THE MATHEMATICAL STRUCTURE OF THE PLEXUS
This page follows the one on Gödel’s incompleteness theorems, where we saw that a mathematical system that supports at least basic arithmetic can generate a true statement, G, which, nevertheless, cannot be proved in that system. Gödel’s explanation for this situation opens up a structure for the Plexus that explains quantum uncertainty and ties up many loose ends.
Why can we see that G is true when the system cannot prove it?
Since G states that it is not a theorem, and since we find that G is, indeed, not a theorem, then, by any reasonable definition of truth, we can see the truth of G. This raises a question, though – the fact that we have satisfied ourselves of the truth of G is another way of saying that we have proved to ourselves that G is true. And yet the mathematical system was unable to prove that G is true – otherwise, we should have included G as a theorem. How can we see it when the rigorous mathematical system doesn’t?
Whenever we encounter a case where neither a formula such as G nor its negation, ~G, can be proved a theorem, we call G (or ~G) an undecidable proposition. In a somewhat enigmatic footnote to his paper (footnote 48a), Gödel said that the presence of such undecidable propositions merely meant that the mathematical system needed to be expanded in order to prove one of the two undecidable propositions (it can’t, of course, prove both, since one is the negation of the other – one of them has to be false).
So how might we expand the system? We could start by labelling the undecidable propositions with the name of the mathematical system in which they can’t be proved. Say our mathematical system is called “Z”. Then the Gödel statement and its negation become GZ and ~GZ. GZ would now read “GZ is not a theorem of system Z”.
Next, we could simply add GZ as an axiom to the other axioms in system Z. That would be technically allowed, because an axiom is true by definition: you can’t prove it using the other axioms and theorems of the system. We could call the new system “Z+GZ”.
However, it would be a remarkably ill-fitting axiom to have in a system whose other axioms include such fundamental statements as “if you add zero to a natural number you get the same natural number” or “if you multiply a natural number by zero you get 0”. These axioms can be written using just a few symbols. However, recalling the convoluted way in which we had to express G in Gödel’s incompleteness theorems, the new system, Z+GZ, would appear to be an exceedingly contrived one.
More seriously, adding GZ to the axioms of Z doesn’t get rid of the problem. After all, we could simply write down the equivalent Gödel statement for the combined system Z+GZ and call this new Gödel statement “GZ+Gz”. So we would still have an undecidable statement, GZ+Gz, which says: “GZ+Gz is not a theorem of system Z+GZ”. This could go on forever.
As we shall see shortly, Gödel had a much more startling resolution in mind when he added his footnote. In order to understand it, we go back to the idea of adding GZ to the system Z, but, instead of calling the expanded system Z+GZ as we did above, let us label the expanded system “W”. In the expanded system W, as well as adding the statement GZ, we also add axioms that were not in Z, along with any theorems that may arise from these new axioms.
Figure 1 illustrates the point. GZ is shown outside of the theorems-and-axioms box of Z because it is not a theorem or axiom of Z. However, the mathematical system W is an expansion of system Z. It includes all of the theorems and axioms of Z as well as additional axioms, which, in turn, generate new theorems in W that could not have arisen in the simpler system Z. GZ is a case in point: it may either be an axiom of W or it may be possible to prove it as a theorem of W with the help of the new axioms that appear in W. The negation of GZ, ~GZ, is still untrue, of course, and must remain outside the theorems and axioms of system W.
So, in answer to the question in the first paragraph – why can we see that G is true and the system cannot? – our own logic is not confined to the axioms of system Z. The trick is to step outside of the system into a higher level.
(Incidentally, some mathematicians would be uneasy about using the word “true” in the context of saying that GZ is true. Well, let’s take it from the Master. In his paper on his incompleteness theorems, Gödel says:
“From the remark that [G] asserts its own unprovability, it follows at once that [G] is correct, since [G] is certainly unprovable (because undecidable).”
In this quotation, the italics are mine, and I have substituted “G” for the more complex expression that Gödel used for his statement – he was certainly not immodest enough to call it “G”! So, while Gödel didn’t in fact use the word “true” here, he used the synonym, “correct”.)
Now this is all very fine, but you may be thinking: we’re not out of the woods yet – couldn’t system W have its own Gödel statement, GW? Exactly! Yes, it can. We can form the Gödel statement “GW is not a theorem of system W” using precisely the same reasoning that we used to derive the original Gödel statement.
Figure 2 shows how this would look. The new Gödel statement, GW, is outside of the theorems-and-axioms box of system W as, of course, is its negation, ~GW. So we seem to be in danger of constructing a continually ascending spiral where each successive expansion of the original mathematical system contains the undecidable propositions of its predecessor either as axioms or theorems generated from new axioms in the expanded system but which, in turn, contains undecidable propositions of its own.
Before I get to the denouement (Spoiler Alert – it is Gödel’s footnote 48a), I should point out that his paper uses a fairly basic mathematical system called “Peano arithmetic” (named after the Italian mathematician, Giuseppe Peano) to serve as a model for more powerful mathematical systems. The paper doesn’t specify what additional axioms and theorems might be contained in any higher mathematical system. However, this leaves the door open for us to add any consistent axioms to higher levels, as long as we retain the core axioms of Peano arithmetic.
Up to this point, we have been thinking of systems Z and W as containing only mathematical, or, to be more precise, arithmetical axioms and theorems. By now, you may be wondering about the relationship between the axioms of arithmetic and those of the universe, the multiverse and the Plexus. The axioms and theorems of arithmetic, after all, are just descriptions of the properties of, and relations between, natural numbers, whereas the axioms and theorems of our universe will tell you things such as the number of space and time dimensions, the limiting speed of information (i.e., the speed of light) and the way gravity distorts spacetime.
I gave a hint of the answer in Self-Awareness in a Toy Universe. Recall that I used the term “isomorphic” to describe the translatability of the structure of the Game of Life, complete with its rules, into the arithmetical statements in Figure 1 of that page. Well, now we can go further. We saw earlier in this chapter that every string of symbols can be translated into a Gödel number, and that every such Gödel number can be translated back into the original string of symbols. In other words, strings of symbols are isomorphic with their Gödel numbers.
So basic mathematical statements about the geometry of our universe, which are essentially statements of general relativity, are isomorphic with their Gödel numbers. These mathematical statements effectively describe the framework of our universe: they are definitions, and don’t follow from any more basic premise. In other words, they are axioms. So these axioms about our universe, being isomorphic with their Gödel numbers, are descriptions of properties of, and relations between, natural numbers, as are the axioms and theorems of arithmetic.
To put it another way – the mathematical structure of our universe is just a mathematical system.
Given that the axioms describing our universe behave just like those of arithmetic, it is legitimate to add these axioms to the collection in system Z, the central box in Figure 2, and to regard system Z as the basic infrastructure of our universe as described by general relativity.
While system Z contained only arithmetical axioms and theorems before it was augmented with the general relativity axioms, it couldn’t be the other way round: you couldn’t have system Z containing only general relativity axioms and theorems without the arithmetical axioms and theorems. This is because the rules of general relativity require at least basic arithmetic – including addition and multiplication – to make it work. No universe even remotely approaching our own can exist without an arithmetical basis. This will prove to be an important point later in this page.
We can continue to build the mathematical structure by adding axioms in higher levels as illustrated in Figure 3.
Gödel’s enigmatic footnote 48a – the denouement
The inspiration for this figure comes again from Gödel’s footnote 48a:
“The true source of the incompleteness attaching to all formal systems of mathematics, is to be found – as will be shown in Part II of this essay – in the fact that the formation of ever higher types can be continued into the transfinite…”
In other words, according to Gödel, the spiral does, in effect, ascend indefinitely until you enter the region of the mysteriously sounding transfinite. This is the term, you will recall from Playing with Infinity, which was invented somewhat coyly by Cantor in order to spare him from ruffling the feathers of the late-nineteenth-century mathematical establishment by uttering the word “infinity”. We shall see in a moment how this fits into the Plexus. (Incidentally, Gödel never did write Part II of his paper on incompleteness.)
At this juncture, I need to crispen-up the definition of the Plexus. You might think of using the word “Plexus” to include only our own universe or multiverse, but then we should have to think of another word for the extension of this structure – for according to the ascending-spiral argument it surely must extend. Instead, I choose to keep to the one name, the Plexus, and regard our universe and multiverse as mathematical substructures within it. The Plexus extends up to, and encompasses, the transfinite.
Figure 3 is relatively simple: it has three boxes, Z, W and the Plexus. However, you could very well imagine a large nest of many such boxes like Russian dolls beyond W before you reach the all-enveloping Plexus box. In such a structure, the box immediately up from W – call it V – would contain GW as an axiom, or would contain new axioms that prove GW as a theorem, but it would, in turn, generate a statement GV that could not be contained within V. The next box up – say U – would contain GV as an axiom or theorem, but generate GU that could not be contained within U, and so on.
As we have just seen, the boxes close to Z, in the foothills, as it were, may well contain axioms that are not only useful for producing solutions to the Gödel statements generated by inferior boxes but also for defining the infrastructure of our universe. (We shall see shortly how such axioms are relevant to quantum phenomena.) Towards the summit, as the layers of nested boxes extend into the infinite – the transfinite – the only difference between any given box and the next one up will be the Gödel statements produced by the lower of the two boxes.
We can tidy things up formally in Figure 3 by including a “final” Gödel statement, GP, as a theorem of the Plexus, actually within the Plexus box rather than outside of it. Furthermore, we can add an axiom, ~GP, again within the Plexus box.
Whoa! How can this possibly be, considering that I ruled out that very scenario in Gödel’s incompleteness theorems? I said that a Gödel statement couldn’t be included as a theorem in a system, and that adding it as an axiom in a system didn’t solve the problem. And apart from anything else, how can it be OK to include both a formula GP and its negation ~GP in the box of legitimate axioms and theorems?
Remember that formulas – strings of symbols – can be translated into Gödel numbers: this means that proofs, or the numbers representing such proofs, can become infinitely long in the transfinite. So, in effect, the clash between the axiom, ~GP, and the theorem, GP, can never be demonstrated in any approach using discrete symbols. We have to accept that, in some cryptic sense, the Plexus tidies off the loose ends at the ultimate level. Of course, ~GW and ~GZ, neither of which is infinite in length, cannot be theorems or axioms of the Plexus at any level.
There is another (heuristic) way to view the resolution to the problem of the ever-ascending nest of Russian-doll boxes. We have seen that each box contains within itself an inner box as well as axioms and theorems that are additional to those in the inner box. These additional theorems include proofs of statements that were undecidable – unprovable – in the inner box. However, this outer box itself generates undecidable statements that will only be proved by axioms in a box yet further out.
So, in effect, each box contains a model of itself. As you ascend upwards through the nest of boxes, more statements are proved by the additional axioms in each box, without which the models at the lower levels were only indistinct approximations to the clarity of the systems at the upper levels. Towards the summit, in the transfinite region, the boxes are essentially identical, differing only by predictable Gödel statements. In effect, then, at the transfinite level, the models contained by the Plexus already contain the complete Plexus itself – the Plexus may then be said to be self-referential!
To look at it another way, as you ascend through the lower levels, new axioms are introduced which, in a sense, “explain” the undecidable propositions of the levels below. However, towards the transfinite levels, no new axioms appear, except for predictable ones like Gödel statements. So, while the lower levels of the mathematical structure contain within themselves approximate models of themselves (lacking in some axioms), as you approach the transfinite, the axioms become predictable and therefore redundant, so that, at that level, the Plexus in effect explains itself!
Figure 4 illustrates the idea. An inner box at the transfinite level is shown as a subset of the Plexus. However, since this box contains, in effect, all of the information of the Plexus itself, it may be said to be a perfect model of the Plexus. A perfect model of a system, though, must contain that system along with the model within the system!
The idea that the Plexus could be contained completely within a model of itself – a model that, after all, is only a subset of itself – seems like the mythological Ouroboros, the snake eating its own tail. Nevertheless, it is a perfectly respectable notion in the mathematics of the transfinite. Look at Figure 5.
At the top of the figure I have drawn the transfinite set of the natural numbers, 0, 1, 2, 3… extending to ℵ0 (aleph-nought – remember I defined ℵ0 in Playing with Infinity as the size of the infinite set of natural numbers). Now you can generate a subset of this set by taking every second number and writing it down, as I have done in the next box down. This sequence begins 0, 2, 4… extending to ℵ0 once again. However, because this subset is transfinite, it contains the original set of all the natural numbers, as you can see just by multiplying every member of the subset by ½, as I have done in the bottom box.
So the concept of the Plexus being its own perfect model – and, consequently, being perfectly self-referential – has a mathematically legitimate provenance.
(In passing, note that this is effectively what Gödel did with his statement. The Gödel statement does the seemingly impossible: it is “large” enough to contain not only itself but also information about itself, namely that it is not a theorem.)
Lighting the blue touch paper
Gödel’s incompleteness theorem showed that there are truths expressible in the mathematical substructure of our universe that cannot be derived from the axioms. These truths do not all have to be Gödel statements: we used these simply to show that higher mathematical subsystems exist than that of our universe. Adding axioms beyond those of our own universe will lead to mathematical subsystems powerful enough to derive the truths that are not provable in our own. It is these additional axioms – those which are not just used to turn a Gödel statement into an axiom or a theorem at a higher level – that allow colour and complexity to emerge within the Plexus and give it meaning (at least to the inhabitants that it creates!). To paraphrase Hawking, it is the addition of these axioms that lights the blue touch paper of the Plexus.
You might think that we could try guessing the axioms that belong to the level above our own universe and thereby prove the undecidable propositions in our own universe. Unfortunately, though, while we might be successful in guessing axioms that will derive these orphan truths, our success cannot be taken to mean that these axioms are indeed those belonging to the higher mathematical subsystem. There will always be facts – undecidable propositions – about our universe that we cannot prove.
This is because, if we were able somehow to prove that the extra axioms were indeed part of the higher mathematical subsystem, then we would be able to prove the orphan truths in our own universe using these extra, proven axioms. But proving a system’s orphan truths – undecidable propositions – is not allowed in that system, as we saw with Gödel’s statement. As a result, we can never do any more than speculate about the structure of mathematical subsystems that are at a higher level than those of our own universe.
With this absolute limitation in mind, you should regard the next diagram, Figure 6, as simply conjecture. Nevertheless, it is designed to accommodate the parallel-universe picture that I showed you in The Origin of Quantum Probability and it reflects the hierarchical character of mathematical systems that we have seen emerge from Gödel’s footnote 48a.
In this diagram I have tried to indicate how the Plexus might be organised down to our “local” level – that is, to the level of our universe. Imagine that we are in Universe 1.1. In this universe, Alice has just measured the spin of her electron to be “up”. I have shown one other universe, Universe 1.2, where she measured it to be “down”. Of course, there are identical versions of Alice in each of these universes as well as in other, parallel universes, but let us concentrate on Universe 1.1.
Notice that Universe 1.1, like Universe 1.2 and all of the other, parallel block universes, is a mathematical system. It is a mathematical system just as the Game of Life depicted in Figure 2 of Self-Awareness in a Toy Universe is a mathematical system. That particular mathematical structure, you may remember, is just based on the axioms of arithmetic with the added axioms in Figure 2 of that page to spice it up a little. In Universe 1.1 and 1.2 here, the axioms are those that define our familiar geometry, like the relations of general relativity (which, in turn, contain special relativity as theorems).
Although Alice can calculate the probability of her being in a universe in which she measures “up”, she cannot work out from the rules of that universe what her result will definitely be. So Alice’s finding “up” is not a theorem of Universe 1.1, unlike, for instance, the fact that time slows down for people moving relative to you, which is a direct result of the theorems (in this case, special relativity) of Universe 1.1. Nevertheless, in Universe 1.1, Alice does, indeed, find “up”, and so it is a true statement of that universe. Since it is a true statement but not a theorem of Universe 1.1 (because it cannot be proved, which would make it a theorem), I have drawn the “up” statement lying outside the box of theorems that constitute Universe 1.1. In Universe 1.1 it is an undecidable proposition.
However, as Gödel wrote in his footnote, such an undecidable statement is merely an indication that there exists a higher system in which the statement is decidable. This higher system corresponds to what we have been calling the multiverse of parallel universes. I have drawn a box around the two parallel universes, Universe 1.1 and Universe 1.2 (representing all of the myriad parallel universes of Multiverse 1) as well as the true-but-unprovable results (“up”, “down”, representing all quantum-mechanical results) that correspond to those parallel universes.
We saw in The Origin of Quantum Probability how the measured probability of a quantum event in a given universe reflects the proportion of occurrences of that event across the multiverse of parallel universes. We can think of quantum events in these parallel universes as theorems generated by the quantum definitions – the quantum axioms – of that multiverse (rather than of any given universe). These quantum axioms determine the number and distribution of parallel universes within the multiverse according to, for instance, the Schrödinger equation or Heisenberg matrix mechanics.
So what I am saying is that, when we carry out a quantum experiment in our universe, although we cannot predict the outcome from within our universe, that outcome is a theorem of the multiverse, manifest by the distribution of parallel, block universes with that quantum outcome as determined by the quantum axioms of the multiverse.
So, to recap, the higher mathematical system called the multiverse (think of it as system W in Figure 3) contains the definitions – the axioms – of quantum mechanics that lead to the generation of parallel universes containing quantum events. A quantum event in any one of these universes (think of each one as a system Z, containing the axioms of general relativity within that universe) is a theorem – a logical consequence – of the axioms of the higher system. However, from within the lower mathematical system that constitutes the universe, that quantum event, while a fact – a truth – is nevertheless undecidable: it cannot be derived from the axioms of that lower system that we call the universe.
Actually, it’s quite easy to see why a quantum event cannot be derived – proved – from the axioms of the universe in which it occurs. Suppose that Universe 1.1 and Universe 1.2 differed only in the outcome of the quantum event – with 1.1 being “spin up” and 1.2 being “spin down”. Now suppose that we could derive a proof within Universe 1.1 that the outcome in Universe 1.1 has to be “spin up”. However, since the two universes are identical, with identical axioms and theorems, apart from the outcome of the quantum event, then the proof derived within Universe 1.1 would equally apply in Universe 1.2. That would be absurd, of course, because that would prove that the outcome of the quantum event in Universe 1.2 was the same as that in Universe 1.1. So there can be no such proof.
In summary, axioms and theorems in the multiverse (equivalent to the higher system W) result in undecidable statements in each individual parallel universe (equivalent to the lower system Z) which appear in each of these parallel universes to be an unpredictable quantum event.
In Figure 6 I have labelled two boxes, “Multiverse 1” and “Multiverse 2”. This is because there are other truths in our universe that can neither be proved from anything else in the universe nor apparently can be derived from the additional quantum axioms of the multiverse. Such truths include “constants of nature” like the masses of elementary particles, the strengths of forces and so on. Just as quantum mechanical events such as the outcome of an electron-spin experiment being “up” in our universe could only be a theorem of a higher system which we call the multiverse, so truths like the strength of gravity that cannot be derived – proved – in the multiverse must be axioms or theorems of a higher system. If there is a multitude of multiverses like Multiverse 1 and Multiverse 2 in the higher system, this allows the possibility of different constants of nature arising in the different multiverses.
Again, there is little point in trying to guess the structure in any sharper detail, because, for instance, it may be that the same multiverse that contains the quantum axioms also contains additional axioms that let you determine the constants of nature. In that case, there would be no need for a higher system than the multiverse, at least, not for the constants of nature. My intuition, though, is that, just as there does appear to be a multitude of parallel universes that explains the probabilities of the outcomes of quantum events, so there may well be a multitude of multiverses, each with a different value for the strength of gravity and so on. Accepting that this is conjecture, I have drawn the two multiverses to represent the multitude of multiverses as part of a higher system which derives constants of nature for each individual multiverse.
At the highest level in Figure 6 some of the theorems and axioms of the Plexus become transfinite. Remember that, of course, all of the axioms and theorems of individual parallel universes, individual multiverses, and the higher-level system containing all the multiverses are still axioms and theorems of the Plexus. However, there will inevitably (from Gödel’s incompleteness theorem apart from anything else) be truths in the Plexus that are not derived in some of the subsystems below. Such truths will be derived from additional axioms of the Plexus that do not appear at a lower level. Nevertheless, as we saw in Figure 3, these remaining truths are derivable in the transfinite, and this includes self-referential, Gödel-like statements, which means that the Plexus can effectively refer to itself. Figure 7 is another, more pictorial, way of looking at the previous figure.
In Figure 7 the Plexus has been split into four levels, starting at the bottom and ascending, according to the arrow, in the direction of increasing numbers of axioms. Notice that, at the level of the “multitude of multiverses” I have suggested that these multiverses may come from “eternal inflation”. The hypothesis of eternal inflation is by no means universally accepted by the physics establishment, but I have mentioned it here just so you can see where it might fit into the Plexus. I shall say a bit more about it in a later page.
Remember, as I pointed out in The Finite Multiverse, the block universe includes not only our observable universe, but all of spacetime, meaning that all of space beyond our observable universe is included, and for all time, both of which are finite.
Axioms at the bottom level of the Plexus define the number of space and time dimensions of our block universe, as well as, presumably, its relativistic geometry. Axioms for other properties of our universe, such as the probabilistic nature of quantum phenomena, must arise, as we have seen, at a higher, multiverse level. Axioms for constants of nature that are common throughout the multiverse, arise, in turn, at a higher level still.
This completes my broad overview of how our universe fits into the Plexus. Some elements are speculative, others less so. The idea that our universe is a block universe is virtually unassailable, although many physicists still balk at the implications. But it is experimentally verifiable, as I described in The Block Universe and the accompanying paper on arXiv. The concept of wave-function collapse was shown to be invalid in Wave Function Collapse, leaving the Many Worlds Interpretation as the most likely candidate to explain quantum phenomena. This is still hotly contested by some physicists, but it has commanded a growing acceptance among the physics population over the past few decades. However, the branching topology of MWI is incompatible with the block universe, and so I proposed a multiverse of parallel, stand-alone block universes in my original arXiv paper and in more detail in Parallel Block Universes. This uses essentially the same mathematics as MWI and so is another interpretation of quantum mechanics.
The principal objection to MWI and, therefore, to a multiverse of parallel block universes is that it seems so wasteful. However, if you regard the universe as part of a vast, purely mathematical structure, that objection melts away. I call that mathematical structure the Plexus. There are other indications that the ultimate infrastructure of our universe is purely mathematical, including fundamental work in string theory.
We can see from the complexity of our own universe that the mathematical axioms of the Plexus must at the least support basic arithmetic. So Gödel’s incompleteness theorems must apply. From his enigmatic footnote 48a, the mathematical system of the Plexus must have different levels. A characteristic of these levels is that there can be true statements – facts – at one level that cannot be explained at that level, but which follow from the logical application of additional axioms at a higher level. Quantum observations in our block universe can be seen as logical consequences of axioms derived from the Schrödinger equation and Heisenberg matrix mechanics operating at a higher level to produce a multiverse of block universes.
This same hierarchical structure can account for other features of our universe, including “constants of nature” like the mass of the electron and the strength of the gravitational field.
In the next page, I shall discuss Max Tegmark’s mathematical structure of a hierarchy of multiverses which, at first acquaintance, seems to have echoes of the Plexus structure but with which, on closer examination, it has little in common (see Where are all the other Universes?).