What can the scientific community do to return to theories grounded in reality, testable against the evidence, and make meaningful strides toward a better understanding of the universe and our place in it? There are ways to test the underpinnings of our physics if we will only try.
Last week, I argued that modern physics has become as faith-based and evidence free as any religion. The real question, however, is why and what the scientific community can do to return to theories grounded in reality, testable against the evidence, and make meaningful strides toward a better understanding of the universe and our place in it. The “why,” of course, is quite complicated. Physics has become an expensive business given the costs to experiment in the age of the Large Hadron Collider or the ever more precise telescopes that enable us to peer further and further into the cosmos with ever more detail. Funding needs to be secured and sustained, which leads to conformity and groupthink. No one wants to stake their career on shaking up the establishment, and so the range of acceptable ideas becomes too limited for radical innovation. Instead, they say essentially the same things in different ways, using ever more sophisticated mathematics to develop ever more insane theories of infinite universes and anti-universes, worlds without time and worlds with ever increasing numbers of dimensions without questioning the current fetish of relying on math for math’s sake.
Math is a powerful tool, one of the greatest creations and discoveries of the human mind in all of history, but it is not the same as the world we see around us and observe every day, nor should we confuse the two because mathematics and reality collide in a fundamental way that is rarely discussed. It is a fundamental principle of mathematics that any formal system can be rewritten in a different yet equivalent way, meaning nothing in our mathematics is uniquely defined or can ever be uniquely defined. The Austrian mathematician and logician Kurt Gödel famously used this principle to devastating effect in 1931, when he presented a proof that mathematics is inherently incomplete. At the time, it was believed that mathematicians could develop one formal system to rule them all. This would include a strictly defined set of mathematical facts, known as axioms, and rules that could be applied that would be able to generate any mathematical proof without contradiction. The hope was that such an edifice could serve as the foundation for an entirely pure mathematics. Gödel, however, showed that such a thing could never be so. He did so by turning math against itself, essentially asking any proposed system to prove its completeness and then showing how that will never be the case.
He achieved this by using the principle of the equivalency of numerical systems, meaning one system can always be translated into another. Quanta Magazine’s Natalie Wolchover covered how this proof works in July 2020, citing Ernest Nagel and James Newman’s streamlined version of the proof from 1958. In their formulation, the proposed axioms are assigned a unique “Godel” number from 1-12. For example, the traditional plus sign for addition was mapped to 11, the multiplication sign to 12. As these are fundamental axioms, the system is rather streamlined. Numbers are defined based on their successor; 2 is the successor of the successor of 0. Variables were then mapped onto prime numbers greater than 12 such as 13, 17, 19, etc. Finally, the formulas and equations that would be derived from these axioms and variables were also assigned a unique number. For example, the statement that zero equals zero is represented by 6,5,6, and the unique number is assigned by taking prime numbers and raising them to the value of the Godel symbols. Thus, 6,5,6 becomes two to the sixth power times three to fifth times five to the sixth. Since a proof is a sequence of formulas, Godel applied a similar method to generate a unique number to represent each proof.
This may seem like a very roundabout and convoluted way to do basic math, but the use of a unique number for every formula and proof enables the output to be easily manipulated. The same process can be applied to false statements as well as true ones, each with their own unique number. This enabled the properties of these statements to be evaluated consistently and more easily than having to work with the complex statements themselves. As Mr. Nagel and Mr. Newman put it, this “exemplifies a very general and deep insight that lies at the heart of Gödel’s discovery: typographical properties of long chains of symbols can be talked about in an indirect but perfectly accurate manner by instead talking about the properties of prime factorizations of large integers.” Moreover, Gödel was able to use this process to ask the system statements about itself and this is where his great insight comes in. To simplify a complex process, Gödel took a true statement, but then substituted one of the variables for the formula of the true statement, essentially saying this formula is true. He then substituted it once more, and developed a statement that asserted it cannot be proved within the system. Because of the beauty of his numbering scheme, this can be presented as an either or: Either G or its opposite is true, but both cannot be. This statement however is not decidable in the system, meaning there are obviously true statements that the system cannot prove as such.
Perhaps needless to say, the implications on mathematics were huge. What was thought to be a complete logical system that can be made whole and in and of itself was shown to be fundamentally incomplete with no way to fill the gap. Physics, however, has been slower to catch up, even as the field relies almost exclusively on math. The first challenge is pretty simple: As far as we can observe, the universe is complete and whole within and of itself. Our experience of it as people and our observation of it with increasingly powerful instruments has never revealed any gaps or flaws. There are things we do not understand, but there are no inconsistencies or hiccups, nor does there appear to be anything “undecidable” within the system. Things either exist or they do not. If they exist, they don’t simply disappear. A past event either occurred or it did not. If it occurred, it doesn’t subsequently not occur. An event in the future either will occur or it won’t, never both. Even in the “spooky” realm of quantum mechanics, we have no reason to doubt the proposition that the universe is complete in and of itself. The rules are different from what we observe at a macro level, but they are still rules that govern what can happen and what cannot. We do not see hybrid statements. The obvious question is: How can a fundamentally incomplete mathematics describe a fundamentally complete universe?
Alas, this question is impossible to answer, but there might well be another way of considering the problem. The universe is complete in and of itself, but it also differs with mathematics in another way that is perhaps more fundamental: Each moment in the history of the universe is unique, different from every other, both before and after. This appears to contradict the property Gödel exploited to translate one system into another, and yet simultaneously conform with the property that every number is unique. Putting this another way, in mathematics nothing is unique except for numbers. Any system can be rewritten in an equivalent manner. Think of it as using a browser on a Mac or a PC. The underlying software is entirely different, but they perform equivalently. The universe, however, isn’t translatable in that manner as far as we can tell. It is unique in its makeup and its history and it couldn’t be any other way; if you were to change one thing in the distant past, the events that come after would all be different as well. Moreover, it is not fully proven that one can necessarily infer the specific past state from the present. Our physics, unfortunately, does not accurately reflect that. Instead, the idea that each moment is relative and readily replicated forward and backward in time is fundamental. It is believed that if you know everything about a system, you can rewind and fast forward however you like. This proposition seems to work on a small scale with a closed system, but we have no real way of knowing if it is true of the universe as a whole. This could be because the assumption that the laws of nature, as codified by mathematics, are fixed and unchangeable is assumed. We are projecting backwards based on how things appear to work today, but what if they worked differently some time in the past, and the unique state of the universe is not simply defined by the matter and forces at play, but also by how they interact at any given instant?
Of course, I am not remotely qualified to ponder things much further than this surface level philosophy, but there are certainly those that are, such as the renowned physicist Lee Smolin who advanced similar ideas in The Singular Universe and the Reality of Time. The challenge for theoretical physics today is that innovative thinking is at a low ebb, while conformity to the supremacy of math is at its peak. The scientific community needs to reverse this trend, and consider a broader range of alternative approaches that begin with how the universe actually works. What makes it singular and unique, different moment by moment from everything else. Rather than exploring the infinite mathematical spaces hidden in models, we should explore the world that is right in front of us, where the arrow of time only runs way, there are only three physical dimensions, and no alternative universes forever beyond our perception and measurement. One way we might do this is to consider whether we can prove experimentally that each moment is truly unique. Modern physics is based on an equivalency between states. If two systems exist in the same state, they will evolve the same way in both the future and the past. It strikes me that this proposition should be testable. First, can two systems truly occupy the exact same state? Second, even if they do, will they truly evolve the same precise way? If we could show that either is not the case, we can establish that one of our key underlying propositions about the universe is incorrect and point the way towards a new approach.
We might not be able to make mathematics complete, but it seems conceivable we could develop a model that tries to incorporate the uniqueness of each and every moment, the same way Gödel exploited number properties. Think of it as a block-chain universe: Each moment bears a unique signature, connecting it to the previous moment and setting the stage for the future.