The Symbolic Transformation

Chapter 5. The Symbolic Transformation

GTR2: the detachment of the convolution from its carrier

[ INSERT FIGURE: the symbolic transformation ]

The Symbolic Transformation.

5. 1. The Boundary of GTR1

The previous chapter left the subject at a threshold. GTR1 gave it space, objects, movement, and time — but all of this is bound to the perceived. The object must be on the map, the goal — in the field of perception, the action — directed at what is present. This binding has a second, harder consequence as well, which is worth stating directly.

The maps of GTR1 are small. As shown in Chapter 4, MP11 is a multimap of many elementary maps of fixed size, and the routes of MP12 with the chains of MP13 are built within one such map; crossing to an adjacent one through a gateway is far more difficult to implement. Therefore the routes and event schemas of GTR1 are physically short — their length is limited by the embryonically fixed size of the elementary map. An animal that thinks only in GTR1 cannot build a path from one city to another: such a path fits into no single map, and stitching thousands of maps together through gateways is unrealistic.

Both limitations — the binding to the present and the limit on length — are removed by one and the same acquisition. This acquisition is the symbol.

5. 2. GTR1 as a Set of Local Maps

To understand what exactly GTR2 convolves, one must clearly picture what lies at its input. A convenient starting point is the geographic information system: a single coordinate system into which features of three geometric types are placed — points, lines, polygons. At first glance GTR1 is arranged the same way:

• MP10 — the coordinate system: the three-dimensional habitat;

• MP11 — points: single objects seated on MP10;

• MP12 — lines: b-vectors and splices stitching subject to object;

• MP13 — event schemas assembled from splices.

But the analogy with GIS breaks off at an important place, and the break is essential. A GIS is one large map: a single coordinate sheet on which all points, lines, and polygons lie side by side and are seen at once. GTR1 is not arranged that way. Here there is no large sheet. MP11 is a multimap of many small elementary maps of fixed size, and MP12 and MP13 live not on top of a single base but inside these small maps. MP13 is not a polygon layer on a common map but a set of separate event schemas (each schema a BLOM, a stitching of “subject–action–object” triplets), and each lies within its own elementary map. GTR1 is not one GIS but a scattering of tiny GISes connected by gateways.

A natural question arises: why such an arrangement at all, if it is so limited — and does it exist in nature, or is it merely a speculative construction? The answer to both questions is given by a device that man built himself and that works every day.

It is cinema. A film can span a long stretch — a year, a decade, a hero’s whole life — but it is shot in scenes, and each scene is limited by the size of the film set. A studio the size of a life cannot be built, and no one tries: instead, limited scenes are shot and edited into a whole. The viewer experiences the hero’s continuous fate without once seeing “the whole map” of his life — because no such map exists. There is only an edited sequence of local scenes.

The correspondence is exact. The film set is an elementary MP11 map. A scene shot on it is an OPRN (MP12): a subject, an action, objects, their movements, all within one field. The shot footage forms a branching graph of episodes — this is MP13. And the finished film is a chosen line through this graph: the editor takes from all the footage a few scenes — as many as fit onto the reel and into two hours — and arranges them into a coherent story that a person can recognize as a whole. Editing does not go beyond the visible scenes; it lays a route through already existing material. This is an operation within MP13, and it belongs entirely to GTR1.

Cinema proves the feasibility of what would otherwise seem a paradox: the experience of an extended life, assembled from admittedly local, size-limited pieces. No large map is needed for this — and in nature such a behavioral multimap is, in all likelihood, the working solution and not a defect. But here too the limitation is visible. The editor is limited by the reel: he will not make a film sixty-five years long — the material will not fit. So too GTR1 is limited by the size of its maps. To encompass something truly large, a different device is needed — and that is precisely what GTR1 does not have.

5. 3. The Limitation of GTR1

This problem has a famous experimental witness. In the 1910s Wolfgang Köhler studied problem-solving in chimpanzees. In a classic experiment he placed a banana on the ground outside the cage, out of reach, and gave the participant — Sultan — two short bamboo sticks, neither of which reached the food; one was hollow at the end, so that the other could be inserted into it to make a long stick. After unsuccessful attempts, Sultan joined the sticks and drew the banana in. Köhler attached particular importance to the fact that the solution arose not by trial and error but suddenly — when all the elements of the task came to be encompassed in a single glance. Where the tool and the goal could not be held in one field of perception, the solution became sharply more difficult.

In Gativus terms this is precisely the limitation of GTR1, outlined with utmost clarity. An OPRN is built only from objects of one elementary MP11 map — from what is in one field right now. As long as both tools and the goal are in this field, GTR1 manages: it assembles them into one action. But what is not in the field does not exist for GTR1 — it is incapable of bringing the absent into the scene. One cannot shoot the missing frame: only what is on the set right now gets filmed.

It is precisely this wall that the next level breaks through. For an absent object to enter the operative window, it must be represented — to place into the scene what is physically not in it. The capacity to bring the absent into operation is precisely the acquisition of GTR2. And it is achieved through the symbol.

5. 4. The GTR2 Convolution: The Birth of the Symbol

The GTR2 convolution takes a fragment of a GTR1 map and convolves it into a single invariant — a symbol. Crucially, the convolution is indifferent to the type of the entity being convolved. One can convolve a point — a single object; one can convolve a line — a separate splice; one can convolve a whole event schema. And all three yield a symbol of the same dimension and the same status.

Convolving the point-object “stone” yields one symbol. Convolving the splice “throw” yields one symbol. Convolving a long chain, “the hunt” — with tracking, preparation, execution, and the division of the prey — also yields one symbol. In the volume of what is convolved these cases are incomparable, but in the form of the result they are the same: the output is always one symbol. To convolve complex structures like a chain on the scale of “the hunt,” the organism must be sufficiently developed — but the operation is in all cases one and the same.

The result of the convolution forms the space MP21. The MP21 symbol is an invariant detached from the scene and from the carrier — the next step of the same principle by which, in GTR1, the object was an invariant detached from a particular projection. There the convolution detached the thing from the angle of view; here the convolution detaches the meaning from the whole scene that gave rise to it.

5. 5. Why GTR2 Arose: One Property, Three Perspectives

It is easy to take the symbol for an instrument of some single task. But GTR2 did not arise for the sake of a particular goal. One new possibility arose — to detach the convolution from its carrier and context — and the known applications are merely different perspectives on it. The three most conspicuous belong to three different observers.

The first view, the spatial one. What is convolved can be operated on outside the original map. Symbols convolved from different, physically unstitchable MP11 maps come to lie side by side in a single symbolic space — and between them a connection can be built that was impossible in GTR1. Thus a route from city to city appears: a gigantic plan not bound to the size of an elementary map.

The second view, the linguistic one. A symbol detached from its carrier can be transmitted to another organism — and that one will unfold it in itself. On this rests language: I utter “hunt,” and in your head unfolds the scene I convolved into the word. Communication is the transmission of a convolution between carriers.

The third view, the mathematical one. A symbol can be manipulated by formal rules without unfolding it back. One can add, substitute, derive, remaining at the level of the signs themselves. On this rest logic and mathematics: operating on the convolved without recourse to the convolved.

The linguist, the mathematician, and the one who builds large maps each see “their own” reason for the appearance of the symbol — and each sees it one-sidedly. The genuine acquisition is one: the autonomy of the convolution from its carrier. Large maps, speech, and reckoning are three ways of making use of it.

5. 6. MP21 — The Single Map

The spatial perspective has a structural basis worth singling out. MP11 was a multimap — a multitude of small maps of fixed size. MP21 is arranged differently: it is one large map.

In this lies the whole breakthrough. Since the symbol is convolved from any elementary MP11 map and detached from it, symbols of different origin coexist on a single MP21. What in GTR1 lived on unconnected maps and could not be joined, in GTR2 lies side by side. The limit on length is removed not because the maps grew larger but because the symbol ceased to be bound to a map at all. The route “from city to city” is short in symbolic space, though immense in spatial space.

For this single map, however, a high price has been paid. The transition from a multimap to a single symbolic space is not a smooth increase but a technological leap: it required a separate hardware platform — the neocortex. The leap is so costly that nature accomplished it in almost no one. In Gativus terms, even in the most developed animals — elephants, dolphins — the presence of a full-fledged symbolic MP21 map remains in question. The GTR1 multimap is widespread; the single GTR2 map is the rarest of acquisitions, not just another step available to all.

5. 7. The Reverse Pass: Imagination

If the symbol is torn away from space, a question arises: is the path back preserved? The answer is yes, and this is the most important property of the GTR2 convolution: it does not lose the data of the lower level. The symbol carries within itself everything needed for the reverse unfolding, and the connection between the levels is bilateral.

It is essential that the reverse pass is typed by the origin of the symbol. The convolution remembers what the symbol was convolved from, and the deconvolution returns it to the same place. A symbol convolved from an object (MP11) unfolds into an MP11 object; one convolved from a splice (MP12) — into an MP12 splice; one convolved from an event schema (MP13) — into an MP13 schema. The reverse pass is not a single channel “down onto the coordinate base” but a family of channels MP21 → MP11/MP12/MP13, isomorphic to the three channels of the convolution. The operational network OP21 carries the capacity for restoration for each type.

Hence an important consequence that dispels an apparent puzzle. An unfolded symbol need not be separately “drawn” onto MP10. It suffices to return it to MP11 — and the connection of MP11 with the coordinate base MP10 already exists within GTR1 and is hardwired: to be an MP11 object means to sit on MP10 coordinates. Therefore the restored object automatically finds itself on the spatial base and automatically becomes available to MP12 and MP13 — for computing routes and building events. Deconvolution does not render a picture; it inserts a full-fledged feature into the living GTR1 system, and all its machinery picks up this feature for free.

The reverse pass is imagination — the capacity to summon the image of the absent by unfolding a symbol into a spatial scene. Hegel called this capacity the power of imagination, Einbildungskraft.

In the Gativus architecture, imagination is not an additional faculty but a direct consequence of the reversibility of a convolution that discarded nothing in convolving. From GTR2 one can influence the composition of the objects of the lower map: a symbol is able to unfold into a scene that is not now before the eyes. It is exactly thus that Sultan’s wall is broken through — the absent object is brought into the operative window by deconvolving the symbol back into an MP11 object.

This accords with the forward and reverse passes introduced for GTR0 in the second chapter. There unfolding unfolded the description into the organism, and convolution convolved the organism into the description. Here is the same principle of reversibility, raised to the level of the symbol: the convolution of a scene into a sign and the unfolding of a sign into a scene. Life at this level too is a continuous forward and reverse pass.

5. 8. The Method of Loci: A Proof of Reversibility

Reversibility has a direct empirical confirmation — the ancient mnemonic technique, the method of loci, also known as the memory palace. The items to be remembered are turned into vivid images and mentally placed along a well-known spatial route; retrieval proceeds as a “walk” along it, picking up from each point the image placed there.

In Gativus terms this is a deliberate activation of the reverse pass. Symbols, normally torn away from space, are forcibly unfolded into image-objects (MP11) and mounted onto the nodes of a familiar route — and the route is an MP13 event schema lying on the coordinate base MP10. Thus the powerful and evolutionarily ancient spatial memory is artificially engaged. The very effectiveness of the technique proves that the data for unfolding are physically present in the symbol and available. Otherwise there would be nothing to unfold and nowhere to place the symbols.

Confirmation comes from neurobiology as well. Memorizing by the method of loci activates the hippocampus and the entorhinal cortex — the very system of spatial memory that serves the terrain map.

One and the same apparatus serves both the spatial anchoring of objects in GTR1 and the reverse projection of symbols in GTR2 — two modes of one organ. The detachment of the symbol from space is the acquisition of GTR2; the capacity to return it to its place is preserved as a legacy of the lower level.

5. 9. MP22 — The Statement

The convolution gave the symbol — a point of the new space. The next transition, the GTR2 splice, connects symbols to one another. In form this is the same stitching as in MP12: subject and object converge around a binding. But the nature of the binding is different. In MP12 the binding was a b-vector — a spatio-temporal displacement. In MP22 the binding is a logical expression relating the symbols of subject and object.

The result is a statement. Not “the subject physically moves the object in space,” but “the subject logically relates to the object through a predicate.” Where MP12 built movement, MP22 builds judgment. This is a step from action to an assertion about action — from what is done to what can be said and thought apart from immediate execution.

5. 10. MP23 — The Narrative

The last transition of GTR2, the chain, builds a sequence over ready-made statements — as in GTR1 the chain built an event schema over splices. The objects of this map are the MP22 statements themselves plus the technique of linking them. The result is a narrative: a coherent schema of statements unfolding as a story, an argument, or a plan.

Here is revealed the same thing as time in GTR1, but at a new level. In GTR1 the order of events generated physical time — the “before” and “after” of observed movements. In GTR2 the order of statements generates narrated time — a sequence detached from immediate observation. One can tell of a past that no longer exists, and of a future that does not yet exist.

5. 11. GTR2 as Foundation

GTR2 repeats the structure of GTR1, as GTR1 repeated the structure of GTR0: four spaces and three transitions, a single forward and reverse pass, elementary maps and a unified one. But the material has again risen a step. GTR0 worked with substance, GTR1 — with space and objects, GTR2 — with symbols. The same transformation, detached one more step from the immediately given.

The acquisition of GTR2 is freedom from the present. The symbol substitutes for the absent, connects what is unconnectable in space, is transmitted from one carrier to another, and obeys formal rules. On this rest language, the memory of what never was, reasoning, and planning.

But here too a boundary remains. The symbols and narratives of GTR2 describe the world — but prefer nothing in it. A story can be built, a judgment uttered, a plan composed — and yet there remains the question of why, out of all possible narratives, this one is chosen. What the subject wants; what is valuable to it; where its will is directed. This missing support is the subject of the next level.