There are "Levels" to Understanding Mathematics

The way it was described to me when I was in high school was in terms of 'levels'.

Sometimes, in your mathematics career, you find that your slow progress, and careful accumulation of tools and ideas, has suddenly allowed you to do a bunch of new things that you couldn't possibly do before. Even though you were learning things that were useless by themselves, when they've all become second nature, a whole new world of possibility appears. You have "leveled up", if you will. Something clicks, but now there are new challenges, and now, things you were barely able to think about before suddenly become critically important.

It's usually obvious when you're talking to somebody a level above you, because they see lots of things instantly when those things take considerable work for you to figure out. These are good people to learn from, because they remember what it's like to struggle in the place where you're struggling, but the things they do still make sense from your perspective (you just couldn't do them yourself).

Talking to somebody two or levels above you is a different story. They're barely speaking the same language, and it's almost impossible to imagine that you could ever know what they know. You can still learn from them, if you don't get discouraged, but the things they want to teach you seem really philosophical, and you don't think they'll help you—but for some reason, they do.

Somebody three levels above is actually speaking a different language. They probably seem less impressive to you than the person two levels above, because most of what they're thinking about is completely invisible to you. From where you are, it is not possible to imagine what they think about, or why. You might think you can, but this is only because they know how to tell entertaining stories. Any one of these stories probably contains enough wisdom to get you halfway to your next level if you put in enough time thinking about it.

What follows is my rough opinion on how this looks in a typical path towards a Ph.D. in math. Obviously this is rather subjective, and makes math look too linear, but I think it's a useful thought experiment.

Consider the change that a person undergoes in first mastering elementary algebra. Let's say that that's one level. This student is now comfortable with algebraic manipulation and the idea of variables.

The next level may come somewhere during a first calculus course. The student now understands the concept of the infinitely small, of slope at a point, and can reason about areas, physical motion, and optimization.

Many stop here, believing that they have finally learned math. Those who do not stop, might proceed through multivariable calculus and perhaps a basic linear algebra course with the tools they currently possess. Their next level comes when they find themselves suffering through an abstract algebra course, and have to once again reshape their whole thought process just to squeak by with a C.

Once this student masters all of that, the rest of the undergraduate curriculum at their university might be a breeze. But not so with graduate school. They gain a level their first year. They gain another their third year. And they are horrified to discover that they are expected to gain a third level before they graduate. This level is the hardest of them all, because it is the first one that consists in mastering material that has been created largely by the student.

I don't know how many levels there are after that. At least three.


As you learn more and more, your comprehension grows and you see more of the big picture. Conversations with people at other levels change as you rise.

Folksonomies: mathematics levels

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Thought (0.979005): dbpedia | freebase | opencyc
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 Mathematicians of Reddit: Is there some point where you see a big picture and everything "clicks"?
Electronic/World Wide Web>Message Posted to Online Forum/Discussion Group:  man_after_midnight, (2013), Mathematicians of Reddit: Is there some point where you see a big picture and everything "clicks"?, Reddit, Retrieved on 2014-02-19
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  • Folksonomies: mathematics levels