Ideas and Machinic Platonism

March 1, 2012 § Leave a comment

Once the cat had the idea to go on a journey…
You don’t believe me? Did not your cat have the same idea? Or is your doubt about my believe that cats can have ideas?

So, look at this individual here, who is climbing along the facade, outside the window…

(sorry for the spoken comment being available only in German language in the clip, but I am quite sure you got the point anyway…)

Cats definitely know about the height of their own position, and this one is climbing from flat to flat … outside, on the facade of the building, and in the 6th floor. Crazy, or cool, respectively, in its full meaning, this cat here, since it looks like she has been having a plan… (of course, anyone ever lived together with a cat knows very well that they can have plans… proudness like this one, and also remorse…)

Yet, how would your doubts look like, if I would say “Once the machine got the idea…” ? Probably you would stop talking or listening to me, turning away from this strange guy. Anyway, just that is the claim here, and hence I hope you keep reading.

We already discussed elsewhere1 that it is quite easy to derive a bunch of hypotheses about empirical data. Yet, deriving regularities or rules from empirical data does not make up an idea, or a concept. At most they could serve as kind of qualified precursors for the latter. Once the subject of interest has been identified, deriving hypotheses about it is almost something mechanical. Ideas and concepts as well are much more related to the invention of a problematics, as Deleuze has been working out again and again, without being that invention or problematics. To overlook (or to negate?) that difference between the problematic and the question is one of the main failures of logical empiricism, and probably even of today’s science.

The Topic

But what is it then, that would make up an idea, or a concept? Douglas Hofstadter once wrote [1] that we are lacking a concept of concept. Since then, a discipline emerged that calls itself “formal concept analysis”. So, actually some people indeed do think that concepts could be analyzed formally. We will see that the issues about the relation between concepts and form are quite important. We already met some aspects of that relationship in the chapters about formalization and creativity. And we definitely think that formalization expels anything interesting from that what probably had been a concept before that formalization. Of course, formalization is an important part in thinking, yet it is importance is restricted before it there are concepts or after we have reduced them into a fixed set of finite rules.

Ideas

Ideas are almost annoying, I mean, as a philosophical concept, and they have been so since the first clear expressions of philosophy. From the very beginning there was a quarrel not only about “where they come from,” but also about their role with respect to knowledge, today expressed as . Very early on in philosophy two seemingly juxtaposed positions emerged, represented by the philosophical approaches of Platon and Aristotle. The former claimed that ideas are before perception, while for the latter ideas clearly have been assigned the status of something derived, secondary. Yet, recent research emphasized the possibility that the contrast between them is not as strong as it has been proposed for more than 2000 years. There is an eminent empiric pillar in Platon’s philosophical building [2].

We certainly will not delve into this discussion here, it simply would take too much space and efforts, and not to the least there are enough sources in the web displaying the traditional positions in great detail. Throughout history since Aristotle, many and rather divergent flavors of idealism emerged. Whatever the exact distinctive claim of any of those positions is, they all share the belief in the dominance into some top-down principle as essential part of the conditions for the possibility of knowledge, or more general the episteme. Some philosophers like Hegel or Frege, just as others nowadays being perceived as members of German Idealism took rather radical positions. Frege’s hyper-platonism, probably the most extreme idealistic position (but not exceeding Hegel’s “great spirit” that far) indeed claimed that something like a triangle exists, and quite literally so, albeit in a non-substantial manner, completely independent from any, e.g. human, thought.

Let us fix this main property of the claim of a top-down principle as characteristic for any flavor of idealism. The decisive question then is how could we think the becoming of ideas.It is clearly one of the weaknesses of idealistic positions that they induce a salient vulnerability regarding the issue of justification. As a philosophical structure, idealism mixes content with value in the structural domain, consequently and quite directly leading to a certain kind of blind spot: political power is justified by the right idea. The factual consequences have been disastrous throughout history.

So, there are several alternatives to think about this becoming. But even before we consider any alternative, it should be clear that something like “becoming” and “idealism” is barely compatible. Maybe, a very soft idealism, one that already turned into pragmatism, much in the vein of Charles S. Peirce, could allow to think process and ideas together. Hegel’s position, or as well Schelling’s, Fichte’s, Marx’s or Frege’s definitely exclude any such rapprochement or convergence.

The becoming of ideas could not thought as something that is flowing down from even greater transcendental heights. Of course, anybody may choose to invoke some kind of divinity here, but obviously that does not help much. A solution according to Hegel’s great spirit, history itself, is not helpful either, even as this concept implied that there is something in and about the community that is indispensable when it comes to thinking. Much later, Wittgenstein took a related route and thereby initiated the momentum towards the linguistic turn. Yet, Hegel’s history is not useful to get clear about the becoming of ideas regarding the involved mechanism. And without such mechanisms anything like machine-based episteme, or cats having ideas, is accepted as being impossible apriori.

One such mechanism is interpretation. For us the principle of the primacy of interpretation is definitely indisputable. This does not mean that we disregard the concept of the idea, yet, we clearly take an Aristotelian position. More á jour, we could say that we are quite fond of Deleuze’s position on relating empiric impressions, affects, and thought. There are, of course many supporters in the period of time that span between Aristotle and Deleuze who are quite influential for our position.2
Yet, somehow it culminated all in the approach that has been labelled French philosophy, and which for us comprises mainly Michel Serres, Gilles Deleuze and Michel Foucault, with some predecessors like Georges Simondon. They converged towards a position that allow to think the embedding of ideas in the world as a process, or as an ongoing event [3,4], and this embedding is based on empiric affects.

So far, so good. Yet, we only declared the kind of raft we will build to sail with. We didn’t mention anything about how to build this raft or how to sail it. Before we can start to constructively discuss the relation between machines and ideas we first have to visit the concept, both as an issue and as a concept.

Concepts

“Concept” is very special concept. First, it is not externalizable, which is why we call it a strongly singular term. Whenever one thinks “concept,” there is already something like concept. For most of the other terms in our languages, such as idea, that does not hold. Such, and regarding the structural dynamics of its usage,”concept” behave similar to “language” or “formalization.”

Additionally, however, “concept” is not self-containing term like language. One needs not only symbols, one even needs a combination of categories and structured expression, there are also Peircean signs involved, and last but not least concepts relate to models, even as models are also quite apart from it. Ideas do not relate in the same way to models as concepts do.

Let us, for instance take the concept of time. There is this abundantly cited quote by  Augustine [5], a passage where he tries to explain the status of God as the creator of time, hence the fundamental incomprehensibility of God, and even of his creations (such as time) [my emphasis]:

For what is time? Who can easily and briefly explain it? Who even in thought can comprehend it, even to the pronouncing of a word concerning it? But what in speaking do we refer to more familiarly and knowingly than time? And certainly we understand when we speak of it; we understand also when we hear it spoken of by another. What, then, is time? If no one ask of me, I know; if I wish to explain to him who asks, I know not. Yet I say with confidence, that I know that if nothing passed away, there would not be past time; and if nothing were coming, there would not be future time; and if nothing were, there would not be present time.

I certainly don’t want to speculate about “time” (or God) here, instead I would like to focus this peculiarity Augustine is talking about. Many, and probably even Augustine himself, confine this peculiarity to time (and space). I think, however, this peculiarity applies to any concept.

By means of this example we can quite clearly experience the difference between ideas and concepts. Ideas are some kind of models—we will return that in the next section—, while concepts are the both the condition for models and being conditioned by models. The concept of time provides the condition for calendars, which in turn can be conceived as a possible condition for the operationalization of expectability.

“Concepts” as well as “models” do not exist as “pure” forms. We elicit a strange and eminently counter-intuitive force when trying to “think” pure concept or models. The stronger we try, the more we imply their “opposite”, which in case of concepts presumably is the embedding potentiality of mechanisms, and in case of models we could say it is simply belief. We will discuss the issue of these relation in much more detail in the chapter about the choreosteme (forthcoming). Actually, we think that it is appropriate to conceive of terms like “concept” and “model” as choreostemic singular terms, or short choreostemic singularities.

Even from an ontological perspective we could not claim that there “is” such a thing like a “concept”. Well, you may already know that we refute any ontological approach anyway. Yet, in case of choreostemic singular terms like “concept” we can’t simply resort to our beloved language game. With respect to language, the choreosteme takes the role of an apriori, something like the the sum of all conditions.

Since we would need a full discussion of the concept of the choreosteme we can’t fully discuss the concept of “concept” here.  Yet, as kind of a summary we may propose that the important point about concepts is that it is nothing that could exist. It does not exist as matter, as information, as substance nor as form.

The language game of “concept” simply points into the direction of that non-existence. Concepts are not a “thing” that we could analyze, and also nothing that we could relate to by means of an identifiable relation (as e.g. in a graph). Concepts are best taken as gradient field in a choreostemic space, yet, one exhibiting a quite unusual structure and topology. So far, we identified two (of a total of four) singularities that together spawn the choreostemic space. We also could say that the language game of “concept” is used to indicate a certain form of a drift in the choreostemic space. (Later we also will discuss the topology of that space, among many other issues.)

For our concerns here in this chapter, the machine-based episteme, we can conclude that it would be a misguided approach to try to implement concepts (or their formal analysis). The issue of the conditions for the ability to move around in the choreostemic space we have to postpone. In other words, we have confined our task, or at least, we found a suitable entry  point for our task, the investigation of the relation between machines and ideas.

Machines and Ideas

When talking about machines and ideas we are, here and for the time being, not interested in the usage of machines to support “having” ideas. We are not interested in such tooling for now. The question is about the mechanism inside the machine that would lead to the emergence of ideas.

Think about the idea of a triangle. Certainly, triangles as we imagine them do not belong to the material world. Any possible factual representation is imperfect, as compared with the idea. Yet, without the idea (of the triangle) we wouldn’t be able to proceed, as, for instance, towards land survey. As already said, ideas serve as models, they do not involve formalization, they often live as formalization (though not always a mathematical one) in the sense of an idealized model, in other words they serve as ladder spokes for actions. Concepts, if we in contrast them to ideas, that is, if we try to distinguish them, never could be formalized, they remain inaccessible as condition. Nothing else could be expected  from a transcendental singularity.

Back to our triangle. Despite we can’t represent them perfectly, seeing a lot of imperfect triangles gives rise to the idea of the triangle. Rephrased in this way, we may recognize that the first half of the task is to look for a process that would provide an idealization (of a model), starting from empirical impressions. The second half of the task is to get the idea working as a kind of template, yet not as a template. Such an abstract pattern is detached from any direct empirical relation, despite the fact that once we started with with empiric data.

Table 1: The two tasks in realizing “machinic idealism”

Task 1: process of idealization that starts with an intensional description
Task 2: applying the idealization for first-of-a-kind-encounters

Here we should note that culture is almost defined by the fact that it provides such ideas before any individual person’s possibility to collect enough experience for deriving them on her own.

In order to approach these tasks, we need first model systems that exhibit the desired behavior, but which also are simple enough to comprehend. Let us first deal with the first half of the task.

Task 1: The Process of Idealization

We already mentioned that we need to start from empirical impressions. These can be provided by the Self-organizing Map (SOM), as it is able to abstract from the list of observations (the extensions), thereby building an intensional representation of the data. In other words, the SOM is able to create “representative” classes. Of course, these representations are dependent on some parameters, but that’s not the important point here.

Once we have those intensions available, we may ask how to proceed in order to arrive at something that we could call an idea. Our proposal for an appropriate model system consists from the following parts:

  • (1) A small set (n=4) of profiles, which consist of 3 properties; the form of the profiles is set apriori such that they overlap partially;
  • (2) a small SOM, here with 12×12=144 nodes; the SOM needs to be trainable and also should provide classification service, i.e. acting as a model
  • (3) a simple Monte-Carlo-simulation device, that is able to create randomly varied profiles that deviate from the original ones without departing too much;
  • (4) A measurement process that is recording the (simulated) data flow

The profiles are defined as shown in the following table (V denotes variables, C denotes categories, or classes):

V1 V2 V3
C1 0.1 0.4 0.6
C2 0.8 0.4 0.6
C3 0.3 0.1 0.4
C4 0.2 0.2 0.8

From these parts we then build a cyclic process, which comprises the following steps.

  • (0) Organize some empirical measurement for training the SOM; in our model system, however, we use the original profiles and create an artificial body of “original” data, in order to be able to detect the relevant phenomenon (we have perfect knowledge about the measurement);
  • (1) Train the SOM;
  • (2) Check the intensional descriptions for their implied risk (should be minimal, i.e. beyond some threshold) and extract them as profiles;
  • (3) Use these profiles to create a bunch of simulated (artificial) data;
  • (4) Take the profile definitions and simulate enough records to train the SOM,

Thus, we have two counteracting forces, (1) a dispersion due to the randomizing simulation, and (2) the focusing of the SOM due to the filtering along the separability, in our case operationalized as risk (1/ppv=positive predictive value) per node. Note that the SOM process is not a directly re-entrant process as for instance Elman networks [6,7,8].3

This process leads not only to a focusing contrast-enhancement but also to (a limited version) of inventing new intensional descriptions that never have been present in the empiric measurement, at least not salient enough to show up as an intension.

The following figure 1a-1i shows 9 snapshots from the evolution of such a system, it starts top-left of the portfolio, then proceeds row-wise from left to right down to the bottom-right item. Each of the 9 items displays a SOM, where the RGB-color corresponds to the three variables V1, V2, V3. A particular color thus represents a particular profile on the level of the intension. Remember, that the intensions are built from the field-wise average across all the extensions collected by a particular node.

Well, let us now contemplate a bit about the sequence of these panels, which represents the evolution of the system. The first point is that there is no particular locational stability. Of course, not, I am tempted to say, since a SOM is not an image that represents as image. A SOM contains intensions and abstractions, the only issue that counts is its predictive power.

Now, comparing the colors between the first and the second, we see that the green (top-right in 1a, middle-left in 1b) and the brownish (top-left in 1a, middle-right in 1b) appear much more clear in 1b as compared to 1a. In 1a, the green obviously was “contaminated” by blue, and actually by all other values as well, leading to its brightness. This tendency prevails. In 1c and 1d yellowish colors are separated, etc.

Figure 1a thru 1i: A simple SOM in a re-entrant Markov process develops idealization. Time index proceeds from top-left to bottom-right.

The point now is that the intensions contained in the last SOM (1i, bottom-right of the portfolio) have not been recognizable in the beginning, in some important respect they have not been present. Our SOM steadily drifted away from its empirical roots. That’s not a big surprise, indeed, for we used a randomization process. The nice thing is something different: the intensions get “purified”, changing thereby their status from “intensions” to “ideas”.

Now imagine that the variables V1..Vn represent properties of geometric primitives. Our sensory apparatus is able to perceive and to encode them: horizontal lines, vertical lines, crossings, etc. In empiric data our visual apparatus may find any combination of those properties, especially in case of a (platonic) school (say: academia) where the pupils and the teachers draw triangles over triangles into the wax tablets, or into the sand of the pathways in the garden…

By now, the message should be quite clear: there is nothing special about ideas. In abstract terms, what is needed is

  • (1) a SOM-like structure;
  • (2) a self-directed simulation process;
  • (3) re-entrant modeling

Notice that we need not to specify a target variable. The associative process itself is just sufficient.

Given this model it should not surprise anymore why the first philosophers came up with idealism. It is almost built into the nature of the brain. We may summarize our achievements in the following characterization;

Ideas can be conceived as idealizations of intensional descriptions.

It is of course important to be aware of the status of such a “definition”. First, we tried to separate concepts and ideas. Most of the literature about ideas conflate them.Yet, as long as they are conflated, everything and any reasoning about mental affairs, cognition, thinking and knowledge necessarily remains inappropriate. For instance, the infamous discourse about universals and qualia seriously suffered from that conflation, or more precisely, they only arose due to that mess.

Second, our lemma is just an operationalization, despite the fact that we are quite convinced about its reasonability. Yet, there might be different ones.

Our proposal has important benefits though, as it matches a lot of the aspects commonly associated the the term “idea.” In my opinion, what is especially striking about the proposed model is the observation that idealization implicitly also led to the “invention” of “intensions” that were not present in the empiric data. Who would have been expecting that idealization is implicitly inventive?

Finally, two small notes should be added concerning the type of data and the relationship between the “idea” as a continuously intermediate result of the re-entrant SOM process. One should be aware that the “normal” input to natural associative systems are time series. Our brain is dealing with a manifold of series of events, which is mapped onto the internal processes, that is, onto another time-based structure. Prima facie Our brain is not dealing with tables. Yet, (virtual) tabular structures are implied by the process of propertization, which is an inevitable component of any kind of modeling. It is well-known that is is time-series data and their modeling that give rise to the impression of causality. In the light of ideas qua re-entrant associativity, we now can easily understand the transition from networks of potential causal influences to the claim of “causality” as some kind of a pure concept. Despite the idea of causality (in the Newtonian sense) played an important role in the history of science, it is just that: a naive idealization.

The other note concerns the source of the data.  If we consider re-entrant informational structures that are arranged across large “distances”, possibly with several intermediate transformative complexes (for which there are hints from neurobiology) we may understand that for a particular SOM (or SOM-like structure) the type of the source is completely opaque. To put it short, it does not matter for our proposed mechanism whether the data are sourced as empiric data from the external world,or as some kind of simulated, surrogated re-entrant data from within the system itself. In such wide-area, informationally re-entrant probabilistic networks we may expect kind of a runaway idealization. The question then is about the minimal size necessary for eliciting that effect. A nice corollary of this result is the insight that logistic networks, such like the internet or the telephone wiring cable NEVER will start to think on itself, as some still expect. Yet, since there a lot of brains as intermediate transforming entities embedded in this deterministic cablework, we indeed may expect that the whole assembly is much more than could be achieved by a small group of humans living, say around 1983. But that is not really a surprise.

Task 2: Ideas, applied

Ideas are an extremely important structural phenomenon, because they allow to recognize things and to deal with tasks that we never have seen before. We may act adaptively before having encountered a situation that would directly resemble—as equivalence class—any intensional description available so far.

Actually, it is not just one idea, it is a “system” of ideas that is needed for that. Some years ago, Douglas Hofstadter and his group3 devised a model system suitable for demonstrating exactly this: the application of ideas. They called the project (and the model system) Copycat.

We won’t discuss Copycat and analogy-making rules by top-down ideas here (we already introduced it elsewhere). We just want to note that the central “platonic” concept in Copycat is a dynamic relational system of symmetry relations. Such symmetry relations are for instance “before”, “after”, or “builds a group”, “is a triple”, etc. Such kind of relations represent different levels of abstractions, but that’s not important. Much more important is the fact that the relations between these symmetry relations are dynamic and will adapt according to the situation at hand.

I think that these symmetry relations as conceived by the Fargonauts are on the same level as our ideas. The transition from ideas to symmetries is just a grammatological move.

The case of Biological Neural Systems

Re-entrance seems to be an important property of natural neural networks. Very early on in the liaison of neurobiology and computer science, starting with Hebb and Hopfield in the beginning of the 1940ies, recurrent networks have been attractive for researchers. If we take a look to drawings like the following, created (!) by Ramon y Cajal [10] in the beginning of the 20th century.

Figure 2a-2c: Drawings by Ramon y Cajal, the Spain neurobiologist. See also:  History of Neuroscience. a: from a Sparrow’s brain, b: motor brain in human brain, c: Hypothalamus in human brain

Yet, Hebb, Hopfield and Elman got trapped by the (necessary) idealization of Cajal’s drawings. Cajal’s interest was to establish and to proof the “neuron hypothesis”, i.e. that brains work on the basis of neurons. From Cajal’s drawings to the claim that biological neuronal structures could be represented by cybernetic systems or finite state machines is, honestly, a breakneck, or, likewise, ideology.

Figure 3: Structure of an Elman Network; obviously, Elman was seriously affected by idealization (click for higher resolution).

Thus, we propose to distinguish between re-entrant and recurrent networks. While the latter are directly wired onto themselves in a deterministic manner, that is the self-reference is modeled on the morphological level, the former are modeled on the  informational level. Since it is simply impossible for cybernetic structure to reflect neuromorphological plasticity and change, the informational approach is much more appropriate for modeling large assemblies of individual “neuronal” items (cf. [11]).

Nevertheless, the principle of re-entrance remains a very important one. It is a structure that is known to lead to contrast enhancement and to second-order memory effects. It is also a cornerstone in the theory (theories) proposed by Gerald Edelman, who probably is much less affected by cybernetics (e.g. [12]) than the authors cited above. Edelman always conceived the brain-mind as something like an abstract informational population; he even was the first adopting evolutionary selection processes (Darwinian and others) to describe the dynamics in the brain-mind.

Conclusion: Machines and Choreostemic Drift

Out point of departure was to distinguish between ideas and concepts. Their difference becomes visible if we compare them, for instance, with regard to their relation to (abstract) models. It turns out that ideas can be conceived as a more or less stable immaterial entity (though not  “state”) of self-referential processes involving self-organizing maps and the simulated surrogates of intensional descriptions. Concepts on the other hand are described as a transcendental vector in choreostemic processes. Consequently, we may propose only for ideas that we can implement their conditions and mechanisms, while concepts can’t be implemented. It is beyond the expressibility of any technique to talk about the conditions for their actualization. Hence, the issue of “concept” has been postponed to a forthcoming chapter.

Ideas can be conceived as the effect of putting a SOM into a reentrant context, through which the SOM develops a system of categories beyond simple intensions. These categories are not justified by empirical references any more, at least not in the strong sense. Hence, ideas can be also characterized as being clearly distinct from models or schemata. Both, models and schemata involve classification, which—due to the dissolved bonds to empiric data—can not be regarded a sufficient component for ideas. We would like to suggest the intended mechanism as the candidate principle for the development ideas. We think that the simulated data in the re-entrant SOM process should be distinguished from data in contexts that are characterized by measurement of “external” objects, albeit their digestion by the SOM mechanism itself remains the same.

From what has been said it is also clear that the capability of deriving ideas alone is still quite close to the material arrangements of a body, whether thought as biological wetware or as software. Therefore, we still didn’t reach a state where we can talk about epistemic affairs. What we need is the possibility of expressing the abstract conditions of the episteme.

Of course, what we have compiled here exceeds by far any other approach, and additionally we think that it could serve as as a natural complement to the work of Douglas Hofstadter. In his work, Hofstadter had to implement the platonic heavens of his machine manually, and even for the small domain he’d chosen it has been a tedious work. Here we proposed the possibility for a seamless transition from the world of associative mechanisms like the SOM to the world of platonic Copy-Cats, and “seamless” here refers to “implementable”.

Yet, what is really interesting is the form of choreostemic movement or drift, resulting from a particular configuration of the dynamics in systems of ideas. But this is another story, perhaps related to Felix Guattari’s principle of the “machinic”, and it definitely can’t be implemented any more.

.
Notes

1. we did so in the recent chapter about data and their transformation, but also see the section “Overall Organization” in Technical Aspects of Modeling.

2. You really should be aware that this trace we try to put forward here does not come close to even a coarse outline of all of the relevant issues.

3. they called themselves the “Fargonauts”, from FARG being the acronym for “Fluid Analogy Research Group”.

4. Elman networks are an attempt to simulate neuronal networks on the level of neurons. Such approaches we rate as fundamentally misguided, deeply inspired by cybernetics [9], because they consider noise as disturbance. Actually, they are equivalent to finite state machines. It is somewhat ridiculous to consider a finite state machine as model for learning “networks”. SOM, in contrast, especially if used in architectures like ours, are fundamentally probabilistic structures that could be regarded as “feeding on noise.” Elman networks, and their predecessor, the Hopfield network are not quite useful, due to problems in scalability and, more important, also in stability.

  • [1] Douglas Hofstadter, Douglas R. Hofstadter, Fluid Concepts And Creative Analogies: Computer Models Of The Fundamental Mechanisms Of Thought. Basic Books, New York 1996.  p.365
  • [2] Gernot Böhme, “Platon der Empiriker.” in: Gernot Böhme, Dieter Mersch, Gregor Schiemann (eds.), Platon im nachmetaphysischen Zeitalter. Wissenschaftliche Buchgesellschaft, Darmstadt 2006.
  • [3] Marc Rölli (ed.), Ereignis auf Französisch: Von Bergson bis Deleuze. Fin, Frankfurt 2004.
  • [4] Gilles Deleuze, Difference and Repetition. 1967
  • [5] Augustine, Confessions, Book 11 CHAP. XIV.
  • [6] Mandic, D. & Chambers, J. (2001). Recurrent Neural Networks for Prediction: Learning Algorithms, Architectures and Stability. Wiley.
  • [7] J.L. Elman, (1990). Finding Structure in Time. Cognitive Science 14 (2): 179–211.
  • [8] Raul Rojas, Neural Networks: A Systematic Introduction. Springer, Berlin 1996. (@google books)
  • [9] Holk Cruse, Neural Networks As Cybernetic Systems: Science Briefings, 3rd edition. Thieme, Stuttgart 2007.
  • [10] Santiago R.y Cajal, Texture of the Nervous System of Man and the Vertebrates: Volume I: 1, Springer, Wien 1999, edited and translated by Pedro Pasik & Tauba Pasik. see google books
  • [11] Florence Levy, Peter R. Krebs (2006), Cortical-Subcortical Re-Entrant Circuits and Recurrent Behaviour. Aust N Z J Psychiatry September 2006 vol. 40 no. 9 752-758.
    doi: 10.1080/j.1440-1614.2006.01879
  • [12] Gerald Edelman: “From Brain Dynamics to Consciousness: A Prelude to the Future of Brain-Based Devices“, Video, IBM Lecture on Cognitive Computing, June 2006.

۞

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