Informality

Although the universe is bounded by its physics, it is flexible enough to accommodate what is essentially a massive number of unimaginable situations. That is a fun way of saying “almost anything can happen”.

So, let us describe that potential as being ‘informal’.

We think there are a set of rules which dictates behavior, but we do understand that there are all sorts of things that can happen “outside” of our known rule set. They may occur at infinitesimally small probabilities, like say once in a billion years, but that doesn’t mean that they can’t happen. They just might not happen in our lifetimes.

Contrast that with mathematics.

We have constructed mathematics abstractly, as a ‘formal’ set of rules and axioms. These explicitly bind any mathematical objects, context, and derived objects, so that anything that occurs outside of them is effectively declared ‘invalid’; not part of the system. Let us refer to that as ‘formal’. We created these objects, the rules, etc. and we can visualize and walk through the mechanics ourselves, and in doing so we strictly enforce the rules.

There are some mathematical objects like ‘infinity’ or concepts like ‘perfection’ that we can use and make part of these formal arrangements, but they do not necessarily derive from our real-world experiences, just from our intellectual capabilities.

In that sense, we can talk about each branch of mathematics as being one or more formal systems, and similarly, we can define our real, existing universe as a really big informal one.

With that framing, we take any mathematical ‘relationships’ and use them to describe things we see in reality, such as physics, accounting, statistics, or even economics. While those relationships mostly fit and explain what we are seeing, the formal math itself is always just an approximation to the underlying informal mechanics. Formal relationships are too rigid to fully express informality.

That is, even though we might describe a physics formula as a ‘law’ since reality is informal, it is entirely possible that while it mostly fits, there could be some unknown and inconceivable circumstance where it does not. If we are unaware of any such circumstance, we tend to believe that the fit is ‘perfect’, but rather obviously if that isn’t a valid real-world concept either then that is a flaw in our understanding, communications, and perceptions.

The rather obvious example of this is gravity. For a long time, people knew it was there as an obvious visible effect, but it took Sir Issac Newton to describe the relationship. Then later Albert Einstein refined it, and they are still trying to corral that within Quantum Physics. Each time, the mathematical relationships got tighter, but also more complex.

So, it gives us a great way of understanding the relationships we are seeing in the world around us. We take formal mathematical relationships and see if they fit informational circumstances. If the fit looks good, we add it to our collective intelligence and build on those foundations. But really, we should know that no formal relationship will ever 100% fit into an informal circumstance, or at least we should always leave room for doubt or more precision.

History is littered with us applying increasingly complex mathematical relationships which do advance our capabilities but later are supplanted by even better, more precise mathematical relationships. It is a never-ending process to just keep improving the precision of our knowledge and expectations.

Now, that happens regardless of what occurs in mathematics. And it is often the case that mathematicians are busy exploring new objects and formal systems long before we ever find a way to relate them back to our world and our lives. My favorite example is Évariste Galois who created Group Theory back in 1829, which was used since the 1980s to help provide scratch resistance in the form of error-correcting codes for encoding music on CDs. There may have been earlier examples of this mathematic branch getting applied in our lives, but certainly this one effect billions of people, most of whom were obvious of what was going on. Sometimes it takes a while to match the formal relationships back to the informal universe.

No doubt, there are formal mathematical relationships that we understand that will never be applied back to the world around us. Just because we can craft a formal system, doesn’t mean it corresponds with some aspect of our informality. That’s the corollary to the fit possibly never being perfect. There is a disconnect there.

What’s really interesting about this gulf between formality and informality is computers.

As the CPU runs, it enforces the formal running of computations, which are Turing-complete. So, the execution of code within a computer is strictly formal. The program runs, or it stops. But, that execution is running on top of hardware, which is bounded by our reality, so it is actually informal. That is, the hardware can glitch, which would throw off the software, breaking the illusion of formality. it doesn’t happen often, but it does happen.

Thus, once we started playing with computers, we rather accidentally found a way to bind our abstract formal systems to our physical informal one. That has a bizarre implication.

Formal systems then are not necessarily just fully abstract things that exist in our minds. Mathematics may not be just the product of our imaginations. It tends to imply that the formality of our abstract systems themselves has some, although limited, binding to the rules that underpin the universe. That is our imaginations as physical constructs may also be bounded by the universe.

In that sense, maybe under all of that informality lies a strictly formal set of rules? Not sure we’ll be able to figure that out in our short lifetimes, but it does seem like there is something below the probabilistic nature of quantum physics and that we haven’t quite reached the bottom yet.

A more obvious consequence of the gulf comes from using computers to solve real-world problems. The core of the software needs to be formal, but the problem space itself is informal. So, we end up mapping that informality onto formalized versions of data and code as solutions, and we often tend to do it statically since that is easier for most people to understand. But of course, both the informal and possibly dynamic nature of the problem space usually throws a wrench into our plans.