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Josh Evans Dec 2011Books, Displays, and Systems TheoryHenry Sussman

A Photographico-phylogenetico-formal Offering:meditations on meaning and structure in the dialogic spirit of Gödel, Escher, Bach: an Eternal Golden Braid:

A metaphorical fugue on minds and machines in the spirit of Lewis Carroll1

Tortoise and Achilles happen to meet one afternoon in the botanical garden.

Achilles: Why hello there, Tortoise. What a pleasant surprise to bump into you here, and on such a fine day!

Tortoise: Indeed. I thought I might take a walk to give my shell a little jolt of sunshine, when I quite absently found myself wandering through the garden gate. What brings you here yourself, Achilles?

Achilles: Well you see, Mr. T, I just obtained this new analog camera, and I’ve been giving it a go in all sorts of different lighting conditions to test its versatility. And I thought, as today was such a blissfully perfect day, that it would prove an excellent opportunity to test my camera in very bright, high-contrast conditions.

Tortoise: I see. How pleasant.

Achilles: Not to mention that after our lovely meeting with Mr. Crab and Mr. Anteater the other week2, and hearing about the remarkable characteristics of Aunt Hillary, I’ve been meaning to suss out some ant colonies to study their patterns and perhaps even try to converse with them, as Mr. Anteater seemed to suggest was not only possible but a perfectly natural and enjoyable way to spend an afternoon.

Tortoise: Quite so. Perhaps I might join you on your expedition?

Achilles: But of course, Mr. T, I would be honoured.

They continue along the path, amid various specimens of native plants, on the lookout for the tell-tale traces of ant activity.

Tortoise: You know, Achilles, your newfound interest in picture-making reminds me of my days as a photographer for the Royal Geographical Society.

Achilles: What an illustrious life you lead, Mr. T! I had no idea. You never fail surprise, do you?

1 Hofstadter, Douglas R. Gödel, Escher, Bach: an Eternal Golden Braid. New York: Basic Books, Inc. (1999).2 cf. GEB, ‘Ant Fugue’, p.310-336

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Josh Evans Dec 2011Books, Displays, and Systems TheoryHenry SussmanTortoise: Quite the opposite, I assure you. In any case, during my time with the Society I had the good fortune of receiving an assignment in Zambia, to document the remarkable ant colonies there, which is why I thought of it just now. What a remarkable experience indeed.

Achilles: Do you still possess any of your prints? I would certainly like to see those!

Tortoise: Oh, no, they are all in the Society’s archive. So perhaps someday, good Achilles. But while I was there I was inspired to carry out a rather different project of my own, which I think you will find interesting.

Achilles: I am sure I will! Please do tell, Tortoise.

Tortoise: Well, in documenting such monolithic superorganisms of staggering scale and complexity, I began to think of how patterns, trends, structure – in other words, meaning – arise out of parts that in themselves are quite meaningless.

Achilles: Certainly – just as Mr. Anteater was describing to us Aunt Hillary’s intelligence, such her ability to converse, even though at the lowest level she is made up merely of individual ants who, by themselves, do not contain such intelligence.

Tortoise: Precisely. But where this realisation took me was somewhere quite different altogether. I began musing on this phenomenon of ‘meaningfulness’ arising out of meaningless parts in relation to the common theoretical technique of analysis in terms a distinction between ‘form’ and ‘content’.

Achilles: As in words out of letters, or the mind out of the brain out of neurons, or…?

Tortoise: Yes, very good. I began to think, sort of differently to many others on the subject, of ‘content’ as the ‘meaningfulness’ that arises out of meaningless formal structures at lower levels. I began to see the world only in terms of form, musing that content or meaning on one level or scale is really just an epiphenomenon arising from the arrangement of form at a lower level, and that therefore all ‘meaningfulness’ or content can be represented in terms of the formal aspects that comprise it.

Achilles: Fascinating. But how is all this related to photography?

Tortoise: Quite elegantly, in fact. Photography has been used for many different purposes throughout its history, from documentation to portraiture, landscape surveying, a method of organising space, a medium through which to comment on the world… but unlike other arts, such as painting, sculpture, and even music,

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Josh Evans Dec 2011Books, Displays, and Systems TheoryHenry Sussmanit doesn’t seem to have undergone a rigourous formal exploration in the same way. That is, photographers are still by and large concerned with the content of their pictures – to varying degrees, to be sure – and I was rather interested in studying how photography could be used purely formally to explore its own nature as medium.

Achilles: Quite deep, as usual, Mr. T. But I’m afraid I’m stuck on one point. Surely these photographers you speak of are not concerned solely with their content, to the exclusion of form? Indeed it seems strange to me that one could exist without the other.

Tortoise: Quite right, as usual, Achilles. They are mutually reliant on the other, and even to make such a disctinction between them is tenuous. Like all theoretical systems, it has is weaknesses – its ‘incompletenesses’ if you will – along with its descriptive and explanatory powers. And it was precisely this set of issues that I was interested in parsing.

Achilles: So what did your project end up looking like? I’m intensely curious to see what you mean by the ‘purely formal’ aspects of photography. I certainly hope those prints didn’t have to be sent straight to the archive too!

Tortoise: Oh no, I keep those at home. I have my own private archive for things like that.

Achilles: I had no idea! Oh the secrets you keep, Mr. T.

Tortoise: It's a habit of age, I suppose.

Achilles: You are an ancient specimen, after all.

Tortoise: Probably as old as some of the trees in this garden, even.

They look around, silently marveling at the high canopy of boughs bifurcating into smaller and smaller, more densely packed branches, ending finally in leaves, too many to count.

Tortoise: In any case, Achilles, I would love to hear what you are taking photographs of, before we rentrer chez moi to show you mine.

Achilles: Ah, I fear my interests are not nearly so lofty or profound!

Tortoise: Nonsense, Achilles. I am sure you are thinking about equally interesting things.

Achilles: Well, one of the reasons I decided to buy this new camera is that I am terribly fascinated by the differences between analogue and digital photography,

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Josh Evans Dec 2011Books, Displays, and Systems TheoryHenry Sussmanand while I have been using my digital camera for a few years now, I realised I had no experience with analogue photography at all! Thus my impetus to learn.

Tortoise: And what sort of differences have you encountered so far?

Achilles: Many! Most generally, it is striking how light is captured so differently by film. And I swoon for how the grain of the film itself can be present in certain pictures, and that it is even possible to process them to bring out this unique, grainy quality. Sometimes I think that analogue and digital are entirely different beasts.

Tortoise: I can see how you might think so.

Achilles: What do you think, Mr. T?

Tortoise: I like grain perfectly well, though I do prefer grass.

Achilles: Hyuk hyuk, Mr T.

Tortoise: Green things are my speciality.

Achilles: Unfortunately for you, today I’m shooting black and white.

Tortoise: Chortle. Well then, now that we’ve each had our fun, I’m curious to hear how you arrived at the notion that film photography is ‘analogue’ and digital ‘digital’.

Achilles: What a funny question. If I didn’t know you as well I might respond with a cheeky tautology. But I suppose it comes down to how the information is stored – whether digitally, in binary code, or analogically, on a piece of film. Isn’t that right?

Tortoise: There is a certain sense in which it may be tempting to think so, yes.

Achilles: Always the man of mystery. What do you mean, Mr. T?

Tortoise: Much in the same way you pointed out the interdependence of form and content, I think a similar principle applies to the relationship between analogue and digital systems.

Achilles: I’m intrigued – do go on.

Tortoise: Let us look at the simpler devices built into the whole device of the camera. Your shutter speed, your aperture, even your light meter – you would agree they are analogue functions, for they operate on scales of ‘more’ and ‘less’, for example.

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Achilles: Yes, certainly.

Tortoise: Yet the same devices also exhibit digital characteristics: they operate on the basis of incremental steps – certain speeds, certain f-stops – fundamentals that cannot be reduced further, that are either ‘on’ or ‘off’, in a certain way.

Achilles: Ah, I see where you’re going with this. And these same devices that comprise my analogue camera are also all present in digital cameras, is that it?

Tortoise: My you’re good. The inevitable layering and mutual embedding of analogue and digital systems is a characteristic of almost all extant systems, both naturally occuring and artificial –

Achilles: which, it seems to me, is itself more of a division out of convenience than necessity.

Tortoise: Indeed. After all, where would we be without each other, dear Achilles?

Achilles: I hardly know. But I do suppose you haven’t finished your enmeshing of the analogue and the digital, have you, Mr. T?

Tortoise: Quite right; certainly not.

Achilles: Well, let me assist, for I have an inkling of where you might go next.

Tortoise: Oh, please yes! I do so love team sports.

Achilles: So: in the same way that there exist certain analogue functions in both ‘analogue’ and ‘digital’ photographic devices, there also exist certain digital functions, don’t there. The actual action of the shutter, for example – it is either closed or open.

Tortoise: Or the film chamber is either full or empty.

Achilles: Or a subject is either in or out of the frame.

Tortoise: Yes, I quite agree. And let’s not stop there – we have only covered those functions layered in the camera device itself. In the making of negatives on film and prints on paper, the degree of light exposed onto the silver salts of the emulsion on the film or paper, the conversion of these silver salts to metallic silver during development, and even the process of development itself – all of these features surely follow analogue rules.

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Josh Evans Dec 2011Books, Displays, and Systems TheoryHenry SussmanAchilles: And to be sure, whether the film or paper are exposed to light, whether the silver is allowed to be converted from salts to metal, and the incremental steps of the development process, all follow more digital rules.

Tortoise: I would be inclined to agree, especially given the recursive complexity revealed in your last statement.

Achilles: In what way?

Tortoise: Simply that analogue and digital functions may be applied to themselves.

Achilles: Ah, you mean how something can be ‘more’ or ‘less’ digital, or ‘more’ or ‘less’ analogue’; and conversely how the binary relation of ‘analogue’ and ‘digital’ is itself a sort of digital function.

Tortoise: I could not have said it better!

Achilles: I highly doubt that, Mr. T. You can always say it better, and often do. Though I don’t know why you felt the urge to shout so loudly just now – I’m right at your side, after all. And people are starting to look at us.

Tortoise:

Achilles: Though perhaps it has something to do with a man talking to a Tortoise.

Tortoise:

Achilles: Or even just a man talking.

Tortoise:

Achilles: Or not talking.

Tortoise: Nonsense.

Achilles: What’s that?

Tortoise: I said ‘Nonsense’. It’s never stopped us from getting on perfectly well in public before. Plus, it’s probably more you than me they’re looking at. After all, a botanical garden is quite a natural home for a tortoise; much less so for a Homeric demigod, being so far from his time and place of origin, and especially one who perhaps only ever existed in the fictional world of the story whence he comes.

Achilles: I daresay you’re right, Mr. T. As per usual.

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Tortoise: Think nothing of it.

Achilles: All this talk of stories and other times and places makes me think of that time we were kidnapped from the top of the ferris wheel at Coney Island by Hexachlorophene J. Goodfortune, Kidnapper-At-Large, and Devourer of Tortoises par Excellence, into his helicopter, from which point we read about ourselves in the story ‘Djinn and Tonic’, pushing into the Escher drawing ‘Convex and Concave’, where I was saved by those kind men after lifting that lamp right from under those lizards’ noses, only to encounter all those Genies and Djinns and metas and GOD that all together granted me my Typeless Wish, which I then used to transport us right into the heart of Tumbolia by wishing “that my wish not be granted” (an affect I did not foresee, I assure you), thus “crashing the system”, as it were, and somehow getting us to turn up in ‘Reptiles’, another drawing by Escher, in which our only option, after I accidentally knocked our popping-tonic off the table, was to read Provocative Adventures of the Tortoise and Achilles Taking Place in Sundry Parts of the Globe in order to push down one level further in hopes of finding an alternative way back out, but that doing so, I remember with both nostalgia and urgency, brought us right into the dark convolutions of the Little Harmonic Labyrinth3, that ominous abode of the Evil Majotaur, out of which, after many musico-logical wanderings, we finally found our way by eating that delicious pop-corn (do you remember?) back up into ‘Reptiles’, to whose plane we moved perpendicularly, somehow popping our way right back into your home. Do you remember?

Tortoise: Almost as vividly as I remember not popping back to the first level of the story.

Achilles: Oh goodness, I had forgot! Does that mean we are still being held hostage by Goodfortune, perpetually awaiting our demise until we pop back up to that level to complete our story? I certainly hope not!

Tortoise: It certainly seems that way, friend. Though I think I am at greater risk, for Goodfortune, if I remember correctly, was a ‘Devourer of Tortoises par Excellence’, not a ‘Devourer of Achilleses’.

Achilles: Too true.

Tortoises: Though it is also conceivable that, as a great warrior, you may be able to stave him off and defend this old soul from harm. In you he may find his Achilles’ heel, so to speak.

Achilles: His my heel, in me find may he? You’re either very brilliant or very mad, Mr. T.

3 cf. GEB, ‘Little Harmonic Labyrinth’, p. 103-126

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Josh Evans Dec 2011Books, Displays, and Systems TheoryHenry SussmanTortoise: Both, I’m sure. Though after that bizarre construction of a sentence, I could say the same of you.

Achilles: Too true.

Tortoise: Thus we twain friends be.

Achilles: Let’s take a photograph.

They pose for the camera, which Achilles has nestled into the fork of a tree.

Wonderful. I’ll print you a copy.

Tortoise: Excellent. Which reminds me, would you like to see my photo project?

Achilles: Ah! Yes! I was waiting to get back to that! All it takes is a perpendicular move, isn’t that right, Mr. T?

Tortoise pulls his limbs into his shell.

Tortoise: If you’ll oblige me, kindly join me inside.

Tortoise pulls his head back into his shell, through which opening Achilles is surprised to pass comfortably as he approaches it.As he passes the threshold, Achilles is astonished at the size of the space before him.

Achilles: Tortoise, I’m simply astonished at the size of your home! From the outside, it doesn’t look nearly so large.

Tortoise: Ah, yes, well, they taught us a few housekeeping tricks back when –

Achilles: Who is ‘they’?

Tortoises: Hmm, well, that’s a story for another time.

Achilles: Very well. You are so full of them, you never seem to run out!

Looks over to the wall, where many different pictures of abtract shapes are hung.

Ah, so these are the prints in question, are they?

Tortoise: Goodness, no, those are just a few simple studies. I keep Project Recourse secure.

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Josh Evans Dec 2011Books, Displays, and Systems TheoryHenry SussmanHe goes over to a large wooden credenza, which he unlocks and from which he pulls a hefty manila folder.

Come over to the sitting room, Achilles, and I’ll show you.

Achilles and Tortoise go over to the sitting room. Tortoise sits on the floor beside a low, wide table of polished wood. Achilles joins him.

Tortoise: Well Achilles, here they are.

Achilles: They’re certainly remarkable – though I’m going to need some help in deciphering what they mean.

Tortoise: But of course. In one sense, they are completely ‘meaningless’. But in another way, they contain nothing but the most meaningful of meaningfulnesses.

Achilles: …

Tortoise: The short explanation is that I derived a recursive program from the medium of film itself, and then ran that program on itself over and over again. Achilles: A sort of self-generating feedback loop, I suppose. And the long explanation?

Tortoise: Well, I suppose it began out of the simple attempt to show how content and meaning arise out of meaningless, formal components on lower levels. It soon became clear that the stymying loops of recursion posed a particularly effective way of translating this idea visually, and things quickly spiralled from there.

Achilles: And these pictures were the result? Please do tell me how they all fit together!

Tortoise: I’m getting there, Achilles. Let’s start at the very beginning then.

He pulls a thick stack of contact sheets and sleeves of negatives from the manila folder.

Do you see this sheet of negatives on top? This is the progenitor of the whole system.

Tortoise shows Achilles the negative sleeve entitled 0.

Achilles: It looks like just a roll of fully-exposed, undeveloped film cut up and put into a plastic sleeve.

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Josh Evans Dec 2011Books, Displays, and Systems TheoryHenry SussmanTortoise: And in fact, that is precisely what it is.

Achilles: Ah.

Tortoise: I can still feel the thrill of ripping open that canister of film in broad daylight. Nothing like it. The impulse to treat with irreverence the medium that usually requires so much care and attention was part of the impetus for the project too, I suppose. But such an impulse is present in formalisms in other artistic media as well, so I didn’t believe I was doing anything too desecratory.

Achilles: I’ll have to try it.

Tortoise: And this is its counterpart, its positive.

He shows Achilles the contact sheet entitled 1.

From each of these, I made exposed one roll, repeating the same image in all thirty-six frames.

He shows Achilles the negative sleeves entitled 0:0 and 1:0.

From these, you may then assume, I printed corresponding contact sheets.

He shows Achilles the contact sheets entitled 0:1 and 1:1.

Etc. I went as far as it took until the first generation of scale, that is, 0 and 1, disappeared into minuteness within the nested frames.

Achilles: Achieving the illusion of infinite recursion.

Tortoise: That was the intended suggestion, yes.

Achilles: Well, Mr. T, you’ve certainly outdone yourself this time, which, as I’m sure you must be at least somewhat aware, is saying something.

Tortoise: But these are just the building blocks of the project – the real interesting ideas arise when one considers the significance of the process. For instance, after a few such cycles of exposing, developing, and printing, I became fixated on how isomorphism functions at each step of the process.

Achilles: Isomorphism – you mean “an information-preserving transformation”.

Tortoise: That’s right.

Achilles: Literally, the quality of having ‘equal form’, from the Greek; or ‘same-shape-ness’, even.

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Tortoise: I suppose, yes.

Achilles: That is to say, in more plainly discursive English, “when two complex structures can be mapped onto each other”.4

Tortoise: Yes, you’ve got it. But who, may I ask, are you so diligently quoting?

Achilles: Ah, some book I read some time ago. Though I can’t for the life of me remember what it was.

Tortoise: Hmm. How curious.

Achilles: Back to these isomorphisms of yours, Mr. T.

Tortoise: Ah yes. Well I imagine it’s quite clear to you now how ridden with different levels of isomorphisms this project became.

Achilles: I never thought about it in this way before, but making pictures by exposing light onto film to make negatives and then exposing light onto paper through the negatives to make positives is nothing if not a very neat system of isomorphisms.

Tortoises: Exactly. And I hadn’t ever thought about it before, but this “neat system of isomorphisms”, as you say, “exposed light”, in a way, onto another, parallel idea – that of “figure-ground relations”.

Achilles: You’ll have to tell me more about those, Mr. T. But who are you quoting in that last phrase? Evidently not me, like the other ones.

Tortoise: If I remembered, dear Achilles, I would have put it in a footnote.

Achilles: Too true. What a reasonable conclusion.

Tortoise: So you haven’t ever heard of figure-ground relations?

Achilles: It rings a very distant bell, as if from another level of reality altogether.

Tortoise: You might remember those Escher drawings we found ourselves in on our meanderings in the ‘Little Harmonic Labyrinth’.

Achilles: Yes, ‘Convex and Concave’, and ‘Reptiles’.

Tortoise: Well, the latter is a particularly good example of figure-ground relations, and how Escher played with them to no end. You remember how the ‘figures’ of

4 GEB p.49

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Josh Evans Dec 2011Books, Displays, and Systems TheoryHenry Sussmanthe Reptiles emerged out of and merged back into the ‘ground’ of the paper? Escher doesn’t let us get away so easily, as the ‘ground’ and the interlocked reptiles in the paper end up becoming just as much a figure as the reptiles out of the paper. Indeed, the reptiles drawn on the paper illustrate that peculiar ‘figure-figure relation’, where each reptile serves both as a figure itself and as the ground for each reptile adjacent to it. In this way, the reptiles are recursive, for alternating figure-ground relations can be run back on each of them in turn indefinitely.

Achilles: But of course! I understand you mean to suggest negative-positive relations in photography as isomorphic to figure-ground relations in general, is that it?

Tortoise: And figure-figure relations in particular. Extremely correct. But not only that – it seemed to me that the isomorphism between the negative and positive renderings of an image exhibited such a purity of parity, in comparison to other types of isomorphisms I’d seen, that there must, in a certain sense, be different types or gradations of isomorphic relations, depending on how closely or distantly the systems map onto each other.

Achilles: And what do you mean by ‘distance’, in this case?

Tortoise: Yes, I guess I should be clear. Take, as an example, the phonograph5. It functions via a fundamental isomorphism – that between physical grooves on a record, and physical vibrations in the air, which we then perceive acoustically as music. Though a textbook isomorphism, it doesn’t carry the same grace of the ‘figure-figure’ relation as does the negative-positive relation in photography.

Achilles: I suppose you’re right.

Tortoise: Which is not to say, I must add, that the basic isomorphism of the phonographic system is in any way inferior to that of the photographic system. But it is nonetheless difficult, in this light, to ascribe to them both the same quality of isomorphism. For their constituent forms are not equal in quite the same way, are they.

Achilles: I suppose not, no.

Tortoise: So, when I invoke ‘distance’, I use it analogically to signify the different degrees of isomorphism that arise through the different processes of information tranformation that occur from one system to another, as well as the different types and quantities of media involved in this process of transformation.

Achilles: So, what I glean from your remarkable theory is that not only may systems be isomorphic or not, but they may exhibit their isomorphism in different

5 cf. GEB, ‘Contracrostipunctus’, p. 75-81; figure 20, p. 84

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Josh Evans Dec 2011Books, Displays, and Systems TheoryHenry Sussmanways. It certainly reminds one of our discussion of the braided nature of the analogue and the digital, wouldn’t you say?

Tortoise: Quite right, Achilles – a wonderful connection I had entirely missed.

Achilles: Speaking of connections, I’d love to see more of these pictures.

Tortoise: All right, Achilles. Be warned, it gets overwhelming fast.

Tortoise empties the rest of the manila folder out onto the table, spreading negative sleeves and contact sheets across the broad wooden surface.

Achilles: Well, I’m certainly impressed, if only for the sheer amount of things you had to create! How did you ever keep track of it all, I wonder?

Tortoise: An excellent question, and one that I think it is the precise time to address. Before I had even finished producing the second round of negatives, I realised that very soon I would be up to my shell in sheets and sleeves of various sorts, and that the only way to keep everything straight in my mind was to find some way to organise all this information I was creating, some sort of structure or system.

Achilles: Hip-hurrah for Tortoise’s maniacally meticulous mind!

Tortoise: Which is when my graduate training kicked in from my days studying number theory in Tokyo, and I thought that perhaps I could devise some sort of formal system6 to represent the ‘phenotype’ of the program I was creating through a more abstract, simple ‘genotype’ that would allow me to effectively predict all possible behaviour within the system.

Achilles: I’m intrigued by that idea, Tortoise, as well as by your use of genetics terminology to describe your thinking. But I suppose we’ll have to come back to that, for I couldn’t bear to make you digress at this point in your story!

Tortoise: I also realised that to find a formal system to define my project would be very fitting, given my initial impulse to study the medium of photography through purely formal methods. So, after a little work, this is what I came up with:

Rules of Inference:(1) If and only if a = 0 or a = 1, then a is a theorem (2) If a is a theorem, then a’ is also a theorem(3) If a is a theorem, then so are aa and aa’(4) a ≠ a’

Axiom:

6 cf. GEB, ‘The MU Puzzle’, p. 33-41

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0 is a theorem.

It seemed that I could derive from this ‘axiom’ – my first premise, if you will, by manually building my first sleeve of negatives – all possible ‘theorems’ of the system – that is, all negative sleeves or contact sheets – using these four rules.

Achilles: Certainly. And did you find this formal system helpful in organising your sleeves and sheets – or theorems, as it were?

Tortoise: In some ways. I did make a couple unexpected discoveries I wouldn’t have made otherwise – for example, it turned out that all possible strings – in this case, all combinations of 0s and 1s – turned out to be theorems of the system. The same is not true of most other formal systems – indeed, one might argue that the whole function of formal systems is to distinguish between theorem and non-theorem strings through their rules of inference. So from the beginning it seemed as though I were approaching a certain tidiness in my project.

Achilles: And what else?

Tortoise: Well, I also noticed that the simple binary code I had set up came to resemble Gödelian numbering in certain ways. But we’ll return to Gödel; he will become more relevant to our discussion later on.

Achilles: All right, I’ll take your word for it. But what about its organisational component? You’ve yet to speak to that.

Tortoise: Well, to be honest, I still wasn’t quite satisfied. My formal system gave me a method of characterising all possible theorems, and of representing the infinite recursion of the project in a finite little bundle. But I still needed a way of mapping out the different sleeves and sheets in relation to each other, so I could keep track of them visually.

Achilles: Certainly. You do have a very visual mind after all, Mr. T. I wonder what sort of visual arrangement you came up with in the end.

Tortoise: One that suited me very well. And it began from the simplest of ideas. Do you remember earlier when I was referring to the various iterations of my project as ‘generations’? That gives it right away – it all stems from when I realised that each negative sleeve and contact sheet – each ‘entity’, if you will – generates two new entities through the corresponding negative and positive processes of film exposure and print-making. From there, the first image that came into my mind was that of the great branching tree of life, that serves as a visual representation of the complex processes of evolution, speciation, hybridisation, etc. Evolutionary biologists call this a ‘phylogenetic tree’, and I knew that was the visual strategy I needed to organise my ever-increasing number of entities being generated by my recursive program.

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Achilles: Of course! What a stroke of brilliance. It sounds almost like a family tree, doesn’t it Mr. T.

Tortoise: Why yes, I hadn’t thought of it that way before, but in a certain way that is an even more apt description. I suppose it depends on whether we view each ‘entity’ as an individual organism passing on its ‘genes’, or as an entire species as an abstract entity evolving over time. A matter of scale, I suppose.

Achilles: And what’s more, it seems as if you’ve inadvertently found a way of creating recursive structures out of isomorphisms – that is, this phylogenetic or family tree, isomorphic to the system by which your photographic entities propogate their traits and evolve over time, is itself built of entities that are isomorphic to other entities – the negative and positive versions of the same image being isomorphic to each other. Isn’t that right?

Tortoise: My, Achilles, you are revealing all of these delectable insights I hadn’t even thought of! You are truly a friend of many gifts.

Achilles: Well, Mr. T, now I would like to see this tree you’ve drawn!

Tortoise pulls another sheet of stiff card from the manila folder, and passes it to Achilles.7

Achilles: That is quite brilliant.

Tortoise: What is most salient about the tree, though, is what it revealed about the limits of the formal system that had led me to it in the first place.

Achilles: Is that so? I can’t imagine what you mean.

Tortoise: Well, quite simply, there were phenomena occurring in my project that were not predicted or allowed by the formal system I had derived from it in the first place.

Achilles: Such as?

Tortoise: For instance, one day, when it was time to shoot 0:0:0, I decided to actually shoot it twice, in different contexts, to see how the different rolls would turn out under different conditions. For the first, I shot 0:0 taped to the light table in the photo lab, a table surface lit from below to enhance the ability to inspect negatives; I named it 0:0:01. For the second, I shot 0:0 taped to my bedroom window, in the morning when the light was soft and diffuse; I named it 0:0:02. I found the first had much better clarity on the negative, owing to the high-contrast

7 see figure ‘phylogeny’

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Josh Evans Dec 2011Books, Displays, and Systems TheoryHenry Sussmanlight of the direct sunlight overhead, compared with the indirect sunlight of the second roll. So I chose the first to continue over the second.

Achilles: Evolution in action, eh Tortoise? Survival of the fittest.

After some thought:

Or at least in this case, survival of those best suited to photographic reproduction.

Further pause…

And now I see why you used the language of genetics and DNA to describe your motivations to create a formal system for your project!

Tortoise: Precisely. And what’s more interesting, as you’ll notice, Achilles, is that neither 0:0:01 nor 0:0:02 are ‘well-formed’; not only are they not theorems of the formal system I’d set up, but they aren’t even strings of the system – they aren’t even possible theorems because they aren’t built using the allowed building blocks of the system – namely, binary shorthand (the colons are purely my own convention).

Achilles: I was wondering why I had found two ‘copies’ of the same ‘entity’ – though I suppose they are actually two separate entities altogether!

Tortoise: I took it as the impossibility of ever fully filtering out content – the conditions under which I was attempting to translate ‘pure’ form through multiple stages of the recursive program inevitably found their way into the representation.

Achilles: Ah, so just as analogue and digital, and figure and ground, are both bound up in each other, so are form and content.

Tortoise: Yes, that’s a nice way of putting it. I’m sure you also noticed how some entities contain blank or partial frames, such as the negatives 1:0, 0:1:0, 0:0:1:0, 0:1:0:0, and their corresponding positives, 1:1, 0:1:1, 0:0:1:1, and 0:1:0:1.

Achilles: Yes, I was wondering about those.

Tortoise: Well, those are aberrations of the photographic process. They are instances where my camera exposed the film only partially, or not at all. I decided to keep them in process to illustrate how the isomorphisms that transform information between positive and negative states are actually not at all as perfect as we might imagine them to be. Though there may be different gradations and types of isomorphisms, there is no such thing as a perfect or unmediated one.

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Josh Evans Dec 2011Books, Displays, and Systems TheoryHenry SussmanAchilles: What an excellent idea. How sensitive to remind us of the imperfection of a system, just as we come to appreciate and even rely on its formal aspects.

Tortyse: I was also piqued by the extra justification these ‘mutations’ gave to my borrowing of the analogy of phylogeny from biology.

Achilees: Ah yes, that is what they are, isn’t it! Information translated imperfectly that is then passed on to successive generations, at lower and lower levels of the recursive structure; or, as smaller and smaller frames nested within the whole. A recursive system evolving in time. Marvelous.

Tortyse: You’ve quite a grasp of it, don’t you! How thrilling, Achiles, I knew you would be the perfect audience with whom to share my work.

Achilees: I’m flattered, Tortyse, really I am. And honoured that you would share such complex stuff with the likes of me, a brawny warrior-cum-sprinter.

Tortyse: You can thank Zeno for that. But there’s one more quiddity I’d like to share with you – it has to do with the formal system we set up earlier.

Rules of Inference:(1) If and only if a = 0 or a = 1, then a is a theorem (2) If a is a theorem, then a’ is also a theorem(3) If a is a theorem, then so are aa and aa’(4) a ≠ a’

Axiom:0 is a theorem.

These rules seem to be necessary to the system, but there is a certain way in which they are merely covering up an anomaly that does not fit that well into so elegant a formal system – much like Euclid’s Fifth Postulate8.

Achilees: Now I really haven’t a clue.

Tortyse: All right, I will explain. (1) says that theorems are only theorems if they are strings made up of 1s and 0s – this rule defines what the building blocks of strings and theorems can be. (3) is our ‘reproductive’ rule – it says that every 0 and 1 can have two offspring by placing a 0 or a 1 to its right. So whence (2) and (4)? Shouldn’t (1) and (3) alone be enough to generate our tree? Unfortunately, no, they aren’t. If we look closely, it becomes clear that with only (1) and (3), we have no way of generating 1; all we are able to do is to assert 0 as our axiom, and then generate all possible offspring of 0. But we have no way of getting to 1 unless we add in (2) and (4), which ensure that two distinctly notated theorems (a

8 cf. GEB, p.88-93

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Josh Evans Dec 2011Books, Displays, and Systems TheoryHenry Sussmanand a’) may not be identical, and thus allow us to establish 1 as a theorem if 0 already is.

Achilees: Sure. So what’s wrong with that?

Tortyse: Well, it doesn’t address the fact that there is a procedural disparity – and therefore, a systemic inequality – between 0 and 1. In the phylogenetic tree we have constructed, we have placed 0 and 1 on the same level – just by the picture, they are both ‘first generation’. But we know that 1 necessarily comes after 0, for the contact sheet is posterior to the sleeve of negatives.

Achilees: Oh, I see. So what does this mean? It certainly makes designing a complete graphical representation of the project very difficult.

Tortyse: You've hit it on the head again, Achilees – the problem is precisely that there is no one isomorphism that by itself maps the system and all of its information completely. This is where Gödel comes back into our discussion, with his famous Incompleteness Theorem. As you know, he proved that there was no such thing as a complete system – that for any system, there would always be some true statement inexpressible within it. Mathematicians and philosophers before and after Gödel have tried in vain to resist this notion, but the fact remains. This is why each system we create inevitably runs up against some salient incapacities – why our formal system doesn’t work quite right, and why our phlyogenetic tree isn’t able to represent the totality of true statements about the system in just one way.

Achilees: And why Euclid’s Fifth Postulate wasn’t fundamental to all geometries.

Tortyse: Exactly – and it was only in recognising our ability to reframe our interpretation of the world in terms of multiple geometries that we appreciated this fact.

Achilees: I do love elliptical geometry; I know you yourself are quite partial to hyperbolic.

Tortyse: They are both necessary. In my view, this chronic Incompleteness of systems is why we need many of them in any instance – it is only through the parallax of multiple systems layered on top of one another, in a vast system of interconnections, that we are able to make sense of reality in any meaningful way. In other words, it is only by embracing the recursion inherent in reality, rather than rejecting it or trying to abstract it away (which is doomed to fail), that we are able to make sense of anything at all. Perhaps not perfect sense, but then again, we know that isn’t possible anyway.

Achilees: That reminds me of Zen Buddhism. When asked a question, Zen masters, instead of giving an answer either way, will regularly unask the question

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Josh Evans Dec 2011Books, Displays, and Systems TheoryHenry Sussmanby saying “Mu”9. It is a way of “jumping outside the system” – or, in the language of our esteemed Mr. Tortyse, a way of recognising the Incompleteness of any system, and the necessity of entering into another one to address these Incompletenesses, even if it means doing so ad infinitum. Such is life in a strange loop, I suppose.

Tortyse: Yes, Achilees, I quite like that. Though I have to admit I still don’t quite understand your Zen Buddhism.

Achilles: It’s rather simple, Mr. T – it’s all about ‘Mu-tation’!

Tortoise: Hm. That’s an idea that definitely requires some more thought.

Achilles: In the meantime, Mr. T, how about we resume our walk? I would love to find an ant colony to photograph and converse with before the daylight fades.

Tortoise: Oh, certainly.

Achilles: Also, Mr. T – I was wondering if you might consider lending me your Project Recourse materials for a while. I would love to look over them more closely.

Tortoise: Oh, yes, of course. Especially after generating such fruitful ideas on the matter yourself, Achilles. Just take good care of them.

Tortoise begins stacking up the sheets and sleeves.

Achilles: Also, I was was curious why you named it ‘Project Recourse’?

Tortoise: Hmm… well the obvious linguistic connection is to ‘recursion’ – but I also wanted some way of expressing that this was my source of help in confronting a difficult problem – and that the ‘problem’ of strange loops also contains the very source of help that alleviates the ‘problem’. So ‘recourse’ seemed like a good title, in those ways.

Tortoise puts the neat stack back into the manila folder, and hands it to Achilles.

Well then, shall we return outside?

Tortoise lets Achilles to leave his home first, before returning his limbs and head to their respective orifices. They meet each other again outside Tortoise’s shell.

Achilles: My, Mr. T, now I think I’ll have to get used to the scale of this world all over again.

9 (無, perhaps: a prefix of negation or absence)

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Josh Evans Dec 2011Books, Displays, and Systems TheoryHenry SussmanTortoise: It shouldn’t take you too long, I think. Oh, look there, Achilles, I think we may be in luck!

Tortoise gestures towards a small line of ants, winding their way across the path ahead into the undergrowth.

Achilles: Let’s follow them!

Achilles bounds ahead of Tortoise, leaping over the bushes and following the trail of ants to a small clearing. Tortoise brings up the rear.

Tortoise, you’re not going to believe this.

Tortoise: What is it?10

Tortoise reaches the edge of the clearing, and looks out over the ant colony.

Achilles: It’s “MU-tation”.

Tortoise: It’s “analog”.

Achilles: It’s “digital”.

Tortoise: It’s “Mutation”.

They gaze in awe for several minutes.

Achilles: I think it’s all those things.

Tortoise: I think you had better take a picture.

Achilles raises the viewfinder to his eye, and releases the shutter. They remain for a while longer, marveling at the constant yet ever shifting image in the settling dusk.

Post scriptum: Achilles writes a letter to Tortoise, returning his project, along with a few notes of his own.

Achilles: Dear Mr. Tortoise,

10 see figure ‘MU’.

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Josh Evans Dec 2011Books, Displays, and Systems TheoryHenry SussmanI must thank you again for the enthralling conversation the other day, as well as your gracious hospitality. What a truly mind-expanding time! I confess I never fail to leave an interaction with you, however long or brief, with many questions and ideas circling round my head for hours afterwards. It is a most pleasant state to be in indeed.I ruminated on your project for a long while after parting, and late into the night I pondered the different concepts and layers of meaning your simple study brings to light. And as I had my camera at hand and was very much in a photographic (or, as it were, meta-photographic (which, I suppose, can be represented simply photographically after all (such is the beauty, I have come to see, of the Gödelian collapsing of language and meta-language))) mood from the course of the day’s events, I found myself trying to make sense of the different entities and trying out different ways of organising them in the frame. I have enclosed for your enjoyment an example of the sort of experiment I was doing.After poring over the various isomorphic methods you used to try to represent the structures of the system of your project, and each of their inevitable Incompletenesses, I thought I might try my hand at making my own. I call this one the ‘master’ sheet of your project: I took one frame of each entity, and compiled them all, in the chronological order of their production, into a sort of catalog of the system, up to the point to which you had explored it. On top are the twelve sleeves of negatives; on the bottom, in an inverse arrangement, are the twelve corresponding contact sheets.I think there are many interesting ideas which arise out of this ‘master’ catalog. One of the most readily apparent facts is the diversity of contexts for each frame – the backgrounds on which each specimen was photographed, the different qualities of light, the various tonal ranges. Each of these, as you mentioned, generates unavoidable ‘content’, even in the pursuit of purely formal expression through the medium itself. The arrangement of the frames also openly recognises this ‘meaningfulness’ – the division between the negatives and the positives, for example. It is by embracing their ‘content’ quality that their formal aspects are allowed to shine.Furthermore, I think including the diagonal line created by the negative space, or the ‘ground’ to the ‘figures’ of the frames, is extremely salient. For me, the diagonal line represents the “jumping out of the system” – it is the first line to exist in both the horizontal and vertical axes, in a system built almost exclusively on rectangular boxes. It collapses the binary of the system’s language, gesturing nicely towards the diagonal lines of your phylogenetic tree, as well as to the ‘softness’ of the otherwise hard binary system of 0 and 1.Finally, you may notice that I decided to include a length of unexposed film as a sort of meridien bisecting the page. I included it both to represent the semi-permeability of the positive and the negative (and the analogue and the digital, no doubt), and also as a gesture towards the axiom of the system – the initial roll of film you tore out of the canister and exposed fully to sunlight. I thought the fully unexposed roll provided a nice sort of Escher-ian counterpoint, don’t you think? To me, it allows the system to fully realise its own strange-loopiness.

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Josh Evans Dec 2011Books, Displays, and Systems TheoryHenry SussmanThese are just a few of the thoughts I have had since our discussion; I am sure you will come up with many more. At the very least I hope this ‘master’ sheet provides an interesting diversion for you from your other projects.Till the next time we meet!I am, your friend,Achilles

ps – I have also included a photograph of that absolutely stunning phenomenon (or should I say epiphenomenon?) we witnessed that afternoon with the ant colony. I still can’t believe this pattern arose simply out of ants scurrying around! Well then, I suppose that’s the point – that it wasn’t just ants scurrying around.In any case, I returned to the ant colony a few days later, at night – and I saw an equally remarkable event, but in an entirely different way. It was this same pattern, but with its figure and ground reversed. Perhaps the colony is nocturnal, and they were all coming out to feed. We’ll have to ask our friend Mr. Anteater.Can you imagine? A huge roiling mass of ants, the only spaces left clear where those fine trails of these selfsame ants ran during the daytime. Perfect counterpoints. I’ll have to take you there some night soon.

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