Three principles to understand cognition

Three principles to understand cognition Devis Pantano DRAFT July 2014 Abstract I think that the main reason why we do not understand the general principles of how cognition works, is not the excessive complexity of cognitive phenomena, but the lack of the conceptual and methodological tools to properly address the problem. In this paper I show three principles that could be proposed as foundation of cognitive processes. I also briefly show some points on the tools that I think it is necessary to develop for this purpose. Are there laws or general principles for cognition? There is a widespread idea that cognition is something inherently complex and for this reason it is not possible to identify a set of fundamental laws, or principles, from which to infer how the cognition itself works. I am able to show that these principles exist and are of such a nature and importance to quickly drive the scientific research towards the understanding of the general functioning of cognitive activity. I also believe that the reason why until now we have missed these principles is due to the lack of appropriate methodological and conceptual tools. To draw a comparison, it is as if you were trying to codify the laws of physics without the mathematical tools needed. Once you have identified and developed appropriate tools, the problem of understanding cognition is greatly simplified. To understand how cognition works, it is essentially necessary to find a way to define with precision, and to manage in an appropriate manner, some important concepts for which, however, usually we have only an intuitive understanding. The highlights of the formulation that I propose are mostly derived by the attempt to clarify the concepts of structure and rule, and to find a way to treat, not in naïve way, the problem of the comparison between the structures of things and phenomena, in order to be able to identify all the emergent rules that can be usefully exploited for cognition. I have developed a methodology to deal with the concept of structure [3]. This notion is used intuitively by almost everyone, but very few have tried to analyze and to specify it, except some mathematicians and philosophers. A delicate point is that it is not enough to simply find a "method" to formalize this concept, but it is important to find the right one. We need a method able to catch, in an efficient and smart way, the key mechanisms of the "phenomenon of structural 1

correspondences", namely correspondences that may exist between the structures of different entities. In my opinion, with the formalizations proposed until now, it is not possible to understand these mechanisms and, for this reason, they are likely to mislead. This is probably due to the fact that the developers of these methods have not understood correctly the role that these phenomena have in cognition. In mathematics, tools have been developed in order to apply the concept of structure to mathematical objects that are far from things of our daily lives. Some philosophers proposed formalizations with little care about their actual usability. They are often unnecessarily complex, lengthy, unsuited to capture and highlight the essential aspects. These also have the defect to divert attention from the important points. I have developed a methodology, actually relatively simple, which allows to work with sufficient accuracy, works well for the objects of our everyday perception (concrete things and phenomena), and is able to capture some essential aspects which until now were not analyzed appropriately. It is just these aspects that allow to perform some important tasks which I call structural derivation. I believe that these steps are critical because they allow to move in a natural way from the most basic representations to more abstract ones. We know that the representations which are formed close to the senses are "pictorial" (they are also called subsymbolic, or analog, by some authors). I think I can show that if the pictorial representation is of good quality, then, by means of these operations you can derive the others. These representations can be abstractions of the original one. It can be shown that these operations can be repeated several times, thus generating a hierarchical stratification, progressively more and more abstract. These steps highlight and "make explicit" some of the properties which are contained implicitly in the original representations. One can show that, in doing this, we get new objects that are still structural representations (and not of a different type). It is very likely that these operations are similar to those used by our mind to analyze and organize the information it receives from the senses. The methodology is particularly effective also for other reasons. With it, it is possible to identify and codify, with sufficient precision, some general principles from which we can derive others, more functional, that lead to the understanding of how cognition itself can work. By this I mean to say that these principles, these laws, allow to understand in detail, both in terms of micro-processes, both in terms of overall management, how it all works and how it can be artificially reproduced. I think we can identify at least three general principles which, combined to the proposed methodology, lead then to identify most of the others. These principles are: 1. What is knowable of the outside world is limited to the structures of objects and phenomena, and to the possible operations on these structures. Beyond these structures, and these operation, there is no knowable external reality. 2

2. Each rule (and each law) consists of a structural prescription of operations that can (or must) be performed, or of the results to be obtained. 3. The emergent properties consist of derivative structures that exhibit some phenomena of functional dependence with others. They are emergent rules the ones that are applied to those emergent structures. With the appropriate tools these principles can be specified in a sufficiently rigorous way. From the first principle it may be deduced that cognition is based on the possibility to build internal representations that have, at least in part, the same structure of the objects of the external reality. These representations must also exhibit behaviors that are always structurally similar to those of real phenomena. These ideas, combined with the principle explaining what the emergent structures are, imply that the information within a cognitive system is organized in a precise manner. They set two important principles ("principle of explicitation," and "principle of convergence of the checks"), which will be described later. They determine the organizational structure of "networks of functions" (in the mathematical sense) that closely resemble the neural networks in the brain. With high probability this is not a case. The second principle, combined with the other two and developed within the proposed methodology, allows to derive many fundamental aspects of rules. It shows that rules are virtually the engine of every cognitive process, from the simplest to the most complex. This principle shows: how rules can be usefully classified to understand how both single microprocesses and more global activities can be implemented; which is the meaning of the hierarchical organization of information; why it is necessary to move from pictorial to more abstract representations; why primary pictorial representations (from senses) have to be translated into others of "better quality"; how the activities of the various cognitive processes can be managed in order to reach specific goals. To these three principles I believe it is important to add another ingredient that I think is one of the "fundamental tricks of intelligence." It can well be argued that intelligence must be opportunist in a similar way as living organisms do. Intelligence has to be opportunistic in the sense that it should take advantage of all the possibilities in order to implement some useful inference processes. A fundamental phenomenon that characterizes the reality in which we live, is that together with the fundamental laws of physics and mathematics, a series of emergent regularities and rules appear. These rules appear at a level of description of reality which is higher than the basic laws. It is just a part of these emergent rules that our brain uses to make inferences, to generate forecasts and to plan actions and behaviors. Here occurs a key phenomenon that, if properly understood, allows to make a major step forward in understanding cognition. 3

I stated that there are operations that allow, starting from good "pictorial" representations, to move towards others, organized by hierarchies, which are abstractions of the first ones. Indeed it is possible to show that emergent rules are connected to these new representations. The fundamental trick of intelligence is that, moving from basic representations to their abstractions, many emergent rules appear that can be used properly. Once again I emphasize the concept that if we can figure out exactly how this phenomenon occurs, we have the key that opens the way towards the understanding of the overall logic of cognition. Below we briefly illustrate some important points on the three principles outlined above. First principle: the fundamental limits of the knowable and the concept of structure In 1902 Henri Poincaré published a treatise on epistemology: "Science and hypothesis." In this work he came to a conclusion that I consider particularly important. According to Poincare "science can only understand the relationship between things, beyond these relations there is no knowable reality!" Today, this conclusion may appear certainly interesting but, at the same, time it may seem quite harmless. It seems to inform us about the limits of science, and it seems to support the formalist approach, which has established itself as the dominant epistemology in some fundamental scientific disciplines. I can show that, by extending the meaning of this conclusion beyond the mere scope of science, and equipped with the appropriate tools, we can extract from it one of the fundamental principles at the foundation of cognition. I think that this thought contains a very important truth which does not refer just to the limits of science, but that actually comprises our fundamental limits to get the knowledge of the reality outside us. As mentioned I believe that the conclusion of Poincaré can be reformulated as follows: Of the external reality it is only possible to know the structure of things and the possible computational operations on these structures. Beyond this structure and these operations, there is no external reality knowable. I think this is one of the fundamental principles of cognition. I am not able to demonstrate its universal validity with rigorous methods, although I can well argue about it. Therefore I propose it as a conjecture. Assuming the validity of this conjecture and of the proposed methodology to describe the concept of structure, many important inferences can be drawn. It can be argued that cognition is based, almost entirely, on the exploitation of the phenomenon of structural correspondences. Representations that are carried out within a cognitive system must have part of their structure which coincide with those of the represented reality. If we study carefully how symbols works, we can understand that actually they need an artifice which it is not necessary with structural representations. Symbols 4

in fact require to build an "artificial" association between the object that serves as a symbol and what it represents. For example it is necessary that the perception of the symbol is connected to the activation of the memories associated to this entity. This artificial combination is not needed with structural representations. In fact, they have natural correspondences with what they represent. These matches consist in the structures they have in common. This observation leads us to think that structural representations are the "first natural basis" for the information contained within a cognitive system. It also encourages to think that the concept of structure is central and that to understand how cognition works it is necessary to describe it accurately. I invite the reader to think over the thought of Poincare. If we accept the idea that we can only know the relations between the objects of the external reality, from this it follows that we can only know the entities that are composed of multiple parts, otherwise we will not have "relations" that we can know. But how do we define what these "relations" are? Which tools do we have to generalize this idea? One of the most intuitive concepts that seems to grasp these ideas is precisely the one of structure. But if we want to understand the deep logic of the phenomenon, we have to find a way to move from an intuitive idea to something which can be expressed in mathematical terms. It is therefore legitimate to ask whether there is some branch of mathematics that deals with this. Unfortunately, the available formalizations can work only for abstract mathematical objects, but they are not good to treat the structures of the objects and phenomena of our everyday life. We need a formulation that is both sufficiently precise and sufficiently flexible. It must also be "natural", i.e. it should correspond to the methods used by our mind. My proposal for this methodology is presented extensively in Chapter 3 of the book [3]. In the following, for the sake of brevity, I will only illustrate some highlights. The general idea is that each static structure (or structures of the first type ) can be accurately described by specifying, with the appropriate instruments (arising, at least primarily, from mathematics and informatics), three groups of information: 1. One that identifies the set of "component parts". These parts are, for the structures, the equivalent of the elements in set theory. 2. One that specifies and describes the "internal properties" of the component parts, and then allows them to be distinguished from each other, from the "internal point of view." 3. One that specifies and describes the "external relations" between the parts. It can be shown that these relationships are what allows us to distinguish the component parts from each other from the "external point of view". For example, in a set of points, which are entities, by definition, with no internal structure and properties, only external relations allow to distinguish points from each other. The strategy of keeping separate the information on the internal properties from the external relations, is particularly important because it allows to easily define 5

some operations that I call "structural derivation", which I think are at the basis of abstraction. We can precisely define when two structures are identical with the notion of isomorphism. Two structures are isomorphic if we can build a correspondence between the respective sets of component parts, and if they match both the internal properties and the external relations. In this methodology it is fundamental, as already mentioned, that some structures can be derived from the others. This possibility imposes a natural hierarchical order between the structural representations. The main operations of derivations are those of portion, of quotient and of morphism. The operations of portion are very simple, consisting simply in considering only a portion of the starting structure. They are important because they can be used in other operations of derivation, and also because, as we shall see, there may be portions of a structure, which are "emergent entities". The operations of quotient consist of a sort of "change of scale". They are obtained by considering a new structural representation, which has, entire portions of the base structure as new component parts. It should be noted that the quotient operation is possible if the internal properties and the external relations between the parts of a structure are treated separately. The operations of morphism are obtained by inhibiting, in a structure, what makes its parts distinguishable from each other. This is achieved by "simplifying" the system of internal distinctions or the complex of external relations. An example of an operation that acts on the internal distinguishability is the one that eliminates the colors in an image. One of the characteristics of the operations of morphism is the loss of information. It can be shown that these operations are very common in our cognitive activity. For example, we perform an operation of morphism every time that we shift the focus from the observation of the individual details of an object, to its global vision: the previous details are now part of other "structural entities" that we consider as individual objects (thus performing a quotient operation). It is very likely that these operations constitute the basis for abstraction. Second principle: the definition of the concept of rule The second principle of reference consists of the definition of the concept of rule. Although the idea of rule is familiar to all, I believe that it has not been well understood, and was not even well understood its importance in cognition. Rules are fundamental: almost every cognitive process is based on the use of rules. In a sense we can say that rules are the engine of cognitive activity. Our brain constantly applies rules: with good probability, inside it, the brain processes in parallel several hundred million of rules (and perhaps many more) for each second. It is likely that an important part of the neocortex is dedicated to the implementation of this type of associative rules. Finding a precise definition, able to grasp the true, deep and universal nature of every rule, can enable us to understand how these work. If we can figure out what the rules are and how we can identify and use them, then 6

we really could understand much about the nature and the profound logic of cognition. Before formulating a definition of the concept of rule, I briefly present some other important operations that we can perform on structures. The first step consists in the observation that it is probably not possible to represent the basic operations of the computation only with structures of first type. So, although I have proposed that the activity of building representations concerns the structure of the objects, and that there is a universal methodology to describe any "static structure", I also propose the idea that there is no possibility to describe unambiguously the basic operations using only those tools. Again, I am not able to demonstrate rigorously the validity of this point and I have to propose it as a conjecture. If we admit that this is true (i.e. that there is no way to represent unambiguously, with only static structures, the basic operations of computation), how can we construct representations for this and for more complex operations? Indeed there are the algorithms, that are representations of operation sequences executed by some computational machine. There must therefore be some possibility to construct representations of sequences of operations! To this purpose, we can use two phenomena. The first is the possibility to use symbols; the other is that the basic computational operations are very few and very simple. It can be argued (Church-Turing thesis) that, by sequences of these, we can reproduce any complex operation. It is known that the basic operations of computing are few and simple. They are the ones which a computational basic machine (a generalization of Turing s one) must perform. It is easy to see how one can associate symbols to physical devices able to perform these operations. We know also that complex operations can be realized by means of sequences of the basic ones. We can then construct representations of these "sequences of basic operations," using sequences of symbols. It is basically what is commonly done when writing a computer program. We can note that the sequences of symbols, in their turn, constitute structures. These objects are in fact composed of a plurality of parts which must be in precise mutual relations. These "static structures" have something in common with the structures of complex operations that they represent. This is a very important point. We can then arrive at an extension of the concept of structure which I briefly explained in the previous section. The idea is to put together the function of symbol and the static structures. What we get is a hybrid entity that I propose to name "structures of the second type". An algorithm is an object that corresponds to this definition, therefore it is a structure of the second type. It can be shown that it is possible to represent, using hierarchies of structures of the second kind, not only the basic operations of computation, but also concrete actions and complex behaviors. Once introduced these points, it is possible to propose a general definition for the concept of the rule in the following way: 7

Each rule (each law) consists of a prescription of the structural operations that can (or must) be performed, or of the results to obtain. In a sense, the rules are constraints that must be respected. All these constraints can be described in structural terms. Many rules can be expressed in a very abstract way. For example, we can formulate rules such as: "you should not put a spoke in the wheels to people capable, honest and willing." At a first glance it may seem hard to believe that rules like this are always attributable to structural descriptions. I think we can actually show that even in these cases it is possible. To explain in detail how this is possible, it is necessary to provide many other explanations. In this introduction, I just mention that our mind uses the strategy of organizing for hierarchical stratifications both,the representations of the world and the possible solutions to problems. The high level representations, which are also the most abstract, refer to other lower-level, more concrete. Going down the hierarchy we pass, gradually, but in a comprehensive manner, from the more abstract representations to representations of real actions and basic operations. There is really a lot to say about rules; for the sake of brevity, I will only refer briefly to some of the most significant points. The concept of rule includes the one of law. For example, even the laws of physics, expressed by mathematical formulas, fall within the definition of rule. These formulas describe the sequence of computation operations that must be performed. In some cases the formulas are more abstract, for example when they are expresses with differential equations instead of explicit formulas, but still fall within the definition given by the concept of rule. Many rules, when applied, give rise to regularities. It is interesting and useful to know that every regularity always corresponds to some structural isomorphism. This point is important because it tells us what we need to identify a rule. Many regularities can be identified from the information we obtain from the senses after performing the following steps: Conversion from raw information to "good structural representations Structural analysis on these good structural representations Gradual construction of a hierarchy of abstract representations To understand how cognition works we need to explore various other aspects on what the rules are, which forms they can take, how they can be used in practice. Third principle: criterion of emergence I think that the logic of the emergent phenomena can be understood using the tools introduced so far. If we admit that we can only know the structure of the objects, this means that the emergent properties relate to the structures and/or to the possible operations on them. I did mention the fact that it is possible to define operations of structural derivation that allow us, starting from some representations, to derive new ones. 8

Then, we can explore the hypothesis that the so-called emergent properties are related precisely to this ability to "derive" new structures. We may think that the emergent properties relate, at least in part, to derived structures. It is not difficult to see that, given a starting structure, not all possible derivation operations generate entities that can be considered emergent. It does not make sense to consider randomly a portion of an object, or to arbitrarily generate a quotient structure by taking casual portions of a more basic one. We need some criterion to distinguish the real derived emergent structures from those which are not. Therefore, we can ask what makes a certain particular portion, or a particular quotient, something which can be reasonably considered as an entity. Is there a general criterion, able to actually grasp the essence of the phenomenon of emergence? I believe that this criterion exists and can be formulated as follows: A derived structure is emergent if there is at least one phenomenon functionally dependent on it. To show the validity of this criterion, let s try to argue by contradiction Suppose that there are no phenomena which are functionally dependent on a certain particular derived structure. In this case there would be nothing able to realize that that structure exists. Hence, this derivative structure cannot belong to any emergent entity because there is nothing that is sensitive to its presence! It is not difficult to see that those entities that we are accustomed to consider single things, are objects or phenomena with effects, either direct or indirect, on other things or other phenomena. They are entities that determine some variation on what is happening, or may happen, or on what we can or we cannot do. Note that if a derivative structure is actually emergent, then it is necessary that it fully participates in determining the effects that can be detected. It must participate with everything that defines it as a structure. For example, if only a portion of a structure determines the detectable effects, then it would be only this portion to be emergent, not the entire structure! Once defined the concept of derived emergent structure, we can also define the concepts of emergent rule: emergent rules are those which can apply to emergent structures. It can be shown that most of the rules used by our brain are emergent and associative. I will return to this point shortly. It can also be shown that, by the criterion of emergence, we can get what we call the principle of explicitation. It states that in order to have processes depending on the presence of a particular structural entity, it is necessary that there is at least a single specific bit (or a single variable, not necessarily binary), whose state depends on the presence of that particular entity. This bit (or this variable) is necessary to make explicit the recognition of the specific entity. The process of explicitation is fundamental to encode and implement rules that react to the presence of a specific entity. From the principle of explicitation, the principle of convergence of the checks 9

derives. It must be taken into account the fact that structural entities are complex objects, thus constituted by a plurality of parts in specific relations. To produce a single bit, whose state depends on the recognition of a particular structure, it is necessary to perform a variety of local checks for the presence of all the individual parts and all their specific relationships. Then everything must converge towards a single bit equivalent (the individual tests must converge in a logical operation of AND, or its equivalent). I invite the reader to meditate on this point. The recognition and the convergence towards a single bit (or towards single variable) is really crucial. It is necessary to proceed in this manner so that a cognitive system is able to verify that the specific structural entity, which is a complex object, is present! Conclusion I believe there are general principles able to explain cognitive phenomena, their overall logic, their deeper meaning. I also think that so far we could not understand and codify them with the required precision, because some of the basic conceptual and methodological tools were missing. It is likely that, on these topics, we are currently in a situation similar to what would be physics without the basic mathematical tools. I believe that to understand cognition we should try to focus our efforts on the development of these tools. It is likely that the necessary tools are the ones that allow to precisely define the concepts of structure, of rule and of emergent property. In this short article I presented some key points on three general principles for cognition. I also briefly shown some points on a methodology to deal with the concepts of structure, structural computation and rule, that I developed [3]. References 1. H. Poincare (1902) Science and Hypothesis. 2. K. Craik (1943) The nature of explanation. Cambridge University Press. 3. D. Pantano Proposta di nuovi strumenti per comprendere come funziona la cognizione (Novel tools to understand how cognition works). - CoRR, 2014 (In Italian) arxiv:1401.1533 10