4 ms·

"Now suppose for all sets of n horses, every horse in the set has the same color" This sounds so obviously false to me, that I initially thought that he must h

"Now suppose for all sets of n horses, every horse in the set has the same color" This sounds so obviously false to me, that I initially thought that he must have meant "now suppose for all sets of n horses such that every horse in the set has the same color". I.e. take all sets of n horses and then filter them to only get the uniformly colored sets. But he doesn't mean that, it is meant as the induction step: "Now suppose a world where for all _possible_ sets of n horses, it turns out that every horse in the set _necessarily_ has the same color." In my mind, the article is a bit poorly worded, but then again, maybe it makes sense for mathematicians.

crdrost 5y ago
I'm not sure any wording helps, because the reason it strikes us so poorly is that it's such an implausible idea. It might help to make it more concrete, to say “assume that we've proven this for say n = 3, all sets of three horses have the same color. Then I can prove it’s true for n+1, because my set {Auris, Brunellus, Camper, Diego} of four horses is the union of two overlapping 3-sets, {Auris, Brunellus, Camper} and {Brunellus, Camper, Diego}. By the transitive property, Diego must have the same color as Auris and the result holds. But obviously this works not just for 3 going to 4, but also 4 going to 5... Any set of size n+1 is made up of two overlapping sets of size n.” Maybe also someone can then intuit that the wool is being pulled over their eyes in this step?
diffeomorphism 5y ago
That is the same sentence just abbreviated and using standard words. Nobody says "suppose a world where", you just say "suppose that". Also, you just say "if ..., then ...", not "suppose if ..., then it necessarily turns out that ...". > In my mind, the article is a bit poorly worded, but then again, maybe it makes sense for mathematicians. That is the point. This example is given to first year students to show them why you need to be careful in your proofs and why properly wording/formulating statements is important.

crdrost 5y ago

I'm not sure any wording helps, because the reason it strikes us so poorly is that it's such an implausible idea. It might help to make it more concrete, to say “assume that we've proven this for say n = 3, all sets of three horses have the same color. Then I can prove it’s true for n+1, because my set {Auris, Brunellus, Camper, Diego} of four horses is the union of two overlapping 3-sets, {Auris, Brunellus, Camper} and {Brunellus, Camper, Diego}. By the transitive property, Diego must have the same color as Auris and the result holds. But obviously this works not just for 3 going to 4, but also 4 going to 5... Any set of size n+1 is made up of two overlapping sets of size n.” Maybe also someone can then intuit that the wool is being pulled over their eyes in this step?

diffeomorphism 5y ago

That is the same sentence just abbreviated and using standard words. Nobody says "suppose a world where", you just say "suppose that". Also, you just say "if ..., then ...", not "suppose if ..., then it necessarily turns out that ...". > In my mind, the article is a bit poorly worded, but then again, maybe it makes sense for mathematicians. That is the point. This example is given to first year students to show them why you need to be careful in your proofs and why properly wording/formulating statements is important.