Trick or Truth: The Mysterious Connection between Mathematics and Physics by Louis Hirsch Kauffman and Rukhsan-Ul-Haq Wani

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.

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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 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 { }.

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!

Dear Chris,

Of course I am modeling. I do not regard this as an epistemological error. In fact I regard the unrestricted identification of physical reality with mathematics as a serious epistemological error.

This does not mean that there are no places where the mathematics and the physical are indistinguishable. When they are indistinguishable, it means that there is an awareness present gluing them together and not distinct from either. I say this because, for me, mathematics is fundamentally conceptual, concept requires awareness/thought, and yet that thought almost universally needs to be grounded in a physical experience. There is another speculation that is very fascinating: that elementary physical entities are purely mathematical. One may find oneself thinking that perhaps the form and the content of an electron are identical. Take care and look at what you really mean. Maybe this is so and maybe you are just living in the model!

You say "I'm not sure you are doing anything more fundamental than "modeling reality"." If you allow that the model could be something like Escher's Print Gallery

http://www.planetperplex.com/en/item/print-gallery/

then I would agree with you.I suggest that a right way to look at the world is to understand that what observes the world is the world. Any attempt to articulate this idea is likely to be incomplete.

Just because one says that the world is observing herself, does not give one license to relax and stop working out whether one's meanings and sayings are well-formed. In fact it makes the articulation much more difficult since one wants to know whether one is finding out what is dependent on the point of view (the way the world is divided into viewer and viewed) or perhaps independent of that. But in in physical science we use all sorts of different viewpoints and we want them to cohere. Bohr and Heisenberg showed us that what were thought to be objective properties of 'physical things' were often the results of the type of splitting (the type of observation) chosen. I wish my understanding were universal. But what universality it has seems to come from hard work!

Ah! Now! Calculus of Indications. I do not say (or did not intend to say) that it is an extension of Boolean Arithmetic. It is a RADICAL COLLAPSE of Boolean Arithmetic.In Boolean Arithmetic there is a firm separation of the operator of negation, ~, and the possible values, T and F. In the Calculus of Indications the marked value { } and the operator (of negation if you will) are IDENTICAL. There is no distinction between the operator and the operand. Processes are Things and Things are Processes (there is only one Thing/Process at this early stage). The remarkable point about the formalism is that even though it is so collapsed, it is still possible to maintain sufficient distinction to capture the patterns of boolean mathematics and unfold them. It is possible to go from the collapsed state to un uncollapsed state by regarding the mark as an operator as in { { } } = (unmarked) and as a value. In fact the most profound equation of the Calculus of Indications is: { } = { }. On the left hand side the mark is seen to operate on the unmarked state. On the right hand side we just have the marked state. They are identical and they are distinct! Just so in observing 'physical reality', we bring forth 'its' properties.

You suggest that I move toward a system without axioms. Maybe. I like the assumptions to be as simple as possible. But you have to ask, if you think that we are getting toward physical modeling, whether starting with very simple structures of distinction will naturally lead to the complexities of physics. Can you tell a good story for this? Stories people tell usually use some accepted complexities. For example, we may take on qubits, and then we have taken on the basics of quantum theory from the beginning. There is a long story from a bit or a mark to a qubit. I think that looking at the mark with its structure of interaction (like a Majorana particle) is a good step to take before letting it become a qubit Qubit means superposition of possibilities and the apparatus of the complex numbers. I want to slow this down and look carefully at the Metaphor of the Imaginary. When you do this you find that the square root of minus one is a clock! But this is too much for one reply.

Lets end in the question of what is possible in 'modeling reality'. We are deeply involved in the models we make of 'reality'. The reality we come to observe and experience is a function of our models and these models (constructions of language and mathematics) are functions of the reality.

It is circular. In physics we demand repeatability and independence of particular points of view.

The insures that this recursive game of producing and knowing will appear as objective as it can. It also insures that there will be surprises. We shall not know it all at once or even in the course of time.

Dear Joe Fisher,

I will certainly read and respond to your essay! I think however that your post is not quite appropriate, since one is supposed to comment on the essay that we wrote in this space. However, first without referring to your essay, I want to examine from our point of view what it would mean to say that the real universe is not mathematical. From my point of view all linguistic constructions accompanied by concepts are mathematical. David Finkelstein once said "Mathematics is a form of literary endeavor

with a very special type of criticism.". When one goes to the basis of mathematics and finds that it is about the properties of distinctions then we see that mathematics always involves awareness/mind/concept. It is not bare formalism or mechanical calculation. Given that stance, it is clear that I cannot say that a world without concepts is a mathematical world. If I believed in a purely material world in this sense, then I would have to say that 'it' knows no mathematics except through the observers of that world and while the universe may apparently follow mathematical rules, that is our description of it and not it itself. But if you take the point of view that there is in the Reality an intertwining of concept/awareness and materiality, then the universe herself could be a great mathematican. Even in this view, the mathematics has to to do with the dialogue of the Universe with herself.

Now I will go read your essay and make comments on it on your site!

Best

Lou K.

Dear Bill,

Thank you for the kind comments. Holocosm is our term for that eternal timeless world,highly creative, that mathematicians imagine as existent. Concepts and ideas are real and living in the holocosm. This can only be metaphorical in relation to common notions of existence and we see the contrast when we examine how we talk about mathematical constructs. Many people imagine that infinity could be real, for example that there could actually be infinitely many parallel universes either in a classical or in a quantum sense. But if you examine how we use the term infinity, you realize that particularly mathematicians do not assume an existence in this sense. They speak as though the real numbers or Hilbert spaces exist, but all that is required is consistency, logical consistency. Even the simple infinity of the natural numbers N = {1,2,3,...} is only a matter of consistency. We do not have access to any but a few of them directly, but we do have access to the concept of adding one and we have access to the concept of expressing numbers (such as 2^{2^{...2^{2}}} with 100 2's in that expression). We structure all this and work with it from a conceptual domain that is very real and supported by physicality, but distinct from physicality.

I call this domain the Holocosm. It is where mathematics is structured and it is where we discover elements of mathematics that seem to correspond to physicality. It is where we find concepts that are useful in the world. There is no multiplicity in the Holocosm, so one might say that it is boolean. Actually, I believe that Spencer-Brown came to a closer understanding of the Holocosm when he realized that negation (action, crossing) and value (such as 0 and 1) are aspects of the same structure of distinction and indication, deeper than the boolean. The moral of this for physics is that one can look at the relationship of the holocosm with our perceptual world and begin to understand the nature of the dialogue that we are creating about the boundary between the conceptual and the physical in a new way. One can begin to see how it is that mathematics and natural science are both exploring the same creation.

Best,

Lou K