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These seem fairly advanced, so I don’t know if I would call them “common”. But maybe I am out of the loop. Are these concepts and very specific sounding belief

These seem fairly advanced, so I don’t know if I would call them “common”. But maybe I am out of the loop. Are these concepts and very specific sounding beliefs actually widely known to math majors or CS majors?

dwohnitmok 5y ago
Yes (although not for CS majors, this is very math heavy). For all of the top answers I went through, I think most math majors have believed it, know someone who has believed it, or could at least plausibly see how someone could believe it. To the question of how advanced these are, with only a few exceptions, the majority of things there are covered in undergraduate analysis, algebra, and topology courses.
Rerarom 5y ago
I have a math degree, and I could understand most of them by the end of my second year.
qsort 5y ago
Many of them are part of a standard CS curriculum. Most of them are part of a standard math curriculum. The "level of difficulty" is very uneven, though. Things like "Many students believe that 1 plus the product of the first primes is always a prime number" (a misunderstandment of Euclid's proof that there exist infinitely many prime numbers) or "If A implies B then B implies A" (difference between a necessary and a sufficient condition) are literally middle-school math. Things like "If 𝑓(𝑥,𝑦) is a polynomial with real coefficients, then the image of 𝑓 is a closed subset of ℝ" are high-school level. The top answer about dimensions of vector spaces is a well-known undergraduate-level "trap". "The Krull dimension of a noetherian integral domain is finite." is, to be brutally honest, not a sentence I understand in the slightest.
- dwohnitmok 5y ago
  The Krull dimension one is one of the few that might not be found in an undergraduate program. You'd probably come across it in an algebra class that straddles the line between upper undergrad and intro graduate coursework (but Noetherian integral domain is something you would usually find in the sophomore/junior undergrad algebra class often taken as a prerequisite for the other hypothetical class). Also > "If 𝑓(𝑥,𝑦) is a polynomial with real coefficients, then the image of 𝑓 is a closed subset of ℝ" I don't think that this is high school level. Secondary school level math rarely touches anything related to multivariate polynomials (and usually doesn't have a notion of "closed sets" beyond closed intervals, but that's not as relevant apart from how the problem is stated).
  - qsort 5y ago
    Yeah, I studied CS, so my background in algebra beyond elementary concepts is very limited. As you said in another comment this is very math heavy. I don't doubt that what you're saying is true, it's just something far outside what I personally am familiar with.
gjulianm 5y ago
For context, mathoverflow is for math professionals, that is, math degree and above, so high level mathematics. I wouldn't expect CS majors to know about topology, complex analysis, advanced algebra... But for math majors, these are common.
- Sharlin 5y ago
  Yep. There's also https://math.stackexchange.com/ https://math.stackexchange.com/ for math questions lower than research level.
MisterSandman 5y ago
Math Majors, yes. CS majors, maybe? It really depends on how the university treats CS. At the University of Waterloo, CS is strictly in the faculty of Math, so I'd expect students to know much of this (I think? I'm in Computer Engineering at UW, which is a distinctively different program). OTOH, I'd imagine things are different in universities where CS is thought of as its own major, with no Software Engineering/Computer Engineering as an alternative.
- seanmcdirmid 5y ago
  Computer science at U Washington (UW also) is hosted in the college of arts and sciences and engineering as two different programs. But the former program has its own faculty and isn’t related to the math department in the closest. The latter program doesn’t have its own faculty at all, sharing with computer science and electrical engineering.
  - MisterSandman 5y ago
    Interesting! I was actually about to go to U of Washington, but ended up avoiding due to the fact that they don't let students directly into CS, but instead have a cut-off system in second year.

dwohnitmok 5y ago

Yes (although not for CS majors, this is very math heavy). For all of the top answers I went through, I think most math majors have believed it, know someone who has believed it, or could at least plausibly see how someone could believe it. To the question of how advanced these are, with only a few exceptions, the majority of things there are covered in undergraduate analysis, algebra, and topology courses.

Rerarom 5y ago

I have a math degree, and I could understand most of them by the end of my second year.

qsort 5y ago

Many of them are part of a standard CS curriculum. Most of them are part of a standard math curriculum. The "level of difficulty" is very uneven, though. Things like "Many students believe that 1 plus the product of the first primes is always a prime number" (a misunderstandment of Euclid's proof that there exist infinitely many prime numbers) or "If A implies B then B implies A" (difference between a necessary and a sufficient condition) are literally middle-school math. Things like "If 𝑓(𝑥,𝑦) is a polynomial with real coefficients, then the image of 𝑓 is a closed subset of ℝ" are high-school level. The top answer about dimensions of vector spaces is a well-known undergraduate-level "trap". "The Krull dimension of a noetherian integral domain is finite." is, to be brutally honest, not a sentence I understand in the slightest.

dwohnitmok 5y ago
The Krull dimension one is one of the few that might not be found in an undergraduate program. You'd probably come across it in an algebra class that straddles the line between upper undergrad and intro graduate coursework (but Noetherian integral domain is something you would usually find in the sophomore/junior undergrad algebra class often taken as a prerequisite for the other hypothetical class). Also > "If 𝑓(𝑥,𝑦) is a polynomial with real coefficients, then the image of 𝑓 is a closed subset of ℝ" I don't think that this is high school level. Secondary school level math rarely touches anything related to multivariate polynomials (and usually doesn't have a notion of "closed sets" beyond closed intervals, but that's not as relevant apart from how the problem is stated).
- qsort 5y ago
  Yeah, I studied CS, so my background in algebra beyond elementary concepts is very limited. As you said in another comment this is very math heavy. I don't doubt that what you're saying is true, it's just something far outside what I personally am familiar with.

gjulianm 5y ago

For context, mathoverflow is for math professionals, that is, math degree and above, so high level mathematics. I wouldn't expect CS majors to know about topology, complex analysis, advanced algebra... But for math majors, these are common.

Sharlin 5y ago
Yep. There's also https://math.stackexchange.com/ https://math.stackexchange.com/ for math questions lower than research level.

MisterSandman 5y ago

Math Majors, yes. CS majors, maybe? It really depends on how the university treats CS. At the University of Waterloo, CS is strictly in the faculty of Math, so I'd expect students to know much of this (I think? I'm in Computer Engineering at UW, which is a distinctively different program). OTOH, I'd imagine things are different in universities where CS is thought of as its own major, with no Software Engineering/Computer Engineering as an alternative.

seanmcdirmid 5y ago
Computer science at U Washington (UW also) is hosted in the college of arts and sciences and engineering as two different programs. But the former program has its own faculty and isn’t related to the math department in the closest. The latter program doesn’t have its own faculty at all, sharing with computer science and electrical engineering.
- MisterSandman 5y ago
  Interesting! I was actually about to go to U of Washington, but ended up avoiding due to the fact that they don't let students directly into CS, but instead have a cut-off system in second year.