connectionism

The central connectionist principle is that mental phenomena can be described by interconnected networks of simple and often uniform units. The form of the connections and the units can vary from model to model. For example, units in the network could represent neurons and the connections could represent synapses, as in the human brain. This principle has been seen as an alternative to GOFAI and the classical theories of mind based on symbolic computation, but the extent to which the two approaches are compatible has been the subject of much debate since their inception.

{{Main|Activation function}}

Internal states of any network change over time due to neurons sending a signal to a succeeding layer of neurons in the case of a feedforward network, or to a previous layer in the case of a recurrent network. Discovery of non-linear activation functions has enabled the second wave of connectionism.

{{Main|Artificial neural networks|Deep learning}}

Neural networks follow two basic principles:

1. Any mental state can be described as a n-dimensional vector of numeric activation values over neural units in a network.
2. Memory and learning are created by modifying the 'weights' of the connections between neural units, generally represented as an n×m matrix. The weights are adjusted according to some learning rule or algorithm, such as Hebbian learning.{{Cite journal|last1=Novo|first1=María-Luisa|last2=Alsina|first2=Ángel|last3=Marbán|first3=José-María|last4=Berciano|first4=Ainhoa|date=2017|title=Connective Intelligence for Childhood Mathematics Education|journal=Comunicar|language=es|volume=25|issue=52|pages=29–39|doi=10.3916/c52-2017-03|issn=1134-3478|doi-access=free|hdl=10272/14085|hdl-access=free}}

Most of the variety among the models comes from:

• Interpretation of units: Units can be interpreted as neurons or groups of neurons.
• Definition of activation: Activation can be defined in a variety of ways. For example, in a Boltzmann machine, the activation is interpreted as the probability of generating an action potential spike, and is determined via a logistic function on the sum of the inputs to a unit.
• Learning algorithm: Different networks modify their connections differently. In general, any mathematically defined change in connection weights over time is referred to as the "learning algorithm".

Connectionist work in general does not need to be biologically realistic.{{Cite web|url=http://www.encephalos.gr/48-1-01e.htm|title=Encephalos Journal|website=www.encephalos.gr|access-date=2018-02-20}}{{Cite book|url=https://books.google.com/books?id=s-OCCwAAQBAJ&q=%22accurate%22&pg=PT18|title=Neural Geographies: Feminism and the Microstructure of Cognition|last=Wilson|first=Elizabeth A.|date=2016-02-04|publisher=Routledge|isbn=978-1-317-95876-5|language=en}}{{cite journal| title=Organismically-inspired robotics: homeostatic adaptation and teleology beyond the closed sensorimotor loop| author=Di Paolo, E.A| url=https://users.sussex.ac.uk/~ezequiel/dp-erasmus.pdf| publisher=University of Sussex| journal=Dynamical Systems Approach to Embodiment and Sociality, Advanced Knowledge International| date=1 January 2003| access-date=29 December 2023| s2cid=15349751}}{{Cite journal|last1=Zorzi|first1=Marco|last2=Testolin|first2=Alberto|last3=Stoianov|first3=Ivilin P.|date=2013-08-20|title=Modeling language and cognition with deep unsupervised learning: a tutorial overview|journal=Frontiers in Psychology|volume=4|page=515|doi=10.3389/fpsyg.2013.00515|issn=1664-1078|pmc=3747356|pmid=23970869|doi-access=free}}{{cite journal| title=Analytic and Continental Philosophy, Science, and Global Philosophy| author=Tieszen, R.| url=https://scholarworks.sjsu.edu/cgi/viewcontent.cgi?referer=&httpsredir=1&article=1015&context=comparativephilosophy| journal=Comparative Philosophy| volume=2| issue=2| pages=4–22| date=2011| access-date=29 December 2023}}{{Cite book|url=https://books.google.com/books?id=uV9TZzOITMwC&q=%22biological%20plausibility%22&pg=PA17|title=Neural Network Perspectives on Cognition and Adaptive Robotics|last=Browne|first=A.|date=1997-01-01|publisher=CRC Press|isbn=978-0-7503-0455-9|language=en}}{{Cite book|url=https://books.google.com/books?id=7pPv0STSos8C&q=%22biological+realism%22&pg=PA63|title=Connectionism in Perspective|last1=Pfeifer|first1=R.|last2=Schreter|first2=Z.|last3=Fogelman-Soulié|first3=F.|last4=Steels|first4=L.|date=1989-08-23|publisher=Elsevier|isbn=978-0-444-59876-9|language=en}} One area where connectionist models are thought to be biologically implausible is with respect to error-propagation networks that are needed to support learning,{{Cite journal|last=Crick|first=Francis|date=January 1989|title=The recent excitement about neural networks|journal=Nature|language=en|volume=337|issue=6203|pages=129–132|doi=10.1038/337129a0|pmid=2911347|issn=1476-4687|bibcode=1989Natur.337..129C|s2cid=5892527}}{{Cite journal|last1=Rumelhart|first1=David E.|last2=Hinton|first2=Geoffrey E.|last3=Williams|first3=Ronald J.|date=October 1986|title=Learning representations by back-propagating errors|journal=Nature|language=en|volume=323|issue=6088|pages=533–536|doi=10.1038/323533a0|issn=1476-4687|bibcode=1986Natur.323..533R|s2cid=205001834}} but error propagation can explain some of the biologically-generated electrical activity seen at the scalp in event-related potentials such as the N400 and P600,{{Cite journal|last1=Fitz|first1=Hartmut|last2=Chang|first2=Franklin|date=2019-06-01|title=Language ERPs reflect learning through prediction error propagation|journal=Cognitive Psychology|volume=111|pages=15–52|doi=10.1016/j.cogpsych.2019.03.002|pmid=30921626|issn=0010-0285|hdl=21.11116/0000-0003-474F-6|s2cid=85501792|hdl-access=free}} and this provides some biological support for one of the key assumptions of connectionist learning procedures. Many recurrent connectionist models also incorporate dynamical systems theory. Many researchers, such as the connectionist Paul Smolensky, have argued that connectionist models will evolve toward fully continuous, high-dimensional, non-linear, dynamic systems approaches.

Precursors of the connectionist principles can be traced to early work in psychology, such as that of William James.{{cite book |last1=Anderson |first1= James A.|last2=Rosenfeld |first2= Edward |date= 1989|title= Neurocomputing: Foundations of Research|publisher= A Bradford Book |page= 1|chapter = Chapter 1: (1890) William James Psychology (Brief Course) |isbn=978-0-262-51048-6 }} Psychological theories based on knowledge about the human brain were fashionable in the late 19th century. As early as 1869, the neurologist John Hughlings Jackson argued for multi-level, distributed systems. Following from this lead, Herbert Spencer's Principles of Psychology, 3rd edition (1872), and Sigmund Freud's Project for a Scientific Psychology (composed 1895) propounded connectionist or proto-connectionist theories. These tended to be speculative theories. But by the early 20th century, Edward Thorndike was writing about human learning that posited a connectionist type network.Edward Thorndike (1931) [http://hdl.handle.net/2027/mdp.39015010306069 Human Learning], page 122

Hopfield networks had precursors in the Ising model due to Wilhelm Lenz (1920) and Ernst Ising (1925), though the Ising model conceived by them did not involve time. Monte Carlo simulations of Ising model required the advent of computers in the 1950s.{{cite journal |last1=Brush |first1=Stephen G. |year=1967 |title=History of the Lenz-Ising Model |journal=Reviews of Modern Physics |volume=39 |issue=4 |pages=883–893 |bibcode=1967RvMP...39..883B |doi=10.1103/RevModPhys.39.883}}

The first wave begun in 1943 with Warren Sturgis McCulloch and Walter Pitts both focusing on comprehending neural circuitry through a formal and mathematical approach. McCulloch and Pitts showed how neural systems could implement first-order logic: Their classic paper "A Logical Calculus of Ideas Immanent in Nervous Activity" (1943) is important in this development here. They were influenced by the work of Nicolas Rashevsky in the 1930s and symbolic logic in the style of Principia Mathematica.{{Cite journal |last1=McCulloch |first1=Warren S. |last2=Pitts |first2=Walter |date=1943-12-01 |title=A logical calculus of the ideas immanent in nervous activity |url=https://doi.org/10.1007/BF02478259 |journal=The Bulletin of Mathematical Biophysics |language=en |volume=5 |issue=4 |pages=115–133 |doi=10.1007/BF02478259 |issn=1522-9602}}

Hebb contributed greatly to speculations about neural functioning, and proposed a learning principle, Hebbian learning. Lashley argued for distributed representations as a result of his failure to find anything like a localized engram in years of lesion experiments. Friedrich Hayek independently conceived the model, first in a brief unpublished manuscript in 1920,Hayek, Friedrich A. [1920] 1991. Beiträge zur Theorie der Entwicklung des Bewusstseins [Contributions to a theory of how consciousness develops]. Manuscript, translated by Grete Heinz.{{Cite journal |last=Caldwell |first=Bruce |date=2004 |title=Some Reflections on F.A. Hayek's The Sensory Order |url=http://link.springer.com/10.1007/s10818-004-5505-9 |journal=Journal of Bioeconomics |language=en |volume=6 |issue=3 |pages=239–254 |doi=10.1007/s10818-004-5505-9 |s2cid=144437624 |issn=1387-6996}} then expanded into a book in 1952.{{Cite book |last=Hayek |first=F. A. |title=The Sensory Order: An Inquiry into the Foundations of Theoretical Psychology |date=2012-09-15 |publisher=The University of Chicago Press |edition=1st |language=en}}

The Perceptron machines were proposed and built by Frank Rosenblatt, who published the 1958 paper “The Perceptron: A Probabilistic Model For Information Storage and Organization in the Brain” in Psychological Review, while working at the Cornell Aeronautical Laboratory. He cited Hebb, Hayek, Uttley, and Ashby as main influences.

Another form of connectionist model was the relational network framework developed by the linguist Sydney Lamb in the 1960s.

The research group led by Widrow empirically searched for methods to train two-layered ADALINE networks (MADALINE), with limited success.pp 124-129, Olazaran Rodriguez, Jose Miguel. [https://web.archive.org/web/20221111165150/https://era.ed.ac.uk/bitstream/handle/1842/20075/Olazaran-RodriguezJM_1991redux.pdf?sequence=1&isAllowed=y A historical sociology of neural network research]. PhD Dissertation. University of Edinburgh, 1991.Widrow, B. (1962) Generalization and information storage in networks of ADALINE "neurons". In M. C. Yovits, G. T. Jacobi, & G. D. Goldstein (Ed.), Self-Organizing Svstems-1962 (pp. 435-461). Washington, DC: Spartan Books.

A method to train multilayered perceptrons with arbitrary levels of trainable weights was published by Alexey Grigorevich Ivakhnenko and Valentin Lapa in 1965, called the Group Method of Data Handling. This method employs incremental layer by layer training based on regression analysis, where useless units in hidden layers are pruned with the help of a validation set.{{cite book |last1=Ivakhnenko |first1=A. G. |url={{google books |plainurl=y |id=rGFgAAAAMAAJ}} |title=Cybernetics and forecasting techniques |last2=Grigorʹevich Lapa |first2=Valentin |publisher=American Elsevier Pub. Co. |year=1967}}{{cite arXiv |eprint=2212.11279 |class=cs.NE |first=Jürgen |last=Schmidhuber |author-link=Jürgen Schmidhuber |title=Annotated History of Modern AI and Deep Learning |date=2022}}{{cite book |last=Ivakhnenko |first=A. G. |url={{google books |plainurl=y |id=FhwVNQAACAAJ}} |title=Cybernetic Predicting Devices |publisher=CCM Information Corporation |year=1973}}

The first multilayered perceptrons trained by stochastic gradient descent{{Cite journal |last1=Robbins |first1=H. |author-link=Herbert Robbins |last2=Monro |first2=S. |year=1951 |title=A Stochastic Approximation Method |journal=The Annals of Mathematical Statistics |volume=22 |issue=3 |page=400 |doi=10.1214/aoms/1177729586 |doi-access=free}} was published in 1967 by Shun'ichi Amari.{{cite journal |last1=Amari |first1=Shun'ichi |author-link=Shun'ichi Amari |date=1967 |title=A theory of adaptive pattern classifier |journal=IEEE Transactions |volume=EC |issue=16 |pages=279–307}} In computer experiments conducted by Amari's student Saito, a five layer MLP with two modifiable layers learned useful internal representations to classify non-linearily separable pattern classes.

In 1972, Shun'ichi Amari produced an early example of self-organizing network.{{Cite journal |last=Amari |first=S.-I. |date=November 1972 |title=Learning Patterns and Pattern Sequences by Self-Organizing Nets of Threshold Elements |url=https://ieeexplore.ieee.org/document/1672070 |journal=IEEE Transactions on Computers |volume=C-21 |issue=11 |pages=1197–1206 |doi=10.1109/T-C.1972.223477 |issn=0018-9340}}

There was some conflict among artificial intelligence researchers as to what neural networks are useful for. Around late 1960s, there was a widespread lull in research and publications on neural networks, "the neural network winter", which lasted through the 1970s, during which the field of artificial intelligence turned towards symbolic methods. The publication of Perceptrons (1969) is typically regarded as a catalyst of this event.{{Citation |last=Olazaran |first=Mikel |title=Advances in Computers Volume 37 |chapter=A Sociological History of the Neural Network Controversy |date=1993-01-01 |volume=37 |pages=335–425 |editor-last=Yovits |editor-first=Marshall C. |chapter-url=https://www.sciencedirect.com/science/article/pii/S0065245808604088 |access-date=2024-08-07 |publisher=Elsevier|doi=10.1016/S0065-2458(08)60408-8 |isbn=978-0-12-012137-3 }}{{Cite journal |last=Olazaran |first=Mikel |date=August 1996 |title=A Sociological Study of the Official History of the Perceptrons Controversy |url=http://journals.sagepub.com/doi/10.1177/030631296026003005 |journal=Social Studies of Science |language=en |volume=26 |issue=3 |pages=611–659 |doi=10.1177/030631296026003005 |issn=0306-3127}}

The second wave begun in the early 1980s. Some key publications included (John Hopfield, 1982){{Cite journal |last=Hopfield |first=J J |date=April 1982 |title=Neural networks and physical systems with emergent collective computational abilities. |journal=Proceedings of the National Academy of Sciences |language=en |volume=79 |issue=8 |pages=2554–2558 |doi=10.1073/pnas.79.8.2554 |doi-access=free |issn=0027-8424 |pmc=346238 |pmid=6953413|bibcode=1982PNAS...79.2554H }} which popularized Hopfield networks, the 1986 paper that popularized backpropagation,{{Cite journal |last1=Rumelhart |first1=David E. |last2=Hinton |first2=Geoffrey E. |last3=Williams |first3=Ronald J. |date=October 1986 |title=Learning representations by back-propagating errors |url=https://www.nature.com/articles/323533a0 |journal=Nature |language=en |volume=323 |issue=6088 |pages=533–536 |doi=10.1038/323533a0 |bibcode=1986Natur.323..533R |issn=1476-4687}} and the 1987 two-volume book about the Parallel Distributed Processing (PDP) by James L. McClelland, David E. Rumelhart et al., which has introduced a couple of improvements to the simple perceptron idea, such as intermediate processors (known as "hidden layers" now) alongside input and output units and using sigmoid activation function instead of the old 'all-or-nothing' function.

Hopfield approached the field from the perspective of statistical mechanics, providing some early forms of mathematical rigor that increased the perceived respectability of the field. Another important series of publications proved that neural networks are universal function approximators, which also provided some mathematical respectability.{{Cite journal |last=Cybenko |first=G. |date=1989-12-01 |title=Approximation by superpositions of a sigmoidal function |url=https://doi.org/10.1007/BF02551274 |journal=Mathematics of Control, Signals and Systems |language=en |volume=2 |issue=4 |pages=303–314 |doi=10.1007/BF02551274 |bibcode=1989MCSS....2..303C |issn=1435-568X}}

Some early popular demonstration projects appeared during this time. NETtalk (1987) learned to pronounce written English. It achieved popular success, appearing on the Today show.{{Cite book |last=Sejnowski |first=Terrence J. |title=The deep learning revolution |date=2018 |publisher=The MIT Press |isbn=978-0-262-03803-4 |location=Cambridge, Massachusetts London, England}} TD-Gammon (1992) reached top human level in backgammon.{{Citation |title=TD-Gammon |date=2010 |pages=955–956 |editor-last=Sammut |editor-first=Claude |url=https://doi.org/10.1007/978-0-387-30164-8_813 |access-date=2023-12-25 |place=Boston, MA |publisher=Springer US |language=en |doi=10.1007/978-0-387-30164-8_813 |isbn=978-0-387-30164-8 |editor2-last=Webb |editor2-first=Geoffrey I. |encyclopedia=Encyclopedia of Machine Learning}}

As connectionism became increasingly popular in the late 1980s, some researchers (including Jerry Fodor, Steven Pinker and others) reacted against it. They argued that connectionism, as then developing, threatened to obliterate what they saw as the progress being made in the fields of cognitive science and psychology by the classical approach of computationalism. Computationalism is a specific form of cognitivism that argues that mental activity is computational, that is, that the mind operates by performing purely formal operations on symbols, like a Turing machine. Some researchers argued that the trend in connectionism represented a reversion toward associationism and the abandonment of the idea of a language of thought, something they saw as mistaken. In contrast, those very tendencies made connectionism attractive for other researchers.

Connectionism and computationalism need not be at odds, but the debate in the late 1980s and early 1990s led to opposition between the two approaches. Throughout the debate, some researchers have argued that connectionism and computationalism are fully compatible, though full consensus on this issue has not been reached. Differences between the two approaches include the following:

• Computationalists posit symbolic models that are structurally similar to underlying brain structure, whereas connectionists engage in "low-level" modeling, trying to ensure that their models resemble neurological structures.
• Computationalists in general focus on the structure of explicit symbols (mental models) and syntactical rules for their internal manipulation, whereas connectionists focus on learning from environmental stimuli and storing this information in a form of connections between neurons.
• Computationalists believe that internal mental activity consists of manipulation of explicit symbols, whereas connectionists believe that the manipulation of explicit symbols provides a poor model of mental activity.
• Computationalists often posit domain specific symbolic sub-systems designed to support learning in specific areas of cognition (e.g., language, intentionality, number), whereas connectionists posit one or a small set of very general learning-mechanisms.

Despite these differences, some theorists have proposed that the connectionist architecture is simply the manner in which organic brains happen to implement the symbol-manipulation system. This is logically possible, as it is well known that connectionist models can implement symbol-manipulation systems of the kind used in computationalist models,{{Cite journal|last=Chang|first=Franklin|date=2002|title=Symbolically speaking: a connectionist model of sentence production|journal=Cognitive Science|language=en|volume=26|issue=5|pages=609–651|doi=10.1207/s15516709cog2605_3|issn=1551-6709|doi-access=free}} as indeed they must be able if they are to explain the human ability to perform symbol-manipulation tasks. Several cognitive models combining both symbol-manipulative and connectionist architectures have been proposed. Among them are Paul Smolensky's Integrated Connectionist/Symbolic Cognitive Architecture (ICS).{{Cite journal|last=Smolensky|first=Paul|date=1990|title=Tensor Product Variable Binding and the Representation of Symbolic Structures in Connectionist Systems|url=http://www.lscp.net/persons/dupoux/teaching/AT1_2012/papers/Smolensky_1990_TensorProductVariableBinding.AI.pdf|journal=Artificial Intelligence|volume=46|issue=1–2|pages=159–216|doi=10.1016/0004-3702(90)90007-M}} and Ron Sun's CLARION (cognitive architecture). But the debate rests on whether this symbol manipulation forms the foundation of cognition in general, so this is not a potential vindication of computationalism. Nonetheless, computational descriptions may be helpful high-level descriptions of cognition of logic, for example.

The debate was largely centred on logical arguments about whether connectionist networks could produce the syntactic structure observed in this sort of reasoning. This was later achieved although using fast-variable binding abilities outside of those standardly assumed in connectionist models.{{Cite journal|last1=Shastri|first1=Lokendra|last2=Ajjanagadde|first2=Venkat|date=September 1993|title=From simple associations to systematic reasoning: A connectionist representation of rules, variables and dynamic bindings using temporal synchrony|journal=Behavioral and Brain Sciences|language=en|volume=16|issue=3|pages=417–451|doi=10.1017/S0140525X00030910|s2cid=14973656|issn=1469-1825}}

Part of the appeal of computational descriptions is that they are relatively easy to interpret, and thus may be seen as contributing to our understanding of particular mental processes, whereas connectionist models are in general more opaque, to the extent that they may be describable only in very general terms (such as specifying the learning algorithm, the number of units, etc.), or in unhelpfully low-level terms. In this sense, connectionist models may instantiate, and thereby provide evidence for, a broad theory of cognition (i.e., connectionism), without representing a helpful theory of the particular process that is being modelled. In this sense, the debate might be considered as to some extent reflecting a mere difference in the level of analysis in which particular theories are framed. Some researchers suggest that the analysis gap is the consequence of connectionist mechanisms giving rise to emergent phenomena that may be describable in computational terms.{{Cite journal|last=Ellis|first=Nick C.|date=1998|title=Emergentism, Connectionism and Language Learning|url=http://www-personal.umich.edu/~ncellis/NickEllis/Publications_files/Emergentism.pdf|journal=Language Learning|volume=48|issue=4|pages=631–664|doi=10.1111/0023-8333.00063}}

In the 2000s, the popularity of dynamical systems in philosophy of mind have added a new perspective on the debate;{{Citation |last=Van Gelder |first=Tim |year=1998 |title=The dynamical hypothesis in cognitive science |journal=Behavioral and Brain Sciences |volume=21 |issue=5 |pages=615–28; discussion 629–65 |doi=10.1017/S0140525X98001733 |pmid=10097022 |url= https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0140525X98271731 |access-date=28 May 2022}}{{Cite journal |last=Beer |first=Randall D. |date=March 2000 |title=Dynamical approaches to cognitive science |journal=Trends in Cognitive Sciences |volume=4 |issue=3 |pages=91–99 |doi=10.1016/s1364-6613(99)01440-0 |pmid=10689343 |s2cid=16515284 |issn=1364-6613}} some authors{{which|date=February 2016}} now argue that any split between connectionism and computationalism is more conclusively characterized as a split between computationalism and dynamical systems.

In 2014, Alex Graves and others from DeepMind published a series of papers describing a novel Deep Neural Network structure called the Neural Turing Machine{{cite arXiv|last1=Graves|first1=Alex|title=Neural Turing Machines|eprint=1410.5401|class=cs.NE|year=2014}} able to read symbols on a tape and store symbols in memory. Relational Networks, another Deep Network module published by DeepMind, are able to create object-like representations and manipulate them to answer complex questions. Relational Networks and Neural Turing Machines are further evidence that connectionism and computationalism need not be at odds.

Smolensky's Subsymbolic ParadigmP. Smolensky: On the proper treatment of connectionism. In: Behavioral and Brain Sciences. Band 11, 1988, S. 1-74.P. Smolensky: The constituent structure of connectionist mental states: a reply to Fodor and Pylyshyn. In: T. Horgan, J. Tienson (Hrsg.): Spindel Conference 1987: Connectionism and the Philosophy of Mind. The Southern Journal of Philosophy. Special Issue on Connectionism and the Foundations of Cognitive Science. Supplement. Band 26, 1988, S. 137-161. has to meet the Fodor-Pylyshyn challengeJ.A. Fodor, Z.W. Pylyshyn: Connectionism and cognitive architecture: a critical analysis. Cognition. Band 28, 1988, S. 12-13, 33-50.J.A. Fodor, B. McLaughlin: Connectionism and the problem of systematicity: why Smolensky's solution doesn't work. Cognition. Band 35, 1990, S. 183-184.B. McLaughlin: The connectionism/classicism battle to win souls. Philosophical Studies, Band 71, 1993, S. 171-172.B. McLaughlin: Can an ICS architecture meet the systematicity and productivity challenges? In: P. Calvo, J. Symons (Hrsg.): The Architecture of Cognition. Rethinking Fodor and Pylyshyn's Systematicity Challenge. MIT Press, Cambridge/MA, London, 2014, S. 31-76. formulated by classical symbol theory for a convincing theory of cognition in modern connectionism. In order to be an adequate alternative theory of cognition, Smolensky's Subsymbolic Paradigm would have to explain the existence of systematicity or systematic relations in language cognition without the assumption that cognitive processes are causally sensitive to the classical constituent structure of mental representations. The subsymbolic paradigm, or connectionism in general, would thus have to explain the existence of systematicity and compositionality without relying on the mere implementation of a classical cognitive architecture. This challenge implies a dilemma: If the Subsymbolic Paradigm could contribute nothing to the systematicity and compositionality of mental representations, it would be insufficient as a basis for an alternative theory of cognition. However, if the Subsymbolic Paradigm's contribution to systematicity requires mental processes grounded in the classical constituent structure of mental representations, the theory of cognition it develops would be, at best, an implementation architecture of the classical model of symbol theory and thus not a genuine alternative (connectionist) theory of cognition.J.A. Fodor, B. McLaughlin: Connectionism and the problem of systematicity: Why Smolensky's solution doesn't work. Cognition. Band 35, 1990, S. 183-184. The classical model of symbolism is characterized by (1) a combinatorial syntax and semantics of mental representations and (2) mental operations as structure-sensitive processes, based on the fundamental principle of syntactic and semantic constituent structure of mental representations as used in Fodor's "Language of Thought (LOT)".J.A. Fodor: The language of thought. Harvester Press, Sussex, 1976, ISBN 0-85527-309-7.J.A. Fodor: LOT 2: The language of thought revisited. Clarendon Press, Oxford, 2008, ISBN 0-19-954877-3. This can be used to explain the following closely related properties of human cognition, namely its (1) productivity, (2) systematicity, (3) compositionality, and (4) inferential coherence.J.A. Fodor, Z.W. Pylyshyn (1988), S. 33-48.

This challenge has been met in modern connectionism, for example, not only by Smolensky's "Integrated Connectionist/Symbolic (ICS) Cognitive Architecture",P. Smolenky: Reply: Constituent structure and explanation in an integrated connectionist / symbolic cognitive architecture. In: C. MacDonald, G. MacDonald (Hrsg.): Connectionism: Debates on psychological explanation. Blackwell Publishers. Oxford/UK, Cambridge/MA. Vol. 2, 1995, S. 224, 236-239, 242-244, 250-252, 282.P. Smolensky, G. Legendre: The Harmonic Mind: From Neural Computation to Optimality-Theoretic Grammar. Vol. 1: Cognitive Architecture. A Bradford Book, The MIT Press, Cambridge, London, 2006a, ISBN 0-262-19526-7, S. 65-67, 69-71, 74-75, 154-155, 159-202, 209-210, 235-267, 271-342, 513. but also by Werning and Maye's "Oscillatory Networks".M. Werning: Neuronal synchronization, covariation, and compositional representation. In: M. Werning, E. Machery, G. Schurz (Hrsg.): The compositionality of meaning and content. Vol. II: Applications to linguistics, psychology and neuroscience. Ontos Verlag, 2005, S. 283-312.M. Werning: Non-symbolic compositional representation and its neuronal foundation: towards an emulative semantics. In: M. Werning, W. Hinzen, E. Machery (Hrsg.): The Oxford Handbook of Compositionality. Oxford University Press, 2012, S. 633-654.A. Maye und M. Werning: Neuronal synchronization: from dynamics feature binding to compositional representations. Chaos and Complexity Letters, Band 2, S. 315-325. An overview of this is given for example by Bechtel & Abrahamsen,Bechtel, W., Abrahamsen, A.A. Connectionism and the Mind: Parallel Processing, Dynamics, and Evolution in Networks. 2nd Edition. Blackwell Publishers, Oxford. 2002 MarcusG.F. Marcus: The algebraic mind. Integrating connectionism and cognitive science. Bradford Book, The MIT Press, Cambridge, 2001, ISBN 0-262-13379-2. and Maurer.H. Maurer: Cognitive science: Integrative synchronization mechanisms in cognitive neuroarchitectures of the modern connectionism. CRC Press, Boca Raton/FL, 2021, ISBN 978-1-351-04352-6. https://doi.org/10.1201/9781351043526

Recently, Heng Zhang and his colleagues have demonstrated that mainstream knowledge representation formalisms are, in fact, recursively isomorphic, provided they possess equivalent expressive power.{{Cite journal |last=Zhang |first=Heng |last2=Jiang |first2=Guifei |last3=Quan |first3=Donghui |date=2025-04-11 |title=A Theory of Formalisms for Representing Knowledge |url=https://ojs.aaai.org/index.php/AAAI/article/view/33674 |journal=Proceedings of the AAAI Conference on Artificial Intelligence |language=en |volume=39 |issue=14 |pages=15257–15264 |doi=10.1609/aaai.v39i14.33674 |issn=2374-3468}} This finding implies that there is no fundamental distinction between using symbolic or connectionist knowledge representation formalisms for the realization of artificial general intelligence (AGI). Moreover, the existence of recursive isomorphisms suggests that different technical approaches can draw insights from one another.

{{div col|colwidth=20em}}

• Associationism
• Artificial intelligence
• Behaviorism
• Catastrophic interference
• Calculus of relations
• Cybernetics
• Deep learning
• Eliminative materialism
• Feature integration theory
• Genetic algorithm
• Harmonic grammar
• Machine learning
• Pandemonium architecture
• Self-organizing map

{{div col end}}

{{Reflist}}

• Feldman, Jerome and Ballard, Dana. Connectionist models and their properties(1982). Cognitive Science. V6, Iissue 3 , pp205-254.
• Rumelhart, D.E., J.L. McClelland and the PDP Research Group (1986). Parallel Distributed Processing: Explorations in the Microstructure of Cognition. Volume 1: Foundations, Cambridge, Massachusetts: MIT Press, {{ISBN|978-0-262-68053-0}}
• McClelland, J.L., D.E. Rumelhart and the PDP Research Group (1986). Parallel Distributed Processing: Explorations in the Microstructure of Cognition. Volume 2: Psychological and Biological Models, Cambridge, Massachusetts: MIT Press, {{ISBN|978-0-262-63110-5}}
• Pinker, Steven and Mehler, Jacques (1988). Connections and Symbols, Cambridge MA: MIT Press, {{ISBN|978-0-262-66064-8}}
• Jeffrey L. Elman, Elizabeth A. Bates, Mark H. Johnson, Annette Karmiloff-Smith, Domenico Parisi, Kim Plunkett (1996). Rethinking Innateness: A connectionist perspective on development, Cambridge MA: MIT Press, {{ISBN|978-0-262-55030-7}}
• Marcus, Gary F. (2001). The Algebraic Mind: Integrating Connectionism and Cognitive Science (Learning, Development, and Conceptual Change), Cambridge, Massachusetts: MIT Press, {{ISBN|978-0-262-63268-3}}
• {{cite journal|title=A Brief History of Connectionism|author=David A. Medler|url=http://www.blutner.de/NeuralNets/Texts/Medler.pdf|year= 1998|volume=1|pages=61–101|journal=Neural Computing Surveys}}
• Maurer, Harald (2021). Cognitive Science: Integrative Synchronization Mechanisms in Cognitive Neuroarchitectures of the Modern Connectionism, Boca Raton/FL: CRC Press, https://doi.org/10.1201/9781351043526, {{ISBN|978-1-351-04352-6}}

{{Spoken Wikipedia|En-Connectionism.ogg|date=2011-11-26}}

• [http://philosophy.uwaterloo.ca/MindDict/connectionism.html Dictionary of Philosophy of Mind entry on connectionism]
• {{cite SEP |url-id=connectionism |title=Connectionism |last=Garson |first=James |author-link=James Garson}}
• [http://srsc.ulb.ac.be/pdp/iac/IAC.html A demonstration of Interactive Activation and Competition Networks] {{Webarchive|url=https://web.archive.org/web/20150703142148/http://srsc.ulb.ac.be/pdp/iac/IAC.html |date=2015-07-03 }}
• {{cite IEP |url-id=connect |title=Connectionism}}
• [https://sapienlabs.org/the-crisis-of-computational-neuroscience/ Critique of connectionism]

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