4 ms·

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

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.

Almondsetat 3y ago
What does it mean to "understand" the square root?
- Jensson 3y ago
  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.
  - Almondsetat 3y ago
    You did not answer the question
  - ChainOfFools 3y ago
    > Such people never understood velocities, they just memorized some rules. These are the ones who always express a special dislike for word problems.
- ChainOfFools 3y ago
  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.

Almondsetat 3y ago

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

Jensson 3y ago
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.
- Almondsetat 3y ago
  You did not answer the question
- ChainOfFools 3y ago
  > Such people never understood velocities, they just memorized some rules. These are the ones who always express a special dislike for word problems.
ChainOfFools 3y ago
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.