One question I was always wondering is: Is Math simply a tool we use for calculations? Or it contains discoveries of something objective such as the Physics Laws we found about our world?
On one hand, we did use Math as a tool to simply our daily tasks. So it first appeared as a human “artifact” (or human creation), something totally subjective instead of “objective”. On the other hand, we have talked about using Math as a “language” to communicate with aliens, since Math describes a pattern that is universal regardless of our thinking process or culture.
In the end, when I think about all the rules we found in Geometry or other Math, I think it is discovering some logical connections in an objective world. In other words, Mathematicians are not much different from Physicists, in the sense that they are all “discovering” instead of “inventing”.
For sure, it seems that there is a lot of “inventing” (or what Mathematicians would call “construction”) in Math, but these inventions/constructions are just a way to facilitate the discovery process.
For example, prime numbers are so useful and essential to the foundation of Math. They are so easy to understand, even to the elementary students, and yet so hard to master, as we still don’t have an easy way to tell what they are or even how many of them exist given a max-bound. Finding the next prime number becomes a constant discovery process not much different from finding the next planet in the galaxy – we are all dealing with an “objective” world that we are constantly exploring.
However, this objective world is certainly not a physical world. In fact, it is totally independent of the physical world. Math has the abstraction built into it from the start. When we talk about numbers, it is an abstraction, since number 2 is just number 2, it could be 2 people or 2 apples. A “point” should be infinitely small and a line should be “infinitely” thin – certainly there is no such “thing” in the real life.
The word “objective” also means it is independent from our “subjective” thinking. In other words, it is independent from our thinking process, and independent from the “spiritual world”.
This is what I would call “the third world“: the “logical world”, which is independent of both “physical world” and “spiritual world”.
Since it is an independent and “self-contained” system, it brings both a bless and a curse. The bless is that we can build a self-contained system without worrying about the physical world and subjective bias. Also, the “abstraction” can work as a tool for simplification, allowing us to search for essential patterns. The curse is that it could easily get detached from the real life and becomes a “distorted” or misleading “description” of the real world.
For example, many professors in academia care too much about the theory than the reality. They want to discover a self-sustained system that is pure and perfect, so it has no crack or flaw in it. Something that is both “simple” and “true”.
Unfortunately, in reality, there is no such thing. When you apply abstraction, you would definitely lose something in the process.
For example, many economists enjoy their theory on “perfect competition” and “commodity industry”. They have some “good” abstraction in place: if a product is commodity, competitors will continue to join in so that eventually all the profits will be competed “away” with nobody making any money as a result.
Sounds so true and scary? Don’t be, since it is something never truly existed in the real life. In the real life, I would argue that there is actually “no” pure commodity, just as there are no infinitely small points or infinitely thin lines.
Businesses have many ways to differentiate themselves even when their products are considered as “commodities”: they could offer a better service (think Cisco), some customization (like Apple), better “presentation” (again, Apple), better “usability” (again, Apple), better “brand” (like Coca Cola selling sugary water), better ecosystem (like Microsoft), or better “availability” (just pay the shelf-space, or build more local stores like Walgreen, or a coal miner build a mine right next to a power plant).
The so-called “availability” isn’t just physical, as some businesses could always do more advertisement or optimize search words to increase its availability to users.
In other words, while professors like to teach their theories about “commodities”, in real world, you can find that things are never so simple. Their theories only described an extreme case which never truly existed at 100%.
So the “logical world” is powerful, useful and yet it could be misleading as well, if we don’t understand its limitations.