THE MYSTERIOUS CONNECTION BETWEEN ...

THE MYSTERIOUS CONNECTION BETWEEN MATHEMATICS AND PHYSICS LOUIS H. KAUFFMAN AND RUKHSAN-UL-HAQ

1. Dialogue This essay takes the form of a dialogue between two participants whose names are “Lou” and “Rukhsan.” Lou is a mathematician with an intense interest in physics, working on physical ideas and concepts from the side of mathematical thinking. Rukhsan is a condensed matter physicist with an intense interest in the nature of mathematics. This essay is inspired by a dream of John Wheeler [15] that the deepest reach of the physical world would be something like logic, like Boolean logic. He expressed this dream in the phrase “It from Bit”. Some have said that this should be modified to “It from Qubit”, and we shall get to that in the course of the essay. We beg the reader to be patient, as the dream is like the Cheshire cat and it is sometimes only possible to see its smile. Along with the references directly cited in the essay, the following references will be useful to a reader who wishes to follow up our themes and ideas [2, 6, 7, 8, 9, 10, 11]. Lou: Commonly, mathematics is regarded as a developing language (including meanings and concepts, not just a formal language) that allows the analysis and classification of patterns (including its own patterns) and the qualitative and quantitative analysis of these patterns. We are all familiar with the remarkable matches between mathematical structures and physical law such as the fit with experiment of Newton’s Law of Gravity which, expressed at the level of acceleration, is said with simple and elegant mathematics and encompasses a vast range of phenomena. The mysterious part of this correspondence of mathematics and Nature is that relatively simple mathematics, or simply stated mathematics, has been successful in describing models for complex aspects of the natural world. We shall address the question “Does physics simply wear mathematics like a costume, or is mathematics a fundamental part of physical reality?”. In his book [1] , “Laws of Form”, G. Spencer-Brown says: “Now the physicist himself, who describes all this, is, in his own account, constructed of it. He is, in short, made of a conglomeration of the very particulars he describes, no more, no less, bound together by and obeying such general laws as he himself has managed to find and record. · · · Thus we cannot escape the fact that the world we know is constructed in order (and thus in such a way as to be able) to see itself. This is indeed amazing. Not so much in view of what it sees, although this may appear fantastic enough, but in respect of the fact that it can see at all. But in order to do so, evidently it must

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LOUIS H. KAUFFMAN AND RUKHSAN-UL-HAQ

first cut itself up into at least one state which sees, and at least one other state which is seen. In this condition it will always partially elude itself.” ([1], Notes to Chapter 12, p. 105) Spencer-Brown reminds us that the Universe is constructed to be able to see itself, and we are constructed to be able to see the Universe. We begin, not with mathematics as a known formalism, or with physics as laws expressed in mathematical form, but with the condition of an observed world, a world in which it is possible to have a division of states into that which sees and that which is seen. One can begin with the idea of a distinction and naturally unfold the elements of mathematics. In set theory the simplest distinction is the empty set, denoted by {} The empty set is not nothing. It is the result of collecting nothing and so has nothing inside it. Sets are referential only to their contents (and not to their complements except indirectly). Two sets are equal exactly when they have the same members. For this reason there is only one empty set, and we can proceed to build more sets by the act of collection. Having created the empty set we can now create the set whose member is the empty set. {{ }} Then we can create the set whose members are the empty set and the set consisting of the empty set. {{ }, {{ }}} These sets are representatives of the numbers 0, 1, 2 since they have respectively zero, one and two members. We write: 0={} 1 = {{ }} = {0} 2 = {{ }, {{ }}} = {0, 1} 3 = {{ }, {{ }}, {{ }, {{ }}}} = {0, 1, 2} The rest of the natural numbers are created by the inductive formula n + 1 = {0, 1, 2, · · · , n − 1, n}. That is, to create n + 1 make a collection of the previously created sets up to n. The number concepts 1, 2, 3 are not identical with sets that have that many members. In fact, to take a case in point, 2 refers to any double. Thus I can say that I have 2 arms. If we use the substitution of {{ }, {{ }}} for 2, then I would say “I have {{ }, {{ }}} arms!” This is comprehensible. It means the same as if I had said “I have [holds up two fingers] arms.” I make reference to an instance of 2 in the course of asserting that my arms are two. In mathematics we often take the elements of this collection hierarchy as the standard

THE MYSTERIOUS CONNECTION BETWEEN MATHEMATICS AND PHYSICS

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instances of the numbers and give to these sets the names of the numbers. It is common to learn a counting system such as the arithmetic we all learned in school and to regard the formalities of this system as the “mathematics.” We work with this and it is part of our understanding. Thus I can find 172 by mental arithmetic by thinking 17 is 10 plus 7. And 10 squared is 100, 7 squared is 49, twice 7 times 10 is 140. So 17 squared is equal to 100 plus 49 plus 140 which is 289. So 17 squared is 289. Mathematical forms can be manipulated like objects. Mathematics is the art of doing this sort of work. But that is not all. After all, we are now confident that a square array of dots 17 × 17 will have 289 marks in the square array. We are sure that if we count the marks in the array in some order then we will get 289. We are confident that the meaning of the numbers includes the regularity of the correspondence with counting and matching that is their meaning. The numbers are not just counters in a game that we play using the rules of arithmetic. Each number has a conceptual meaning in relation to all possible multiplicities of that order (to which it can be mapped by a one-to-one correspondence), and it is this conceptual meaning that allows us the use arithmetic in the world. In fact, without the formalities and the concepts of arithmetic we would not be able to work with multiplicities in the world. One could say that without the language of arithmetic and the understanding of its meanings, the world (perceived by us) would not have multiplicity. A world without multiplicity would have quite a different physics than the one we know. Now you can begin to see how physics and mathematics are related. As we grow mathematics, we expand the possibilities for physics. As we make new observations, we grow new possibilities for mathematics. Counting in the social realm came from physical necessity. Persons needed to count their sheep and not just to go to sleep! But I must talk about non-numerical worlds. These days everyone is familiar with the formalities of Boolean algebra. It is the mathematics of computers, of patterns of logic, patterns of sets, the properties of switching circuits and transistors. It underlies all of our practical mathematics. In the Boolean arithmetic that everyone knows there are two values 0 and 1 and an operator that switches them: ¯0 = 1 and ¯1 = 0. There are two ways that the elemental opposites 0 and 1 interact, called plus (+) and times (×) and we have that 0 times anything is 0 while 1 times anything is that thing. So multiplication behaves just like the old numerical values of zero and one. Addition is almost the same. 0 plus anything is that thing. But one plus one is one! 1+1=1 Now you see how the world of Boolean arithmetic has no multiplicity beyond one. It is the world of one only. Try to count beyond one, and you are returned to 1, returned to unity! Of course we understand this from the examples. Two closed switches in parallel are no different that one closed switch. Either closed switch will carry the current. Two calls of your name are the same as one call of your name. The other is redundant. But go further! Live in the Boolean world! You have logic, reason, either, or, and neither. But you have no time, and you have no ability to count beyond 1. Lewis Carroll in his classics “Alice in Wonderland” and ”Through the Looking Glass” played with his heroine Alice,

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projecting her into just such worlds. Worlds where you have logic, you have the Boolean, but you do not have the customary multiplicities and regularities. Mathematicians are people who live half their lives in the eternal timeless world. For them it is the Reality, the holocosm from which emerges (just as we have see above with the multiplicity of numbers coming from the distinctions of set creation) at once and inevitably worlds of unimaginable multiplicity. Because mathematicians are exploring the holocosm beyond multiplicity, when they do create multiplicity that multiplicity is not constrained by any prejudice from the physical world. Thus you hear mathematicians working with higher orders of infinity, and even hoping to approach the meaning of absolute infinity. All this is possible by working from the holocosm where there are no numbers, no multiplicities, no infinities and no finity other than zero and one, nothing and something. This is the freedom of mathematical thought and to the mathematician, the question of the possible relation of mathematics and “physics” is utterly inevitable. By leaving the holocosm for worlds of possible multiplicity, the mathematician understands that he may meet examples of these phenomena. If they can be imagined they may be encountered, and if so we will point to an amazing correspondence between mathematics and physics. Rukhsan: You are trying to say too much! Let me slow you down and ask some questions. First of all, what do you mean by the “holocosm”? Lou: The holocosm is a metaphor for the conceptual arena where a mathematician thinks and understands with little or no calculation. For example, I understand that there are infinitely many primes by the famous argument of Euclid that asks one to consider the number N = 1 + p1 × p2 × · · · × pn where {p1 , p2 , · · · pn } is any finite set of primes. One sees that N is not divisible by any of these primes and so N must itself either be a new prime or have a divisor that is a new prime. Euclid’s argument shows that no finite set of primes can be all primes. There is no calculation in Euclid’s argument. It is all form and so it applies to all sizes of numbers if we must consider the sizes of numbers. Rukhsan: So you are saying that mathematical proofs like this are really happening in a conceptual domain distinct from calculation? Lou: Yes. And I call this domain the holocosm. It is a metaphor for the place of thought in mathematics and I shall use it that way as we speak. Rukhsan: You speak of non-numerical worlds. And you use the Boolean world as an example. Surely this is not all. Is not geometry a fundamentally non-numerical world? Lou: Yes and no! All our concepts of number can be related to the measurement of geometrical figures. And as you know the emergence of irrational numbers in geometry was a scandal for the ancient Greeks. But much of geometric understanding is certainly not tied to such measurement. We understand the Pythagorean Theorem via the relationships in the geometry and not by just measuring the areas of the squares on the sides of a right triangle. The proof of the theorem applies to any size of right triangle. It does not use


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anything but the properties of such triangles. Rukhsan: Alright. Now, you quote Spencer-Brown as saying that “Thus we cannot escape the fact that the world we know is constructed in order (and thus in such a way as to be able) to see itself. This is indeed amazing.”. This makes the world appear as though it were the result of some great magic trick, or perhaps the illusion of stage magician, who could also make it disappear at will. Spencer-Brown laid no ground at all for this conclusion of intelligent design that he cannot escape. Darwin escaped it. The world evolved the ability to see itself because structures that can see reproduce more than those that cannot. From my point of view as a physicist, it is perfectly possible to have a world that is not aware of itself. Most of the parts of the world that I study have no self-awareness. Lou: Ah Rukhsan, you are taking those words in a complexity that is not intended by Spencer-Brown, at least as I read him. The sense of world perceived inevitably entwined with the perceiver of that world is an enthusiastic and child-like direct view of world. It is clear that world and observer appear together in the perception that also perceives itself. The question that comes later – how does this happen? – leads to the evolutionary theories of the rise of consciousness from the inanimate. Our first world, in the eye of direct perception, is an animate world where no object is really fully separate from us. We have evolved in language, action and technology to see the world as separated more substantially into a part that sees (ourselves) and a part that is seen (as in the world of physics). Our evolution in this regard is significant and an important part of science, but we must remember that it is all enfolded in the condition that each of us can indeed perceive. This is part of the tension between the timeless (object of mathematical deduction like Euclid’s) and the temporal (like the process of physical induction of Francis Bacon). Rukhsan: I see what you mean, but indeed the advent of the evolutionary hypothesis, obviating intelligent design, is a great advance in science and freedom of investigation. If we accept it, we have to ask how indeed our intelligence can, if it can, be the result of an evolutionary process. Now you say the mathematician has no use for time, but what about recursions and calculations. They all take a lot of time and seem to need multiplicity. Lou: Yes. I said that the mathematician spends half his life in the timeless holocosm of the non-numerical Boolean world. The other half is spent “out here” in the world where we can build computers and calculate and remember and take our time to tease out the consequences. But you will notice how it is done. Each bit of mathematics that seems to have something to do with time is expressed as structure and can be comprehended in a definition that does not require any definite amount of time. For example. I say that the Mandelbrot set M is the set of complex numbers z such that the iteration of the function f (w) = w 2 + z starting at w = 0 goes to infinity. This is a statement that is understandable in the timeless eternity of the holocosm. No calculations are necessary to grasp the definition, once it is grasped. Oho! You almost caught me there. I heard you

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about to object!! Yes it takes time to get straight what the definition says. You have to read and reread it. But it is all a matter of logic, and once you have it you have it in eternity! That is why mathematicians are skittish about computers. They want to take it with them back into the holocosm, and if a result depends on temporal computation and cannot be eventually instantly understood, then it is suspect. But oh my, there is a wealth of phenomena in that Mandelbrot set that we need the computer to see. It is all implicitly there, but we are as curious as the next person and so we are pulled out of the realm of pure logic and find ourselves glued to computer screens. Rukhsan: So would you say that when you are not in the eternity of the holocosm, then you are in the physical world? Lou: When you leave the holocosm, you enter a realm where multiplicities can be instantiated. It is not obvious that such a world would exist, but it does and we experience that world. We understand from a number of points of view in the holocosm how such worlds may be structured. For example, we have the forms of recursion in the holocosm. In the holocosm, we can write an equation that refers to itself, an equation of higher degree. For example we can write G = F (G). And in the holocosm, this is just an equation (actually to write it at all, we are peeking through into a world of memory but let that be and understand the amphibian nature of the mathematician!). There is no need to resort to recursion here. But in the outer world we are can recurse and then G = F (G) = F (F (G)) = F (F (F (G))) = F (F (F (F (G)))) = F (F (F (F (F (G))))) = · · · . At once the exit from the holocosm releases not only some multiplicity, but a potential infinity that cannot be fulfilled in the present known world. From the mathematician’s point of view there is no reason why the phenomenal world might not allow an “actual” passage to infinity. Some cosmologists imagine that the physical world does indeed contain passages to infinity or infinitely many parallel worlds! There is no evidence for such things, but this does not bother the mathematician. From his point of view, just leaving the holocosm, anything is possible. Rukhsan: You did not answer my question. What do you think is the composition of the physical world? Is it made out of mathematics? Lou: I was not talking about the composition of the physical world, except indirectly. The question of the composition of the world requires us to back up and ask what every word in that question means. Look carefully at what you know and at your actions. When we say that this wall is composed of bricks, we know it because we know how to build the wall by taking entities called bricks and putting them together in a certain way. But if I say that numbers are made by forming certain sets, this is obviously just a way to get to the concept of a number. And the concept of a number is made from thought. Thought is not composed of pieces like bricks. Mathematics involves concepts, particular languages that embody those concepts and many examples, some conceptual and some physical. The physical examples are still conceptual, but they involve actions in the world that we regard as physical. Archimedes weighed an irregular object by noting how much


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water it would displace. Here mathematics and actions in the world combine to give us new informatiion. But this does not indicate that the world is composed of mathematics! In fact after all this saying just now, I have to admit that I do not know what it could possibly mean to say that a ‘physical object’ is ‘composed of’ mathematics! Rukhsan: I still feel that you are avoiding the question. As a physicist, I work with the notion that substances are composed of atoms, atoms are composed of electrons and protons, protons are composed of quarks and so on. Furthermore, as we delve more and more deeply into elementary particles we find that they seem to be characterized entirely by their mathematical properties. It would seem that in the end, all that is left of matter is mathematics! Lou: If we start with Mathematics as what mathematicians do, we find something very different: mathematicians describe structures, and they prove things about them. Now, it is a simple theorem – call it E – that the sum of two odd numbers is always even. We have found that we can embed arithmetic in devices, in Babbage’s brass or Intel’s silicon. Whenever a device adds two odd numbers it does deliver an even number (or an overflow). But where, in this, is theorem E? Does the device contain it, embody it, or only (in some sense) obey it? We embed the rules of arithmetic in silicon, but not the theory of those rules. Physics looks much more like the algorithms that act in our computers, than like the proofs that fill the books about those algorithms. Rukhsan: I see. You are again concerned with the distinction between the holocosm and the outer world. In the holocosm, we see the proof (so simple) that the sum of two odd numbers is even. Our proof has nothing to do with any specific calculation. If we instantiate arithmetic in a physical process, then this rule will be obeyed, because it follows from the properties of arithmetic. But I imagine that elementary particles are completely characterized by their properties. In this sense, they are like entities in mathematics, and sometimes I imagine that they are no different than mathematical forms. Perhaps at the most elementary level, the physical world and the Platonic world of forms are identical! Lou: One could look for pure form in a physical object. I imagine Michelangelo felt that his David embodied the pure form of the rock of which it was composed and was not just an imposition of the David form on this particular block of stone. Physicists hypothesize that all electrons are the same and that the mathematical form of the electron is the electron. But by the time you get to speaking of electrons you are actually talking about certain modes of observation that sometimes exhibit particulate properties (e.g. dots on a phosphor screen) and sometimes other less local properties. A host of ideas and mathematical ways of geometrizing are combined to make the concept of the electron useful and matching with the actions and observations of experimentalists. Simple localized objects have disappeared. It is as if Michelangelo had carved the David so well out of rock that all the rock vanished and there was still something left to appreciate. I can give you an example that is closer to home. Consider a closed loop of rope that is knotted. The knot is not the rope but the rope forms a knot in space. The very same knot can be woven

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in many different types of rope. A mathematician studying the topology of knots (as opposed to friction and other matters) does not care about the substrate of the knot. He cares only about the form of the knot. And the form can be represented by any or all ropes that are sufficiently long and sufficiently flexible. But you have never seen a knot that was not tied in something, even if that something was an imaginary curve of the sort that mathematicians like to sketch on a blackboard. But for the mathematician that knot exists in the eternal holocosm of non-numerical forms. There is a desire to make this holocosm the basis of the physical world. I cannot assent to that unless we explore how ideal entities like numbers and knots are related to our experiences. Rukhsan: So you would identify the physical world with our experiences. Is not mathematics, thought and concept the basis of our experience? Lou: We have experiences and we have thoughts and concepts. They happen together, but thought is singular in that thought can examine itself. It is this that lets the mathematician proceed from the holocosm, but he still needs experiences. Each experience is accompanied by some concept. I look at the light in my room and ‘see’ a lamp. The concept lamp fits that light and fits a web of explanation about it going through all I know about electricity and too much to say. We meet experiences that do not meet their best concepts until later. It took a long time to understand the laws for falling bodies or the unity of electricity and magnetism. The concepts that unify bring us closer to the core of our experience. I take experience to be a mystery that we explore and it is through concept and particularly mathematical concepts that we can come to terms with that experience. In this sense, mathematical concepts are the basis of our experience. Rukhsan: Certainly there are experiences that are beyond mathematics. Shakespeare says “Juliet is the Sun.”. Surely we understand this metaphor, but we do not pretend that it is mathematics. Oh. Well. We also do not pretend that it is physics. Maybe physics is exactly that part of our experience that can be understood with mathematics. If that is so, then our discussion would seem to be over. Lou: Even if physics were that part of our experience that can be understood with mathematics, we shall have to look at that word “understood.” Some who like the calculational side of mathematics best, have said that a few pages of code should suffice to simulate the universe. Suppose we had this simulation in hand. Would we be satisfied? No! To use only the simulation would require ‘running the Universe’ and seeing how ‘it’ behaved. We would have lost the unification that conceptual work gives. We would not have understood. The understanding that we seek is in the interface of the conceptual, non-numerical holocosm and the realms of experience, calculation and simulation. This is a continuing mystery. Rukhsan: You started by characterizing the holocosm as the place best described by logic alone, or thought alone or perhaps the Boolean arithmetic. I recall that long ago John Wheeler suggested that the universe could emerge from just that. He said “It from


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bit.” Since then we have seen much work on informational bases for quantum theory and it may be that the Wheeler motto has evolved “It from Qubit.”. Comment? Lou: Look at the Boolean arithmetic. It has two values, zero and one. And it has an operator that switches these values. In logic the operator is negation, ∼, and the values are true (T) and false (F). Ludwig Wittgenstein [16] had this to say about negation: “How can the all-embracing logic which mirrors the world use such special catches and manipulations? Only because all these are connected into an infinitely fine network, to the great mirror.” ([16], Tractatus 5.511) ‘‘ ... the sign ∼ corresponds to nothing in reality.”. ([16], Tractatus 4.0621). I prefer that the mirror be seen as the amphibian boundary between the logical timeless world and the world of experience and action. Rukhsan: How will you modify the already very simple Boolean arithmetic. It seems you want to remove the separation between the operator of negation and the two states of 0 and 1. Lou: Yes. I want to do that. And I have to give up something, but it is something that I really do not mind giving up! There are two symbols for the two Boolean states. Lets give up one of them! We can call that one the unmarked state. If we want to name it, we will use the letter U, but remember that state is actually not indicated by anything. The other state will be called the marked state and denoted by M. These states can be 0 and 1 or F and T, or whatever distinct states you have in mind. Rukhsan: And this will make a difference in the boundary structure? You will include the great mirror in the world itself and also in the holocosm? Lou: Yes. Here I will explain the discovery of the Calculus of Indications of G. SpencerBrown [1]. I will take the following notation for negation. ∼a= a . Then we have U =M and M = U. But now you have to take me seriously and allow the unmarked state to be unmarked! Then we have =M and M = .

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Something incredible and wonderful has happened here! We see that the marked state can be identified with the operator symbol itself! And so we can further write = . What is on the right is unmarked and we have a fundamental equation for a deeper version of Boolean arithmetic where the operator symbol is seen as a distinction itself, as the marked value in the arithmetic and as the operator that switches between the arithmetic values. There is one more fundamental equation. It is =

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This is just the statement that “A mark with its name (mark) appended is still marked.” or (with a slide of metaphor) ”A rose is a rose is a rose.” It is most economical to also give up the sign for “or” and use “AB” for “A or B.” Then stands for “Marked or Marked.” Rukhsan: Now the negation operator is also the marked value in the arithmetic. The two ways of seeing this are right there when you examine the symbol. This is a mark itself making a distinction in the plane. This is an operator, operating on the unmarked state to produce the marked state. This is both of those notions at once. The negation sign is not separate from the world. The Great Mirror is more like a Great Permeable Membrane or cell boundary, with logic,mathematics and the timeless holocosm on one side and the world of calculation and experience on the other. From here we can begin to explore anew how physics and mathematics are entwined. Lou: Yes. And I will give you a hint about how we can proceed. The mark can be seen as a “logical particle” whose counterpart in the mathematical physical world is a Majorana Particle [3, 13, 14]. A Majorana particle is the simplest of possible particles. It is its own anti-particle and it interacts with itself either by annihilating itself as in −→ or interacting with itself to produce itself as in −→

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Rukhsan: I see that you have used a process arrow in this description of the interactive properties of our logical particle. The arrow has shifted us from the holocosm of timeless form to the world of process. But are there any real Majorana Fermions? It seems that this may be so. The mathematics of electrons suggests that each electron is a linked pair of Majorana particles, and recent experiments [5] appear to confirm this indirectly. Also, vortices, collectivities of electrons confined to flat surfaces as in the quantum Hall effect [4, 12] can be understood by viewing the collectivities as quasi-particles that are Majorana particles. These quasi-particles are just like our logical particle and they have non-trivial phase relationships in the plane when then move around one another. This means that there is significant braiding of the world lines of these quasi-particles spinning around one another, and it is hoped that these systems will produce the first topological quantum computers. Up to density all finite dimensional unitary transformations can be


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produced from the phase relationships of these vortices. Topological phases of matter are described by mathematics that is just this side of the holocosm! is a logical Majorana particle. It is the “bit” Lou: It is very simple. The mark from which the pre-geometry of topological matter will emerge. But in the emergence that “bit” becomes a “qubit”, a quantum Majorana particle that can be in superposition with the unmarked state and undergo quantum interaction. Phases and braiding fill in the picture and start to give us all the background for quantum physics and topological quantum computing. Perhaps we are witnessing the very thing we have been discussing – new physics and new mathematics arising through the permeable boundary between the conceptual worlds and the worlds of experience. Wittgenstein’s Great Mirror has been replaced by that permeable boundary and we are at the beginnnings of a great adventure. Rukhsan: Again we are going too fast here. For one thing it is a big leap to consider superpositions of states as in quantum mechanics, and to use the physicists’ model of observation, where a superposition of marked and unmarked states would be observed to be either marked or unmarked with some probability. All this is a lot of structure to just suddenly add on to the very pure interactions of the mark as logical particle. Lou: I agree with you. I do not know how to simplify the foundations of quantum physics so that they are as simple as the mark itself. I would like to find my way to such a place. I am only making a jump to the pre-logical domain of the mark and pointing out that it has a pattern of elementary particle interaction! We can keep thinking about this. I certainly will think about it. Rukhsan: As we move away from the purity of the marked and unmarked states, complexities occur. The braiding of Majorana particles is a first step into pre-geometry (topology). We want our models to match the observed complexities of condensed matter physics. We hope to be able to use this knowledge to extend calculation into the quantum domain to create topological quantum computation. Neither all the mathematics, nor all the concepts have yet come forth. It is work in progress that will change our understanding of the world, our understanding of mathematics and our understanding of ourselves. Lou: This is an adventure at the very foundations of physics, mathematics and the roots of thought. The physicist is inseparable from the Universe herself. It is the Universe that studies herself through the articulations of mathematics and the observation of experience. The key lives in that moving permeable boundary of the holocosm, allowing the world to come to awareness of itself.

2. Epilogue Early in this essay, we adrressed the question “Does physics simply wear mathematics like a costume, or is mathematics a fundamental part of physical reality?”.

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The two possibilities are stated metaphorically, but they are not truly mutually exclusive nor together exhaustive because they (and Spencer-Brown and Wheeler) tacitly assume that mathematical theories are exact. Kolmogorov even said that probability has no place in an exact science. The idea that mathematics is a fundamental part of Nature is a form of the mystical doctrine of intelligent design, and dates from the Logos of Parmenides. Costumes evolve and need not fit exactly. Perhaps the costume metaphor leaves room for statistical inexactness, like quantum theory. Perhaps we should readily accept that Nature has no exact description, since all descriptions are small parts of Nature. We speak to exactness. Indeed mathematical theories are exact. This is because mathematics is a form of fiction and it occurs in language. In language we can have the words marked and unmarked. We can state axiomatically that a distinction has two sides, one marked and one unmarked. What is not unmarked is marked. What is unmarked is not marked. The language and the axiomatics leave no room for middle ground. If we then want middle ground, we legislate that as well. We go on and axiomatize number and geometry, sets and categories and all the structures in the panoply of mathematics. The fiction has a reality of its own, the reality of what is allowed by the acts of definition and the axioms. Nowhere in the worlds of experience and measurement does such exactness occur. The exactness is a fiction that has a reality in the games played by the rules that have been accepted by the players. We play these games. We use such games to model the way Nature behaves, as we measure her. We are part of Nature and in that sense, Nature does play these games. But, unaided by our models, does Nature play the games we have found? Does Nature solve the differential equations of Newtonian and Quantum Mechanics? This we cannot know. In this essay we have focused on a singular place in the interaction of mathematics and the world. We have focused on the role of the sign/operator of negation ∼ . This singularity is worth appreciating since we use the logic of negation ubiquitously in handling our descriptions of the world. And it is worth focusing because Ludwig Wittgenstein took great pains to place the negation sign outside reality and at the same time said “The limits of my language mean the limits of my world.” ([16], Tractatus 5.6) Thus, for Wittgenstein, negation and the logic of negation, the fiction of the logical pivot, was at the limits of the world or, in our terms, the interface of world and the holocosm of concept and thought. And yet we have reconsidered negation in the form of the mark in the calculus of indications where ∼a= a = = . Now the negation sign as mark takes on the pattern of a logical particle that can interact with itself to produce itself or to annihilate itself. The mark no longer appears as only part of the weaving of Wittgenstein’s Great Mirror. It is the descirption of a Majorana particle. The mark partakes of language that can weave it with fundamental physical process. We have illustrated that no matter how carefully we attempt to create language that exists only at the limits of the world, that language inevitably is also related, if only


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in our fiction, to the world itself. References [1] G. Spencer–Brown, “Laws of Form,” George Allen and Unwin Ltd. London (1969). [2] L. H. Kauffman, , Knot Logic, In Knots and Applications ed. by L. Kauffman, World Scientific Pub. Co., (1994), pp. 1–110. [3] E. Majorana, A symmetric theory of electrons and positrons, I Nuovo Cimento,14 (1937), pp. 171– 184. [4] G. Moore and N. Read, Noabelions in the fractional quantum Hall effect, Nuclear Physics B360 (1991), 362–396. [5] V. Mourik,K. Zuo, S. M. Frolov, S. R. Plissard, E.P.A.M. Bakkers, L.P. Kouwenhuven, Signatures of Majorana fermions in hybred superconductor-semiconductor devices, arXiv: 1204.2792. [6] L. H. Kauffman [2002], Biologic. AMS Contemporary Mathematics Series, Vol. 304, (2002), pp. 313–340. [7] L. H. Kauffman [1991,1994,2001,2012], Knots and Physics, World Scientific Pub. [8] L. H. Kauffman and H. P. Noyes [1996], Discrete Physics and the Derivation of Electromagnetism from the formalism of Quantum Mechanics, Proc. of the Royal Soc. Lond. A, 452, pp. 81–95. [9] L. H. Kauffman and H. P. Noyes [1996], Discrete Physics and the Dirac Equation, Physics Letters A, 218 ,pp. 139–146. [10] L. H. Kauffman [2004], Non-commutative worlds, New Journal of Physics 6, pp. 2–46. [11] L. H. Kauffman, Non-commutative worlds and classical constraints. in “Scientific Essays in Honor of Pierre Noyes on the Occasion of His 90-th Birthday”, edited by John Amson and Louis H. Kaufman, World Scientific Pub. Co. (2013), pp. 169–210. [12] L. H. Kauffman and S. J. Lomonaco, q-deformed spin networks, knot polynomials and anyonic topological quantum computation. J. Knot Theory Ramifications 16 (2007), no. 3, 267–332. [13] L. H. Kauffman, Laws of Form, Majorana Fermions and Discrete Physics, In “The Physics of Reality” edited by R. L. Amoroso, L. H. Kauffman, Peter Rowlands. World Scientific Pub. Co., pp. 1–18. [14] L. H. Kauffman, Knot Logic and Topological Computing with Majorana Fermions, (to appear in “Logic and Algebraic Structures in Quantum Computing and Information”, Lecture Notes in Logic, J. Chubb, J. Chubb, Ali Eskandarian, and V. Harizanov, editors, Cambridge University Press). [15] C. W. Misner, K. S. Thorne and J. A. Wheeler [1970], “Gravitation”, W. H. Freeman Co., San Francisco. [16] L. Wittgenstein, “Tractatus Logico – Philosophicus”, New York: Harcourt, Brace and Company, Inc., London: Kegan Paul, Trench, Trubner and Co. Ltd. (1922).