Is Mathematics Real?

It has been proposed by Max Tegmark most recently and earlier by others in other ways that Reality is fundamentally Mathematics. I say Mathematics and not mathematical because that is apparently the proposal – that everything, nay Everything is just mathematics! Well lets see what this could mean. We know, that all the mathematics that can be articulated at the present time is built from sets and definitions about sets. I could ask for a further foundation for sets, but lets postpone that. If I were teaching a class about foundations of mathematics, I would likely introduce them to the empty set {  } (it has no members) and to the notion that if I have already constructed some sets, then I can form the set whose members are these formerly constructed sets. Thus I can form { { } }, the set whose member is the empty set. This is the first set born after the appearance of the empty set. Then I have the sets { }, {{}} and we can now arrive at { { { } } } and { { } , {{}} }, the next two sets in the process of creation. I would also tell my students that two sets are equal if and only if they have the same members. Now we are finitely on our way to making more and more in a big bang from the empty set into the vast set theoretic universe.

Before proceeding, it is good to be assured that the empty set is unique. It is so! For if there were two empty sets they would have exactly the same members — none!

And we are finding multiplicity for { } has none and { { } } has one and { { }, { { } } } has two distinct members. There is a recursion here. Let S(X} = X U { X } be the set obtained from a set X  by making X itself a member. This yields a new set S(X). For example  S({ }) = {{}} and S({{}}) = {{},{{}}} and so on. The sequence of sets {}, S({}), S(S({}))),… give sets with 0,1,2,3,… members and the natural numbers are born. So it goes, all from nothing. Every mathematical structure that we know can be defined in a few lines as a creation from the empty set. And so we are brought inescapably to the  conclusion that the Universe is constructed from nothing but emptiness.

The notion that the Universe comes from nothing is echoed in other traditions, but the sense of that emptiness, that Nothing, is quite different from the mathematical nothing of the empty set. This becomes apparent as soon as we inquire just how did we get that empty set at the beginning of the mathematical process?

Look again. The empty set was represented by us as a container with no contents {  }. I grant you that we used language and the definition of set equality to show that {  } is unique, and so we could have notated it any other convenient way, but this is to assume that mathematics is ‘just’ what comes from some set of rules. As soon as there are rules, we get to ask how do these rules fit a context, where to they come from, how do we know how to follow them? In the case of the empty set we needed a concept of emptiness, and now we are in an open realm for discussion.

If you want to say that the Universe IS mathematics, you will have to explain what is mathematics and you will find that you cannot explain and understand even the simplest mathematics without looking into the context in which it occurs. If you believe that this context is itself more mathematics, then you believe that mathematics is fundamentally circular in nature, and wraps around itself to produce itself. I am willing to believe this! I can even demonstrate it with a bit of mathematics. Let G be the following operator. When G meets an entity X, it produces two copies of X and places them in brackets.

GX = {XX}

For example GA = {AA}. Pretty innocuous!

But GG = {GG}. So GG sits inside itself and by instruction will fall down the rabbit hole in an endless  recursion

GG = {GG} = {{GG}} = {{{GG}}} = {{{{GG}}}} ={{{{{GG}}}}}={{{{{{GG}}}}}} = …

The mathematical entity GG wraps right around itself.  Just so does our language and apparent existence wrap around itself and give us the possibility that we are ‘nothing more’ than our own description of our own description, a kind of illusion that generates its own illusion.

To see how we are propelled into the imaginary, define another operator R by RX = ~XX, the Russell Operator (apres Bertrand Russell) and INTERPRET: XY as “Y is a member of X”. Then the equation

RX = ~XX

is the statement that “X is a member of R exactly when X is not a member of X.” And we have the curious fixed point of negation

RR = ~RR.

In the Universe that expands from the  Russell Operator, there are imaginary logical values, neither true nor false. RR is such a value and being a mathematical construct, it exists in the Russell Universe. Perhaps you do not want to live in the Russell Universe, then being a Form of Mathematics you can devise rules to follow (Mathematics following the rules that it creates in order to follow its own rules) that will avoid the jubjub bird and shun the frumious bandersnatch.

In this note I have pointed out that if you take the Universe to be Mathematics, then you have to face that “it” is generated from emptiness and that this emptiness is very full in the capacity to wrap right round itself and seem to be inside and outside itself at the same time. If the Universe is Mathematics, then Mathematics may not be what she seemed to be. We are down the rabbit hole.

38 thoughts on “Is Mathematics Real?

  1. Dear Marius,
    Indeed it would be interesting to compare with Plato. The notion that behind each concept or idea there is a soul or being is a longstanding notion in philosophy that would bind the apparently abstract world of thought to a more living reality.
    Lou

  2. A self-generating illusion – I like it, very Buddhist (or maybe Taoist). Are you familiar with the everything list, and in particular the work of Bruno Marchal?

    1. Ah! Thank you for the everything list. Self-generating illusion is a bit satirical, but very close to performatives in language such as “This bill is legal tender.” or
      “With this ring I thee wed.”. Declaration in language can create the real, through thought and agreement. The set X that is its only member, X = {X}, comes into existence by its own declaration (as read by us in the context that surrounds it). The concept of “All Sets” comes into being with the saying, and will not go away just because there is no set of all sets. This does not mean that the context of thought and imagination is not a kind of mathematics. It is an undiscovered universe.

  3. Mathematics is a very large term, and it is not well defined, but if we assume computationalism, then the arithmetical reality is enough, and it can be shown than assuming more cannot add anything. In fact it has been proved that if computationalism is correct, then the belief in physical reality must be derivable from arithmetical self-reference (à-la Cantor, Russell, Gödel, Löb, Solovay), and some quantum logic and an arithmetical quantization have bee derived. It is still an open problem to derive a quantum computer from it. The key point is the first person self-indeterminacy: the simple fact that no machine can predict where it will find itself after a (self) duplication. Eventually such approach provides an arithmetical interpretation of Plato and Plotinus indeed. See “sane 2004 origin of the physical laws” for more on this. The quantum is the digital “seen from inside”. The paper on Plotinus is on my front URL page:
    http://iridia.ulb.ac.be/~marchal/

    1. Rather than take on the hypothesis that thinking is a form of computing, I prefer to examine just what is required that mathematics and other forms of thinking come about. The hypothesis that thinking is a form of computing is … interesting … but is not as interesting as watching how concepts and formalisms intertwine. So I do not reject the possibilities inherent in stances such as computationalism, but I am more interested in how concepts arise and how they become related to percepts and how we manage to relate formalisms and concepts. Let me give a concrete example. Consider the glider gun in Conway’s game of Life. We are all aware that it could appear spontaneously from some random Life configuration on a large board, although no one has ever seen this happen. What is more interesting is that the glider gun was discovered through the interaction of concept and experiment by thinking beings in a creative loop with the automaton. Here in miniature is how mathematics comes into being. What I have called thinking is a process that happened in the context of that creative feedback loop of formal recursive game and observer/participants. So far we have many experiences and examples of thinking, but no direct evidence that thinking is computing. If thinking is computing, then we do not yet know what is computing.

  4. That is an interesting approach, and it might be complementary to the “more theological” computationalist approach. I am not defending that computationalism is true, only that it is testable. I am a mathematical logician working on the mind-body problem. By the way I truly love your book “Knots and Physics”, and I appreciate very much your papers on the Temperley-Lieb couplings. For computability, I have no problem with the thesis of Church, but I agree we don’t have such a thing for the more complex notion of computing. Fortunately, for the mind body problem (that I reduced to the problem of belief in bodies arising in arithmetic) I have succeeded to derive key testable points (I think) using only computability and provability, which are better known and simpler to manage. Many thanks for your clarification and blog.

  5. Dear Louis,
    I am talking about the “deductions” from the second and third part of the dialogue, see this entry of Stanford Encyclopedia of Philosophy, but better is to read a good translation of the original (such a subject makes me wish to learn ancient greek). Thing is that the modern set theory appears as a try to solve in some sense these deductions, but there is not a lot of fundamental advance in these matters, perhaps the only exception is that we have now a more clear notion of recursion. Plato, and in particular Parmenides, is hard because it is very much alike (if not even a form of) mathematics. One does not read hundreds of lines of a hard math proof like a novel.
    Coming back to the matter, for example the construction of an infinity of sets from the empty set is just one argument used in one of the deductions. I wonder how much from Parmenides can be formalized somehow, without caring that the conclusions are (as they should!) contradictory (another reason why Plato is great, he offers rigorous proofs of contradictory conclusions and lets the reader to follow from there).

  6. Is mathematics like the concept of sptrezatura the art of being totallly authentic when its totally false. Being totally true when its totally false just depending in the context. Is CH true or false, true & false, or if true then false or iff true then false?

    1. In my view, mathematics arises for us when we start to explore and create distinctions. Usually this occurs in the middle, from the point of view of later formalizations. Thus counting arose by 1-1 correspondence (notches on sticks, pebbles, knots on a rope …) and words came along and then many patterns were perceived in this structure that snowballed of its own accord as we worked with it.
      There is no reason to suppose that in making constructions, one might not run around a path, come back to where one started, and find out that it had changed!
      This is not yet inconsistency. In order to have consistency and inconsistency one needs to have an expectation that a certain distinction will stay put. This also arose out of the necessities of counting, since one wanted to count and track possessions (another notion of immutable distinction — what is mine). But distinctions are always mutable and never perfect except in the imagination (whence comes the notion that mathematics is entirely in the mind!). So this means that many systems of mathematics will become inconsistent when one tries to extend them to larger domains. And it means that they may already be inconsistent in their own domains. Thus it was thought (by Frege and others) that
      every concept had, associated with it, the set of those entities that satisfy the concept. But this Russell showed to be inconsistent. Do you worry that your everyday mathematics is inconsistent? Perhaps you do not worry, but if you are engaged in making new mathematics it is quite possible that your proofs are wrong and your definitions are mutually incompatible. You keep checking and at least get things into a framework where the logic is working and the examples are non-contradictory. Maybe you get it perfect compared to a known structure like
      standard ZFC set theory. But is ZFC consistent? You mentioned continuum hypothesis. The earlier case of independence is part of general mathematical culture — the multiplicity of geometries, Euclidean and non-Euclidean. The geometric models are well-known to us. The non-standard models of set theory are of great interest and not so universally known. They depend crucially upon the fact that there are countable models for set theory. But I digress! The point is that it is up to each person to come to terms with what he or she understands and does not understand. Mathematics expands so that what appears contradictory is a new way of thinking in a larger context.

  7. Mr. Kauffman, I’m a big fan of your work which I’ve managed to find around the web, and in particular your article on ‘virtual logic’ (I think that’s what it was called) inspired by G. S.-B’s “Laws of Form”. I came to those ideas intuitively and have been wrestling with them ever since, but I particularly wanted to thank you for this article because you pointed out my one peeve which drives me up the wall every time I see it: the ‘null set’ /isn’t/ true ‘nothingness’! It’s not even zero! Zero isn’t zero* for cryin’ out loud… The {} is too-often neglected as the first distinction – ‘not’, or as I call it, the “zero-dimensional reflection”… and that leads to a whole head-ache of wonderful epistemology. So thank you 🙂

    *as ‘most people’ think of it…

    1. Dear Tom,
      Thank you for your comment. It seems that in mathematics we go in and out about recognizing the power of nothingness. For example, in algebra we have the “empty word” and it is very important in making presentations of algebraic structures. For example, I can define the integers (multiplicatively) as all symbols of the form
      empty string, a, b, ab, ba, bb, aba, aaa, abababa, bbb,… (all possible finite strings of a’s and b’s including the empty string) plus the relation ab = ba = empty string.
      So then we have aaabb = aa b = aab = a. And so on. Of course you can make a symbol for the empty string such as 1 and then x1y = xy and ab = ba = 1. But it is still important to appreciate that the strings are written on an empty background and that background can also be regarded as a value. In the case of Spencer-Brown’s Laws of Form, the emptiness is crucial to the formal system since we have
      < > = , and = and we interpret as a marked state that is the result of “crossing from” the unmarked state that is inside it, while we interpret
      < > as the unmarked state that is the result of crossing from the marked state that is inside the outer mark. There are other places where an unmarked state is important in mathematics. A favorite of mine from three dimensional topology is th Kirby Calculus where a three dimensional space with no framed links in it represents itself, while a three dimensional space with various framed links in it represents the manifold obtained by surgery on those links. A whole head-ache of wonderful mathematics and wonderful epistemology!
      Best,
      Lou K

      1. I FORGOT THAT THE WORDPRESS PROCESSOR DOES NOT FOLLOW MY INSTRUTIONS WHEN I USE SHARP BRACKETS. LETS TRY AGAIN.
        [ ] REPRESENTS THE MARKED STATE OBTAINED BY CROSSING FROM THE UNMARKED STATE ON ITS INSIDE. [[ ]] REPRESENTS THE MARKED STATE OBTAINED BY CROSSING FROM THE MARKED STATE ON ITS INSIDE.

      2. Can LOF derive Category Theory?
        All discussions here and more are in what I call the finite Pseudo-Infinity set (the set of different pictures that a given digital camera can take); which can be generated with a simple/short bit add program.

      3. Do cats eat bats? Do bats eat cats? It depends on what you mean by LoF and what you mean by derive. The formal system of LoF is very tiny. One symbol, the mark , and two rules of replacement ~ and <> ~ . There is a category or two here that you could define instead if you wished. But the mark stands for a distinction and the formal system stands for the concept of building mathematics consciously on the basis of distinctions. You can then look at all sorts of mathematics and tell or retell its story in this way. For example, the empty set { } is a first distinction between nothing (its contents) and something (the empty set itself). And you can work along and see how each mathematical construction and concept is brought forth via certain significant distinctions. In that sense Category Theory is about the possibility of distinguishing objects and relations (morphisms) in a certain pattern that we call the axioms of category theory. The importance of the full level distinction or construction that category theory makes is a matter of long mathematical experience with these kinds of patterns. I could say to you that yes categories do follow from certain situations. For example if you have a directed graph G, then there is the category Cat(G) of the paths on G. This is a natural category and it satisfies the axioms. You can if you are so inclined regard this as the key generating example for a category. But of course there are many examples such as the categories of sets or algebras or topological spaces. And functors are a real motivating factor for making categories (cohomology coming right to mind). All these structures can be seen as made from the possibility of forming certain distinctions. In that sense they can all be seen as extensions of LoF. I hope this helps with that question!

      4. In my opinion mathematics is the form of any “content “. What does that mean? It’s like art or photography but unlike art and photography where form is restricted to the composition that makes that composition. Not math it can take on any Composition

  8. Hi Lou, me again. I hope this finds you well.

    I’m turning to you here because, with your topological experience, combined with your familiarity with GSB’s work, I think you have a flexible-enough mind to ‘visualize’ and properly intuit my latest quandary – not to mention I don’t know who else to turn to (who’d be able to ‘get’ it):

    George Spencer Brown is _wrong_: The distinction is NOT ‘perfect continence’ as he says…

    This has very troubling consequences in my mind, and I really need your help (or someone’s!) to get past this realization.

    I thought it best to properly explain this on my own blog (as opposed to in a ‘reply’ here) and so I beg of you a moment of your busy schedule for some help in elucidating this matter. Here’s the link: http://wp.me/p544A5-3B

    I can’t begin to express my gratitude for your attention in this matter…

    Kind regards,

    Tom

  9. Dear Tom,
    It is a matter of definition. Distinctions can be perfect by declaring them so.
    Thus I declare { } to be the empty set, characterized by having no members.
    In this way mathematics begins. Distinctions are imaginary. They are fictions.
    But they are very powerful and useful fictions that allow us to explore and calculate and find out about the world. As models of the world the distinctions and mathematical structures that we form are always approximate. In this way distinctions can be perfect and in mathematics we explore the possibility of that perfection. Of course with any human invention it is also possible that there are mistakes. But mathematics is founded on the idea of a construction that has the perfection of the imaginary distinction.
    Best,
    Lou Kauffman

    1. Dear Lou,

      Thank you for taking the time to reply – however it seems you understood my contention to be with the wrong word: it was not ‘perfection’ which poses a problem (in that respect, I agree with you regarding the constructive approach). No, my contention is with ‘continence’ – as being able to discern whether something is ‘inside’ as opposed to ‘adjacent’. It is in this respect that the LoF is wrong – especially if “we take as given the ideas of distinction and of indication, and that one cannot make an indication without drawing a distinction” because it rests entirely upon our human capacity to reason – so if one is to remain faithful to that premise, then one cannot ‘distinguish’ one space as being ‘inside’ another… So even if it were a ‘matter of definition’, one cannot ‘define’ ‘continence’ given only the distinction (which is, as it turns out, defined to ‘enclose’ as space wholly such that no other space can be reached without crossing the distinction).

      Do you understand the issue I’m raising here and the ramifications of such an initial error of assumption? So as an axiom it is ill constructed, it seems…

      Thanks again and best regards,

      Tom

      1. P.S. I encourage you to read the post I made explicitly for this conversation as it seems you have not yet done so, which could explain the misunderstanding since initial my comment was sparse, at best:

        http://wp.me/p544A5-3B

        Thank you 🙂

  10. Dear Tom,
    Matters of adjacency involve more distinctions and can be handled by further formalisms. Spencer-Brown explicitly points out that many terms just refer to distinction in the first place or simplest place. To say a distinction is continent is to say that it is determinate whether an entity is on “one side” or the “other side” of that distinction. But “sides” just refers to the fact that the distinction makes a distinction.
    For example, we create sets so that a given element is either inside or outside the set. The notion of boundary or adjacency comes later. In LOF Spencer-Brown illustrates his concepts by using a topological/geometrical notation and this is fine, but it is understood that at least at first we are ignoring the structure of the boundary except as it is relevant to distinguishing. Once one starts a specific representation then such conventions must be discussed. The use of notation does involve human perception, just as does the use of language. The idea of a distinction is intended to be the ideal of the distinction! Just so in sets we have { } represents the empty set. The brackets are only an indicator of that concept.

    I will read your blog, but not right now as I am in Singapore, leaving for Lisbon and getting my taxi to the airport in a few minutes!

    See

    Click to access LOFTopology.pdf


    for a discussion of mine on some of these issues.
    Best,
    Lou Kauffman

    1. Lou,

      Excellent! And thank you thank you for your time! I’m reading your discussion now. Have a safe flight and I’m eager to hear your input on the site when you have the time.

      Kind regards,

      Tom

      1. Dear Lou,

        I hope your time in Lisbon was (is?) pleasant. I spent a coupla days there and it was wonderful – and LOVED the salt-crust baked Bacalhau, I highly recommend it.

        I hate to be a nag, and I know it was only two weeks ago, but have you had a moment to see my blog entry and the problem of adjacency?

        I have read your transcript of the topology discussion (thank you for that btw), and while there are many interesting topics that were brought-up, none quite address my current conundrum.

        I’m guessing it could be exam-time around now, so just let me know and I’ll shut up and leave you in peace, otherwise, any assistance would be greatly appreciated!

        (I only have this as a means of contacting you – if you’d prefer these ‘comments’ not appear on your site, just let me know how else might be better to reach you and delete these, no problem 🙂 )

        Many thanks again and kind regards,

        Tom

  11. Dear Tom,
    I have looked at your site.
    http://wp.me/p544A5-3B
    You are engaged in producing some very nice structures that arise when one considers the boundaries of distinctions. This reminds me of the algebraic topology where we consider (for example) simpliicial complexes where each basic n-simplex has a boundary that is a union of (n-1) simplices. Then there is a hierarchy of distinction making that occurs in both the adjacenies in a given dimension and the ascent and descent among dimensions. The key formula d^2 = 0 (the boundary of a boundary is zero (empty)) governs the structure and has a role similar but not the same as GSB’s Law of Crossing. I could go on here and talk about many structures that involve adjacency and boundary. Once one includes the boundary even in a simple distinction such as a circle in the plane, the distinction becomes a multiple entity – there is the inside, the boundary and the outside. Thus one is not, in articulating the boundary in this way, talking about the simplest possible idea of a distinction. The simplest idea of a distinction is a two-valued one. Marked or unmarked. Even or odd. Inside or outside. I am sure that we can both agree that (at least in the context of somewhat more complex situations) simple distinctions do occur and that we idealize them in mathematics. Thus when I let P be the set of all prime natural numbers (with the convention that 1 is not prime), then every natural number is either prime or it is not prime. This is a simple mathematical distinction.

    Even with a simple mathematical distinction such as the primes, one may argue and discuss boundary issues. For example, I can ask if there are infinitely many Fibonacci numbers that are prime numbers. We do not know the answer to this question and so the relationship of P and the set of Fibonacci numbers is problematical. Nevertheless, it is a finite computation to determine if a given number N is prime and so (theoretically) we regard the matter of primality as a simple distinction.

    If I want to regard disks in the plane as giving simple distinctions then I can either ignore the boundary of a given disk, or decide to include it on the inside of the distinction. This is what we do when we think of the disk as a closed set in planar topology. The the complement of the disk is an open set and it does not contain its boundary. For any given closed disk in the plane, every point in the plane is either inside the disk (this includes the boundary) or outside of the disk.

    Spencer-Brown takes the idea of a distinction to be a zero-one notion. One stands either inside or outside the distinction at the beginning of this idea. The idea is not something that is right or wrong. It is an idea, and one can follow it along to see what happens with it. Thus in set theoretic mathematics we start with exactly this idea and we say that a given set S either does or does not have another set T as a member. Aha! You say. But this is known to lead to paradox such as the Russell paradox of the set R of all sets that are not members of themselves. Is R a member of R? The apparent perfect continence of R has been controverted by R itself!!
    How do we take this situation? These days, people are rather sanguine about it.
    They make rules to that can keep culprits like the Russell Set out of their theories so that it would seem that all the sets in standard ZF set theory have “perfect continence”. But it takes some study to learn how to talk the set theoretic language that keeps you from paradox.

    Spencer-Brown suggests in his Chapter 11 another way out of the Russell paradox.
    He says that the solution is in a temporal dimension. If R is all the sets we know right now (time t) that are not members of themselves, then we find that once R is created, then R is not a member of itself, and so at time t+1 there must be a new R’
    that has the old R as a member. But now the new R’ does not belong to itself and so we must add it in and this process goes on forever, time after time. Language can be performative. An act of language can create an on-going process and our perfect continences will appear to leak.

    Let me repeat. We start with the ideal of perfect continence and then we find out how far we can go with it. We always find that it starts to fail and then we construct new mathematics to keep it present. It is not wrong to ask for perfect continence.
    But you have to understand that mathematics is the deliberate creation of ideal worlds, conceptual worlds, where such simple distinctions can live Mathematics is conceptual fiction aiming to create logical worlds. Such worlds are like the tangent lines to complex curves. They always shoot off from ‘reality’ and the way they do tells us much about the ideal worlds and about the ‘realities’ from which they have sprung.

    1. Dear Lou,

      Thank you so much for the time you took to answer. May I have your permission to copy your answer as a comment on my blog post? I will, naturally, link it to here.

      Though I understand your point, I hope you can understand how dissatisfied I can be with such an approach: That mathematics is allowed such a ‘hand-wavey’ pass from reality – saying it’s just so because I said so – just seems like cheating. I guess, like Leibniz, I am more deeply convinced that _this_ is already an ideal world – necessarily-so, and so feel that our human attempts to break from reality does not address the real beauty of what mathematics could be (Thankfully, this finally ties-back into your own OP 🙂 ).

      With Godel’s Incompleteness Theorems, we’ve seen that Set Theory is necessarily incomplete – for the sake of consistency. For some reason it seems that mathematicians have lauded consistency over completeness, where I think the opposite should be more praise-worthy (leave nothing un-proved, and contend with inconsistency). You recognize the temporal nature of the Liar Paradox – this is just one ‘useful’ out from the problem, there are others too. We humans can be very creative when posed with such problems and it will push us towards a more ‘real’ mathematics. I personally feel that inconsistency is what imparts our reality with the very possibility of change – so gives rise to the very rich and varied world of mathematics, our language to describe change. That is why I haven’t allowed myself to ‘dictate’ that the distinction is continent. I guess I’m trying for a (IMO) ‘more honest’ theory of Forms…

      I am not discouraged, and will drudge on. (Churchill’s quotation applies well here: “If you find yourself going through hell, _keep_ going!”)

      Once again, many many thanks and kind regards,

      Tom

  12. Dear Tom,
    You can certainly use my comment or link to this blog. I remind you that I did not write Reality. I wrote “Reality”. From my point of view, all realities we know are relative to the way we make or perceive them. There may be an absolute reality in back of all of that, but in mathematics we explore various idealized realities that are are brought forth via our axiomatics and conceptual structures. These mathematical realities can be explored via logic and intuition. Many of them have deep historical roots, such as the natural numbers and classical geometry. Some of them are quite new and under investigation for the first time (by us!). This is the way it is for us.
    Our perceptions of the world are not absolute. In fact, what we perceive in physics and other natural sciences are our own summaries and constructions of ‘what there is’. I do not know any way to carry on a mathematics that could be regarded as the ‘absolute truth’. Most of us mathematicians who were brought up in the 20th century regard this discarding of the notion of absolute truth in mathematics as a great liberation. We continue to believe in the power of human reason and keep looking to see what may come forth. In the same vein the incompleteness results tell us that specific formal systems cannot capture all of (relative) mathematical truth. This is to be celebrated. All of this is one important reason why I believe that it is very important to distinguish mathematics from the ‘physical world’. That way one can keep open the possibility that physical world shows more than we will ever find by pure thought and that conceptualization may be an enterprise that goes beyond the physics that we know now.
    Best,
    Lou

    1. To assume the Digital Mechanist hypothesis requires a minimum of “arithmetical realism”, i.e. we have to believe proposition like P(x) v ~P(x) for P decidable. From this we can extract the existence of all computations, and explain the appearance of the physical reality from a internal statistics on all computations. Up to now this works, and the “burden of the proof” is on the side of those who believe in a primitive (irreducible) physical universe, which now needs a non Mechanist theory of mind. I can give references where this is proven. Unfortunately, many people still miss the fact that elementary arithmetic (its standard semantic) emulates all computations (and that the notion of computation is not physical, but arithmetical). Galaxies, bosons, and pizzas … it is all in the head … of the universal number/machine…

      1. It would be interesting if you would give a self-contained explanation of your assertions that all appearances in so called physical reality can be explained in terms of the Universal Turing Machine. I think that you assume that the form of our observations of even formal systems can be fully emulated by the UTM. This is in some sense your mechanistic hypothesis. Thus in your view the human observer is equivalent to the structure of the UTM observing its own operations. Under those circumstances it would seem that whatever we do observer we are observing the operations of ourselves and that it equivalent to a UTM. This makes your mechanistic hypotheses very huge and I, for one, would not accept it out of hand. On the other hand, there are technicalities that you claim that I would very much like to see on the basis of your hypothesis. For example Quantum Mechanics can be summarized by saying that a physical state corresponds to a vector in a Hilbert space, that physical processes correspond to the vector undergoing unitary transformations, and that observation projects to real eigenspaces so that if {|e_i>} is a measurement basis (orthonormal) then the probability of measuring |e_k> is equal to the absolute square of its coefficient in the state vector ( of unit length). Can you explain how this model arises from the self-observation of the UTM?

      2. It is weird as I see your comment in my mail, but not here (where my early comment is said that it is awaiting moderation). The Digital Mechanist Hypothesis is mainly the Church-Turing thesis and the “yes-doctor” bet (to accept a digital body prosthesis). It is weaker than many version in the literature, because I do not bound the “resolution” of the substitution level, it could be “the entire observable universe at the level of string theory with 10^100 decimal exact (!). That makes the mechanist bet non practical, but the reasoning I propose still works on. Now, this predicts already the “many-histories” view on the physical reality, which comes from the fact that the elementary arithmetical reality (even just its sigma_1 complete part) emulates all (infinitely) computations going through my local state (the one saved by the doctor in case I accept his proposition to replace by a digital device). To get the quantum interference is less easy, and we get them by the ([]p & t, i.e. provable(‘p’)-and-consistency modes of self-reference (adding consistency correspond to the fact that a bet suppose the existence of a reality by default). This works in the the sense that the logic of the “observable predicate” (the one corresponding to the mode []p & t) gives a minimal orthomodular quantum logic, close to the system describe by Louisa Dall Chiara and others. Similarly we get an intuitionist logic for the first person mode (the Theaetus’ knower: []p & p), and an intutionist quantum logic for the “sensible” mode ([]p & t & p). The adding of “& p” leads to first person (intuitionist) logics, and adding “& t” leads to quantum-like logics. The beauty here is that Solovay G* logic proves the arithmetical (extensional) equivalence of all modes, but G (what the machine can prove) does not prove any of those equivalence. It makes the truth unique, yet necessarily seen in many different possible ways by the machine (and we get a distinction between the non sharable qualia and the sharable quanta, in presence of collection of interacting universal machines. What is harder to derive is space, and the Hamiltonian/Lagrangian, or the notion of energy. But we do get already the fact that the core of the physical laws have to be very symmetrical. I use some results by Goldblatt relating quantum logic with the modal logic B (the “Brouwersche system). The logic of the observable appears to be B^- (B, without necessitation rule) extended by some awkward but meaningful formula. The main axioms of B are symmetry (p -> []p, and reflexivity []p -> p).

  13. Dear Bruno,
    Lets take the following simple axioms for quantum theory. For the computational hypothesis we can take the underlying field to be the set of numbers F of the form
    a + bi where i^2 = -1 and a and b are in Z/pZ for p a large prime. I can dispense with real numbers as long as I am allowed to sometimes increase that prime p. Then a physical state is a vector |psi> in a complex (see field above) vector space that I am willing to take as a large tensor product of V = the two dimensional qubit space over F of the form z|0> + w|1> and we take it so that |z|^2 + |w|^2 = 1. And generally the basis for H = tensor product of many V, is orthonormal. We take the basis elements of H as indexing possible observations. If |e_i> is a basis for H, then observing |psi> gives |e_i> with probability |z_i|^2 where z_i = coefficient of |e_i> in |psi> (|psi> has unit length). A physical process is a unitary transformation of H. Now, the questions.
    Certainly we can imagine that some UTM “observes itself” according to these patterns. But
    1. What constitutes self-observation for the UTM?
    2. By what argument, as direct as possible, do we see that these postulates are
    inevitable for the UTM?
    I am hoping for a more direct explanation than the one you outlined in your response.
    Very best,
    Lou Kauffman

    1. Dear Lou,
      Thank you very much for the reply and the interest. Note that I am not defending the Digital Mechanist hypothesis, but I show it to be experimentally testable.
      My problem with your answer is that it seems to already assumes the notion of qubit, or some amount of quantum physics. In fact I have two main point, in the form of
      1) an argument that if we assume Digital Mechanism, then physics is reduced to a theory of machine’s consciousness, in fact the machine’s own theory of consciousness. This is given by an informal reasoning, which I usually provide in eight steps. Digital Mechanism is defined by the belief that we can survive with an artificial body made of digital components. Here the body is defined by the portion of the observable world that I need to emulate, at some level of description, on a computer, to survive the substitution. It could be the entire observable universe, as long as that is emulable at some level of description, but it is easier to assume it is the biological usual brain to proceed in the thought experiment, and then explain why the size and the resolution does not matter as long as that level of description exist (which is the Digital Mechanist Hypothesis). In that case we are duplicable (at that level) and this introduces “first person indeterminacy” (step 3 in my presentation Sane04(*)). If I am “read and cut” in Helsinki (say) and they copy-pasted simultaneously in Washington (W) and Moscow (M), in Helsinki, I am uncertain of my first person future. I can predict only that I will feel to have survived in W *or* in M, as I am sure that I will see only one city (assuming that I survive this of course). In fact, if I am told that my mind is processed by two identical computer in different room, I cannot know in which room I will feel to be in case I will be given some clues. It can be shown that the speed processing, or the time of the reconstitution is not available from the first person point of view. Then, we know that the arithmetical reality (in fact any arithmetical reality, i.e. any model of Robinson arithmetic, Peano arithmetic without the induction axiom) is Turing-complete, and that all computations are emulated in such models. There is an infinity of computations (in such models) going through my state, and this entails a general first person indeterminacy based on this infinity of computations. To make an exact prediction on any first person experience, like reading where is a needle on some experimental device, I should, in principle, take all computations going through my state into account. This eventually reduce physics to a statistics on those computations, with the proviso that they are to be considered from some internal perspective (I call them “histories” in that case). That was my fist prediction in physics: if we look at ourselves or our environment close enough, we should find the trace of those many histories, and at the time I discovered this, I concluded that Mechanism was false, but I was ignorant of quantum Mechanics and Everett at that time. So I change my mind on Mechanism when I discovered Everett Quantum Mechanics (QM without collapse of the wave).
      Then it took me 30 years to do this in a more formal way. I discovered by myself the soundness of the logic of provability G (for the machine probable part) and G* (for the entire true part), and I was very amazed that they are both complete at the propositional level (Solovay 1976 Theorems). I don’t need that completeness to get the formalism of the observable, but it makes everything simpler and more handy.
      The key point is twofold, using p, q, r… for the sigma_1 (partial computable) arithmetical sentences, “[]” for Gödel’s beweisbar arithmetical predicate, and t and f for the boolean constant true and false. “p” is “~[]~p”:
      1) G* proves the equivalence of p, []p,([]p & p), ([]p & t), ([]p & t & p). All those notions are variants of “[]p”, and are extensionally identical: they prove the same arithmetical proposition. yet
      2) G does not prove any of those equivalence, making them obeying quite different intensional (modal) logics.
      The thought experiment, or the reading of Plato, suggest to define believable by []p, knowable by []p & p, observable by []p & t, and sensible by “[]p & t & p”. I expected some intuitionist logic and quantum logics to appear. Normally it was expected that the adding of “& p” leads to intutionist logics, and the adding of “& t” (consistency) should lead to quantum logic, and eventually I show this to be the case, at least in some minimal sense.
      I am still far from being able to justify very “simple” physical notions, or why to use real or complex numbers, why the physical bottom seem linear, where tensor products come from, etc.
      Unfortunately I have to go. I have the “exams of September”, and things are complex due to the confinement. I hope this helps a little bit. I will be busy up to the end of October. Meanwhile, you might take a look at my 2004 presentation in Amsterdam:
      B. Marchal. The Origin of Physical Laws and Sensations. In 4th International System Administration and Network Engineering Conference, SANE 2004, Amsterdam, 2004.
      http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHALAbstract.html
      I can give link to more detailed accounts.
      I think that with digital Mechanism, the burden of proof is in the hand of those who believe in some ontologically primitive physical universe, given that mechanism asks us to postulate arithmetic (to just define “digital”) and then we get all computations, and their statistical interference, which seems to be quantum-like through the logics of the intensional variants of Gödel’s “rational-belief” (provability) predicate.
      Kind Wishes,
      Bruno

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