It’s well known that different areas of mathematics have varying levels of difficulty and complexity. It’s also firmly established that math is infinite. There is no end to it. No matter how far we push back the boundaries of mathematical knowledge, there are always new areas, branches, and regions to reveal and explore. But how complex can math get? If math is truly infinite, it’s reasonable to assume that it can also be infinitely complex.
Picture mathematics as an endlessly tall tower, with an infinite number of floors. As you climb the tower, each floor increases in complexity and abstraction. And this remains true forever. Math is infinite, so there is no top floor to this tower.
Now our first question is, what floor are we currently on? Any answer I give would be arbitrary, but for the sake of this article, let’s say we are currently on the fortieth floor. We are a lot farther along than the ancient Greeks, but we can still see the ground from here. The natural follow up question is, how high can we climb?
There are two ways to look at this. If you recall, each subsequent floor is more mathematically complex than the last. Let’s assume that one always ascends to the next floor via a 20 step staircase. Despite the growing mathematical complexity of each successive floor, moving between any two floors requires only these same 20 steps, eliminating the need for any significant cognitive leaps, no matter which floor you might find yourself on.
Alternatively, you can say that as each level increases in complexity, you reach a point where human cognitive ability hits a ceiling. Let’s say that occurs at floor 100. At that point, despite only being 20 steps away, floor 101 is forever out of reach because the mathematics on that floor requires a level of intelligence beyond what biological humans are capable of. We CANNOT get beyond floor 100 because our brains just aren’t up to the task. On these higher floors are problems whose very nature might require a level of abstract thought, pattern recognition, or information processing that our current biological intelligence cannot achieve.
The underlying mathematical structures or the complexity of the relationships involved might be so far removed from our intuitive understanding that we lack the cognitive architecture to truly comprehend them, let alone solve them.
These problems might involve concepts that are as alien to us as, perhaps, advanced quantum field theory is to a caterpillar.
But this is where computers and AI come in. As technology improves, as AI improves, it will reach a point where it is more capable than us. As such, it will be able to reach floors beyond 100. Then, via biotech, we can merge with our machines and thereby augment our own intelligence to the point that we can reach those higher floors ourselves.
So in this future, where our cognitive functions are artificially enhanced, how high can we climb? Let’s say floor 500. Much better than floor 100, right? But still nothing compared to the infinite height of this mathematical tower. So then, how do we climb higher? Let me ask a slightly different question. How high can ANY intelligence in our universe climb?
What do I mean by this?
Whether it's the biological architecture of a brain or the silicon and wiring of a computer chip, intelligence is a physical phenomenon. It arises from the organization and interaction of matter and energy according to the laws of physics.
Because intelligence is physical, it must be subject to the constraints and limitations imposed by physical reality. There are upper bounds on factors like information processing speed, memory capacity, energy efficiency, and the complexity of interconnectedness that can be achieved within the framework of our universe's laws.
Therefore, it logically follows that even the most advanced intelligence allowed by the laws of physics would eventually reach a ceiling in its ability to comprehend and solve increasingly complex mathematical problems. The sheer vastness of the mathematical landscape will inevitably extend beyond the cognitive reach of any physically realizable intelligence.
This suggests that there are mathematical truths and solvable problems that will forever remain beyond the grasp of any intelligence that can exist within our universe, simply because the level of cognitive power required to understand them exceeds the physical limits of what's possible.
So now let’s say an intelligence, upon reaching this limit, would find itself able to access the 1,000th floor of our mathematical tower. What about floor 1,001 or higher? These floors would forever remain out of the reach of any intelligence in our universe.
But surely there MUST be a way to surpass this limit. All that unexplored math on those higher floors are just begging to be explored. The answer… maybe. But to explore this idea, we must now enter the realm of the highly speculative.
Solution 1: Wormholes to parallel universes where the fundamental constants are different
Ignoring the difficulty of getting to another universe (if such alternate universes even exist), it's conceivable that in another universe, the fundamental constants governing the interactions of particles and the structure of spacetime could be different. These differences might theoretically allow for a higher maximum density of stable matter or different ways of encoding information at a fundamental level, potentially leading to a higher information density limit. This in turn could allow for intelligence levels beyond what is possible in our universe. In such a universe, perhaps an intelligence could access floors higher than 1,000.
Granted, such a parallel universe, having its own laws, would also have an upper limit to intelligence. By definition, a universe is a physical system governed by some set of laws and composed of matter and energy (or their equivalents). All physical systems we understand are subject to limitations. And whether it's a brain or a machine, intelligence needs a physical medium to exist and operate. The capabilities of this medium will be bound by the physical laws of the universe it inhabits.
Given that any universe would have physical limitations, and intelligence is a physical phenomenon requiring physical resources, it logically follows that the level of intelligence achievable within any conceivable universe must also be finite. There would ALWAYS be an upper bound, even if that bound is vastly higher in some universes than in others.
So this chips away at the higher floors in our Math tower, but what are a few more floors in the face of infinitely more floors? This would get us closer to the “top floor” of the tower. But it would ultimately fall short.
What next? How do we get to the top? Time to jump into an even more speculative area of thought.
Solution 2: Decouple Reality from Physicality
The latest obstacle we have encountered is that intelligence is an emergent property from physical reality, and by its very nature, there are always fundamental limits to any physical reality one can conceive of. Ergo, there are limits to intelligence. Which has us falling far short of climbing this infinite tower of mathematics.
So what about a non-physical reality? Is such a thing even conceptually possible or is the very idea nothing but hogwash?
From a purely scientific standpoint, envisioning a reality entirely decoupled from the physical is exceptionally difficult because our very understanding of "reality" is built upon our sensory experiences and the laws we've observed governing the physical world. Our tools for understanding – our brains and our scientific instruments – are themselves physical.
However, some physicists and philosophers propose that information, rather than matter or energy, is the fundamental building block of reality. In this view, the physical world we perceive might be an emergent phenomenon from an underlying realm of information. While this doesn't necessarily mean a non-physical reality, it shifts the emphasis away from tangible matter.
Now if we entertain the idea that information is the fundamental substrate of reality, and our physical universe is just one emergent property of its organization and dynamics, then it's certainly conceivable that other emergent properties could exist, potentially leading to entirely different types of realities or domains.
Here's how we might envision this:
The way information is structured, interacts, and flows could be different in other emergent phenomena. Just as specific patterns of information give rise to particles, forces, and spacetime in our universe, other patterns might give rise to entirely different fundamental entities and laws.
Some of these emergent properties might not manifest as what we would traditionally consider "physical" in our sense of the word. They could be realms of pure information, mathematical structures that become self-sustaining, or even the substrates for consciousness in ways we don't currently understand.
If other emergent properties of information exist, some of them might provide substrates or environments that are far more conducive to the development of intelligence than our, or any, physical universe. They might allow for higher information processing capabilities, or perhaps different forms of consciousness or awareness. Or even the ability to explore concepts and solve problems in ways that are fundamentally limited by the physics of a physical universe.
It's also possible that these emergent properties could themselves give rise to further levels of emergence, creating a vast and complex hierarchy of realities or domains, each with its own characteristics and laws.
In that same vein, what’s to say that our own universe is not the physical substrate to even higher and more complex levels of emergence?
A common characteristic of emergent systems is that the emergent properties at a higher level can often exhibit greater apparent complexity than the fundamental components at the lower level. And as we mentioned previously, greater complexity could mean greater upper limits for intelligence levels.
And in the "upward" direction, towards more complex emergent properties, there doesn't seem to be an inherent reason why there must be a limit to the number of layers. Each emergent level could, in principle, become the foundation for even more complex structures and behaviors at the next level. This aligns with the idea of infinite mathematical complexity – you can always build more intricate structures upon existing ones.
So how do we keep up with mathematics that reach infinite complexity?
By continually "ascending" through this infinite hierarchy of increasingly complex realities, intelligence could theoretically evolve or adapt to match and comprehend ever-greater levels of mathematical complexity.
In such a scenario, there would be no absolute limit to the complexity of mathematics that could be understood, as there would always be a "higher" reality with the potential to support the necessary level of intelligence. The limitation would then become the ability to traverse these realities and for intelligence to evolve or adapt within them.
If such an infinite hierarchy of increasingly complex emergent realities exists and if we could somehow navigate it and adapt our intelligence, then the potential for mathematical understanding could indeed be limitless.
The tower of mathematics is infinitely tall, so we would never truly reach the “top floor,” but in this scenario, we would also never stop climbing.
No matter how far we delved into the mysteries of mathematics, no matter how many truths we uncovered, no matter how much beauty we revealed, there would always be more to discover.
