Mathematic constraints as cause of emergent properties

[KALSTER]

This could be difficult and tedious to follow, so please bare with me. 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 , where [math]x > 0[/math]; [math]x[/math] is Distance and [math]y[/math] 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/minimal 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”)?

[Buffy]

I think you might be trying to bite off too much in this post. You might want to think about it a bit more and break it up.

 

To answer your primary question--can mathematical "constraints" cause emergent properties--I think there are so many examples that you already have your answer. My favorite of course is the Mandelbrot Set equation:

[math]z_{n+1}=z_n^2 + c[/math]

 

What you have left out of your problem is any specification of the randomness of the points in the plane. As such, a simplification of your problem is to reduce it to one dimension and consider the infinite number of integers on an infinite line. These are distributed of course with perfectly even distribution and if you wanted to examine if "clumping" would occur with your function (which I assume you chose because it is similar but slightly simpler than Gravity's inverse square relationship ([imath]y=\frac1 {x^2}[/imath])), the answer would be no, because there is perfect balancing symmetry of the distances.

 

If you want to relate this to Cosmology, one of the purposes of the Inflation Theory is that it explains an *initial condition* of slight clumping which then can cascade given gravity's inverse square relationship, resulting in "emergent" structures.

 

Man is equally incapable of seeing the nothingness from which he emerges and the infinity in which he is engulfed, :phones:

Buffy

[KALSTER]

I think you might be trying to bite off too much in this post. You might want to think about it a bit more and break it up.

I'll do what I can.:hihi:

 

To answer your primary question--can mathematical "constraints" cause emergent properties--I think there are so many examples that you already have your answer. My favourite of course is the Mandelbrot Set equation

That is simply a mathematical construct though, isn't it? Fractals that infinitely reduce and stay self-similar don't exist in nature, do they? Even if they did, and even ones that terminate, they are produced by some underlying process (molecular structure, etc.)AFAIK. The reason I brought up the (perhaps defunct) question of a mathematic relation not simply describing, but directly causing behaviour, is that if you reduce anything enough, you'd arrive at a point were no underlying processes can be identified (Well, only AFAIK, because you might tell me in a minute that those processes (strings and such) are products of the conditions at the big bang and that the reduction only stops at the start of the formation of the superforce or something:shrug:). But anyway, it is not really essential to my question (despite my poorly chosen title), so we can assume that some further underlying process are responsible.

 

What you have left out of your problem is any specification of the randomness of the points in the plane.

I provided an analogy on another forum that might give you an idea of what I have been pondering:

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). If you rolled the dice, it would have a larger probability of landing on a side with a small number on it. 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 between 1 and 50 and 2 from between 51 and a 100.

That is the type of behaviour I want my graph to represent, only that the dice would be infinitely large (any value is possible, however improbable) and there would be no side with a 0 on it. I hope that makes a bit more sense?

 

These are distributed of course with perfectly even distribution and if you wanted to examine if "clumping" would occur with your function (which I assume you chose because it is similar but slightly simpler than Gravity's inverse square relationship ()),

You know, I just chose the function because of the shape of the graph, which is the type of distribution I wanted, but you make an interesting point/suggestion here. I am trying to consider (in my own naïve way I guess) some possibilities for a dynamic medium that might be able to exhibit behaviour that could give rise to the emergent properties we are part of (fundamental forces, matter, etc). Since I am not a physicist, I know that it is way too simplistic and naïve, but I learn a lot as I read up on different fields and enjoy it a lot. Anyway, Gravity's inverse square relationship would still give the same shaped curve I think, only it would lean much more heavily towards a preponderance of low numbers. Since gravity is a curvature of space-time, the curvature could be represented by density differentials in the point space I am attempting to construct.

 

If you want to relate this to Cosmology, one of the purposes of the Inflation Theory is that it explains an *initial condition* of slight clumping which then can cascade given gravity's inverse square relationship, resulting in "emergent" structures.

Is this the same type of thing I am investigating? :naughty:

 

Anyway, we do not need to discuss my crackpot ideas. I’d really like to work on this point space construct though. Did I provide enough information?

[Buffy]

That is simply a mathematical construct though, isn't it? Fractals that infinitely reduce and stay self-similar don't exist in nature, do they? Even if they did, and even ones that terminate, they are produced by some underlying process (molecular structure, etc.)AFAIK.

Sure they're based on some "underlying process," but what you're seeing is that the underlying process *implements* the mathematical construct!

 

And where did that underlying process get it? Well obviously we drill down to the quantum level and have to stop there, and the funny thing is that when you get that simple you see almost nothing but fundamental mathematical relationships!

 

They just bubble up, sometimes transmogrifying (I found Roger Penrose using the term in his latest book, so I can too and still be totally scientific!) along the way.

 

The really cool thing about Mandelbrot and other fractals is that the math is so darned simple, its easy to see why the physical structures implement it! This is actually one of the main topics of the aforementioned Roger Penrose book "The Road to Reality" where he goes through the justification for mapping ideal mathematical concepts onto the physical world (chapters 1-4), but that's for another thread...

 

But anyway, this is to make the point that your statement:

...if you reduce anything enough, you'd arrive at a point were no underlying processes can be identified (Well, only AFAIK, because you might tell me in a minute that those processes (strings and such) are products of the conditions at the big bang and that the reduction only stops at the start of the formation of the superforce or something:shrug:).

misses the obvious dust bunnies now that you've lifted the rug and looked under it: the mathematical functions are quite simple and easy to generate from any three-dimensional (or heck, up to 11 if you want to throw them all in!) physical structure. Continuing to look for a mystery when the cause is so obvious is what gets in the way of most people when they try to think about this stuff. For some reason, the perception is that it's *got* to be more complicated than that.... :rolleyes:

 

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). If you rolled the dice, it would have a larger probability of landing on a side with a small number on it. 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 between 1 and 50 and 2 from between 51 and a 100.

There ya go! *That* is a *distribution* function though, which is *not* the same thing as the *process* function that might *cause clumping*.

 

To get clearer in defining what you're talking about you need to separate these two concepts, because they interact, but they have totally different roles in seeing how the system works.

 

The distribution function is the one that specifies the initial conditions of the system. In the example I gave if the distribution of the "particles" is the same as integers along a number line, then their distribution is perfectly ordered and no matter whether your *process* function is [imath]\frac 1 x[/imath] or [imath]E=mc^2[/imath], the process will have no effect because the perfectly regular spacing will cancel out the process effects, assuming that the process function operates on each particle in exactly the same way. OTOH, if your distribution function is [imath]\frac 1 x[/imath], you'll find most of the particles already clumped around some center (you may want to consider the math behind [imath]\frac 1 x[/imath] where x<0 because you have a "disjunction" as we call it at the origin of the graph where you jump from [imath]\infty[/imath] to [imath]-\infty[/imath] instantaneously, something that does not happen with [imath]\frac 1 {x^2}[/imath]). That kind of distribution is also not terribly random, and its got you already at a highly clumped state that will collapse to an infinitely large distribution at the origin PDQ!

 

This is where the much more random clumping observed in the Cosmic Microwave Background Radiation makes things much more interesting. Given that data, which shows clumping but is highly disordered, how did we end up with the beautiful galaxy shapes we see? Well, because gravity uses that inverse square law, and any slight pre-existing direction will influence a curved path around a center of gravity, you get spiral motions, resulting in very pretty--and ordered emergently--structures! All from [imath]\frac 1 {x^2}[/imath], because that's how gravity works at the most fundamental level!

 

So coming back to this distinction:

You know, I just chose the function because of the shape of the graph, which is the type of distribution I wanted...

Are you sure? because I think you really do want to use it as your process and not your initial distribution. [imath]\frac 1 x[/imath] is a wonderful process to work with, but for your initial distribution--going back to your 2D plane--it just means that when you start, 90% of everything is piled up in the middle, and the stuff 2, 3, 4 standard deviations out will get there eventually if the same function is *also* your process definition, and do so in a very straightforward--and I'd argue, uninteresting--fashion.

 

Anyway, we do not need to discuss my crackpot ideas. I’d really like to work on this point space construct though. Did I provide enough information?

Its fun to discuss cool ideas, crackpot or not, as long as you're open to thinking in different ways! :cheer:

 

Think about distribution vs. process: its the crux of your confusion--or at least your ability to explain what you're doing--at this point.

 

Speak your mind and fear less the label of 'crackpot' than the stigma of conformity, :phones:

Buffy

[KALSTER]

Ok. I wanted to work from a position where this point space had always existed and where it is infinitely large. So it is infinite, both in space and time. So there aren’t such a things as t = 0, or a centre. It can only have a t = 0 if you choose it and a centre if you consider surrounding points relative to a single one. So, at arbitrarily chosen t = 0, if you were to take a sample of billions of neighboring points in a volume and plotted them, where y is the number of points and x is relative distance to its neighbor, you would end up with a graph like this:

 

 

This distribution (?) is the product of a previous instance (unit of time) of process (?) and is homogeneous for any sufficiently large data set. Now, the process, I was thinking, would be the dice. It is rolled once for each unit of time for each point. So if you were to watch a movie of an area of points you’d see a Brownian motion kind of thing going on. Similarly a movie of the plots after each time interval would have the line becoming fuzzy. I am not sure if it really matters whether the distances instantly switch to a different value or if they gradually change to the new value, perhaps in the time it takes for a new roll of the dice.

 

Did I make sense?

 

Why would the Gravity graph not have a disjunction? If you substituted x = 0, y would still be undefined, no? Anyway, x <= 0 is not even considered, since I am only interested in the scalar value for now. Clumping could then occur as a result of a remote statistical eventuality or some pre-existing disturbance affecting a new area, no?

[CraigD]

(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 , where [math]x > 0[/math]; [math]x[/math] is Distance and [math]y[/math] is Probability (The first quadrant half of a Hyperbola).

To be a valid probability density function, the area under the function [math]y=f(x)[/math] must equal 1 (that is, [math]\int_{\infty}^{\infty}\,f(x)\,dx=1[/math]). You can multiply a function by a constant [math]\frac1{k}[/math]where [math]k = \int_{\infty}^{\infty}\,f(x)\,dx[/math] to satisfy this condition, but only if [math]k < \infty[/math]. (Well… you could get all transfinite-y and say

[math]p(x) = \frac1{\infty} \frac1{x} = \begin{cases} 1 & x=0 \\ 0 & \mbox{otherwise}\end{cases}[/math]

, but that’s a pretty boring function (other than for pissing off math purists ;)) that can generate nothing more interesting than a single point)

 

The function [math]\frac1x[/math] over the range [math]x>0[/math] can’t be made into a probability density by multiplying it by a constant, because [math]\int\,\frac1x \,dx = \ln x[/math], so [math]\int_0^{\infty}\,\frac1x \,dx = \ln \infty - \ln 0 = \infty[/math].

 

So you’ll need to limit the range (something like

[math]y= \begin{cases} 0 & x < 1 \\ \frac1{100x} & 1 \le x \le e^{100} \\ 0 x > e^{100} \end{cases}[/math]

would work, but be rather “clipped”) or base your probability distribution on some other function (something like

[math]y=\frac1{(x+1)^2}[/math]

is pretty terse, and already normalized)

 

Any normalizable function you chose can be used to generate points (with a bit of elaboration, in a space of any number of dimensions), which I suspect would show pretty “emergent” structures. Continuous functions are, however, a pain to compute, and can practically only be approximated, so a graph like this using them will suffer from precision and pseudorandomness artifacts. It also lacks a built-in change-over-time feature.

 

A neat way to avoid both these shortcomings this is by using discrete algorithm, such as cellular automata like Conway's Game of Life or a Wolfram rule. There’s also a lot of literature about these, some of it mind-blowingly deep.

[Buffy]

 

I think you're still not completely clear on the distinction between initial conditions and the process that changes this initial condition over time. The dice mechanism you're referring to is indeed a "process" but its just the process for picking the distribution of the initial condition, and that's different than the change in the initial condition over time.

 

Note that this has nothing to do with there being a literal "t=0" if for some reason you really want infinite time scale (something you really haven't explained yet): the "zero time" can be any arbitrary point at which the initial condition is met.

 

What you describe here is a mixing of the two concepts:

This distribution (?) is the product of a previous instance (unit of time) of process (?) and is homogeneous for any sufficiently large data set. Now, the process, I was thinking, would be the dice. It is rolled once for each unit of time for each point. So if you were to watch a movie of an area of points you’d see a Brownian motion kind of thing going on.

In this scenario, the initial distribution--that is simply the state after the first time unit rolls for all of the points--is completely independent of all subsequent time units. It's basically saying any particular point blinks on in t=n, and then it goes away. What happens in t=n+1 is a completely different set of independent probabilistic events. You're not describing a "change process" here, simply a succession of initial conditions that are completely unrelated. They may indeed "look like Brownian Motion" but even Brownian motion is a description of a process that has a direct causal relationship to the previous state.

Why would the Gravity graph not have a disjunction? If you substituted x = 0, y would still be undefined, no?

Well, the thing is that with [imath]\frac 1 {x^2}[/imath] you have [imath]\infty = \infty[/imath] at x=0 whereas with [imath]\frac 1 {x}[/imath] [imath]\infty \neq -\infty[/imath] which is a disjunction. Its useful in many math problems to consider what the negative numbers *mean*, which is of course why bringing in Calculus like Craig does above is very useful in this case since you're trying to do this in 3 dimensions!

 

So,

Clumping could then occur as a result of a remote statistical eventuality or some pre-existing disturbance affecting a new area, no?

The way you have stated this "process" no, clumping would never occur except for what is defined by your initial condition distribution function because there is no *feedback*--that is, linkage between states--that would cause the distribution to change over time; just that "Brownian" jitter as points flash in and out of existence in exactly the same distribution.

 

Make sense?

 

There is occasions and causes why and wherefore in all things, :phones:

Buffy

[KALSTER]

Hey guys, I haven't abandoned this thread yet, I just have to work through some of the info you have given me. Thanks a lot for your detailed replies! They are very much appreciated. I'll be back....