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.

30 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.

  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:

    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!
      Lou K


  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:

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

    Kind regards,


  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.
    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,


      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:

        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!

    for a discussion of mine on some of these issues.
    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,


      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,


  11. Dear Tom,
    I have looked at your site.
    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,


  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.

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