Essay Abstract

The essay is in the form of a dialogue between the two authors. We take John Wheeler's idea of "It from Bit" as an essential clue and we rework the stucture of the bit not to the qubit, but to a logical particle that is its own anti-particle, a logical Marjorana particle. This is our key example of the amphibian nature of mathematics and the external world. We emphasize that mathematics is a combination of the conceptual and the calculational. At the conceptual level, mathematics is structured to be independent of time and multiplicity. Mathematics in this way occurs before number and counting, and can be described by the world of logic and boolean arithmetic. From this timeless domain, mathematics and mathematicians can explore worlds of multiplicity and infinity beyond the apparent limitations of the physical world and see that among these possible worlds there are matches with what is observed.

Author Bio

Louis Hirsch Kauffman is a Professor of Mathematics at the University of Illinois at Chicago. He is a topologist working on knot theory and the structure of form. He is the author of books on knot theory and physics and is the originator of state summation models for knot invariants that relate these invariants to partition functions in statistical mechanics. Rukhsan-Ul-Haq is presently a phd student working in the field of strongly correlated electron systems in JNCASR Bangalore India. He is very fascinated by the relation between physics and mathematics. He is also interested in the foundations of physics and mathematics.

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Dear Lou and Rukhsan,

I like the idea pointed out by Spencer-Brown, that "the Universe is constructed to be able to see itself, and we are constructed to be able to see the Universe". This is probably because otherwise we couldn't adapt and survive. It is interesting the notion of "distinction", and its expression in terms of sets. Also, you emphasize the role of numbers, but also of non-numerical worlds, which don't have multiplicities. I like the idea of "holocosm", where mathematicians spend half their time, and when understanding is timeless. As opposed to the physical world, where things are instantiated, composed, and multiplicity rules. While many try to answer the question whether the world is made of mathematics, your answer is a honest one "I have to admit that I do not know what it could possibly mean to say that a 'physical object' is 'composed of' mathematics!" "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."

> "it may be that the Wheeler motto has evolved 'It from Qubit.'."

Wheeler's 'it from bit' was based on quantum. But by 'it' he meant the outcomes of measurements, which are bits, rather than qubits. In this context, by 'it from bit' he meant to reconstruct everything from the outcomes, and that everything is indeed made of just what we observe. But 'it from qubit' means that everything is actually composed of qubits (which is true in some sense).

I liked that "Wittgenstein's Great Mirror has been replaced by that permeable boundary and we are at the beginnnings of a great adventure."

"The key lives in that moving permeable boundary of the holocosm, allowing the world to come to awareness of itself."

I wish you success with the Majorana fermion, and topological computing based on it!

Best wishes,

Cristi Stoica (link to my essay)

    Dear Lou,

    It is always a real pleasure reading your well written and ambitious papers. In your essay with Rukhsan, you succeed in relating, in simple words and concepts, the Majorana particle concept to the permeable boundary between physics and maths "the very foundations of physics, mathematics and the roots of thought.".

    I use a different approach motivated by my continuous interest for the Kochen-Specker theorem. Since the braid group B3 modulo its center is also the modular group, there are clearly connections to my last paragraph. I also know that the general braid group is connected to dessins d'enfants in a highly non trivial way.

    Best wishes

    Michel

      Hi Lou, it has been a few years.

      Glad to see you are still expanding on G-Spencer Brown's Laws of Form. Such a curious thing, creating something out of nothing.

      However I would like to ask a question and I can think of no one better to provide an answer. I am not sure this is the forum to do so but I thought I might try. My question is about your example of an array of dots. Is this not a classic example of something that mathematics can't model?

      You claim the answer is 289d (d=dots). You can't be assuming that 17d horizontally times 17d vertically because that would mean, mathematically that the answer should be 289 d2. Another reason is that it seems that 17d 17d should be 33d unless you want to add one twice.

      Or are you making the logical argument that 289 dots has a relationship to a group of 17 dots and the group of 17 dots has a relationship to 1 dot thus logically 289 dots should have a relationship to 1 dot, which symbolically could be encoded as follows: {(289d/17d) x (17d/1d) = (289d/1d)}. This matches up very nicely with your 172 = 289. However it leaves the curious question as to what 17 17 might be, as one encodes the Whole/group relationship and the other encodes the group/dot relationship.

      cheers

      Rob MacDuff

      Lou, i do have an answer but I would like to see your comments.

        Thanks Dear Cristi Stoica for your comments.

        What is quite amazing is that the urge in a physicist/mathematician to understand the universe is not his own(because he is just a part of the whole) rather it comes from the universe herself.Universe seeks to understand herself.It is this urge which is the base of the quest in physics and mathematics.

        Now the curious thing is that for universe to understand herself she needs an observer and that is where "distinction" comes.As observer makes distinction and universe get created and with that "universe of discourse" also takes birth.The formal properties of distinctions are same as that of projection operators in quantum mechanics which are operators for making measurement.Mathematics follows from these distinctions.Both Boolean logic and arithmetic can be derived from the laws of distinctions/laws of form.But what is mots amazing is that algebra of Majorana fermions and fusion rules of Fibonacci anyons also arise from the same ground.So symbolically what happens in a distinction is that a mirror get created between observer and the observed on which unfolds the recursive play of forms and out of this created duality time is born.The eternal thing is that there is unity and duality is just emergent.Observer and observed are tied in a knot.

        Accurate writing has enabled me to perfect a valid description of untangled unified reality: Proof exists that every real astronomer looking through a real telescope has failed to notice that each of the real galaxies he has observed is unique as to its structure and its perceived distance from all other real galaxies. Each real star is unique as to its structure and its perceived distance apart from all other real stars. Every real scientist who has peered at real snowflakes through a real microscope has concluded that each real snowflake is unique as to its structure. Real structure is unique, once. Unique, once does not consist of abstract amounts of abstract quanta. Based on one's normal observation, one must conclude that all of the stars, all of the planets, all of the asteroids, all of the comets, all of the meteors, all of the specks of astral dust and all real objects have only one real thing in common. Each real object has a real material surface that seems to be attached to a material sub-surface. All surfaces, no matter the apparent degree of separation, must travel at the same constant speed. No matter in which direction one looks, one will only ever see a plethora of real surfaces and those surfaces must all be traveling at the same constant speed or else it would be physically impossible for one to observe them instantly and simultaneously. Real surfaces are easy to spot because they are well lighted. Real light does not travel far from its source as can be confirmed by looking at the real stars, or a real lightning bolt. Reflected light needs to adhere to a surface in order for it to be observed, which means that real light cannot have a surface of its own. Real light must be the only stationary substance in the real Universe. The stars remain in place due to astral radiation. The planets orbit because of atmospheric accumulation. There is no space.

        Warm regards,

        Joe Fisher

          oops, I forgot to paste the first part of the letter.

          Dear Professor Kauffman,

          You wrote: "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."

          All that needs to be done then is to remove the zero from mathematics and the abstract "nothing" from physics and good sense will emerge.

          My point of view is that observer and distinction arise together. You can ask how this comes about,

          but at the bottom this is ones experience. Halls of mirrors and complexity arise naturally enough once there appear to be distinctions. But distinctions are, like coordinates, telling a tale from a particular point of view. You have to start with particular points of view and work hard to find out what is common to them. Spencer-Brown says "We take as given the idea of distinction and the idea of indication and that one cannot make an indication without drawing a distinction. We take therefore the form of distinction for the form." This makes "the form" into a circularity or a quest for "the form of distinction"! It is a cybernetic point of view to show a meaning through a circularity. I particularly like Heinz von Foerster's sentence "I am the observed link between my self and observing myself.".

          All such circularities are invitations to enter into the spiral experience of working through the circle and returning at a different place. Differential geometry measures curvature by just that, and here we are looking at the holonomy of the cybernetics of cybernetics.

          Dear Michel,

          Thank you! I shall look at your papers. I am particularly fascinated by how a representation of the

          circular Artin braid group arises from Majorana operators. Suppose you have $c_1, ..., c_n$ with

          $c_{i}^{2} = 1$ and all different pairs anti-commute. Let $\sigma_{i} = (1 + c_{i}c_{i+1})/\sqrt{2}.$

          Then the $\sigma_{i}$ satisfy the braiding relations. This is a fundamental representation of the braid group that is just underneath the usual representation for Fermions. Perhaps you have your own point of view on this.

          Best,

          Lou K.

          I meant

          $\sigma_{i} = (1 + c_{i+1}c_{i})/\sqrt{2}.$ with $\sigma_{n} = (1 + c_{1}c_{n})/\sqrt{2}$ to make it

          wrap around and become a representation of the circular braid group.

          Dear Rob,

          We usually interpret 3 x 3 as 3 rows of three dots. That makes 9 dots altogether.

          ***

          ***

          ***

          What are you thinking here? It would seem to be something else!

          Why is 2 x 3 = 3 x 2? A good answer in this model is to rotate the rectangle by ninety degrees.

          Concrete arithmetic is a good place to examine how distinctions that have some memory associated with them generate arithmetic. When we start working more abstractly and want to talk about

          numbers like 2^{2^{2^{2^{2^{2^{2}}}}}}}, then there is no way to keep using dots unless you have an ideal notion of dots. So the principle of mathematical induction takes over.

          Best,

          Lou

          Dear Joe,

          I think that nothing will come from attempting to remove nothing. Nothing is our most practical concept. It stands for those clearings that we find or create that allow us to work, construct and perceive. I really appreciate that sense of uniqueness that you so beautifully express about every perception and every perceived phenomenon. I fell that way when I "look" at the empty set.

          So perfect and unique it is. And there can be only one empty set. For if two sets were empty, then they would have exactly the same members, namely none! And two sets are equal exactly when they have the same members.

          Best,

          Lou K.

          Dear Louis and Rukhsan,

          I think your essay is very relevant, which is not to say that I entirely agree with everything you say.

          I think that physical reality has an "inner dynamic" quality, and that aspects of the dynamic RELATIONSHIPS of physical reality could be symbolized by a Not operator. But I can't quite see that this symbol "can be seen as a "logical particle" whose counterpart in the mathematical physical world is a Majorana Particle".

          I like your idea of a "distinction":

          "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 I also liked the following passages:

          "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...I can give you an example that is closer to home...for the mathematician [a] 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. "

          "In this sense, mathematical concepts are the basis of our experience. "

          "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..."

          I have a quote from Louis' article "What is a Number?" in my essay "Reality is MORE than what maths can represent" - using it in connection with one of my arguments: that numbers represent fundamental physical structures, but sets don't.

          Cheers,

          Lorraine Ford

            Dear Lorraine Ford,

            Let me explain what I mean by saying that the mark is a logical particle. For typographical purposes lets use < > for the mark. Then its formal properties are < > < > = < > and = . Here the = sign means "can be replaced by" and the blank space is a blank space. We could use # to stand for a blank space. Then # would have the formal properties ## = # and # = and = .

            Thus IF < > is thought to represent a particle and # the absence of that particle, then < > can interact with itself in two ways: either to produce itself as in -------> < >, or to produce the neutral state # as in < < > > ----------> #. This is all symbolic of course. But it is a symbolism that describes the so-called fusion algebra for a Majorana particle. Having a symbolism for an algebra does not imply the existence of that pattern in the world of physical particles.

            But what we do see is that this symbolism can describe the Majorana particle's fusion algebra. We do not know if Majorana particles exist! This was the speculation of Ettore Majorana long ago when he studied real solutions to the Dirac equation. We are struck by a number of things.

            1. The formalism of the mark is a way to write the 'arithmetic' behind Boolean logic and it is from

            this point of view at a very fundamental place in mathematics. The mark itself stands for and 'is' a

            distinction. That is, the mark < > is a mark, a symbo on a page, but it also is a physical instantiation of a distinction that you imagine.

            2. A simple fundamental place in mathematics corresponds, as a pattern, with a fundamental place in possible physics.

            When we start to talk about how mathematics may be related to physics it is always in the form of a correspondence of patterns. Mathematics studies patterns not things, and so when mathematics and physics come together it is through mathematical patterns being observed in

            Nature. When they are observed (as in the Eight Fold Way and its relationship with representations of SU(3)) we are happy, surprised and we have to wonder what is in back of that.

            I do not wish to say that mathematics and physical world are identical. I hope to say that they arise from the same source and, just so, that Mind and Nature arise from the same source.

            Source is a source of metaphor here, and many come a cropper on attempting to speak this way.

            But this way of thinking is natural for me and I prefer it all the way back to the thought that any distinction carries with it an awareness as a side of the dividing of the universe into what sees and what can be seen. It is not a sharp division and therein comes the metaphor of the permeable boundary.

            Right at the beginning a typo!

            < > < > = < >

            =

            And we can take = #

            where # is a symbol to stand for the absence of any symbol.

            The typo was more serious than I thought.

            In this word processor a mark inside a mark automatically vanishes, or so it would seem.

            I want to let M = < > and write < M > = #. But when I write it directly it vanishes. Lets try and experiment. Use { } for the marked state. Then the interactions would be

            { } { } = { }

            {{ }} =

            or

            {{ }} = #.

            All this illustrates how issues of language and symbols always interact with attempts to express mathmatics.

            Ok. So if I refer to distinctions and marks again, I'll use { } for the marked state or the logical particle.

            This extends the metaphor anyway. After all, you can think of { } as the empty set. And this is to be

            distinguished from the void (represented here by # or an empty word). Note also the underlying long-ago history of the symbols { and }. They are left and right-handed representatives for a bifurcation from a point. The form of a cusp. There is the instantiation of the metaphor of a branching from a

            'source' into the two parts of a distinction. Then the making of the mark has a back-story of two mirrored bifurcations making a container { }.

            Dear Louis,

            Irrespective of a Majorana particle, if I understand correctly, what you are saying is that: "the Mark operator is equivalent to the Marked state, if and only if the initial state is Unmarked".

            1. I would agree that the Mark, or indeed any type of operator or relationship, is a distinction that is as important as the state itself. This must be "true" in any type of system. Do you conclude this?

            2. If the initial state is Unmarked, then how do you account for the ex nihilo appearance of the Mark i.e. the distinction?

            3. I would conclude that, in physical reality, the initial state is necessarily Marked i.e. that it is (subjectively) distinguishable. I would consider that this is a "first principle": that there is something rather than nothing. The nothing, the Unmarked state, exists only by comparison to the overall something (i.e. the universe). What do you say to that?

            Cheers,

            Lorraine

            This is a reply to Lorraine Ford's post of March 14, 2015.

            1. The mark as a sign such as { } makes a distinction in its own form. If it refers to a physical state or

            to a mathematical state, then it may refer to something quite different from itself. For example the sign for two, 2, refers to pairs, but 2 as a sign is connected and not a pair! If I decide that the numbers are represented by |, ||, |||, ... then each sign has the property of the number to which it refers. This iconic nature of some signs is important, but we cannot insist upon it. Thus the sort of

            distinction that a sign makes is important but may have a different importance from its referent.

            2. How can something arise from nothing? All stories we tell turn the 'nothing' into a something from which other somethings can arise. For example. Consider a smooth flat woven price of cloth. This can be the 'nothing'. Then folding the cloth or crumpling it gives rise to many remarkable forms. These forms arise from the nothing of the cloth. Nothing means no thing and that means no object relative to some way of discerning or creating objects.

            3. I prefer to take the initial state (if there is one) as not distinguished and the states that arise 'from it' as the result of processes of discrimination. This is the mathematical point of view where we start from very little and construct mathematical universes by making definitions. Nobody says that Nature does this except when she is being mathematical!

            Lou

            The point that I was attempting to make is that mathematics, since it is mainly based upon ordered sets of symbols, is limited in the types of structures it can naturally describe without being extended by adding units or alternate types of numbers. It seems to me, that mathematics uses two different types of numbers indiscriminately: cardinal and relationship. To easily see this requires extending mathematics by including units. 3 in three rows is a cardinal number and 3 in the number of dots per row is a relationship number.

            3r x 3d/1r = 3r x (3/1)(d/r) = 3r x (9/3)(d/r) = 9d.

            The standard proof that 3 x 3 = 9 assumes 3 x 3 = 3 3 3. However using both cardinal and relationship numbers 3 x (3/1) = 3 x (9/3) = 3/3 x 9 = 1 x 9 = 9. So we can interpret 3 x (3/1) as 3 rows where there are three dots per row and thus 9 dots in total. I would say that this models the array in a more precise way.

            Is the distinction between cardinal and relationship numbers useful? It matches very nicely with the way in which science uses operations, especially as multiplication can't be interpreted as multiple addition but many can as a relationship times a quantity. Relationship numbers enable the encoding of multiplicities but they also act as recursive functions that map numbers back onto themselves. In other words they encode how numbers are related to one another. It also puts multiplication and division on their own foundation rather than being tied to addition. If multiplication did not introduce a tacit alternate type of number then 3 3/1 would make sense within the context of the problem.

            Lou, it seems as if this would allow multiplication in your set of numbers. 3 X (3/1) to get 9, where 3 = {{ } {{ }} {{{ }}}} 3/1 = {{ } {{ }} {{{ }}}} /(( }}.

            Cheers

            Rob