Mathematical Exploration into Ontology

Problem:

Premise 1: Pi is an infinitely complex number.

Premise 2: Pi manifests naturally in nature at base reality.

Conclusion: Base reality is infinitely complex as opposed to simple.

Contradiction: Emergence requires the simple emerging towards the complex, rather than the complex reducing towards the simple.

So, I was talking with a friend the other day about the basal nature of math, you know your everyday type of conversation. He made the bold claim that 2+2=5 when you switch from base-10 to base-3. Struggling to remember everything I ever learned about all the math I haven't used in a decade, we puzzled through this claim together. In the end, it seemed like we came to the conclusion that he was off a bit. 2+2 still equals 4 in base-3, but 4 is represented by 11 instead.

But while he was teaching me how to think about switching bases, he brought up a typical analogy that catalyzed further thought. The analogy was of a pie, but to avoid confusing this with 'pi' (π), he termed it as pizza. Each base represents the number of ways you slice the pizza. Base-2 slices the pizza into 2 pieces. Base-3 slices the pizza into 3 slices. Base-10 into 10 slices. Base-12 into 12 slices.

https://www.youtube.com/watch?v=y_QBDrBlbds&t=3s&ab_channel=SmartbyDesign

When you increment your numbers through the pizza, you always start with zero. Then you have 1 slice. For binary, this 1 is equivalent to 1/2 of the pizza. The next increment after 1 is the entirety of the pizza. The entirety of the pizza is always equivalent to 10, in any base. So, in binary, we go, 0 (0/2), 1 (1/2), and 10 (2/2). In base-3, we go 0 (0/3), 1 (1/3), 2 (2/3) and then 10 (3/3). To finish up the 2+2 example in base-3, we would just add 1 to our 10 and get 11 (4/3).

But somehow this discussion sparked a deeper thought. The mention of pie started pushing me towards the actual concept of 'pi' (π). π was always a very interesting number. Despite seeming to be very simple (circles that even children can understand), and ontologically ubiquitous (central to the laws of physics as the particle building blocks of reality are fundamentally based on π for their spheroid nature),  π was an extremely complex mathematical construct that formed an infinite series of seemingly random numbers that result in a fundamentally irrational number.

Something about this sparked important to me. It was as if my friend had reminded me of a mechanism for performing a mathematical paradigm shift, and simultaneously pointed me towards something that might need a mathematical paradigm shift in order to understand better. And given how central the concept of  π was to reality, it seemed important to understand.

Our mathematical paradigm seemed largely anthropocentrically foolish, as our math was based on our number of fingers. Of course, we couldn't understand fundamental reality when we are trying to force it to match our number of fingers!

I started to wonder if there was another mathematical base that we could translate π into such that it would suddenly become a coherently understandable number. After a bit of investigation, it started to seem like π was bound to be an unintelligibly irrational number in any base. No matter what base you translated it into, it would form this infinitely long random pattern of unintelligible nonsense. But that wasn't a satisfactory endpoint at all. How could something as simple and ubiquitous as a circle be so mathematically incoherent? We MUST be doing something wrong.

Imagine if there was a much easier way to look at mathematics that we have been missing this whole time due to our anthropocentric bias! What if the big problems in physics could be simply solved if only we had the right mathematical paradigm - the one that nature was using?

The Wikipedia for π had already attempted to do the very thing I was looking for - convert π into different bases! Yet, none of the mathematical bases on offer seemed to simplify the concept of π. Then I wondered.... π is connected to the structure of a circle, and these mathematical bases form the structure of a circle. There has got to be a link here. What if the proper mathematical base is π itself! Base-π... a very confusing paradigm. How do you incorporate decimals into a mathematical base? Even worse, how to incorporate an infinite number of decimals?? Counting from 1/3.14159 to 2/3.14159 to 3/3.14159 then presented us with an awkward step... we now had an extra slice of pizza left over, the size of 0.14159. The pizza was not being cut properly. Each step was supposed to be an equivalent slice.

I was struggling to intuit how this should work. My first guess was that 3.14159 wasn't going to be helpful. I needed to multiply by a factor of zeros. I decided to just try to conceptualize base-314 and see where that would take me. I imagined the pizza. I imagined slicing it up into 314 tiny slices. We could count our way up to 314 and then we would have finally completed our pizza. But this only further elucidated the incomplete nature of our circle. We were drawing a circle with 314 line segments. While we could simulate an imperfect circle this way, it would never be perfectly round. We would need to cut the pizza into more slices to make this a more perfect circle. But base 314159 wouldn't be good enough either, because once you zoom in enough, there are too many rough edges. This is why π is infinite in nature, the circle is never perfect enough at any level of analysis. The more you zoom in, the more imperfections you find in its curvature. So there are an infinite number of numbers after the decimal in order to infinitely perfect this curvature at every level of analysis. So a actually trying to implement base-π suddenly seemed impossible, since we would necessarily be counting to infinity before we got to our first complete pizza.

I started complaining about this to a different friend, who also has a deep philosophical perspective on math (I know, I have the best friends).

TP: "How does nature manifest  π if it is so weird?"

Friend: "Maybe it's not nature that's doing the manifesting? Maybe something (e.g. mathematics?) precedes nature?"

TP: "Sure, but my intuition would want me to believe that things are most simple at rock bottom, and emerge into complexity. π seems complex. If  π is at rock bottom, then it would seem like complexity is at rock bottom?"

Friend: "Maybe if you think about π within its place within a complete and consistent system of mathematics or mathematic principles, you might find your simplicity there?"

This conversation only reinforced my desire to get to the bottom of this. How could complexity be at rock bottom? There must be a better more intuitive way of framing these concepts! We discussed some different equations together.

While minorly helpful, none of these seemed to provide the intuitive grasp on what π actually was in its conceptual simplicity. Suddenly, I came across an equation that seemed to make immediate intuitive sense to me.

Basel problem

In looking at this, I intuited (π^2)/6 as a circle (I know that this is not the exact mathematical form for a circle, but it seems close enough to use as an intuition pump for these purposes).

In looking at the sigma series to the right of the equals sign, I began intuiting the denominator as area within the circle (x^2 typically gives you the area of a square).

This infinite series of ever-increasing denominators was creating ever larger squares of area. But the numerator was DIVIDING itself by this ever-increasing base. So each of the 1s in the numerator could be thought of as a pixel. Each of the denominators could be thought of as ever-increasing resolution (a 500x500 image is less resolution than a 1000x1000 image). So, what was happening was we were adding 1 low resolution pixel, then 1 higher resolution pixel, then 1 even higher resolution pixel, and incrementing the resolution to infinity.

So, the way I began to think about it is as a low resolution pixelated attempt at a circle. We can calculate the area of a rough circle with a rough truncated version of π (like 3.1). We get an area of an imperfect circle as shown in the image below. We see a part of the circle's curve as an obviously imperfect edge. A perfect circle would not have square "stairs" to step through. But due to the pixelation, we are incrementing from one square to the next, creating a hierarchy of stairs to step through to approximate this curvature.

How is one to fix this? Well, with π, we can increase the number of digits we use in our calculation. We can go to 3.14 for the next layer of resolution. Then 3.141, then 3.1415, and on and on. In the image below, we can see how zooming in on different layers of resolution gets us a closer approximation of the perfect curvature, but it never is quite perfect, since each layer is still pixelated.

So back to the equation:

Basel problem

What it seemed like this equation was doing was constantly incrementing the pixels outwards. It's as if there is a perfect line in our abstract minds, and we are using math to constantly push towards it. We push towards it with big pixels. Then we zoom in and push towards it again with smaller pixels. Then we zoom in again and again, constantly repeating this pattern of attempting to push towards the ideal.

This iterative pattern reminded me a lot of fractals. The main issue is that fractals imply the recreation of its own shape at smaller layers of analysis. The curve of the circle was in a sense, recreating itself, but in another sense, it was just refining itself. After more discussion with my friend, I figured that "fractal dimension" was a sufficient descriptor for the behavior of the π within a curve.

As defined on the Wiki,  "A fractal dimension is an index for characterizing fractal patterns or sets by quantifying their complexity as a ratio of the change in detail to the change in scale."

For the circle, we are definitely seeing an increase in detail as we increase the scale at which we are viewing the curve. So, then the question becomes - what does it mean to say that reality produces fractal dimensions? What does it mean to say that fractal dimensions are normal, simple, fundamental, and easily produced in nature? How is infinitely detailed normal, simple, fundamental, and easy?

Perhaps it's not! Perhaps our mathematical ideal for a circle is just that - an ideal. It isn't natural. When atoms create a spherical shape, they are not true spheres at the level of infinite detail, but rather chaotic approximations. When gas clouds form into spherical shapes like stars or planets, they are not perfect spheres, but rather chaotic approximations.

The fractal dimension problem (as I am coining it now) is the problem of ontologically instantiating infinite detail. One solution is to posit that there is infinite detail beneath the atom. The other solution is to posit that there is no such thing as a perfect circle/sphere in nature, because there is no such thing as infinite detail, ontologically.

Interestingly, this fractal dimension problem should apply to a straight line as well. The ideal stright line is not supposed to have any pixelated width, just as our ideal curve is not supposed to have pixelated steps. To approach the ideal straight line, we have to infinitely reduce our resolution to get rid of the width of our pixels. But we can never succeed at this, so we must reduce our pixels infinitely, proving the same exact problem.

There is no way to build a perfectly straight line in nature, because the building blocks are not perfect. If you lined up a bunch of atoms, even they would have too much width for our ideal. If you increase your resolution beyond the atom, you enter quantum territory where the building blocks become less stable. How can you build a perfectly straight line if the materials are chaotically bouncing around?

So, the end result of this investigation seems to be pointing me towards "ideal math skepticism", or the idea that our abstract concepts of perfect circles and perfect lines have no ontological correlates in reality. We can't concretize these concepts. We can only attempt mere approximations.

So then what is to be said about the fact that nature seems to like to build approximate spheres? What is it about approximating π that has such a grip on nature? I am beginning to think that the approximation is just due to the nature of the forces. If the forces are such that they push outwards, then they will naturally form spheroids - this is the mathematical pattern in the Basel problem that was solved by Leonhard Euler. We see how the perfect circle is just the sumation of all the 1s pushing outwards. If at base reality within particles, we have a force generator that pushes outwards in all directions, it will approximate a sphere.

So what do you think? Do you agree that the idealized form of curves and lines are fractal dimensions with infinite detail and complexity? Is nature full of infinite detail and complexity, or does it stop at a certain point and just give us rough approximations? What does this say about nature or the mind of God if you are a theist?