Maybe I was unclear. The part that I meant a constructivist would disagree was the subjective part: "This kind of proof (existence of a counterexample) provides more insight into the problem than simply finding a counterexample."
I find it hard to believe that a constructivist would agree that across the board, proof of existence of X is more interesting than a construction of X. Isn't that the point!?
To roughly repeat myself from a sibling comment: I may have been unclear. I didn't claim that the only way to disprove something in a constructive system is to produce a counterexample.
The part that I was referring to was the last statement from the OP: that "a proof of existence of a counterexample necessarily provides more insight than a counterexample". I can't imagine a constructivist would agree with that in general.
Those are all good questions, but I don't really understand what alternative you or the OP are looking for actually resolving an untrue conjecture, besides a counter example.
Are you thinking of proof by contradiction, which is rejected by constructionism?
[Dis]proof by counterexample is the most straightforward way to show a statement to be false. What better way is there to disprove a general statement like 'all x are y' than finding an 'x' that isn't 'y'?
It's even worse than implicit casting, if the value can't be cast to the the column's type, it's just inserted without casting. Eg. into an integer column, '10' -> 10 and '1O' -> '1O'
An intuitive motivation for the solution in the article (2n choose n). For an n*n grid you have to you will take 2n steps, n "over" and n "down". All that matters is the order of the steps. So if you think of there being 2n "slots", you have to pick n to be "over", and the rest are forced to be "down". So it's n choose 2n indeed.
You can also think of it another way, without using the formula combinations, and only the fact that there are n! permutations of n objects. We can think of this a permutation of 2n items, made up of two groups of n identical items each. Using (2n!) will overcount, due to the fact that each of the "over" steps are identical, and similarly for the "down" group. We have cut down our answer by dividing out all of the repeated sequences. There will be n! redundancies for all the ways we can permute the "over" group and, the same for the "down" group. So this results in (2n!) / (n! * n!), which is exactly equal to 2n choose n. See [1] which explains permutations with repetion this in general. [Note: We pretty much re-derived the formula for combinations!]
Preferences vary on both sides of the Atlantic. Another comment on this post complains that Americans pour beer wrong because they _do_ pour with a head.
> Also in the US (probably due to lack of training and the customer too embarrassed to complaining) tend not to fill it the brim (and so not even 16''). I've seen 2-3 inch heads and asked them to top it up. They look at me as if I've just insulted George Washington
I was skeptical about the claim that 80% of soldiers refuse to fire their weapons, so I did a little reading and it seems like the original source has been pretty much debunked. This 2011 article sums it up: https://scholars.wlu.ca/cmh/vol20/iss4/4/ but it's been doubted for decades.
Overall I agree with the approach, but just wondering, why do the first point if you are already doing the last two?
> * Use the issue id at the start of every commit message for that issue
> * Use a single branch per issue, whose name also starts with the issue id
>* Use a single PR to merge that branch and close the issue
To me the noise at the start of every message is unnecessary, and given a lot of interfaces only display 80 chars of the message by default, it's not negligible.
> What we tend to forget is that even with the catastrophic effects of climate change, the Earth is still vastly more inhabitable than other planets in the solar system.
Speak for yourself. I have never forgotten that Earth is more inhabitable than Mars or Jupiter