##### Disclaimer

^{The following blog post touches many different scientific fields like Maths, Physics, Anthropology and Computer Science, which the author has little (or no) idea about. Neither the author nor the publisher shall be held liable or responsible to any person or entity with respect to any loss or incidental or consequential damages caused, directly or indirectly, by the information contained herein. Read at your own risk.}

## Background

It all began a couple of months ago when I started watching some videos^{[1]} from Bartosz Milewski about Category Theory (highly recommended for any software developer). Bartosz has also recently given a really interesting talk^{[2]} at the Øredev Conference named "The Earth is Flat: Exploring the Limits of Science". In his talks Bartosz approaches Category Theory and the way the human brain works from a philosophical perspective. This blog post is heavily influenced/inspired by that.

## Composability in programming

Programming is about forming a "structure" and formalising a bunch of operations that we want a computer to execute for us. Good code is considered the one that is easy to maintain and extend, one that is easy to reason about. But how do we translate a very complex problem, or some really complex business logic into code? Well it is all about "divide and conquer". We split the complex problem into smaller problems/modules, and then we combine all the modules together to create the system we wanted in the first place. So it boils down to creating components/solutions that can be easily composed. This is the reason we used design patterns, in the object oriented era and the Gang of 4 frenzy, although that didn't work very well with concurrency. This is one of the main reasons functional programming is so hot these days: Functional programming allows us to split a problem and combine its solutions. We can use Monads and everybody knows that Monads compose!

This way of tackling complexity however permeates all human activity.

## Mathematical beauty

In mathematics, some solutions/theorems are considered "beautiful". Many people would argue that beauty stems from simplicity, some other from symmetry. What would make Paul Erdős say:

"Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is."

In any case, it is definite that there is a certain bias towards a certain way of thinking, towards certain type of models. Is that bias a feature of the universe, or is it a feature of the human brain?

Unarguably, many "beautiful" theories have been contradicted throughout history. One very famous example of that is David Hilbert's theory that logic is complete: "Every true statement is provable, every provable statement is true". This was devastatingly proved as impossible by Kurt Gödel in 1931, with the well-known Incompleteness theorems.

But is Mathematics discovered or invented after all? This question has been troubling mathematicians and philosophers for many years now. There are actually numerous theories around that, in the whole spectrum of invented-discovered^{[3]}.

Maybe certain fields of Mathematics that have to do with proof systems and logic are a projection of how the human brain works and this is why they look similar. This actually has been proven with the famous Curry-Howard-Lambek Isomorphism^{[11]}: There is a one-to-one correspondence between three seemingly distinct fields of Mathematics: Constructive logic, Typed Lambda calculus and Cartesian closed categories, ie. they are identical.

*Correspondence of types (top) and formulas of logic (bottom) by Luca Cardelli.*

Is this actually an ultimate truth of the Universe that we discovered from 3 different routes? Maybe not.

## Physics - Maybe we are still far from understanding the Universe

We have tried to apply the exact same logic, combining simpler models into more complex ones, into the way we intepret the world. Physics however is not like philosophy or mathematics. Every theory has to be verified by experiments.

Throughout our history, we have come up with some really "beautiful" models of the world. Ideas that were simple, but we hoped that would explain a big part of the universe. Celestial spheres, the 4 classical elements and Aether are some examples, just to name a few. The truth is however that the universe is much more complex. Richard Feynman, one of the most famous physicists of all time, would say^{[10]}:

"I think I can safely say that nobody understands quantum mechanics."

*The Standard Model of particle physics is the theory describing three of the four known fundamental forces (the electromagnetic, weak, and strong interactions, and not including the gravitational force) in the universeSource: Wikipedia*

We believed that matter is built from atoms and the atoms are built from elementary particles that compose nicely. We wanted to consider elementary particles as points, but we then realised that they sometimes behave like waves. This confused us and is an indication that maybe we still don't understand what's happening. Even in the case of a 2 particle system, it cannot be broken down to simpler 1 particle systems due to phenomena like quantum entanglement^{[6]}.

Finally gravity is another example of a non-decomposable "problem": it cannot be decomposed into quanta. Gravity is actually something that we still fail to understand, as we cannot renormalise it^{[7]} (at this point, I have no idea what I am talking about).

## But humans are awesome!

Maybe if we look at humans and human evolution from an anthropological perspective, we can get a better understanding.

The human brain is a product of evolution over 2 million years, since the first humans (ie. mammals of the genus "homo") appeared on Earth. We - Homo Sapiens - are the epitome of creation. We evolved in a very distinct way, being the best "model" of humans. Previous "models" like Homo Ergaster, Homo Erectus and Neanderthals were worse, less perfect versions of us. We are a unique creature that have little in common with the rest of the animal kingdom. Well, the truth is that there are a lot of fallacies in the previous statements.

Homo sapiens is just another species of the genus Homo. Neanderthals were not worse models of us, on the contrary they were bulkier and had bigger brains. We share a common ancenstor but we didn't evolve in a straight line from Homo Erectus or Neanderthals - they are just different species of the same genus, just like horses and donkeys. From about 2 million years ago until around 10,000 years ago the earth was inhabited by several different human species that co-existed^{[4]} and had a lot of common characteristics. Even today the genetic difference between humans and chimps is less than 2%^{[5]}.

So maybe after all, we are not that unique.

## Conclusion?

The truth is that there can be no definite conclusion, but rather just questions. Is our brain capable of ever understanding the universe? Will computers and AI helps us towards that task? Or is it that maybe the universe is so complex that cannot ever be modeled by any computer? Well, at least in that case we can be confident that we don't live in a simulation...

## References

[1] Category Theory 1.1: Motivation and Philosophy

[2] The Earth is Flat: Exploring the Limits of Science

[3] Philosophy of mathematics

[4] Sapiens: A Brief History of Humankind

[5] Human evolutionary genetics

[6] Quantum entanglement

[7] Quantum gravity - Nonrenormalizability of gravity

[8] "Propositions as Types" by Philip Wadler

[9] Mathematical beauty

[10] Richard Feynman - The Character of Physical Law

[11] Curry–Howard–Lambek correspondence