Saturday, October 31, 2009

Vicipeadia

There's something about Latin. It has, to me, a distancing effect. There's not a joke on the page on the Iraq War, but I can't help laughing. Latin is so distant and dead, but here it is pretending it is contemporary. It's hysterical!

You know, I suppose this may seem like a good time to write about the wiki concept. The contrast between the dynamic modern idea of a wiki seems to contrast the old language. The fact is that the Library of Alexandria was smaller and held more errors, but we are allowed to celebrate it. Some people find it astonishing that there exist imperfect sources of information - it would be more rational to remember that there occasionally exist good sources of information. If Wikipedia is less reliable that, say, Encyclopedia Brittanica it must be remembered that the wiki dwarfs the print work. The Wikimedia community is an enormous source of information, and not just encyclopedia articles. Paintings, pictures of animals, court documents, plays, recorded music, and much more. If used in a mature way, Wikipedia is even better than good, it is useful.

It is pleasant to live in a world where there is such access to information that we can afford to be snobbish about sources. Could you imagine the reaction of Diderot to the French Wikipedia?

Saturday, October 24, 2009

Limericks

I've been addicted to limericks these past few days. They are simple fun, so I can write three in a lunchbreak! Because I'm not compelled by any standard of quality, here are a few of my pieces. Special thanks to Aaron Philby and The Medicated Cartoonist for their excellent work.

The mathematician Euler
was near blind far-sighted toiler.
He wrote with great haste,
but without making waste,
his mind as hot as a broiler!

alternate poem for english speakers:

Sitting the with his ruler
is the mathematician Euler.
Invisible to his eye
are gamma, sigma, and pi,
but to me there isn't anything cooler!
Euler's Student Lagrange,
Also covered a range.
His work in mechanics
cleaned up the antics
of many a model most strangeRichard Phillips Feynman,
liked to wine and dine them.
He'd meet a nice girl,
they'd go for a twirl,
digging those intellectual mines inMurray Gell-Mann discovered the quark,
Using the time honored method "hard work".
He now studies complexity
and adaptivity-
He certainly has made his mark!

I've fallen into a rut;
my poems just don't make the cut.
I start with a name
then play a game,
and never do anything butThere lives a man named Eddie Fizgerald,
a cartoonist/philosopher/actor/and/herald.
Well maybe he's never acted in a movie,
but he's done something equally groovy:
he's always been Eddie FitzgeraldIn Britain, on medication,
lives a maker of animation
He draws a cartoon
as he stares at the moon,
and thinks about emigration

This next poem was a tough one.
But pi is just way too much fun!
The number is gentle,
but quite trancendental
and the rhyming's a son of a gun.

The π sits a mite amazingly,
on adders shock and typey.
Ignoring beginners,
for to see againers
find novels of brains idly

I bid my readers adieu,
though I'll soon get back to you.
I have no gift for rhyme
and even less time!
And I'm trying to start posts anew!

Sunday, October 11, 2009

What two posts on words in a row?

W.C. Minoralso had a preoccupation with defining words partly because he was a madman.


In this post I'm actually going to define a word though. That word is: "deterministic". Why? I was asked to offline.

Given that at least one state of a process is known and that there are no further operators on that process, the process is "deterministic" if and only if this makes future states known with certainty.

Example of a deterministic process: A ball is dropped from a building on a given planet from a given height. Because we understand gravitational acceleration and air resistance, we can find all the information about the ball we need until it hits the ground.

Why does the process become non-deterministic when it hits the ground? Because the ground acts as an operator, altering the behavior of the system. If I added information about the ground it might become deterministic.

Example of a non-deterministic process: If a list of names is sorted, and two people have the same name which person goes first? The right answer ("Who Cares?") does not remove the non-determinism.

A process can be deterministic if all information is known about it, but non-deterministic with ignorance. A process can be non-deterministic if all information is known about it, but deterministic with ignorance. Can a process can be non-deterministic is all information is known about it but deterministic in ignorance?

The answer can be found by restating the question: Can statistical processes be deterministic? The answer to that question is Yes. For instance, in a bizarro world where you know the state of every quantum molecule in a system, you still cannot make good deterministic predictions about that system. However, by using statistics to turn dumb information into interesting information one can find some information with certainty! For instance, you know the macroscopic system will comply with the laws of thermodynamics whatever the behavior of individual microstates are. Since we happen to live on the thermodynamical level most of our lives, that the process is deterministic in many cases has been well established.

I hope that helps with word. As Confucius said:"If language is not correct, then what is said is not what is meant; if what is said is not what is meant, then what must be done remains undone; if this remains undone, morals and art will deteriorate; if justice goes astray, the people will stand about in helpless confusion. Hence there must be no arbitrariness in what is said. This matters above everything."

Confucius, incidentally is the all time master of the slippery slope. So to lighten things up, here is a funny picture of David Ben-Gurion using the Feldenkrais Method:

Saturday, October 3, 2009

How Do You Define A "Definiton"?

Still working on that larger idea, but here is something a little smaller to hold you over.

Defining words is rather difficult, especially in natural language. The rub of defining is there is no escape. Words are defined in words. Words are also non-monotonic, one often uses complex ideas to define ideas seen as being simpler!

Let's begin by going over a few intuitive solutions.

There is the Representation School. This theory is most associated with the analytic school, though few of them held a pure version (AJ Ayer held a very interesting version of this that I don't want to get into yet). In it's simplest most naive formulation this formulation holds that words represent things. If I wanted to define "art" for instance I could just say:or if I wanted to define comedy I'd utter:and indeed Moe Howard owned an encyclopedia which used an illustration of The Three Stooges illustration of the definition of comedy! However, it did not only use the illustration. It contained many words.

They say a picture contains a thousand words, but that isn't nearly the size of the average language (indeed, some natural languages allow recursive words making them theoretically infinite!). The Representation Theory isn't quite the worst thing out there. I would think that most people believe (including myself) that at some low level, most words are defined relative to an external object.

But it is incomplete. One thing it cannot do is define words like "like", "define", "is", "do", "cannot", and "and". What would you point to in order to demonstrate the idea of "to" to a strictly representational martian? Some have advocated the use of a English semi-variant called E-Prime that excludes "to be" from the language (for unrelated philosophical reasons). Is it possible that words like "these" are syntactic sugar capable of being removed from the language? As an exercise, try to write this paragraph using only words you can illustrate.

What the Representational School leaves out is imprecision. Not imprecision in the grammar school sense, but advanced engineering imprecision the kind that makes things possible. The imprecision that makes generality possible. Since this imprecision is a real and necessary thing (otherwise we'd live in a world of endless special cases) this is a flaw with the Representational School that has not a few implications. The biggest is the observation that We can take certain ideas as being atomic. I will explain that in a moment.

We have observed that there exist flaws in the Representational School. But perhaps we can plug the hole by doing another obvious tactic: defining words in words. In this view there are some words that are atomic and others that are built up. This is sometimes compared to a pyramid, though I find that the definitions almost always asks the reader to do something. In this way, words are much like Daniel Dennett's idea of Cranes. For instance, we can use textual substitution to get around some words like "is". E' claims that "is" is a synonym for "seems", but this doesn't really capture the flavor of the word - and purposefully so in that case. One can use context and substitution to rigorously define "is" as it is used in most, but that's not my purpose here.

What is the subject this time in is the existence of words which cannot be evaded by textual substitution or illustration, atomic words like "to do", "or", and "and". There's a finite set of atomic words, which is good because natural language is atomic. Since multiple words can use the same spelling (like the philosopher's concept of "to be" and it's more common use as a coupla and auxiliary verb) it is possible for a letter sequence to be simultaneously atomic and non-atomic, but by fiat we will define a "word" as having a unique definition.

The biggest debate in this school is "What do we get for free?" In other words, what are atomic concepts? Some, like Chomsky, go as far as to claim that the atomic words form a language (albeit not an easily spoken one) while others claim that nothing is free and thus all language is formed by experiences.

I personally am in the latter school, a newborn baby is incapable of organizing action and if one is not capable of formulating an idea like "do this" in what sense does one have language? I have read that newborns do not even have their senses under control yet, which would make understanding concepts of the outside world impossible.

Do we now have a good definition of "definition"? Is a word defined when it can be decayed into simpler concepts and experiences? I would say that is a definition, but not a good one. As humans have limitless ability for turning finery into pattern, I have little difficulty imagining that we are able to use the powerful machinery of the mind to understand words without difficulty.

Why? Because as Doug Hofstadter points out, we associate things with words that may not be in a strict definition. Because such a thing would be a language, but not English as it is written. Hofstadter uses the idea of a probability cloud to illustrate this.At the center is the tiny nugget that is meaning in the above dense sense, meaning that can be decayed to (sub)atomic ideas/experiences (much like how protons and neutrons are composed of quarks, "define" means to break down into component concepts) while the electron cloud is the flavor of the word, what it is associated with (just as the electron cloud decides how an atom reacts chemically, the associations cloud allows us to use flavorful concepts like "associations cloud" and "flavorful concepts"). Also like the atom, the two are fundamentally different things when separated. The whole was the object of this study.

Well, that's about all I have to say about defining. We all know from experience what tedious work it is to correct misunderstanding and how long it can take to explain what one means. I recently told a mildly dirty joke to an audience of non-native speakers and had to resort to a very embarrassing explanation. The composition of ideas in the joke was profane, even though the text was the kind of joke an uncle could tell a nephew. It was that experience that led me to write all this!

Finally, I want to acknowledge that not all meaning is mediated by words with a wonderful piece by Scriabin played by Vladimir Horowitz.

Wednesday, September 30, 2009

Mnemonics

I haven't posted in a bit since I have had a couple large ideas for future posts. Working on them two of them really had legs and I'll post it soon. Until then, enjoy these mnemonics.

The Golden Ratio = φ = 1.61830988... = I primed a triangle ... but Sir Fibonacci finished.

Primes = 2, 3, 5, 7, 11, ... = We can write anyone's biographies.

Conway's Constant = 1.3035772690342963912570... = I see ... may agile escaped crewman ax rubber wrappings? But with an authority around few foolishly, I am, admit analogs. (Punctuation except commas represent zeros)

Fibionacci Sequence - 1, 1, 2, 3, 5, 8, ... - A "A" to the right child.

2^(n-1)*(2^n - 1) - 1, 6, 28, ... - A little hepaticocholangiogastrostomy?

This one isn't mine, but it is the best number mnemonic I have seen:

e = 2.718281828459045235360287471352662497757... = We present a mnemonic to memorize a constant so exciting that Euler exclaimed: '!' when first it was found, yes, loudly '!'. My students perhaps will compute e, use power or Taylor series, an easy summation formula, obvious, clear, elegant. ('!' represents 0)

Thursday, September 17, 2009

Modern Cryptographic History

"Whitfield Diffie, a key figure in the discovery of public-key cryptography, traces the growth of information security through the 20th century and into the 21st. In the 1970s, the world of information security was transformed by public-key cryptography, the radical revision of cryptographic thinking that allowed people with no prior contact to communicate securely. "Public key" solved security problems born of the revolution in information technology that characterized the 20th century and made Internet commerce possible. Security problems rarely stay solved, however. Continuing growth in computing, networking, and wireless applications have given rise to new security problems that are already confronting us." - The Computer History Museum

I don't know the general tech-savvy, or history-savvy, of the people who read this blog. If you have any questions, post them in the comments section and I'll answer them as soon as possible.

Monday, September 14, 2009

Alan Turing, Britain Asks Forgiveness

Britain has come to it's senses in at least small way and has issued a formal apology to Alan Turing for it's what it did to him - actions which led directly to his suicide at the age of 41. The apology can be found here.

It is happy that this apology has taken place. I hope that the British and American governments both apologize for the harsh treatment of homosexuals in various psych wards until 1973. Dr. Turing's death was a great blow to mathematics and computer science, yes, but it is the larger issue that takes precedence. It is not just sad that Turing was destroyed because he was so bright a light, it is sad because so many suffered and have become forgotten. I don't mean to cast aspersion on the British Government. Indeed, the Prime Minister recognizes this in the above linked statement.

However, I feel that I ought to explain Dr. Turing's work, in order to have people comprehend him as a man of incredible insight and rare genius rather than as a victim of society. To do this will require some scene setting. In 1900 David Hilbert (above) proposed a list of 23 problems, the solutions of which he proposed should take approximately a century of search. One of these questions (the second) asked for mathematicians to prove the axioms of arithmetic consistent. He was, in fact, referring to the work of Gottlob Frege, whose Begriffsschift had seemingly embedded the Peano Arithmetic in rock solid ground. All that remained to Hilbert was the asked for proof - and a century seemed plenty of time to write even such a non-trivial thing out.In 1901, however, the question changed. Bertrand Russell developed a formal paradox in Begriffsschrift. This paradox comes about through self-reference, and the importance of strange loops in these things is an interesting study that is beyond our scope. The point is that Begriffsschrift was as solid a foundation as a rollicking sea. The question altered suddenly in 1910. Luckily, a new system was written by Russell himself free of the paradoxes of the old kind.Unfortunately (or perhaps not) the mathematician Kurt Gödel proved that there were contradictions in Russel's work and these contradictions come not out of weakness but strength! Indeed every sufficiently strong mathematical system must bear the Gödellian problem.

Godel had proved that No Mathematical System Strong Enough To Hold The Counting Numbers Could Be Consistent And Complete. At this point, the only thing that remained of Hilbert's naive 2nd problem was this: the Entscheidungsproblem ("Decision Problem"). Could we build a mathematical device to simply throw out all the Godelian chaff and keep all the number theory wheat?
Enter Alan Turing, whose memory we are attempting to honor.

Dr. Turing was a student of the great logician Alanzo Church, who also did work on the Entscheidungsproblem (in fact, Church solved the problem first - but no one figured it out except people who had studied his λ-calculus extensively. Turing's beautiful and intuitive proof is the better and thus the wider known) and had figured out the basic problem was that no one really knew what mathematical machinery looked like. There were the systems of Russell, Zermelo, and Peano - but these were abstract set pushers. The sentences they made were dead, no one had ever bothered to define how someone could build a process in these languages.

Well, no one except Church and Turing. Turing's method is beautiful and simple, relying on only one even simpler idea. That idea is Finite State Machines.A Finite State Machine is a machine that when fed an input, changes state. The above, for instance, is fed an arbitrarily long binary number and bounces between states until it lands in S1 or S2. If it lands in S1, then the number has an even count of zeros and the machines accepts. Otherwise, the machine rejects. Try it and see!

Many "languages" (groups of acceptable strings) can be defined with finite state machines. But try to build one that can check if parenthesis are balanced. It is impossible, the machine needs some kind of memory in order to know how many layers deep it is!

And Turing began thinking about memory. What is a mathematician? A mathematician has a finite number of states (an enormous number, but finite) in his head and all the paper in the world to write on. And what is a Turing Machine? A finite state machine with all the memory in the world.

Anything in the world that can be done mechanically can be done with a Turing Machine. So Truing subtly re-wrote the question. He began to consider whether if one made a Turing Machine made to look over mathematical statements it could be the thresher aforementioned. His conclusion involved an insight into mathematics hidden deep in Godel's proof, an insight his explicit drawing has enriched every mathematician since.

This insight is deceptively simple "Turing Machines are a language," - more broadly "Metamathematics is a language." Since Turing Machines can define a language, then could not one be made to build Turing Machines? One could, this is the Universal Turing Machine. One can feed a string in to a UTM (we now call this "programming computers") and it can output any Turing Machine, which itself can do wonders. Dr. Turing's simple conceptions had struck to the heart of mathematics, which was so moved that it split and the new creature was dubbed "computer science".

And the Entscheidungsproblem? It's solution is simple: "No." Turing simply built an anti-Gödellian thresher and showed that it could not even thresh the program for an anti-Gödellian thresher! Yes, the Turing Machine turned the Entscheidungsproblem from a fearsome German phrase to a well-understood part of computation. Speaking of which, the Turing Machine becomes vital and powerful in the study of computational complexity; it gives the basis for the development of Artificial Intelligence while offer clarification on the intellegence of ourselves; his work on "Unorginized Machines" has given way to Neural Networking and Genetic Algorithms; his work crystallized the mathematical underpinnings of modern cryptography.These are perhaps Turing's deepest developments - the ones he will be remembered for when schoolchildren have to be reminded that Hitler and Attila lived at different times. These amazing feats did not slow Dr. Turing down as, famously, the writing of Principa Mathematica had done to Bertrand Russell. He remained a force of mathematical nature, working on ideas of practical, abstract, and computational nature until the end of his life. His service during WWII involved the most successful signals intelligence corp in history, and it is - as the Prime Minister of Britain notes - no coincidence that he was on the winning side.

In learning the story of the life of Dr Turing one becomes aware that the only essential thing Turing lacked was a society capable of accepting him, a society that would allow him to flourish. Instead he got one that took the best of his work and destroyed him in a fit of false moralistic madness. The society -not government true, but the society- that saved von Braun had destroyed Turing. If anything should be learned by the world from Turing's life (there is too much that must be learned from his work) it has nothing to do with homosexuality. What must be learned is the acceptance of deviation from the mean, the promotion of talent even if it comes with peculiarity, and the end of the wicked practice of the categorization of humans in a hierarchy of "moral correctness" that underpins all prejudice.