMORROW:
Write up a report type document evaluating the whole project - pros and cons, things I could have improvedon etc.
I need to print out all of my GANTT charts and a bibliography/webliography (is webliography a word?)
And I think that might be it!!

Woow....it's been hard work but I'm so glad I did this project. I'll miss blogging every other day.

Evaluation of Does God Play Dice?

Does God Play Dice? by Ian Stewart is a book that discusses the mathematics of chaos. He was professor of mathematics at the University of Warwick, so I knew that the book would be reliable to useful to my project.

I found the book helpful because Ian Stewart talks about Chaos in a way that allowed me to have a clearer understanding of it. I also learnt about the connection between Quantum Theory and Chaos Theory and so, I now have more evidence to support the argument of randomness existing in the world.

6 days to go

I have 6 days until all my work must be submitted and I think that I'm going to abandon my GANTT chart. The GANTT chart has helped me alot during this whole project, but now I'm just going to list the remaining things that I need to do...and do them!

SO:
Write Chaos/Quantum Theory section
Write Unpredictability section
Conclude all remaining sources
Conclude my research
Write up the the audience's evaluations of my presentation
Proof read/correct any parts of dissertation

Presentation

I did my presentation yesterday evening. It went okay. Members of the audience filled out an evaluation sheet so I could use this in my own evaluation. I could possibly represent the results on pie charts...?
I haven't blogged much because I was focusing on the presentation. Really need to prioritise writing the essay, as the deadline for the project is the 8th december.

Enough of reading for now...

I have 4 days to make my presentation, seeing as it is on Monday. I don't really know what I've been doing that has stopped me from already planning it, but that's life, I guess.

I want my presentation to be as interactive as possible. Clarity is extremely important too, as my topic can get a bit complicated and difficult to explain. I am debating whether to make my presentation a summary of my dissertation, or if it should be more about what I have done to get to my presentation. I may try to ask Mr Wright for advice, seeing as I missed the meeting about extended project presentations when I had a uni interview.

VERY rough structure of my presentation and helpful sources to go with it:

Introduction to myself and my project.
Should talk about why this topic interests me, what I wanted to get out of it. Structure of my project?

What is randomness?
"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."
Breif history about randomness?

How I structured the project
My aims, Blog, GANTT chart etc.

What I've researched? (this will be a big part of my presentation)

Probability, Quantum Theory, Chaos, Random number generators... basically all the different parts of my dissertation. Examples of each of these concepts. Reference to books etc.

What I've learnt?
Conclusion. My personal opinion on randomness after doing this project.


OKAY. I definitely need to get cracking on the presentation now.

Does God Play Dice?

"When you roll a die, any one of the six possible faces may end up on top. The result of throwing a die is like an eigenstate - it is a special state that is selected by the measurement process. [...] A Copenhagenist would say that the presence of a table mysteriously causes a die to 'collapse' to one of the states 1, 2, 3, 4, 5, 6, an that the rest of the time it is in a superposition of those eigenstates - which is, of course, a mathematical fiction with no intrinsic physical meaning. Bohm would say that it does have a physical meaning but you can't observe it - at least with any conventional apparatus, and perhaps not at all. Diosi-to-Percival would say that as the die rolls along the table its state randomly jiggles, and eventually settles down to one of 1, 2, 3, 4, 5, or 6. [...]
The imgae of dice suggest that they might all be right -
- and all wrong.

Chaos teaches us that anybody, God or cat, can play dice deterministically, while to naive onlooker imagines that something random is going on. The Copenhagenists and Bohm do not notice the dynamical twists and turns of the rolling die as it bounces erratically but deterministically across the table-top. [...] Diosi-to-Percival notice the erratic jiggles of the die [...], not realising that underneath they are actually deterministic.
Nobody tries to write down the equations for a rolling die. [...] One good reason is that they think it can't be done."

This extract helps me to understand the different ideas around chaos and quantum theory. It is also the first extract that has made me see a connection between the two concepts. It isn't possible to determine the state of a die before it lands on a number - this is the same idea of a decaying nucleus. Only when it actually happens, can we determine its state.

The last part of this extract supports the point that I made about randomness just being events that we cannot mathematically model because of its complexity.

Reading through Does God Play Dice?

After yesterday's meeting, I have begun to crack down on reading the remaining books and specifically picking out the parts that I believe will be most relevant to me. Chapter 16 of "Does God Play Dice?" by Ian Stewart is called "Chaos and Quantum" and may be exactly what I need to help me with understanding these 2 concepts for my project.

Ian Stewart discusses the 'cat in a box' experiment. I came across this experiment in the book Quantum by Jim Al-Khalili but I don't think I blogged about it:
"Imagine a box that contains a source of radioactivity, a Geiger counter to detect the presence of radioactive particles, a bottle of (gaseous) poison, and a (live) cat. These are arranged so that if a radioactive atom decays and releases a particle, the then Geiger counter will detect it, set off some kind of machinery that crushes the bottle and kill the unfortunate cat. From outside the box, an observer cannot determinw the quantum state of the radioactive atom; it may either have decayed or not. So [...] the quantum state of the atom is a superposition of 'not decayed' and 'decayed' - and so is that of the cat, which is part alive and part dead at the same time. Until, that is, we open the box. At this instant the wave function of the atom instantly collapses, say to 'decayed', and that of the cat also correspondingly collapses instantly to 'dead'.

This extract has helped me understand quantum theory a bit more. It shows that it is impossible to know what state something is in, and therefore it can be said that they are in both states.

Today's supervision

My concerns before the meeting:
Whether I will have enough time to finish reading all my books and finish the essay before the deadline.
I missed the meeting about the presentation because I had an interview with Manchester university, so I wanted to ask if I missed anything important.

What we talked about during the meeting:
If I really feel like I am running out of time, I should choose to read chapters of books that I believe will help me, and the conclusion.
I may have been getting confused between the presentation date and the final deadline for my project. I do not need to have finished my project by the 29th (my presentation).

Things I learnt from the supervision:
Forget about the essay right now-focus on reading the books.
Begin preparing the presentation.

Supervision has been rescheduled to Thursday.

RIGHT...

I spoke to Mr Wright yesterday and the date of my presentation is the 29th November. My supervision was supposed to be last Tuesday but it got rescheduled to today, in about 2 hours time.

Here's my updated GANTT Chart:


It is obvious that I need to read the rest of the books, as they are what is stopping me from being able to write my dissertation.

So, reading through Quantum Theory:

John Polkinghorne talks about Paul Dirac, an English theoretical physicist who was one of the founders of quantum mechanics, and one of his lectures on quantum theory.

"He took a piece of chalk and broke it in two. Placing one fragment on one side of his lectern and the other on the other side, Dirac said that classically there is a state where the piece of chalk is 'here' and one where the piece of chalk is 'there', and there are the only two possibilities. Replace the chalk, however, by an electron and in the quantum world there are not only states of 'here' and 'there' but also a whole host of other states that are mixturs of each possibilities - a bit of 'here' and a bit of 'there' added together.

Quantum theory permits the mixing together of states that classically would be mutually exclusive of each other."

He then goes on to describe the double slit experiment, which I have already come across in Quantum by Jim Al-Khalili.

So far this book is really helping me to understand the basic concepts of quantum theory. It is a world of the unknown - which is evidence to why randomness exists.

Reading through Quantum Theory

The book begins with the history of the discovery of quantum theory. John polkinghorne talks about the roles of different professionals in this discovery. This includes people such as Planck, Einstein, Young, maxwell and Bohr. Although this is interesting, I don't think that the history of quantum mechanics is terribly important in my dissertation. What is relevant is why quantum mechanics is evidence that randomness does exist.

I realised that my GANTT chart says that I should be reading An Introduction to Quantum theory, not Does God Play Dice?. So I have begun reading that instead. My next supervision is on Tuesday.

Writing "The history of randomness"

I have begun to write this section. I have collected most of the relevant quotes and sources - now I just need to sort them out.

Here is what I have so far (At the moment, it's pretty much just a bunch of sources)
The debate of whether randomness exists has been around for a very long time. The Chinese were perhaps the earliest people to formalize odds and chance 3,000 years ago. Chance mechanisms have been used since antiquity and people took advantage of its unpredictability to make quick decisions events such as settling disputes among neighbours and choosing which strategy to follow in the course of battle. In her book, Randomness, by Deborah J. Bennett states that "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 extract coincides with Albert Einstein’s famous quote – “God does not play dice”. To humans, it may seem as though there is no explanation for a random event, but in reality, it is just too complicated for our minds to understand.

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.
Another quote from Randomness:"[...] 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."
Another quote: "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."

Wikipedia - "Some theologians have attempted to resolve the apparent contradiction between an omniscient deity, or a first cause, and free will using randomness. Discordians have a strong belief in randomness and unpredictability. Buddhist philosophy states that any event is the result of previous events (karma), and as such, there is no such thing as a random event or a first event.

A quote by Jim French (physics PhD) : "The concept of being able to predict and describe the future behaviour of anything we wanted dates back to the 18th century and is typically associated with the French mathematician Pierre-Simon Laplace and something called Laplace's Demon, which is some hypothetical creature that he thought of as (in principle and possibly way into the future) possessing sufficient knowledge of all the positions and speeds of all particles in the universe as to be able to perfectly predict the entire future evolution of the universe. This concept of causal or Newtonian determinism seemed unavoidable at the time, though it had some uncomfortable questions for the nature of free will, until the beginning of the twentieth century, when it ran into the twin problems of quantum theory and chaos theory. The latter presents no conflict with the principle of determinism, but it does in a practical sense. The physics of chaotic systems are still governed by underlying deterministic processes, but what was realised in the middle of the last century was that putting any knowledge of initial conditions into an actual prediction was far harder than previously realised. A small lack of precision in our knowledge of an initial state can quickly lead into huge uncertainty in the state of the system at some later time."

A quote from wikipedia: "In ancient history, the concepts of chance and randomness were intertwined with that of fate. Many ancient peoples threw dice to determine fate, and this later evolved into games of chance. Most ancient cultures used various methods of divination to attempt to circumvent randomness and fate.

. The Greek philosophers discussed randomness at length, but only in non-quantitative forms. It was only in the sixteenth century that Italian mathematicians began to formalize the odds associated with various games of chance. The invention of the calculus had a positive impact on the formal study of randomness. In the 1888 edition of his book The Logic of Chance John Venn wrote a chapter on "The conception of randomness" which included his view of the randomness of the digits of the number Pi by using them to construct a random walk in two dimensions.

The early part of the twentieth century saw a rapid growth in the formal analysis of randomness, as various approaches for a mathematical foundations of probability were introduced. In the mid to late twentieth century ideas of algorithmic information theory introduced new dimensions to the field via the concept of algorithmic randomness.

Although randomness had often been viewed as an obstacle and a nuisance for many centuries, in the twentieth century computer scientists began to realize that the deliberate introduction of randomness into computations can be an effective tool for designing better algorithms. In some cases such randomized algorithms outperform the best deterministic methods."

I have just emailed my suppervisor to arrange another meeting somtime next week.

Probability

Here are the 954 words that make upthe section of my dissertation titled "Probability".

I'm not 100% satisfied with it, but it'll have to do for now. There are too many tasks that need to be done.


“Probability is humanity's attempt to understand the uncertainty of the universe,to define the undefinable. A probability is a quantitative measure of the likelihood of a given event." - Amir D. Aczel

In this section, I will talk about how probability helps us to understand the concept of randomness.

Sometimes, when I think of a random event, I imagine something that is totally impossible to predict or understand. Probability shows that some random events are not as unpredictable as people assume. In July, I attended a maths taster day at Queen Mary University of London, called “Inevitable patterns in mathematics”. There were about 50 of us at the taster day. The lecturers, Ben Green and Imre Leader,called out all the months of the year and asked us to raise our hand when he said the month that our birthday was in. The most popular month was September, so they made all the students who had birthdays in September stand up. This included me. They then asked us what date in September we were born in, and if no one else shared that birthday, the person would sit down. When they came to me, I stated that my birthday was on the 17th of September and funnily enough, another girl who was standing up said that she was also born on that day.
I was very surprised that out of a group of 50 people, it was possible that two people shared the same birthday. As Imre Leader later explained, it was almost certain that two people would share the same birthday: The probability that there is a repeated birthday in a group of 50 people is 1-(365/365)(364/365)(363/365)(362/365)...(316/365) which is around 0.97.
Amir Aczel also talk about a similar situation in the book, Chance- 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.

Now, let us take the classic example of a rolling of a die. Although one can not predict the next number to show up on the die, we know that after 60 rolls, each number should show up roughly ten times. This is because each number on the die has a 1 in 6 probability of being shown. Obviously, this doesn’t mean that every number will actually appear 1 in 6 times. There is a chance that if a die was rolled 60 times, the number 1 would appear every single time, but the probability of that happening is very small, so we know that this event is extremely unlikely. To conclude, it is possible to roughly predict the amount of times a number should appear on a die if the die has been rolled many times. However, it is impossible to predict what the next outcome on the die will be. Or is it?

I asked some physics and maths graduates: “Do you believe with enough research, one will be able to predict a random event (e.g. the outcome of a dice roll) in the future?” Laura Wherity, a maths graduate, answered: “Rolling a die may appear to be random, but in fact it depends on your starting conditions. For example, if you could control the experiment such that the die is always rolled from the same height, at the same angle with the same forces etc. then it should be possible to achieve the same outcome each time. What would appear to be random actually depends on the starting state. Extending this idea, it may be possible to control the starting conditions of other events aswell, so in this sense events that appear 'random' at present may become more predictable in the future as we understand the conditions in more detail.”

Jonathan Wright, another mathematics graduate, said “If we roll a dice in exactly the same way 100 times, 100 times it would give us the same result. If we model the roll of the die, given the starting conditions we could predict the outcome every time.”

So technically, rolling a die isn’t a random event. It is our lack of knowledge of the mechanics of the die which forces us to call it random, although it actually isn’t. Theoretically, it may be possible to one day understand all the conditions of a random event such as a dice roll, and therefore predict the outcome. This may be able to be applied to other such events, and therefore it will be proven that randomness does not exist.


"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."

I feel as though I had something else to say, but I can't remember it...

Anyway, TASKS FOR THE WEEKEND/UPCOMING WEEK:
Read Does God Play Dice
Finish writing Random Number Generators and/or The History of Randomness
Email supervisor tomorrow about next meeting.

Back from half term

I didn't manage to blog during the half term because I found myself with a lot of other work to do. My maths admissions text for Oxford univeristy was this morning, so, with that out the way, I can now try to focus on this project. During the half term I managed to begin writing the probability AND history of randomness section of my dissertation. I don't think I am going to read Chaos - I have already got many quotes from it and I am quickly running out of time.

I need to arrange a supervision with my supervisior soon, probably next week.

I've just realised that I have forgotten to put "Read Choas" in my new GANTT chart. I am already fully aware of the fact that I may not have enough time to read all of the books. I have decided that, because I have so many quantum books to read, I will not read the second Quantum book (by Manjit Kumar) and that slot on my GANTT chart will be dedicated to reading Chaos.

Next week is half term and so hopefully I will have a lot of time to do extended project work. However, my Oxford maths admissions test is on November the 3rd (2 weeks time) so I will be doing a lot of preparation for that next week. Luckily I dont't have too much school work to do so I will make extended project a main priority.

Probability section

I have begun collecting all sources and quotes in preparation for writing the Probability section of my dissertation.

Evaluation of Randomness

Deborah J. Bennett is assistant professor of mathematics at Jersey City State College, New Jersey. This shows that she is definitely a reliable source. Reading this book has helped me with my project in many ways.
Firstly, before reading the book I did not know much about the history of randomness, but now I do. This encouraged me to do more research on the history of the deabte of randomness, which I did not think of doing before.

Bennett made all the mathematical concepts (probabilities) very easy to understand and quite enjoyable. I liked how Bennett investigated a wide range of ideas about randomness. There were quotes about probability, random number generators and other topics that will be extremely useful to me when writing my dissertation.

"In this very entertaining little book, simply written but intended for careful readers, some of the most common mistakes people make about chance are carefully analyzed." - J.A.Rial, American Scientist

"Deborah J. Bennett's book is a useful survey of an often misunderstood topic. Randomness is deceptively complex -- in particular, as Bennett points out, because aspects of it are counterintuitive. " www.complete-review.com

I will definitely be using quotes for this book in my dissertation.

More from Quantum

"Consider a million identical radioactive nuclei that are unstable and will, sooner or later, spontaneously 'decay' by emitting a particle and changing into a more stable form. While quantum mechanics enables us to calculate something called the half-life (the time after which half of the nucleus will have decayed)it cannot tell us when any particular nucleus will decay. [...] We can calculate the probability that a nucleus will have decayed after any given time, but the fact that we cannot do any better that this is not due to our ignorance.[...] What we are lacking is a deeper understanding of Nature whereby we are able to predict exactly when any given nucleus might decay, just a fuller knowledge of all the forces involved in a toss of a coin would allow us to predict its outcome."

This is an example of quantum mechanics. This implies that although theoretically, if we knew everything about the conditions of the atom, we could predict when it would decay, it is impossible for us to actually know the conditions. It could then be argued that randomness does exist.

What am I doing right now?

At the moment, I am reading Quantum by Jim Al-Khalili. I am also writing the Random Number Generators section of my dissertation. I don't seem to have much to really say about number generators, despite the research I have done on them. I have typed up roughly 300 words, and don't really know where to go with it. I think I will leave this section for a while and begin another part, because it is stopping me from progressing.

I was just about to update my GANTT chart when I realised that I have nothing to add to it. I want to redo my GANTT chart again, but I feel as though I shouldn't keep doing this. The wholepoint of a GANTT chart is that it is there for you to stick by it, but if I keep changing it whenever I fall behind, I'm not really progressing, am I?

Having said that, I really think that a new GANTT chart is what I need right now. But I promise that this will be the last GANTT chart that I make!!



I am fully aware that I have a lot of books to read in a short amount of time. I will aim to always be ahead of this GANTT chart.

Things I need to do this week:
Evaluate Randomness
Finish reading Quantum - Jim Al-Khalili

"Isaac Newton believes that every particle in the Universe should obey simple laws of motion subject to well-defined forces. This mechanistic view - one that was still shared universally by scientists and philosophers more than two centuries later - states that no matter how complex the workings of nature are, everything should be ultimately reducible to interactions between the fundamental building blocks of matter. [...] if we could know the precise position and state of motion of every particle in a given system, no matter how many are involved, then we should be able to predict, through Newton's laws, how these particles will interact and move, and hence how a system will look at any given time in the future. [...]

Of course in practice such determinism is impossible for all but the simplest systems."

This is similar to what I have researched previously; randomness may just be a lack of knowledge. It could be a pattern that is just too complex for us to be able to understand right now.

Really interesting example of quantum mechanics from the book

This is quite a long extract so I'll try to cut it down to the most relevant parts. There are some really useful diagrams that I may take pictures of and blog them because they explain this very well. I cannot upload photos onto this blog via my mobile though, so will have to do it another time.

"First, a beam if light is shone on a screen with two narrow slits in it that allow some light to pass through to a second screen where an interference pattern is seen. This is a sequence of light and dark bands that are due to the way the separate light waves emerging from the two slits spread out, overlap and merge before hitting the back screen."

I remember being shown this experiment in a gcse science lesson.

"Next, a similar experiment is carried out using sand. This time the second screen is placed below the one with the slits and gravity does the work. As the sand falls onto the first screen, separate piles gradually build up on the lower one beneath the two slits. This is not surprising since each grain of sand must pass through one or the other of the two slits; we are not dealing with waxes now and there is no interference. The two piles if sand will be if the same height provided the two slits are of the same size and the sand is poured from a position above their mid-point."

"Now for the interesting part: repeating the trick with atoms. A special apparatus - let us call it an atomic gun for want of a better name - fires a beam if atoms at a screen with two appropriately narrow slits. On the other side, the second screen is treated with coating that shows up a tiny bright spot wherever a single atom hits it. [...] First, we run the experiment with just one slit open. Not surprisingly, we get a spread of light spots on the back screen behind the open slit. [...] Next, we open the second slit and wait for the spots to appear on the screen. If I asked you now to predict the distribution if the bright spots that build up you would naturally guess that it would look like the two piles on sand. [...]

Instead, we see an interference pattern of light and dark fringes just as we did with light. [...]
With a detector in place that records which slit each atom passes through, the interference pattern disappears. It is as though the atoms do not wish to be caught in the act of going both ways at once, and only travel through one slit of the other. Two bands form on the screen adjacent to the slits as a result of particle-like behaviour, similar to what happens with the sand.

With the detector turned off we now have no knowledge of the route taken by each atom. Now that their secret is safe, the atoms revert to their mysterious wave-like behaviour and the interference pattern comes back!"

I find this so fascinating! I dont even know what to say in order to evaluate this extract.

"How can we assess the legitimacy or truth of an account of a phenomenon that we can never, even in principle, check? As soon as we try, we alter the outcome. [...] Physicists have been forces to admit that, in the case of the double slit trick, there is no rational way out. We can explain what we see but not why. However strange you may find the predictions of quantum mechanics, it must be emphasized that it is not the theory - mankind's invention - that is strange, but rather Nature herself that insists on such a strange kind of reality on the microscopic scale".

My laptop has been sent off to get fixed and hopefully will be back on Monday. I need it back as soon as possible because although I can blog through my mobile, I haven't been able to finish the "random number generators" section, which is on word. Also, I've just realised that there could be a chance that the people who are fixing the laptop may have to reboot it. If this happens, I will lose a lot of my extended project work such as my GANTT charts. Virtually all of my research and work is in this blog but I need to GANTT charts in excel form, not just a print screen which is what they are in this blog. Anyway, hopefully this isn't going to happen.

I am reading through "Quantum" but Jim Al-Khalili.
"Quantum mechanics is remarkable for two seemingly contradictory reasons. On the one hand, it is so fundamental to our understanding of the workings if our world that it lies at the very heart of moat of the technological advances made in the last half a century. On the other hand, no one seems to know exactly what it means!"

"Quantum mechanics accurately predicts and explains the behaviour of the very building blocks of matter - not just the atoms, but the particles that make up the atoms - with incredible accuracy. It has led us to a very precise and almost complete understanding of how subatomic particles interact with each other and connect up to form the world we see around us, and of which we are of course a part."

Jim Al-Khalili is a physicist and has been studying quantum mechanics for over twenty years. The quotes that I find in this book should therefore be very reliable.

Supervision

I had a meeting with my supervisor a couple of hours ago. I discussed where I am in terms of my GANTT chart. I have now completed my ucas application and although I will be preparing for the Oxford maths admissions test, I should still be able to get some extended project work done. My supervisor said that I have an impressive range of sources but I need to make sure that I evaluate all of them and talk about how they all relate to each other. In order to fulfil the criteria for the assessment objectives, I need evidence to show that I have used technology. This could mean using Microsoft excel for my GANTT chart, using the Internet and discussing how a website works, for example RANDOM.ORG, the random number generator.
From this supervision, I learnt that I am on the right track at the moment and just need to make sure that I motivate myself to read all the books and write my dissertation.

I have begun reading "quantum theory: a very short introduction"
I doubt that I will be making much progress in my project during the next few weeks because I will be preparing for the maths admissions test for Oxford university. Doing this and also keeping on top of school work may mean that I will have to not do as much extended project work as I would like to. I'll jus have to see how it goes. I need to arrange another meeting with my supervisor because the last meeting did not take place.

Half way through Random Number Generators

Here is what I have so far:
"One day, I was sitting in my maths class fiddling with my calculator when I came across a button called Ran#. I realised that if I typed in a number and pressed this button, the calculator would randomly produce a number between zero and the number that I typed. But how could this be? Calculators use algorithms to perform functions so does this mean that there is an algorithm to produce random numbers? If so, random must not exists because the fact that there is an algorithm to produce a random sequence completely undermines the concept of randomness.

After some research, I came across a website called Random.org which claimed to be a “True Random Number Service”. The website discussed how computer and calculator programs use “pseudorandom number generators” which is less effective than this website because they use the randomness from “atmospheric noise”. Wikipedia defines atmospheric noise as “radio noise caused by natural atmospheric processes, primarily lightning discharges in thunderstorms”. Pseudorandom number generators are algorithms that are not entirely random but produce a set of numbers that pass all tests for randomness. So, there are certain properties that a sequence must have for it to be random. The fact that people are aware of this and it is possible to create sequences that could be classed as random shows that although one cannot predict a random sequence, a sequence can be made that is identified as random.

American mathematician Robert R. Coveyou stated that “The generation of random numbers is too important to be left to chance.” This is an incredibly intriguing quote. "

I'm finding it very difficult to word things well in this section. Also, I'm not too sure of the message that I actually what to get across in this section. I have all the quotes and sources but what do I actually want to acheive?
Anyway, as I keep saying, HOPEFULLY this will be finished soon. I have a meeting with my supervisor in about half an hour so will blog about how that goes soon.

Writing "Random Number Generators"

I am just writing about algorithms and came across a relevant quote that I found a while ago. The quote itself doesn't matter at the moment, but the website that it is from is called scholarpedia.org.uk. I decided to Google the website just to see if it had any connection to wikipedia and the actual description for scholarpedia is:

"Scholarpedia feels and looks like Wikipedia -- the free encyclopedia that anyone can edit. Indeed, both are powered by the same program -- MediaWiki. Both allow visitors to review and modify articles simply by clicking on the edit this article link.

However, Scholarpedia differs from Wikipedia in some very important ways:

Each article is written by an expert (elected by the public or invited by Scholarpedia editors).
Each article is anonymously peer reviewed to ensure accurate and reliable information.
Each article has a curator -- typically its author -- who is responsible for its content.
Any modification of the article needs to be approved by the curator before it appears in the final, approved version."

This implies that scholarpedia is a more accurate and reliable version of wikipedia. For future references, I will be using scholarpedia rather than wikipedia to look up things.

Atmospheric Noise (a method of generating random numbers)

http://en.wikipedia.org/wiki/Atmospheric_noise

"Atmospheric noise is radio noise caused by natural atmospheric processes, primarily lightning discharges in thunderstorms.

Atmospheric noise is mainly caused by cloud-to-ground flashes as the current is much stronger than for cloud-to-cloud flashes. On a worldwide scale, eight million lightning flashes occur daily. This is about 100 lightning flashes per second.

The sum of all these lightning flashes results in atmospheric noise. It can be observed[1] with a radio receiver in the form of a combination of white noise (coming from distant thunderstorms) and impulse noise (coming from a near thunderstorm). The power-sum varies with seasons and nearness of thunderstorm centers.

Atmospheric noise and variation is also used to generate high quality random numbers. Random numbers have interesting applications in the security domain."

I have just borrowed the remaining three books from the school library! They have allowed to to keep them until December (when the Extended Project will end). Hopefully I have everything I need to just work and work on my project and finish everything.
I am in the school library right now, and my lesson starts in 10 minutes but I will try and write more of the Random Number Generator section now.

HOPEFULLY I will be able to get it finished soon!

Right...

Up to date GANTT chart:



OKAY.
Tasks for this week:
1) Write Random Number Generators section.
2) Don't read the quantum physics books just yet - focus on starting to read Chaos again (as I stopped reading this over the summer so I could get other tasks done).
3) Take out Does God Play Dice?, Quantum Theory and Introduction to random time and quantum randomness from the school library.

So I haven't managed to get much done this week, which is extremely disappointing. My laptop is still not working and it seems as though I haven't had time to anything other than schoolwork. On top of that, I've been doing my university application that needs to be submitted in a couple of weeks. As a result of this, my extended project has not been high in my list of priorities. I have begun reading Quantum (well, the chapter that seems most relevant) but there's my much to comment on right now. On a brighter note, my school library have bought the remaining books that I need.

Tomorrow, my plan is to go to muswell hill library and use their computer to write the random number generators section and update my GANTT chart (which I am definitely behind on). I hope to get this project back on track and really work hard to get all the research done.

Plan for my dissertation.

Because all my quotes and sources are scattered around this blog, I think it would be helpful to have one blog with all my sources so far, each libked to a section of my dissertation. This will make it easier for me when it comes to writing each section because I won't miss anything out.

The history of randomness - why is it important?:

Randomness by Deborah J. Bennett: "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."
Another quote from Randomness:"[...] 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."
Another quote: "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."

Wikipedia - "Some theologians have attempted to resolve the apparent contradiction between an omniscient deity, or a first cause, and free will using randomness. Discordians have a strong belief in randomness and unpredictability. Buddhist philosophy states that any event is the result of previous events (karma), and as such, there is no such thing as a random event or a first event.

A quote by Jim French (physics PhD) : "The concept of being able to predict and describe the future behaviour of anything we wanted dates back to the 18th century and is typically associated with the French mathematician Pierre-Simon Laplace and something called Laplace's Demon, which is some hypothetical creature that he thought of as (in principle and possibly way into the future) possessing sufficient knowledge of all the positions and speeds of all particles in the universe as to be able to perfectly predict the entire future evolution of the universe. This concept of causal or Newtonian determinism seemed unavoidable at the time, though it had some uncomfortable questions for the nature of free will, until the beginning of the twentieth century, when it ran into the twin problems of quantum theory and chaos theory. The latter presents no conflict with the principle of determinism, but it does in a practical sense. The physics of chaotic systems are still governed by underlying deterministic processes, but what was realised in the middle of the last century was that putting any knowledge of initial conditions into an actual prediction was far harder than previously realised. A small lack of precision in our knowledge of an initial state can quickly lead into huge uncertainty in the state of the system at some later time."

A quote from wikipedia: "In ancient history, the concepts of chance and randomness were intertwined with that of fate. Many ancient peoples threw dice to determine fate, and this later evolved into games of chance. Most ancient cultures used various methods of divination to attempt to circumvent randomness and fate.

The Chinese were perhaps the earliest people to formalize odds and chance 3,000 years ago. The Greek philosophers discussed randomness at length, but only in non-quantitative forms. It was only in the sixteenth century that Italian mathematicians began to formalize the odds associated with various games of chance. The invention of the calculus had a positive impact on the formal study of randomness. In the 1888 edition of his book The Logic of Chance John Venn wrote a chapter on "The conception of randomness" which included his view of the randomness of the digits of the number Pi by using them to construct a random walk in two dimensions.

The early part of the twentieth century saw a rapid growth in the formal analysis of randomness, as various approaches for a mathematical foundations of probability were introduced. In the mid to late twentieth century ideas of algorithmic information theory introduced new dimensions to the field via the concept of algorithmic randomness.

Although randomness had often been viewed as an obstacle and a nuisance for many centuries, in the twentieth century computer scientists began to realize that the deliberate introduction of randomness into computations can be an effective tool for designing better algorithms. In some cases such randomized algorithms outperform the best deterministic methods."


Probability:

Laura Wherity: "Rolling a die may appear to be random, but in fact it depends on your starting conditions. For example, if you could control the experiment such that the die is always rolled from the same height, at the same angle with the same forces etc. then it should be possible to achieve the same outcome each time. What would appear to be random actually depends on the starting state. Extending this idea, it may be possible to control the starting conditions of other events aswell, so in this sense events that appear 'random' at present may become more predictable in the future as we understand the conditions in more detail. Another good example of this are the weather models used for predicting the weather in forecasts. The better we can get at determining the initial conditions, the better our models will become. Of course in certain situations there may be a limit to the accuracies involved, and thus exact predictions or modelling may not be possible."

Jonathan Wright: "As an applied mathematician, all physical situations can be modelled mathematically, and as such we can predict all possible outcomes. If we roll a dice in exactly the same way 100 times, 100 times it would give us the same result. If we model the roll of the die, given the starting conditions we could predict the outcome every time."

Randomness - Deborah J Bennett
Also Chance and Reckoning with Risk

Short Introduction to Chaos and Quantum Mechanics:

Jim French: "However, in quantum theory (at least in the Copenhagen interpretation), it is meaningless to speak of a property of a particle (such as its position) before we go in and measure it. The particle is not sitting there, waiting for us to shine a light on it, revealing its location. All we can talk of is the probability of observing it in one place and not another. The Heisenberg Uncertainty Principle is related to this concept and it states that our certainty in predicting a particle's velocity is limited by our certainty in measuring its position (and vice versa). The more precisely we know where a particle is, the less precision we can have in knowing how fast it is moving. To be clear, this is not an engineering limitation, something that will be overcome in a hundred years’ time with improved technology; it is a fundamental property of nature. Before we have made a particular measurement, it is meaningless to talk of a particle’s position etc, since such properties simply do not exist. This has implications for predictability and randomness, since if a particle’s position (or velocity etc) does not objectively exist, it is impossible to predict precisely what that position will be measured to be and what the subsequent evolution of a system of particles will be."
Another quote: "...quantum mechanics gave predictions which couldn’t be explained by any locally real theory (that is, any theory which pictured particles as having objectively real, well-defined properties). Various experiments have demonstrated that nature does indeed obey the rules of quantum physics and we must therefore adopt this peculiar view of nature based probability and abstraction, rather than concrete realism."

Jonathan Wright: "Quantum theory on the other hand, may also appear to be random, but similarly I think it is just not fully understood. We may not know the exact position of electrons in an atom, so instead we give electrons a 'probability' of being in certain positions or states. This doesn't mean that the electrons are in a random place, just that we are unable to observe their exact position. (In fact, and here is where you should ask a physicist, I think the very process of looking into an atom changes the states of the electrons..So we dont know.) But does this make it random?"

Randomness: "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"

Wikipedia: "According to several standard interpretations of quantum mechanics, microscopic phenomena are objectively random. That is, in an experiment where all causally relevant parameters are controlled, there will still be some aspects of the outcome which vary randomly. An example of such an experiment is placing a single unstable atom in a controlled environment; it cannot be predicted how long it will take for the atom to decay; only the probability of decay within a given time can be calculated. Thus, quantum mechanics does not specify the outcome of individual experiments but only the probabilities. Hidden variable theories are inconsistent with the view that nature contains irreducible randomness: such theories posit that in the processes that appear random, properties with a certain statistical distribution are somehow at work "behind the scenes" determining the outcome in each case."

Chaos by James Gleick:
"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."

John Polkinghorne - Quantum Theory: A very Short Introduction
Introduction to random time and quantum randomness - Kai Lai Chung
Quantum: A guide for the perplexed - Jim Al-Khalili.
Quantum - Manjit Kumar


Random number generators:

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

Randomness - "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".

http://www.scholarpedia.org/article/Algorithmic_randomness: "Algorithmic randomness is the study of random individual elements in sample spaces, mostly the set of all infinite binary sequences. An algorithmically random element passes all effectively devised tests for randomness."


Uncertainty and Unpredictability.

A quote from Jim French: "Essentially, yes, in principle, if we knew enough about all the degrees of freedom (at its finest point, the positions and momenta of all particles in the system, though I suspect this could be done on a classical level, without recourse to quantum considerations), we could predict the result. Practically speaking, though, no. We would either need to set up such a precisely controlled system or know so much about the system under consideration, that it would be impractical and/or would require the power of a supercomputer that has better things to do with its time. At the very least, predicting the behaviour of a die with well-measured properties would actually be pretty trivial and it tells you little about whether or not truly random things do actually exist."

Jonathan Wright: "However, with both the roll of the dice or in predicting the weather, it is this 'knowing' of the starting conditions which creates the randomness that we experience I every day life. In the weather models, if your temperature measurement is off by 0.01 degrees, eventually, perhaps in hours, days or weeks time, the predictions made by the model will become drastically different from those you experience. In fact, this was how chaos was discovered; a seemingly well understood piece of theory, when run on a computer on two occasions, gave two drastically different answers with seemingly the same starting values. The difference was attributed to a difference in the 6th decimal place of the starting values.."

Randomness: "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.'"

Reckoning with Risk

Conclusion

Okay, I didn't manage to include every single relevant quote in this, but I have written the titles of books that I may need to refer to.
Making this plan has made me feel a lot more confident about writing my dissertation.

Things to do this week:
Update GANTT Chart
Write Random number generator section of dissertation!!!

Final reply to the email

Yesterday, Jim French replied to my email:

"Hi Zainab,

Randomness is difficult to define. With regard to your first question, I think it's fair to say it's not clear that it is a proper question - that is to say, if we equate randomness with unpredictability, then of course we wouldn't be able to predict any random event (by definition). If you mean could we predict the behaviour of systems typically described as random (such as the roll of a die), the answer would be... it depends. Essentially, yes, in principle, if we knew enough about all the degrees of freedom (at its finest point, the positions and momenta of all particles in the system, though I suspect this could be done on a classical level, without recourse to quantum considerations), we could predict the result. Practically speaking, though, no. We would either need to set up such a precisely controlled system or know so much about the system under consideration, that it would be impractical and/or would require the power of a supercomputer that has better things to do with its time. At the very least, predicting the behaviour of a die with well-measured properties would actually be pretty trivial and it tells you little about whether or not truly random things do actually exist.

The concept of being able to predict and describe the future behaviour of anything we wanted dates back to the 18th century and is typically associated with the French mathematician Pierre-Simon Laplace and something called Laplace's Demon, which is some hypothetical creature that he thought of as (in principle and possibly way into the future) possessing sufficient knowledge of all the positions and speeds of all particles in the universe as to be able to perfectly predict the entire future evolution of the universe. This concept of causal or Newtonian determinism seemed unavoidable at the time, though it had some uncomfortable questions for the nature of free will, until the beginning of the twentieth century, when it ran into the twin problems of quantum theory and chaos theory. The latter presents no conflict with the principle of determinism, but it does in a practical sense. The physics of chaotic systems are still governed by underlying deterministic processes, but what was realised in the middle of the last century was that putting any knowledge of initial conditions into an actual prediction was far harder than previously realised. A small lack of precision in our knowledge of an initial state can quickly lead into huge uncertainty in the state of the system at some later time. Plenty of physical systems are not chaotic and (fortunately) perfectly predictable, but this is not so for many other systems (such as the weather, where even with huge computing power at our disposal, we struggle to make good, accurate and specific predictions more than a week ahead).

The most important concept at play in the existence (or lack thereof) of true randomness is quantum theory. Causal determinism assumes a realist view of the world - objects in it have definite, objective properties that are true regardless of our having measured or observed those properties (the moon does exist and it is it not made of cheese and this remains true whether or not I choose to taste a mouthful). However, in quantum theory (at least in the Copenhagen interpretation), it is meaningless to speak of a property of a particle (such as its position) before we go in and measure it. The particle is not sitting there, waiting for us to shine a light on it, revealing its location. All we can talk of is the probability of observing it in one place and not another. The Heisenberg Uncertainty Principle is related to this concept and it states that our certainty in predicting a particle's velocity is limited by our certainty in measuring its position (and vice versa). The more precisely we know where a particle is, the less precision we can have in knowing how fast it is moving. To be clear, this is not an engineering limitation, something that will be overcome in a hundred years’ time with improved technology; it is a fundamental property of nature. Before we have made a particular measurement, it is meaningless to talk of a particle’s position etc, since such properties simply do not exist. This has implications for predictability and randomness, since if a particle’s position (or velocity etc) does not objectively exist, it is impossible to predict precisely what that position will be measured to be and what the subsequent evolution of a system of particles will be.

Of course, it could be objected that this is only according to quantum theory and that theory may be incorrect. Indeed, the theory was not (and I suppose is still not) uncontroversial and its most famous detractor was Albert Einstein. He helped found the subject, but came to reject the theory as it was developed and pursued his own independent (and ultimately unsuccessful) line of research. He disliked the interpretation of nature as probabilistic at heart and famously declared “God does not play dice”. He developed various different thought experiments to try to show that quantum mechanics, as formulated at the time, was incomplete and led to contradictions and paradoxes. None of these convinced the mainstream, but one of the most intriguing was called the Einstein-Podolsky-Rosen (EPR) paradox. The details of the proposal aren’t important here, but they led a British physicist called John Bell to formulate Bell’s theorem. This showed that quantum mechanics gave predictions which couldn’t be explained by any locally real theory (that is, any theory which pictured particles as having objectively real, well-defined properties). Various experiments have demonstrated that nature does indeed obey the rules of quantum physics and we must therefore adopt this peculiar view of nature based probability and abstraction, rather than concrete realism.


In answer to your questions, then: yes, randomness does exist in nature and it is found in quantum processes. Radioactivity, for example, is governed by quantum physics. There is simply no way to predict when a radioactive nucleus will decay and it may be considered genuinely random.

There are all sorts of books out there that deal with the subjects of randomness, chaos theory and quantum physics. The standard popular exposition of chaos theory is Chaos by James Gleick. A good recent book that deals with randomness is The Drunkard’s Walk: How Randomness Rules Our Lives by Leonard Mlodinow. Having not read any pop science books about quantum theory for years, I can’t give many recommendations, but any library or book shop will be stuffed with many that all cover the same ground. One that was particularly important to me as a teenager (though it is about a particular aspect of quantum physics rather than a general introduction) was QED: The Strange Theory of Light and Matter by Richard Feynman. You just need to google things like the EPR paradox and the uncertainty principle to find out about them (and they are fascinating subjects). Wikipedia has large articles on them.

If you have any more questions, I’d be happy to answer them."

So, to conclude, all the graduates believe that for most random events, it is possible to predict the outcome. However, there are concepts such as chaos and quantum theory in which the outcome cannot be predicted and so pure randomness is present.

Great news! and some bad news.

I've finally finished reading randomness! The final chapter is called "paradoxes in probability" and was about the probability related problems that I have come across before. This included the probability of two people sharing the same birthday and the monty hall problem.

The next book that I am going to read is Quantum by Manjit Kumar. This is the first book that I've read about quantum mechanics so it should be interesting.

Unfortunately, my laptop is broken and I'm not sure when it will be fixed. Luckily I can blog via my mobile, but a lot of the work that I do now will have to be done in a library. This is the main reason why I haven't finished writing the "random number generators" section of my dissertation yet. This will be a priority for this week though.

Another reply to my email.

Yesterday Laura Wherity, a maths graduate, replied to the email I sent out:

"Question 1:

Rolling a die may appear to be random, but in fact it depends on your starting conditions. For example, if you could control the experiment such that the die is always rolled from the same height, at the same angle with the same forces etc. then it should be possible to achieve the same outcome each time. What would appear to be random actually depends on the starting state. Extending this idea, it may be possible to control the starting conditions of other events aswell, so in this sense events that appear 'random' at present may become more predictable in the future as we understand the conditions in more detail. Another good example of this are the weather models used for predicting the weather in forecasts. The better we can get at determining the initial conditions, the better our models will become. Of course in certain situations there may be a limit to the accuracies involved, and thus exact predictions or modelling may not be possible.



Question 2:

I liked your comment that randomness is subjective - this may well be true, mainly depending on people's understanding of the models. This links into the comments in the previous paragraph - the idea of the randomness of rolling a die depends on how well informed a person might be. To some it may be random, to others it may be predictable.

An area of maths associated with randomness is chaos theory - a book that has been recommended (although I have not read it) is James Gleick - chaos. I am afraid I do not know much about chaos, although the basics of it are that a small change to the starting conditions can start to spiral out of control and lead to large changes. The book appears to be an overview of ideas regarding to chaos, with some maths in it although only a small amount.


I hope some of this helps.


Good luck with the project,

Laura Wherity"



The "Question 1" section to Laura Wherity's email is similar to that of Jonathan Wright's. They both believe that a random event can be calculated if we know all the conditions of the situation. This seems very logical. In the Mechanics 1 module of my Further Maths AS level, I learnt a number of equations that involve the conditions of an object in which you can find out things about it such as its mass or speed. If we have enough information, we can mathematically figure out the way in which an event will happen.

Laura Wherity then recommends the book Chaos by James Gleick. I have already bought that book, so at least I know that my bibliography is on the right track. Basically all of my books will be relevant to my project, which is reassuring.

What have I done today?

I have just discovered that I can blog via my mobile phone! This is the first blog that I am doing in this way and I can tell that this will definitely make it easier for me to document my project.

Today I gave Mr Wright my grant proposal. He talked to the librarians in my school and they are going to buy the books that I listed so that I can borrow them and after that, they will be available to all students in my school. The books should be arriving next week, which will mean that I will be behind on my GANTT chart once again. Hopefully I can get some more stuff done this week to make up for waiting for the remaining books.

I began writing the "random number generators" section to my dissertation today. So far so good, should be done by Friday if I really push myself.

Updated Bibliography

BOOKS

oxford english dictionary
John Polkinghorne - Quantum Theory: A very Short Introduction
Chance - Amir Aczel
Reckoning with Risk - Gerd Gigerenzer
Chaos by James Gleick
Randomness by Deborah J. Bennett.
Does God Play Dice? - Ian Stewart
Introduction to random time and quantum randomness - Kai Lai Chung
Quantum: A guide for the perplexed - Jim Al-Khalili.
Quantum - Manjit Kumar

WEBSITES

http://en.wikipedia.org/wiki/Randomness
http://www.igs.net/~cmorris/index_subject.htm
http://en.wikipedia.org/wiki/Pseudorandom_number_generator
RANDOM.ORG
http://www.fortunecity.com/emachines/e11/86/random.html
http://t1.gstatic.com/images?q=tbn:ANd9GcQk5Bw3BdGRlboekIxXWw8YvWhsHzVQFK8tM8s7vSRCWEUaOsE&t=1&usg=__CxFNmGp1uPBw86HSTTmeF9oBhEw=
http://www.goodreads.com/book/show/441215.Chance
http://ezinearticles.com/?Book-Review---Chance,-by-Amir-D-Aczel&id=3507603
http://www.faqs.org/docs/qp/chap01.html
http://www.scholarpedia.org/article/Algorithmic_randomness


PEOPLE

Ben Green and Imre Leader
Einstein
Tony Hillerman
Robert R. Coveyou
Jonathan Wright

Improved structure of my dissertation

After having my supervision with Ms Caroussis, I realised that I need to change my dissertation slightly so that it includes a part on the history of the debate of random and why randomness is important.
My previous structure was:

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)


My new structure will be:

Introduction: Includes definitions of random and examples of random in everyday life. (approximately 500 words)

The history of randomness - why is it important? (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? (500 words)

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


I have reduced the "Uncertainty and Unpredictability" section from 1000 words to 500 words, so that the section about the history of randomness and its important can be included, at 500 words. I chose to cut down that Uncertainty and Unpredictability part because in comparison to he other "chapters", I don't think it is as important.

A reply to the email

Last night I got a reply from the email I sent out to the maths and physics graduates. Jonathan Wright, a maths graduate, said:

"In my opinion, there is no such thing as a random event. As an applied mathematician, all physical situations can be modelled mathematically, and as such we can predict all possible outcomes. If we roll a dice in exactly the same way 100 times, 100 times it would give us the same result. If we model the roll of the die, given the starting conditions we could predict the outcome every time.

Predictions are made using models such as this all the time, a classic example being those used by weathermen every day to make their forecasts. Granted, the models can always be improved, but in theory if we had every piece of information (all temperatures, all pressures all over the world etc), and the perfect model, we could exactly predict the future weather.

However, with both the roll of the dice or in predicting the weather, it is this 'knowing' of the starting conditions which creates the randomness that we experience I every day life. In the weather models, if your temperature measurement is off by 0.01 degrees, eventually, perhaps in hours, days or weeks time, the predictions made by the model will become drastically different from those you experience. In fact, this was how chaos was discovered; a seemingly well understood piece of theory, when run on a computer on two occasions, gave two drastically different answers with seemingly the same starting values. The difference was attributed to a difference in the 6th decimal place of the starting values..

And here lies the problem, at some point you reach the limit of your accuracy. If you know the starting position of your dice to 1000 decimal places, the difference in the 1001st decimal place coupled with a similar error in your understanding of the spin the die is thrown with etc may result in a different outcome. A good book on this is by Ian Stewart, Does God Play Dice? The Mathematics of Chaos

Quantum theory on the other hand, may also appear to be random, but similarly I think it is just not fully understood. We may not know the exact position of electrons in an atom, so instead we give electrons a 'probability' of being in certain positions or states. This doesn't mean that the electrons are in a random place, just that we are unable to observe their exact position. (In fact, and here is where you should ask a physicist, I think the very process of looking into an atom changes the states of the electrons..So we dont know.) But does this make it random?

Similarly, is an earthquake a random event?..To someone capable of studying the inner fluid dynamics of the earth, and properties of it's crust etc, then it is not random, in fact with the right models predictable. But will humans ever be able to actually predict such events precisely?..Probably not, because we will never be able to know the starting conditions to the required accuracy.."

So, to conclude, Jonathan Wright believes that pure randomness doesn't exist because probability has allowed us to be able to accurately predict so-called "random" events. Also, he discusses the fact that a random event could be subjective - it really depends on how knowledgeable a person is. Therefore, things that seem random for us right now may just be things that are too mathematically complex for us to understand, or things that we do not have much information about.

This reply has helped me look at randomness for a mathematicians point of view. Jonathan Wright touches upon the different areas of randomness that I will be covering in my dissertation and so all of this will be very helpful to me.

More books.

Today, I wrote up my proposal for a grant. I couldn't find Mr Wright today, so I need to make sure that I give it to him tomorrow. Time is of the essence!
I went to Muswell Hill Library after school to look for any of the books that I need to buy. None of the books that I need were in that library, however, I found two books on quantum theory that I borrowed:
1. Quantum: A guide for the perplexed - Jim Al-Khalili. Chapter 2 is called "Probability and Chance" and I am sure that this will be useful to me.
2. Quantum - Manjit Kumar. The 4th part of the book is named "Does God Play Dice?" so this will most probably be relevant to my project.

GANTT chart

Tomorrow will be September 20th. The main task that I need to complete in order to keep progressing through my GANTT chart is to buy the remaining books. I have written out a grant proposal for Mr Wright so I must make sure that I print it out at school tomorrow and give it to him. All that is left for me to do is finish reading Randomness and I will be back on track.

So, priorities for this week are:
FINISH READING RANDOMNESS.
Get the remaining books.
And perhaps start researching quantum mechanics and chaos theory, to prepare for writing that part of my dissertation. One way of doing this is to talk to physics teachers in my school.

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.

More research

http://en.wikipedia.org/wiki/Randomness

About the history of randomness:
"In ancient history, the concepts of chance and randomness were intertwined with that of fate. Many ancient peoples threw dice to determine fate, and this later evolved into games of chance. Most ancient cultures used various methods of divination to attempt to circumvent randomness and fate.

The Chinese were perhaps the earliest people to formalize odds and chance 3,000 years ago. The Greek philosophers discussed randomness at length, but only in non-quantitative forms. It was only in the sixteenth century that Italian mathematicians began to formalize the odds associated with various games of chance. The invention of the calculus had a positive impact on the formal study of randomness. In the 1888 edition of his book The Logic of Chance John Venn wrote a chapter on "The conception of randomness" which included his view of the randomness of the digits of the number Pi by using them to construct a random walk in two dimensions.

The early part of the twentieth century saw a rapid growth in the formal analysis of randomness, as various approaches for a mathematical foundations of probability were introduced. In the mid to late twentieth century ideas of algorithmic information theory introduced new dimensions to the field via the concept of algorithmic randomness.

Although randomness had often been viewed as an obstacle and a nuisance for many centuries, in the twentieth century computer scientists began to realize that the deliberate introduction of randomness into computations can be an effective tool for designing better algorithms. In some cases such randomized algorithms outperform the best deterministic methods."

This extract implies that randomness has been a part of our world for a very long time. People have learnt to accept randomness and there are many uses for it now.

About Quantum Mechanics:
"According to several standard interpretations of quantum mechanics, microscopic phenomena are objectively random. That is, in an experiment where all causally relevant parameters are controlled, there will still be some aspects of the outcome which vary randomly. An example of such an experiment is placing a single unstable atom in a controlled environment; it cannot be predicted how long it will take for the atom to decay; only the probability of decay within a given time can be calculated. Thus, quantum mechanics does not specify the outcome of individual experiments but only the probabilities. Hidden variable theories are inconsistent with the view that nature contains irreducible randomness: such theories posit that in the processes that appear random, properties with a certain statistical distribution are somehow at work "behind the scenes" determining the outcome in each case."


About Religion:
"Some theologians have attempted to resolve the apparent contradiction between an omniscient deity, or a first cause, and free will using randomness. Discordians have a strong belief in randomness and unpredictability. Buddhist philosophy states that any event is the result of previous events (karma), and as such, there is no such thing as a random event or a first event.

Martin Luther, the forefather of Protestantism, believed that there was nothing random based on his understanding of the Bible. As an outcome of his understanding of randomness, he strongly felt that free will was limited to low-level decision making by humans. Therefore, when someone sins against another, decision making is only limited to how one responds, preferably through forgiveness and loving actions. He believed, based on Biblical scripture, that humans cannot will themselves faith, salvation, sanctification, or other gifts from God. Additionally, the best people could do, according to his understanding, was not sin, but they fall short, and free will cannot achieve this objective. Thus, in his view, absolute free will and unbounded randomness are severely limited to the point that behaviors may even be patterned or ordered and not random. This is a point emphasized by the field of behavioral psychology."

While searchng through Google, I can across a film called "Chaos Theory".

This may be a useful film to watch. I read some summaries about the film and it doesn't seem extremely relevant to my project, but perhaps I could watch it if I have the time.

I emailed the maths and physics graduates that my supervisor introduced me to:
"Thanks.
Hi everyone.
First question: Do you think that one day, with enough research, we will be able to confidently predict a random event (for example, the rolling of a dice)?
Also: Do you believe that pure randomness exists? It can be argued that randomness doesn't exists because it is subjective - something that may seem random to one person may seem like an obvious pattern to others. On the other hand, there are some mathematical (and physics related) concepts such as chaos and quantum theory, that show that randomness is very much a part of our world?
To conclude, I would just like your opinions on the subject of randomness and any suggestions for books or certain topics that may be useful to me.
Thanks,
Zainab Kwaw-Swanzy"

Supervision

I had my first supervision with Ms Caroussis since last term. We discussed the work I did over the summer. I feel a lot more confident about my progress now, but I really need to make sure that I dont fall behind on my GANTT chart.
I talked about possibly obtaining a grant and my supervisor told me that I need to write a proposal to Mr Wright. This must include:
Project Brief
Detailed costs
Why I need the money to pay for these things
Why is it important to get the grant for my project

I asked about how large my range of sources should be and so, next supervision, I must bring a detailed, up to date bibliography so we can discuss other sources that I may need. I haven't managed to get opinions from any professionals so my supervisor started up an email with a few maths and physics undergraduates that she knows. I could also talk to physics and maths teachers in the school.
Possible questions that I should ask them:
- Whether they believe that randomness really exists
- Will we ever be able to predict the next number to show on a dice?
- Any book recommendations?

I can't think of any other questions at the moment but I need to come up with more so I can make the most out of having maths and physics related contacts.

During the supervision I thought about other things that I may want to put in my dissertation that I had not thought about before. My supervisor suggested that I wirte a bit about the importance of the topic - why should people be interested in randomness? And also possible a history of the debate of randomness.

The supervision helped me find out more about getting different types of sources to add to my list. I am now confident that my research will become very detailed and varied.

Tasks for this week:
FINISH READING RANDOMNESS!!
Write up random generators chapter.

GANTT Chart Update

This week I need to focus on finishing Randomness and writing the "Random Generators/mechanisms" chapter of the dissertation. This means that I must complete my research on that topic because at the moment, I may not have as many sources as I would like that will help me with writing that chapter.

I also need to enquire about possibly receiving a grant so I can buy the remaining books.

Useful quote

"From where we stand the rain seems random. If we would stand somewhere else, we would see the order in it."
Tony Hillerman - american author.

Suggests that randomness is subjective - doesn't truly exist.

Random generators

http://www.scholarpedia.org/article/Algorithmic_randomness

"Algorithmic randomness is the study of random individual elements in sample spaces, mostly the set of all infinite binary sequences. An algorithmically random element passes all effectively devised tests for randomness."

An algorithm is a set of instructions that will solve a problem.(I have learnt this from my further maths AS level in a topic called Decision Maths 1). Algorithms are mainly used for computers.



I've begun to realise that a lot of the information about random mechanisms is very complex. I have two choices:
1. Don't go into much detail about the random mechanisms and it is very complicated and a very vast subject. This will make writing that part of my dissertation much easier.
2. Research random mechanisms IN DEPTH so that my dissertation will be more informed. I will then learn a lot more which is what this project is all about.

Number 2 it is.

This is an extract from the same webpage as the above quote:

"The theory of algorithmic randomness tries to clarify what it means for an individual element of a sample space, e.g. a sequence of coin tosses, represented as a binary string, to be random. While Kolmogorov's formalization of classical probability theory assigns probabilities to sets of outcomes and determines how to calculate with such probabilities, it does not distinguish between individual random and non-random elements. For example, under a uniform distribution, the outcome "000000000000000....0" (n zeros) has the same probability as any other outcome of n coin tosses, namely 2-n. However, there is an intuitive feeling that a sequence of all zeros is not very random. This is even more so when looking at infinite sequences. It seems desirable to clarify what we mean when we speak of a random object. The modern view of algorithmic randomness proposes three paradigms to distinguish random from non-random elements.

Unpredictability: It should be impossible to win against a random sequence in a fair betting game when using a feasible betting strategy.
Incompressibility: It should be impossible to feasibly compress a random sequence.
Measure theoretical typicalness: Random sequences pass every feasible statistical test.
It is the characteristic feature of algorithmic randomness that it interprets feasible as algorithmically feasible. "

This indicates that there are certain properties that a sequence must have for it to seem random. The fact that people are aware of these properties and can recreate them in things such as a random number generator shows that although one cannot predict a random sequence, a sequence can easily be indentified as random and it is possible to produce sequences that can easily pass as random.

I haven't blogged in a few days so here we go...

Bibliography so far:
BOOKS
Chance by Amir D. Aczel
Reckoning with Risk by Gerd Gigerenzer
Chaos by James Gleick
Randomness by Deborah J. Bennett
Introduction to random time and quantum randomness by Kai Lai Chung
Quantum Theory: A very Short Introduction by John Polkinghorne
Does God Play Dice? by Ian Stewart

WEBSITES
http://www.fortunecity.com/emachines/e11/86/random.html
RANDOM.ORG
http://www.scientificamerican.com/article.cfm?id=how-randomness-rules-our-world
http://en.wikipedia.org/wiki/Pseudorandom_number_generator
http://www.igs.net/~cmorris/index_subject.htm
http://ezinearticles.com/?Book-Review---Chance,-by-Amir-D-Aczel&id=3507603
http://www.goodreads.com/book/show/441215.Chance
http://www.faqs.org/docs/qp/chap01.html

I should be having a meeting with my supervisor very soon, now that I am back at school. I need to ask if this is a good amount of sources considering my stage in the project.

Reading through Randomness:
The next chapter is about dice rolls and how the probability of each number coming up will change depending on the dice (e.g. having 2,2,3,4,5,6 instead of 1,2,3,4,5,6)
Bennett then talks about rolling two dice.

"[...] let's imagine using coloured dice, one red and one green. For each of the 6 possible throws on the red die, 6 are possible on the green die, for a total of 6 x 6 - 36 equally possible throws. But many of those yield the same sum. To make things even more complicated, different throws can result in the same two numbers. For example, a sum of 3 can occur when the red dice shows 1 and the green die shows 2, or when the red die shows 2 and the green die shows 1. Thus the probability of throwing a total of 3 is 2 out of 36 possibilities, or 2/36. A sum of 7, on the other hand, can be thrown 6 different ways - when red is 1 and green is 6; red is 6 and green is 1; red is 2 and green is 5; red is 5 and green is 2; red is 3 and green is 4; red is 4 and green is 3. Therefore the probability of throwing a 7 is 6/36"

This extract implies that if you throw two dice at random many times and writed down the total of the dice, one can predict that a total of 7 will occur more than a total of 3. This quote, like many others that I have found on the subject of probability, show that is isn't absolutely impossible to predict a random event - we havea rough idea of what the occurence of numbers should be when a dice (or 2) are thrown a certain number of times.