Fluency in math

Fluency in language is relatively easy to measure: you can give a talk, keep up in a coversation and write jarry, more or less gramatically coherent texts. Math is trickier: most people struggle with it, some people seem to be better at it naturally. I have no idea why is that the case, but there is an interesting observation.

Math is language we describe universe with, because words aren’t suited well for this purpose. There are a few major concepts that are tough to describe in plain language, like limits in calculus and tensors in algebra. And math is a weird language, mistakes are punished way more than ever, infact, one wrong symbol renders the entire “text” meaningless. This breeds frustration.

Fluency in math, in a particular parts of it, consists of two things. Firstly, the ability to derive new relations and transform existing ones effortlessly and without mistakes. No, there is no “I know this, I’m just so inattentive” when you skipped a minus sign. Mistakes show gaps in either knowledge or skill, they are a signal for you to get some more practice.

Secondly, the internalization of concepts. It boils down to the Feynman rule: you only understand it if you can explain it. The only way to internalize a concept is to link it to existing knowledge: think of the knowledge as a map, and your competence grows in a tree-like shape all over it, creating nodes and lines. As long as there are enough nodes near something new, you can learn it. If you struggle - roll back and explore the area around, maybe go slightly sideways or practice what you already know.

@academicalnerd how the hell do you misinterpret that quote so freaking badly? Who do you think you explaining it to? Yourself? Explanation implies that you are doing it to someone else, anyone else. Mathematics is not about describing the universe, that’s physics, mathematics is about communication. It’s the universal language, not by coincidence, but by definition. Any mathematics that is somehow inherently inaccessible to anyone honestly willing to learn is failed mathematics, and any mathematician who practices that is a failed mathematician. The whole point of mathematics is to maximize understanding in communication, and in the context that you bring up - to prove fluency. Meanwhile fluency in natural languages is fuzzy at best, since it’s a natural phenomenon and is studied as such.


misinterpret that quote

Yeah, I may have butchered Mr. Feynman a bit there, my bad. As for fluency in math - it is a methaphore and I don’t mean it to be absolute truth.

It’s the universal language, not by coincidence, but by definition

I’d argue with you on mathematics being the universal language: there are a few things mathematics fails at. The more complex systems that are called “chaotic”, if I recall.

mathematics that is somehow inherently inaccessible to anyone honestly willing to learn is failed mathematics

I am not by any means talking about failed mathematics or mathematicians. The concepts that are far away from what you already know are usually inaccessible, not inherently, but because one lacks prerequisites.

Donn’t take this too seriously, anyways. I’m a random stem student ranting on the web.

@academicalnerd and I rant back, don’t take it too seriously yourself, student, hmph!

You are now talking about lack of experience, which is far from the initial “most people just can’t math”. Not everyone is a linguist or a writer or a plumber or a baker either, cause not everyone wants to be. Nothing oh so special about mathematics there. Most everyone speaks a language however, and most everyone these days can multiply large numbers, and while there is always debate on the meaning of the simplest words and phrases, I don’t think I ever saw anyone debate on a result of a multiplication.

My point is precisely the opposite: everyone can do math, given the incremental learning and some patience. Probably a miscommunication on my part.


Well, @namark’s reaction is worded a bit harsh for my taste, but I agree with it. Mathematics is just a communication tool, not a language of its own. And indeed it’s such by definition. It’s a purely human construct, not a natural phenomenon. Maybe a good way to think about math language is to observe that definitions are just abbreviations for complex relationships capturing concepts we want to use as nouns and verbs and then we just relate them as in normal language. At the bottom are axioms which are (relatively) arbitrary symbolic statements which just so happen to “make sense to us”. Math does not exist on its own. That’s why it can be “pure” and therein lies its beauty and appeal to certain type of minds.

As a side note, when you speak about the tree-like structures of statements growing out of axioms (of e.g., set theory - which BTW is just a specific choice), you won’t capture the most interesting parts of what can be said with that language: self-referencing/recursive definitions concepts (e.g., tree width) and non-constructive relationships, such as the proof by contradiction.



Yeeeah, I’m not into pure math for the most part. It’s interesting, but I need to ba practical, or rather my major requires me to. And studying extra stuff is resourse-intensive, I can’t have this luxury as of now.

And the best example for how arbitrary the entire math is would be Godel’s theorem, I suppose. There was a video on it recently by Veritasium, I remember you liked the guy.


@academicalnerd @namark It’s not arbitrary man! It’s a major result with vast practical implications on e.g., software development.

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