3 ms·

>Step 3. Memorize the exact wording of the definition. Huh. Any mathematicians who want share their own opinions and experiences about this? This pretty much

>Step 3. Memorize the exact wording of the definition. Huh. Any mathematicians who want share their own opinions and experiences about this? This pretty much goes completely against my experience with other grad school level neuroscience/ML You don't want to be so familiar with stuff as to make it second nature but NOT from memorization. That, at least an other areas, leads to surface level recognition Does the author mean internalize and not memorize?

wisty 2y ago
The author also said to internalise it, but the simple fact is that in a proof-y math test you can't just say "well it should converge if everything is well behaved", the edge cases matter and definitions set the boundaries.
vector_spaces 2y ago
The reasoning is that, similar to memorization of times tables, being able to recall a definition or a theorem and its context / assumptions "automatically" without needing to use your brain frees you to worry about higher order activities. Being able to apply a theorem over and over again ultimately builds mastery and internalization. Counterintuitively, mathematicians like being "brain-off" as much as possible -- you want to be able to read a phrase like "closed convex subset of a Hilbert space" and effortlessly think to yourself "oh! there's a unique norm minimizer" -- if you have to piece that together from scratch every time, you're going to have a hard time -- reading papers and learning new fields becomes a dreadful slog, similar to how math in general becomes a slog for kids who don't memorize their times tables.
kmill 2y ago
The author really does mean memorize. To engage with pure mathematics, you must know the definitions, since the definitions are the bedrock of the subject. If you don't know the axioms of a topology, how can you check for yourself whether something forms a topological space? Or without knowing the exact definition of continuous, how can you know whether a proof of continuity is correct? Without knowing the definitions, you can't really know mathematics. To be clear, this does not mean memorizing all the theorems. Getting to know the theorems (and solving problems) is what helps you internalize the subject. Math is the art of what's certain, and knowing exactly what the objects of the subject are is necessary for that. Theorems are derived from the definitions, but definitions can't be derived. In my experience with a math (undergrad and PhD), I realized I had to know definitions to feel competent at all. In my teaching, it's hard to convince students to actually memorize any definitions — so many times students carry around misconceptions (like that "linearly independent" just means that no vector is a scale multiple of any other vector), but if they just had it memorized, they might realize that the misconception doesn't hold up. Math is weird in that the definitions are actually the exact truth (by definition! tautologically so), so it does take some time to get used to the fact that they're essential.
- xanderlewis 2y ago
  > the definitions are actually the exact truth (by definition! tautologically so) It’s easy to forget that non-math people find this — the idea that the definition is its own ‘model’ rather than an approximation of something more ‘real’ — somewhat hard to stomach. Outside of pure mathematics the idea is that mathematics is a tool for (usually lossy) modelling of reality, not a collection of already perfectly well-motivated objects to be studied in their own right.
  - gowld 2y ago
    More generally and symmetrically: When you are studying science and technology, and the math theorem doesn't match experiment, the theory is probably wrong (or incomplete, missing factors), so you can discard it or try to improve it. When you are studying math, and the intuition doesn't match the theorem, the intuition is probably wrong (or incomplete).
    - kmill 2y ago
      Things get a bit messier once you're doing research mathematics — definitions don't just come from nothing, and a good definition is one that serves its theorems. Definitions can be "wrong" (they might be generalizable, they might have unexpected pathological examples, etc.), and it's the result of lots of hard work by lots of mathematicians throughout history that we have the definitions we enjoy the use of today. But yeah, while studying math, I think it's similar to learning programming — don't blame the compiler for your mistakes, it's a well-tested piece of software.
      - xanderlewis 2y ago
        Yes. It’s slightly disingenuous of me to suggest that any definition is as valid and ‘real’ as any other. Obviously, mathematicians care about some ideas more than others.
        kmill 2y ago
majikaja 2y ago
Maybe not the exact wording but the exact meaning...
Skeime 2y ago
I think they mean memorize. Most students simply lack the mathematical experience to internalize a definition correctly without simply memorizing it. They will forget an "only if", or accidentally swap two quantifiers around, or conflate two variables that need to be kept separate, etc. This is okay! They're students on the way to gain that experience. At some point you can and will go over to internalizing, instead. But as advice to students just starting out, memorizing is the way to go.
jyounker 2y ago
Mathematical definitions are precise. If you miss one part of a definition, then you cannot actually understand what the definition actually means. You have to internalize the meaning also, but you must know the definition precisely.
- Koshkin 2y ago
  It is also useful to know all alternative definitions, if any exist (they often do).

wisty 2y ago

The author also said to internalise it, but the simple fact is that in a proof-y math test you can't just say "well it should converge if everything is well behaved", the edge cases matter and definitions set the boundaries.

vector_spaces 2y ago

The reasoning is that, similar to memorization of times tables, being able to recall a definition or a theorem and its context / assumptions "automatically" without needing to use your brain frees you to worry about higher order activities. Being able to apply a theorem over and over again ultimately builds mastery and internalization. Counterintuitively, mathematicians like being "brain-off" as much as possible -- you want to be able to read a phrase like "closed convex subset of a Hilbert space" and effortlessly think to yourself "oh! there's a unique norm minimizer" -- if you have to piece that together from scratch every time, you're going to have a hard time -- reading papers and learning new fields becomes a dreadful slog, similar to how math in general becomes a slog for kids who don't memorize their times tables.

kmill 2y ago

The author really does mean memorize. To engage with pure mathematics, you must know the definitions, since the definitions are the bedrock of the subject. If you don't know the axioms of a topology, how can you check for yourself whether something forms a topological space? Or without knowing the exact definition of continuous, how can you know whether a proof of continuity is correct? Without knowing the definitions, you can't really know mathematics. To be clear, this does not mean memorizing all the theorems. Getting to know the theorems (and solving problems) is what helps you internalize the subject. Math is the art of what's certain, and knowing exactly what the objects of the subject are is necessary for that. Theorems are derived from the definitions, but definitions can't be derived. In my experience with a math (undergrad and PhD), I realized I had to know definitions to feel competent at all. In my teaching, it's hard to convince students to actually memorize any definitions — so many times students carry around misconceptions (like that "linearly independent" just means that no vector is a scale multiple of any other vector), but if they just had it memorized, they might realize that the misconception doesn't hold up. Math is weird in that the definitions are actually the exact truth (by definition! tautologically so), so it does take some time to get used to the fact that they're essential.

xanderlewis 2y ago
> the definitions are actually the exact truth (by definition! tautologically so) It’s easy to forget that non-math people find this — the idea that the definition is its own ‘model’ rather than an approximation of something more ‘real’ — somewhat hard to stomach. Outside of pure mathematics the idea is that mathematics is a tool for (usually lossy) modelling of reality, not a collection of already perfectly well-motivated objects to be studied in their own right.
- gowld 2y ago
  More generally and symmetrically: When you are studying science and technology, and the math theorem doesn't match experiment, the theory is probably wrong (or incomplete, missing factors), so you can discard it or try to improve it. When you are studying math, and the intuition doesn't match the theorem, the intuition is probably wrong (or incomplete).
  - kmill 2y ago
    Things get a bit messier once you're doing research mathematics — definitions don't just come from nothing, and a good definition is one that serves its theorems. Definitions can be "wrong" (they might be generalizable, they might have unexpected pathological examples, etc.), and it's the result of lots of hard work by lots of mathematicians throughout history that we have the definitions we enjoy the use of today. But yeah, while studying math, I think it's similar to learning programming — don't blame the compiler for your mistakes, it's a well-tested piece of software.
    - xanderlewis 2y ago
      Yes. It’s slightly disingenuous of me to suggest that any definition is as valid and ‘real’ as any other. Obviously, mathematicians care about some ideas more than others.
      - kmill 2y ago

majikaja 2y ago

Maybe not the exact wording but the exact meaning...

Skeime 2y ago

I think they mean memorize. Most students simply lack the mathematical experience to internalize a definition correctly without simply memorizing it. They will forget an "only if", or accidentally swap two quantifiers around, or conflate two variables that need to be kept separate, etc. This is okay! They're students on the way to gain that experience. At some point you can and will go over to internalizing, instead. But as advice to students just starting out, memorizing is the way to go.

jyounker 2y ago

Mathematical definitions are precise. If you miss one part of a definition, then you cannot actually understand what the definition actually means. You have to internalize the meaning also, but you must know the definition precisely.

Koshkin 2y ago
It is also useful to know all alternative definitions, if any exist (they often do).