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There's one habit in particular that he didn't really touch on that has been one of the most impactful parts of getting my math degree on my thinking. There se

There's one habit in particular that he didn't really touch on that has been one of the most impactful parts of getting my math degree on my thinking. There seems to be a gap between formal definitions and what we feel actual definitions are. Every once in a while, a professor would prove something that was clearly right, but felt like a violation of some unstated implicit part of a definition (for example, that the set of all vector spaces is itself a vector space). Despite feeling wrong, I couldn't come up with a particular reason that it wasn't right (short of falling back on some weasel word like "technically that's correct"). Math education forces that gap out into the open, and makes you pay more than lip service by building more and more theorems on top of it. The real world effect is to make you examine whether there's some actual value in this definitional gap (which can then be explicitly stated) or whether it's just fallacy. I know far too many people who are comfortable with stopping at "this doesn't feel correct despite me having no counterargumemt to its 'proof' ". I've never been inclined that way but my math education certainly sharpened my ability to avoid this.

SatvikBeri 10y ago
My coworkers and I use a related technique all the time, e.g.: "I think X, Y, and Z are true." "If those are true, then A should also be true. But A seems clearly false." "Hmmm...I agree A is surprising, but not clearly false. What would distinguish a world in which A was true or not?" "Well, if A was true, we'd expect B, otherwise C." "Ok, I agree C seems much more likely than B, so that suggests one of X, Y, or Z is wrong."
gunnihinn 10y ago
> for example, that the set of all vector spaces is itself a vector space Hmm... under what operations? It's a semigroup under direct sum, and probably something under the tensor product, but I'm having trouble imagining what the scalar multiplication should be. (Or was that a "fictional" example? :)
- woopwoop 10y ago
  Unless I'm mistaken, The totality of vector spaces is not even a set.
  - gunnihinn 10y ago
    There's one for every cardinality (up to isomorphism). I forget my set theory, but if the cardinalities form a set, then the totality of vector spaces does too. If not, we can amuse ourselves by looking at the set of finite-dimensional vector spaces.
- wutbrodo 10y ago
  Sigh, teach me to write a comment quickly without proofreading on HN, on a topic of any complexity. As written, my comment is wrong: I meant to write "vector space functions"[1] but left out a word. What I was getting at was the unintuitiveness that arises from thinking of vector space elements as number-like things (in the sense that we're familiar with them from basic math: numbers or coordinates) to being "anything that meets the definition", including functions. Seriously though, thanks for pointing that out, part of what I still love about the HN community is getting called out for even esoteric incorrectness. [1] which was my concise but imprecise way of expressing "the set of functions from a set over a field to a vector space over the same one", with + and * defined as you'd expect.
adrusi 10y ago
When a conclusion feels wrong despite being derived logically, that usually means that the definitions are flawed. In math the defintions of structures don't have to reflect anything real, it's all about starting with arbitrary rigorously stated definitions and exploring their implications. So when it feels like definitions are wrong in math, that's just a reflection of that "definitional gap". But the virtue of always siding with the logical-but-unintuitive isn't as valuable outside of mathematics, so it's maybe not the best point to bring up in an article that's about skills math teaches that are useful in "real life". In philosophy, definitions aren't arbitrary and rigorous, they're meant to be reflections of reality. So when a conclusion feels unintuitive, that's a decent argument against the definitions it was derived from. If mathematics teaches people to just pay attention to the definitional gap, that might be useful enough, but isn't that covered by "discussing definitions"?

SatvikBeri 10y ago

My coworkers and I use a related technique all the time, e.g.: "I think X, Y, and Z are true." "If those are true, then A should also be true. But A seems clearly false." "Hmmm...I agree A is surprising, but not clearly false. What would distinguish a world in which A was true or not?" "Well, if A was true, we'd expect B, otherwise C." "Ok, I agree C seems much more likely than B, so that suggests one of X, Y, or Z is wrong."

gunnihinn 10y ago

> for example, that the set of all vector spaces is itself a vector space Hmm... under what operations? It's a semigroup under direct sum, and probably something under the tensor product, but I'm having trouble imagining what the scalar multiplication should be. (Or was that a "fictional" example? :)

woopwoop 10y ago
Unless I'm mistaken, The totality of vector spaces is not even a set.
- gunnihinn 10y ago
  There's one for every cardinality (up to isomorphism). I forget my set theory, but if the cardinalities form a set, then the totality of vector spaces does too. If not, we can amuse ourselves by looking at the set of finite-dimensional vector spaces.
wutbrodo 10y ago
Sigh, teach me to write a comment quickly without proofreading on HN, on a topic of any complexity. As written, my comment is wrong: I meant to write "vector space functions"[1] but left out a word. What I was getting at was the unintuitiveness that arises from thinking of vector space elements as number-like things (in the sense that we're familiar with them from basic math: numbers or coordinates) to being "anything that meets the definition", including functions. Seriously though, thanks for pointing that out, part of what I still love about the HN community is getting called out for even esoteric incorrectness. [1] which was my concise but imprecise way of expressing "the set of functions from a set over a field to a vector space over the same one", with + and * defined as you'd expect.

adrusi 10y ago

When a conclusion feels wrong despite being derived logically, that usually means that the definitions are flawed. In math the defintions of structures don't have to reflect anything real, it's all about starting with arbitrary rigorously stated definitions and exploring their implications. So when it feels like definitions are wrong in math, that's just a reflection of that "definitional gap". But the virtue of always siding with the logical-but-unintuitive isn't as valuable outside of mathematics, so it's maybe not the best point to bring up in an article that's about skills math teaches that are useful in "real life". In philosophy, definitions aren't arbitrary and rigorous, they're meant to be reflections of reality. So when a conclusion feels unintuitive, that's a decent argument against the definitions it was derived from. If mathematics teaches people to just pay attention to the definitional gap, that might be useful enough, but isn't that covered by "discussing definitions"?