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My go-to example of this is PCA. If you know how to diagonalize a 5x5 matrix by hand, then you "know the math" behind PCA. But this gives you absolutely no understanding of what PCA is, what it does, and why it works. You need higher-level mental models.
This is almost universally true: to understand something, you need the *right* mental models, that capture what *actually matters* about that thing, not just the lowest-level mathematical description you can find. In most cases, the two are completely orthogonal
The same is true of backprop in deep learning -- knowing how to code up backprop by hand gives you no useful knowledge wrt deep learning, and inversely, developing powerful mental models for deep learning does not in any way require knowing the algorithmic details of backprop
(coming from someone who had to implement backprop a lot in the past, first in C, then in Matlab, then in Numpy)
In addition, if you have the right mental model for something, it is generally easy to work out the algorithmic details on your own when you need them, at least down to a level where you can roll out a working implementation (& it becomes trivial if you can just look up details)
Similar to how, say, you can always reinvent the Pythagorean theorem on the fly if you think about geometry through the lens of vector products, or how you don't need to memorize the quadratic formula if you understand what an equation is and the general process for solving them
I’m going to assume that your use of the phrase “completely orthogonal” was intentional.
This tweet basically sums up my current frustrations with my Graduate Algorithms class. Lots of "here's how to transform from A to B" without a higher-level mental model.
Actually, this is one of the first things you learn when you start going deeper into mathematics: that you really ought to care about understanding subjects the "right way" or knowing where characterizations come from ( their motivations, etc.) and I think no one outside realizes
In other words, a good story
I am reminded of a quote by John Tate on Grothendieck: "He just had an instinct for the right degree of generality ... Not generalization for generalization’s sake but the right generalization."
Finding the model or generalization that captures *what matters* and forgets all the details that don't matter ... that is hard, and also is exactly what makes a great mathematician (such a Grothendieck!). Finding that "right" degree of generality should always be our goal.
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