We have finally come to the last of the multiverse ideas that are presented in the book “The Hidden Reality” written by Brian Greene a few years ago. This idea is called the ultimate multiverse – also known as the mathematical universe hypothesis – for a particular reason. It is really all-embracing. If this Multiverse actually exists, the number of universes within it would be ginormous. This would even be, in a sense, a Multiverse of multiverses. This is pretty much Max Tegmark’s idea, who is a professor at the Massachusetts Institute of Technology. The idea is mostly philosophical, but so was the simulated multiverse idea that we were talking about in the previous article, so let us delve a bit into this one as well. Today we shall be talking about Tegmark’s point of view that led him to this idea, mathematical Platonism, Gödel’s Incompleteness theorem and those kinds of things, but first let us look at what mathematics is for us.
There is an interesting philosophical question concerning what mathematics actually is. Are mathematical truths invented or discovered? I mentioned one of my friends in the previous article, who stands for the first. According to her point of view, everything we, humans, deal with is invented either by our hands or in the process of evolution. So everything human mind is capable of accomplishing is actually generated right there from this point of view, and actual truth just does not exist. Whilst this is certainly true that a lot of our notions and everyday experience are generated inside our minds, this might not be true for everything we perceive as truth. I stand against the previous idea myself, being under the impression that there is actual truth lying outside of our minds. Particularly, I am among those who stand for Platonism in the philosophy of mathematics.
Mathematical Platonism is the point of view which declares the abstractness of mathematical objects and their independent existence of our language, thoughts, practices and, ultimately, of our existence. Just as far away galaxies exist independently of us, so do numbers and sets. Therefore, mathematical truths are discovered, not invented. Consequently, there is a world consisting of such abstract objects ‘out there’ that we address anytime we are mathematically reasoning through something. Thus, although this world consists of only abstract objects, we do interact with it but cannot assert influence on its constituents in any way. If such a world somehow exists then mathematics is as real as the monitor or smartphone that you are looking at right now.
This takes us into the main idea of this article – the ultimate multiverse. What does this idea mean? Well, if we recall the main goal of Einstein that he followed throughout his career, it was of searching for a unified theory which would be capable of describing all of physics within a single framework. Although Einstein has never succeeded, many physicists continue to pursue this goal seeking such a unified theory. But if we eventually find such a physical theory, we will have to ask ourselves a question as to why this particular theory instead of any other? What is so special about it that makes it prominent among others? The answer from the ultimate multiverse point of view would be – nothing. Any theory adequately constructed on various mathematical concepts contains physical reality. Moreover, we use mathematics to define various aspects of physical reality, but there is a vast number of mathematical concepts that aren’t considered to be applicable to physical reality at least in our universe. A reader might argue that sometimes sophisticated mathematical ideas are kept out from physical theories for some time but eventually they find their place in the realm of physics. The most famous examples are Riemann geometry, which was put forward by Bernhard Riemann in the nineteenth century and later became the very basis of Einstein’s General theory of Relativity; and complex numbers notion that was first put forward by Italian mathematician Gerolamo Cardano in 1545 and remained a purely mathematical concept for centuries, before, at last, quantum theory was developed where complex numbers play one of the most crucial roles. We can’t also forget that the existence of anti-particles was first proposed by Paul Dirac and that proposal was based on mathematical equations that contain two solutions one of those corresponds to the same particle, e.g. electron, but with opposite electric charge. The existence of these anti-particles was experimentally confirmed in 1932.
While this is definitely true that some mathematical ideas are put aside only for a period of time, we can imagine some mathematical systems which we can definitively know not to depict any real process in our universe. For example, does your mathematical structure have multiplication? It does not have to. Our own mathematical structure gets us to think in its terms since we are used to it. This structure certainly has multiplication in it and quite a lot of subjects of our mathematical system are based on multiplication, e.g. exponents and logarithms to name but a few. But in principle, your mathematical structure does not have to contain multiplication. And it would be as good as any other.
The question is: does only our mathematical structure correspond to physical reality or do also others? From the mathematical multiverse point of view all of those structures are physically real. They all correspond to certain universes, but the very notion of universe is not clearly defined in this case. Some of those universes might not be universes in our notion of this word. One might not contain gravity, another one might not contain space and time and yet another one might consist of nothing.
You might have heard a famous quote of Gottfried Leibniz where he asks, “Why is there anything at all rather than nothing whatsoever?” This question is certainly very important since the existence of nothing would be completely logically probable. And we are not talking here about empty space whose existence is also impossible in our universe because of quantum mechanics. But we can at least imagine such an empty space. Yet, our concern here is complete nothingness, not an empty space. The existence of such nothingness will make everything very simple. No physical laws in operation, no matter in our concern, neither space nor time whatsoever. Leibniz’s question takes its central place in the above reasoning; why would not nothingness exist?
And here we return to our mathematical universe hypothesis in accordance with which nothingness would exist. And not only would it exist, but a universe, consisting of nothing, must also be included in our Multiverse since its existence is logically consistent. There would not be imbalance between something and nothing in this model; universes consisting of both something and nothing would be parts of this all-embracing conglomerate.
The ultimate multiverse idea is in marked contrast to all the other previous models that we have covered. The main reason behind it is not even in its immensity, but rather in the fact that it was initially proposed as a multiverse model without being constructed upon an existing physical theory. All the previously considered models were developed using various existing theories such as General Relativity, Quantum Mechanics and Superstring theory, rather than being independent. But the ultimate multiverse idea is not based on any existing theory, it was proposed absolutely independently. All the other models suggest answers to some important questions, for example, Quantum Multiverse might solve the problem of quantum measurement; Cyclic Multiverse might address the question of the origin of time; Brane Multiverse might explain why gravity is so much weaker than all the other fundamental forces; Landscape Multiverse might explain the value of the cosmological constant. But these supplements are secondary. Quantum Mechanics was developed for describing the behavior of the micro-world; Inflationary cosmology was introduced for explaining the observed features of the cosmos; Superstring theory – for combining General Relativity with Quantum theory. Conversely, the Ultimate Multiverse idea does not have any meaning except for the very idea of the Multiverse.
Now imagine a pair of friends – Jessica and Robert – who spend a great weekend together in the countryside. While they are discussing various philosophical topics Jessica says that, to her own way of thinking, matter is nothing but a figment of the mind. Hearing that Robert stands up, gets to the nearest tree, spurns it and says “this is how I can deny your statement.” But here we can recall our previous article concerned with simulated universes. According to the idea of simulated universes, everything, including the tree, Robert himself, his thoughts that led him to spurn the tree, his feelings after that etc, might be just an illusion provided by the mathematical operations performed by a powerful computer.
But we can go even further if we assume that the computer isn’t necessary at all, being just some kind of a bridge between abstract mathematical world and physical world that we perceive as real. Mathematics itself – by means of its notions, their relations, inner connections and transformations – contains Robert, his thoughts and emotions. You do not need a computer. Robert himself is inside mathematics!
Tegmark himself describes his mathematical universe hypothesis in the following way. The fundamental description of the Universe should not contain notions whose meaning is based on human experience. Actual reality lies outside our experience hence it should not depend on human-constructed ideas. Tegmark conceives that the very mathematics is the language that is devoid of harmful human influence. But what then can differentiate mathematics itself and a universe that its structure governs? Tegmark thinks that the correct answer to this question would be – nothing. If some characteristic that does distinguish between mathematics and physical reality would exist – it would certainly be nonmathematical; otherwise it can be included into mathematical structure, hence loses its value. However, from this point of view, if it is not mathematical in nature – it contains fingerprints of human intervention, hence cannot be fundamental. Therefore, there is no difference between the mathematical description of reality and its physical materialization, they are identical. Consequently, in this sense mathematics is reality!
Why does Tegmark like this? Well, it seems partly that he does because it is, in a sense, final. Since we can have so many mathematical structures, the idea of ultimate multiverse tells us that the set of all of them is itself a mathematical structure of a higher level, and all of the members of this set contain physical reality. All of them! So this is the main idea behind this speculative and highly controversial model. But partly, it is so controversial among physicists because, as I’ve mentioned, it is mostly philosophical. We can in no way experimentally confirm or disprove this idea, and it is hardly imaginable that we will ever have a chance to do this.
But there are also a couple of problems with this model that I am going to mention. On one level it is not clear that the set of all mathematical structures is well defined. On another level we have a challenge by Gödel’s Incompleteness theorem which says that within a mathematical formal system there are some statements which are unassailably true, but can’t be proven to be true inside this particular system. And since our mathematical multiverse is pretty much based on these formal systems, there seems to be some sort of weakness there.
In fact, Tegmark has lessened his statements to the point where only Gödel-complete mathematical structures are physically real. To understand why this theory is a problem let me briefly explain what it is about in a way that will, hopefully, be comprehensible for readers with no mathematical background. Let me first mention what mathematical formal systems are. I am not going to dig deep into this notion since it is quite involved, but it will be sufficient for our discussion if I say that a formal system takes a number of axioms at start and provides all the possible theorems constructed by the means of these axioms, probably with an enormous number of branches. But eventually, such a system will include all the possible theorems of this system. It was hoped at the early 1900s that such a formal system will be found, but suddenly, one of the best logicians of all times, Kurt Gödel, provided his very sophisticated and logically consistent theorem which shows that the notion of formal systems has to overcome its greatest challenge first.
It has to do with mathematical incomputability and takes a very good analogy in computer science. Computer programmers will surely understand that I am talking about infinite loops. That is, you often need such a method inside your program that will perform some actions and return to its initial state, then starting the process again. Usually, a programmer uses such functions that will allow the program to compute everything it needs for giving you a certain output. But sometimes the programmer could use an incomputable function so his loop will never terminate.
For example, suppose you program a new universe but you’ve done it a thousand times already so you are a bit bored. This time you’ve decided to create quite a funny universe consisting of just one hall where there are several dozens of people and a cook who needs to prepare food for everybody who doesn’t cook for himself. While our cook is grilling, frying and steaming he starts to get hungry. The question is: who is going to prepare a meal for our cook? Give it a thought and your head will spin. The cook cannot do the cooking for himself since he must prepare food only for those who don’t cook for themselves, but if he does not prepare food for himself then he is among those people for whom he must cook! Don’t worry, computers will solve this problem no better than you do. This function would be incomputable so it will never terminate.
With that said, we can now understand that computable mathematical functions are free of Gödel’s Incompleteness theorem, they are Gödel-complete. Therefore, according to Tegmark’s suggestion, only such mathematical structures are physically real. But through our reasoning we’ve come to that question again – why only Gödel-complete mathematical structures? What is special about them? Is it possible that physics also has to be incomplete in such a sense that some of the real world’s properties will never be accessible for a mathematical description? From the Gödel-complete point of view the answer to this question is negative. Computable functions are by definition within the realms of mathematics. These are the functions that computer, following certain procedures, can find an output for. Thus, a multiverse consisting of members constructed upon computable mathematical structures would successfully overcome Gödel’s theorem.
We’ve come to the end of the story about the ultimate multiverse idea. I want to mention once again that it is very controversial, perhaps the most controversial of all the multiverse models, and you might find it either really exciting or completely daffy. Honestly, I am among the first group. If you’ve been following this series of articles or if you are acquainted with me either on Facebook or in real life this may be quite surprising for you, but this is the multiverse model that excites me more than any other. I am among those who are interested in ‘in principle’ questions more than in ‘in practice’ ones. This is why I am so interested in the question of Multiverse. A lot of physicists find this question to be in the realms of philosophy, rather than of physics. They would argue that physics is the discipline working with such predictions or models that contain some testable consequences, which would then be tested to see whether or not the model under test holds true after these experiments have been performed. However, to my own way of thinking, physics and philosophy are getting closer and closer nowadays and such questions that were treated as strictly philosophical are gradually taking their place in the realms of physics. Who knows what physics will look like in the next decade, we will see.
Thank you very much.