Horizons

Introduction: like "The Fractal Structure of a Dispositional Universe," this is another of my papers from my university days. I will be publishing several such works on this blog over time, so this is the latest entry into the online catalog of my previous work. Some minor stylistic and linguistic edits have been done for this representation.

This essay was written in the Fall of 2002 as part of a course on Hermeneutics which I attended with Dr. Morny Joy at the University of Calgary. It offers an interpretation and analysis of the word 'horizon' as it pertains to our experiences of such a thing both in the world and in our modes of interpretation. Perhaps by synchronicity, it also corresponds to a digital art project I am currently working on for release in the coming days.

Enjoy.

Horizons

  

Sometimes when we don't understand something we will remark that we are "in the dark" about this or that, or that the matter in question is "over our heads." In the former, we might say that we lack the "light" of comprehension--there is something in our interpretation of things that is obscured in shadow, that is unknown. In the latter, we may feel that the topic is too "airy" in the sense that it is unfixed and drifting, avoiding the grasp of our intellect or perhaps our interest. We might feel that our incomprehension is what keeps us chained, but perhaps also sure-footed, on the ground, while the interpretation that would promote some kind of understanding drifts aloft, soft, defined with fuzzy edges, and open to the fancies of imagination--a cloud in the sky. Off in the distance from where we stand, where the earth stops and the sky begins, we find a unique series of points which forms a line so special that it has its own name--the horizon. It is the horizon that traces the boundary between life amongst humans and life amongst the birds. We fly unaided only in our dreams.

An interesting thing occurs on the horizon, a trick of perspective: parallel lines intersect. We can imagine ourselves standing on a railroad track which stretches off into the distance, built with careful hands in a straight line over a vast and flat plain. Looking down (and assuming our basic understanding--our preunderstanding--of railways, trains, and tracks), we see that the two rails are side by side--never to meet--and we know that they will remain that way the whole length of the track. They are as two soulmates whose paths are entirely complementary, but are destined to remain forever unable to hold one and other in a loving embrace. Yet, when we look towards the horizon, we see that the rails come together--the hopelessly separated lovers merge into one line, they do get to embrace one and other after all! At, or perhaps in, the special location known as "the vanishing point" the two parallel lines become the same line. this unique point occurs on, or perhaps within, the horizon. It exists as a part of the edge of our world, the edge where the sky and the ground caress one and other.

The horizon is not merely responsible for invoking the paradox of intersecting parallel lines, it is also the prime and unique location of the rising and setting of celestial bodies. In particular, the sun brings its light each day, sweeping out an arc from east to west across the sky, but it begins its journey by rising out of the horizon and ends its travels by sinking into the horizon. The moon also traces its path from east to west. It too comes from and returns to the horizon each day. It is at this point, but only in passing, that we may note that occasionally the sun is associated with reason, and sometimes the moon is associated with our intuition. Of course, the two bleed into one and other--our intuition can sometimes be clearest in broad daylight, and our reason can keep us up late, working with precision and logic under the silvery light of the moon--and like our parallel lines, reason and intuition, in their distinctness, share a loving embrace.

Day breaks on the horizon, chasing away the twilight which was yielded by the dark (but later it is the dark which makes the twilight begot from the sun scurry off). The beauty of a sunrise is comparable only to that of a sunset, and the glory of both events is interwoven in the distant horizon. If there was no horizon, then there would be neither rising nor setting sun. There would not be a day separate from night, and there would be no subtle interim between the two that cloaks itself in greens and blues as stars fade from or into sight. The horizon is woven into the cycles of life, and appears to be a necessary aspect of our very being. If there were no earth as distinct from and complement to the sky, and if there were no sky as distinct from and complement to the earth, then there would be no life for humans: we would not be around to understand or misunderstand anything! Without a definition of the horizon, without a line between the earth and the sky, there would be no individuals who would utter, when they misunderstood something, "that's over my head."

Which returns us to our exclamation of misunderstanding that we began with before we embarked on a metaphorical, paradigmatic plane shifting journey provoked by thoughts on the horizon. But such a whimsical romp was not without purpose. In chapter two of his book Hermeneutics, Richard E. Palmer introduces us to the notion of a horizon.ⁱ He suggests that it is the meeting of horizons of understanding between an individual and, in the context of his discussion, a text, which gives rise to the "hermeneutical problem" (26). However, we could easily substitute 'other' for 'text,' ⁱⁱ and we would now have a better idea of the range of hermeneutical inquiry. Hermeneutics appears to be concerned with the "dynamic complexity of interpretation" (ibid.) that arises when the self encounters the other, and it seeks to study the ins and outs of understanding, which is generated out of this complexity. It is suggested here that our reading of Palmer's "horizon" would do well to include the shifting sense of our previous meanderings.

Our parallel lines might be framed as the self and other, polarized points of view, or perhaps any sharply defined dichotomies: appearing in the foreground as distinct and separate, but merging into a unity at a point on the horizon. Or perhaps the lines are like two differing interpretations. Our contrast between sky and earth could mark the distinction between our future--up in the air--and our past--roots firmly planted in the ground--with the horizon tracing out the meeting of these in the moment of the now. It is in the present moment that we live, and we live within our understanding of the world. The sky could also stand in for the unknown, the strange, the foreign which drifts beyond the reach of our understanding in vast expanses, while the earth could represent the known, the common place, the familiar things which make up our everyday comprehension of our world. Again, it is the horizon where these two contraries meet and mingle, blending together with subtlety, one embracing the other. Likewise, the cycles of day and night, the rise and setting of sun and moon, could be read as the alternating cycle of reason working in tandem with our intuition. It is these two aspects of our human lives that support and enhance one and other, and it is a relation of mutual respect and necessity for one and other which generates a broader understanding. We could also see these cycles as related to the hermeneutical circle, the interplay between part and whole within which understanding is generated "by a dialectical process [where] a partial understanding is used to understand further still..." (Palmer 25). The light of day being our understanding which must succumb to the night's darkness--our incomprehension--which will itself dissolve back into understanding when the sun returns from the horizon again during the dawn: the dawning of our comprehension. Again, it is the horizon which marks out distinctions, and without it there would be no separation of one from the other, but, at the same time, it is the horizon which blends and shades one into the other, and, as such, the horizon is a sort of paradox, but it is necessary for our lives. These possible readings of the elements of our journey are only part of an interpretation which might open up vast expanses of sense and meaning--the layers of the horizon are as intricate and bountiful as the complexities of human interaction.

It is my hope that these thoughts on the horizon point to a deeper and more meaningful understanding of Palmer's use of the word with respect to hermeneutics. Moreover, we can see that "the meeting of horizons" creates a further horizon: the boundary between our circles and cycles of understanding, and the circles and cycles of an other's understanding. This horizon is the turbulent edge where self and other are united, where experience is generated, where interaction and embrace gives rise to human interpretation and mutual understanding.

---

i) Palmer, Richard E., Hermeneutics; Northewest University Press, 1969.
ii) We do this in the spirit of David E. Klemm's opening remarks, on page one, in his Hermeneutical Inquiry Volume 1 The Interpretation of Texts; The American Academy of Religion, 1986.

Observations From A Fellow Traveler

I have been hard at work, again, lately--finally--at pushing, probing, and examining the limits of my conceptual models. I have, in other words, been testing the precepts of my reality. From a slightly different perspective, I have been reflecting self-referentially on my perceptions and looking for the doors with some intent to step through them.

In order to accomplish this with any sort of actuality, I need to engage in discourse that lies outside of any sort of private language in a Wittgensteinian sense. This is to say, I need to take the strings of my language that also have a sense and reference in the world, and compare and contrast them to the sense and references of other people. I am required to fit my concepts as best as possible to those of Others who also share in the use of these concepts. Again from a Wittgensteinian sense, I need to play within the boundaries of our language games in order to investigate these same boundaries. With reference to Alan Watts and in the context of my previous statement, I recognize I am playing a game that necessarily requires that I am not playing it alone. Thus, I am playing the game with not only myself, but with Others as well.

Some people believe, and I don't feel that they are wrong, that "work" on One's self is necessarily work on the world. Put differently, when we engage in playing the game in the milieu of self and Other referencing, we are necessarily also engaged with the conceptual models that Others use to play the game from their own perspective and experiencing. If we think closely about this, then we realize that when we are playing the game in such a way as to push at, question, and/or dismantle or otherwise alter our own conceptual models--our altars of being--then we are also necessarily, pushing at, questioning, and dismantling or altering the altars of Others. To put it more plainly, work at changing ourselves not only changes our world, but also changes the worlds of Others.

A problem we will sometimes encounter directly because of this necessary relationship between ourselves and Others is that while we are free to engage in the game from this sort of strategy, and they are also free to engage in the game from this sort of strategy, Others may not necessarily choose or desire to use such a strategy at this time. It is less that we, as investigators into our own concepts, are encroaching on the freedoms of Others, and much more that we are necessarily engaged with attempts to liberate them from the shackles of their concepts simply because we are also attempting to liberate ourselves from the shackles of our own conceptual limits. Put differently, work towards liberation of One's Self from the self, entails work towards liberating the Self from the self of Others. And to some not prepared or otherwise willing of their own free choice to join us in this process of liberation, in this potentially perilous journey, it can also sometimes make these same unwilling or unprepared Others become defensive in the protection of their self from the Self. People often tend towards deep emotional investment in the altars of their existences.

And this is where it can become somewhat tricky or otherwise problematic for us to proceed.

We are, as travelers wrapped up in notions of self necessarily dependent on our concepts and precepts of not only self but also Other, attempting to recognize and communicate with that greater archetype of Self. We seek to relate to the Other in a way that is explicated by Martin Buber as relating to our experiences and interpretations of our experiences not from a position of I and It, but from a position of I and Thou. Such a position seeks to recognize, acknowledge, and embrace the inherent divinity or absolute mystery wrapped up in the core of any particular thing and at the core of everything. Put differently, we are, in seeking to shed our current self images and instead reformulate them as informed by the Self, necessarily engaging with the Self as it manifests to us in instantiations of the Other. And, as noted previously, if the Other is not able to recognize this sense of Self within their self, well, our engaging with them from this perspective can create much friction.

We are, as travelers in this manner, necessarily moving in a direction that is neither perceivable nor understandable from our common, everyday, "mundane" movements through time and space. Indeed, it can also be difficult for our own self to understand and perceive such movement and the direction along which this motion proceeds. It is precisely our need to represent such motion towards our Self to our self in a way that becomes comprehensible to our self that we must necessarily engage with the common language game and the existential game that we are all playing: we are all pieces embedded in an interpenetrating and binary cross self-referencing existential and linguistic milieu. So, when we seek to manifest the Self in instances of self, we are also manifesting the Self in the Others we are necessarily related to.

The Self, as entirely empty in itself, but full of all the Other possible archetypes as potentiae, is an awesome thing: it meets us in our own sense of awe. However, its very emptiness, its absolute Otherness in absence of any specific set of rules governing either our language games or our conceptual framings of our existential games, is also terrifying. As Rudolph Otto described it, it is the manifestation and experience of the mysterium tremendum et fascinans. It is, so much as we can describe it in language that must necessarily limit its unlimitedness--giving form to its empty formlessness--the meeting of the divine, the numinous, in our self and the Other; put differently, it is the meeting and recognition of the numinous from itself, to itself, and with itself.

Put differently still, it is a greeting and acknowledgement that we are, in fact, playing a game with our Self: it is breaking the fourth wall of reality where we as players come to see and communicate with our selves as the audience, and we as the audience come to see and communicate with ourselves as the players. Translation: we position the Self to acknowledge its Self as identical to both players and audience. And this is a threat, in some ways, to maintaining the illusion of the game. If the Self desired to maintain only a relationship to its Self, then it would not have ever bothered to begin the process of wrapping its Self in Self deception as manifest in the dual paring of One and Other.

Now, all the above simply to get to the notes of this particular traveler at this particular point in time at this particular point in space, so this particular traveler would like to thank you for coming along with me on this linguistic journey thus far: thanks, thanks to you.

To the notes, then.

As a human traveler we relate to the members of three general classes of Otherness in the world:

1) Members from the class of all things we commonly call "inanimate matter,"
2) Members from the class of all things we call "animate or biological matter" that excludes other humans, and
3) Members of the class of all things we call "other humans."

It is my opinion and experience that when we are pushing our own boundaries in order to move in an unnameable and incomprehensible direction that our relations to the instances of Others from class (1) and (2) relate to us from a position of joy. As Others that are not invested in, and shackled by, a human based sense of self, when we recognize our Self in them, then they can not help but be joyous in the manifestation of I and Thou. We, in relating to One and Other lift ourselves towards the direction of the divine, an embracing of the mystery that manifests the reality of One and Other. It is relating to the third class of Otherness that can be problematic.

There are, I think, I feel--I experience--three ways that relating to this class of Others tends to manifest in relation to our self:

1) Acceptance and mutual recognition.
2) Indifference.
3) Rejection of a mutual recognition.

Relating to those of type (1) is a positive experience of I and Thou. It can be as simple as an exchange of sincere smiles, including, perhaps, a warm and heartfelt greeting between strangers fully engaged in the fleeting moment of transitional recognition of Self. This will almost always be joined with at least a moment of direct and intimate eye contact, usually several such moments.

Relating to those of type (2) can also be a positive experience in terms of our own interpretation of that experience. Again, with reference to strangers, we offer a sincere smile and perhaps include a heartfelt greeting, and we receive in return something less than that. We might simply be ignored, we might get a forced or partial smile, we might get a mumbled greeting, we might get some formal "hello." This will sometimes involve fleeting eye contact, but not necessarily any eye contact at all.

Relating to those of type (3) is where the potential danger lies. We might simply get the very bottom of the barrel form of indifference, and this is the best we can hope for in such an encounter. We might also get a sense of possible negative outcomes, a sense of potential violence, a sense of a self acting out to openly defy the recognition of Self within itself. And, in the worst case scenario, we, in fact, get the manifestation of actual violence. It very much brings to mind a scene in the movie Inception:

It's obviously not as fantastical as all that. We aren't manipulating the very structures of existential experiences, but we are subtly imposing upon the Self, subtly beckoning to the Self, to acknowledge its Self, and the Self in some instances of self will do whatever it can to avoid acknowledging and recognizing its own entrapment within the game that it has created.

Sometimes, fleetingly, I get the sense of this last sort of interaction from strangers when I am out and about and engaged with the world. Rarely have instances of confrontation over acceptance ever spilled over into actual violence: perhaps never. However, sometimes, like recently, when I am really expending energy and training my focus intensely upon recognizing the Self in my self and Others, I get messages--brief, but clear--that from where I am and the motion I am attempting along that indescribable and incomprehensible direction that if only I take a few more steps, if only I am not very careful about the steps I take, then that Self that is inside the Others but also wrapped up so tightly as to deny its own existence, well, I feel like I am on a precipice where a simple transition or slight flutter leading to catastrophe is entirely possible and some kind of actual violence is a distinct outcome.

To conclude my notes on these matters, I will point out that the best way to banish such feelings--and it is best to banish such feelings swiftly and completely so as to not risk undesired and counterproductive consequences for our own mental and emotional well being--is invoking laughter. Laugh at One's self, laugh at One's Other, and laugh within the context of playing an incredibly complicated game of hide and seek. Because what we are encountering is not only, or merely, the resistance of Others to their recognition and embracing of Self in self within the context of I and Thou, but the remnants, the fragments, the slivers of our own resistance to acknowledging and embracing the Self in our self.

One day, perhaps, we will all be ready, each of us as manifestations within each of the three classes mentioned above, and we will call out with a sense of relief, a sense of sadness and mourning, and a sense of joy in our accomplishments and imaginations:

"Come out, come out wherever you are!"

And it is at that moment we will all cease our journeys and all return home: eternal in our undifferentiated Self, and yearning to once again become travelers in the next instance of the game.

Informal list of references (not necessarily referenced directly, but instead mentioned intuitively) for interested fellow travelers:

Wittgenstein and Language Games.

Alan Watts and the Game.

Carl Jung and The Self.

Martin Buber and I and Thou.

Rudolph Otto and mysterium tremendum et fascinans.

The Numinous. 

Catastrophe theory.

See also:

Warning.

Śūnyatā.

Pratītyasamutpāda.
The Holy Moment.

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.