Logical foundations and analogies of Anglo-American cognitive science

Ip₁: Cybernetics is logically based on Formalism

We establish some temporal, historical, and intensional ¹ evidence to demonstrate Ip₁. While the first two sources of evidence are easily demonstrable, the third one needs a more detailed exposition. We begin with a temporal argument: Formalism precedes Cybernetics by about 40 years. In fact, while Formalism arose between the late 19th and early 20th centuries, and dominated the 1920s, Cybernetics arose in the 1940s. From a causal perspective, Formalism can only be an antecedent or a causal factor of Cybernetics. This first assumption has an historical confirmation. In fact, Wiener, the father of Cybernetics, was a student of Hilbert, the father of Formalism, and Russell, the father of New-Logicism, which is strictly related to the same Formalism. This cannot be enough, however, to counter the eventual criticisms of this consideration. We must analyse the common arguments to further strengthen the initial hypothesis. In fact, Cybernetics and Formalism increase the symbolic dimension, deductive method, and the general axiomatic setting in similar ways.

Cybernetics arose from Wiener’s works. During the Second World War, he collaborated with the military on a secret project to realise an anti-aircraft tracking system for predicting the movement of enemy aircraft. He studied feedback processes and goal-directed behaviours. In the same period, Wiener collaborated with Cannon and Rosenblueth on the study of heart behaviour. Here, he found it useful to apply the same concepts introduced with the anti-aircraft tracking system project. So he concluded that these concepts were universal for describing processes and behaviours both in living beings and in human-made entities. Their study became the subject of a new science called Cybernetics (from the Greek kybernétes: “helmsman”). Wiener (Rosenblueth, Wiener, & Bigelow, 1943; Wiener, 1948) argues that there is a new analytic level, beyond physics and based on unobservable entities such as control, information, and feedback. These are the principles of intelligent, intentional, and goal-directed behaviours. Cybernetics explains human behaviour and creates intelligent artificial devices on the basis of purely abstract rules of signal manipulation and control.

First Formalism arose in Germany in the late 19th century. It reaffirmed and redefined the Euclidean foundational rule for all the logical-mathematical sciences. The first Formalism founds logical and mathematical practices on the human mind’s capacities to abstract and use symbols: the internal constitution of the human mind is, therefore, the real foundation of mathematics and logic.

Hilbert’s Formalism represented the peak of these tendencies and dominated logic research until the beginning of the 1930s, when Gödel destroyed it with his innovative theorems. A New Euclidean perspective (Hilbert & Bernays, 1934–1939/1978) does not assume that reality is clear and evident, but that it is focused on the mutual logical relations between axioms, or rather between the implicit definitions of concepts and primitive terms. The mathematician starts from problems that she tries to solve with the given axioms and, if she fails, she introduces new axioms to solve the problems. In an attempt to formulate theories without delineated models, Hilbert’s axiomatic method defines the fundamental mathematical structures. Primary are the individual theories given in an axiomatic form as statement systems and not the idea of a unitary system, where all the mathematical principles are connected. Specific contents or fundamental intuitions do not prevail in the system; instead, the pure logical structure prevails as a matrix that connects statements with statements and theories with theories. The foundation of mathematics coincides with the foundation of its single theories and the demonstration of their non-contradictory nature. According to Hilbert (Hilbert & Bernays, 1934–1939/1978), there are deep differences between the various mathematical statements. The correctness of some statements is immediately knowable, because they refer to concrete objects; they are based on material and learnable contents. Instead, the correctness of other statements is not immediately knowable, because they are based on abstract and unlearnable contents. Only concrete objects are demonstrably included in a finite mathematics, where the problem of non-contradictory nature does not exist, because the totality is reduced to an inventory of meaningless formulas. The theories that do not refer to these concrete objects are ideal and can be verified in the same finitist mathematics only later. So Hilbert’s Formalism prefers to use symbolic languages: purely syntactic; coherent, complete, and decidable; accompanied by rules for the formation and demonstration of formulas; and deductively structured by primitive axioms. The use of these logical languages permits the election of logic as the basis of mathematics: an ideal of indubitability and certainty.

Some intensional analogies between Cybernetics and Formalism are therefore evident: (a) a basilar logic framework is important to both, (b) symbolic abstractionism is dominant in both, (c) both Cybernetic signals and Formalism symbols are related to each other by clear and evident rules, and (d) Cybernetics considers intelligent behaviours as logical systems that are coherent in their internal equilibrium but mutable in relation with the external environment, while Formalism similarly studies coherent and complete theoretical systems that are always based on mutable axioms.

Ip₂: Computational Cognitivism is logically based on all the Formalistic currents, including the newer versions of New-Formalism

We establish some temporal, historical, and intensional evidence to demonstrate Ip₂. There is a strong time lag of about 20–30 years between the birth of New-Formalism and the birth of the first Cognitivism. In fact, while New-Formalism arises in the early 1930s, First Cognitivism arises only at the end of the 1940s. Historically, some facts show direct links between the members of Computational Cognitivism and the various formalistic tendencies. Chomsky (1955/1975, 2000), for example, follows the philosophy of his professor Goodman (1978; Goodman & Elgin, 1988), who draws on the work of the late new-logicist Quine (1969, 1953/1980). Instead, the Logic Theorist program, created by Newell, Shaw, and Simon (1957), tries to demonstrate computationally some of the most important theorems of the Principia Mathematica (Whitehead & Russell, 1910–1913/1962). New-Logicism is, however, linked to Formalism. The first Cognitivism also draws further inspiration from Formalism, in addition to its direct derivation from Cybernetics, Information Theory, and AI that we have already understood to be derived from Formalism itself. Now we must show the possible links that are active between First Cognitivism and New-Formalism. In fact, some intensional schemas seem to unite these two tendencies. Gentzen’s idea of a natural deduction seems to recur in the implementation modalities of Cognitivism that supplants, in a short time, Behaviourism. It begins as a gradual and shared realisation of dissatisfactions with the dogmas and limits of American psychology of that time. The rediscovery of the mind, as a complex mechanism that elaborates the information of reality, refers to unobservable inner human reality until then denied by Behaviourism. The first phase of Cognitivism is called Computational or HIP (Human Information Processing). It dominates American Psychology during the 1960s, establishing the metaphor of the computer (Bechtel, 1988; Bechtel, Abrahamsen, & Graham, 1998). Both mind and computer are, in fact, information processors. The computer becomes the inspiration for theoretical hypotheses on the functioning of the human mind and it also allows scientists to verify them experimentally. Specifically, cognitivists investigate rules that determine psychic processes and the computer seems to mirror these processes. The mind can, in fact, be studied without considering its organic basis–the brain–through the analysis of the organisation and functioning of its representations and processes. Similarly, the computer operates at highly symbolic level–the software–although it is a physical machine composed of circuits mechanically connected–the hardware. Computational Cognitivism is typically abstract and symbolic.

Some important events contributed to the rise of initial Cognitivism (Bechtel et al., 1998; Piccinini, 2004). First of all, during the Hixon Symposium in 1948, Lashley expresses the need to go beyond Behaviourism. Only one year later, while the ethologist Lorenz (1949/1961) accuses Behaviourism of having excluded innate factors from the explanation of human and animal behaviour, Hebb (1949) creates the first cognitive model. He shows how the associations between stimuli and responses are mediated by a set of neurophysiologic processes. These processes promote close links between neurons and give shape to cellular assemblies, where the information travels faster and is retained for a longer time. Some years later (September 11, 1956), during a seminar at the Massachusetts Institute of Technology on Information Theory, three different conferences were held: (a) Chomsky on language, (b) Newell, Simon, and Shaw on the Logic Theorist, and (c) Miller on short-term memory. Miller (1956) presented some concrete experimental results on the informational capacity of human memory. Chomsky (1957) held, instead, that the superficial structure of the phrases of a particular language derives from a universal and innate deep matrix, through a culturally determined series of transformational rules. Chomsky’s popularity subsequently increased thanks to the publication of a book review refuting Skinner’s verbal learning theory (Chomsky, 1959). In 1959, McCarthy created the LISP programming language that allows a computer to execute symbolic elaborations rather than only numerical ones (Minsky, 1968). In 1960 at Harvard, Miller managed the first cognitive studies centre. In collaboration with Galanter and Pribram, he described behaviour in terms of action plans created in order to obtain specific results (Miller, Galanter, & Pribram, 1960). In 1962, Gagnè (1962a, 1962b) applied the principles of conditioning in military training and revealed their ineffectiveness. A few years later, Neisser (1967) developed the first theoretical formulation of Cognitivism: the science that considers mental activities as processes of elaboration and construction (from sensorial information to cognitive schemas). Eight years later, Fodor (1975) proposed the Representational Theory of Mind, ³ which is the philosophical basis for First Cognitivism. Finally, Lindsay and Norman (1977) introduced the concept of Human Information Processing.

New Formalism arose in Germany in the 1930s. It represented a formalistic reaction to Gödel’s results and to the developments of Intuitionism. Gentzen (1969) is the most important exponent of New Formalism. As Hilbert did, he deals with the mathematical foundational theme and tries to solve the antinomies, always referring to the concept of infinity. Considering infinity as active in logical-mathematical languages leads inevitably to paradoxes. It is necessary to give a finitist and constructive meaning to the sentences with references to infinity, considering this quality as exclusively potential and highly dynamic but not formalised. In fact, Gentzen observes informal reasoning and notes that deductions do not derive from a few basic logical formulas (axioms) and do not develop through a few inferential rules. On the contrary, natural deductions use many assumptions and rules that are finite in a formal sense, but infinite in a dynamic sense.

Similar to Formalism in general, HIP is a purely symbolic approach. In fact, cognitively, a symbol is an entity with a complete meaning on a macroscopic level, which is phenomenal and mental. In experiential terms, it is an immediately evident meaning and so not further analysable. The interest is focused on the macroscopic functioning of manipulation programs of symbols. As in Formalism, the cognitive and symbolic functioning of the human mind is quite elastic, because it is based on revisable assumptions. Thus, in First Cognitivism, as in New Formalism, the emphasis on a general basic symbolic modality is integrated with the highlight of changing macroscopic structures and the tendency toward implementation and modification. Finally, worthy of note, especially considering Hebb’s assemblies, is the foundational value of logical set theory for First Cognitivism, as well as before for formalistic approaches.

Ip₃: Connectionistic Cognitivism is logically similar to Intuitionism that precedes it in time

We establish only some intensional and argumentative evidence to demonstrate Ip₃. In fact, we assume that the second phase of Cognitivism is a reaction to Computational Cognitivism, because of the emergence of some important dilemmas within it. The idea that cognitive functioning is purely symbolic leads to significant problems (Pessa & Pietronilla Penna, 2000). Symbols are, for example, potentially unlimited, as are the abstractions that produce them. This would require an improbable unlimited capacity of human memory. In addition, symbols produce useful results only if they are correctly related within ordered operations. A small error would cause disastrous effects. The sensitivity to interference, noise, and malfunction is, therefore, high. Finally, symbols are exclusive: they can only refer to other symbols. So, to establish their meaning, it is necessary to refer to other cognitive dimensions and therefore cognition cannot only be symbolic. Non-symbolic cognitive aspects play a primary role and they are not reducible to symbolisation. Similarly, about 40 years earlier, ⁴ Intuitionism represented a profound reaction to logicist and formalist approaches accused of losing their intuitive and creative origins because of the exaltation of symbolic dimensions. Also in Intuitionism, a new dimension arises over mere formalisation. Logic maintains a simple communicative role. It is a purely formal game full of contradictions that cannot guarantee unambiguous languages for mathematics. Connectionism and Intuitionism break the traditional notions, develop innovative principles, and modify the meanings of classical principles. Furthermore, Connectionism denies the logical foundations of previous computational phases that are, on the basis of Ip₂, formalistic tendencies already opposed to Intuitionism.

During the 1970s and 1980s, several scholars began to ask questions about the exhaustiveness of computers as a metaphor for the human mind. The unresolved issues created, inside Cognitivism, different afterthoughts (Bechtel et al., 1998). The use of rules, for example, would seem to explain the initial phases of learning but not the later ones, in which intuition takes on more importance. Moreover, the structure and functioning of the nervous system show how neurons operate more slowly than microcircuits, even if in a way more complex, dynamic, and multilinear. This explains why computers have a higher processing speed, but less flexibility and elasticity. The Cognitivist paradigm seems to capsize. While during the HIP phase the computer constituted an ideal metaphor for neuronal and mental function, the latter became a model for the creation of an ever more powerful computer. The nervous system appears as a complex and intricate network of neurons variously interconnected through the mutual exchange of electrical pulses. The information is distributed and elaborated in parallel, at the same time, in the brain. Various elements of the nervous system are working simultaneously on different parts of the same problem. Humans become organisms who elaborate information, but in a distributed manner and by assigning specific meanings to aspects of this information.

The second phase of Cognitivism, Parallel Distributed Processing (PDP), arose in this period but has older roots (Aizawa, 1992; Bechtel et al., 1998; Walker, 1990). This phase contributed to the study of sub-symbolic models of cognitive processes and in the use of the Neural Networks model, whose theoretical bases were set by McCulloch and Pitts (1943). Neural Networks, initially considered as logic and abstract models of cerebral cortex function, soon became an excellent model of human cognition. They allow the comparison between cognitive and neuronal functioning that is considered a complex configuration constituted of elementary units (micro-characteristics) interconnected in an intense exchange of information. The aforementioned Hebbian model shows the basilar elements of connectionist structures (Hebb, 1949). Years later, Rochester, Holland, Haibt, and Duda (1956) realised the first simulation of neural network behaviour in the manner provided by Hebb. Rosenblatt (1958) created, instead, the Perceptron, the first neural network model able to learn from experience. Later, Minsky and Papert (1969) considered this model incapable of also solving simpler problems of classification. In 1976, a book by Neisser once again represented a moment of revision and criticism of Cognitivism and showed its experimental and procedural limits. Computationalism explains the mind as a closed cover of symbols and thus able to lose the undeniable value of the external environment. Thus, some new ecological trends emphasised the active role of the mind in its living environment (Gibson, 1979). Hopfield (1982) described a new model of a neural network able to work as an associative memory. He used, with Feldman and Ballard (1983), the term “Connectionism” to denominate the new sub-symbolic approach. Finally, McClelland and Rumelhart (1986) published one of the best books on Connectionism.

Intuitionism arose in the Netherlands in the early 20th century. It comes from Poincarè’s Semi-Intuitionism, according to which, the paradoxes do not affect pure mathematics, but only mathematics contaminated by logic. Poincarè (1902/2001) considered that, on an intuitive basis, the only way to obtain new entities from old entities is to define the latter in a substantively sensible manner: it is possible to construct mathematical objects only by recourse to given entities and without reference to totalities. Mathematical objects are independent of our thought, because their existence depends on a process of construction and identification.

Beginning in 1910, Brouwer, the father of Intuitionism, explained mathematical contents as products of constructive thought based on intuition. Brouwer (1976a, 1976b) created a new mathematics without any logical contamination. The use of demonstration is not based on logic, but on intuition, or else on over-intuition of the continuous flow of time. This is the capacity to consider separately concepts and constructions that, outlining indefinite succession, are involved in habitual thinking. While logic studies words, mathematics is instead a construction without words. In this way, Brouwer rejects the traditional laws of logic, such as the Excluded Middle that can be valid only in the finite domains of human knowledge. These laws do not have universal validity. So conclusions do not derive from the use of predetermined and crystallised logical sets. Classic logic, which uses language valid for each formalistic orientation, is not acceptable, because it is subjected to the concrete work of mathematics.

Ip₄: Emergentist Cognitivism is logically similar to Gödel’s work and New-Intuitionism, both of which precede it in time

We establish some intensional and argumentative evidences to demonstrate Ip₄. In fact, the third phase of Cognitivism is a reaction to previous phases (Computational and Connectionistic), because they are considered incomplete and problematic. Similarly, several years before, ⁵ Gödelian and New-Intuitionistic logics represent a deep correction to the approaches from which they derive (Formalism and Intuitionism). In addition, Emergentist Cognitivism is a reaction to previous phases in that it reviews their logical foundations, which, on the basis of Ip₂ and Ip₃, are the same formalistic and intuitionistic tendencies.

In the late 1980s and 1990s, the third and latest phase of Cognitivism, named Emergentist, arises. Emergentism shows the limits both of Computationalism, considered too abstract, and Connectionism, considered, on the contrary, completely free of any abstraction. In fact, while Computationalism refers only to the symbolic and macroscopic dimension of cognitive functioning, Connectionism refers instead only to the sub-symbolic and microscopic dimension of neuronal functioning. A theoretical integration between these two levels becomes necessary so as to define a model of reciprocal causality that is, at the same time, top-down and bottom-up. In an integrative study, we must not prefer one of the domains to another, but we must make explicit their mutual causal links (Bechtel, 1988). The neuronal level is elementary and basic, but the cognitive and symbolic level, which emerges from the neuronal level, is higher and different. Hierarchy is structured as an emersion of increasingly complex dimensions that are irreducible to their underlying levels. The cognitive architectures assume a hierarchical shape with higher capacities, such as reasoning, and lower capacities, such as sensory perception. While the first rely on symbolic processing systems, working through logical and mathematical algorithms, the latter are implemented by neural networks.

The birth of this new phase is due to theorists of skills. In fact, they study a type of learning that is not new but complex, and barely explainable with only computational or connectionistic models (Bechtel, 1988; Sawyer, 2002). Motor learning implies behaviours that are highly structured, hierarchical, non-linear, and non-serial (Lashley, 1951). It is impossible to isolate all their control parameters, because their degrees of freedom are very high (Bernstein, 1967). The use of explicative mathematical models, which include a large number of variables that change in function over time, becomes necessary. In fact, over time, the status of the system changes in a dynamical and unpredictable way.

In 1951, Gödel (1951/1995b) believed that the human mind is always able to find strategies to solve any problem. It goes beyond the power of each finite machine. Truly insoluble problems do not exist, because the mind is not static; the mind continuously evolves. The number of its states, although finite in individual evolution, is potentially infinite. Similarly, Emergentism rejects the claim to reduce every ascending system to formal models that are increasingly complex; this claim is in fact supported by the only emerging idea of a utopian nature—that each level can be represented by a closed and rigid symbolisation. Instead, emergence brings a continuous and unpredictable innovation that finds a direct parallelism in quantum physics and in the science of complexity (Climatology, Chaos Theory, Self-Organisation Process Theory, Artificial Life, etc.), but it is in the logic of Gödel that it finds its essence of infinitary opening. In the work of Gödel, in fact, we can see the infinity logic of each system. Also, if we decide to add the undecidable statement g between the axioms of the system, thus creating a more powerful system, this new system will have its own undecidable statement. Here we can realise the infinite principle: (a) it is established that S=γ, where the system (S) is identified with its undecidable statement (γ); (b) γ becomes an axiom of another powerful system (S₁); (c) also in this case, the presence of another undecidable statement (γ₁) would be proved (S₁=S+γ₁); and (d) it is possible to proceed further and the results would delineate a regularity (S₂=S₁+γ₂, S₃=S₂+γ₃, S_n=S_(n-1)+γ_n). If the undecidability of a formal system is postponed ad infinitum, however, the human mind cannot see its possible coherence (infinity of decidability where S_∞=γ_∞).

Emergentism shows the infinite meaning of Gödel’s logic: a perspective of definition of hierarchical horizons anchored to empirical and organic data. From an intuitive point of view, however, this hierarchy could be infinite. Does a last level exist? Or can an emergency develop with a new level? Is the emergence phenomenon ready to run out? Or is it always latent and ready to wake as a nature law is? Does each hierarchical level preserve an evolutionary and unpredictable emergency factor from which a higher and more complex level could arise? This emergency factor seems to refer logically to Gödel’s factor (g). In fact, emergence seems to keep the coherence and completeness of each hierarchical level in check.

New-Intuitionism arises in the Netherlands in the 1930s. It tries to expand Brouwer’s Intuitionism into logical formalisation. This approach does not reject logic itself, but rejects only classical logic. Despite the fact that Brouwer does not consider the logical formalisation of intuitionistic mathematics, Heyting (1956) considers this a useful operation, but he reiterates that mathematics is a pure mental process and that every language, even formalised, is only a mnemonic aid to communication. In fact, thought cannot be reduced to a finite number of rules. The intuitionist does not seek the rigour of language, but the rigour of mathematical thought. The axiomatic method has a purely descriptive value of accommodation and abbreviation; it has no creative value. Each proposition of this language hides the intention of a mathematical construction that has to satisfy some determined conditions.

Similar to New-Intuitionism, where the logic dimension is a level that emerges from the basic level of intuitive mental construction, Emergentism considers the symbolic dimension as a level that emerges from the basic level of neuronal and sub-symbolic functioning. While New-Intuitionism relegates the logical level to a minor and secondary dimension, new Cognitivism interprets each level as equal: this is undoubtedly an important difference.

Emergentism has different relations with logical tendencies that we have described. In fact, while Gödel’s logic seems to be a foundation of Emergentism, New-Intuitionism is instead simply similar to it. Some evidences can prove this statement: Hofstadter (1979), for example, one of the fathers of new philosophical Emergentism, is extremely interested in the study of the mind. He is a follower of Gödel, however he interprets his works in philosophical concepts and applies them to mental problems. Therefore, Ip₄ is more properly a composed hypothesis, both analogical and foundational.

Ip₅: Computationalism and AI are logically identified with von Neumann’s Computational Logic

On the basis of the historical evidence, we can sustain that von Neumann is one of the greatest exponents of Formalism, which is a foundation for AI and First Cognitivism (Ip₁, Ip₂). This important affinity requires more in-depth argumentation. On the basis of the second intensional evidence, it seems that, effectively, the Computational Logic of von Neumann, AI, and Computationalism share the same primary argumentation. In fact, the idea of a possible simulation on a computer of intelligent behaviours is common to all three approaches. Intelligence is a processing system that follows simple rules and manipulates symbols thanks to a large memory. As in a computer, the intelligence is in the software, not the hardware. Just as we can implement the same software on different hardware, intelligence can be implemented on different hardware too, not only purely biological. The brain is a physical machine; therefore other physical machines, such as computer hardware, may perform any cognitive function. Each cognitive function is also similar to a symbolic process of calculation, governed by appropriate codifiable rules, just as the software of a computer. In a formalistic way, the mind is identified with an algorithmic and finite machine, just like the machine idealised by von Neumann (1958/2000).

Ip₆: Connectionism is primarily related to Turing’s Computational Logic and, secondly, to Gödel’s logic

TM constitutes, at the same time, the algorithmic demonstration of logical calculus, which Gödel (1963/1990) uses to expand the applicability of his theorems, and the demonstration of paradoxical nature implicit in the possibility of logical systems explaining their own functioning. In fact, Turing (1937) shows the antinomy of his machine known as the halting-problem: given a word processing program and an input, we cannot prove that the program will terminate computation. Computers can enter into an endless loop. This does not happen, however, in real minds, because the human body has some abilities called “oracular” (Turing, 1939) that intervene in the process of elaboration from outside. They establish a cognitive constraint on computation that allows the mind to change codes of elaboration and reach a final solution. Computers remain a useful tool for the study of the human mind, but now they are simple instruments that imitate neural functioning. It is possible to create programs capable of reproducing neural functioning and checking the comparability between virtual and real results. PDP rejects von Neumann’s heteronomous and linear logical theory. The brain does not seem to have a CPU, but countless neuronal units that are interrelated, multilinear, and complex.

Ip₇: Emergentism is logically identified with Computational Logic

Since the 1950s, without any distinctions, the works of Turing, von Neumann, and Prigogine (1980, 1997; Prigogine & Nicolis, 1977, 1989; Prigogine & Stengers, 1984) have stimulated the birth of a first mathematical theory of processes of self-organisation able to explain the emergence of order from disorder and the morphogenetic evolution of living beings. Inside Cognitivism, Emergentism is nothing more than an attempt to integrate Connectionism and Computationalism on the basis of a strong theory and shared foundations.