[HN Gopher] Should proofs include a third "context" column?
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Should proofs include a third "context" column?
Author : abrax3141
Score : 15 points
Date : 2022-01-12 11:37 UTC (11 hours ago)
(HTM) web link (matheducators.stackexchange.com)
(TXT) w3m dump (matheducators.stackexchange.com)
| [deleted]
| dqpb wrote:
| I love reducing a problem to its minimal formalism.
|
| I hate documentation that only includes the minimal formalism
| without examples.
|
| It suggests the author never developed "Theory of Mind".
| abrax3141 wrote:
| "It suggests the author never developed "Theory of Mind"."
|
| Interesting...Can you explain this more?
| 13415 wrote:
| No, they should be written in comprehensible and elegant prose
| that is as short as possible while allowing an informed reader to
| check each step.
| dwohnitmok wrote:
| I disagree. Two column proofs of the sort introduced in American
| middle school and high school mathematics programs are really
| just an attempt to introduce students to the beginnings of formal
| mathematical proofs and show that there is a world beyond prose
| proofs. No mathematicians actually use two column proofs (and
| even formal proof systems don't use two column proof notation).
| The state of formal mathematical proof technology is not
| currently at the stage where it makes sense for even professional
| mathematicians to be writing formal proofs, hence it's not even
| clear that two column proofs have a place in a future world where
| (hopefully) formal proofs become the norm in theoretical
| mathematics, depending on how formal proofs ultimately become
| structured. And the "third column" is effectively just comments,
| which are very easy to bolt on to any system after the fact.
|
| As it is right now, it is enough to know merely how in theory a
| prose proof could be reduced to a formal proof. And besides most
| middle school and high school students aren't going to be able to
| write even a rigorous prose proof. Trying to stack more stuff on
| the edifice of two column proofs seems like just saddling
| students with even more (from their point of view) tedious make-
| work.
| dnautics wrote:
| > (and even formal proof systems don't use two column proof
| notation)
|
| Might be interesting to see a crack at that, though.
| qsort wrote:
| Is this ``two-column'' proof style an American thing?
|
| I'm not a mathematician but I've written my fair share of proofs
| (incl. math competitions) and I don't ever recall being taught or
| having used something like this.
| jacksnipe wrote:
| In the US, it is typically a large part of the High School
| Geometry curriculum. This is exacerbated by Euclid's Elements
| being a required part of the syllabus in many states (I'm
| familiar with southern ones, but I'm sure there are more).
| alisonkisk wrote:
| hjorthjort wrote:
| I don't know at what level this is used, but I have only ever
| seen it in my graduate level logic and mathematics courses. (I'm
| European.)
|
| I like it. Part of a good proof is a rigorous derivation. There
| is plenty of thing sto be said about what intuition can lead us
| towards such a proof. But at the end of the day, a good proof is
| ideally almost machine-checkable. Most proofs (much like code)
| lives two lives: the one in which they are written, and the one
| in which they are read. It should be almost trivial for a student
| to go back, on their own, days or weeks later, and see that each
| step is valid. And I think prose interspersed between column-
| style proofs do a better job than what margin notes do. The
| reason two columns work well is that the step and it's
| justification are of both condensed and almost always of similar
| length. Justification (much like code comments) may or may not
| need a good chunk of space. And surely nothing is stopping anyone
| from inserting little margin notes here and there in a two-colum
| proof, right?
| jacksnipe wrote:
| Graduate level logic? That's so interesting. I only took
| undergrad, but did an ambitious independent study in logic
| once; we did not use two-column proofs, which seem mostly
| useful as a pedagogical tool to make the student see where the
| logical steps in the proof are, and less as a tool for serious
| work, since prose is really the best tool for communicating the
| insights that led to the proof.
|
| Definitely did plenty of "one column" symbolic proofs though. I
| think part of the lack of need is that seeing two steps right
| next to each other generally makes it very clear what was done
| to get from one step to another.
|
| Unless it invokes the axiom of choice, in which case fuck me.
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