Math, Thoughts

My Learning grows like Math

I don’t think it’s innate, but I discern a pattern through the way I learn stuff in a unkown environment.
This is not a discrete process but more some fuzzy steps adding to a global context. So here’s the way I get the grip on this unknown environment :

  • When I don’t know, I’m just following orders and establishing rules based on induction and questioning
  • When I start figuring out the rules, I’m aiming for the limits implied by the rules and I observe what constrains are behind
  • When the constraints are established, I observe their reasons by the way I could go around them and overpass the limits
  • Finally, figuring out constraints and limits, I established a structure of the topic, I made the first tools to think on this unknown environment; seeding a knowledge I could grow the now-known environment by finding other ways around in different cases

What it seems to me is; the first two steps are hard to acquire. The two last parts can be iterated to populate an environment. When you know the limits, you get generally stuck. But, adding a hypothesis, there might be a work around. The nature of this assumption makes a specific context of your environment.

I like math, and I could picture it the same way it develops, in a sense.
When you end with something impossible. For instance, you haven’t heard about irrational number and you try to measure the diagonal of a one unit square.

You have a ruler of one unit to make the square, but the diagonal is a bit longer than your ruler. So you decide to put a mark after one unit and break the ruler into 10 pieces. Then you put pieces after your mark and you can fit a little more than 4 of them.
So you break a piece into 10 more pieces (it was a really huge ruler. Like skyscraper tall) and put a piece while still having to break another into 10. You figure out it can go for pretty long before you reach something.

Damn that’s not cool ! My goal was to find the way to measure the diagonal but there’s no limit to this breaking system.
As an engineer (yeaaaah! I finally got the title ffs!), I will have tools to make it (my ruler) but that have a given limit of “breakability”. So I assume I’m making an error.

You broke  your ruler 2 times into 10 pieces. If you had done that for every piece, you would have 10^2=100 of them. Also you calculated there is 1 unit and 4 subunits and 1 subsubunit and some subsubsubunits undefined. That gets a value of 141/100 but it’s not quite acurate.
We’ll give a margin to this value, it will account for the part we haven’t measure due to the precision. This part is less than a subsubunit, so we define a 1/100 error margin.

So I could express my diagonal as 1.41 (+/- 0.01) units. That’s cool and accurate enough for an engineer. Things are defined in a context; it’s somehow a local approximation of larger possibilities. I don’t need more accuracy to make it works.

But the mathematician won’t accept the engineer approximation. He doesn’t work in the real world, with its fuzzy noisiness, but in this thought world made of pure ideas where everything is perfectly defined and is also abstraction. Therefore, he won’t be satisfied with this idea and will try to find a logical way to express it.

You seem now desperate to measure the diagonal of a one unit square (you got some silly hobby though). To get a grip on this idea, you decide to make the context evolve by adding some stuff.
If you use this diagonal with others of the same length to make a new square, you observe its diagonal fitting 2 rulers exactly. That is an interesting property. So you decide to define this length as a square root of 2. By showing you can use your unit back and forth on this property, you opened crazy ideas that’ll lead to years of headaches. Now people will extend this “square root of” idea to crazy extreme like making a square root of 3, or even defining the 2 unit square diagonal as two times the square root of 2. But who really has an exact square root of 2 ruler ?

Well, that’s the idea and the generalisation. It can grow on different context while the rule still apply. Even better as there are some surprising square roots like 9 going back to 3 units, but you can navigate this acquired knowledge with this handy tool.
By adding other ideas, like negative numbers, you end up with new limits. The reason being; length are always positive. If you have a negative, it’s not a length but a distance from some point, and a positive direction have to be given. (leading to imaginary numbers and complex numbers allowing rotation in the plane)
That’s really something you can grow in tremendous way to establish a richer context; the spatial geometry, vectors, matrices and so on.

 

While my point here is reached on the comparison between how to grow math and how I think I approach unknown environment, I just thought of something really interesting !

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