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.
Sunday, December 30, 2007
Christmas break
Admittedly I have been lax in my blog posting duties. If anyone out there is still listening (doubtful I know) then I promise you to double the number of my posts inside of the next month. I realize that this isn't much of a promise since that averages out to like a post every two days... and I can and might do them all at once if I get too far behind but I am going to be taking considerably fewer hours of school this semester. I haven't really had time to post during the break because my dad has been in the hospital and that combined with the regular holiday madness has consumed my time.
Monday, November 5, 2007
Late Start
If I want to have any chance of being able to achieve the nanowrimo word count of 50,000 then I had best get started soon. I still haven't written anything for it. If I had been writing every day then I would have had to average 1666 words a day but if I start now then I will have to average a nice round 2,000 words a day. I have decided to go with the spark thing and I will start it today. Right now I will work with the following general outline of the story order.
1. scene featuring the antagonists performing some sort of misdeed and foreshadowing the danger to our hero (~8 pgs)
2. ordinary life of our hero (~15 pgs)
2.3 include information of area of study of protagonist
2.5 subpoint in normal life introduce hero romance interest
3. botched assassination/kidnapping attempt on protagonist (~4 pgs)
4. introuduce the good guys (may or may not have helped prevent #3) (~5 pgs)
5. flight and hiding (~15 pgs)
6. betrayal or failure of some sort (~8 pgs)
7. secondary flight (~5 pgs)
8. capture of either protagonist or love interest (~2 pgs)
9. Revelation of betrayers identity (~2 pgs)
9. confrontation of main protagonist with a powerful villain ending in failure for protagonist (~5 pgs)
10. horrible fate averted/ redemption of betrayer (~3 pgs)
11. Escape (~2 pgs)
All in all that should carry me to the 50,000 mark and a bit beyond hopefully it won't be total crap. Ok I have wasted enough time not writing it I had better start.
1. scene featuring the antagonists performing some sort of misdeed and foreshadowing the danger to our hero (~8 pgs)
2. ordinary life of our hero (~15 pgs)
2.3 include information of area of study of protagonist
2.5 subpoint in normal life introduce hero romance interest
3. botched assassination/kidnapping attempt on protagonist (~4 pgs)
4. introuduce the good guys (may or may not have helped prevent #3) (~5 pgs)
5. flight and hiding (~15 pgs)
6. betrayal or failure of some sort (~8 pgs)
7. secondary flight (~5 pgs)
8. capture of either protagonist or love interest (~2 pgs)
9. Revelation of betrayers identity (~2 pgs)
9. confrontation of main protagonist with a powerful villain ending in failure for protagonist (~5 pgs)
10. horrible fate averted/ redemption of betrayer (~3 pgs)
11. Escape (~2 pgs)
All in all that should carry me to the 50,000 mark and a bit beyond hopefully it won't be total crap. Ok I have wasted enough time not writing it I had better start.
Saturday, November 3, 2007
nanowrimo
So november is national novel writing month and I am definitely going to make it to the 50,000 word count this month... even though I haven't started yet and actually probably have even less time to do it than I have had other times. I still haven't decided what kind of book it is going to be. I was thinking either science fiction or fantasy if it was going to be a science fiction I was going to do some sort of story about a set of superhuman genetically engineered creatures who are essentially immortal and each of which has a very different view on the prospect of being immortal. That may sound a bit fuzzy and that is because it is a bit fuzzy I'm not really sure where to go with that one.
On the other hand if it was going to be a fantasy story I think I would do something rather similar to the backstory of my current dnd character which runs something like this. There is a powerful mage who is collecting the memories of people from accross the planes of existence and our hero is earmarked to have his memories consumed by this being and he would rather like to avoid it. It could be fun to write but just exactly how our hero defeats the godlike magus is not clear to me but that is ok... I don't have to end the story I only need to drag the story out to 50,000 words.
Lastly though I have been thinking about a story that is more of the sci-fi/fantasy mix thing it would be set in a modern day setting (maybe a few years in the future for fun) and the main character would be a skeptical physicist (yes I know it is tacky to base ones main character on ones self but he isn't me he would just be posessed of certain qualities of me and how else can a writer work but to put bits of themselves on the page?) This physicist would be relatively ordinary except he would happen to have a soul. It would not be called a soul in the book it would be called something more subtle like a "spark" and then be immediately likened to a soul something that is the essence of the person who has it. In the context of the story the idea is that not everyone has a soul or rather not everyone has a "spark". The book wouldn't make it clear what the nature of this spark object was there would be multiple groups with different theories about it. There would be a religious group who believe the spark is the physical manifestation of the soul and that those people who have the spark are something akin to a godly army on earth. But there would also be more secular groups who would simply be researching the properties of this relativley newly discovered "spark" of course our hero much more readily takes to the idea that this spark can be explained with the sciences of the day. At any rate the plot of the story runs something like this sparks give super powers to people who have them and there is something very unusual about our hero's spark that makes him very desirable to a large number of groups the good guys get to him first but then he is on the run and has to find out what is going etc.
if you feel like it you could post comments on the story ideas I rather think I won't have anything set in stone for the next couple of days.
On the other hand if it was going to be a fantasy story I think I would do something rather similar to the backstory of my current dnd character which runs something like this. There is a powerful mage who is collecting the memories of people from accross the planes of existence and our hero is earmarked to have his memories consumed by this being and he would rather like to avoid it. It could be fun to write but just exactly how our hero defeats the godlike magus is not clear to me but that is ok... I don't have to end the story I only need to drag the story out to 50,000 words.
Lastly though I have been thinking about a story that is more of the sci-fi/fantasy mix thing it would be set in a modern day setting (maybe a few years in the future for fun) and the main character would be a skeptical physicist (yes I know it is tacky to base ones main character on ones self but he isn't me he would just be posessed of certain qualities of me and how else can a writer work but to put bits of themselves on the page?) This physicist would be relatively ordinary except he would happen to have a soul. It would not be called a soul in the book it would be called something more subtle like a "spark" and then be immediately likened to a soul something that is the essence of the person who has it. In the context of the story the idea is that not everyone has a soul or rather not everyone has a "spark". The book wouldn't make it clear what the nature of this spark object was there would be multiple groups with different theories about it. There would be a religious group who believe the spark is the physical manifestation of the soul and that those people who have the spark are something akin to a godly army on earth. But there would also be more secular groups who would simply be researching the properties of this relativley newly discovered "spark" of course our hero much more readily takes to the idea that this spark can be explained with the sciences of the day. At any rate the plot of the story runs something like this sparks give super powers to people who have them and there is something very unusual about our hero's spark that makes him very desirable to a large number of groups the good guys get to him first but then he is on the run and has to find out what is going etc.
if you feel like it you could post comments on the story ideas I rather think I won't have anything set in stone for the next couple of days.
Thursday, November 1, 2007
odd perfects
I did a research project a while ago on Odd perfect numbers (as in they actually paid me to do original research. Turns out my research wasn't so very original basically everything I thought up has been thought up before. While they aren't actually paying me for it anymore I have been continually trying to figure out some sort of novel approach to the problem. Today I was thinking about odd perfects instead of working on my electronics homework and the following equation found its way onto my page,
ln(2) = sum(ln(Pi-(1/Pi)ai)-ln(Pi-1))
Which is actually a rather unremarkable equation since in the end it ultimately says something rather akin to the sum of the reciprocals of the divisors of the OPN is equal to 2. The reason putting this equation together so intrigues me is that it brings up a question of how difficult it can be to determine whether or not a given equation has any solutions at all. This more complicated equation made me realize that I don't even know when the equation sum(1/xi)=c has solutions. What series of reciprocals can I add together to get 5? does there exist any set of reciprocals that will give me 5? Without having even begun to think about proving it I think that any rational number can be expressed in terms of a sum of reciprocals of different integers. After all the sum of the reciprocals diverges so there is no number too large to be approximated by them and they converge to zero so there is no number that they would skip over so to speak. Of course that isn't a proof and there very well may be some sort of bizarre number that cannot be written as a sum of reciprocals for some strange reason just like there are ratios of quantities that can't be written as the ratio of two integers (the ratio of the diagonal of a square to its side being the famous one of course). Assuming for the moment that we can express any rational number as a sum of reciprocals of integers then the natural question arises is there more than one way to do it? I know that 1/2 + 1/3 = 5/6 but can I break 5/6 into a combination of the reciprocals of any other two or more numbers? Furthermore how does one go about finding out how many reciprocals is the minimum necessary to compose some number? How many reciprocals does it take to add up to 19/3? or can you at all?
ln(2) = sum(ln(Pi-(1/Pi)ai)-ln(Pi-1))
Which is actually a rather unremarkable equation since in the end it ultimately says something rather akin to the sum of the reciprocals of the divisors of the OPN is equal to 2. The reason putting this equation together so intrigues me is that it brings up a question of how difficult it can be to determine whether or not a given equation has any solutions at all. This more complicated equation made me realize that I don't even know when the equation sum(1/xi)=c has solutions. What series of reciprocals can I add together to get 5? does there exist any set of reciprocals that will give me 5? Without having even begun to think about proving it I think that any rational number can be expressed in terms of a sum of reciprocals of different integers. After all the sum of the reciprocals diverges so there is no number too large to be approximated by them and they converge to zero so there is no number that they would skip over so to speak. Of course that isn't a proof and there very well may be some sort of bizarre number that cannot be written as a sum of reciprocals for some strange reason just like there are ratios of quantities that can't be written as the ratio of two integers (the ratio of the diagonal of a square to its side being the famous one of course). Assuming for the moment that we can express any rational number as a sum of reciprocals of integers then the natural question arises is there more than one way to do it? I know that 1/2 + 1/3 = 5/6 but can I break 5/6 into a combination of the reciprocals of any other two or more numbers? Furthermore how does one go about finding out how many reciprocals is the minimum necessary to compose some number? How many reciprocals does it take to add up to 19/3? or can you at all?
Tuesday, October 23, 2007
forshadowing
So I really haven't had much time lately to do much of anything which is not all that surprising considering I am taking 21 credit hours this semester. I thought I could spare a quick minute to make a post though mostly because I am having trouble with this calculation for number theory and I am taking a break.
So far I have been trying to make it clear how I have been ariving at results and giving a bit of background explanation to most things that I have been talking about. I am going to stop doing that for a second and just give you some straight results without talking about how they were derived.
If Jupiter were turned into a black hole space time immediately around the event horizon would be moving circularly at about 168 times the speed of light.
I estimate that the core densities and temperatures necessary to create a black hole are around 1028 g/cm3 and 1019 K which would need to be sustained for only 10-20 sec or so...
If a system of atomic bombs could be used to send pressure waves into Jupiters core the minimum necessary pressure could be provided by about 107 one megaton atomic bombs.
The method of creation using equally distributed h-bombs is not very economical but is the least invasive as it would cause a minimum of damage to the moons and surrounding systems.
I'll fill you in a bit on how I got these estimates later now its back to hw.
So far I have been trying to make it clear how I have been ariving at results and giving a bit of background explanation to most things that I have been talking about. I am going to stop doing that for a second and just give you some straight results without talking about how they were derived.
If Jupiter were turned into a black hole space time immediately around the event horizon would be moving circularly at about 168 times the speed of light.
I estimate that the core densities and temperatures necessary to create a black hole are around 1028 g/cm3 and 1019 K which would need to be sustained for only 10-20 sec or so...
If a system of atomic bombs could be used to send pressure waves into Jupiters core the minimum necessary pressure could be provided by about 107 one megaton atomic bombs.
The method of creation using equally distributed h-bombs is not very economical but is the least invasive as it would cause a minimum of damage to the moons and surrounding systems.
I'll fill you in a bit on how I got these estimates later now its back to hw.
Friday, October 19, 2007
finitaphobia
The Greeks abhorred the infinite thinking that anything that was infinite was illogical. Zeno's famous paradox is a good example of how introducing infinity can cause logical trouble. Without allowing ourselves to use the infinite though we cannot arrive at calculus which uses the continuity of numbers as a necessary part of its operation. In some sense of course the Greeks allowed countable infinities grudgingly since they didn't think that there was a "last number" so to speak but allowing for unboundedness is a long way off from allowing the entity of infinity to exist as a philosophically analyzable entity. Allowing for unboundedness allows the Greeks at least partial access to the first order of infinity the countable infinities. For those of you who aren't terribly familiar with the concept of transfinite numbers there are different sizes of infinity the infinity that is the number of the counting numbers is called countable infinity because anything that is countably infinite can be put in a one to one correspondence with the counting numbers. That is countable infinities can be numbered while uncountable infinities are too large to be counted. If that doesn't make sense to you go find some material on cantor set theory.
The Greeks had a pretty good understanding of the rational numbers which are countably infinite. Saying that the Greeks were really only afraid of the real numbers (which are uncountable) is being too nice. They abhorred the existence of numbers like the square root of two since it could not be expressed as a rational number. The Greeks were quite happy to deal with the rational numbers as discrete objects but they would have frowned on talking about all of them as a whole. There is nothing disturbingly infinite about 2/3 but you cannot then haul off and start talking about all the numbers a/b that are between 0 and 1. The later the Greeks would have viewed as beyond the realm of logic because of its association with the infinite.
This inability to talk about questions which touched on the infinite or to use any sort of methods which invoked the infinite limited Greek mathematics. The reason of course for this abhorrence of the infinite goes deeper than the possibility of logical inconsistency, not everyone agreed with Zeno. The real reason is that in mathematics, logic, and geometry the Greeks did not just see disciplines of abstract thought. The profession of logician and geometer was to deal with the world as it really was. The Greeks took philosophy and mathematics as a way of uncovering deep truths about their own physical and metaphysical reality. The abhorrence of the infinite was rooted in the belief that the physical world cannot support the infinite and so any infinite argument or object does not have any reality and so does not deserve to be thought about.
I am afraid that in the intervening time physicists may have become afraid not of the infinite but of the finite. Mathematicians also though to a lesser extent tend to distance themselves from problems that are inherently discrete. We have learned how to deal with the infinite in a logical and rigorous way and we have found the tools that this dealing with the infinite has given us so useful that we have become at least partially unable to work without them. The assumption of the continuity of space is so basic to the world of physics that is essentially completely unchallenged in all of physics. Even in quantum mechanics where things become discrete instead of continuous the underlying space is still thought to be continuous. String theory is even worse since it assumes a completely flat subspace in which the strings are to interact. If you don't see the connection between continuity and the finite then I will just take a moment to point out that continuity is a concept that cannot be achieved without the use of something similar to the real numbers and that means the introduction of uncoutable infinities. Countable infinities are what you get when you allow discrete systems to be unbounded. So intrinsically countable systems are discrete. Since countable systems are necessarily discrete (even though any two rational numbers can approach each other as close as you like every set of rational numbers is disconnected)then that implies that countable systems are necessarily made up from finite objects.
What I really am talking about when I say that physicists have a fear of the finite is that they have a fear of methods that are based in discrete objects. Calculus with its dependence on uncountable infinity and the continuity of things that are being examined is totally indispensable to modern physics. I don't necessarily think this is a bad thing in the sense that calculus is a wonderfully robust and interesting tool and we should be not afraid to use it. However physicists should have a way around calculus no matter how painful and difficult there ought to be a way to do physics that is fundamentally discrete but there is no such thing. Mathematicians and computer scientists can play with the discrete but physics remains fearful of it.
The Greeks had a pretty good understanding of the rational numbers which are countably infinite. Saying that the Greeks were really only afraid of the real numbers (which are uncountable) is being too nice. They abhorred the existence of numbers like the square root of two since it could not be expressed as a rational number. The Greeks were quite happy to deal with the rational numbers as discrete objects but they would have frowned on talking about all of them as a whole. There is nothing disturbingly infinite about 2/3 but you cannot then haul off and start talking about all the numbers a/b that are between 0 and 1. The later the Greeks would have viewed as beyond the realm of logic because of its association with the infinite.
This inability to talk about questions which touched on the infinite or to use any sort of methods which invoked the infinite limited Greek mathematics. The reason of course for this abhorrence of the infinite goes deeper than the possibility of logical inconsistency, not everyone agreed with Zeno. The real reason is that in mathematics, logic, and geometry the Greeks did not just see disciplines of abstract thought. The profession of logician and geometer was to deal with the world as it really was. The Greeks took philosophy and mathematics as a way of uncovering deep truths about their own physical and metaphysical reality. The abhorrence of the infinite was rooted in the belief that the physical world cannot support the infinite and so any infinite argument or object does not have any reality and so does not deserve to be thought about.
I am afraid that in the intervening time physicists may have become afraid not of the infinite but of the finite. Mathematicians also though to a lesser extent tend to distance themselves from problems that are inherently discrete. We have learned how to deal with the infinite in a logical and rigorous way and we have found the tools that this dealing with the infinite has given us so useful that we have become at least partially unable to work without them. The assumption of the continuity of space is so basic to the world of physics that is essentially completely unchallenged in all of physics. Even in quantum mechanics where things become discrete instead of continuous the underlying space is still thought to be continuous. String theory is even worse since it assumes a completely flat subspace in which the strings are to interact. If you don't see the connection between continuity and the finite then I will just take a moment to point out that continuity is a concept that cannot be achieved without the use of something similar to the real numbers and that means the introduction of uncoutable infinities. Countable infinities are what you get when you allow discrete systems to be unbounded. So intrinsically countable systems are discrete. Since countable systems are necessarily discrete (even though any two rational numbers can approach each other as close as you like every set of rational numbers is disconnected)then that implies that countable systems are necessarily made up from finite objects.
What I really am talking about when I say that physicists have a fear of the finite is that they have a fear of methods that are based in discrete objects. Calculus with its dependence on uncountable infinity and the continuity of things that are being examined is totally indispensable to modern physics. I don't necessarily think this is a bad thing in the sense that calculus is a wonderfully robust and interesting tool and we should be not afraid to use it. However physicists should have a way around calculus no matter how painful and difficult there ought to be a way to do physics that is fundamentally discrete but there is no such thing. Mathematicians and computer scientists can play with the discrete but physics remains fearful of it.
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