Main Article Content
The interaction between syntax (formal language) and its semantics (meanings of language) is one which has been well studied in categorical logic. The results of this particular study are employed to understand how the brain is able to create meanings. To emphasize the toy character of the proposed model, we prefer to speak of the homunculus brain rather than the brain per se. The homunculus brain consists of neurons, each of which is modeled by a category, and axons between neurons, which are modeled by functors between the corresponding neuron-categories. Each neuron (category) has its own program enabling its working, i.e. a theory of this neuron. In analogy to what is known from categorical logic, we postulate the existence of a pair of adjoint functors, called Lang and Syn, from a category, now called BRAIN, of categories, to a category, now called MIND, of theories. Our homunculus is a kind of “mathematical robot”, the neuronal architecture of which is not important. Its only aim is to provide us with the opportunity to study how such a simple brain-like structure could “create meanings” and perform abstraction operations out of its purely syntactic program. The pair of adjoint functors Lang and Syn model the mutual dependencies between the syntactical structure of a given theory of MIND and the internal logic of its semantics given by a category of BRAIN. In this way, a formal language (syntax) and its meanings (semantics) are interwoven with each other in a manner corresponding to the adjointness of the functors Lang and Syn. Higher cognitive functions of abstraction and realization of concepts are also modelled by a corresponding pair of adjoint functors. The categories BRAIN and MIND interact with each other with their entire structures and, at the same time, these very structures are shaped by this interaction.
Adámek, J., Lawvere, F. and Rosický, J., 2003. On the duality between varieties and algebraic theories. Algebra Universalis [Online], 49(1), pp.35–49. Available at: https://doi.org/10.1007/s000120300002 [visited on 3 November 2020].
Awodey, S., 2021. Sheaf Representations and Duality in Logic. In: C. Casadio and P.J. Scott, eds. Joachim Lambek: The Interplay of Mathematics, Logic, and Linguistics. Springer International Publishing.
Awodey, S. and Forssell, H., 2013. First-order logical duality. Annals of Pure and Applied Logic [Online], 164(3), pp.319–348. Available at: https://doi.org/10.1016/j.apal.2012.10.016 [visited on 3 November 2020].
Batterman, R., 2016. Intertheory Relations in Physics. In: E.N. Zalta, ed. The Stanford Encyclopedia of Philosophy (Fall 2016 Edition) [Online]. Stanford: Metaphysics Research Lab, Stanford University. Available at: <https://plato.stanford.edu/archives/fall2016/entries/physics-interrelate/> [visited on 3 November 2020].
Borceux, F., 1994. Handbook of Categorical Algebra. 3: Categories of Sheaves, Encyclopedia of Mathematics and its Applications vol. 52. Cambridge: Cambridge University Press.
Ehresmann, A., 2017. Applications of Categories to Biology and Cognition. In: E. Landry, ed. Categories for the Working Philosopher. Oxford, New York: Oxford University Press, pp.358–380.
Ellerman, D., 2015. On Adjoint and Brain Functors. arXiv:1508.04036 [math] [Online]. Available at: <http://arxiv.org/abs/1508.04036> [visited on 3 November 2020].
Fu, Y., 2019. Category Theory, Topos and Logic: A Quick Glance [Online]. Available at: [visited on 28 February 2019].
Gómez, J. and Sanz, R., 2009. Modeling Cognitive Systems with Category Theory: Towards Rigor in Cognitive Sciences [Online]. Autonomous Systems Laboratory, Universidad Politécnica de Madrid. Available at: <http://www.aslab.org/documents/controlled/ASLAB-A-2009-014.pdf> [visited on 4 November 2020].
Halvorson, H., 2016. Scientific Theories. In: P. Humphreys, ed. Oxford Handbook of Philosophy of Science [Online]. Vol. 1. Oxford: Oxford University Press, pp.402–429. Available at: https://doi.org/10.1093/oxfordhb/9780199368815.013.33 [visited on 3 November 2020].
Halvorson, H. and Tsementzis, D., 2017. Categories of Scientific Theories. In: E. Landry, ed. Categories for the Working Philosopher. Oxford, New York: Oxford University Press, pp.402–429.
Healy, M.J. and Caudell, T.P., 2006. Ontologies and Worlds in Category Theory: Implications for Neural Systems. Axiomathes [Online], 16(1), pp.165–214. Available at: https://doi.org/10.1007/s10516-005-5474-1 [visited on 3 November 2020].
Hodges, W., 2018. Tarski’s Truth Definitions. In: E.N. Zalta, ed. The Stanford Encyclopedia of Philosophy (Fall 20218 Edition) [Online]. Stanford: Metaphysics Research Lab, Stanford University. Available at: <https://plato.stanford.edu/%20archives/fall2018/entries/tarski-truth/> [visited on 3 November 2020].
Johnstone, P.T., 1982. Stone Spaces, Cambridge studies in advanced mathematics 3. Cambridge: Cambridge University Press.
Koch, C., 1997. Computation and the single neuron. Nature [Online], 385(6613), pp.207–210. Available at: https://doi.org/10.1038/385207a0 [visited on 3 November 2020].
Leinster, T., 2014. Basic Category Theory, Cambridge Studies in Advanced Mathematics 143. Cambridge: Cambridge University Press.
Mac Lane, S. and Moerdijk, I., 1992. Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Universitext. New York [etc.]: Springer-Verlag.
Makkai, M., 1987. Stone duality for first order logic. Advances in Mathematics, 65, pp.97–170.
McCulloch, W.S. and Pitts, W., 1943. A logical calculus of the ideas immanent in nervous activity. The Bulletin of Mathematical Biophysics [Online], 5(4), pp.115–133. Available at: https://doi.org/10.1007/BF02478259 [visited on 3 November 2020].
Mizraji, E. and Lin, J., 2011. Logic in a Dynamic Brain. Bulletin of Mathematical Biology [Online], 73(2), pp.373–397. Available at: https://doi.org/10.1007/s11538-010-9561-0 [visited on 3 November 2020].
nLab, 2011. Coherent Functor. Available at: <https://ncatlab.org/nlab/show/coherent+functor> [visited on 3 November 2020].
nLab, 2017. Internal Logic [Online]. Available at: <https://ncatlab.org/nlab/show/internal+logic> [visited on 3 November 2020].
nLab, 2020a. Doctrine [Online]. Available at: <https://ncatlab.org/nlab/show/coherent+functor> [visited on 3 November 2020].
nLab, 2020b. Substitution [Online]. Available at: <https://ncatlab.org/nlab/show/substitution> [visited on 3 November 2020].
nLab, 2020c. Syntactic Category [Online]. Available at: <https://ncatlab.org/nlab/show/syntactic+category> [visited on 3 November 2020].
Rosaler, J., 2018. Inter-Theory Relations in Physics: Case Studies from Quantum Mechanics and Quantum Field Theory. arXiv:1802.09350 [quant-ph] [Online]. Available at: <http://arxiv.org/abs/1802.09350> [visited on 3 November 2020].
Rosen, R., 1985. Organisms as Causal Systems Which Are Not Mechanisms: An Essay into the Nature of Complexity. Theoretical Biology and Complexity: Three Essays on the Natural Philosophy of Complex Systems [Online]. New York [etc.]: Academic Press, pp.165–203. Available at: https://doi.org/10.1016/B978-0-12-597280-2.50008-8 [visited on 3 November 2020].
Sher, G., 1999. What is Tarski’s Theory of Truth? Topoi [Online], 18(2), pp.149–166. Available at: https://doi.org/10.1023/A:1006246322119 [visited on 3 November 2020].
Simmons, H., 2011. An Introduction to Category Theory. Cambridge [etc.]: Cambridge University Press.
Tarski, A., 1933. Pojęcie prawdy w językach nauk dedukcyjnych, Prace Towarzystwa Naukowego Warszawskiego. Wydział III: Nauk Matematyczno-Fizycznych 34. Warszawa: nakładem Towarzystwa Naukowego Warszawskiego, z zasiłku Ministerstwa Wyznań Religijnych i Oświecenia Publicznego.
Tsuchiya, N., Taguchi, S. and Saigo, H., 2016. Using category theory to assess the relationship between consciousness and integrated information theory. Neuroscience Research [Online], 107, pp.1–7. Available at: https://doi.org/10.1016/j.neures.2015.12.007 [visited on 3 November 2020].
Woszczyna, A. and Heller, M., 1990. Is a horizon-free cosmology possible? General Relativity and Gravitation [Online], 22(12), pp.1367–1386. Available at: https://doi.org/10.1007/BF00756836 [visited on 3 November 2020].