A Category Theoretic Approach to Metaphor Comprehension: Theory of Indeterminate Natural Transformation
Miho Fuyamaa,∗, Hayato Saigob, Tatsuji Takahashic
aWaseda University, 2-579-15 Mikajima, Tokorozawa, Saitama, 350-1192, Japan bNagahama Institute of Bio-Science and Technology, 1266 Tamura, Nagahama, Shiga, 526-0829, Japan cTokyo Denki University, Ishizaka, Hatoyama, Hiki, Saitama, 350-0394, Japan
Abstract We propose the theory of indeterminate natural transformation (TINT) to investigate the dynamical creation of meaning as an association relationship between images, focusing on metaphor comprehension as an example. TINT models meaning creation as a type of stochastic process based on mathemat- ical structure and defined by association relationships, such as morphisms in category theory, to represent the indeterminate nature of structure–structure interactions between the systems of image meanings. Such interactions are formulated in terms of the so-called coslice categories and functors as struc- ture-preserving correspondences between them. The relationship between such functors is “indeterminate natural transformation,” the central notion in TINT, which models the creation of meanings in a precise manner. For instance, metaphor comprehension is modeled by the construction of inde- terminate natural transformations from a canonically defined functor, which we call the base-of-metaphor functor. Keywords: Meaning, Morphism, Metaphor, Category theory, Analogy
∗corresponding author Email address: [email protected] (Miho Fuyama)
Preprint submitted to Biosystems July 17, 2020 1 1. Introduction
2 Our minds are filled with various images, visual or auditory, concrete or
3 abstract, verbal or non-verbal: the image of a cup, snow, mothers, a girl
4 called Naomi, red, cold, a series of sounds like “kiki,” the God, Pegasus, the
5 number 3, and so on. For each image, several meanings are being created in
6 our minds. Meaning creation is one of the essential abilities of humankind
7 and also one of the fascinating topics for cognitive process research.
8 In this paper, we investigate the process of creating meaning, especially 1 9 for novel metaphors as typical examples, as the meaning of an image rep-
10 resents its relationships to other images. As a framework, we propose a new
11 theory of novel metaphor comprehension, the theory of indeterminate natural
12 transformations (TINT ).
13 The aim of TINT is to describe and explain how the meanings of the 2 14 source and target of a metaphor interact with each other to create novel
15 meanings based on the structure of the relationships among images. TINT
16 is based on fundamental concepts in category theory, which is a branch of
17 mathematics that investigates various mathematical structures in terms of
18 relationships, called morphisms (or arrows). We extend the notion of cat-
19 egory, functor, and natural transformation to the context of a stochastic
20 process to model the indeterminate nature of meaning creation.
21 The remainder of the paper is organized as follows: In Section 2, we
22 briefly explain the intuition of the TINT without using the terminology of
23 category theory. Then, we introduce the basic concepts of category theory
24 required for formulating the TINT in Section 3. In Section 4, we formulate
25 the TINT based on category theory and introduce a stochastic process. The
26 central notion is “indeterminate natural transformation.” Section 5 provides
27 an example of TINT applications to metaphor comprehension. In the last
1We do not distinguish similes from metaphors, for our focus is on the general structures not dependent on this classification. 2“Target” and “source” are widely used terms in the metaphor and analogy studies [1]. For example, in the metaphor “T is S” or “T is like S,” T is a target and S is a source. In general, a target is relatively unknown while a source is relatively known. Holyoak and Stamenkovi´c(2018)[1]defined a target as what is being talked about and a source as the concept used to characterize the target. Although some studies have used the terms “tenor” and “topic” instead of “target” and “vehicle” and “base” and “focus” instead of “source,” we chose the more commonly used terms “target” and “source” to explain metaphor comprehension.
2 28 section, we discuss some essential problems of metaphor comprehension based
29 on TINT and future research directions.
30 2. Summary of TINT
31 First, we define the “meaning” of an image. Under TINT, the meaning of
32 a certain image is defined as the entire associative relationship of the image
33 to other images. Assume that some sort of event evokes image A and image A
34 evokes other images, B, C, and so on. We then define the meaning of A as the 3 35 entire association from image A to images B, C, and others . For example,
36 when an image of “love” evokes other images such as “warm,” “children,”
37 “lover,” “suffer,” “passion,” “heart,” we regard the meaning of “Love” as its
38 association to all these images.
39 Some readers may be puzzled by our terminology of “meaning of image.”
40 If one conceives the “meaning” as the “object” itself, which a symbol such
41 as a word refers to, our terminology will seem somewhat strange. However,
42 in the semiotics of Saussure, Pierce, and others, meaning is always thought
43 of in terms of “relations.” Pierce in particular defined the concept of the
44 symbol as follows.
45 Namely, a sign is something, A, which brings something, B, its
46 interpretant sign determined or created by it, into the same sort
47 of correspondence with something, C, its object, as that in which
48 itself stands to C(Peirce 1902, NEM IV, 20–21 [2]).
49 According to the definition by Peirce, everything that is the starting point
50 of a relationship from there to other things is a sign. Therefore, not only
51 phonetic or textual symbols such as words but also multimodal images can
52 be called symbols in the sense that they evoke other images. Furthermore,
53 the meanings of the various images as symbols should be taken as the total
54 structure of the relationships from them to the various other images, as with
55 other symbols.
3The boundaries of association are decided by the strength of associative relationships between images that decay with increasing steps of association, and they are also con- strained by the capacity of cognition. In TINT, this decay is modeled by the probabilistic weight, which is decided based on the strength of association. We discuss this point in 4.2.
3 56 Accordingly, creating new meanings can be assimilated to creating new
57 relationships between images. Therefore, the simplest case of meaning cre-
58 ation is one associative relationship from image A to image B excited, which
59 changes the meaning of A and creates new meanings for the other image.
60 An example of this process is the comprehension of a novel metaphor,
61 in which a new connection between the target and source of the metaphor
62 is excited. For example, if the metaphor of “love is like drinking water” is
63 new, an associative relationship from the image of “love” to the image of
64 “drinking water” is made. Then, the meaning of “love” is changed by this
65 new relationship. Therefore, we can recognize the process of comprehending
66 novel metaphors as a basic process of creating new meanings.
67 Furthermore, each image of the target and source has associations to other
68 images prior to understanding the new metaphor. Therefore, when compre-
69 hending a metaphor, we will not only make the associative relationship from
70 “love” to “drinking water” but also create a correspondence of some part
71 of the associative relationships from “love” to some part of the associative
72 relationships from “drinking water.” These associations for each structure
73 enrich the meaning of “love,” which becomes like drinking water, making it
74 necessary for living, feeling good, cleanliness, transparency, coldness, and so
75 on.
76 In this paper, we use the concepts of category theory, which is a fun-
77 damental “relationalist” framework in mathematics, especially the concept
78 of “coslice category,” to formulate the concept of meaning and the process
79 of metaphor comprehension from the relationalism point of view. From our
80 viewpoint, the most important process of metaphor comprehension is thus
81 constructing relationships between part of the associative structures of the
82 target and the source. In this context, TINT represents this structure-to-
83 structure interaction as a construction of indeterminate natural transforma-
84 tion. The next section is devoted to formulating this concept based on the
85 fundamental notions of category theory
86 Our perspective of metaphor comprehension can be basically placed in
87 “analogical position” and “conceptual mapping position” among studies of
88 metaphor comprehension. Holyoak and Stamenkovi´c(2018)[1]conducted a
89 broad survey of studies of metaphor comprehension and identified three the-
90 oretical positions: analogy, categorization, and conceptual mapping. It seems
91 there are no conclusions or agreements on which position is better than oth-
92 ers in cognitive psychology. The analogy position claims that metaphors are
93 based on analogies that find and exploit similarities according to relationships
4 94 among entities rather than solely on the entities themselves [3]. On the other
95 hand, the categorization position argues that metaphors are interpreted as
96 category statements in which a target concept belongs to a source concept [4].
97 While these first two positions mainly propose information-processing mod-
98 els, the conceptual mapping position suggests a novel perspective wherein
99 metaphorical linguistic expressions represent realization in language of un-
100 derlying metaphorical patterns in thought [5]. Therefore, metaphor compre-
101 hension is not merely a kind of verbal cognition but more general and basic
102 cognition of thought and understanding. It is worth noting that, although
103 the conceptual mapping position emphasizes embodiment cognition, this po-
104 sition seems consist with the other two, especially the analogy position [1].
105 Since TINT represents the process of constructing relationships between
106 part of the associative structures of the target and the source, TINT is re-
107 garded as a kind of analogical position. In addition, we postulate that the
108 linguistic expressions of metaphors are realizations of underlying metaphor-
109 ical patterns in thought using multi-modal images. In this sense, TINT is a
110 similar idea as the conceptual mapping position. We discuss the relationships
111 between TINT and other studies of metaphor comprehension in more detail
112 in section 6.
113 Whichever position is chosen, some representation of meaning of words
114 or concepts is needed to compute metaphor comprehension. One meaning of
115 a word/concept seems to have specific similarity with other meanings. For
116 example, “hot” and “sun” have relatively closer meanings than “hot” and
117 “moon.” Operation between two meanings also seems to be required to rep-
118 resent conceptual operation such as conceptual combination (e.g., portable
119 plus phone becomes mobile phone). Many studies have attempted to con-
120 struct this kind of semantic space as a kind of metric space [6, 7]. However, in
121 some cases, symmetric property between meanings can be broken [8]. These
122 results suggest that the constraints of metric space are too strong to represent
123 the semantic space.
124 In TINT, we propose using a category with indeterminate arrows to rep-
125 resent quasi-metrics and compositionality (one of the simplest operations)
126 to represent semantic space. Since not much is known about the semantic
127 space of human, and category is the weakest structure that can manage quasi-
128 metrics and compositionality, the modeling of semantic space using category
5 4 129 theory seems appropriate .
130 3. Category-theoretic concepts for TINT
131 The fundamental concepts of category theory are “category,” “functor,”
132 and “natural transformation” (for more details, see Mac Lane (1998)[9]).
133 A category is, roughly speaking, a network formed by composable “mor-
134 phisms” that intertwines “objects.”
135 0. A category is a system consisting of objects and morphisms, satisfying
136 the following four conditions:
137
138 I. Each morphism f is associated with two objects, dom(f) and cod(f),
139 respectively called domain and codmain. When dom(f) = X and cod(f) = Y , we denote
f : X −→ Y
or
f X −−→ Y.
140 It is not necessary to limit the direction of the morphism from left to right, if
141 it is convenient to consider it, for example, from bottom to top or from right
142 to left. A subsystem of a category is built based on these morphisms and
143 objects are called diagrams.
144 II. If there are two morphisms, f, g such that cod(f) = dom(g), we have
g f Z ←−− Y ←−− X,
4A category is a directed graph with a structure of composition. In this sense, categories are special cases of directed graphs. On the other hand, when a directed graph is given, we can construct some categories from the directed graph. For example, we can construct a category whose objects are vertices and arrows are paths. This is called “the free category generated by the directed graph.” Another example is a category whose objects are vertices and whose arrows are ”accessibility” by paths (two vertices are connected by an arrow if and only if they are connected by some path), which we make use of in the present paper. Note that, however, as a mere collection of elements in a vector space and the vector space itself with linear structure are different, a mere directed graph and some categories constructed from it are different.
6 and there is a unique morphism
g◦f Z ←−−− X
145 called the composition of f, g.
146 III. We assume an associative law. For the diagram,
o h◦g W[[8 Y \8 88 ¨¨ 88 8 ¨ 8 f 88 ¨¨ 88 h 8 Ó¨¨ g 8 Z o X, g◦f we assume
(h ◦ g) ◦ f = h ◦ (g ◦ f).
147 As above, when all morphism compositions are equal, we call the diagram
148 commutative.
149
150 IV. The last condition is the unit law. For any object X there exists a 151 morphism 1X : X −→ X called the identity of X such that the diagram
¦X\88 ¦¦ 88 f¦¦ 818X Ó¦¦ 8 o f Y [88 ¦X 88 ¦¦ 88 ¦¦ 1Y 8 Ò¦¦ f Y
is commutative for any f : X −→ Y . In other words,
f ◦ 1X = f = 1Y ◦ f.
152 By the natural correspondence from objects to their identities, we can identify
153 the objects with identities. In other words, we can consider the objects as
154 special morphisms. Sometimes, we can adopt this viewpoint without notice.
7 155 Definition 1. A category is a system composed of two types of entities,
156 called objects and morphisms, which interrelate through the notion of do-
157 main/codomain, which is equipped with composition and identity, thus satis-
158 fying the associative and unit laws.
159 A functor is defined as the structure-preserving correspondence of two
160 categories as follows:
161 Definition 2. Let C and D be categories. A correspondence F from C to D,
162 which maps each object/morphism in C to a corresponding object/morphism
163 in D, is called a functor if is satisfies the following conditions:
164 1. It maps f : X −→ Y in C to F (f): F (X) −→ F (Y ) in D.
165 2. F (f ◦ g) = F (f) ◦ F (g) for any (composable) pair of f, g in C.
166 3. For each X in C, F (1X ) = 1F (X).
167 In short, a functor is a correspondence that preserves diagrams or, equiv-
168 alently, categorical structure. A functor is a universal concept. Previous
169 studies have suggested that some representation, modeling, and theorization
170 can be interpreted as the creation of functors [10, 11]. In this article, we
171 suggest that metaphor comprehension, which is a kind of cognitive process,
172 can also be interpreted as a functor. Then, what is the morphism between
173 functors? The answer is natural transformation.
174 Definition 3. Let F,G be functors from category C to category D, a corre-
175 spondence t is called a natural transformation from F to G if it satisfies
176 the following conditions:
177 1. t maps each object X in C to the corresponding morphism tX : F (X) −→ 178 G(X) in D. 2. For any f : X −→ Y in C,
tY ◦ F (f) = G(f) ◦ tX .
179 For the natural transformation, we use notation t : F =⇒ G. The second
180 condition above is depicted as follows:
8 f Y o X
F (f) F F (Y ) o F (X)
t tY tX GG(Y ) o G(X) G(f)
181 The upper-right part denotes the morphism in C and the lower-left part the
182 morphism in D. The second condition in the definition of natural transfor- 5 183 mation means the diagram above is commutative.
184 4. Formulation of TINT
185 In Section 2, we defined the meaning of an image as the entire associative
186 relationship from that image to other images. Novel metaphor comprehen-
187 sion can be regarded as the process of constructing new local/partial corre-
188 spondence between a part of the associative structure between two images:
189 the target and source. Now, we can formulate this idea as the TINT using
190 concepts from category theory.
191 First, we describe our theory without indeterminacy. As previously men-
192 tioned, the idea of metaphor comprehension is formulated in terms of cate-
193 gory theory. Second, we introduce stochastic processes to describe alteration
194 of categories and functors from moment to moment as a process of metaphor
195 comprehension. Finally, we sum up the TINT axioms.
196 4.1. Formulation of TINT using category theory without indeterminacy
197 First, we define the category of images. The objects of Category of
198 images C are images (visual or auditory, concrete or abstract, verbal or
199 nonverbal) and the morphisms of C are associations between them.
200 For simplicity, here, we let the number of morphisms between two given
201 objects be at most one. That is, the morphism is considered as “the pos-
202 sibility of association,” although it is fruitful to relax this condition to the
5Note that in the setting of this present paper (i.e., there exists at most one morphism between any two objects), this condition is automatically satisfied when the functors are well-defined, although we will consider more general cases in future studies.
9 203 general case (i.e., types of associations are distinguished). This general case
204 is used to discuss the stochastic process in the next subsection.
205 Then, the meaning of an image can be represented using coslice category.
206 A coslice category is defined as follows.
207 Definition 4. The coslice category A\C : Let A be an object in C. Ob-
208 jects in A\C are morphisms in C from A. The morphisms between two ob- 209 jects f1 : A −→ X1 and f2 : A −→ X2 is the triple of (f1, f2, g), where 210 g : X1 −→ X2 and f2 = g ◦ f1. That is, the objects are morphisms from A 211 (something “for A”) and the morphisms are commutative triangle diagrams
212 (relationships for A).
213 When C is the category of images, we identify category A\C as the cate-
214 gories of meanings of A based on our idea in Section 2.
215 Next, a metaphor can be considered as a functor representing the corre-
216 spondence between structures. Metaphor “A is like B” can be interpreted
217 as functor from a coslice category B\C to a coslice category A\C, where C
218 is a category of images. For example, the metaphor “from atomic worlds to
219 celestial worlds” is considered as a functor from the categories whose objects
220 are protons, electrons, electromagnetic forces, etc., and morphisms are asso-
221 ciations between them and stars, planets, gravitational forces, etc., and the
222 associations between them.
223 The example above is a conventionalized metaphor [1]. The main topic
224 is thus how such metaphors are constructed. To model the construction, we
225 propose a “base-of-metaphor” functor between coslice categories as follows:
226 Base-of-metaphor functor between the categories of meanings is de-
227 fined as follows (see also Figure 1). Let us consider a given morphism
228 f : A −→ B, the association induced by a (new) metaphor expression as
229 “A is like B.” Then, a functor f\C := (·) ◦ f : B\C −→ A\C (i.e., “(·) for B”
230 for A )is canonically defined. We call it the base-of-metaphor functor.
231 For example (Figure 1), A\C and B\C is a coslice category that represents 232 the meaning of A and B in this order. In A\C, a1, a2, a3, ... are the objects and 233 (a1, a2, h1), (a2, a3, h2)... the morphisms. The new association f : A → B is 234 induced by a new metaphorical expression, “A is like B.” Then, the base-
235 of-metaphor functor, f\C, is canonically created by these morphisms using
236 composition.
237 Our computational hypothesis is that the construction of meaning is a
238 new functor (metaphor functor) from base-of-metaphor functor. Of course,
10 Figure 1: The example of inducing a base-of-metaphor functor and constructing another functor that represents a more natural interpretation of the metaphor with a natural trans- formation. A\C and B\C is a coslice category. In A\C, a1, a2, a3, ... are the objects and (a1, a2, h1), (a2, a3, h2)... the morphisms. These coslice categories represent the meaning of A and B in this order. The new association f : A → B is induced by a new metaphor- ical expression, “A is like B.” Then, the base-of-metaphor functor, f\C, is canonically made by these morphisms using composition. Finally, a more natural interpretation of the metaphor is made as another functor F by a natural transformation from f\C. This natural transformation t is made by a family of morphisms in A\C so that tb1.
239 the base-of-metaphor functor itself is a functor and provides indirect mean-
240 ings such as “X for B for A.” As such, our minds tend to construct other new
241 associations from it such as “Y for A: X is like Y (for A).” The construction
242 should be considered as a “morphism” from the base-of-metaphor functor to
243 some other new metaphor functor. This “morphism” between functors is a
244 natural transformation conducted by each morphism in A\C .
245 Continuing previous example, this situation is represented as follow (Fig-
246 ure 1). Finally, a more natural interpretation of the metaphor is made as
247 another functor F by a natural transformation from f\C. This natural trans- 248 formation t is made by a family of morphisms in A\C so that tb1.
11 Figure 2: The base-of-metaphor is canonically constructed from morphism f and metaphor functors can be constructed based on this base-of-metaphor by searching for the natural transformation.
249 In sum, Figure 2 shows the scheme of this process by evoking morphism
250 f : A −→ B to construct a new metaphor functor based on base-of-metaphor
251 functor and its natural transformation.
252 4.2. Toward indeterminacy with introducing stochastic process to category
253 In the previous subsection, we have introduced the basic concepts of cat-
254 egory theory to formulate meanings and metaphors in mathematical terms.
255 However, the most important topics are not the completed metaphor func-
256 tors and natural transformation, but the process of creating them. As all
257 of us understand that the possible interpretation of a metaphorical expres-
258 sion in daily life is by no means unique or complete and that the process of
259 creating a new interpretation for a metaphor cannot be considered a deter-
260 ministic process as in classical mechanics. Here, we introduce the concept
261 of indeterminate category, indeterminate functor, and indeterminate natural
262 transformation to address this type of indeterminacy on the basis of category
263 theory.
264 The intuitive ideas behind an indeterminate category, indeterminate func-
265 tor, and indeterminate natural transformation are quite simple. They are just
266 a fluctuating, growing, and co-existing family of (subsystems of) categories,
267 functors, and natural transformation, respectively, constructed through an
268 indeterministic processes. To model this simple concept, we need to introduce
12 269 some stochastic concepts over categories, such as the weight of morphisms,
270 excitement of morphisms, and excitation/relaxation processes. For simplic-
271 ity, we consider that morphism has two-level states: excited or relaxed.
272 A stochastic category (C, µ, R) is the triple of a category C, the proba-
273 bilistic weight µ (not normalized) on morphisms, and excitation/relaxation
274 rules R referring to µ. We assume R contains the following rules.
275 • (Basic rule). The morphism composed by two excited morphisms is
276 excited.
277 These rules do not depend on µ. Based on the notion of a stochastic 278 category, a “stochastic process” of category Cexc,t as the (subsystems of) cat- 279 egories of C containing all the excited morphisms in step t (Figure 3). We 280 consider Cexc,t the mathematical realization of the concept of “indeterminate 281 categories.” An indeterminate functor or indeterminate natural transforma-
282 tion are defined by (a family of the subsystems of) functors and natural
283 transformations between indeterminate categories, constructed in terms of
284 stochastic categories.
285 When one morphism can be made by a composition of two morphisms,
286 probabilistic weight µ on its morphism is not necessarily determined by mul-
287 tiplying their probabilistic weights. As described in the previous section,
288 there can be more than one morphism between two objects. Therefore, its
289 morphism can be constructed by another composition, meaning the proba-
290 bilistic weights should be determined by the sum of the weights among all
291 possible compositions.
292 4.3. Axioms of TINT
293 Based on the above arguments, we propose the axioms of TINT as a
294 working hypothesis:
295 • The system of all images (visual or auditory, concrete or abstract, ver-
296 bal or non-verbal) and all association between them can be modeled by
297 a category C. The stochastic aspect is provided by a weight on µ that
298 represents the strengths of associations and the excitation/relaxation
299 rule introduced below. (We call this stochastic category of images the
300 latent category of meaning):
301 • The system of meanings of an image A and the relationships between
302 meanings can be modeled by coslice categories and its weight is natu-
303 rally induced by µ:
13 Figure 3: Latent category of meaning and indeterminate category of meaning. The latent category of meaning is a stochastic category and its µ is decided based on a strength of association. The excited category of meaning is created from the latent category of meaning and a metaphor expression according to stochastic rules. In this diagram, in the latent category of meaning, bold morphisms have relatively large µ and fine morphisms have relatively small µ. In the excited category of metaphor, the morphisms that have large µ in the latent category are relatively excitable.
6 304 • A metaphor “A is like B” excites morphism f : A → B. It causes the
305 construction of natural transformations based on the excitation/relaxation
306 processes described below, which are the basic dynamics of development
307 of coslice categories:
308 • Excitation processes occur (relatively fast) under the stochastic rules
309 as follows (see also Figure 4):
6When the weight of the morphism is (almost) zero, “creates” is more suitable than “excites.”
14 310 – 0. (Basic rule). The morphism composed of two excited mor-
311 phisms is excited. The identities of the domain and codomain of
312 excited morphisms are excited. This rule represents the composi-
313 tionality of morphism.
314 – 1. (Neighboring rule). The morphisms whose domain is the
315 codomain of an excited morphism tends to be excited. The exci-
316 tation probability is provided by µ. This rule represents an asso-
317 ciative relationship between images.
318 – 2. (Fork rule). For the pairs of morphisms sharing domains, mor-
319 phisms between their codomains (which may be equal to a com-
320 position of many morphisms) are searched and tend to be excited.
321 The search criteria and the excitation probability are provided by
322 (the localization of) µ. This rule and the Anti-fork rule represent
323 the compositionality of morphism, which is especially needed in
324 the coslice category and while searching for a natural transforma-
325 tion.
326 – As a special case of rules 1 or 2, the inversely directed morphisms
327 of excited morphisms tend to be excited.
328 • Relaxation processes occur (relatively slowly) under the rule below (see
329 also Figure 4):
330 – (Anti-fork rule). For the pairs of morphisms sharing domains,
331 if the morphisms between their codomains (which may be equal
332 to some composition of many morphisms) are not excited, these
333 morphisms are relaxed.
334 • As a result, an excited category of meaning is defined and a family
335 of excited morphisms becomes an indeterminate natural transformation
336 (i.e., a family of subsystems of natural transformations) from (a subsys-
337 tem of) base-of-metaphor functor f\C to some indeterminate functor
338 (i.e., metaphor functor). The INT (indeterminate natural transforma- 7 339 tion) provides the meaning of the metaphor.
7In this article, we basically discuss novel metaphors. In this case, this understanding of the meaning of metaphors can be regarded as the creation of new meanings since excitation of this associative structure is novel (See also section 2). On the other hand, in the case
15 Figure 4: Rules of the excitation and relaxation processes. When the blue morphisms are excited, the red morphisms will be excited. The solid line represents the deterministic process and the dashed one the stochastic process with probabilistic weight µ. For the anti-fork rule, the blue morphisms will be banished because of the absence of the necessary composition.
340 • When the INT above become stable (long-term survival), comprehen-
341 sive (the domain subcategory of functor defined by INT is large) and
342 the weight of the components of the INT is increased. In other words,
343 it will provide the change of weight µ: a change in our world-view. In
344 other words, this process can be interpreted as feedback and learning,
345 such as Hebbian learning.
346 5. Exemplification of TINT
347 Here, we exemplify an application of TINT to explain metaphor com-
348 prehension. Note that this example is a thought experiment for a better
349 understanding of TINT. In another study, we are now conducting a psycho-
350 logical experiment to test TINT empirically where we are attempting to es-
of a conventionalized metaphor, because this excitation often happens, constructing INT can be regarded as a remembering of the meaning of metaphor.
16 351 timate the strengths of association µ between images and simulate metaphor
352 comprehensions using TINT. Please see section 6.2 for more information.
353 We take the example of a metaphor from a Japanese poem “Tsuchi” (Soil)
354 [12], written by Tatsuji Miyoshi. The free translation of this poem is:
355 An ant
356 pull a wing of a butterfly.
357 O
358 It is like yachting.
359 For simplicity, we abstract the metaphor “Wing is like a sail” from this
360 poem to determine the dynamical creation of meanings. Although there is
361 no explicit word “sail,” we (the authors) discussed this poem and came to
362 an agreement what the most salient metaphor in this poem is: “Wing is 8 363 like a sail.” Figures 5 and 6 represent the comprehension of this metaphor
364 according to TINT. Before the poem comes to mind, each “sail” and “wing”
365 have made the coslice category “Sail\C” and “W ing\C”. When we read the
366 poem, a morphism f from “wing” to “sail” is excited. Then, the base-of-
367 metaphor functor f\C := (·) ◦ f : Sail\C −→ W ing\C (i.e., “(·)for Sail” is
368 mapped to “(·)for Sail for W ing”) is canonically constructed through the ex-
369 cited f. Given the dynamics of excitement/relaxation based on the strengths
370 of association µ, the base-of-metaphor functor causes the construction of the
371 metaphor meaning as an indeterminate natural transformation. For example, in Figure 5, according to the basic rule, “W ing to Y acht” is excited.9 By the fork rule, the morphism from Y acht will be searched as excited. There are lots of candidate morphisms such as “Y acht to Butterfly” to “trigonal” or to “black” but, based on the probability weight of µ in this context, we can assume that the morphism “Y acht to Ant” is excited. Although the weight of the morphism is not large in the total µ, it is natural to think its weight will be evaluated as some local maxima, since it has a relatively natural “transit point” similar to the image of “moving on some
8In our empirical studies to test TINT that we are currently conducting, we will use simpler metaphors. However in this article, we chose this example to showcase the potential of TINT. 9Here, we naturally assume that the morphism “Sail to Y acht is already excited be- cause of the expression of the poem.
17 Figure 5: “Tsuchi (Soil)”: the base-of-metaphor functor f\C is canonically constructed through the excited f. Then, the morphism from yacht will be searched according to the fork rule and “yacht to ant” will be excited. The red morphisms (objects in the coslice categories) belong to the functor f\C, the blue morphisms belong to functor F , and the purple morphisms correspond to the natural transformation section.
plane entity.” The fork rule will excite many morphisms from Sail to W ing,10 including a rather trivial morphism from “trigonal to trigonal.” Generally, excitement/relaxation dynamics make these newly constructed morphisms part of natural transformations t from the base-of-metaphor functor to some (subsystem of) another functor F :
o i b2 b1
o i f\C b2 ◦ f b1 ◦ f
t tb2 tb1 F a o a 2 h 1
\C \C 11 372 Here, b1, b2, and i are in Sail and a1, a2, h, i, tb1 , tb2 are W ing . Note
10Since the fork rule is symmetric, the inversely directed morphisms may also be excited. Here, we focus on the “Sail to W ing” direction for simplicity. 11 i (together with b1 ◦ f and b2 ◦ f) can be considered a morphism in W ing\C.
18 Figure 6: “Tsuchi (Soil)”: the “hidden” image of the sea is excited according to the fork rule and probabilistic weight of µ. These processes can contribute to associate other images such as “black,” “blue,” “death,” “life.”
373 that, by the anti-fork rule, the family of morphism satisfying non-trivial
374 commutative diagrams above tends to survive more frequently compared with
375 that with only trivial commutative diagrams. As a result, these morphisms
376 construct a family of (subsystems of) natural transformations from the base-
377 of-metaphor functor f\C to a new functor F (metaphor as a functor): the
378 indeterminate natural transformation from the base-of-metaphor functor as
379 the creation of the meanings of metaphor.
380 The meaning creation explained above also includes a fairly non-trivial
381 process. Let us focus on the pair of “W ing to Soil” and f. By the fork rule,
382 some morphism “Sail to Soil” will be searched to be excited. Although it is 383 not easy to find a direct path, it will be searched using some indirect path b2 ◦ 384 f = tb2 ◦b2 , with the “transit point” Sea. As a result, the “hidden” image of 385 the sea is excited and morphism tb2 becomes part of the indeterminate natural 386 transformation. In other words, the indeterminate natural transformation 12 387 makes a non-trivial fusion of the world of soil and that of the sea. Maybe
388 these processes continue to make non-trivial fusions of “black” and “blue” or
12The aesthetic importance of this non-trivial fusion caused by the “hidden” image of the sea is discussed in Saigo(2005) [13], for example, where the essence of metaphor comprehension is analyzed in terms of the interrelationship between various images without any mathematical formulation.
19 389 “death” and “life,” for example (see Figure 6). This continuation phenomena
390 will expand and enrich our meaning of this metaphor. In short, TINT can
391 also explain the function of a metaphor as a method of creating “profound”
392 meanings.
393 6. Discussion
394 6.1. Characteristics of TINT
395 In this paper, we proposed the TINT as a new theory of metaphor compre-
396 hension. Several studies have been conducted to reveal the cognitive process
397 of metaphor comprehension [3, 14]. We here summarize the characteristics
398 of the TINT and discuss its contributions to metaphor studies. The charac-
399 teristics of the TINT can be summarized as follows:
400 • TINT represents the meanings of images and metaphors as a structure
401 of images using coslice categories.
402 • TINT represents the structure-to-structure interaction between a target
403 and a source as the construction of indeterminate natural transforma-
404 tions based on the base-of-metaphor functor.
405 • TINT explains the process of comprehension of novel metaphors as
406 co-existing with various interpretations by its indeterminacy.
407 • TINT models both the static meaning space by the weight of mor-
408 phisms and the dynamical processes of metaphor comprehension by
409 excitement/relaxation dynamics.
410 The most important process of novel metaphor comprehension is the con-
411 struction of novel meanings by the interaction of the meanings of the tar-
412 get and source. These meanings are determined by the relationship among
413 images, which can be regarded as the structure of images. Therefore, it is
414 natural to explain the interaction between the target and source by structure-
415 to-structure interactions.
416 TINT can treat this structure-to-structure interaction as the construc-
417 tion of indeterminate natural transformations based on the base-of-metaphor
418 functor. Metaphor comprehension can be regarded as searching or construct-
419 ing a meaningful functor between the target and source. However, when we
420 search such functors one by one, we should search all possible correspondences
20 421 of objects and morphisms between the coslice categories of the target and
422 source and also check their compositionality. However, this process implies a
423 large amount of computation. The idea of the base-of-metaphor functor and
424 its natural transformation can relieve this problem. The base-of-metaphor
425 functor gives us one structure corresponding to the source structure; then,
426 we can search more meaningful structures by shifting its structure using the
427 natural transformation on the target coslice category. Since a natural trans-
428 formation is actually made of morphisms of the coslice category of the target,
429 it is easier to search for an appropriate functor. Furthermore, this process
430 can be aborted during optimization. Even then, we can comprehend some
431 meanings of the metaphor by constructing a better functor.
432 We treat novel metaphor comprehension from the analogy position (cf.
433 Holyoak and Stamenkovi´c,2018[1]), which involves the construction of a
434 mapping from the source to the target (i.e., a functor between the coslice
435 categories generated from the source and target). However, the search of a
436 mapping may be computationally intractable (see, e.g., Wareham and van
437 Rooij, 2011[15]). TINT enables an efficient search of an analogical map-
438 ping because it starts with a trivial mapping (base-of-metaphor functor) and
439 shifts gradually to a more meaningful mapping through the construction of
440 a natural transformation. The behavior of the TINT algorithm is similar to
441 an anytime algorithm in that it returns an output (natural transformation)
442 anytime it is interrupted. However, it is in stark contrast with the structure-
443 mapping engine [16], which is an implementation of the structure-mapping
444 theory [17], where the computational stages are clearly divided into the enu-
445 meration of mappings, evaluation of mappings, and construction of global
446 mappings.
447 Previous studies do not treat this structural interaction effectively. Al-
448 though Bowdle & Gentner (2005) [3] tried to introduce structure-to-structure
449 interactions to study metaphors using the structure-mapping engine (SME)
450 [17, 16], they eventually relied on one-to-one image relational mapping and
451 did not consider the structural interaction itself. Their algorithm also seems
452 not able to avoid the large computational effort. Furthermore, when SME
453 searching the best mapping between target and base, they use knowledge
454 representations tailored to the problem at hand [18]. The knowledge ap-
455 propriate to find analogical mapping is represented as predicate logic. such
456 as “REVOLVE(planet, sun),” and this process of representation seems un-
457 clear. TINT needs only probabilistic weights between images and eases this
458 problem of formatting the representation.
21 459 Other studies modeled the process of metaphor comprehension as a com-
460 position of vectors of the target and source represented in a semantic space
461 with metrics. For example, Kintsch (2001)[14] proposed the predication algo-
462 rithm to model metaphor comprehension in a structural manner. Predication
463 algorithm represents the meaning of words as a vector in a high-dimensional
464 semantic space using latent semantic analysis. Then, in short, the vectors
465 most relevant to the target and source are selected and combined with the
466 centroids of the target and source. Although it seems important that the
467 predication algorithm tries to model the context of the meanings of the tar-
468 get and source, this algorithm compresses the information on contexts and
469 interaction between meanings as the composition of vectors, despite their
470 philosophy. To us, vector composition seems too simple to represent the
471 structural interaction, at least for explaining the novel construction of pro-
472 found meanings discussed in the previous section for constructing indetermi-
473 nate natural transformation. In short, the composition of vectors compresses
474 structures, while the natural transformation preserves them.
475 6.2. Future research directions
476 We assume TINT can be used as a new platform of the interplay between
477 meanings, metaphors, and morphisms. This paper proposed the fundamental
478 concepts and working hypotheses of TINT as a first step.
479 As a next step, we will have to sophisticate the “indeterminate” process
480 in a mathematical and cognitive manner. Furthermore, we should attempt
481 simulations based on TINT using some corpus that reflects the space of im-
482 ages and semantic structures to show the adequacy of TINT. This line of
483 thinking is compatible with the idea of word embedding, meaning we can
484 approximately use these semantic spaces with metrics that can be calculated
485 by appropriate algorithms, such as in word2vec [7, 19]. Since the TINT algo-
486 rithm is quite simple, it will probably be easily implemented and simulated
487 if we obtain the appropriate semantic space from these corpora and soft-
488 ware. We are also planning to conduct experiments employing subjects to
489 verify this theory. For example, we can control the excitation/inhibition of
490 images using the priming technique and check changes in the interpretations
491 of metaphors.
492 Let us sketch the computational/empirical studies for TINT to come. As
493 TINT is an algorithm, it has inputs and outputs. As a cognitive model,
494 we prepare the input from human experiments and evaluate the output in
495 terms of its explanatory power on the data on metaphor comprehension by
22 496 humans. The input is the latent category that has arrows weighted by µ,
497 the strength of association. For a specific metaphor “S is like T ”, we can 498 pick a few words S1,S2, ..., Sk and T1,T2, ..., Tl that are associated with S 499 and T, respectively, to form their coslice categories. The complete directed 500 graph with the node as the set of words {S, S1, ..., Sk,T,T1, ..., Tl} can then 501 be given µ by asking human participants how strong they would associate w ′ 502 with w . This way, we determine the toy semantic space as the latent category
503 on which TINT performs metaphor comprehension. The latent category can
504 also be partially determined using the cosine similarity and/or distance in the
505 vector embedding by word2vec or other algorithms. The output of TINT, the
506 functor F that represents a metaphor comprehension constructed by BMF
507 and INT, can then be evaluated for its appropriateness by comparing it to
508 how humans would form the correspondences from the coslice category of S
509 to the coslice category of T as a result of their metaphor comprehension. The
510 output of TINT is a single distribution of correspondences made by several
511 random seeds. In this article, we propose TINT as a novel theory of metaphor
512 comprehension. The axioms and rules are constructed based on the concept
513 of TINT as a working hypothesis. We will test and improve these axioms
514 and rules through these verifications.
515 One theoretical significance of the dynamic development of categories is
516 that we can make the category theory more dynamic so that it can be applied
517 to a broader range of phenomena. In fact, our idea of indeterminate natural
518 transformation is nothing but a kind of a “random walk” on a category whose
519 objects are functors and where morphisms are natural transformations. The
520 notion of a random walk on a category seems not to have been investigated,
521 likely because of the social distance between the category theory community
522 and the probability theory community. However, it is the fusion of two
523 theories that we actually need for the dynamical understanding of metaphor
524 comprehension.
525 Acknowledgement
526 The authors would like to thank Shohei Hidaka and Shunsuke Ikeda for
527 helpful discussions. This work was partially supported by JSPS KAKENHI
528 Grant Number 17H04696.
529 [1] K. J. Holyoak, D. Stamenkovi´c,Metaphor comprehension: A critical
530 review of theories and evidence., Psychological bulletin 144 (6) (2018)
531 641–671.
23 532 [2] C. S. Peirce, The Carnegie Application, L 75 (1902) 13–73.
533 [3] B. F. Bowdle, D. Gentner, The Career of Metaphor, Psychological Re-
534 view 112 (1) (2005) 193–216.
535 [4] S. Glucksberg, B. Keysar, Understanding metaphorical comparisons:
536 Beyond similarity., Psychological review 97 (1) (1990) 3–18.
537 [5] G. Lakoff, M. Johnson, Metaphors we live by, University of Chicago
538 Press, Chicago, 1980.
539 [6] W. Kintsch, Metaphor comprehension: a computational theory., Psy-
540 chonomic bulletin & review 7 (2) (2000) 257–266.
541 [7] T. Mikolov, K. Chen, G. Corrado, J. Dean, Distributed representations
542 of words and phrases and their compositionality, in: Advances in neural
543 information processing systems, 3111–311, 2013.
544 [8] A. Tversky, Features of similarity., Psychological review 84 (4) (1977)
545 327–352.
546 [9] S. Mac Lane, Categories for the Working Mathematician, 2, Springer,
547 New York, 1998.
548 [10] F. W. Lawvere, Functorial semantics of algebraic theories and some al-
549 gebraic problems in the context of functorial semantics of algebraic theo-
550 ries, Ph.D. thesis, Columbia University, reprinted in: Theory and Appli-
551 cations of Categories, http://www.tac.mta.ca/tac/reprints/index.html.,
552 1963.
553 [11] M. Makkai, R. Par´e,Accessible Categories: The Foundations of Cate-
554 gorical Model Theory: The Foundations of Categorical Model Theory,
555 vol. 104, American Mathematical Socety., Providence, R.I., 1989.
556 [12] T. Miyoshi, Nansoushu, Chikuma-shobou, Tokyo, 1932.
557 [13] T. Saigo, Topics and Subjects: Forms of Poetry and Nature of
558 Metaphor(in Japanese), Mitsumura Tosho Pubilishing, Tokyo, 2005.
559 [14] W. Kintsch, Predication, Cognitive Science 25 (2) (2001) 173–202, ISSN
560 03640213.
24 561 [15] T. Wareham, I. Van Rooij, On the computational challenges of analogy-
562 based generalization, Cognitive Systems Research 12 (2011) 266–280.
563 [16] B. Falkenhainer, K. D. Forbus, D. Gentner, The structure-mapping en-
564 gine: Algorithm and examples, Artificial Intelligence 41 (1989) 1–63.
565 [17] D. Gentner, Structure-mapping: A theoretical framework for analogy,
566 Cognitive science 7 (2) (1983) 155–170.
567 [18] D. J. Chalmers, R. M. French, D. R. Hofstadter, High-level perception,
568 representation, and analogy: A critique of artificial intelligence method-
569 ology, Journal of Experimental & Theoretical Artificial Intelligence 4 (3)
570 (1992) 185–211.
571 [19] T. Mikolov, K. Chen, G. Corrado, J. Dean, Efficient estimation of word
572 representations in vector space, arXiv preprint arXiv:1301.3781 .
25