REALITY AND SUPER-REALITY

 

The universe is purely a mathematical structure

If you have read my blog of June 2019, you will know that my favourite moment of the film, The Thirteenth Floor, is Douglas staring into the distant horizon and seeing not mountains, but a green wire-frame structure marking the boundary of his world.  As I put it in the blog, he had discovered the literal edge of his reality.  Setting aside for a moment, though, the apparent clumsiness of the programmers who created Douglas’s computer-simulated world (leaving visible scaffolding at the edge of the simulation was always going to be a give-away!), in the final analysis, did such a revelation somehow invalidate Douglas’s reality?

As Douglas says to his girlfriend, Jane Fuller, “none of this is real…you pull the plug: I disappear.  How can you love me?  I’m not even real!”.  (To her credit, Jane replies: “you’re more real to me than anything I’ve ever known”.)

The flaw in Douglas’s argument, of course, as Descartes could have told him, is the very fact of Douglas doubting his own existence: this means that he is thinking, and so he exists – he is, indeed, real!

We came across an equivalent situation in Self-Awareness in a Toy Universe, where a simple mathematical structure was nevertheless complex enough for a substructure (which we called “Ant1”) to be aware of its environment and itself, so that, for Ant1, the mathematical structure in which it was embedded was real.  For this to be relevant to our own universe, we come back to our claim that our universe is ultimately purely a mathematical structure.

There are several ways to examine this claim, but they all lead to the same conclusion.  In Self-Awareness in a Toy Universe, we imagined that we had already discovered the Theory of Everything and we asked ourselves what language it would be written in.  It couldn’t be written in terms that depend upon our physical universe.  For instance, there would be no point in explaining fundamental particles in terms of other fundamental particles or bundles of energy, because we then have to explain those fundamental particles and bundles of energy in turn.  This strongly suggested that the ToE might instead be purely a mathematical structure.

Another way to approach the question is to start by saying that our universe is nothing more and nothing less than the sum of its quantum fields.  According to quantum field theory, every kind of particle is an excitation of a quantum field, including those particles that transmit forces.  An electron is an excitation of the quantum electron field, the top-quark is an excitation of the top-quark field, and so on.  Quantum fields themselves, though, are purely mathematical structures.  So, again, this indicates that the ultimate structure of our universe is a purely mathematical one.

In yet another approach, some physicists are starting from a purely mathematical structure (using what is technically called Hilbert space – the space is a mathematical space, and not our familiar three-dimensional one) and their equations indicate that gravity may emerge from this mathematical structure.  Since gravity is a property of the structure of spacetime, these physicists may be effectively demonstrating the emergence of spacetime itself.  (Two relevant papers are arXiv:1606.08444 and arXiv:1605.03942.)

With this background, it is difficult to imagine that our universe is not fundamentally a pure mathematical structure.  In Self-Awareness in a Toy Universe, we saw that a Universal Turing Machine running in a simple mathematical structure (i.e., the UTM which we placed at the heart of Ant1) can, in principle, be aware of its own existence.  In other words, the mathematical structure, in which Ant1 is embedded, is real, at least to itself: the mathematical structure is its own reality!

 

Patterns of reality

We also saw that another UTM, which Ant1 was running, contained the program and data for Ant2.  Since Ant2 didn’t share the same parameters as Ant1, Ant2 couldn’t be displayed on the computer screen along with Ant1.  Nevertheless, Ant2 perceived itself to be real just as much as did Ant1.  However, since the mathematical structure containing Ant2 was embedded within the mathematical structure containing Ant1, we could say that Ant1’s mathematical structure was, in some sense, superior to that of Ant2.

So, for Ant2 and Ant1 (and for Douglas and Jane Fuller), there is a hierarchy of reality.  The question is, can this be generally true for mathematical structures?   To try to answer this, we start at the top of Figure 1, where I have written the number 33874822719 and expressed it as a binary sequence of 35 bits:

 

 

Clearly, these bits are arranged in a pattern.  If there were no pattern at all, in other words, if the sequence of bits were truly random, then there is no way that the sequence could be made any shorter than 35 bits.  Notice that this sequence of bits is – as is any sequence of bits – just a mathematical structure, albeit not a particularly complex one.

Since 35 is the product of two prime numbers, 7 and 5, we can divide the sequence into seven sets, r = 0 to 6, each containing five elements, s = 0 to 4.  Now we can see that five of the seven sets, namely r = 1 to 5, contain identical sequences of bits, namely 10001.  The remaining two sets, r = 0 and r = 6, are identical to each other, each containing the sequence 11111.

So, the 35-bit sequence can be described as a function, or structure, R (r, s), that contains only “0”s except where indicated in Figure 1.  Now, to be fair, it would probably take more than 35 bits to describe the structure R as written down in Figure 1, and so there would seem to be no advantage in using R.  However, look at the bottom right of Figure 1, where the seven sets are arranged in a matrix, and the pattern is obvious: it is a rectangle of “0”s enclosed by a wall of “1”s.  Suppose that the sequence had been much longer than 35 bits – say 3,599 bits, which is the product of primes 61 and 59 – but with the same pattern of “0”s enclosed by a wall of “1”s.  In that case, R would hardly be any greater, but the potential saving in using R instead of 3,599 bits is now obvious.

Notice that the alternative of arranging 35 bits into five sets rather than seven, each containing seven elements rather than five, would not have yielded such a neat algorithm for R, or such a striking image when arranged in a matrix.  So, the original sequence of 35 bits contains within it a pattern that enables it to be expressed most succinctly if it is divided into seven sets of five elements.  It is the property of being compressible into the algorithm R that self-selects the two dimensions and leads to the image (the matrix): the image is the result of the intrinsic pattern of the bits, and it is this pattern that naturally resolves into two dimensions.  When we encounter an algorithm – a mathematical structure – that can be expressed more succinctly in parameters like r and s, we shall call such parameters emergent parameters, because they were not put into the structure explicitly but emerged naturally from its form.

Now look at Figure 2, which shows a sequence of 3,293 bits, which I have displayed for convenience as a 37 x 89 matrix.  Standing out from the background of “0”s, you can see two rectangles of “1”s, each of which is the same size (in terms of bits) as the rectangle in Figure 1.  It is only by dividing the 3,293 bits into 37 sets, each containing 89 elements, that this pattern of two rectangles will be revealed.  Since we already have a structure R that can succinctly describe the rectangle in Figure 1, we can use it, modified appropriately as written in Figure 2, to express the full sequence in less than 3,293 bits.  As for the structure R, this may not in practice be the shortest way to express the sequence, but you see the principle – for larger backgrounds and more complex patterns, resolving the sequence into sets may become the most succinct way to express the sequence.  (Notice, incidentally, that the background is not exclusively “0”s – I have included a “1” at the positions [(r = 0 or 37) and (s = 0 or 89)] ).

Since the “1”s in the sequence of 3,293 bits are most efficiently described when we use the parameters r and s, we can regard r and s as emergent parameters, as before.  As an aside, this also highlights the fact that the sequence of 3,293 bits is simply a mathematical structure.

 

Now compare Figure 3 with Figure 2.  Once more, we have a sequence (i.e., a mathematical structure) containing 3,293 bits which I have again resolved into r = 37 sets of s = 89 elements and presented them as a matrix.  In this figure, you can see two vertical lines of “1”s standing out against the background of “0”s.  Once again, r and s are emergent parameters because they allow the sequence to be described more efficiently through the function L (r, s) written at the bottom of the figure.

Although L lists the sequence of 35 bits individually for each line, it is still more efficient to use L in describing the 3,293 bits than just to list them as a long, single string.  This is because, for the arrangement in Figure 3, in each of the two vertical lines, we only need to mention the parameter s in L once (because it is constant – either 29 or 59 – for each of the Ls), whereas, if we had used the other way of resolving the bits (89 sets of 37 elements), then the “1”s would not form a neat line and so s would not be constant, and therefore it would take more bits to express the function L.

Since the two lines of “1”s and “0”s are expressed most efficiently when we use the parameters r and s, we can once again label r and s as emergent parameters for the 3,293-bit structure.

 

 

A hierarchy of reality

In Figure 3, though, there is an added wrinkle when compared with Figure 2.  Look at the sequence of 35 bits that comprise each of the Ls, and that are listed in order at the bottom of Figure 3.  This sequence happens to be the same sequence of 35 bits that we used in Figures 1 and 2.  So we know that we can use the same structure R to describe them that we used before.

However, we cannot use the parameters r and s for the dependence of R, because these parameters are already used for the 37 x 89 matrix.  If the function R depended upon r and s, then R would describe a rectangle drawn in the 37 x 89 matrix, which clearly isn’t there – there are only two vertical lines.  Nevertheless, we have seen that these particular sequences of 35 bits do form rectangular structures described by R.  We can get around the difficulty by using new parameters p and q in place of r and s in R, and I have written the structure R in terms of these new parameters at the bottom of Figure 4.  This is the same structure as in Figure 1, with p and q substituted for r and s.

Notice that, in the expression R (p, q) in Figure 4, there is no indication of where it is embedded in the matrix of Figure 4.

So, to summarise: the mathematical structure of 3,293 bits can be displayed in two dimensions using emergent parameters r and s, and this reveals two lines L₁ and L₂, which are each expressible as a mathematical structure R (p, q) using emergent parameters p and q – which have no relation to the parameters r and s.

This is an instance of a general property that, if a mathematical structure, S (in this case, the sequence of 3,293 bits), contains embedded structures (in this case, the two lines, L₁ and L₂) which, themselves, can be expressed more simply in terms of independent, emergent parameters (in this case, the structure R, which depends upon p and q), then S, with its embedded structures (L₁ and L₂), may be regarded as a mathematical superstructure.  Since mathematical structures are their own reality, then, by analogy, if we consider the structure R as its own reality, then we should consider the structure S as a super-reality!

So, returning to Ant1 and Ant2, we say that the mathematical structure in which Ant2 finds itself is Ant2’s reality, whereas we can say that Ant1’s reality can be considered to be a super-reality in which Ant2’s reality is embedded.  Notice that Ant1’s block universe is dependent upon emergent parameters which we can call p, q and h (referring back to Self-Awareness in a Toy Universe), but that these are not part of Ant2’s reality, which will be dependent upon a completely unrelated set of parameters, say r, s and t, which will emerge from the structure of the input to Ant1’s computer.

 

The universe is absolutely unpredictable

So, how does this translate to our own universe, when viewed as a purely mathematical structure?  To answer that, first I need to convince you that the quantum world is truly random.  By that, I mean that the randomness that we observe is not just a matter of not being able to see down to a small enough scale: our universe is absolutely random.

I have copied Figure 5 from the Quantum Entanglement page and reproduced it as Figure 5 here.  It shows two electrons moving in opposite directions from their point of creation (the sunburst).  When Alice measures the spin of her electron, she can choose the orientation of her measuring apparatus, which is the Stern-Gerlach spin filter that we described in the Quantum Spin Filter page.  Here, she has chosen her spin filter to point vertically upwards.  Suppose she detects a signal: then she knows that the spin of her electron is vertically upwards – if there is no signal, the spin must be vertically downwards as we saw in the Quantum Spin Filter page.  If Bob also sets his spin filter with a vertical orientation, then he will find in this case that the spin of his electron is vertically downwards.

Now I have said throughout the website that quantum outcomes are purely random.  However, thinking back to the beginning of this page, where I said that the universe can be considered as the sum of its quantum fields, you might have a lingering suspicion that, if only we could “see” the individual wavelets of the electron quantum field, then we might be able to see the wavelets coalescing before the moment of measurement to produce Alice’s observation of spin-up.  In other words, while the outcome – spin-up or spin-down – is random, might this randomness be due simply to the particular shapes in which we happen to find the broiling wavelets of the electron quantum field at the moment of measurement?  If that were the case, then quantum outcomes would, in principle, be predictable.

The knee-jerk reaction to that statement is to appeal to entanglement: if Alice and Bob decide only at the last moment on the orientations of their respective spin filters, then we still find, when the spin filters are in alignment with each other (both vertical or both 120⁰ etc.), that Alice’s and Bob’s outcomes are always exactly opposite.  If the outcomes are produced by the random shapes of the quantum electron fields at the moment of each of the two measurements, then the outcomes would not be exactly opposite when the spin-filters are aligned – there would be no correlation between the results.

However, it is possible to argue that, as the electrons each proceed towards their respective spin-filters, so, too, do accompanying wavelets of the quantum field.  Furthermore, since these wavelets were produced at the same instant and proceeded in the opposite directions to each other, you might be tempted to say that, when they arrive at the spin-filters, they are exactly in anti-correlation.  In other words, if the wavelets of the quantum electron field conspire to produce spin-up at Alice’s spin filter, then the wavelets of the quantum field at Bob’s spin filter, being exactly opposite in phase etc. to those at Alice’s instrument, will produce spin-down for Bob.

Nice try, but we already know that can’t work!  That is because the wavelets at Alice’s and Bob’s spin filters would effectively be hidden variables.  However, hidden variables have been ruled out by exhaustive experiment, as we saw in the Hidden Variables page.

So, if Alice’s and Bob’s measurement outcomes were the results of random, broiling wavelets at each of their measuring instruments, there would be no correlation between their results, because any such correlation would be the result of the broiling wavelets acting as hidden variables, which we have ruled out.  However, since their results are correlated (or, to be pedantic, anti-correlated), and, since that cannot be due to anti-correlation of the wavelets, then we cannot attribute Alice’s and Bob’s measurement outcomes to the random fluctuations of the quantum electron field.  In other words, the detailed state of the quantum electric field prior to a quantum measurement gives no clue as to the outcome: quantum outcomes are truly, absolutely unpredictable within our universe.  Quantum outcomes of individual quantum events are intrinsically random.

Figure 6 tries to make this point by showing two identical pictures of the supposed quantum electron field at the moment that an electron is detected at a screen beyond the barrier of a two-slit experiment.  This is an experiment where a particle (traditionally a photon, but it works with electrons, too) is fired at a barrier with two vertical slits that are cut into the barrier close to each other. 

If we fire a single electron at the slits and repeat the experiment many times, we find that the points where the electrons are detected on the screen are bunched into an interference pattern of vertical bands.  The density of the detected points within the bands is simply calculated by assuming that the electrons have a wavelength that is inversely proportional to their momentum.  Details of the calculation are not important here: what is important is that, for any given electron, it is impossible to say where it will land on the screen.

In Figure 6(a), the electron has landed toward the edge of the screen, and, in Figure 6(b), it has hit the screen close to the centre.  However, the quantum electron field is exactly the same in both cases (it has to be, from our discussion above – the quantum electron field can give no clue as to where the electron will be detected).  So, what makes the electron hit the screen as in (a) rather than in (b)?

There are only two ways in which our block universe could be structured to make such a choice.  (We have proposed the solution widely throughout this website, but it is illuminating to approach the question from this different perspective.) 

The first possibility that might occur to you is that there is some kind of random number generator that provides the quantum output, and its probability of occurring, for every quantum event.  However, this won’t work, because the random number generator can be one of only two kinds.  If it is the first (most common) kind, then random numbers are generated from algorithms (equations) that are, by definition, predictable.  So, such random number generators are ruled out by our conclusion above, i.e., that quantum events cannot, even in principle, be predicted.  If the random number is of the second kind, which uses “quantum noise”, such as the randomly varying electric field in some lasers, then that is ruled out, too, because you cannot explain quantum uncertainty using quantum uncertainty itself!  It would be hopelessly circular!

 

Parallel block universes

Since the explanation for quantum randomness cannot be some kind of random number generator, what remains?  The only other possible way to explain how the choices are made for the outcomes of quantum events in our universe is to propose that our universe is but one of many parallel block universes, many of which are very similar to ours, but with the alternative outcomes embedded within them, so that quantum uncertainty arises from our not knowing in which particular universe we happen to find ourselves in.

This is the scenario that we first encountered in the page Parallel Block Universes, but we have arrived at it here by approaching from a different direction.  It should, by now, be easier to accept the unimaginably large number of parallel universes in this scenario, since, as we saw above, we can regard our universe as a mathematical structure – a pattern – which can be duplicated over and over again at no cost.

 

The dilemma of uniqueness

As we saw in the Parallel Block Universes page, some parallel universes not only have to be very similar to ours – they have to be identical.  This is because quantum events with, for example, a 75% probability of a spin-up outcome could only happen if there are at least three identical universes with that outcome (and another “copy”, except that, in this “copy”, the outcome is spin-down).

However, this presents us with a problem.  You cannot have two or more identical mathematical structures!  To explain, suppose we take the first 10 prime numbers, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.  These numbers constitute a mathematical structure.  There is no other identical mathematical structure which also comprises the first 10 prime numbers.  Even if you write them in a different order – say, 5, 29, 17, 2, 19, 23, 11, 13, 3, 7 – they are still the same numbers and, therefore, the same structure.  Therefore, the mathematical structure is unique.

The problem is that we have said that quantum uncertainty depends upon there being identical parallel universes – that is, identical mathematical structures.  But I have just said that you can have only single, unique mathematical structures.  So, it seems that you can’t have two identical parallel universes.

As it turns out, we have the resolution of this dilemma already: you can see it in Figure 3.  In that figure, there were two structures, L₁ and L₂, which were expressed using parameters r and s.  We were able to distinguish between L₁ and L₂ because, in L₁, the parameter s was 29 whereas, in L₂, s was 59.  Nevertheless, when L₁ and L₂ were expressed in the emergent parameters p and q as R(p, q) in Figure 4, the expressions were identical.

So that is what we need to do for our universe!  Think of our universe as a mathematical structure embedded within a larger superstructure, which we can call the multiverse.  Then our universe is analogous to the structure L₁, and its structure, in turn, can be expressed in terms of the emergent parameters (analogous to p and q) that are needed to define the totality of all of the quantum fields that make up our block universe.  There can be identical parallel universes to ours: so, for instance, there can be a universe L₂, which, when expressed in terms of the same emergent parameters as our own, turns out to be identical with our own.  Identical, that is, from our own perspective, since our two universes only use the emergent parameters for their component quantum fields.  From the perspective of the multiverse superstructure, however, our two universes are not identical: they are distinguished by different values of parameters in the multiverse just as L₁ and L₂ are distinguished when the parameter s takes different values, in this case, either 29 or 59.

In Figure 7, I have tried to show you how I think of our universe in the multiverse by highlighting it as a block universe drawn beside an identical universe.  The picture in this figure is from the Mandelbrot set (as visualised in 3D on Youtube).  Here, the Mandelbrot set is supposed to represent the mathematical superstructure that is our multiverse, which contains (very) many identical copies of our universe along with even more universes that are just similar to ours and even more that are wildly different.  The numbers of these universes, as we have seen in The Finite Multiverse page, are distributed according to the quantum probabilities of the quantum outcomes within the universes.  The parameters in the multiverse that determine this distribution are not shared with our own universe.  (Just a note on the model – remember that, while the Mandelbrot set is infinite, our multiverse, although unimaginably large, is finite, as we saw in The Finite Multiverse page.)

So, while the individual block universes – each containing past, present and future – are part of the super-reality of the multiverse, the super-reality of the multiverse cannot be accessed by any individual universe, except for that universe’s own reality.

 

Questions

Referring back to The Mathematical Structure of the Plexus, you may be wondering whether the super-reality of the multiverse is the very same higher level that is implied by Gödel’s enigmatic footnote 48a in his paper on the incompleteness theorems.  I think it probably is.  However, we weren’t able to say anything about the higher level implied by Gödel’s observation except that, in order to explain Gödel’s observation, such a level must exist.  On the other hand, our alternative derivation of the super-reality tells us a little more about its structure and how our universe might relate to it.

In particular, the structure of our super-reality – our multiverse – must, as we noted above, contain the basic structure (i.e., the relations) of quantum mechanics, which allows the appropriate distribution of universes.  I have no idea why our multiverse has that structure, though.  If there is an infinite number of different multiverses, then there is nothing to explain – the quantum structure is bound to arise in an infinite sea of different structures.  (Note that I have not contradicted myself here – the number of universes is finite, and our multiverse is finite, as we saw in The Finite Multiverse page, but the number of multiverses – or other mathematical structures and superstructures – may not be finite: I shall have to think about that some more!)

Another question that may occur to you is: considering that components of the multiverse (our universe, for instance) contain self-awareness, can self-awareness be a property of the super-reality of the multiverse, too?  Again, I don’t know, but, if there were such a self-awareness in our super-reality, it would be literally unimaginably different from the self-awareness that we know.  That is because self-awareness in our universe depends, among other things, upon the passing of time.  If time did not pass, then models in our brains could not be tested against other models, and so on.  In our block universe, our brains are in different states at different times, but we only perceive these states one at a time, in an ordered chronology.  However, from the perspective of the multiverse, all times are present, and the day of my birth and the time preceding it, and the day of my death and the time succeeding it, are all equally accessible to the super-reality.  The super-reality is beyond human comprehension.

You can see a version of this web page in this paper, taken from the mathematical and philosophical journal Axiomathes.  (Note that the text of the paper is not exactly the same as that of the one in PhilSci archive, referenced in my blog of June 19, 2019, because I had to re-touch my paper for Axiomathes to satisfy the referees.)