Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 1 | P a g e DRAFT Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer A Topological Analysis of Space-Time-Consciousness: Self, Self-Self, Self-Other Kim, Hye Young, PhD Institut Jean Nicod Ecole Normale Supérieure hye.young.kim@ens.fr This paper attempts to explore a possibility to visualize the structure of time-consciousness in a knot shape. By applying Louis Kauffman's knot-logic, the consistency of subjective consciousness, the plurality of now's, and the necessary relationship between subjective and intersubjective consciousness will be represented in topological space. For a long time, I believed that time must flow like this, without much doubt: Figure 1 But, does it? What if time's arrow doesn't flow straight forward towards the future? I couldn't be sure if time doesn't flow like this, for example, in the shape of a knot: Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 2 | P a g e Figure 2 What is time? What do we understand by 'time'? Is time different than my time-consciousness? In other words, is there objective and subjective time? If my time-consciousness refers to subjective time, what is objective time? Can I understand time beyond my perception of time? 1 Let's say now is 19:13, June 8, 2018 at Gare de l'Est in Paris, France. It takes 30 minutes to go to the Notre Dame from here by bus. Is this objective time? I remember yesterday when I was walking home in the rain, and I think about what to have for dinner in about 10 minutes. Is this subjective time? Here I take my timeconsciousness as time and time as my time-consciousness. Then, why time? Or why time-consciousness? If consciousness refers to the perception of the subjective experiences by an autonomous individual, consciousness is, or is based on, time-consciousness. And our time-consciousness is spatial. This is an attempt to prove that the structure of time is not a straight arrow that flows forwards. In doing so, I will show how an identical subjective consciousness is possible based on time-consciousness and reveal the necessary relationship between subjective and intersubjective consciousness. The problem of 'now' is in the center of the discussion. In my discussion, I deal with two knot models: one is trefoil and the other is Borromean rings. A trefoil knot is the simplest example of a nontrivial knot, which means that it is not possible to untie this knot in three dimensions without cutting it. It is one seamless line, i.e. one ring. Figure 3 1 In this sense, the question of time is basically in the same structure as the question of being. What is being? Can I understand 'being' beyond the realm of my understanding of my own being? Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 3 | P a g e Borromean rings consist of three topological circles which are inter-linked with each other. Figure 4 Knot Logic: Linking as Mutuality Let's name the rings A, B, and C. They are woven in the way that ring A surrounds ring C, ring C surrounds ring B, and ring B surrounds ring A. There are six crossings in this model where the three individual rings are connected and separated. Figure 5 Each crossing, i.e. the link between the rings, manifests the mutual relationships between the rings. Here I apply Louis Kauffman's Knot Logic, which is a variant of set theory that allows mutual relationship (Kauffman 1995). Kauffman presents knot set theory as a diagrammatic alternative to Venn diagrams that models a nonstandard set theory and explains its relationship. Venn diagram shows the possibility of a logical comprehension of the connective tissue in topology and geometry. Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 4 | P a g e Figure 6 Each marker (where A intersects B and B intersects C) is placed in two regions and indicate that at least one of these regions is not empty. Therefore, some A are B and some B are C (Kauffman 2015, 33). Kaufmann articulates its logical connection in relation to theory of knots and links in three dimensional space. Let's say there are undefined objects 'a,' 'b' and a notion of membership is denoted 'a Є b,' which means 'a belongs to.' It will be possible for 'a' to belong to itself or 'a belongs to b' while 'b belongs to a' (Kauffman 1995, 32) Objects will be indicated by non-self intersecting arcs in the plane and a given object may correspond to a multiplicity of arcs, which will be labelled with the label corresponding to the object. And membership is indicated by the diagram as below (Kauffman 1995, 32-33): Figure 7 Here we see that 'a belongs to b.' The arc 'b' is unbroken, while 'a' labels two arcs that meet on opposite sides of 'b,' which represents that 'a passes under b' according to the convention of illustrating one arc passing behind another by putting a break in the arc that passes behind. This pictorial convention is important for the logic and the relationship with three dimensional space (Kauffman 1995, 33). Kaufmann shows this with the example of von Neumann construction of sets of arbitrary finite cardinality that starts with an empty set Φ={ } and building a sequence of sets Xn with X0={ } X1={{ }} X2={{ },{{ }}}. By using overcrossing convention for membership, one can draw a diagram of this construction (Kauffman 1995, 33): Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 5 | P a g e Figure 8 The three rings of Borromean rings mutually belong together and the crossings between the rings show their belongingness to each other. I marked the three crossings on the outer edge as a, b, c, and the other three inside as α, β, and γ. At the crossing 'a,' where the ring C is placed under the ring A, C belongs to A. At each crossing, there is a membership relationship. See figure 5. The mutuality of the three rings are as below at each crossing: crossing belongingness a C Є A B = {A, C} b A Є B A = {B, C} c B Є C C = {A, B} α A Є B A = {B, C} β B Є C B = {A, C} γ C Є A C = {A, B} The mutual membership relationship between different rings is as below, for example, in the Borromean rings: Figure 9 These sets are the sets that are members of each other. But there are sets that are members of themselves as well (Kauffman 1995, 34): Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 6 | P a g e Figure 10 These diagrams indicate sets that may have a multiplicity of identical members (Kauffman 1995, 34): Figure 11 Instead of regarding the multiplicity of identical members as all equivalent to one another to condense them ({...a, a...} = {...a...}), Kauffman suggests another solution, in which identical members cancel in pair. Thus {...a, a...} = {... ...}, which is {a, a} = { }. And it looks like below in diagram (Kauffman 1995, 3435): Figure 12 The rule of cancellation of identical members is fundamental to knot set theory, as in the second Reidemeister move: Figure 13 Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 7 | P a g e These are the three Reidemesiter moves: Figure 142 A knot consists in a single closed curve and a link may have many closed curves and a tangle has arcs with free ends. Reidemeister proves that any knot or any link in three dimensional space can be represented by a diagram containing only crossings of the type indicated as in the figure 7. And the first Reidemeister move allows the creation or cancellation of self-membership in the corresponding knot set. But if the loop is a physical loop in a rope, "the cancellation of the loop in the first move must be paid for by a corresponding twist in the rope" (Kauffman 1995, 38). Figure 15 2 Figure 14: (Kauffman 1995, 36) Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 8 | P a g e According to Kauffman, one can regard it as an exchange rather than an elimination or creation of the loop. If this is applied to the diagram that represents a twisted band, the self-membership is not lost as we move to the topology (Kauffman 1995, 39). "Any knot set has a representative that is a member of itself and the states of self-membership and non-self membership are equivalent" (Kauffman 1995, 40). Therefore, a radical knot set is a member of itself and only if it is not a member of itself (Kauffman 1995, 40). And this knot set gives a way to conceptualize non-standard sets without recourse to infinite regress through a twist in the boundary. The self-membering set is represented by a curl, where the observer on the curl itself goes from being container to being a member. This shows that membership becomes topological relationship (Kauffman 1995, 41). The same logic works in a Mobius band as well: A = -A. Figure 16 Kauffman shows that this self-membership solves Russell's paradox. Let R be the set that contains all the sets that do not include themselves. If R contains itself, R does not include itself, but if R does not include itself, R should be a member of itself. But we can solve this paradox in the domain by having every set as a member of itself and not a member of itself (Kauffman 1995, 40). One could regard it not as 'solving' of the paradoxical contradiction but as rather resolving the contradiction.3 Figure 17 3 Based on the comments from Atocha Aliseda (Professor of Philosophy at the Institute for Philosophical Research at UNAM, Mexico) for the author's lecture "Knotted Space-Time-Consciousness: Intersubjective Subjectivity" at Instituto de Matemáticas, UNAM, Mexico (Oct 9, 2018). Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 9 | P a g e Belongingness: Not-I, Knot-I What is interesting about the two-dimensionalized model of the Borromean rings is that inside the Borromean rings, a trefoil knot is nestled, i.e. in the inner structure of the mutual relationships of the rings is a trefoil knot with the outer edges open. Figure 18 In a trefoil knot, there are three crossings as in the inner structure of the Borromean rings: α, β, γ. This shows the self-mutuality, because a trefoil knot consists of one seamless ring. Therefore, a trefoil knot represents a stable self-mutuality in three loops about itself: a = {a}. Figure 19 I apply this logic of self-mutuality, i.e. self-membership to explain the consistency of a subjective consciousness throughout the change of time. Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 10 | P a g e How can a subjective 'I' at each different temporal locus a, b, and c be identical? This means that different 'I's at each point of now is the same 'I' with continuity. 'I' at 'a: now (-1),' 'I' at 'b: now (0),' and 'I' at 'c: now (1)' mutually belong together. Figure 20 'I' at each moment has to be able to observe 'I' at another moment which, as the observed, is not the same 'I' from the observant's point of view; instead these different 'I's conform to an identical 'I' that understands this connectivity and the continuity between different moments. To see how this works more precisely, I will place the knot on a plane with axes of time and space. Let's say that vertical lines indicate temporal (y), horizontal lines spatial locations (x). Figure 21 By having a plane of both vertical and horizontal axes of time and space, we can understand the spaceconsciousness which is necessarily attached to my time-consciousness. Time-consciousness is always Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 11 | P a g e spatial. If time is perceived through change which is caused by movements, it always presumes the space that is either the (back)ground of the movement to 'take place' or the act of movement itself creates (results in) space. A linear time-consciousness does not manifest the dimension of space on its single-linear structure. But when we move, either actually moving (our body) from here and there, or reconstructing our memory from then or pre-grasping our future from now, we always and necessarily locate ourselves spatially. When my time-consciousness at each now is constructed through the modification4 of my memory and anticipation, it is impossible for me to recall my experience or expect the experience in the future without its spatial location because these experiences 'take place.' Now, place a trefoil knot on this plane: Figure 22 The point 'o' is placed on the point where x and y meet. At 'o,' I am now here: x(0): y(0). 4 Edmund Husserl uses this term as well to explicate his theory of inner time-consicousness. Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 12 | P a g e Figure 23 I move – time flows clockwise. Starting from 'o' the flow passes countless moments on the line including 'p, q, r, s, t, u, v' Figure 24 If we understand 'now' as a point of 'now,' which is 'o,' from there one sees oneself in different temporal and spatial locations – in other words, I 'locate' myself. Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 13 | P a g e Figure 25 The most interesting act of seeing happens between the points of 'o' and 's' at the crossing, because they are overlapped they share the same temporal and spatial location. Figure 26 Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 14 | P a g e The act of seeing – I (o) see my (present) self (s) – is the basis of the formation of 'self-ness.' In other words, I have to be able to see that I am there as I, for me to be able to perceive myself as I. The I that is observed is I but at the same time not the I, the observer. The subjective understanding of (my)self is based on the distinction of I and not-I and its identity. We can solve the paradox of I = -I with the knot logic of self-mutuality: not-I as knot-I. Figure 27 This is what happens when one sees oneself at different points from where they are standing now, here – spatial/temporal locations. We can call this process 'modification.' This is, however, not only the modification process of time-consciousness but the formation process of consciousness. 5 The reason that I chose "Vorstellung" instead of "imagination" is to avoid the strongly connected connotation of the word related to "image" or "imago." And the "vor-" in "Vorstellung" could be interpreted as "pre-" or "fore-," expressing the "fore-grasp" of the future. Vorstellung5 I imagine myself being here/there then. Beobachtung I see myself being here now. Erinnerung I remember myself being here/there back then. Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 15 | P a g e A trefoil knot as a whole is each moment of now, in which there are different now's (the past and the future), i.e. different temporal and spatial location (self-location) in a continuous structure. Figure 28 This connection between now's at each now indicates the modification process and this modification creates the self-ness (subjective) of the consciousness. Conclusion 1. one trefoil knot represents the modification of each now (point of time). The generation of consciousness (self-ness) is based on one's awareness of temporal and spatial location which happens through the modification of multiple points of now's. My consciousness is, therefore, my spatial and temporal consciousness. Self-Mutuality as Mutuality: Mutuality as Self-Mutuality The mutual relationship in the Borromean rings between three different rings could possibly represent the mutual relationship between different 'now's of different subjects: my 'now,' your 'now,' hers, his, etc. Figure 29 Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 16 | P a g e I have already shown that the center of the Borromean rings forms a trefoil knot. This trefoil knot in the Borromean rings has individual sides formed from two different rings that surround it. Therefore, in the self-mutual structure of the identical A, we can find the mutual relationships between three individual rings of the Borromean structure. crossing Belongingness α A Є A / A Є B A = {A} = {B, C} β A Є A / B Є C A = {A} = {C, A} γ A Є A / C Є A A = {A} = {A, B} My (subjective) 'now' is composed of different 'now's, not only mine, but also others' as well. At the same time, the self-mutual (subjective) now is placed in the heart of the (intersubjective) 'mutual' now. The Borromean rings with a trefoil structure within itself manifests the fundamental and necessary relation between mutuality and self-mutuality. Conclusion 2. The logic of mutuality and self-mutuality in the Borromean rings and the trefoil knot models visualizes the fundamental inter-relation of the subjective (self) and intersubjective consciousness and their structure, through which we understand how we are aware of myself and others at the same time. Consciousness 'happens' through the interaction of the perception of self and the perception of not-self (including other objects and observed self). This double perception occurs simultaneously. Consciousness = Self (-consciousness). Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 17 | P a g e Plurality of Now's Traditionally (in philosophy), we have understood the 'flow of time' as the line of points of now, a seamless succession of the now moments. Each now is believed to have its own plane. Figure 30 On the plane of each now, spatial locations are marked. If time is to be understood as the fourth dimension that explains spatial changes, e.g. my being here now and then, my being there now and then, then the flow of time is explained by connecting the points of here and there at each plane of now, and my memory acts as the mediator of connections, i.e. the connecting lines between different points on this and that plane of now's. Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 18 | P a g e Figure 31 These multiple lines between different planes of now's can be crossed or paralleled, but these lines do not describe the 'backward' or 'repeatable' movements of temporal comprehension (modification of consciousness) between the points which modify one's construction of time, understanding it as a flow. At each point of now, there can be multiple 'I's as the subject of time-consciousness. I not only remember being just now, or before a certain amount of time, but I conceive my being in the very next moment through my imagination and expectation. On each plane of now, there are countless points that represent 'now'-ness of my time-consciousness. Firstly, different 'now's of different subjects, and secondly different 'I'-ness of the same subject on the same plane of now. In this context, I raise two questions: a) Are the 'I' on the P and the 'I' on the P′ the same 'I's? b) Can 'I' be here and there at the same time? If so, how? If the second hypothesis is true, there can be multiple 'I's of the identical subject on each plane. On the plane of time and space above, the point 'o' and 's' represent the identical here and now but at the same time different here and now, because there is space for the act of observation between the different points of 'o' and 's.' I will come back to this 'space' between 'o' and 's' more specifically. One might be able to regard each plane as each subject's plane of now. However, neither the multiplicity of now that are scattered here and there on the identical plane of each now, nor their inter-connection between different planes, is explained in the model with paralleled planes of now. The problem of paralleled planes of now is that on each sperate plane, my understanding of 'now' in relation to the past and the future, i.e. my continuous time perception of each moment of now is not fully clarified. In short, here we ought to be able to explain a) how the planes belong together Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 19 | P a g e b) how multiple different points of now on a plane belong together. But, what if the planes of 'now' at each crossing would rather look more like this, if there is a plane at all: Figure 32 Let's 'sew' a seamless string – continuous flow of the time-consciousness of an identical subject into these planes, then it would look like below: Figure 33 And there can be many of them, because there are a great number of individuals. Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 20 | P a g e Figure 34 Then if we see it from above – it would look like these tangled strings: Figure 35 Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 21 | P a g e And there are knots! and therefore we see crossings and links. Figure 36 The links of these countless knots and their mutual relationships could be explained by Borromean space (connective space): Figure 37 Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 22 | P a g e Conclusion 3. The multiplicity of now's and their inter-relations can be proven in the knots and their connective space. Time flows in the shape of knots. The connection between now's at each now indicates the modification process, and this modification creates the self-ness (subjective) of consciousness. The generation of consciousness (self-ness) is based on one's awareness of temporal/spatial location which happens through the modification of multiple points of time, which were, are, and will be all now's. When we understand the temporal modification at each now between the past and the future in a knot form, we can explain the identity of the same 'I's on different planes of now. I belong to myself, yet I don't belong to myself. This act of self-observation generates 'space' in itself. For example, in the diagram of the trefoil model of time-consciousness, the 'space' between the point 'o' and the point 's' which are the same present moment is not explicable with the model of paralleled planes of 'now.' This space between I and not-I is explainable through the links and their diagrams of the knots, because "links and their diagrams encode three dimensional manifolds" (Kauffman 1995, 111). The process of self-perception is always three-fold, which creates three-dimensional space.6 Conclusion 4. Time-consciousness is spatially constituted and perceived. Seeing myself, i.e. locating myself (objectively) necessarily requires a third person view-point, even if it is after all only 'I' who plays the third person role by observing myself in the 'not-I' position. Self-observation is the process of self-othering. As the necessary relation of the mutuality and self-mutuality logic in the Borromean rings and the trefoil knot, self-consciousness, which is subjective, is always, at the same time, intersubjective, for the intention of my self-perception is based on and points at my othered self. In this sense, the perception of my self-consciousness takes place simultaneously with my perception of not-self and vice versa. Figure 38 6 Johann Gottlieb Fichte introduced the equation of "I=-I." Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 23 | P a g e Conclusion 5. The phenomenon of the necessary inter-relation of the consciousness of self and the consciousness of the other is visualized, i.e. zur Erscheinung kommen, in knots and their links as seen above, because "knots and links form a calculus that is inherently self-referential and mutual" (Kauffman 1995, 112). Music And now why music? Because music is the most intuitive but at the same most analyzable way of our understanding of time. Once I wrote that we listen to music like this:7 Figure 39 But, in fact, this is how we 'listen' at each moment: Figure 40 We hear the flow, but we listen to each moment as a whole. Each moment of music not only comprises the whole music but itself is the whole music. The note A then, the note B now, and the note C afterwards become one in the moment, creating space where they are interwoven as in the Borromean rings. Another important aspect of music lies in the relationship between the performer and the listener of the music. This always includes the self-mutual relationship because the performer is always a listener too. The performer and the listener mutually belong together at each moment of music. This is a temporal unity, but one which is always spatial because of the distance between the performer and the listener. For a performer, either as solo or with other performers, performing music is the process of creating and experiencing each moment as a whole, where the whole universe falls into the present moment. This experience is not explained through the theory of successive notes and the flow of the melody, but it is 'creating' each moment as a whole, where the (spatial) movement of my body turns into one note, and each 7 Kim, Hye Young. "Music as Medium of Time-Consciousness: Augustine, Husserl, and Heidegger." Glimpse. Vol. 18: 75-82, 2017. Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 24 | P a g e note constructs the whole (temporal) structure of music; my consciousness as a performer and as a listener are separated (difference), but become one (identity) again. Kauffman's Universe and HYK's Self Kauffman has his knotted version of Wheeler's Universe, and here you go, my version of knotted 'Self' (consciousness) Figure 41 And this knotted 'consciousness' is placed in the eye that observes the very beginning (arche) – big bang. Figure 42 The riddle of identity–difference and subjectivity–intersubjectivity seems to have one variant of its answers in knots. Why knot. Forthcoming in: When Form Becomes Substance. Power of Gesture, Grammatical Intuition and Phenomenology of Space, Basel: Birkhäuser-Springer Hye Young Kim, IJN, ENS hye.young.kim@ens.fr 25 | P a g e *Reference Kauffman, Louis H. 1995. "Knot Logic." Knots and Applications. Series on Knots and Everything Vol. 6. (ed.) Kauffman, L. World Scientific. Kauffman, Louis H. 2015. "Knot logic: logical connection and topological connection" in Mind in Mathematics: Essays on Cognition and Mathematical Method, M. Bockarova, M. Danesi, D. Martinovic and R. Nunez (eds.): 33-57. Kim, Hye Young. "Music as Medium of Time-Consciousness: Augustine, Husserl, and Heidegger." Glimpse. Vol. 18: 75-82, 2017.