Monday, March 27, 2006

so we didn't make any progress on that topic... at least not yet, so maybe another will spark some interest. i recently saw the movie Primer, which if you have not you should... i will see it again so we can watch it together if you want. the movie is about time travel, that's all i'll say, but it had me thinking about the consequences of time travel, at least theoretically, and it's relation to a book I read some time ago, The End of Time by Julian Barbour. It's definitely one of the best books i have ever read, and if i wasn't moments from going to sleep, i would extrapolate more on it alone. anyway, so that i don't corrupt your own ideas (or the movie), i won't say much at this point, but here goes. so consider the idea of traveling back in time (whether or not this is possible doesn't matter of course, and we'll consider traveling forward later, but it's interesting as well). the first question that comes to mind for me is, IF you travel back say 1 week, can you see yourself? You see, an interesting thing happens right... a cyclic, perhaps paradoxical line of thought is imminent. we assume that at some particular point in time, we travelled back in time, for the first time. but consider the following scenario, where all days refer to the same week Sun Mon Tue Wed Thur Fri Sat:

You exist and live Sun through Sat, but then on Sat afternoon you finally discover for the very first time in history how to travel back in time. so you decide to travel back in time, for the very first time ever of course, and you decide to go back to Wed of that week (remember we're only talking about a single week, so all days are in the same week). when you go back to Wed, the first question is, can you see yourself? this may be a choice, OR it may turn out that one way will lead to an irreconcilable paradox. well the first question that comes to mind is, if you do go "back in time" do you arrive in that time, as a). the person you were originally at that time, or b). the person you were when you decided to travel back in time, so that you are really (at least on Wed having travelled back) a different person on Wed. Well, here's the paradox, and it's not just with reconciling whether you are the same person when you travel back or you are a different person. it has to do more fundamentally with our preconceived notion of time itself. you see, IF you are a distinct, separate person looking at your younger self on Wed (having travelled back to Wed from Sun), then what is troubling is that, IF you are truly going back to that moment in time, that very same Wed, then even BEFORE you travelled back you must have been there, with your younger "double" counterpart on Wed the FIRST time Wed happened. the idea here, is that if you are really going back in time to that Wed, then when you go there, you are in the same point in time as when that Wed happened "the first time around." Thus, when that Wed happened the first time around, you had "already" traveled back in time to it. This leads to one of two things:

Either

a). it can't happen, because it contradicts the idea of travelling back for the first time, OR
b). there is no first time

now, at first glance, it would seem that b). makes the most sense on multiple levels. first, we're assuming that you can travel back in time, which in and of itself supposes that you can arrive at the same point in time multiple times, which implies that there is no longer such a thing as a first time. but we like to think of the future, as time that hasn't happened yet. so it would seem that we should be able to pinpiont the "first" time when the time travelling mechanism was discovered. but what we're saying is that there is no longer such a thing as a first time, as a single instance which exists and cannot be changed. consider the Wed if it's not clear. the "first time" Wed came around, there was only 1 of you in it... you were just a few days away from discovering the time travelling mechanism. but, on Sat when you achieve this, you travel back to that Wed, and now all of a sudden there are 2 of you in Wed... how can that be? Well, if you can indeed travel back in time, it means that the past is dependent on the future, and not just the other way around as we are used to thinking. indeed in that case, what happens on Wed seems to be dependent on what happens on Sat. but we still have a bit of a paradox, or something here. because now we must ask the question, WHICH Wed is or was the real Wed? did both Wed's exist or is there only 1 Wed in that week? i think we hope there is just 1 Wed in that week based on conventional thought, and in that case we indeed have to admit that the first time Wed came around, you must have been there already, having travelled back in time to that Wed from Sat. thus, what happened on Wed depends on what happened on Sat in the same week! well, let's try the unconventional thought... that there are multiple Wed's the question is how many? well, we're still assuming we can travel back in time, and now once the time travel mechanism is discovered, what would stop anyone in the future from travelling back to that Wed from any point in the future, and therefor effectively potentially changing that Wed infinitely many times over... presumably nothing (as far as we are discussing at this point) and we are seemingly lost... this seems to be an irreconcilable conclusion, that leaves Wed hopelessly unstable and changeable by all points in the future that are after the time travel mechanism's discovery. add the possibility of taking a copy of the time travel mechanism or machine back to that Wed, and now we back up the time of creation of the thing, and you are no longer the first time creator... well, we already did away with "first times" i guess under this scenario. but now we are left with the conclusion that there are infinitely many Wed's... which really means there are infinitely many of everyday... maybe we can try finding the size of that infinity, but it turns out i think that the premise of time in and of itself if the culprit. that is, actually, julian barbours text reconciles this in a beautiful way, doing away with time altogether, and only considering static configurations of the universe. but there's even a very subtle problem with his view (which i haven't discussed in enough detail to really make the connection here), because he considers basically that all that has ever and will ever exist are static configuratoins of all the particles in the universe, and that life is really just a journey through the configuration space which represents all possible configurations of all the particles in the universe. we should really talk about this, and i'll discuss the 3 particle universe case in detail, afterwhich it will become clear how to extrapolate to N dimensions (although the picture of the configuration space as i've called it, and as he called it, will not be clear because it will be more than 3 dimensional). but to get back to my point, the sublte problem i think is that if we are only talking about static configurations of the universe, then a very interesting thing happens. first, assume that there are finitely many particles in the universe, and that this number doesn't change. well, if you travel to a different configuration of the universe which is exactly the first Wed, only now YOU are there again (as though you travelled back in time) and your double is there too. well, YOU take up particles... you have a body and therefore you consume or exist only as a bunch of particles. This is startling, because (and i don't think julian barbour draws this kind of conclusion, but it's been a while since i read his text) that means that it is impossible for there to be a Wed and also EXACTLY a Wed + YOU, because that adds up to too many particles. Wed is a static configuration of the universe, which has finintely many particles, but Wed + YOU is YOU too many particles! anyway, this means, i think at least, that julian's model (which we will have to talk a lot more about) actually leads to something we are all very comfortable with, that the "moments" we live are entirely, 100% unique and immutable. while julian states this much about the moments, what i am emphasizing here is the fact that under that model, the kind of time travel i discussed above is impossible, because Wed + YOU is YOU too many particles! anyway, the other possiblity, that there are NOT finintely any particles in the universe, or at least that the finite number is NOT constant, changes things a bit, because then there can exist a Wed + YOU. But where does that live? what i mean by that is, as you'll see when we talk about his model in detail, THAT universe doesn't live in the same configuration space as just Wed, because Wed + YOU has more particles (this will make sense when we define the configuration space). this has it's own cool and interesting consequences i think... some are running in my head but in all honesty i really need a shower since returning from the gym, so i'm gonna roast on that and post them later. plus, before i go farther, i think there's some value in us having a good discussion about this, if you're interested. anyway, what's also nice about all this is that in the last few posts we were talking already about dimensions, and perhaps defining them differently. we're also talking about infinities and how many there are. and somehow, i think it all may play an important role for discussing the rather intrisically interesting philisophical (and physical) possibilities of time travel (for now just to the past), or as i ended, travel between static configurations of the universe with no time, or even travel between entirely different configuration spaces defining infinitely many universes each, again with no concept of time required.

ok, that's all for now, i think all of this needs some formalization and thought, and perhaps we'll discover some interesting results under various hypotheses.

hope to chat soon,

jazz

Thursday, March 02, 2006

hey again... so i had some trouble posting the last time, but there was something else i thought was actually the most interesting part of the link in my previous post. basically, there was a claim there that is has been proven that in fact it cannot be proven whether or not there exists an infinity between what they call alpha0 and alpha1. Thus, it is something of a choice, where of course if you choose to take either view as an axiom in your system, you get different results. Well, on the one hand, that's a bit unaesthetically pleasing... but on the other hand, it's good news regarding your intuition. even if we weren't aware of everything there was to consider about infinities, and even after i claimed that there was some good formalization of this matter, it appears there IS room for a better definition... at least better in my opinion. perhaps there's a way to think of cardinality (where equivalence is not perhaps strictly defined by bijections) where this ambiguity in the number of infinities does NOT occur. in this case, we would of course be starting with our own kind of "axioms" or decisions about what fundamentally makes the most sense to define the size of infinite sets, and we would then be able to prove, under that system, that there's no such choice of the number of infinities... a more precise number would arise, whether itself is an infinity or not, we'll have to consider.

all for now... i'm tempted to start reading some real mathematics again... to sharpen... if i come across something interesting i'll let you know :)

many many,

jls

Wednesday, March 01, 2006

Wow, it's been a while again. I didn't even remember my username and password... pretty bad huh. anyway, i was thinking about this last night, and i started reading around the web... not about the dimensions topic, but more specifically the infinities... here's a very interesting blurb about the topic on another blog or posted email stream or something:

http://at.yorku.ca/cgibin/bbqaforum=ask_an_analyst;task=show_msg;msg=0168.0001

in particular, "Now, Cantor does not prove that P(N) = aleph_1. Indeed he famously doesn't do this, as this is precisely the continuum hypothesis!" so it appears as though our claim that the cardinality of the power set of the naturals (let's refer to this as P(N), where P(N) is the power set of the naturals) is not equal to cardinality of the reals ( or simply R). What got me searching about this topic was the following thought on the matter:

suppose i take any irrational, say pi = 3.14159........... the question that came to mind, and seems to have an obvious answer, is "Is there a rational which has the same digits but just in a different order?" Well, by construction, i think of course that's the case... how about just trading the first 1 and 4... so we have 3.41159............ that will do it. So, given what i quoted above, how can it be that a non-trivial subset (not the null set and not the whole set) of P(N) is sufficient for being the same size as R. I think the answer lies in the fact that we can break that infinitely long string of digits in infinitely many ways... here's what i mean. consider the following 2 elements of P(N) (remember elements of this set are themselves sets... they are subsets of N):
Note: the order we list elements in a subset doesn't matter... any set only cares about it's elements, and not how we list them inside... the fact that we can order them is an independent consideration

first element to consider:
--------------------------
{3, 14, 15, 9, ...} - we see from this element we can get 3.14159...

second element to consider:
---------------------------
{3, 141, 59, ...} - we see from this element, we can ALSO get 3.14159...

So, at least under that construction where we use the pattern above to identify an element of P(N) with an element of R... we will end up with many representations of a given irrational (in fact, infinitely many... which size of infinity that is, is another good question). of course, this is just one construction, and although it may be an obvious one, perhaps another will prove this one to be not so dramatic. but for now, i can at least "see" how it's possible that P(N) > R. I mean, in the same way that at first we may think that the naturals are bigger than the even naturals... of course we know they have the same cardinality because the map f(n) = 2n and it's inverse g(e) = (1/2)*e prove that a bijection exists between them (any function f is a bijection if and only if it has an inverse function).

Ok, that's all i'll say for now, but there's more...

Monday, February 13, 2006

It's good to finally have something up we can use. You have some very interesting questions there about the way we "describe" dimensions. I use the word describe because I believe that we do not build dimensions per say, but rather, we build descriptions of the dimensions. The problem I believe is that we mix description with described object. Dimensions allow us to be able to have something we can imagine about space [not the 3 D space]:)

The real question which you raise is, can we have a different description that works as well or maybe better? Another way of asking the same question is: what is wrong with the current description? Does it for example allow us to clearly define infinities? I think not and this ties back to our last conversation about the set of "all" infinities. What is it? how is it constructed? Does such a set even exist?

PS: Adding to your list of some of the things we establish last time, we "proved" that there are as many rational numbers as there are irrational numbers by creating injections both ways.

We did not quite reach a conclusion on the number of infinties as we tried to define what the set of infinities could look like. One of the ideas was that it is itself an infinite set...puzzling...


SLL

Monday, February 06, 2006

I'm here - it's taken too long, but now is now :) Anyway, something struck me today while I was driving... however profound or mundane it is, and that was, is our present construction/conception of physical dimension necessarily correct. I mean, we start with a 0 dimensional point, build a 1 dimentional line by "lining up" points (although not necessarily in a line... consider a circle), then build 2 dimensional surfaces by "swiping" those "lines" (in quotes because they are not necessarily lines as noted), then 3 dimensional objects by "pushing" surfaces... what bothered me was that we seem to build fundamentally from those constructions (which are admittedly ok for me intuitively)... the question that came was... can we start in a different way and construct "dimensions" which are fundamentally different from those we now use intuitively? I have a feeling this is either something subtle and interesting, or simply nothing to consider at all... perhaps the current construction encapsulates all of what even I am thinking of as dimensionality... but something about the limitation of starting with points and getting to "lines" and building from there seems to be something of a choice... are there other creative choices for "starting" which yield parallels to what we now think of as dimensions? I think either there are OR the concept of dimension is something more fundamental as an idea, than simply to serve as an important construction for physics.

PS - we still have much more to discuss regarding the number of infinities which exist :) At our last meeting, we discussed the following:
1. things that are the size of the naturals (0, 1, 2, 3, 4, ...) - also same size as integers.
2. things that are the size of the reals (which is like the power set - set of all subsets - of the naturals)

We discussed much more, and I want to find the link to the formalism on that matter we found last time... there was a conjecture there that there were only "so many" infinities if I remember correctly.

Talk soon,

JLS