An Effective Recipe for Mathematical Modelling

By: Danny Geisz | July 20, 2020

Project: Orchid

Sup readers. Prepare yourselves for another disjointed post. I’m basically using my blog as a repo for and log of my thought processes.

K, so basically a big thing that I’ve learned about myself is that even though I really enjoy learning about pretty much anything technical (math, physics, CS, etc) it’s immensely hard for me to stay motivated to continue working on a project unless I can very easily discern why the project benefits humanity. That’s part of the reason why I’m really enjoying my current project.

Oh wait. I forgot that I haven’t written a single blog post about my current project yet. RIP. I’ll do that soon. Basically, I’m trying to build a social media platform that psychologically incentivizes healthy, non-violent conversations about traditionally controversial and polarizing topics. I’m bringing this up simply because I perceive the goal of this project as being of upmost importance for the future health of the purportedly self-healing super graph that is humanity.

Anyway, back to Orchid. The reason I bring all this up is because in past months, I haven’t really been able to come up with a compelling reason why building Orchid would be beneficial to humanity, which has meant that it’s somewhat difficult to remain motivated to work on it, which in turn means I haven’t prioritized working on it, which means I haven’t worked on it all that much.

Orchid, however, has been dancing around the back of my head for the past couple of months simply because I think it would be super cool if it worked. If nothing else, it would give me a way to work on learning things like QFT without the overwhelming frustration of working with ginormous equations with pencil and paper. While I personally don’t see learning QFT as a particularly meaningful pursuit at this juncture of my life, the equations that govern quantum fields are super, super juicy, and it’s fun to play around with them.

Anyway, in trying to come up with ways that Orchid could benefit the humanity graph, I’ve had a couple ideas. This first one kinda showed its head in my last post, but I’ll flesh it out a bit more here. Basically, if Orchid works as I intend, it will essentially create a method for storing mathematical entities of all varieties in well-defined computer data structures. Neat. Additionally, the rules governing mathematical manipulations of these particular mathematical entities will also be stored in similarly well-defined computer data structures. The reason why this is compelling is because computers would then have the ability to work with and manipulate these structures on their own, without human interference. Obviously, the algorithmic approach the computers would take in manipulating the mathematical structures would be defined by humans, but it would essentially allow computers to perform the same sorts of computations humans do all by themselves.

Why is this potentially cool? Because up until this point, most mathematical software is essentially super-powered calculators. Mathematica, for example, has been built over the last 30 years, and during that time, human beings have translated all varieties of mathematical processes into code. There’s a reason it takes up like 20 GB. It’s gigantic. But if you want to numerically solve erf, like I did back in high school, well then boy is Mathematica for you. How is orchid different? Instead of having a bunch of hard-coded algorithms for solving and manipulating previously defined mathematical expressions, Orchid allows you to define entirely new mathematical entities and the rules that govern them. This, furthermore, allows users to have a central location for creating a manipulating all varieties of mathematical entities.

Not only that, but the computer itself would have a way of creating and manipulating exotic mathematical structures. This means that computers would be able to do what previously could only be done with pencil and paper, or on a blackboard. Instead of slaving away doing a bunch of predefined calculations, computers could actually be “doing the math” in the same way that humans have for the last several centuries.

I imagine it might not be immediately clear why computers doing pencil and paper mathematics would be beneficial to humanity. Let me explain as clearly as I can. Mathematics has provided humanity with perhaps the most robust and effective method for quantitatively understanding the world around us. And when humans better understand the world, they can make better predications about the state of the world. As I’ve discussed in previous posts, whenever a structure exhibits stable characteristics and behaviors, that structure can be effectively utilized in the creation of another, larger stable structure. In this context, that means that whenever humans are able to understand the stable characteristics and behaviors of a system, they are frequently able to use that system in a manner that increases humanity’s overall fitness.

Theoretical physics, as a field of study, is perhaps the best and most well-established example of what I’m describing. The phone in your pocket, the missile defense system that keeps you safe, and the lightbulb that illuminates your house are each examples of the staggering amount of technologies that fundamentally would not exist without developments in theoretical physics.

And what is theoretical physics? It is the practice of modelling physical systems in terms of well-defined mathematical structures, and then making predictions about the systems themselves using well-defined mathematical manipulations on those structures. In case you are wondering, experimental physics is the practice of designing and carrying out experiments to gather data on physical systems. Analyzing these data sets is what allows theoretical physicists to determine the degree to which their models accurately describe reality. So, if you’ve ever wondered what those nerds up in national labs, universities, or particle accelerators are doing, it’s that.

Physics is all fine and good, but I’ve found that much of what goes on in modern day physics doesn’t actually seem to affect many people. Some of it certainly does, but a good deal of it is documented in papers that 99.99% of humanity can’t understand and don’t really do people any good.

However, there are plenty of other real-world systems that could be analyzed using the aforementioned methodology. Data scientists are basically the people attempt to understand large data sets using machine learning or other statistical methods. From what I understand of current practices in data sciences, there really isn’t a large amount of coming up with novel mathematical models for understanding data sets in a manner similar to theoretical physics. There certainly is some, but nothing compared to physics.

And there’s a very good reason for this: coming up with appropriate mathematical models for different systems requires a huge amount creativity and intellectual capacity. In other words, it’s really, really hard. Ever wonder why everyone thinks Einstein was so smart? It’s because he came up with extremely precise mathematical frameworks for describing relativistic systems. He did a bunch of other stuff too, but he’s celebrated mainly for his ability to create compelling mathematical models for physical systems. And let me tell you, General Relativity is no small pill to swallow. There are a ton of moving parts and different mathematical structures at play, but my gosh, the resulting theory is objectively gorgeous.

Anyway, I could fawn on about GR all day long, but that’s not why we’re here. The reason Orchid is compelling is because it would drastically simplify the process of coming up with mathematical models to describe different systems. Not only that, with the right set of algorithms, the computer could by itself create mathematical models to describe data sets.

That last sentence probably didn’t have the desired effect on you, so let me rephrase. Orchid could potentially allow a computer to do the same thing Einstein did that made him famous as one of the smartest people in history.

IMHO, that’s pretty dank. But even if that last bit doesn’t happen, Orchid would still provide humanity with a framework for easily constructing and manipulating mathematical entities, which could in turn be used to understand the world around it with a degree of precision previously unattainable.

Well, maybe that’s being overly optimistic. We’ll see.

Anyway, when I started this post, I wasn’t going to talk about Orchid, I was going to give the recipe for consistently and efficiently modelling a system mathematically. Here goes.

  1. Determine which parts of the system in question that you would like to understand.
  2. Create a set of mathematical structures that quantitatively describe the interesting characteristics of the system in question.
  3. Clearly lay out the method that can be used to extract observable quantities from the structures in part 2.
  4. Describe all factors that effect the system in question as transformations on the mathematical structures from part 2.
  5. Use the defined mathematical entities to make predictions about the system in question. Queso (K, so) I think this is, if nothing else, a reasonable starting point for understanding a system in terms of mathematical structures. It also gives me the spiritual reassurance that Orchid is actually a tool that could be used to improve and optimize the humanity graph.

Just hit page 7, which is about 5 pages longer than I thought this post was going to be, so I’m going to wrap this sucker up. I guess as one last thing, if this stuff interests you, shoot me an email. After leaving Berkeley, I haven’t talked to a lot of people interested in this stuff.

May your socks always come bedecked with cool designs. Peace.