Cognitive Science Mathematics for Cognitive Science --Manuscript Draft-- Manuscript Number: Full Title: Mathematics for Cognitive Science Article Type: Letter to the Editor Keywords: Category Theory; Cognition; Functorial Semantics; Representation Corresponding Author: Venkata Rayudu Posina, B.Tech Independent Scientist Hyderabad, Telangana INDIA Corresponding Author Secondary Information: Corresponding Author's Institution: Independent Scientist Corresponding Author's Secondary Institution: First Author: Venkata Rayudu Posina, B.Tech First Author Secondary Information: Order of Authors: Venkata Rayudu Posina, B.Tech Order of Authors Secondary Information: Abstract: That the state-of-affairs of cognitive science is not good is brought into figural salience in "What happened to cognitive science?" (Núñez et al., 2019). We extend their objective description of 'what's wrong' to a prescription of 'how to correct'. Cognitive science, in its quest to elucidate 'how we know', embraces a long list of subjects, while ignoring Mathematics (Fig. 1a, Núñez et al., 2019). Mathematics is known for making the unknown to be known (cf. solving for unknowns). This acknowledgement naturally raises the question: does mathematical knowing inform knowing in general? Here we show that drawing parallels to mathematical knowing can facilitate the advancement of cognitive science (Lawvere, 1994). Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation To, Professor Richard P. Cooper Executive Editor Cognitive Science Dear Professor Cooper, I am herewith submitting my manuscript "Mathematics for Cognitive Science" to be considered for publication as a 'Letter To The Editor' in your journal Cognitive Science. My Letter was motivated by the unsatisfactory contemporary state-of-affairs of cognitive science highlighted in the recent study: Núñez, R. et al. What happened to cognitive science? Nat. Hum. Behav. 3, 782-791 (2019). Based on the kinship between mathematics [in particular and science in general] and cognition (cf. cognition is science writ small; Daedalus 135: 86, 2006), I make a case for mathematical abstraction as a pathway for the advancement of cognitive science. It may be noted here that the parallels between mathematical knowing and knowing in general, which I bring into focus, provide a means "to connect cognitive science theories to computational foundations" (Nat. Hum. Behav. 3: 782, 2019). Cover Letter I outline the mathematics of calculating representation(s)-a fundamental notion in cognitive science. I felt that it is important to explicitly state that the notion of 'representation' figuring in the foundational tenet-"cognition is computation of representations" (Nat. Hum. Behav. 3, 782, 2019)-of cognitive science is indispensable in theorizing about cognition. However, we may need to replace computation with calculation [of representations] so as to move past the computer metaphor and bring the insights of functorial semantics to bear on cognitive science. Representation (or model), according to functorial semantics, is an interpretation of a theory into a background category (Reprints in Theory and Applications of Categories 5, 8-11, 2004). The category of all models of a theory T in a background B is a functor category B^T of all functorial interpretations T --> B of the theory T in the background B. Changing theories (T1 --> T2) induces contravariant changes in representations (B^T2 --> B^T1), while changing backgrounds (B1 --> B2) induces covariant changes in representations (B1^T --> B2^T). I discuss these mathematical insights into representation in a manner readily accessible to the multidisciplinary audience of your journal. This discussion can help insure against: throwing the baby (representation) with bathwater (hexagon; Fig 1a in Nat. Hum. Behav. 3, 782, 2019). In closing, my manuscript brings out the reach of functorial semantics [of calculating representations] into sharper focus so as to facilitate ready recognition of the relevance of functorial semantics for the development of cognitive science. If I may, the following may be considered for reviewing my manuscript since they are experts on cognitive science and category theory. Professor Michael A. Arbib (arbib@usc.edu) Professor Andrée C. Ehresmann (ehres@u-picardie.fr) Professor F. William Lawvere (wlawvere@buffalo.edu) Professor Giuseppe Longo (Giuseppe.Longo@ens.fr) I earnestly hope that you will find my manuscript suitable for publication in your journal Cognitive Science. I sincerely thank you for your kind consideration of my manuscript and I eagerly look forward to hearing from you. Thanking you, Yours truly, Venkata Rayudu Posina Affiliation: Independent Scientist Google Scholar Profile: https://scholar.google.co.in/citations?user=cnMxV9MAAAAJ&hl=en&oi=ao ORCID ID: http://orcid.org/0000-0002-3040-9224 Address for Correspondence: Venkata Rayudu Posina, 101B2 Swathi Heights, A. S. Rao Nagar, Hyderabad 500062, Telangana, India Email: posinavrayudu@gmail.com, Tel: +91-963-222-4686 1 Title: Mathematics for Cognitive Science 1 Article Type: Letter To The Editor 2 Word Count: 1586 3 4 Author: Venkata Rayudu Posina 5 6 Affiliation: 7 Independent Scientist 8 Google Scholar Profile: 9 https://scholar.google.co.in/citations?user=cnMxV9MAAAAJ&hl=en&oi=ao 10 ORCID ID: http://orcid.org/0000-0002-3040-9224 11 Address for Correspondence: 12 Venkata Rayudu Posina, 101B2 Swathi Heights, A. S. Rao Nagar, Hyderabad 500062, 13 Telangana, India 14 Email: posinavrayudu@gmail.com, Tel: +91-963-222-4686 15 Manuscript 2 That the state-of-affairs of cognitive science is not good is brought into figural salience in "What 16 happened to cognitive science?" (Núñez et al., 2019). We extend their objective description of 17 'what's wrong' to a prescription of 'how to correct'. Cognitive science, in its quest to elucidate 18 'how we know', embraces a long list of subjects, while ignoring Mathematics (Fig. 1a, Núñez et 19 al., 2019). Mathematics is known for making the unknown to be known (cf. solving for 20 unknowns). This acknowledgement naturally raises the question: does mathematical knowing 21 inform knowing in general? Here we show that drawing parallels to mathematical knowing can 22 facilitate the advancement of cognitive science (Lawvere, 1994). 23 What is cognition? One scientific approach is to answer: what is cognition good for? 24 Science-reconstructing reality from appearances-is the signature product of human cognition. 25 Since a product retains traces of the process that gave rise to the product, a declarative 26 understanding [but not merely procedural knowledge] of the scientific reconstruction of reality 27 from planned perception constitutes the foundations of the science of cognition. The 28 mathematical basis of scientific reconstruction across disciplines is: comparison of the observed 29 variation with constancy (Lawvere & Rosebrugh, 2003, pp. 125-126, 148-152). Reality is 30 inferred from appearances by establishing an isomorphism between perceived (generalized 31 points) and actual (points). The relationship between points and generalized points involved in 32 scientific reconstruction is analogous to the relationship between stimuli and percepts: 33 reconstruction of the causes that gave rise to sensation (Albright, 2015). The process of going 34 from stimuli to percepts involves a two-step process of sensation followed by interpretation 35 (Croner & Albright, 1999), which is reminiscent of the double-dualization involved in going 36 from points to generalized points. Therefore, we can consider generalized points of scientific 37 3 reconstruction of reality as an abstraction of conscious percepts of ordinary cognition (Lawvere, 38 2004a). 39 With conscious experiences as representations (Chalmers, 2006) in the foundational tenet of 40 "cognition is computation of representations" (Núñez et al., 2019, p. 782), a comprehensive 41 theory of cognition naturally subsumes conscious experiences. Consciousness, the totality of 42 conscious experiences (Koch, 2018), can be construed as a mathematical category of conscious 43 experiences along with their transformations (Lawvere & Schanuel, 2009, p. 21). Beginning 44 with a theory of conscious experiences, say, 'conscious experience is an interpretation of 45 sensation' (alluded to earlier) we obtain a category of two-sequential functions as the category of 46 models of conscious experiences (Posina, Ghista & Roy, 2017). The theory 'conscious 47 experience is an interpretation of sensation' is also a mathematical category consisting of three 48 component structural objects (stimuli, neural codes, conscious percepts) and two component 49 structural maps (sensation, interpretation; Lawvere & Schanuel, 2009, pp. 149-150). More 50 broadly, for each abstract theory of conscious experiences we obtain a corresponding category of 51 models. With every theory of the category of conscious experiences construed as a graph or a 52 small category (Lawvere, 2016; Lawvere & Schanuel, 2009, p. 149, 199-203), we obtain a 53 category of models, which is a functor category (a category whose objects are functorial 54 interpretations of a theory category into a background category). The category of all functor 55 categories subsumes every possible value of every property of each object of the category of 56 conscious experiences, i.e. constitutes an adequate characterization of consciousness (Lawvere & 57 Schanuel, 2009, pp. 370-371). This category of all functor categories is the space of all possible 58 mathematical answers to the question: What is consciousness? (This is analogous to the scenario 59 where the set N = {0, 1, 2...} of natural numbers can be thought of as the set of all answers to 60 4 the question: What is the size of X, where X is any set?) Furthermore, since representations of a 61 given category of particulars are functorial interpretations of a theory into a background 62 category, representations of particulars vary contravariantly with changes in theory and 63 covariantly with background changes (Lawvere, 1994, p. 46; Lawvere & Rosebrugh, 2003, pp. 64 120-122). In cognitive terminology, conscious experiences (representations) vary as a function 65 of both mental concepts (theories) and intuitions (backgrounds); functorial semantics of 66 mathematical knowing spells out how, i.e., provides a mathematical account of calculating 67 representations of mathematical objects (Posina, Ghista & Roy, 2017; Lawvere, 2004b). 68 Returning to reality, if reality were the category of sets, then three-valued properties guarantee 69 the recovery of "reality" from perception (Lawvere, 2004a). Of course, reality is not a set; it's 70 much more structured than the structure-less sets. For instance, reality is self-representing since 71 its representations: collective science and individual perception are parts-reflective parts-of 72 the reality (Lawvere & Schanuel, 2009, pp. 84-85). So, reality can be modeled as the self-73 representing mathematical category of categories, wherein a mathematical category (e.g. 74 reflexive graphs) is represented in its discrete-and-constant subcategory (Lawvere, 2004b, p. 12; 75 see also Lawvere, 2003, p. 215, 217). This correspondence suggests calculation of the basic 76 types-of-knowing (objects analogous to the three-element set in the category of sets) in 77 mathematical categories that are reflective of reality such as the category of categories. In doing 78 so, we can address two related foundational questions: 79 1. What is the nature of reconstructable reality (the structure of reality that can be reconstructed 80 from appearances)? 81 5 2. What is the nature of revealing appearances (the structure of appearances that is conducive for 82 reconstructing reality)? 83 More fundamentally, reality is often analyzed into the categories of Being and Becoming, 84 which is not satisfactory since reality consists, as noted above, of parts-individual cognition 85 and collective science-reflective of the reality. Hence we need a mathematical category of 86 Reflecting in addition to the categories of Being and Becoming in order to bridge the two 87 categories of objective reality on the one hand and its subjective reflections on the other. A 88 mathematical category of Reflecting can be objectified along the lines of the objectification of 89 Being and Becoming as mathematical categories of reflexive graphs (exemplifying unity) and 90 dynamical systems (change), respectively (Lawvere, 1991, 1992, 2007). The mathematical 91 category of Reflecting makes room for the basis of science-human cognition-in the scientific 92 representation of reality. 93 In closing, it must be noted that the foundational tenet-"cognition is computation of 94 representations" (Núñez et al., 2019, p. 782)-is still viable provided we shift our focus from 95 'computation' to 'representation', move on from the computer metaphor along with its attendant 96 conceptual baggage (cf. hardware vs. software), and build on the definitive mathematical 97 understanding of calculating representations (Lawvere, 2004b). Furthermore, however exciting 98 they may be, we also need to shift our focus away from the excitement of winning games such as 99 Go, especially because of the anti-scientific faith that they demand (Editorial, 2016) and start 100 focusing on the development of theory, as the Cognitive Science Society correctly decided: 101 "greater effort must be made to connect cognitive science theories to computational foundations" 102 (Núñez et al., 2019, p. 789). Here 'computational foundations' can be understood as 103 'foundations of calculating representations', i.e. functorial semantics (Lawvere, 2004b). 104 6 Summing it all, we suggest that functorial semantics is to cognition what calculus is to physics, 105 i.e. the mathematics needed for the development of cognitive science (Lawvere, 1994, p. 43, 55; 106 Lawvere, 1999, p. 412). 107 7 References 108 Albright, T.D. (2015). Perceiving. Daedalus, 144, 22-41. 109 Chalmers, D.J. (2006). The representational character of experience. In B. 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