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I'm trying to understand the details and differences between saber, saber de, and conocer a little bit better.

Here's my understanding of what these words mean in various contexts. Are these translations all accurate? Which ones could be improved and how?

  • A person
    • Sé a David. Not used.
    • Sé de David. I'm knowledgeable about David (but I haven't necessarily met him).
    • Conozco a David. I've met David; David is an acquaintance of mine.
  • A place
    • Sé Colorado. Not used.
    • Sé de Colorado. I know a lot of things about Colorado (but I haven't necessarily been there).
    • Conozco Colorado. Colorado is a familiar place; I've probably been there.
  • A topic
    • Sé matemáticas. I know the entirety of mathematics (which is impossible).
    • Sé de matemáticas. I know a lot of mathematics; I'm knowledgeable about mathematics.
    • Conozco matemáticas. I've seen a lot of mathematics before; seeing or hearing about mathematics is a familiar experience (but I don't necessarily understand anything about mathematics.)
  • A language
    • Sé español. I can speak Spanish; I have the knowledge and skills needed to speak Spanish.
    • Sé de español. I'm knowledgeable about the Spanish language (but I'm not necessarily able to speak it).
    • Conozco español. Spanish is a familiar language; I've had a lot of exposure to Spanish (but I don't necessarily have any knowledge about Spanish).
  • An action
    • Sé nadar. I can swim; I know how to swim.
    • Sé de nadar. I know various things about swimming.
    • Conozco nadar. (I'm not sure what this would mean. I know what it's like to swim? I've seen a lot of swimming happen?)
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Sé matemáticas doesn't mean you know the entirety of mathematics; it's perfectly OK to use it.

Sé de matemáticas and Sé de nadar both sound a bit off; with a quantifier (Sé algo de matemáticas, Sé bastante de nadar) it might be better. In any case, saber de doesn't necessarily mean you know things; it often means just that you've heard of them or read about them, maybe with a bit more certainty than mere hearsay.

Conozco matemáticas sounds positively weird, as does Conozco español. In general I'd say conocer goes with people and places, not with skills. And it never goes with an infinitive: Conozco nadar is ungrammatical.

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    Although, adding an article makes more sense: conozco las matemáticas (= some of it); conozco el español (= not much but it rings a bell)
    – Schwale
    May 3, 2017 at 1:06
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You are correct for all the sentences marked "not used."

Some of the other sentences are okay theoretically, but sound weird to my ear.

The following work great:

Sé algo/un poco/mucho acerca de (topic) OR Sé algo/un poco/mucho sobre (topic, e.g. las matemáticas, la economía)

Conozco a (person); conozco (place)

¿Sabes (verb, e.g. hablar español, nadar)?

For a couple of your sentences, I'll suggest some more natural equivalents:

No one knows everything there is to know about mathematics: Nadie domina las matemáticas completamente

I was exposed to Spanish where I lived as a child: Se oía frecuentemente el español donde yo vivía de pequeño

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