Thank you for such a thorough, thoughtful post! Really enjoying this discussion.
Rather than mathematics, my background is that of a writer / story-teller. I am also very interested in philosophy, psychology, and evolutionary biology, so a lot of my opinions and frameworks come from those backgrounds. However, it’s interesting that the same problems and questions come up - i.e. “are we finding truth or inventing it?” or “are we seeing reality as it really is or are we fundamentally limited to seeing what we are expecting to see?”
If one studies comparative philosophy long enough, something quite unsettling becomes clear: while any philosophical viewpoint allows one to gain insight into the universe in some particular way, it will necessarily blind the viewer in another way. As in your example above - things that make perfect explanatory sense from the Platonist point of view become nonsensical from the formalist view.
There is also the problem that - regardless of our philosophical background - our conceptual frameworks are innately, inescapably human. They are based on human experience and try as we might, we cannot avoid painting our reality with the colors and metaphors gained from that experience. And so our religious mythologies become filled with stern fathers, loving mothers - we make arguments for or against abstract concepts based on what our experience has trained us to expect (i.e “it can’t be turtles all the way down” or “God doesn’t play dice with the universe” etc.)
The philosophical framework of Zen gives us some interesting clues and tools in this regard, in that it asks us to explicitly focus on how our mental structures draw limits on what we are ready to accept, and urges us to actively break through those limits. However, just like any philosophical framework, Zen also has boundaries - in that - if you’re not using those tools to approach a particular goal (i.e. detachment / going beyond ego) then you’re “doing it wrong”
So, given that we are hobbled by our human perspective in all these different ways, is the whole process of making sense of the universe doomed? I don’t think it is - and again - I use a framework from evolutionary biology. At any one point there is a dialectic between our approaches and the reality outside of us. If our approach doesn’t fit - if it fails to explain the reality outside of us, we have to discard it, adapt, and evolve something else.
The point isn’t to get to some final answer at the back of the book, the point is to continuously evolve our frameworks, approaches, and answers to the reality before us because Reality is That Which Does Not Go Away When We Stop Believing In It - which is pretty darn close to the Scientific Method. I like this approach because - at any given point - we acknowledge our limitations. We aren’t seeing Reality as it really is - we acknowledge that it might be beyond our current ability to measure with the instruments we have at hand, or might even be more complex and strange than we can wrap our human minds around - but at least we are constantly building and discarding frameworks rather than being stuck in any one particular framework.
I think about these kinds of things a lot. How do you take something so huge and abstract and make it concrete? We look out at the darkness between stars and we can tell ourselves “That goes on forever”* - but again, we are limited by our human experience. Since nothing in our living experience is infinite, we are reduced to abstractions.
(*and yes, I know that our universe may not in fact be “infinite”: https://phys.org/news/2015-03-universe-finite-infinite.html)
For me, it helps to think about much smaller infinities like repeating decimal fractions. I know that 1/3 = 0.333333333… is in some ways just an artifact of base10 math, but there is something there about those 3’s going off into infinity that helps me make the abstract more concrete. The same thing goes for Pi - on one hand, I know that this irrational number goes on forever, on the other hand I know that they all fit in to that little tiny chunk of red between 3 and Pi.
It’s the same with fractals. Using a good computer program and diving deep into the Julia set, one gets that awe-inspiring feeling that it is bottomless. Now, some part of me knows that this is an illusion - that the software I’m using is only rendering so many iterations of the equation, and that my monitor is only able to display only so much granularity of that image, because of the limits of the pixel sizes. However, the thing that works for me isn’t the actuality of infinite depth but the POTENTIAL for infinite depth - that given these particular conditions - one could perform the calculations and they would theoretically go on forever.
So maybe that is the most useful way for me to think about infinity - something that has the potential for being an infinite process because there is no natural limit to keep it finite.
So, so back up and bring this around full circle - the idea I started with was that once the capacity for embodied awareness evolved, it began adding additional layers of complexity to the universe in ways that
- accelerated the universe’s natural process of adding those layers
- made novel resources and products available for complexity to use and
- created a situation where the process of adding additional layers of complexity had the potential to become an infinite process.
So from this point of view - if you think about the state space of the universe as not only the positions of the atoms in the universe, but also the motion of those atoms, and the procedural/interactive information that is made possible by all the processes and forces acting on those atoms - then you approach a framework where new ideas really do expand the state space of the universe.
What do I mean? Well for one - the presence of living beings has reshaped the chemistry of our planet many times, the most famous being the Oxygen Holocaust:
Once that great chemical change occurred, and a more energetic element was available as an energy source (i.e. oxygen metabolism) that then led to all these other evolutionary and metabolic pathways evolving, increasing the overall complexity of the system.
Or, let’s look at something like the hormone Oxytocin:
At one point in biological evolution, you had sexual reproduction, but no pair bonding, no parents taking care of offspring, etc. Along comes this new hormone (note: from an evolutionary perspective, this first evolved in sea birds, see this book for more: https://www.goodreads.com/book/show/262728.The_Naked_Neuron)
and suddenly you have these internal reward pathways that lead animals to develop new behaviors to nurture their young, form pair bonds, form larger societies of animals that are capable of more complex group behaviors than any individual. This increase in behavioral complexity then circles back and becomes the evolutionary pressure which leads to development in language complexity, group dynamics, and eventually the beginnings of theory-of-mind, etc.
Fast forward to modern times and we have scientists filling in the Table of Elements by creating new elements that were once only theoretical. We can even misuse those things, and create huge nuclear disasters (Chernobyl / Fukushima) that create long-term physical changes in our world. We are sending our machines and information about ourselves out into space. And now, we are working towards technology that is taking our embodied/physical awareness and transferring those processes into machines. Will we ever get a strong/general AI? I don’t know! But I do see it as something likely, and if we ever do succeed, it will create new configurations of matter, energy, and information - it will open up entire new situations where complexity can take root and grow - it will create its own new by-products and secondary/tertiary effects that we cannot predict.
So - yes - I do see our ideas as increasing the available state spaces which are possible in the universe in a very real / physical sense. I also see this entire process as something that has the potential to be infinite - even though I cannot hold the actuality of that infinity in my hand. Of course there are many factors that might limit it - our potential to destroy ourselves - the finite lifespan of our local star - even the potential heat death of the universe. But again, I can stipulate that this might be an infinite process because it has the property of constantly jumping beyond its limitations - creating new configurations of matter, energy, and information - making new forces, new materials, and new ideas available.