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.
Saturday, March 22, 2008
Thursday, March 20, 2008
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.
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.
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.
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.
Tuesday, February 26, 2008
Terminal Length
It has been a thought in the past that there must be some limiting hair length beyond which it would be infeasible for a person to be able to grow their hair. Because your hair has random breakages and can get pulled out by its roots and of course cut. Lets see if we can make our model predictive of some difficult to measure thing. A reasonable model of the hair length of someone who never cuts their hair (me for instance) is probably a good place to start. For what follows let L represent the average hair length and l represent the length a strand coming from a particular follicle. To begin with though we should put down some assumptions to simplify the model first the hair has a constant growth rate meaning d2l/dt2 = 0, which I think is a pretty reasonable assumption. Secondly lets assume that d/dr * d/dt * l = 0 meaning that the rate of growth is the same over the entire head, admittedly this is simply not true but it is still not a horrible assumption. The solution to our system without any breakage or shortening mechanism is simply linear growth both for l and L. I proposed three modes of shortening, breakage, pulling, and cutting. Cutting is an uninteresting case as it is merely setting the initial length distribution of the hair and so the models I develop for the other two cases can easily be applied to cutting to obtain solutions. Breakage is easier to model than cutting so I will begin there. A breakage model of hair will look something like (dl/dt)*t - E*t*(dl/dt) = l. I've broken it into two terms to make it clear what I am doing the first term is the growth term and represents the total length in the absence of breakage the second term is the breakage term where E is a randomly determined factor between 0 and 1. The equation in a more compact form is t*(1-E)(dl/dt) = l. Obviously E must be a monotonically increasing function to make sense. Otherwise breakage would sometimes make the hair grow longer. Though there are many ways one might structure E to have the right kinds of properties I am going to make it a randomly increasing recursive function. in a unit time E has a certain probability M of increasing. if the current value is E0 the value of E when it increases will be E1 = E0 + (1-M)(1-E) admittedly this isn't a terribly accurate model of breakage since the amount of breakage each time is fixed but it does have some nice properties for instance the more probable we take the breakage to be the smaller the breakage and the actual length of hair likely to be lost increases with increasing length (though not directly as a virtue of our function of E) I haven't worked out the function L as a consequence of the model yet and it doesn't look pretty more to come in future.
Wednesday, January 30, 2008
The stories of my return were greatly exaggerated.
So... apparently I am not back to blogging... in fact I suppose I have never really been that much of a blogger, I suppose I don't have the time at the moment to be a really good blogger. I have been applying for internships and doing homework and on occasion I admit I have just plain been goofing off. I haven't made a single new calculation on the prospect of collapsing Jupiter into a black hole... Sad but true. I have been making some other interesting sorts of computations... by that I mean I have been doing homework problems and calculating things like change in entropy the probability of spin states the distribution of electric fields that sort of thing. I hate the fact that I can do the homework even though I feel totally incompetent when it comes to tackling real problems. I am double majoring in mathematics and yet any physics problem that requires me to do a Fourier transform makes me cringe. This summer when I go to my internship (sounds like going to my rest a little doesn't it?) I think I will require myself to do a great deal of drill. That has always been my weak point I have always been able to understand the concept after doing like 2 problems and then I just go and take the test and if I do well then I don't need to do all the homework problems. So I end a math class having done a very small number of the problems and because the homework problems are easier than the real problems and because the test problems are even easier I can get by that way. Isn't that disgusting?
Wednesday, January 16, 2008
Towards an applet
I think I am going to try and write and then post an applet on my blog see I figure people like things that are interactive and or things that they can just watch. I could make the applet something that is actually interactive... but then that might be too much to ask for my first one to actually put up. I think instead it will just be something like an applet that does something like show a simulation of some aspect of the plan to collapse Jupiter. Later I could make it dynamic so that you can set parameters and watch it do cool stuff but... maybe that is asking a bit much. does anyone know if you can actually even put up applets on a blogger controlled site? I know how to embed applet tags into the html but I don't know how to upload content... uhh... any content at all actually... ok maybe I should think about putting up a picture before I try to put up something as dynamic as a program.
Sunday, January 13, 2008
The Conditional Pascal Wager
For those of you not in the know pascals wager is an argument that runs something like this. Let us analyze belief in god as a gambling wager. The expected return of a wager is simply the sum of the possible returns multiplied by their probabilities. We have two wagers that we can place we can either believe in god or not believe in god. Say there is some probability P(G) that god exists. Lets say we place the wager of believing in god then if god does exist we go to heaven and enjoy eternal bliss, an infinite payoff, if we believe and god doesn't exist then we get nothing. Calculating our payoff we have our average payoff is P(G)*infinity + (1 - P(G)) * 0 = infinity regardless of what the probability of god existing is. On the other hand if we don't believe in god and god exists we go to hell and are eternally tortured which is a sort of negative infinity and if we don't believe in god and he doesn't exist then we again get squat. So according to the pascal argument the expected return of believing in god is infinite while the expected return of not believing in god is negative infinity. That was strong enough reason for pascal to decide to become devout and ditch mathematics. Here is a post on Booms blog expanding on pascal's wager. I disagree with the argument in general but it is the impetus for this post so I figured I should at least link to it.
The biggest problem in the reasoning of Pascal's wager is that it is assumed that the probability that god exists P(G) is independent of the event of believing in god. I have used some standard notation up to this point without declaring it but after this point I am going to need a lot more notation so I had better introduce it formally. An event is the set of all outcomes which share some property. For instance if we were to talk about the event that the third flip in a series of coin flips comes up heads then the event that corresponded to this would be all possible coin flips in which the third flip is heads. The function P(X) is the probability of the event X. The notation P(X | Y) denotes the probability of the event X given that Y occurs. This conditional probability is equivalent to the probability of the event X intersect Y divided by P(Y) So the conditional probability only makes sense when Y has a non zero probability. Let the event of believing in god be denoted B and let the event that god exists be denoted by G also let the complement of any event X be denoted by X*. The complement of an event is the set of all possible outcomes in which X does not occur. Finally let the reward function be R(X | Y) which is the return we get upon the event X occurring when we have made the wager Y.
If two events have no bearing on each other then the probability of the combination of their outcomes is equal to the product of their probabilities. If all this doesn't make perfect sense to you here is a wiki article that might help.
In pascal's wager we take the expected return of believing in god to be P(G)*R(G | B). This calculation implicitly assumes however that the probability of gods existence is independent of our choice to believe or not believe. In other words P(G) = P(G | B) otherwise the calculated return should be P(G | B) * R(G | B). Of course the argument would run something like that whether god exists or not is an aspect of physical reality and therefore it is not a probabilistic event and so our belief or non belief could not possibly alter the existence of god one way or another. Either god exists or he does not and believing one way or another cannot change that so the assumption of this independence of belief and actuality seems consistent. However Pascal's wager is a deterministic partial information game not a complete information probabilistic game but pascal's wager is a probabilistic analysis. Chess is a good example of a perfect information game meaning that there is nothing about the game that is hidden from the players. A partial information game is where each player has only part of the picture such as in a game of poker. Poker is not probabilistic because once the cards have been dealt there is no shuffling or random influence in the game you don't know what cards your opponent has or what is on top of the deck but it is not randomly determined during the game. A game of craps is an example of a complete information probabilistic game because at each stage of the game you know everything there is to know about the state of the game but the outcome is randomly determined.
Even though poker is not a random game but rather a partial information game we treat it as a random game because the laws of probability apply because one can use probability to see how likely any particular initial set up is. Thus because pascal is applying the rules of probability to this partial information game if we are to be consistent with our analysis we must treat the existence of god to be something that is truly random. Just like in poker we let anything that is not yet known to be something that is randomly determined during the course of the game not something that has already been determined otherwise our probabilistic analysis would be inconsistent or at least incomplete.
Now let us consider the probability P(G | B). If there is indeed no interaction between god and the universe meaning that the universe would be exactly the same if god existed as it would be if he did not exist. If the existence of god makes no difference in the universe then our belief must certainly be independent of his existence and therefore the original assumption of the wager holds that P(G | B) = P(G) So in essence the pascal wager considers the condition that gods existence has no effect on the world whatsoever. While this might work for some deists I think most people would like to think god has at least some effect on the universe you know a few miracles here and there and whatnot at the very least resurrect a Jesus or two.
If god does have an effect on the universe then things get a little bit more complicated for the argument. Lets say that our belief in god is a function of a number of independent factors at least one of which is the actual existence of god. Since the case where the existence of god and our belief are in no way correlated under the original pascal wager we will assume that the correlation between our belief and gods existence is greater than zero. Since not everyone believes in god we also know that the correlation is strictly less than 1. Since the existence of god does not change from time to time or from person to person that means that the probability of a person believing in god is affected in a constant way by the existence of god. In other words because the existence of god does not vary any particular persons belief in god can be treated entirely as a function of variables other than the existence of god.
So here we have an experiment, We keep the existence of god constant and vary everything else, race, class, gender, sexuality, intelligence, etc. When we factor in all of these conditions and then look at the belief of these people as a cross section then we can use a statistical technique whose name at the moment escapes me in order to see what the effective dimension of the distribution is or in other words how many things the belief is dependent on. This isn't at all where I was thinking I would end up with this but I like that so I am going to stop there for a bit and I will give you the rest of the tour of the original thought I had in a later blog post.
The biggest problem in the reasoning of Pascal's wager is that it is assumed that the probability that god exists P(G) is independent of the event of believing in god. I have used some standard notation up to this point without declaring it but after this point I am going to need a lot more notation so I had better introduce it formally. An event is the set of all outcomes which share some property. For instance if we were to talk about the event that the third flip in a series of coin flips comes up heads then the event that corresponded to this would be all possible coin flips in which the third flip is heads. The function P(X) is the probability of the event X. The notation P(X | Y) denotes the probability of the event X given that Y occurs. This conditional probability is equivalent to the probability of the event X intersect Y divided by P(Y) So the conditional probability only makes sense when Y has a non zero probability. Let the event of believing in god be denoted B and let the event that god exists be denoted by G also let the complement of any event X be denoted by X*. The complement of an event is the set of all possible outcomes in which X does not occur. Finally let the reward function be R(X | Y) which is the return we get upon the event X occurring when we have made the wager Y.
If two events have no bearing on each other then the probability of the combination of their outcomes is equal to the product of their probabilities. If all this doesn't make perfect sense to you here is a wiki article that might help.
In pascal's wager we take the expected return of believing in god to be P(G)*R(G | B). This calculation implicitly assumes however that the probability of gods existence is independent of our choice to believe or not believe. In other words P(G) = P(G | B) otherwise the calculated return should be P(G | B) * R(G | B). Of course the argument would run something like that whether god exists or not is an aspect of physical reality and therefore it is not a probabilistic event and so our belief or non belief could not possibly alter the existence of god one way or another. Either god exists or he does not and believing one way or another cannot change that so the assumption of this independence of belief and actuality seems consistent. However Pascal's wager is a deterministic partial information game not a complete information probabilistic game but pascal's wager is a probabilistic analysis. Chess is a good example of a perfect information game meaning that there is nothing about the game that is hidden from the players. A partial information game is where each player has only part of the picture such as in a game of poker. Poker is not probabilistic because once the cards have been dealt there is no shuffling or random influence in the game you don't know what cards your opponent has or what is on top of the deck but it is not randomly determined during the game. A game of craps is an example of a complete information probabilistic game because at each stage of the game you know everything there is to know about the state of the game but the outcome is randomly determined.
Even though poker is not a random game but rather a partial information game we treat it as a random game because the laws of probability apply because one can use probability to see how likely any particular initial set up is. Thus because pascal is applying the rules of probability to this partial information game if we are to be consistent with our analysis we must treat the existence of god to be something that is truly random. Just like in poker we let anything that is not yet known to be something that is randomly determined during the course of the game not something that has already been determined otherwise our probabilistic analysis would be inconsistent or at least incomplete.
Now let us consider the probability P(G | B). If there is indeed no interaction between god and the universe meaning that the universe would be exactly the same if god existed as it would be if he did not exist. If the existence of god makes no difference in the universe then our belief must certainly be independent of his existence and therefore the original assumption of the wager holds that P(G | B) = P(G) So in essence the pascal wager considers the condition that gods existence has no effect on the world whatsoever. While this might work for some deists I think most people would like to think god has at least some effect on the universe you know a few miracles here and there and whatnot at the very least resurrect a Jesus or two.
If god does have an effect on the universe then things get a little bit more complicated for the argument. Lets say that our belief in god is a function of a number of independent factors at least one of which is the actual existence of god. Since the case where the existence of god and our belief are in no way correlated under the original pascal wager we will assume that the correlation between our belief and gods existence is greater than zero. Since not everyone believes in god we also know that the correlation is strictly less than 1. Since the existence of god does not change from time to time or from person to person that means that the probability of a person believing in god is affected in a constant way by the existence of god. In other words because the existence of god does not vary any particular persons belief in god can be treated entirely as a function of variables other than the existence of god.
So here we have an experiment, We keep the existence of god constant and vary everything else, race, class, gender, sexuality, intelligence, etc. When we factor in all of these conditions and then look at the belief of these people as a cross section then we can use a statistical technique whose name at the moment escapes me in order to see what the effective dimension of the distribution is or in other words how many things the belief is dependent on. This isn't at all where I was thinking I would end up with this but I like that so I am going to stop there for a bit and I will give you the rest of the tour of the original thought I had in a later blog post.
Lets analyze for a moment shall we the idea of physics as geometry. Now most successful physical theories have at their heart been geometries. For instance Newtonian mechanics was based deeply on euclidean geometry and the deep success of Newtonian mechanics is at least related to how well it sits with Euclidean geometry. Electromagnetics is an area where the connection is less strong but because the laws are so well described by fields the theory has become very much a geometric one though in its inception it was not necessarily so. General relativity is entirely dependent on geometry not only for its success but also for its explanation. Rather the reverse phenomenon occurred in the others where the geometry was a side effect of the explanation but with general relativity the geometry was the explanation. Quantum mechanics however is less than strictly geometrical because although we use geometry to describe the wave function of a quantum mechanical object the theory itself is not a geometrical object we merely use geometry as a useful means of calculating transitions. The wave function is only a probabilistic guide to reality any actual event requires a collapse of the wave function which makes the idea of viewing the wave function as the geometry of the real physical situation at the very least quite difficult. It may seem an odd sort of thing to require of the world but I like to think of the world as being a place that can be described by a static geometry. The uncertainty and indeterminism of quantum mechanics are such that this sort of description makes the description of our world by a static geometry impossible if you keep to the usual tenets of such representations. I think that it is possible that one could describe the world using a static geometric reperesentation if one is willing to accept the parallel universes postulate which is I think a very good argument for it.
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