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Counting sheep

It takes me about forty minutes on the train to get to work in the morning. Which is a great opportunity for sleep. But today there where ladies in the seat ahead of me talking, despite it being the quite carriage.

So to distract me from those inconsiderate women I counted sheep just like any other insomniac. And because I'm a an unabashed furry I anthropomised my sheep into cute bipedal sheep girls.

So I sat in my seat imagining these sheep girls parading past and as they stepped in front of me I would give them an ear tag. The first numbered with zero because as well as a furry I'm a computer scientist and we always number from zero.

But what was I to do with ovine friends after they received there numerical earring? Unlike other sheep counters I couldn't house them in pastures it would be inhumane and degrading. So I mentally constructed a grand hotel for them. It was simple, sheep 0 would he housed in room 0, sheep 1 in room 1, and so on. Of cause there are only as many rooms as there are room numbers because having a hotel room without a number would just be confusing.

So I continued to do this for a while, but the amusement was wearing thin. So each as each girl passed I took half the time to imagine giving her her ear-ring and sending her to her room as the girl previous.

Quite soon all the rooms in my hotel where filled. For each natural number there was a sheep girl and each girl was in her room. So what was I going to do with the next sheep that was looking up at me ever so cutely and bleating at me for a room and for that matter what was I going to write on her ear-tag?

Given the cute ω look on her face I decided then and there that I was going to name the time compressed event ω and then named this lamb ω + 1, the sheep after ω.

But where was I to house ω+1? Fortunately I knew a trick with rooms in grand hotels. I would simply move everyone into the room twice there current room number. Now we have plenty of space, so much I fired up the time compressor and filled the hotel once more.

So my next sheep was ω*2+1 and I realized that these ω events could be treated like the sheep themselves and I could pull my time compression trick on them.

So I spent the rest of my trip doing that, working out sequences and compressing them. When I got off at my train station I was left wondering. If I continue with this behavior is there any point that I will have more sheep then rooms to house them?

The cosmic horror within mathmatics

At its core lovecraftian horror is not about tentacles and gibbering things. Its about the fear that the universe is fundamentally beyond human understanding. The time scales too grand, the distances so vast, that our intuition and our reason are inadequate to the task.

I have for a quite a long time had an intuitive understanding that in many fields of mathematics there are hierarchies where each member of the hierarchy has a greater power then the one below it. However there is a point in the hierarchy where intuition starts to break down, where maths no longer behaves in a manner that one would expect.  Maths starts to be about not what one can know but the limitations of that knowledge.

Within the Chomsky Language hierarchy for example that boundary happens between the recursively enumerable and the context sensitive.  With the theory of computation that happens when we shift to Turning
machines. The location of the boundary is not coincidental as the links between language acceptance and computation are well known.  This is the boundary where the halting problem takes hold.

Likewise in logic when one transitions from first order logic to second, and when one creates a formal system strong enought to model the natural numbers one passes over that threshold again. In this case the boundary is governed by Godel's incompleteness theorem.

Going a little deeper there are connections between the halting problem and Godel's incompleteness, such that you can prove one from the other and visa versa.

However we can look at these boundaries from a different perspective.  Defining a property within a set of elements is effectively creating a subset of that set. It follows that there are 2^n non-redundant properties (where n is the number of elements in your set).

However when you deal with the natural numbers while there are 2^(aleph_null) properties there are only aleph_null sentences to describe those properties. Basically there are more properties then we can talk

My intuitive feeling of "weirdness" is a reflection of the fact that the vast majority of mathematics at that point becomes embedded in incomprehensibility. If one can't form sentences about a subject then one can't use the tools of logic.

Mathematics is an island of sanity and rationality floating on a maddeningly uncountably infinite ocean of arational paradox. Mathematics is a lovecraftian cosmic horror.

An answer to a question about maths.

About a week ago I was discussing mathematics and philosophy when I was asked. In the equation "1 + 2 = 3" is the "1 + 2" part of it the same entity as the "3"? It was agreed that they where clearly equivalent and equal but
where they the same thing.

I've been thinking this over and now feel confident answering the question, unfortunately I have two answers.

The first is that the question can't be answered because it contains a type error. Distinct existence, that is to say having separate instances, is a property of concrete objects. Numbers being pure abstractions can't be considered in the same way.

However this answer isn't that satisfying. And while abstractions don't have separate existences in the concrete sense there may be some property that distinguishes the "1 + 2" abstraction from the "3" abstraction.

I'll dismiss first the trivial, clearly the string "1 + 2" is not the same as the string "3". But the strings are just labels that allow us to talk about the abstractions and not the abstractions themselves.  It is perfectly possible for two strings to refer to the same entity.

In both mathematics and philosophy its important to get your definitions in place, or otherwise you can end up discussing different things in a confused manner. One of things we first have to establish is what are we talking about when we say "Number". Since in the initial discussion we where talking about beer coasters and how many of them there where, I'm going to use cardinal numbers that is the numbers representing the quantity or count of a collection.

Each cardinal number can be defined semi-formally as the class of sets which can be placed into a bijective relation with one another.  Where a bijective relation is a relation that pairs off every element of one set with every element of the other.

The cardinality class containing the empty set is called 0. We can also define a function.

s(X) = {{}} union {x element of X| {x}}

This function returns a set with an additional empty set as an element and wraps all the elements within the argument set inside there own sets.

We can also create a mapping

S: cardinality class -> cardinality class
S(c) = c' is the cardinality of s(x_c)
where x_c is a set of within c

We call S(0) one. The mapping S is called succession and can be thought of as returning a number one more the the number given to it.

I've got most of the definitions in place all that remains is addition which I define recursively as follows.

x + 0 = x
S(x + y) = S(x) + y

So we are back to "1 + 2 = 3" which in our language of classes and succession is "S(0) + S(S(0)) = S(S(S(0)))". Which we can step through.

S(0) + S(S(0)) => S(0 + S(S(0))) => S(S(S(0)))

Which spelt out says any set with one element additively combined with a set of two elements will return a set with three. In other words both sides of the equation "1 + 2 = 3" are talking about the class of sets that can be put in bijection with (for example)  {{},{{}}, {{},{{}}}}. It is the same class.

So the answer to the question is, yes both sides of the equation are the same thing.

More on my binary tree structure

The structure is a system of tiers, each tier alternates between a references to a tiers one order deeper or to complete balanced binary trees of depth n, where n is the level of the tier.

With the rules that if a change would make two trees be next to each other those two trees are merged into a tree or depth n+1 and promoted to a tier one deeper.

And that if two tier references are next to each other the tiers are concatenated together.

Binary trees

I like balanced binary trees.  But you can only have a balanced binary tree when the number of items you have is 2^n.  However I thought that you can get the same effect if you have an "overflow" bbt that takes up the nodes that are more then is in your tree, I'll start on some code later so it makes scene. 

The question the devil can't answer

 When I was in high school I recall a joke along the lines of "Three stereotypes are in hell the devil approaches them and says 'If you ask a question of my that it can't answer correctly then I will grant you your freedom'"

Now the jokes punch line is a stupid bit of anti-intelectionalism but taking the question as a brain teaser is better.  My best answer so far.  "What is the question and the answer to that question that the devil can't answer?".
 I can hear and read the speech and writing of the smartest, most entertaining and funny.
The entire world's knowledge is being condensed processed and presented in a form that is accessible to all.
The love of reason is on the rise pushing back superstition and magical thinking.
I truly believe that we are in the midst of a second enlightenment.   

Kodomo no Jikan Reviewing the reviewers

As I said in a previous entry I came across KnJ by reading a site that called it the worst anime of 2000. I've since finished watching the anime and read the manga to this date. When searching I've also come across discussion and reviews of KnJ. However many of them have been of very poor quality.

The other thing that draws me to do this is the controversy that surrounds this anime. Controversy is a type of conflict and from conflict is the source of all drama. If I was more post modern I would be critiquing the controversy itself as a text but that would be silly. Instead I'm going to critique the reviews instead, at least they have some form of concrete instantiation.

Review is hereCollapse )

The madness of imaginary crimes.

Have we as a society grown so irrational and psychotic that we have to be reminded that there is a boundary between the reality outside our heads and the fantasies within? Are we in danger of drifting into a stupefied slumber where the figments of our dreams can claim the rights that we have previously reserved for concrete entities?

The characters of fiction, written down as literature, drawn as manga or anime, expressed in the multitude of means technology has granted to us, lack independent agency. By hijacking our mind's ability to empathize and reason about other minds they create the cognitive illusion that they exist.

However we must remember that without sentience, without true moral agency, these mental phantasms whose pseudo-existence is purely within the mind of their hosts can make no ethical claims to the concrete world outside. I can imagine the vilest of crimes within my head without harming a single creature of ethical consequence. I can commit fictional atrocities whose magnitude and horror eclipse the worst tyrant's genocide but still act and be in every way a moral, ethical contributor to society.

This fact should be obvious to every sane adult, indeed learning to distinguish between the imaginary and the real is one of the prerequisites for being an adult. I should not need to devote three paragraphs of text to hammering this point but unfortunately I feel I must. People have been charged, prosecuted and punished for this style of thought crime.

If such actions were mistakes of a more censorious past one could look back with the knowledge of lessons learned. However not only are real people being prosecuted for crimes against fake people in the present, laws are being passed to make it easier to prosecute this in the future. 

Exposing children to Homosexuality

The argument I was posting to youtube which had to be cut dramatically short was in response to a comment along the lines of "Its OK if gay people exist as long as children don't find out."  Or words to that effect.  Indeed homosexuality is an "Adult concept" in the eyes of our censorship board.

However I have come to the view that the presence of out GLBT in the community and in the media where children may see them is a good thing.  Most people realize there sexual orientation during early adolescence and a significant number report knowing there orientation while they where in primary education.

Hiding homosexuality from children just means that the population of nonhetrosexual children and teenagers who feel isolated, that they are the only ones like this and they have no role models at all.  Indeed part of the way we learn how to navigate the difficulties of adolescence is  by seeing how others have dealt with or are dealing with simler struggles.