Because machines can remember without understanding and do just fine solving just about any problem you might find in undergrad math? A calculator can find the square root of 7 without knowing what a square root is, or what a square is, or what a number is.

People who have a knack for memorizing long lists of arbitrary if-then tables can excel in rote mathematics (up to say multivariable calculus) without needing a philosophically deep understanding of what's going on, for the same reasons.

What does it mean to "understand" the square root?

I can't speak for everyone because I don't know that there is a universal axiomatic understanding, but one way to "understand" finding the root of a given quantity would be that you are peeling off a dimension from a base unit (of, say, area) to arrive at a lower dimensional base unit in the same numbering system.

Another aspect of understanding is _why_ you are doing this, where does it fit into the programme of necessary compression of infinite information density (i.e. the number line is infinitely "dense") so it may be accessible despite the material confines of a human brain and its limited, discretizing capacity for dealing with multiple elements in a single operation. So, philosophically, different lower dimensional spaces integrate to form higher dimensional spaces, in order to facilitate the description of change from one thing into another thing. One linear dimension changing into another linear dimension requires a transition through a quadratic space, from which we get a curve.

You connect it to your intuition. Square root is the side of a square with that area. You can memorize that description, which doesn't help. Or you can connect that to your intuitive understanding so that your intuition understands it, then we say you understand.

A good example is velocity. Many people who passed math classes can't answer "how long does it take to drive 80 miles if you are going 80 miles an hour". Such people never understood velocities, they just memorized some rules. Memorizing rules wont help you solve that question well, you need to make your intuition understand the relationship between velocities and distances.

As you learn more you start to build a net of intuitive connections between all the things, that should be your goal when learning these things. That net will last you a lifetime. Word based or symbol based memorization is mostly worthless in comparison, doesn't help you apply it to other subjects and takes more effort to build.

> Such people never understood velocities, they just memorized some rules.

These are the ones who always express a special dislike for word problems.

You did not answer the question

Because they're different things? There's obviously interplay, but from my experience...

I have only vague recall of most CLI things, but I can get up to speed again very quickly (when a situation demands it) because I've outsourced remembering arcane command line options to "man" and just need a system with docs installed and remember(!) that there are commands called "cat", "ls", "awk", etc.

A similar thing applies with math... the analogy strains a bit, but "cat", "ls", "awk", etc. are the 'understanding' which underpins everything else. I did a recent thing with force calculations and symmetry which apparently impressed some of my engineer friends' colleagues... but the details are unimportant. Just knowing what a sin/cos curve looks like and what an integral fundamentally is got me there... but the fundamentals were enough to get there, is the point :)

Can you demonstrate how to understand and remember word genders in gendered languages, for instance? A table is masculine in one (German) and feminine in another (French).

Language learning is a perfect example indeed.

You memorize so much stuff that at a certain point the language "clicks" and you can infer meaning and rules even if you don't know them yet.

Remembering is understanding, understanding is remembering.