Recursive Messages

2010-06-07T10:36:00.001-07:00

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The Twin Slits Experiment

2010-06-06T05:10:00.001-07:00

The Twin Slits experiment is mentioned in the text several times.
I had intended to provide an overview of the experiment here, however there are so many other excellent sites the exercise seemed pointless.
A Google search using the string ‘Twin Slits experiment’ will provide a number of explanations at every level.
A fascinating and very readable account of the Twin Slits experiment and its implications is given in John Gribben’s book “Schrodinger’s Kittens”

Maximum Capacity

2010-06-06T05:09:00.003-07:00

Here are some dictionary definitions of Capacity:

The ability to hold receive or absorb
The maximum amount that can be contained
Ability to perform or produce
The power to learn or retain knowledge
Innate potential for development
Position or Authority to do something

According to the Canadian Building Digest “Elevator cars are rated according to maximum allowable weight and number of passengers… Usually, the maximum loading of an 1150 kg elevator will not exceed 13.”
So a 1150 kg elevator will have a maximum capacity of 13 people.
What about a computer? How many human minds can we fit in a computer?
It all depends on how you define a mind. You might start by trying to model a brain
A human brain contains around 100 billion cells. If we imagine that each cell is either on or off (firing or not firing) we could say that each cell could be represented by a bit, or 1 or 0.
A human brain would therefore need 100 000 000 000 bits or 12 500 000 000 (12.5 billion) bytes of memory to be stored on a computer. This is the same as about 11.6 Gb. In other words, by this model, I could fit two brains on my 30Gb iPod. (I have written this in August 2005. No doubt in a couple of years time the capacity of my iPod will appear laughably small.)
How many human minds could be stored on an iPod? Is the answer still 2?
No. In fact, to be honest the previous answer is not really 2. It assumes an awful lot about the structure of a brain, not least the fact that a frozen snapshot of the states of a set of neurons captures all the data stored therein. It also says nothing about the processing performed by the brain that gives rise to a mind.
For example, each neuron is connected to thousands of other neurons, via trillions of synaptic junctions. Modeling this bumps up the numbers of bits required tremendously, and blows the possibility of iPod storage out of the water.
Current estimates of brain capacity are expressed in Terabytes ie 1 x 10^12 bytes. Coincidentally, that is about equal to the current estimate for the number of people who have ever lived. So that means we would need about 1x10^24 bytes to store the combined experience of the human race, assuming each brain has a maximum capacity of 1 terabyte.
That’s a lot of memory. But suppose we apply Moore’s law to memory storage, and assume that capacity doubles roughly every two years.
That suggests that by the year 2095 one could purchase an iPod capable of storing the memories of everyone who has ever lived.
That’s certainly something to think about.

Next: The Twin Slits Experiment

Wafuku

2010-06-06T05:09:00.001-07:00

Judy’s appearance was inspired by a piece of clipart, of all things. Her dress style was elaborated on reading Liza Dalby’s two books Geisha and Kimono. Both give a fascinating insight into these aspects of Japanese Culture.
Wafuku means native Japanese clothing, as distinct from yofuku- western clothing.
The main item of wafuku is the kimono. This made from a length of cloth cut up into several pieces and sewn to fit the wearer. The cloth may be plain, or very elaborately decorated (and consequently very expensive). The resulting garment is fastened with a sash or obi. Women’s sashes in particular can be very wide and again very elaborate. Consequently, a kimono and obi combination can be very expensive, costing literally thousands of pounds/dollars/euros.
Judy keeps a white kimono in her chest. It is possible she is saving it for her wedding day, as other 23rd Century wafuku followers do, however, as a confirmed virgin, this is unlikely. More probably she is saving it for her death, when her corpse will be dressed all in white, the overlap of the kimono being reversed to right over left, as is traditional.
Oddly, the digital Judys habitually overlap their kimonos right over left for everyday use. Possibly this is a subtle joke, seen only by other followers of the wafuku fashion.

Next: Maximum Capacity

Mahler's Symphony Number Eight in E flat Major (1907)

2010-06-06T05:08:00.001-07:00

This symphony is quoted in various ways throughout Capacity. The following gives a little background on the piece and answers some of the questions raised in the book.

Mahler was born in 1860 in the Bohemian village of Kalište, the son of an Austrian-Jewish tavern keeper. Later in life he was baptised as a Christian. The eighth symphony is generally supposed to have been the last written whilst Mahler retained that faith. His works after this reflect his gradual disillusionment with religion and growing awareness of the reality of death.

The eighth symphony itself can be said to be a cry for illumination from both a religious and humanistic point of view. It consists of two parts, the first a setting of the medieval Catholic Pentecost hymn Veni Creator Spiritus (Come Creator Spirit); the second is based on the last scene of Goethe’s Faust, the scene of Faust’s redemption.

Both parts feature in the novel: for example the musician sings the main theme at the close of the book. The landscape at the end of Justinian’s story was inspired by the desolate land described by the music in the opening of the second movement. There are other references hidden away in the book if you have the time (or the inclination) to look for them.

About half way through Capacity, Zinman quotes the closing lines from the second movement. Helen asks what it means and Judy suggests that she looks it up on her console. I have included the text and a translation below. To my mind, Zinman has taken the text entirely out of context.

All things transitory

are but parable;

here insufficiency

becomes fulfillment,

here the indescribable

is accomplished

Judy also appears to be familiar with the work. Possibly this is because she is trying to understand Zinman, or maybe she just likes Mahler. Either way, Judy replies with the final two lines of the text

The Eternal Feminine

draws us heavenward

Next: Wafuku

The Novel

2010-06-06T04:53:00.002-07:00

Recursion in the Novel

Many novels are recursive: the key character sets out to achieve some objective and meets other lesser objectives on the way that must also be resolved. If the story is not to be too linear, these objectives will be met and resolved in a nested fashion.
I don't intend to discuss the recursive games I played in the novel here: they are there for the reader to discover. I will, however, draw attention to one point.
Towards the end of the novel Constantine says “Examine any artefact of intelligence, and you will see the threads of childhood running through it. Like sixteen sheep, walking in their sleep...”
This is the topmost level of one recursive thread. If you enjoy the novel, you may like to try and find where it leads...

Recursive Functions

2010-06-06T04:53:00.000-07:00

What is a Function?

In mathematics, a function describes a relationship between two things. For example, there is a function that relates the weight of a chicken to its cooking time. (My mother taught me that function is 20 minutes per pound plus an extra twenty minutes.)

Function Notation

Mathematicians use various ways to represent functions. One of the most useful was devised by Euler. In Euler's notation, the time taken to cook a chicken depends on the weight of the chicken, so we could say
f(w) = 20 * w + 20 where w is the weight of the chicken in pounds
This function converts Fahrenheit to centigrade
c(x) = (x-32) * 5/9
and this one converts it back again
f(x) = x *9/5 +32

Recursive Functions

You can form recursive functions, functions that call themselves, quite easily.
For example: f(x) = 1 + f(x-1)
You have to be careful, of course, the above function calls itself for ever. It’s a little like the meaning of the recursive acronym, GNU: GNU’s Not Unix, or (GNU’s Not Unix)’s Not Unix. Or ((GNU’s Not Unix)’s Not Unix)’s Not Unix. Or...
Recursive functions are only usually useful if they have an end. We can define and ending quite easily; in the example above we could say for example, f(0) = 0.
So
f(1) = 1+ f(0) =1
f(2) = 1 + f(1) = 2
f(3) = 1 + f(2) = 3
Which is not very interesting...
Here's another function
f(x) = f(x-1) + f(x-2); f(0) = 1, f(1) =1
So
f(0) = 1
f(1) = 1
f(2) = 2
f(3) = 3
f(4) = 5
f(5) = 8
f(6) = 13
This is the Fibonnacci sequence which describes everything from the way rabbits breed to the pattern on pine cones.
The next function comes from the book Godel Escher Bach, by Douglas Hofstadter, a book which contains all manner of information about recursion and much, much more.
The Q function is defined as
Q(n) =Q(n - Q(n-1))+ Q(n-Q(n-2)) for n>2
Q(1) = Q(2) = 1
Here are the first few terms:
1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, ...
You work out the next term as follows
1) Look at the last term you found, in this case 8, and then count that many places back from the three dots to get the number 3
2) Repeat the process with the second from last term, 6, to get the number 5
3) Add together 3 and 5 to get the next term, 8.
This is similar to the Fibonacci sequence, except the results are chaotic. After a relatively orderly start, there seems to be no logic to the sequence produced. There would seem to be no way of predicting the next term, except of course by using the deceptively simple recursive function above.

Next: Uses of Recursion

Uses of Recursion

2010-06-06T04:29:00.001-07:00

Binary Search

A simple recursive routine used by many people is the binary search.
When asked to guess a number between one and a hundred, most people would start by guessing fifty. If you are told that the number is higher, you would probably guess seventy five. If you were told lower, you'd then guess a number roughly half way between fifty and seventy five.
Here, you are applying the principal of recursion, defining the number you are searching for by repeatedly applying the rule "Find the number half way between x and y"

Language

Linguists point out that Recursion is used in sentence construction. For example one might say
"The book is very interesting"
One could embed a sentence inside the first
"The book (that I read last week) is very interesting"
and again
"The book (that I read (on the bus) last week) is very interesting"
The process could go on forever...

Do you know what I'm thinking?

Do you realise that I know what that you're thinking what I'm thinking?
Guessing what others are thinking is a basic human trait. How far can we recursively map others thoughts?

Intelligence

It has been suggested that the ability to think recursively is a necessary component of intelligence
Recursion allows a limited number of "rules" to be applied recursively to create an infinite number of states.
For example, the game rock paper and scissors can be described by a finite number of states
(rock rock) draw
(rock scissors) player 1 wins
(rock paper) player 2 wins
(scissors rock) player 2 wins
...etc.
It would take an awful lot of space to write down all the states, or possible board layouts, of a game of chess. Programming a computer to play chess by putting in all possible states in the manner of the rock paper scissors game would take a very long time.
Recursion means that a few rules (say the rules of chess) can be laid down that, applied over and over again, effectively work out all the states of the game as they are required.

Next: Recursion the Novel

Recursion Defined

2010-06-06T04:25:00.001-07:00

Why is this page here?

Recursion is a common concept in the fields of Maths, IT and Linguistics. Nonetheless, I have been surprised by the number of people who, on hearing the title of the novel have asked "Is that a real word?" For those who are interested, here is a brief overview. It is not by any means intended to be comprehensive. If you want to find out more, get Googling!

Recursion Defined

An internet search gives lots of definitions for recursion, depending on what recursion is applied to. For the moment, however, a couple of working definitions could be:
Defining something in terms of itself
or
Repetition of concept within itself

Discussion

A simple example of recursion can be seen on The Simpsons, when Bart and Lisa sit down to watch Itchy and Scratchy on the TV. Here you sit in the real world and mentally drop down to the Simpson level to watch Bart and Lisa, who drop down to the Itchy and Scratchy Level to watch their cartoon. (On some episodes Itchy and Scratchy watch TV too, dropping down a further level.) Here the concept of watching TV is repeated within itself.
We keep a mental stack of where we are as we watch the program. If we don't, we might get confused as to what is going on. For example, we probably realise that Bart Simpson does not live in the same world as we do, it is also not too difficult to realise that Itchy and Scratchy live in a different world to Bart and Lisa.
However, If we were to extended the level of recursion, it is likely that we would run the risk of forgetting who goes where.
We can represent this recursive relationship as follows:
You(Bart and Lisa(Itchy and Scratchy)))
A computer programmer may try to make things a bit clearer by writing this out as follows
You
(
            Bart and Lisa
            (
                        Itchy and Scratchy
                        (

                        )
                )
)

Next: Recursive Functions

Writing

2010-05-31T11:36:00.000-07:00

First Post

2010-05-31T11:32:00.001-07:00

Just Testing