Math Seems To Be Counterintuitive And Hard

Math is pretty inaccessible to most people, I’ve spent a fair bit of time studying it, but I’m not foolish enough to say I really know any. I sincerely doubt that, even if I was given an infinite amount of time, I could have ever discovered the simplest proofs one learns through in high school on my own.

This graph is an attempt to describe theorems that can be conveyed in short natural language prose that most (English speaking) people understand. You would think it would be hard to whittle things down to such a small list, but some googling suggests that many would come up with something similar. One easy way to find these is to look for lists of the “top 100 theorems” or theorems that people have tried to make expository books for.

Outside In – Finding a way to get at the sweet
purple insides of a yellow ball.

There are other short, interesting theorems not in this graph but they require things extra qualifying statements or background knowledge (e.g. the abc conjecture or the Robbins conjecture). All of the theorems in the graph can be explained fairly easily; would love to hear any suggestions you might have of other theorems that should be up here.

Just because we can state a theorem (even one that sounds simple), doesn’t mean we know it is true. Truth is asserted by the existence of a deductive set of steps that most experts agree is correct i.e. a proof. Most interesting theorems require many years of exceptionally capable people’s time to prove that they are true. A great deal of interesting mathematical machinery is invariably constructed as a result of (dis)proving a long standing theorem. It is interesting that there are a few simple ideas, which most of us take for granted (like the Jordan curve theorem) which took some smart people a long time to actually prove as true. On the other side, I think that Cantor diagonalization is one of the most accessible mathematical ideas there is which leads to a profound result (transfinite numbers). This is perhaps no surprise, given how often it is explained in layman’s books.

Some interesting ideas come from this graph about the set of all interesting theorems:

  • There are relatively few interesting mathematical theorems that can be conveyed adequately in short natural language prose.
  • There does seem to be a trend of more counterintuitive ideas having more difficult proofs.
  • There are plenty of exceptions to this trend, e.g. the proof for the four color problem was generated by a computer; no one thinks a short concise proof exists (but who knows, maybe there is).
  • It seems that few people are interested in proofs of things that are only just past intuition and are difficult to prove (this is perhaps a tautology).

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