Images of Math
images-of-math.tumblr.com
Images of Math
1–10 of 30 posts
Re: Images of Math
#2I like how it only goes article by article, it brings more focus and attention to these short posts and leaves more space for the mind to wander.
Re: Images of Math
#3 You never need more than four colors to color every country
on a map a different color from its neighbours. This was
proved in the 20th century — but nobody knows why it is true.
Re: Images of Math
#4You never need more than four colors to color every country on a map a different color from its neighbours. This was proved in the 20th century — but nobody knows why it is true. Heh. I suppose we've proved it, but this author doesn't accept this proof as an explanation for why it's true.
> Heh. I suppose we've proved it, but this author doesn't accept this proof as an explanation for why it's true.
I think that this is a reasonable philosophical position; given the heavily computer-based nature of the proof, even mathematicians who might accept that (say) the proof of the solubility of odd-order groups explains 'why' they are soluble are reluctant to accept this (the 4-colour proof) as a 'true' explanation.
Even if you have no problem with computer-generated proofs, I think that there is a big gap, for professionals and amateurs alike, between proof and explanation.
Re: Images of Math
#5I like how it only goes article by article, it brings more focus and attention to these short posts and leaves more space for the mind to wander.
Agreed. Here's the short attention span version though, http://images-of-math.tumblr.com/archive
Re: Images of Math
#6Missed "Aryabhata" , http://en.wikipedia.org/wiki/Aryabhata
Re: Images of Math
#7Suggestion: allow us to use left/right button to navigate.
Re: Images of Math
#8You never need more than four colors to color every country on a map a different color from its neighbours. This was proved in the 20th century — but nobody knows why it is true. Heh. I suppose we've proved it, but this author doesn't accept this proof as an explanation for why it's true.
The "proof" you're most likely talking about doesn't show why this is true. Merely shows that it is true.
Exhaustion doesn't qualify as a real argument, in my opinion.
Re: Images of Math
#9You never need more than four colors to color every country on a map a different color from its neighbours. This was proved in the 20th century — but nobody knows why it is true. Heh. I suppose we've proved it, but this author doesn't accept this proof as an explanation for why it's true.
The "proof" you're most likely talking about doesn't show why this is true. Merely shows that it is true. Exhaustion doesn't qualify as a real argument, in my opinion.
Proof by exhaustion is considered a real argument as long as there are not too many cases to exhaust. One of the most dramatic examples is the proof of that result that goes... if RH is true, then the result is true. If RH is false, then the result is true. Therefore, the result is true!
So, why make the cut-off point of when exhaustiong is an explanation at some finite number of cases?
Re: Images of Math
#10You never need more than four colors to color every country on a map a different color from its neighbours. This was proved in the 20th century — but nobody knows why it is true. Heh. I suppose we've proved it, but this author doesn't accept this proof as an explanation for why it's true.
The "proof" you're most likely talking about doesn't show why this is true. Merely shows that it is true. Exhaustion doesn't qualify as a real argument, in my opinion.
People often talk about the proof as if they just enumerated every possible planar graph and checked. Which is true in a sense, but disregards the work that went into making it possible to enumerate the cases in the first place.
There are infinitely-many possible planar graphs, so you of course need to find some invariants that allow you to bring the necessary number of graphs to check to be finite. Beyond that, the techniques in the 4CT which make it actually possible to reduce the number of cases to a feasibly enumerable number are very clever and require some insight.