This could be difficult and tedious to follow as a result of the lack of proper terms on my part , so please bare with me. :o I’d like to know if it is possible for a mathematical constraint to be the cause of emergent properties or self organisation.

(1)Lets say that you have an infinite number of points on a boundless 2D plane. The distance between any two points is a 1D line/string, and is related by a probability curve. This curve is http://upload.wikimedia.org/math/a/4/8/a489398c1987e8d4b7dc9387b9b9175d.png, where x > 0; x is Distance and y is Probability (The first quadrant half of a Hyperbola).

It is my understanding that there can be structure within infinity. An example of this is that in a hypothetical, infinite and unbounded universe, it would contain an infinite amount of matter, whether it consists of one proton every light year or if it is one continuous gas cloud. Similarly I am thinking that such an arrangement can exhibit structure as well, that is, areas of varying density.

I am not sure how exactly to frase this. :?

OK. If you start with any given physical system, you can analise the workings of such a system by observing behaviour and then trying to find the causes behind each occurrence. Eventually one would start to see patterns emerging, patterns that could be described by equations/formulas. Each pattern could be further analysed until the cause and effect relationships between constituent participants in the pattern can be deduced. This process can be repeated again and again, further reducing the system to a larger number of constituent predictable processes each time, but then eventually a limit is reached, which can be the limit of computing power, etc. I am wondering, after sufficiently reducing the system, if one could eventually reach a point where a simple mathematic expression can be the direct cause of all the macro observed effects.

Take the setup at (1). The distance between any two points tend toward 0.
If the formula is valid for an infinite plane, could a clumping of points, that is nearer to each other than surrounding points, actually directly cause the surrounding points to be further apart from each other in order for the formula to stay exactly valid? That is, if a clump of higher density points are formed locally, that the “violation” of the formula could cause an equal amount of deviation in the opposite direction (lower density) in the surrounding point space, starting from a maximum deviation at the clump’s border and petering out to zero deviation.

Would this require base/minimum units of distance and/or time to be possible?
How (if at all) would other areas of higher average density be affected by areas of lower density it might be passing into? Would further parameters be required for one high density area to affect another (by way of the low density “aura”)?

I think this might be better suited for philosophy.

However; I will attempt to answer what I believe you are intending to ask.

I could be mistaken, but I think you're having a misunderstanding of simple principals of mathematics that are leading your mind to wander into philosophical possibilities. You need to realize that the nature of numbers is not directly connected to the physical world, that the physical world rests on top of numbers. In the physical world 2 points can interract, this is not true in numbers.

I think what you are attempting to discuss is something that slips my mind, it was used to prove Fermats last theorem though. That equations of numbers behave in ways (according to a number system) that create patterns, these patterns can be formed into 3 dimensional shapes and then infinite numbers of sets can be utilized in a simple system. However, this has never been used for anything but proving something doesn't exist. Also it's important to note that the patterns derived are a direct result of the number system not the numbers themselves, in any other base system the sets would be drastically changed. Thus moderately proving that the numbers themselves don't interract.

As for your last question; if my previous explanation didn't null this question...
An infinite set abiding by logical rules has finite constraints.
An infinite set abiding by illogical rules has infinite constraints.
No example of the latter comes to mind.

This curve is http://upload.wikimedia.org/math/a/4/8/a489398c1987e8d4b7dc9387b9b9175d.png, where x > 0; x is Distance and y is Probability.
I'm afraid this doesn't make a lot of sense - your "probability" can be greater than one.

I'm afraid this doesn't make a lot of sense - your "probability" can be greater than one.

I think you're to assume x>1

But why should all distances be greater than one?

But why should all distances be greater than one?

Perhaps I am not understanding his question. But the idea of breaking down the sub-patterns infinitely to me implied an ideal situation (in the gas cloud). Therefore anything between 1 and 0 would not be ideal...unless you constrained your minimum distance and defined it as 100% probability.

I'm afraid I don't follow.

I'm afraid this doesn't make a lot of sense - your "probability" can be greater than one.Well, that because I have close to zero math knowledge.;) Would I make sense if I said it was frequency? That is, the distances are more probable to occur the smaller they get.


I think this might be better suited for philosophy.

However; I will attempt to answer what I believe you are intending to ask.Thanks for the effort.:)


I could be mistaken, but I think you're having a misunderstanding of simple principals of mathematics that are leading your mind to wander into philosophical possibilities. You need to realize that the nature of numbers is not directly connected to the physical world, that the physical world rests on top of numbers.I understand and concur with this bit. Numbers and equations are simply our way of describing and predicting what happens.


In the physical world 2 points can interact, this is not true in numbers.The two points in the physical world have attributes, such as charge, etc, that effect the interaction. But if you keep looking deeper and deeper (the cause of the cause of the cause, etc) you would eventually reach a point where the cause is no longer obvious. I was thinking that maybe eventually a sort of basic truth or mathematic relation could be the cause behind all emergent properties (the universe). I am not thinking of some kind of existential thing like universal mind or such nonsense. String theory, for example, employ concepts like super symmetry, that one could argue as being an abstract human concept as well, but it is duly deemed a basic property of space-time. Processes are deemed impossible if it would brake certain symmetries. That is the same kind of thing I am wondering about. A basic law of constrained probability that is the ultimate cause of the entire universe.

But anyway, the above is not really essential to the question. Lets for convenience’ sake say that some even further underlying property of this hypothetical space-time is responsible for this behaviour. If my equation were to be followed closely, i.e. infinite space is filled by an infinite number of points with non-zero distances between them. These distances tend towards zero, but can never reach it. The fluctuations governed by the graph could maybe even exhibit Brownian motion-like behaviour. So, if the equation is to remain true for the infinite space, any clump of matter needs to have a balancing opposite effect, e.g. the “halo” of further spaced particles, kind of analogous to a planet (clump of closer spaced points) with a reverse atmosphere (lower density at the surface).

Hope that makes a bit more sense.

Well, that because I have close to zero math knowledge.;) Would I make sense if I said it was frequency? That is, the distances are more probable to occur the smaller they get.
I'm not entirely sure what you mean. Frequency and probability mean fundamentally different things! I had previously stopped at this point in your post, but since then I've read a little more and feel there are bigger problems about:

If you start with any given physical system, you can analise the workings of such a system by observing behaviour and then trying to find the causes behind each occurrence. Eventually one would start to see patterns emerging, patterns that could be described by equations/formulas. Each pattern could be further analysed until the cause and effect relationships between constituent participants in the pattern can be deduced. This process can be repeated again and again, further reducing the system to a larger number of constituent predictable processes each time, but then eventually a limit is reached, which can be the limit of computing power, etc. I am wondering, after sufficiently reducing the system, if one could eventually reach a point where a simple mathematic expression can be the direct cause of all the macro observed effects.
You seem to have a strong feeling that if we look at a system closely enough and for a long time, we will be able to see a pattern from which deduce a model. Then you want to run this model backwards, and find what the state of the system was at the beginning. There are huge mathematical problems with your conjecture: namely that it's false.

Let's reduce the problem somewhat: suppose we already know the equations governing the evolution of the system. Your idea is to take a snap-shot of the system, run the equations in reverse and figure out what the system looked like originally. The problem with this is our "snap-shot" bit: here we take a measurement, and being mere mortals we're unlikely to get this bang on (I don't really want to talk about quantum level things, but the same would be true in this case). So what we'll have is a measurement X, and let's say we're clever enough to gauge how "wrong" our measurement is. That is to say, we can choose our measurement so that we guarantee that:

\| X - X^* \| < \epsilon

where \epsilon >0 is some small number and X^* is the actual state of the system when we measured it. For example, if we were measuring temperature, we might measure 175K, but knowing that our apparatus is only so good, we might say that the actual temperature is within 1K. So in our previous notation:

\| 175 - X^* \| < 1

So now we run our equations backwards, and if we want to gain any sort of decent information from the results, it must be the case that the errors in our measuring apparatus don't affect the results too much. One might say, that if we make the error in a measurement small, then the corresponding error when we run the system back must also be small (how small would depend on our measuring error). In similar terms as we had before, we'd hope that:

\|f(X)- f(X^*) \|< \delta

where \delta>0 is some number that depends on our previous error, \epsilon and f is our "function" that runs the whole thing back. The idea should be that if we can make our measurement arbitrarily accurate, then we can make the corresponding results when we run the system backwards, also arbitrarily accurate. The condition you want, therefore, is that your system be backwards well-posed***. That is to say, if you only make small errors in your measurement, then the results of the system "played back" will only differ slightly from the actual ones. This property does not hold in most cases.

If you'd like to learn about this kind of thing, then I would recommend googling things like "dynamical systems". This is an area of mathematics which deals with similar problems.

***Those of a more mathematical background might notice that the conditions we've pointed out look suspiciously like those which would impose continuity on the "function" f. This is pretty much the case, though in general f is a map between function spaces, and the "initial states" are functions. So we need some sort of notion of "distance" or "closeness" between functions. This is no problem though.

It seems to my uneducated self that you are talking about trying to work backwards from a chaotic system and trying to work out the initial cause. That is not exactly what I am after. I am simply assuming that the initial conditions exist, proposing a governing formula and trying to see what exactly will happen if this formula is to be satisfied. I am not considering measurements at all, simply how the system behaves given the constraints. I guess I am trying to predict the behaviour of the system in the short term at least. The actual values of each x are totally random, but simply tend towards 0 on avergage.

The initial conditions I am choosing is the setup I have at (1) in the OP.

From this let me paste my questions again:

The distance between any two points tend toward 0, but can never equal it.
If the formula is valid for an infinite plane/volume, could a clumping of points, that is nearer to each other than surrounding points, actually directly cause the surrounding points to be further apart from each other in order for the formula to stay exactly valid? That is, if a clump of higher density points are formed locally, that the “violation” of the formula could cause an equal amount of deviation in the opposite direction (lower density) in the surrounding point space, starting from a maximum deviation at the clump’s border and petering out to zero deviation.

Would this require base/minimum units of distance and/or time to be possible? How (if at all) would other areas of higher average density be affected by areas of lower density it might be passing into? Would further parameters be required for one high density area to affect another (by way of the low density “aura”)?

The initial conditions I am choosing is the setup I have at (1) in the OP.
But how are these "initial conditions"? You haven't given any sort of dynamical system, you've simply introduced a function y=1/x, said "this is a probability curve". You need to make sure you have a well-defined question. The passage I quoted:

If you start with any given physical system, you can analise the workings of such a system by observing behaviour and then trying to find the causes behind each occurrence. Eventually one would start to see patterns emerging, patterns that could be described by equations/formulas. Each pattern could be further analysed until the cause and effect relationships between constituent participants in the pattern can be deduced. This process can be repeated again and again, further reducing the system to a larger number of constituent predictable processes each time, but then eventually a limit is reached, which can be the limit of computing power, etc. I am wondering, after sufficiently reducing the system, if one could eventually reach a point where a simple mathematic expression can be the direct cause of all the macro observed effects.
seemed to refer to the material I addressed. Perhaps I misunderstood?

To give another example of Guest's point, consider the planets motion. We have long known about the equations of motion of gravity. Infact, we discovered Neptune by the effect it was having on Uranus, though we couldn't see Neptune! It was only after we worked out where the planet would be and looked there that we saw it.

And yet, despite the advent of supercomputers, NASA is unable to compute where the planets will be in say 1000 years. Why? We know the equations of motion, thanks to Einstein. So what's the problem? Can't we just measure where they are, their motion and then just get crunching? Well yes, except the solar system (infact any gravitational system with more than 2 objects!) is 'chaotic'. The definition of that is that if you are wrong in your initial measurements, even the slightest bit, eventually your calculations, no matter how much computer power you throw at it, will predict nonsense.

By that I mean that you might calculate that in a million years, on July 17th at 2.34pm (GMT) all the planets will line up. But because there's a tiny tiny error in your measurements of the motion and position of the planets today (say you get the motion of Titan wrong by 50m/s) and the system is chaotic, the error will spread through your calculations like a virus. Titan affects Jupiter. so after 100,000 years your calculations about the position and motion of Jupiter is wrong by 10,000 km. Then, at 200,000 years, your calculations for the Sun is wrong by an entire solar radius (since Jupiter is a big player in the solar system). 500,000 years and you're wrong about the position and motion of ALL the planets and moons and then at 1,000,000 years you actually find that instead of all lining up, as your model predicts, the planets are all scattered around like they usually are. The tiny errors spread, faster and faster, bigger and bigger. That's what a 'chaotic' system is.

In the case of just 2 objects gravitationally interacting, if you're wrong by say 1%, then in 1,000,000 years your errors will still be about 1%. They don't get worse. Same equations used to derive the motion. Same instruments measuring to the same accuracy levels. Same computers doing the calculations. But totally different mathematical substructure.

Too many people (mostly cranks) think that 'chaotic' means 'complicated'. It doesn't. Some horrifically complicated systems are not chaotic and they can be solved analytically. Some very simple systems are chaotic and nightmarish to describe.

I am sorry for making such half-arsed statements that need a lot of explaining.:o All of this is done in my head and I, more often than not, do a poor job of fully describing all the particulars of the mind experiment. Thanks for keeping with it thus far.


But how are these "initial conditions"? You haven't given any sort of dynamical system, you've simply introduced a function y=1/x, said "this is a probability curve". You need to make sure you have a well-defined question.It is a curve that represents the distances between points, with the probability increasing as distance gets smaller. So, lets for arguments sake say that the curve touches the y-axis at some arbitrarily small value of x. If you where then to consider the distances of a large number of points to their nearest neighbours, you would be left with a very large number of distances that would be dominated by zero's, with real numbers for x becoming less and less common as they get larger. One could then, I suppose, get an average, non-zero value for x. The higher density area is where the average distance between points comprising the area are less than the average of the plane.

In my example, though, two points being separated by zero distance would be exactly the same as one point and so need not be considered, giving rise to the condition x > 0. So if the average distance between neighbouring points over the entire plane/volume are to remain exactly the average, any local density needs to have an adjoining halo of lower-than-average density. I hope that makes more sense.


But how are these "initial conditions"? You haven't given any sort of dynamical system, you've simply introduced a function y=1/x, said "this is a probability curve". You need to make sure you have a well-defined question.Sorry again. That part was simply to demonstrate that I am trying to look at the primary cause for all emergent properties of space-time. But it need not even be considered for the purposes of this discussion, which is how such a system would behave and if any additional parameters need to be added for it to even make sense or for it to be able to be successfully modeled.

To add to AlphaNumeric's post: the phenomenon we are speaking of is known as "sensitive dependence to initial conditions". For a dynamical system to be termed "chaotic" there are another couple of technical conditions that need to be specified. That is to say, there are systems which have sensitive dependence on initial conditions, but are not chaotic in the sense that mathematicians refer to.

I think the thing to do here is for you to make sure the terms you're using are well-defined. After we've done this, you might be able to formulate a question that makes a little more sense!

It is a curve that represents the distances between points
You have defined a function y=f(x) - a function of one variable, that hopes to represent the distance between points. If I were to ask you "what's the distance between two points", you'd have to ask what two points? I need to know two bits of information. This means you need to provide a function of two variables. A familiar example would be the usual notion of "distance" in the plane between the points y=(y_1, y_2), \, x=(x_1, x_2).

d(x,y) = \sqrt{ (x_1- y_1)^2 + (x_2 - y_2)^2}

Do you see why this makes sense to measure "distance"? For instance, what's the distance between x and x? Well let's see:

d(x,x) = \sqrt{ (x_1- x_1)^2 + (x_2 - x_2)^2} = 0

Which is what you'd expect! You can ignore this next bit for the moment, but perhaps later you might like to google "metric" and you'll see some things that might look familiar.

with the probability increasing as distance gets smaller.
With the probability of what? You need to be a lot more specific!

Ok then. It looks like you are talking about using Pythagoras’ theorem on a Cartesian plane. The distance is only a scalar quantity and only a relation between two points, so I guess one point would be in the origin, while the second one would be defined by (x;y). So I guess that in that way you would have the two variables, x and y. But my graph only deals with the hypotenuse, i.e. the distance between the points, r, the solution of the equation. Can I do that or do I still have to provide the equation?


With the probability of what? You need to be a lot more specific!The probability of the size of the scalar value of the distance between two points. That is, it is more likely for the value to be small than large; the smaller, the more probable and the bigger, the less probable, as described by my graph.

The distance is only a scalar quantity and only a relation between two points, so I guess one point would be in the origin, while the second one would be defined by (x;y). So I guess that in that way you would have the two variables, x and y. But my graph only deals with the hypotenuse, i.e. the distance between the points, r, the solution of the equation. Can I do that or do I still have to provide the equation?
This paragraph doesn't make an awful lot of sense. However, I'll assume the point you're trying to get across is that you want to define "distance" between two points x, y by:

d(x,y) = \frac{1}{\sqrt{ (x_1 - y_1)^2 + (x_2 - y_2)^2}}

If this indeed the case, could you explain why this makes a sensible definition of "distance"? For instance, surely we should have d(x,x)=0, but in this case, d(x,x) is undefined!

The probability of the size of the scalar value of the distance between two points.
This doesn't make any sense I'm afraid. If I asked you "what's the probability of the height of a table", you wouldn't be able to give me any sort of sensible answer. Sorry to be a pain, but can you think over your post again and give it another go?

This doesn't make any sense I'm afraid. If I asked you "what's the probability of the height of a table", you wouldn't be able to give me any sort of sensible answer. Sorry to be a pain, but can you think over your post again and give it another go?You know, this might not come as a surprise to you, but that happens to me quite regularly. I am fully aware that the confusion is entirely my fault.;) Thanks for your patience!

Ok. Think of a large dice with hundreds of sides. Each side has a number on it representing the distance between the two points. The size of the sides depend on the number printed on it. The smaller the number, the larger the area of the side (probability, or what I have been calling probability:o). If you rolled the dice, it would have a larger probability of landing on a side with a small number on it.

How is that?

Think of a large dice with hundreds of sides. Each side has a number on it representing the distance between the two points.
By construction, you're assuming there are only "hundreds" of possible distances between two points. Since you're working in the Euclidean plane, this is clearly false.

Yes, I agree. The dice was the only thing I could think of. For any real number to be possible (which I assume is Euclidean space), the dice would need an infinite number of sides and according to my graph, it would have no 0. But, if one were to assume a minimum base unit, say a Planck length, then the dice would have a finite number of sides if it represented a finite number of points, but still infinite in the case of an infinite and unbounded plane/volume. I also want to clarify that the values on each side is assigned totally randomly to each point. The values that are printed on the sides (and the corresponding area of the side) are spread according to the graph, that is, after, say, 100 throws you'd have a list of maybe 98 numbers from 1 to 50 and 2 from 51 to 100.

Did I make it clear enough, or do I still have to define some things?