Title: Buddhist Thought on Emptiness and Category Theory 1 Authors: Venkata Rayudu Posina and Sisir Roy 2 ORCID IDs: 0000-0002-3040-9224 and 0000-0002-5328-2713 3 Manuscript type: Article (original research paper) 4 Running title: Buddhist Thought and Category Theory 5 Keywords: Cantor; Contradiction; Emptiness; Essence; Figure; Functor; Nagarjuna; Natural 6 Transformation; Object; Property; Reality; Relation; Set; Shape; Structure; Structure-respecting 7 Morphism; Truth Value Object; Yoneda Lemma; Zero. 8 Word count: 8431 9 10 Affiliation: Consciousness Studies Programme, National Institute of Advanced Studies 11 Address for correspondence: Professor Sisir Roy, Consciousness Studies Programme, National 12 Institute of Advanced Studies, Bengaluru 560012, Karnataka, India 13 Email: sisir.sisirroy@gmail.com; Tel: +91-80-22185000; Fax: +91-80-22185028 14 Acknowledgment: One of the Authors (SR) is indebted to Homi Bhabha Trust, Mumbai for 15 financial support to perform this work. VRP is grateful for the NIAS-Mani Bhaumik and NIAS-16 Consciousness Studies Programme Fellowships. 17 2 Abstract 18 Notions such as Sunyata, Catuskoti, and Indra's Net, which figure prominently in Buddhist 19 philosophy, are difficult to readily accommodate within our ordinary thinking about everyday 20 objects. Famous Buddhist scholar Nagarjuna considered two levels of reality: one called 21 conventional reality and the other ultimate reality. Within this framework, Sunyata refers to the 22 claim that at the ultimate level objects are devoid of essence or ―intrinsic properties‖, but are 23 interdependent by virtue of their relations to other objects. Catuskoti refers to the claim that four 24 truth values, along with contradiction, are admissible in reasoning. Indra's Net refers to the 25 claim that every part of a whole is reflective of the whole. Here we present category theoretic 26 constructions which are reminiscent of these Buddhist concepts. The universal mapping 27 property definition of mathematical objects, wherein objects of a universe of discourse are 28 defined not in terms of their content, but in terms of their relations to all objects of the universe is 29 reminiscent of Sunyata. The objective logic of perception, with perception modeled as [a 30 category of] two sequential processes (sensation followed by interpretation), and with its truth 31 value object of four truth values, is reminiscent of the Buddhist logic of Catuskoti. The category 32 of categories, wherein every category has a subcategory of sets with zero structure within which 33 every category can be modeled, is reminiscent of Indra's Net. Our thorough elaboration of the 34 parallels between Buddhist philosophy and category theory can facilitate better understanding of 35 Buddhist philosophy, and bring out the broader philosophical import of category theory beyond 36 mathematics. 37 3 Introduction 38 Buddhist philosophy, especially Nagarjuna's Middle Way (Garfield, 1995; Siderits and Katsura, 39 2013), is intellectually demanding (Priest, 2013). The sources of the difficulties are many. First 40 it argues for two realities: conventional and ultimate (Priest, 2010). Next, ultimate reality is 41 characterized by Sunyata or emptiness, which is understood as the absence of a fundamental 42 essence underlying reality (Priest, 2009). Equally importantly, contradictions are readily 43 deployed, especially in Catuskoti, as part of the characterization of reality (Deguchi, Garfield, 44 and Priest, 2008; Priest, 2014). Lastly, reality is depicted as Indra's Net-a whole, whose parts 45 are reflective of the whole (Priest, 2015). The ideas of relational existence, admission of 46 contradictions, and parts reflecting the whole are seemingly incompatible with our everyday 47 experiences and the attendant conceptual reasoning used to make sense of reality. However, 48 notions analogous to these ancient Buddhist ideas are also encountered in the course of the 49 modern mathematical conceptualization of reality. These parallels may be, in large part, due to 50  experience' and  reason' that are treated as the final authority in both mathematical sciences and 51 Buddhist philosophy. Here, we highlight the similarities between Buddhist philosophy and 52 mathematical philosophy, especially category theory (Lawvere and Schanuel, 2009). The 53 resultant cross-cultural philosophy can facilitate a proper understanding of reality-a noble goal 54 to which both Buddhist philosophy and mathematical practice are unequivocally committed. 55 56 Two Realities 57 There are, according to Buddhist thought, two realities: the conventional reality of our everyday 58 experiences and the ultimate reality (Priest, 2010; Priest and Garfield, 2003). In our 59 4 conventional reality, things appear to have intrinsic essences. It is sensible, at the level of 60 conventional reality, to speak of essences of objects, but at the level of ultimate reality there are 61 no essences, and everything exists but only relationally. There is an analogous situation in 62 mathematics. On one hand, mathematical objects can be characterized in terms of their relations 63 to all objects, in which case the nature of an object is determined by the nature of its relationship 64 to all objects. In a sense, there is nothing inside the object; an object is what it is by virtue of its 65 relations to all objects. The objects of mathematics are, as Resnik (1981, p. 530) notes, 66 ―positions in structures‖, which is in accord with the Buddhist understanding of things as ―loci in 67 a field of relations‖ (Priest, 2009, p. 468). However, there is another level of mathematical 68 reality, wherein we can speak of essences of objects (e.g. theories of objects; Lawvere and 69 Rosebrugh, 2003, pp. 154-155). For example, one can characterize a set as a collection of 70 elements or ―sum‖ of basic-shaped figures (1-shaped figures, where 1 = {•}), with basic shapes 71 understood as essences (Lawvere, 1972, p. 135; Lawvere and Schanuel, 2009, p. 245; Reyes, 72 Reyes, and Zolfaghari, 2004, p. 30). Similarly, every graph is made up of figures of two basic-73 shapes (arrowand dot-shaped figures; Lawvere and Schanuel, 2009, p. 150, 215). This 74 characterization of an object in terms of its contents i.e. basic shapes or essences (Lawvere, 75 2003, pp. 217-219; Lawvere, 2004, pp. 11-13) can be contrasted with the relational 76 characterization, wherein each and every object of a universe of discourse (a mathematical 77 category; Lawvere and Schanuel, 2009, p. 17) is characterized in terms of its relationship to all 78 objects of the universe or category (see Appendix A1). The relational nature of mathematical 79 objects, as elaborated below, is reminiscent of the Buddhist notion of emptiness-an assertion 80 that objects are what they are not by virtue of some intrinsic essences but by virtue of their 81 mutual relationships. 82 5 83 Emptiness 84 According to Buddhist philosophy, everything is empty and the totality of empty things is empty. 85 Here, emptiness is understood as the absence of essences. Things, in the ultimate analysis, are 86 what they are and behave the way they do not because of [some] essences inherent in them, but 87 by virtue of their mutual relationships (Priest, 2009). This idea of relational existence has 88 parallels in mathematical practice. Mathematical objects of a given mathematical category (e.g. 89 category of sets) are what they are not by virtue of their intrinsic essences but by virtue of their 90 relations to all objects of the category. For example, a single-element set is a set to which there 91 exists exactly one function from every set (Lawvere and Schanuel, 2009, p. 213, 225). Note that 92 the singleton set is characterized not in terms of what it contains (a single element), but in terms 93 of how it relates to all sets of the category of sets. In a similar vein, the truth value set Ω = 94 {false, true} is defined in terms of its relation to all sets of the category of sets. The truth value 95 set, instead of being defined as a set of two elements  false' and  true', is defined as a set Ω such 96 that functions from any set X to the set Ω are in one-to-one correspondence with the parts of X 97 (ibid, pp. 339-344). To give one more example, product of two sets is defined not by specifying 98 the contents of the product set (pairs of elements), but by characterizing its relationship to all 99 sets. More explicitly, the product of two sets A and B is a set A × B along with two functions 100 (projections to the factors) pA: A × B → A, pB: A × B → B such that for every set Q and any pair 101 of functions qA: Q → A, qB: Q → B, there is exactly one function q: Q → A × B satisfying both 102 the equations: qA = pA ◦ q and qB = pB ◦ q, where  ◦' denotes composition of functions (ibid, pp. 103 339-344). The universal mapping property definition of mathematical constructions brought to 104 sharp focus the relational nature of mathematical objects. It conclusively established that ―the 105 6 substance of mathematics resides not in Substance (as it is made to seem when ∈ [membership] 106 is the irreducible predicate, with the accompanying necessity of defining all concepts in terms of 107 a rigid elementhood relation) but in Form (as is clear when the guiding notion is isomorphism-108 invariant structure, as defined, for example, by universal mapping properties)‖ (Lawvere, 2005, 109 p. 7). More broadly, Yoneda lemma (Lawvere and Rosebrugh, 2003, pp. 249-250; Appendix 110 A1), according to which a mathematical object of a given universe of discourse (i.e. category) is 111 completely characterized by the totality of its relations to all objects of the universe (category), is 112 an unequivocal assertion of the relational nature of mathematical objects. Yoneda lemma, as 113 pointed out by Barry Mazur, establishes that ―an object X of a category C is determined by the 114 network of relationships that the object X has with all the other objects in C‖ (Mazur, 2008). 115 Thus the Buddhist idea of emptiness or relational existence finds resonance in mathematical 116 practice, especially in terms of universal mapping properties and the Yoneda lemma. 117 However, note that according to the Buddhist doctrine of emptiness, not only is 118 everything empty, but the totality of empty things is also empty (Priest, 2009). In other words, 119 even the notion of relational existence is empty i.e., emptiness is not the essence of existence; 120 emptiness is also empty. This idea of emptiness being empty is much more challenging to 121 comprehend. When we say that objects are empty, we are saying that objects are mere locations 122 in a network of relations. But when we say that the totality of empty things is empty, we are 123 asserting that the existence of totality is also relational just like that of the objects in the totality. 124 What is not immediately clear is how are we to think of relations especially when all we have is 125 the totality i.e., one object. Within mathematics, note that the totality of all objects (along with 126 their mutual relations) forms a category. More importantly, categories are objects in the category 127 of categories (Lawvere, 1966), and hence the totality of objects i.e. category is also empty or 128 7 relational as much as the objects of a category. Thus the idea of Sunyata (everything is empty) 129 resonates with the relational nature of objects and of the totality of objects (within the 130 mathematical framework of the category of categories). 131 Equally importantly, Nagarjuna's Middle Way, having gone to great lengths to 132 distinguish two realities (conventional essences vs. ultimate emptiness) identifies the two: ―there 133 is no distinction between conventional reality and ultimate reality‖ (Deguchi, Garfield, and 134 Priest, 2008, p. 399). Contradictions (such as these) within Buddhist philosophy, on a superficial 135 reading, are diagnostic of irrational mysticism. However, as we point out in the following, 136 contradictions also figure prominently in the foundations of mathematical modeling of reality. In 137 light of these parallels,  contradiction' may be intrinsic to the nature of reality, which is the 138 common subject of both Buddhist and mathematical investigations, and not a sign of faulty 139 Buddhist reasoning. 140 141 Contradiction 142 Within the Buddhist philosophical discourse, one often encounters contradictions and these 143 contradictions are treated as meaningful (Deguchi, Garfield, and Priest, 2008; Priest, 2014). 144 There is an analogous situation in mathematics. Though not every contradiction is sensible, 145 there are sensible contradictions such as the boundary of an object A formalized as  A and not A' 146 (Lawvere, 1991, 1994a, p. 48; Lawvere and Rosebrugh, 2003, p. 201). More importantly, within 147 mathematical practice, it is now recognized that contradictions do not necessarily lead to 148 inconsistency (an inconsistent system, according to Tarski, is where everything can be proved; 149 Lawvere, 2003, p. 214). Of course, admitting a contradiction invariably leads to inconsistency in 150 8 classical Boolean logic. In logics more refined than Boolean logic contradiction does not 151 necessarily lead to inconsistency. This recognition is very important, especially since 152 contradiction plays a foundational role in mathematical practice. Briefly, Cantor's definition of 153 SET is, as pointed out by F. William Lawvere, ―a strong contradiction: its points are completely 154 distinct and yet indistinguishable‖ (ibid, p. 215; Lawvere, 1994a, pp. 50-51). Zermelo, and most 155 mathematicians following him, concluded that Cantor's account of sets is ―incorrigibly 156 inconsistent‖ (Lawvere, 1994b, p. 6). Lawvere, using adjoint functors, showed that Cantor's 157 definition is ―not a conceptual inconsistency but a productive dialectical contradiction‖ (Lawvere 158 and Rosebrugh, 2003, pp. 245-246), which is summed up as the unity and identity of adjoint 159 opposites (Lawvere, 1992, pp. 28-30; Lawvere, 1996). 160 A related notion is catuskoti, which is routinely employed in Buddhist reasoning (Priest, 161 2014; Westerhoff, 2006). To place it in perspective, in the familiar Boolean logic, any 162 proposition is either true or false. Put differently, there are only two possible truth values, and 163 they are mutually exclusive and jointly exhaustive. Unlike Boolean logic, in Buddhist reasoning 164 more than two truth values are admissible. In the Buddhist logic of Catuskoti, a proposition can 165 possibly take, in addition to the familiar truth values of  true' or  false', the truth values of  true 166 and false', or  not true and not false'. Given a proposition A, there are four possibilities: 1. A, 2. 167 not A, 3. A and not A, 4. not A and not not A. Here contradiction is admissible, i.e.  A and not 168 A' is a possible state of affairs, which is reminiscent of the boundary operation and the unity and 169 identity of adjoint opposites in mathematics, alluded to earlier. Moreover, double negation is not 170 same as identity operation as in the case of, to give one example, the non-Boolean logic of 171 graphs (Lawvere and Schanuel, 2009, p. 355). Note that if not not A = A, then the fourth truth 172 value of catuskoti is equal to the third. 173 9 As an illustration of how the four truth values of catuskoti could be a reflection [of an 174 aspect] of reality, we consider the category of percepts. Perception involves two sequential 175 processes of sensation followed by interpretation (Albright, 2015; Croner and Albright, 1999). 176 So, we define the category of percepts as a category of two sequential functions of decoding after 177 coding. The truth value object of the category of percepts has four truth values (Appendix A2). 178 Thus the objective logic of perception, with its truth value object of four truth values, is 179 reminiscent of the Buddhist logic of catuskoti (see Linton, 2005). 180 181 Indra's Net and Zero Structure 182 Another important concept in Buddhist philosophy is the idea of Indra's Net, wherein reality is 183 compared to a vast network of jewels such that every jewel is reflective of the entire net (Priest, 184 2015). In abstract terms, reality is characterized as a whole wherein every part is reflective of 185 the whole. Admittedly, this Buddhist characterization of reality sounds mystifying, but there is 186 an analogous situation, involving part-whole relations, in mathematics. 187 How can a part of a whole reflect the whole? First, note that mathematical structures of 188 all sorts can be modeled in the category of sets (Lawvere and Schanuel, 2009, pp. 133-151). Sets 189 have zero structure (Lawvere, 1972, p. 1; Lawvere and Rosebrugh, 2003, p. 1, 57; Lawvere and 190 Schanuel, 2009, p. 146). Negating the structure (cohesion, variation) inherent in mathematical 191 objects, Cantor created sets: mathematical structures with zero structure (Lawvere, 2003, 2016; 192 Lawvere and Rosebrugh, 2003, pp. 245-246). In comparing his abstraction of sets with zero 193 structure to the invention of number zero, Cantor considered sets as his most profound 194 contribution to mathematics (Lawvere, 2006). Sets, by virtue of having zero structure, serve as a 195 10 blank page-an ideal background to model any category of mathematical objects (Lawvere, 196 1994b; Lawvere and Rosebrugh, 2003, pp. 154-155). However, structureless sets are a small 197 part-the only part-of the mathematical universe which reflects all of mathematics. It seemed 198 so until Lawvere axiomatized the category of categories (Lawvere, 1966; Lawvere and Schanuel, 199 2009, pp. 369-370). Along the lines of Cantor's invention of structureless sets, Lawvere defined 200 a subcategory of structureless (discrete, constant) objects within a category by negating its 201 structure (cohesion, variation; Lawvere, 2004, p. 12; Lawvere and Schanuel, 2009, pp. 358-360, 202 372-377). Thus, within any category of mathematical objects, there is a part, a structureless 203 subcategory, which is like the category of sets in having zero structure, and hence serves as a 204 background to model all categories of mathematical objects (Lawvere, 2003; Lawvere and 205 Menni, 2015; Picado, 2008, p. 21). Modeling a category of mathematical objects requires, in 206 addition to the subcategory with zero structure, another subcategory objectifying the structural 207 essence(s) of the objects of the category, i.e. the theory of the given category of mathematical 208 objects. Finding the theory subcategory also depends on the structureless subcategory, by way of 209 contrasting or negating the structureless subcategory (Lawvere, 2007). Once we have the 210 subcategory with zero structure and the subcategory objectifying the essence (theory) of a given 211 category, interpreting the theory subcategory into the structureless subcategory gives us models 212 of the given category of mathematical objects. Thus, thanks to the recognition of significance of 213 Cantor's zero structure, every mathematical category can be modelled in any category of the 214 category of categories. 215 If we compare the category of categories to Indra's net, then categories within the 216 category of categories would correspond to jewels in Indra's net. Just as in the case of Indra's 217 net, wherein every jewel in the network of jewels is reflective of the entire network, in the 218 11 category of categories every category (part) of the category of categories (whole) reflects the 219 whole. For example, the category of dynamical systems is a part of the category of categories. 220 Within the category of dynamical systems, we have the constant subcategory (obtained by 221 negating the variation) of dynamical systems (wherein every state is a fixed point), which is like 222 the category of sets, and within which any category can be modeled. Similarly, the category of 223 graphs is another part of the category of categories. Within the category of graphs there is the 224 discrete subcategory (obtained by negating the cohesion) of graphs (with one loop on each dot), 225 which is also like the category of sets, and hence can model every category. Thus, we find that 226 within the category of categories, every part is reflective of the whole, which is reminiscent of 227 the Buddhist depiction of reality as Indra's Net: a whole with parts reflective of the whole. 228 229 Conclusion 230 There are similarities between Buddhist philosophy and mathematical practice, especially with 231 regard to essence vs. emptiness, contradictions, and part-whole relations. These similarities 232 might be a natural consequence of identical objectives-understanding reality and commitment 233 to truth-and identical means-experience and reason-employed towards those ends. It is in 234 this respect that the practices of the two-mathematicians and Buddhists-can be compared. Our 235 exercise, on one hand, can help better appreciate the rationality of Buddhist reasoning. 236 Oftentimes, admission of contradiction (as in catuskoti) tends to be equated with irrational 237 mysticism. However, as we have seen, contradictions are also an integral and indispensable part 238 of the mathematical understanding of reality. On the other hand, in drawing parallels between 239 12 Buddhist thought and mathematical practice, we hope to have brought out the broad 240 philosophical import of category theory beyond mathematics. 241 13 Appendices 242 243 A1. Yoneda lemma 244 We begin with an intuitive introduction to the mathematical content of Yoneda lemma (Lawvere 245 and Rosebrugh, 2003, pp. 175-176, 249). With simple illustrations of figures-and-incidences 246 (along with [its dual] properties-and-determinations) interpretations of mathematical objects, we 247 prove the Yoneda lemma (Lawvere and Schanuel, 2009, pp. 361, 370-371). Broadly speaking, 248 Yoneda lemma is about [properties of] objects [of categories] and their mutual determination. 249 First, let us consider a function 250 f: A → B 251 We can think of the function f as (i) a figure of shape A in B, i.e., an A-shaped figure in B. For 252 example, in the category of graphs, a map 253 d: D → G 254 from a graph D (consisting of one dot) to any graph G is a D-shaped figure in G, i.e., a dot in the 255 graph G. We can also think of the same function f as (ii) a property of A with values in B, i.e., a 256 B-valued property of A (Lawvere and Schanuel, 2009, pp. 81-85). For example, with sets, say, 257 Fruits = {apple, grape) and Color = {red, green}, a function 258 c: Fruits → Color 259 (with c (apple) = red and c (grape) = green) can be viewed as Color-valued property of Fruits. 260 14 Now let us consider two figures: an X-shaped figure in A 261 xA: X → A 262 and a Y-shaped figure in A 263 yA: Y → A 264 Given a transformation from the shape X to the shape Y, i.e. an X-shaped figure in Y 265 xY: X → Y 266 we find that the X-shaped figure in Y (xY) induces a transformation of a Y-shaped figure in A 267 into an X-shaped figure in A via composition of maps 268 yA ◦ xY = xA 269 (where  ◦' denotes composition) displayed as a commutative diagram 270 271 272 273 showing the transformation of a Y-shaped figure in A (yA) into an X-shaped figure in A (xA) by 274 an X-shaped figure in Y (xY) via composition of maps. 275 As an illustration, consider an object (of the category of graphs) i.e., a graph G (shown 276 below): 277 278 X Y A xA = yA ◦ xY xY yA 15 279 280 281 282 283 and a shape graph [arrow] A with exactly one arrow  a', along with its source  s' and target  t', 284 as shown: 285 286 along with an A-shaped figure in G 287 aG: A → G 288 displayed as: 289 290 291 292 with, say, 293 aG (a) = a1 294 This A-shaped figure in G, i.e. the graph map aG maps the [only] arrow  a' in the shape graph A 295 to the arrow  a1' in the graph G, while respecting the source (s) and target (t) structure of the 296 d3 G a1 a2 d1 d2 A a s t d3 G a1 a2 d1 d2 A a s t aG 16 arrow  a', i.e., with arrow  a' in shape A mapped to arrow  a1' in the graph G, the source  s' and 297 target  t' of the arrow  a' are mapped to the source  d1' and target  d3' of arrow  a1', respectively. 298 Next, consider another shape graph [dot] D with exactly one dot  d' 299 300 along with a D-shaped figure in A 301 dA: D → A 302 with 303 dA (d) = s 304 i.e., the graph map dA maps the dot  d' in the graph D to the dot  s' in the graph A, i.e. the source 305 dot  s' of the arrow  a', as shown below: 306 307 308 This graph map dA from shape D to shape A induces a transformation of the (above) A-shaped 309 figure in graph G 310 aG: A → G 311 into a D-shaped figure in G 312 dG: D → G 313 via composition of graph maps 314 dA D d s A t a D d 17 dG = aG ◦ dA 315 i.e., dG (d) = aG ◦ dA (d) = aG (s) = d1 316 as depicted below (Lawvere and Schanuel, 2009, pp. 149-150): 317 318 319 320 321 In general, every X-shaped figure in Y transforms a Y-shaped figure in A into an X-322 shaped figure in A i.e., every map 323 xY: X → Y 324 induces a map in the opposite direction (contravariant; Lawvere, 2017; Lawvere and Rosebrugh, 325 2003, p. 103; Lawvere and Schanuel, 2009, p. 338) 326 A xY : A Y → A X 327 where A Y is the map object of the totality of all Y-shaped figures in A, A X is the map object of 328 the totality of all X-shaped figures in A, and with the map A xY of map objects defined as 329 A xY (yA: Y → A) = yA ◦ xY = xA: X → A 330 d3 G a1 a2 d1 d2 dA D d s A t a dG aG 18 assigning a map xA in the map object A X to each map yA in the map object A Y . Thus, the figures 331 in an object A of all shapes (all X-shaped figures in A for every object X of a category) along 332 with their incidences 333 A xY : A Y → A X 334 induced by all changes of figure shapes 335 xY: X → Y 336 (i.e. every map in the category) together constitute the geometry of figures in A, i.e., a complete 337 picture of the object A. Summing up, we have the complete characterization of the geometry of 338 every object A of a category in terms of the figures of all shapes (objects of the category) and 339 their incidences (induced by the maps of the category) in the object A (Lawvere and Schanuel, 340 2009, pp. 370-371). 341 Let us now examine how figures of a shape X in an object A are transformed into figures 342 of the [same] shape X in an object B. We find that an A-shaped figure in B 343 aB: A → B 344 induces a transformation of an X-shaped figure in A 345 xA: X → A 346 into an X-shaped figure in B 347 xB: X → B 348 via composition of maps 349 19 X xB = aB ◦ xA A B xA aB aB ◦ xA = xB 350 displayed as a commutative diagram 351 352 353 showing the transformation of an X-shaped figure in A (xA) into an X-shaped figure in B (xB) by 354 an A-shaped figure in B (aB) via composition of maps. Thus, every map 355 aB: A → B 356 induces a map in the same direction (covariant; Lawvere and Rosebrugh, 2003, pp. 102-103, 357 109; Lawvere and Schanuel, 2009, p. 319) 358 aB X : A X → B X 359 where A X is the map object of all X-shaped figures in A, B X is the map object of all X-shaped 360 figures in B, and with the map aB X defined as 361 aB X (xA: X → A) = aB ◦ xA = xB: X → B 362 assigning a map xB in the map object B X to each map xA in the map object A X . Thus, the totality 363 of maps aB X of map objects (for all objects and maps of the category) induced by a map aB from 364 A to B constitutes a covariant transformation of the figure geometry of object A into that of B, 365 i.e., specifies how figures-and-incidences in A are transformed into figures-and-incidences in B. 366 Putting together these two transformations: (i) the covariant transformation of X-shaped 367 figures in A into X-shaped figures in B induced by an A-shaped figure in B, and (ii) the 368 20 A Y X B xY xB xA yB aB yA contravariant transformation of Y-shaped figures in A into X-shaped figures in A induced by an 369 X-shaped figure in Y, we obtain the diagram (Lawvere and Schanuel, 2009, p. 370): 370 371 372 373 374 from which we notice that there are two paths to go from a Y-shaped figure in A (yA) to an X-375 shaped figure in B (xB): 376 Path 1. First we map the Y-shaped figure in A (yA) into an X-shaped figure in A (xA) along the 377 X-shaped figure in Y (xY) via composition of the maps 378 yA ◦ xY 379 and then map the composite X-shaped figure in A (yA ◦ xY) into an X-shaped figure in B along 380 the A-shaped figure in B (aB) via composition 381 aB ◦ (yA ◦ xY) 382 Path 2. First we map the Y-shaped figure in A (yA) into a Y-shaped figure in B (yB) along the A-383 shaped figure in B (aB) via composition of the maps 384 aB ◦ yA 385 and then map the composite Y-shaped figure in B (aB ◦ yA) into an X-shaped figure in B along 386 the X-shaped figure in Y (xY) via composition 387 21 (aB ◦ yA) ◦ xY 388 Based on the associativity of composition of maps (Lawvere and Schanuel, 2009, pp. 370-371), 389 we find that 390 aB ◦ (yA ◦ xY) = (aB ◦ yA) ◦ xY 391 i.e., the two paths of transforming a Y-shaped figure in A 392 yA: Y → A 393 into an X-shaped figure in B give the same map 394 aB ◦ yA ◦ xY = xB: X → B 395 Since the associativity of composition of maps hold for all maps of any category (Lawvere and 396 Schanuel, 2009, p. 17), we find that every A-shaped figure in B induces a covariant 397 transformation of the figure geometry of A into the figure geometry of B. More explicitly, each 398 A-shaped figure in B 399 aB: A → B 400 induces a commutative diagram (of maps of map objects) 401 402 403 404 satisfying 405 A X B X aB X A xY B xY aB Y A Y B Y 22 aB X ◦ A xY = B xY ◦ aB Y 406 for every map in the category, and hence a natural transformation (compatible with the 407 composition of maps) of the figure geometry of A into the figure geometry of B. To see the 408 commutativity, consider a Y-shaped figure in A, i.e. a map yA of the map object A Y and evaluate 409 the above two composites: 410 aB X ◦ A xY (yA) = aB X (yA ◦ xY) = aB ◦ (yA ◦ xY) 411 B xY ◦ aB Y (yA) = B xY (aB ◦ yA) = (aB ◦ yA) ◦ xY 412 Again, according to the associativity of the composition of maps 413 aB ◦ (yA ◦ xY) = (aB ◦ yA) ◦ xY = aB ◦ yA ◦ xY 414 and hence both composites map each Y-shaped figure in A (a map in the map object A Y ) 415 yA: Y → A 416 to the X-shaped figure in B (a map in the map object B X ) 417 aB ◦ yA ◦ xY = xB: X → B 418 Since we have the above commutativity for every shape (object) and figure (map), i.e. for all 419 objects and maps of the category, we conclude that an A-shaped figure in B corresponds to a 420 natural transformation (respectful of figures-and-incidences) of the figure geometry of A into the 421 figure geometry of B. 422 Now we formally show that every A-shaped figure in B 423 aB: A → B 424 23 of a category C can be represented as a natural transformation 425 n aB : C (–, A) → C (–, B) 426 from the domain functor C (–, A) constituting the figure geometry of the object A to the 427 codomain functor C (–, B) constituting the figure geometry of the object B, which is the core 428 mathematical content of the Yoneda lemma (Lawvere and Rosebrugh, 2003, p. 249): ―maps in 429 any category can be represented as natural transformations‖ (Lawvere and Schanuel, 2009, p. 430 378). Since natural transformations represent structure-preserving maps between objects, the 431 domain (codomain) functor of a natural transformation represents the domain (codomain) object 432 of the structure-preserving map. 433 Let us define the (domain) functor 434 C (–, A): C → C 435 as: for each object X of the category C 436 C (–, A) (X) = A X 437 where A X is the map object of all X-shaped figures in A 438 xA: X → A 439 and, for each map 440 xY: X → Y 441 of the category C 442 C (–, A) (xY: X → Y) = A xY : A Y → A X 443 24 where A Y is the map object of all Y-shaped figures in A, and with the map A xY of map objects 444 defined as 445 A xY (yA: Y → A) = yA ◦ xY = xA: X → A 446 assigning a map xA in the map object A X to each map yA in the map object A Y . Thus the functor 447 C (–, A): C → C 448 in assigning to each map 449 xY: X → Y 450 (of the domain category C) its [induced] map [of map objects] 451 C (–, A) (xY: X → Y) = C (–, A) (Y) → C (–, A) (X) = A xY : A Y → A X 452 (of the codomain category C) is contravariant, i.e. a transformation of a shape X into a shape Y 453 induces a transformation (in the opposite direction) of Y-shaped figures in A into X-shaped 454 figures in A (Lawvere and Rosebrugh, 2003, pp. 236-237). 455 Now, we check to see if C (–, A) preserves identities, i.e. whether 456 C (–, A) (1X: X → X) = 1C (–, A) (X) 457 for every object X. Evaluating 458 C (–, A) (1X: X → X) = A 1X : A X → A X 459 at a map 460 xA: X → A 461 25 we find that 462 A 1X (xA: X → A) = (xA ◦ 1X) = xA: X → A 463 (for every map xA in the map object A X ). Next, evaluating 464 1C (–, A) (X) = 1AX: A X → A X 465 at the map 466 xA: X → A 467 we find that 468 1 AX (xA: X → A) = (xA ◦ 1X) = xA: X → A 469 (for every map xA in the map object A X ). Since 470 A 1X = 1 AX 471 i.e. 472 C (–, A) (1X: X → X) = 1C (–, A) (X) 473 for every object X of the category C, we say C (–, A) preserves identities. 474 Next, we check to see if C (–, A) preserves composition. Since C (–, A) is contravariant, 475 we check whether 476 C (–, A) (yZ ◦ xY) = C (–, A) (xY) ◦ C (–, A) (yZ) 477 where yZ: Y → Z. Evaluating 478 26 C (–, A) (yZ ◦ xY) = A (yZ ◦ xY) 479 at any map zA in the map object A Z , we find that 480 A (yZ ◦ xY) (zA) = zA ◦ (yZ ◦ xY) 481 Next, we evaluate 482 C (–, A) (xY) ◦ C (–, A) (yZ) = (A xY ◦ A yZ ) 483 also at the map zA 484 (A xY ◦ A yZ ) (zA) = A xY (zA ◦ yZ) = (zA ◦ yZ) ◦ xY 485 Since 486 zA ◦ (yZ ◦ xY) = (zA ◦ yZ) ◦ xY 487 by the associativity of the composition of maps, we have composition preserved 488 C (–, A) (yZ ◦ xY) = C (–, A) (xY) ◦ C (–, A) (yZ) 489 Having checked that 490 C (–, A): C → C 491 with 492 C (–, A) (X) = A X 493 C (–, A) (xY: X → Y) = A xY : A Y → A X 494 where A xY (yA) = yA ◦ xY, is a contravariant functor, we consider another contravariant functor 495 27 C (–, B): C → C 496 with 497 C (–, B) (X) = B X 498 C (–, B) (xY: X → Y) = B xY : B Y → B X 499 where B xY (yB) = yB ◦ xY. 500 With the two functors C (–, A) and C (–, B) representing the [figure geometry of] objects 501 A and B, respectively, we now show that every structure-preserving map 502 aB: A → B 503 is represented by a natural transformation 504 n aB : C (–, A) → C (–, B) 505 More explicitly, given a map aB, we can construct a natural transformation n aB . A natural 506 transformation n aB from the functor C (–, A): C → C to the functor C (–, B): C → C assigns to 507 each object X of the domain category C (of both domain and codomain functors) a map 508 aB X : A X → B X 509 (in the common codomain category C) from the value of the domain functor at the object X, i.e. 510 C (–, A) (X) = A X to the value of the codomain functor at X, i.e. C (–, B) (X) = B X ; and to each 511 map xY: X → Y (in the common domain category C), a commutative square (in the common 512 codomain category C) shown below: 513 514 28 515 516 517 518 519 satisfying 520 aB X ◦ A xY = B xY ◦ aB Y 521 (Lawvere and Rosebrugh, 2003, p. 241; Lawvere and Schanuel, 2009, pp. 369-370). We have 522 already seen that with the composition-induced maps (of map objects): 523 A xY (yA) = yA ◦ xY 524 aB X (xA) = aB ◦ xA 525 aB Y (yA) = aB ◦ yA 526 B xY (yB) = yB ◦ xY 527 the required commutativity: 528 aB X ◦ A xY (yA) = aB X (yA ◦ xY) = aB ◦ (yA ◦ xY) 529 B xY ◦ aB Y (yA) = B xY (aB ◦ yA) = (aB ◦ yA) ◦ xY 530 is given by the associativity of the composition of maps 531 A X B X aB X A xY B xY aB Y A Y B Y 29 aB ◦ (yA ◦ xY) = (aB ◦ yA) ◦ xY = aB ◦ yA ◦ xY 532 Thus, each A-shaped figure in B (aB) is a natural transformation (n aB ; homogenous with respect 533 to composition of maps) of the figure geometry C (–, A) of A into the figure geometry C (–, B) 534 of B. 535 Furthermore, we can obtain the set |B A | of all A-shaped figures in B based on the 1-1 536 correspondence between A-shaped figures in B and the points (i.e. maps with terminal object T 537 of the category C as domain; Lawvere and Schanuel, 2009, pp. 232-234) of the map object B A . 538 This 1-1 correspondence, which follows from the universal mapping property defining 539 exponentiation, along with the fact that the terminal object T is a multiplicative identity 540 (Lawvere and Schanuel, 2009, pp. 261-263, 313-314, 322-323), involves the following two 1-1 541 correspondences between three maps: 542 543 544 545 Yoneda lemma says, in terms of our figures-and-incidences characterization of objects, 546 that the set |B A | of A-shaped figures in B 547 aB: A → B 548 is isomorphic to the set |C (–, B) C (–, A) | of natural transformations 549 n aB : C (–, A) → C (–, B) 550 of the figure geometry of A into that of B. The required isomorphism of sets 551 T → B A T × A → B A → B 30 |B A | = |C (–, B) C (–, A) | 552 follows from the 1-1 correspondence between A-shaped figures in B and the natural 553 transformations (compatible with all figures and their incidences) of the figure geometry of A 554 into that of B, which we have already shown (see also Lawvere and Rosebrugh, 2003, p. 104, 555 174). 556 Dually, a map 557 A → B 558 viewed as a B-valued property on A induces a natural transformation 559 C (B, –) → C (A, –) 560 of the function algebra of B into that of A (Lawvere and Rosebrugh, 2003, p. 249). Here also the 561 proof of Yoneda lemma involves two transformations: (i) Contravariant: a map from an object A 562 to an object B induces a transformation of properties of B into properties of A, for each type 563 (object) of the category, and (ii) Covariant: a map from a type T to a type R (of properties) 564 induces a transformation of T-valued properties into R-valued properties, for every object of the 565 category. The calculations involved in proving Yoneda lemma in this case of function algebras 566 are same as in the case of figure geometries, except for the reversal of arrows due to the duality 567 between function algebra and figure geometry (Lawvere and Rosebrugh, 2003, p. 174; Lawvere 568 and Schanuel, 2009, pp. 370-371). More specifically, function algebras and figure geometries 569 are related by adjoint functors (Lawvere, 2016). 570 31 A2. Four Truth Values of the Logic of Perception 571 Conscious perception involves two sequential processes of sensation followed by interpretation: 572 Physical stimuli → Brain → Conscious Percepts 573 (Albright, 2015; Croner and Albright, 1999), which can be thought of as 574 X – coding → Y – decoding → Z 575 and objectified as two sequential processes: 576 A – f → B – g → C 577 Without discounting that the processes of sensation and interpretation are much more structured 578 than mere functions, and with the objective of simplifying the calculation of truth value object, 579 we model percept as an object made up of three [component] sets C, B, and A, which are sets of 580 physical stimuli, their neural codes, and interpretations, respectively, and two [structural] 581 functions f and g specifying for each interpretation in A the neural code in B (of which it is an 582 interpretation) and for each neural code in B the physical stimulus in C (of which it is a 583 measurement), respectively (see Lawvere and Rosebrugh, 2003, pp. 114-117). The logic of [the 584 category of] perception, whose objects are two sequential functions is determined by its truth 585 value object (Lawvere and Rosebrugh, 2003, pp. 193-212; Lawvere and Schanuel, 2009, pp. 586 335-357; Reyes, Reyes, and Zolfaghari, 2004, pp. 93-107). The truth value object of a category 587 is an object Ω of the category such that parts of any object X are in 1-1 correspondence with 588 maps from the object X to the truth value object Ω. Since parts of an object are monomorphisms 589 with the object X as codomain, for each monomorphism with X as codomain there is a 590 corresponding X-shaped figure in Ω. 591 32 In order to calculate the truth value object, first we need to define maps between objects 592 of the category of percepts. A map from an object 593 A – f → B – g → C 594 to an object 595 A' – f' → B' – g' → C' 596 is a triple of functions 597 p: A → A', q: B → B', r: C → C' 598 satisfying two equations 599 q ◦ f = f' ◦ p, r ◦ g = g' ◦ q 600 which make the two squares in the diagram 601 602 603 604 605 606 commute, i.e. ensure that maps between objects preserve the structural essence of the category 607 (Lawvere and Schanuel, 2009, pp. 149-150). 608 p A A' f' f q B B' g' g r C C' 33 Now that we have maps of the category of percepts defined, we can calculate its truth 609 value object. The truth value object of a category is calculated based on the parts of the basic 610 shapes (essence) constituting the objects of the category. In the category of sets, one-element set 611 1 (= {•}) is the basic shape in the sense any set is made up of elements (see Posina, Ghista, and 612 Roy, 2017 for the details of the calculation of basic shapes, i.e. theory subcategories of various 613 categories). Since the set 1 is the also the terminal object (i.e. an object to which there is exactly 614 one map from every object; Lawvere and Schanuel, 2009, pp. 213-214) of the category of sets, 615 and since every set is completely determined by its points (terminal object-shaped figures), we 616 can determine the truth value object of the category of sets by determining its points, i.e. maps 617 from 1 to the (yet to be determined) truth value object. According to the definition of truth value 618 object, 1-shaped figures in the truth value object are in 1-1 correspondence with parts of 1. Since 619 the terminal set 1 has two parts: 0 (= {}) and 1, the truth value set has two points (elements). 620 Thus, the truth value object of the category of sets is 2 (= {false, true}). 621 Along similar lines, let us calculate the terminal object of the category of percepts. Since 622 there is only one map from any object (two sequential functions) to the object T (two sequential 623 functions from one-element set to one-element set): 624 1 → 1 → 1 625 the terminal object of the category of percepts is T. Since parts of the terminal object T 626 correspond to the points of the truth value object, let's look at the parts of the terminal object. 627 The terminal object T 628 1 → 1 → 1 629 has four parts: 630 34 Part 1 (0: 0 → T) 631 632 633 634 Part 2 (01: 01 → T) 635 636 637 638 Part 3 (02: 02 → T) 639 640 641 642 Part 4 (1: T → T) 643 644 645 646 These four parts correspond to the four points (global truth values) of the truth value object, 647 which means that the component set (of the truth value object) corresponding to the stage of 648 1 → 1 ↓ 1 → 1 ↓ 1 → ↓ 1 ↓ 0 1 ↓ 0 1 ↓ 1 → 1 0 1 ↓ 1 → 1 ↓ 1 → 1 ↓ 0 1 ↓ 0 1 ↓ 0 1 35 interpretations is a four-element set 4 = {0, 01, 02, 1}. Since objects in the category of perception 649 (two sequential functions) are not completely determined by points, we look for all other basic 650 shapes that are needed to completely characterize any object of two sequential functions. The 651 other basic shapes, besides the terminal object T, are: domains of the parts 02 and 01 of the 652 terminal object T, i.e. shape 02 653 0 1 → 1 654 and shape 01 655 0 0 1 656 Since the basic shape object 02 has three parts (0, 01, and 1), there are three 02-shaped figures in 657 the truth value object, and since the object 01 has two parts (0 and 1), there are two 01-shaped 658 figures in the truth value object, which means that the component set (of the truth value object) 659 corresponding to the stage of neural coding is a three-element set 3 = {0, 01, 1}, while the 660 component set (of the truth value object) corresponding to the stage of physical stimuli is a two-661 element set 2 = {0, 1}. Putting it all together we find that the truth value object of the category 662 of percepts is: 663 4 – j → 3 – k → 2 664 We still have to determine the functions j and k, which can be done by examining the structural 665 maps between the basic shapes 666 01 – c → 02 – d → T 667 which as a subcategory constitutes the theory (abstract essence) of the category of two sequential 668 functions. More explicitly, the incidence relations between the three basic-shaped figures in the 669 36 truth value object are calculated from the inverse images of the parts of the basic shapes (01, 02, 670 and T) along the structural maps (d and c). The inverse images of each one of the four points (0, 671 01, 02, and 1 corresponding to the four parts of the terminal object T) along the structural maps 672 decoding d and coding c give for each one of the four global truth values 4 = {0, 01, 02, 1} its 673 value in the truth value sets 3 = {0, 01, 1} and 2 = {0, 1} of the previous stages of neural codes 674 and physical stimuli. For example, the global truth value 02 corresponds to the part 02 of the 675 basic shape T, and its inverse image along the structural map d: 02 → T is the entire basic shape 676 02, which corresponds to the truth value 1 (of stage 3); and the inverse image of the entire object 677 02 along the structural map c: 01 → 02 is the entire basic shape 01, which corresponds to the truth 678 value 1 (of stage 2). 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