The Fractal Structure of a Dispositional Universe

Introduction: I wrote this essay many years ago--in 2001, as a matter of fact (gah, how the years go by!). I had made it available online, also many years ago, and it received what I'd consider to be some small degree of exposure. However, due to circumstances a few years ago I had to cancel the service where the paper was being hosted, so it has, until now, been unavailable online. As a result of some friendly exchange with a couple people--one in particular expressing interest in reading--I have decided to again make this essay available online. Thanks to those two people (you know who you are) for the motivation!

I'd also like to add that there was a bit of, er, "tom foolery," let's say, with regards to formatting the image placement that required digging into the HTML, which is not necessarily my specialty--more like not at all my specialty. It looks fine in the preview, but if the reader should find the images display in less than a tidy manner, well, blame that on my lack of coding skills. If there are problems in this regard, I hope it doesn't detract from the presentation too much.

So, without further explanations, I offer for your perusal:

The Fractal Structure of a Dispositional Universe

by

b.e. hydomako

There is no thing that exists which is not contained within the structure of the universe: a thing which exists must necessarily be extended in space-time. It does not make any sense to talk of a thing existing without it occupying a location of space during an interval of time¹. Thus, all things that are exist as part of the universe. However, we do not really know how space-time is structured; that is, we only know the order of the universe as it is received through our senses and then interpreted by our minds. This situation leads us to believe that reality is made up of various levels. We have the sub-atomic, the atomic, the chemical, the molecular, the biological, and the psychological-the order of the universe appears to us as if it is some sort of pyramid scheme! This way of dividing up the world leads directly to complication. We become puzzled as to how the levels interact, and as to how specific systems in one level work to influence systems of another level. "Where is the work done," we ask, "do the micro-systems supervene on the macro-systems, or vice-versa?" We are led to asking all sorts of questions like this, and they do not appear to have definite answers. This is simply because interpreting the world as divided up into levels is a dead-end, a dry well. However, there is a way of looking at the universe's structure that is a well spring-the view dissolves or resolves these sorts of questions (and others). A better way of seeing the world is not as divided up into levels, but as realizing that the whole of the universe manifests on one level-the base-and one level only. Such a notion is found in CB Martin's dispositional ontology. We will use his theory in this examination in order to understand that all of the apparently discrete things in the universe are actually a complex whole which has its manifestation within the boundaries of a base level: we will come to see that a fractal model best describes the structure of the dispositional universe, and this will allow us to clearly see that the work of the universe takes place within the only existing level of reality.

To begin we will look at our current notions regarding the various levels of reality. The goal is to arrive at an understanding of how the various levels we see are actually all part of the same level. It is important that we recognize that the purpose here will not be confined to illustrating only that the parts we see are of a whole (indeed, some of us who agree with the current division of the world into levels would also assent to the fact that the levels work together to produce a whole). The task here will be to understand that, as there is only one level to reality's manifestation, the work that appears as divided (e.g. that the interactions which occur on a chemical level work to produce actions on a biological level) is not divided at all; that is, there is no division of labour into these artificially constructed levels: there are only partnerings of dispositions which are identical to manifestings, and all of this occurs within the same level-the level within which reality finds its various expressions.

As humans we have developed tools and skills which allow us to delve into reality in a manner which goes deeper than the macro-systems which commonly and readily manifest in conjunction with us. Examining what is generally thought to be the smallest level of reality we have the discipline of physics. This way of assessing the "deep structure" of the world around us leads to two of the initial levels in the pyramid scheme model: the sub-atomic and the atomic. These two levels of reality are often asserted to be the foundation upon which further more complex structures are built. We can suppose that the pictures of these levels of reality look like the illustrations A and B respectively². Notice that in A and B the picture appears as unique: there is nothing in A to suggest that it is the same as B; that is, in both A and B we see that each picture seems to imply that the two levels of reality are somehow separate from each other in that their constituents appear as different from each other. It is important to recognize that while A and B are not meant to adequately capture the real picture of the atomic and sub-atomic domains the differences in the illustrations do reflect the fact that we perceive the levels as qualitatively distinct-we experience the qualia of one domain as different from the qualia of the other.

Illustration A.

Illustration C.

Illustration B.

To carry this further we will now refer to illustration C which will stand in for the next level which is the domain of chemistry. In this level we commonly think that interactions in the previous two levels of A and B have worked to produce complex systems which are in turn differentiated from the systems found in the levels which A and B represent. Again, we see that the picture of reality on this level is somehow unique and separate from the two previous pictures. We feel that the chemical level then follows rules which are distinct to this domain, and thus, interact in ways which produce more complex systems of molecules (pictured in D). These molecules then go on to work to provide picture E-standing in for the biological level, and then we get, through interactions within E, the level which the psychologists study. This level of our cognitive capacities is represented by F.

Illustration D.

Illustration F.

Illustration E.

Each of these pictures represents a unique level of our world with respect to this analysis of the current popular conception regarding the workings of our universe. There is cross-level action; that is, the interactions on one level are seen to spur events on another level-in this pyramid scheme of a picture we have work being done all over the place! This does not appear as a sensible way to view the world's workings. It offers an interpretation where there is wasted energy through redundant actions betwixt the levels. Martin says of this, "if you have levels of being which really be, [and they] do the work over again, then [we must] provide an account of how there is work to be done!" (beh 1). This is where our nagging unanswerable questions come from. We look at this sort of structure and can not help but asking, "how do the interactions within A work to produce a different order of interactions within D?" As well, this is what leads us to the inclination to reduce, say, the cognitive capacities of human beings down to the mere chemical interactions within our brains. Then there is the further inclination to reduce the work of the chemicals to mere atoms swerving in the void. It is clear that once we are on this path we remove any notion of what it is to be human from our account: atoms swerving in a void possess no emotions or anything else remotely human! As Martin is fond of saying (quoting physicist Richard Feynman), "We have to have a way of putting it all back together again!" (ibid.). In other words, the reductionist tale we tell must account for how things like emotions and colour (a mere two examples of the many kinds of qualia) end up occurring in the world when the world itself seems to be composed of bits of things which do not partake of these qualities at all. There are many other problems which come out of our misconceived notions regarding the structure of our universe, but we are not here to list them all. We are here to see that this pyramid scheme of work on multiple levels is a mistaken way to represent the actual structure and workings of the universe.

In the above we saw that we have the notion that there are various levels to reality, and we noted (through the illustrations) that each apparent level looks different, acts different, and produces separate conceptions of the systems found within the universe. We will now show that even though these separate pictures of the world appear to be radically different from each other they are all in fact part and parcel of the same system; that is, there are not different and unique levels which stack-up in some fashion in order to produce the world. All of our perceptions regarding the variety of systems found on different levels are in fact sub-systems of a singular over-arching system occurring within one level of being. To understand this in a visual way we will now look at illustration G.

Illustration G.

Illustration G (above) is a picture of a mathematical structure called a fractal. In particular it is a picture of the Mandelbrot set which has been called, "...a universal object." It has received this title because, "[s]ets remarkably similar to the Mandelbrot set occur in many other [fractals]" (Devaney and Keen 77). Due to the fact that the Mandelbrot fractal seems to be the only set which has this ubiquitous nature we will use it as the paradigmatic example of the fractal model which is being asserted as the best conception of the structure of our universe (this is not to say that the Mandelbrot set is the structure of the universe, but that the fractal structure of the universe would also have this same ubiquitous nature-its occurrence would be found in all the sub-systems within the manifested world).

G illustrates that what has been previously seen as separate and unique pictures are in fact all parts of the same picture. We can see that illustrations A, B, C, D, E, and F are all contained within the larger structure of the Mandelbrot set. Recall though, it is not merely our goal to see the parts as a complex whole, but to account for why there is not a division of labour amongst the ill-conceived levels of reality. In order to understand this we will first take a detour into why the fractal model fits like a glove to the hand of CB Martin's dispositional ontology, and we will also come to understand specific qualities of a fractal structure. It is through the latter that we will arrive at an account of how the work that occurs within the universe can only occur within a base level.

Composition

In Martin's theory of dispositions he claims that reality is, "a composition or constitution of simple/simpler parts..." (Martin 196). The simplest parts are known as base dispositions, and they are the smallest dispositions that exist-they are atomic. Martin asserts that the manifestation of reality occurs as a product of the partnerings of these base dispositions. This is summed up on page 202, "[c]apacities are capacities for (dispositions for) their exercises or fulfillments (manifestations) under certain conditions (reciprocal disposition partners)." This is saying that the base dispositions have the ability to undergo composition with other dispositions, and it is in this composition that manifestation occurs. It is important to note here that regardless of whether or not a given disposition has a partner for a manifestation, it is the dispositions which are real in the sense that the dispositions are actual.

The fractal model is also a composition. A typical equation for generating the Mandelbrot set relies on what is known as limit cases. Dr. Clifton Cunningham has described limits as, "...a game that requires two players" (beh 2). In the mathematics of fractals, a given number (the input) is then compared with another number. If the numbers "agree" then a limit does exist, and the result of this process gets mapped within the fractal. If the numbers do not "agree" then there is no existing limit, and then the point gets mapped outside the boundary of the fractal³. In other words, for any number, the number only gets mapped into the fractal if it is able to undergo composition with another number.

We see that this is an analogous process to the dispositional account: the solitary number (like the lone disposition) does not get mapped within the fractal (has no manifestation) without first partnering with some other number. In this account we see that the whole of the fractal picture is manifest through both those things which have partners and those things which do not. The fractal itself is the structure of the manifestation occurring through partnerings, and the edge of the fractal is defined by those things (numbers qua dispositions) which do not have partnerings. Both the partnered dispositions and the unpartnered dispositions must be present in order to define the whole: it is impossible to create the picture without having both the manifest and the unmanifest.

Iteration

We will now turn our attention to the process by which the universe becomes manifest under Martin's ontology. In Martin's theory the partnerings of dispositions are the manifestation⁴. This is summed up in his equation "D + P = M (where D is a disposition, P is the disposition's reciprocal partner, and M is the manifestation)" (beh I). The important aspect of this is that for any M the manifestation itself becomes a disposition for further manifestations. Martin says of this on page 207, "...it could be tempting to treat manifestations as themselves disposition-free, or at least not to raise the question of their dispositionality. This would be a gross error." What we have then, on Martin's account, is an iterative process. The output of the initial partnerings become the input for the next occurrence of partnering in a further mutual manifestation. There is nothing that is novel above and beyond the dispositional constituents; that is, the dispositions, even in their partnerings, never become anything other then more complex dispositional systems: there is never a manifestation that is not "...ready to go" (beh 1) for further partnerings.

It should come as no surprise that this is exactly how the generation of a fractal occurs. When composing a fractal we rely on an iterative process where, "the result of one cycle... becom[es] the starting point of the next" (Marshall and Zohar 211). The result that is generated by plugging in a specific number into the equation of the fractal becomes the input of the next step in determining the constituents of the given fractal.

We recognize that what is occurring in the generation of the fractal is exactly what is going on in Martin's dispositional theory. The output of the partnerings is a manifestation which is itself the input for the next potential partnering. The fractal, as noted above, is created by exactly the same process. We now see that an iterative process of composition creates both the dispositional universe and the fractal model.

Deterministic Yet Unpredictable

On Martin's account of the ontology of the universe he asserts that, ultimately, the universe's construction is completely determined; that is, the entire universe, when taken as whole and complete-from start to finish-is completely mapped out and could not have been any other way then how it is. However, he does not think, at any one time or over some length of time, that we can exactly predict which partnerings will occur for the next manifestation. Martin says, "we can't begin to keep up with the mutual manifestations-my God!" (beh 1). By this he means that the interactions of all the dispositions and their partners for the mutual manifestation of the world are so complex that we don't even have a chance to consider all the partnerings which occur at any given moment. This is the epistemic realization that we can't know it all (ibid.). We are only able to have some useful (but incomplete) generalizations about what we expect to occur; however, we can never know with absolute certainty what will manifest next. This is due to the fact that, "we cannot examine an entity without including all its interconnections and interrelations with other entities" (ibid.); in other words, to properly predict the course of the manifestations we would already have to know everything about every manifestation that currently is-this seems well nigh impossible!

The fractal model is, of course, exactly like this as well. The fractal, in its whole and complete form is entirely deterministic. There is not one point that occurs inside or outside the fractal which can not be accounted for by its predecessors. In other words, the entire fractal can be deconstructed to find with certitude what went where and why. However, this is not possible from within the system itself; that is, during the process of the fractal's construction we can not tell, for a given input, if the result is going to have a limit which exists or not until the input has been run through the equation of the fractal's generation. This means that we are unable to predict a priori whether any given unit undergoes composition or not.

Again, we see a startling and obvious connection between Martin's dispositional theory and the fractal model: the entire structure in finished form is absolutely deterministic, but while the structure is being formed there is no way to exactly predict what will happen next. As Benoit Mandelbrot puts it, "The notion of [a] deterministic yet unpredictable [system] is a surprise to most people" (Hall 122).

Self-Similarity⁵ 

A key feature of a fractal is the quality of self-similarity. This concept is best illustrated by Benoit Mandelbrot's example of a cauliflower. In the video, "Fractals: an Animated Discussion," Mandelbrot holds up a sizable cauliflower and asks us to examine its shape and structure. He then proceeds to break off a chunk of the cauliflower and then asks us to examine this piece of the whole. He notes that it looks very much like the original, but only smaller. He then breaks off another bit from the smaller part of the whole. Again, we are asked to observe that this even smaller bit looks like the piece it came from, and that it mirrors the structure of the whole: the point being that this same sort of self-similarity is an essential aspect of any fractal.

The Mandelbrot set displays this self-similarity, but as Bodil Branner informs us, "M [the Mandelbrot set] contains small copies of itself which in turn contain smaller copies...ad infinitum...every mini-Mandelbrot set has its very own pattern of external decorations, every one different from every other" (Devaney and Keen 76). So it is not an identical self-similarity, but only a rough self-similarity; however, there is no doubt that the Mandelbrot is a fractal!

Now in a similar sense, we see that our world has this feature: things in the world are self-similar, but not strictly identical. We see that one person exists as a similar system to another, but that does not make them the same person. The same can be said of birds, houses, trees, whatever. There seems to be in our universe this same sort of self-similarity among its constituents where each complex dispositional system, "...has its very own pattern of external decorations, every one different from every other."

Now since the universe on Martin's account is composed out of dispositions, it is reasonable that we can look to the manifestations around us and recognize that these complex dispositional systems share this sort of self-similarity with our paradigmatic example of the Mandelbrot set. Everything that exists appears to have its unique pattern in the world, but there is also much that is similar amongst the differing kinds of expressions.

The Base Level (The Only Level)

It is in this section that we will come to understand why there is not work being done across levels, but only interactions and interrelations all happening on one level.

On Martin's account the only work that is being done occurs at the only level of reality. "What it comes down to," Martin says, "is the interactions and interrelations amongst the quarks of the base level"⁶ (beh 1). From what we have seen of the theory this far we understand that there can be no other level above and beyond the level in which the dispositions interact and interrelate. To think otherwise is to think incorrectly about what is occurring in Martin's dispositional ontology. We have seen that dispositions partner for mutual manifestation. The manifestations can become very complex sub-systems of the overall whole, but simply because they are complex does not somehow bump them up into another level of being⁷. On Martin's account there is the substratum of space-time, and this is where the manifestations take place. Space-time provides the canvas upon which the manifestations are the paint. Granted, we may perceive layers of paint, but this is a product of our interpretation of the manifestations in space-time: the paint of the dispositions is whole and complete and only ever occurs upon the surface of one canvas! There are not several canvasses layered one on top the other (and this would be a peculiar way to try and paint a picture-with canvasses obscuring the work done on previous canvasses)-it is but a single substratum on which all manifestations occur.

On the fractal model we have an analogous situation occurring. In the case of the Mandelbrot, the fractal occurs on the complex plane. This is a two-dimensional surface. In extending the model to account for the universe we understand that the fractal which models our world would be composed in a three dimensional space. The surface (or space) is the substratum upon which the creation of the fractal occurs. While it might seem to some that we could designate the dimensions (i.e., length, width, and height) as different levels this is merely a semantic mistake: the dimensions themselves are not levels, but instead they exist together to form the singular space in which the fractal finds its expression over time. In other words, the fractal is contained and constructed on only one level. All the mathematics that goes into generating the fractal results in points being mapped into one space and one space only.

Now from this we find Martin's explanation of the base level not only plausible, but also mandatory under the fractal model interpretation. From what we have developed over the course of justifying that the fractal model fits like a second skin to Martin's description of the universe-that is, through showing that the generation and perpetuation of both the fractal and the dispositional universe is identical-we find that it no longer makes any sense to think of a division of labour occurring within a hierarchy of levels. The fractal model illustrates clearly that there is only one level in which manifestation occurs, and thus, there is only the base level within which the dispositional universe finds its expression.

One Model and One Object

We have come to understand that a fractal model best captures the notions contained within CB Martin's dispositional ontology. A result of this analysis was discovering that Martin is correct in the assertion that reality plays out within a singular base level. It also appears that the fractal model can account for much more with respect to the workings of the dispositional universe⁸. Between the fractal and the non-existing limits, for instance, is a "crumpled boundary." It is called this because the defining edge of the fractal is in a constant state of turbulence. Under the dispositional theory this is likely where we will find true emergence taking place: the boundary of the fractal is in a perpetual state of novel manifestation, and this reflects the idea found in Martin's ontology that new properties can emerge from the composition of existing dispositions-none of which posses the property before the partnering. As well, when asked how many things there are in the universe, Martin responded, "It is a one object universe" (beh 1). We see this due to the fact that we can not examine any dispositional system without considering its ties to other dispositional systems. This creates a vision of reality where there are really only complex dispositional sub-systems. These sub-systems all interact together to produce a singular complex dispositional system which is the whole of the manifested universe. When we compare this to the fractal we see more strong evidence to tie the two models together: the fractal is a complex whole that contains within it an infinite number of sub-fractals. Under both descriptions we find that we can not help but take the idea of "the oneness of everything" to heart.

End Notes

1. We realize that this is controversial to some people when it comes to so-called "mental" phenomena; however, we recognize, if nothing else, that thoughts, ideas, and feelings must also have extension in space and time even if we are unwilling to call such extension "physical."

2. All illustrations in this paper appear on page 318 of Michael Barnsley's "Fractals Everywhere"; Acadmeic Press, Inc.; San Diego, CA, USA; 1988.

3. In G the fractal is the black structure. All that is outside this structure are the limits which do not exist.

4. Here we will note that this is the key feature of Martin's theory which deals with the problems of causality; however such problems are not the focus of this analysis, and so, we will not delve into this aspect of Martin's theory here.

5. While self-similarity is key feature of fractals, it is not a key feature in Martin's dispositional ontology; however, Martin is not obligated to discuss self-similarity, but in the case being developed here we have to account for this feature in the universe. That is to say, if we desire to see the fractal model as the best model of a dispositional universe, then we have to illustrate that self-similarity exists in the universe. To provide a strong and convincing argument for this would be a paper of its own; thus, the argument here is only a brief sketch of this notion.

6. At this point (since it has not yet been mentioned) we note that Martin uses the word "quarks" to designate the atomic dispositions; thus, "quarks" is being used here to refer to the most basic dispositions.

7. We can see here an analogy with the archaic notion of the shell model of the atom. It was at one time supposed that the electrons of a given atom circled the nucleus in tidy orbits with levels which expanded ever outward from the nucleus. We now know that this was an overly simplistic way to represent the workings of the atom. Modern physics declares that the electrons could be anywhere, and that there are not these neat shells which the electron gets "bumped into" due to its becoming more or less energetic. In a similar sense, we find that there are not tidy well-defined levels of reality that more "energetic" (i.e., complex) dispositional systems get knocked into. We find that there is only an increasing complexity contained in one level with respect to both the orbits of electrons and dispositional systems.

8. It was thought upon initial conception that this paper would deal in more detail with at least the idea of how our "readiness lines" (as Martin calls them) work to create a "dispositional tunnel" through which we interact with the rest of the fractal system. This would provide an explanation of why we see the world as divided up when really it is not. Due to spatial limitations the concept of the dispositional tunnel will have to be reserved for a later exposition.

Works Cited

beh 1. Notes from "The Physical and the Mental". Collected during CB Martin's Philosophy 589.45 class at the University of Calgary, Winter semester 2001.

beh 2. Conversations with a Mathematician. Interviews conducted with Dr. Clifton Cunningham over the course of the Winter Semester of 2001 at the University of Calgary.

Devaney, Robert L. & Keen, Linda, eds. Chaos and Fractals The Mathematics Behind the Computer Graphics.American Mathematical Society, USA; 1980.

Fractals: an Animated Discussion. Peitgen, H. O., H. Jürgens, D. Saupe, & C. Zahlten. Videocassette. New York: Freeman, 1990.

Hall, Nina, ed. Exploring Chaos A Guide to the New Science of Disorder. W.W. Norton & Company, New York, USA; 1993

Marshall, Ian & Zohar, Dana. Who's Affraid of Schrodinger's Cat?. Quill, William Morrow, New York, USA; 1997.

Martin, CB. On the Need for Properties The Road to Pythagorianism and Back. Handout for Martin's 589.45 class, reprinted from the original, but source unknown. Pages numbers correspond to original source.