Saturday, March 22, 2008

I can has comment?

yes I understand the vile thing I am doing in introducing lolcat syntax into my blog but I wants me some comments. Tell me how horrible I am for making such an attrocious mistake. Slander pickles and grapefuits or people who read new scientist. Just please leave me some comments. Of course if they happen to be vaguely topical that would be lovely and if it is a question (preferably one I know the answer to) awesome! I know you people are reading my blog... well I know that at least one or two people visit it a month... so this is a cry for some feedback. Please leave me some comments! Plus if I can trick you people into investing yourself in the blog enough to leave a comment that has to lead to increase future reading (and also increased posting), which is probably bad for you but good for me.


I define an ordering on a set S as a relation < defined on S such that for any a and b that are members of the set if a =\= b then either a < b xor b < a (xor is meant to make it explicit that this is an exclusive or) and such that if a < b and b < c then a < c. You could basically make any ordering you choose to care about for a particular set. We tend to take 3 < 4 but the definition of an ordering would still work just as well if 4 < 3 with all the other appropriate changes made (for instance if 4 < 3 then that would be a fine ordering relation but it would make it damn near impossible to make ordering work with a nice definition of addition) I will define a listing of a set as a injective mapping from the set of objects to the positive integers. If you think about any list you have ever seen you know that you can always number the items on the list 1, 2, 3.... and so on. While many lists don't have this numbering actually done that is not important, here I am simply looking for the quality of what makes something listable. Now for the paradox, the set of real numbers is not listable no matter how you try to list them any list you make will always leave out some real number. You can see this from the cantor diagonalization argument So there exist sets of things which can be ordered but which cannot be listed.
Intuitively it seems that ordering something and listing something are more or less the same. Taking a pile of rocks and putting them in order seems much the same process as taking those rocks and writing them down in order on a piece of paper. In fact it would seem that ordering is an even stronger requirement than listing since if you put the rocks in order along a line then the job of listing the rocks is done for you already. But from the example of the reals we know that this is not in fact the case apparently you can have orderability without having listability. Clearly any finite or even countably infinite set of objects is both orderable and listable, since by definition any countable set can be mapped onto the integers and so we can use the ordering relation on the integers to define one on any countable set. But orderability and listability become different when we move into the realm of sets with the cardinality of the reals. Let us go back and define listable and orderable in a little more detail. Listable means that given a set of objects the set is listable iff there exists an algorithm for reading through the set one element at a time so that you will eventually encounter every element in the set. The definition of orderability also changes slightly. A set is orderable if there exists an algorithm that has basically the same properties as an ordering. If the algorithm is given two numbers the algorithm must be able to tell if one number is less than the other in a finite number of steps.

Here is the fun part an ordering algorithm can be allowed to run for an infinite number of steps if it is given two numbers that are the same. The requirement is only that if the two numbers are not the same number that the algorithm will recognize that in a finite amount of time. Whereas the listing algorithm must conclude in a finite time for all elements in the set no matter what. So if you know something about the chomsky hierarchy that means that in some sense at least things that are orderable are recognizable languages whereas things that are listable are decidable languages. That is the difference between a list and an ordering.

Thursday, March 20, 2008


My blog is worth $1,129.08.
How much is your blog worth?

I figured it was worth about nil... odd...

semi-pure intelligences (humans)

A "pure intelligence" is something an intelligence that is only intelligence a mind without extra bits. What exactly constitutes "extra bits" stuff that is not absolutely necessary to the existence of a mind is certainly not something anyone can offer a definitive answer on. Clearly no human could ever be a pure intelligence because we have a bunch of extra stuff which we cannot go without our bodies for instance. Whether or not emotions are necessarily part of a pure intelligence I don't know. They cannot be easily be dismissed since emotions in large part are the motivational force behind human actions and one might argue that computation without motivation cannot be intelligence.

Perhaps such a thing as a "pure intelligence" cannot, even in principle, exist. After all every intelligence must have some physical basis in order to have a place in our universe so even a computer intelligence must have a "body" in some sense and therefore have some aspects of its existence which are not essential to what really makes a mind. When I say that humans cannot be pure intelligences because we have bodies I do not mean that having a physical shell is what makes our intelligence less than "pure". What I mean is that our bodies motivate us in ways that are outside of what is essential to our intelligence. One might argue that self preservation and reproductive urges could be essential consequences of real intelligence (though personally I find that hard to believe) but clearly many things that are important to human beings are simply visceral in their nature and nothing more. While sex might have the benefit of reproduction and one could eve say that there can exist a purely intellectual interest in it, people do not engage in sex primarily as an intellectual activity. This is generally true of a great many pass times snowboarding for instance is in part an intellectual activity (plotting your course and avoiding trees etc) but this is only a secondary reason why the activity is fun, the actual physical sensation is tremendously important to the activity.

Saying that this means that a pure intelligence can't have outside stimuli is perhaps going a bit too far. One could imagine a very powerful intelligence simulating an external world and a lesser intelligence interacting with it. One could of course say that the simulated intelligence is then not pure and the greater one because it is self contained is but the distinction is unnecessary in my opinion. Furthermore I am not willing to rob a pure intelligence of the ability to dream or imagine.

Human beings were not made for thinking but for hunting and surviving. Hunting in particular requires pattern recognition, planning and flexibility. Survival requires adaptability either in the form of adapting to ones environment or adapting ones environment to suit ones self. Adapting requires creativity you need to be able to understand your environment and imagine a way to change either yourself or the environment. The combination of pattern recognition and imagination with a sufficiently interesting database is probably sufficient for "intelligence".

Whatever the qualities necessary for intelligence human beings are not terribly good at them. We recognize patterns most readily that would have helped in tracking down food and other patterns that proved useful to our survival. Obviously not every pattern that humans are good at recognizing falls into this narrow view but I imagine a disturbingly large portion of them do. Because all of this overhead of pattern recognition and imagination comes at a cost of energy and development it wouldn't make sense for evolution to make us too good at it. Our brains already take up a large amount of the energy our bodies produce and when it comes to tracking prey once you can tell how a rabit is going to run there is no point to further increasing the power of the ability further and further.

Ultimately I find it rather annoying the degree to which things that are essential parts of who I am are not essential to what constitutes my "intelligence". I admit to being a visceral being. I like sunsets and not because they are interesting or elegant but just because I like to look at them. I like being warm and the way that I seek physical comfort from others and what kinds of ice cream I like are parts of me that are not important ultimately to what really makes my mind but they can be important to what makes me who I am. Sometimes I wish I could divorce myself from the physical aspects of myself and exist as a pure intelligence. More often I think that I would like to deepen my physical experience and experience more. I want to know what it is like to be human in the most general sense. I want to grow old and I want to be depressed and happy and I wish I could be female as well as male at some point (somehow I doubt I am alone in that). I want to know what it is like to be bald and to have long hair I want to break some bones sometime just to know what it is like. This deeply visceral aspect of my nature puzzles me but I suspect that it is an important part of what it means to be the semi-pure intelligence that is human.

Wednesday, March 19, 2008

MAA conference

I just attended my first real mathematical conference this past weekend and gave my first real presentation of my work (read, in a forum not catering entirely to undergraduate research). There are a lot of new things to think about that I picked up at the conference. This post is going to be basically just a long list of stuff that I thought about while at the conference rather stuff that was sparked by the conference. I want to put some of these thoughts down somewhere so that I won't just forget about all of them. Maybe in a year or two I will look back at this post and pick something I had forgotten about up again, Who knows.

I met Thomas Garrity from Williams College and had what I would like to call a conversation with him. This is really rather exciting for me because I do mean conversation, I do not mean that I asked him a question at the end of his lecture and got brief expository answer from him. I mean that there was actually dialogue. That is not to say that I think I was an equal partner in these discussions I fully realize that the flow of information was primarily from one to the other. But what characterizes a conversation (at least one part of the characterization) is that both parties shape the course of the discussion not just one or the other. In a lecture the one person chooses the subject and the other listens and even perhaps asks questions but they do not really have any real control over what is discussed or how. Especially in the atmosphere of conferences and lectures and the like it can be difficult for a lowly audience member to have a real conversation with a high up. When I went to Lisa Randall's lecture on campus during the science and literature symposium I found it essentially impossible to have any more than completely superficial interaction with her. She certainly didn't seem interested with interacting with one of the swarm. Admittedly the mathematics conference was a very different sort of atmosphere since most of the people attending were not physicists.

Over the course of the two days of the conference I talked to Garrity a number of times and there are four things in particular that I will have to give some greater thought.

The most obvious one is what he gave his lecture on, at the end of the lecture he posed an open problem which is does there exist a way to express real numbers as a sequence of integers so that the sequence is periodic if and only if the number being represented is cubic. Cubic here means that somehow it contains a cube root. Continued fractions do it for square roots and decimal expansions are periodic for rational numbers which is basically the impetus for thinking about the question.

More interesting though were the things that we discussed outside of a lecture structure. First off he made me question whether my recent push to find a method of imposing a coordinate structure onto a fractal space is possible at all. I had always assumed that it was of course possible and even had a vague set of arguments why it should be possible. Something I hadn't considered is that many (maybe all?) fractal spaces are not decidable. By that I mean that probably in order to parameterize a space you probably need to be able to decide what points are in the space and what points are not in the space in a finite number of steps. There is no finite time algorithm for computing the mandelbrot set by which I mean that given any general point in the complex plane you cant always in a finite number of steps decide if the point is in the set or not (assuming that the operations on the reals such as addition and multiplication have a finite computational value which you just make 1 step for convenience). This sort of points out to me that I really don't know exactly what parameterization really is. On the superficial level that I was thinking about it a parameterization was simply some function which mapped the set of points of some space onto an ordered set of real numbers. Clearly such a procedure must exist because the cardinality of the sets are the same and so therefore a one to one mapping exists. At the very lowest level that means that you can construct such a map from one to the other in a direct manner just by looking at all the points and making the map one set of points at a time doesn't it? Actually the answer is no, it doesn't mean that you can make such a map because you cant make a list of the points and so if you cant find a way to characterize all of the points in your set then you clearly can't make the map. So in at least one well defined sense of what you mean by "generalized coordinate system" there is a clear reason that there cannot exist a generalized coordinate system for the space of the mandelbrot and probably most fractal spaces. I don't really have any reason to think that this result would generalize to any fractional dimensional space but I do. There might be very nice fractal dimensional spaces that are decidable but I doubt it. It just so happens that this piece of the puzzle fits with what I have been thinking for a while now, that fractal dimensional spaces cannot be handled within the framework of first order logic

This brings us to the next thing we talked about which I will just mention briefly. That is the suggestion by a philosopher of the name Hintikka that the usual first order logic needs to be extended by game theoretic ideas. I don't know if I agree yet or not but it certainly sounds like something that I should look at.

Lastly, he talked about the applications of the partition function to abstract objects in sets instead of to thermal physics. The partition function being something that I am only just really becoming familiar with in my thermal physics course this semester I was of course intrigued. Apparently the properties of the partition function can be very useful in the analysis of completely abstract collections of things.

I also attended a talk on the ordering dimension of partially ordered sets. I personally thought that the talk was not of particularly high quality and the idea was not very exciting since it was merely a slight further generalization of a set of objects called generalized crowns. However the talk did introduce me to the concept of order dimension and made me think a little bit more about what a dimension really is. While the talk didn't actually go into any real detail about the nature of partially ordered sets or posets since they basically assumed that everyone in the audience was familiar with them (bad assumption). I still don't know in a formal way what a partially ordered set is but basically a partially ordered set is one in which there is a partial ordering relation. Lets say you have 5 objects in the set and let @ denote an ordering of some kind of two objects like for instance a @ b shows that in some sense a < b. A partial ordering on the five objects a, b, c, d, e might say something like a @ b, c @ d, e @ d. in which case we would know nothing about the "ordering" of a and d so the set is only partially ordered. Thinking about this made me think of possible connections to second order logics and also more to the point non-integer dimensional spaces. I don't know if this really sticks but it would make sort of intuitive sense that one requirement for a set to have a countable cardinality is for there to exist a complete ordering on the set. This condition is related to being able to write the set as a list since a list constitutes an ordering on the set (the things that come before something on the list is "less" than it and things that come after are "greater" than it). But it is intriguing to notice that having an ordering relation is not the same as the capacity to write a set as a list. For instance the reals have a complete ordering but they cannot be written as a list. I wonder if perhaps things with the cardinality of the power set of the reals cannot have complete orderings. At any rate any set with at least two elements can have a partial ordering on it (as I understand it) since you can just relate two elements and leave all the others unrelated. Seeing how this is related to non integer dimensions is perhaps a bit of a stretch and this post is primarily meant to write down ideas not expound on them in detail (and it is running long as it is) so I will just leave that one alone for now.

Sunday, March 9, 2008

Back in Black

Ok so I haven't been raving about black holes enough lately so lets get a nice good rave about turning Jupiter into one theoretically that is what this site is all about after all. Since in my nanowrimo novel I am using a society which has become an interstellar one by use of a Jovian singularity it wouldn't hurt to think about its implications a bit more.

One interesting thought is that the existence of a Jovian singularity implies that humanity became very certain at some point that it needed one. For some reason humanity can do certain very important things with a singularity that would be impossible without one. This means that before the black hole was created we knew pretty well what we could do with it. If we weren't really very certain then we would almost certainly not have been willing to sacrifice an entire planet just to satisfy curiosity.

Of course that also might say something about the government that would be willing to sacrifice a planet for whatever reason. The chances of a government being willing to get rid of Jupiter though are helped out by the fact that really ultimately Jupiter is valuable only as a research tool and as a gravity well that keeps its moons in orbit. As a research tool Jupiter is an interesting study of fluid mechanics and weather patterns and I'm sure of other things as well. As a black hole though it would provide an unbelievable basic physics laboratory. Plus as an added bonus the gravity well would be unchanged and so all the moons could still hang out in orbit. Of course probably collapse into a black hole of Jupiter would send the moons flying. I'm not sure how much energy would be released from the remains of the planet getting sucked in. If the collapse was quick enough it might just be a brief radiation burst and that's the end of it. But my bet is that it would take quite a while. If the moons of Jupiter house any sort of life (or come to by way of human habitation) then most likely we would not be willing to nuke them with the radiation even if the moons would still stay in orbit. But as far as a sci-fi like setting goes having a government that is willing to blow up Jupiter probably means either some sort of really crazy totalitarianism or plutocracy etc. I doubt a real democracy would ever get close to doing something so destructive but then... there are lots of ways people might be motivated to it. For instance if the earth has been thoroughly trashed and an 18 billion population earth is looking for a new place to trash.

Or maybe we want to go all doctor who and say that our future society is extremely power hungry and they think that a black hole would be a nifty way to get lots of energy at will. Actually this is probably the surest perk to having a local black hole. Since a Jovian singularity would be rather small actually it would put out quite a large amount of energy in the way of hawking radiation. But sucking away the rotational energy would probably be more efficient and if we wanted to we could even feed the black hole some gas and get the energy from the radiation which is probably the best way to go about things.