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Comment by nothis | Hacker News Reader

That's a very elegant way of saying: "Communication in math is just as fucked up as you'd think."

As an outsider who only ever needs to use maths as a "tool" for very specific things, maths books and general attitude by professors and teachers is just nightmarish. Somewhere behind that curtain of inaccessibility, it has to be their fault. Especially, since once you actually do understand certain mathematical concepts you suddenly realize they're easily explainable with a few words of plain text or --gasp-- a "childish" drawing to illustrate.

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I'm working on something at the moment to try to explain exactly why what you've said here is, to a large extent, simply wrong. It's too long to include here, and it's not yet ready to "publish". You're about two weeks too early.

But let me ask you this. It's easy to cut a square into identical pieces so that all the pieces touch the center point.

In slightly more detail, the pieces are disjoint sets such that their union is the whole square. The pieces are identical except perhaps for details as to the boundaries. To say that they all "touch" the center point means that every non-zero radius disk centered at the center contains some points from each piece.

So now, how many ways can this be done? No, it's not five. And no, it's not six either.

When you start trying to work it out you find that the details matter, and they can't just be covered by a "childish" drawing to illustrate.

Details matter, and some of them are hard.

Yes, most math teaching is atrocious. We all know that. But it's not always just the teacher's fault. Sometimes it's at least partly the fault of the readers expecting everything to be made simple and immediately accessible with neither work nor effort.

So now, how many ways can this be done? No, it's not five. And no, it's not six either.

An infinite number of ways this can be done.

Yes, but there's more that you can say. In particular, there is more than one infinite family. How many are there? And do all solutions belong to an infinite family? Or are there isolates/sporadics?

Can you characterise the solutions? How many pieces do they contain? Some solutions have two pieces. Some have four. Are there other possibilities?

I don't know what an infinite family, isolate, or sporadic is. A quick Google search didn't make it obvious.

I think 8 pieces also works, but I think that may be it.

An infinite family seems to refer to an infinite collection of solutions that are all related in some sense. An isolate or sporadic solution is a solution that is unique and unrelated to other solutions. They are both terms used to categorize solutions.

Thanks nocipher for the definitions. From this I see no sporadics. And two infinite families, at least how I'd classify a family.

nocipher's definition are correct.

And I have, so far, 5 infinite families with, respectively, 2, 4, 8, 16 and 32 pieces.

But details matter, and you might start to question what it really means when I say "piece." Does the definition I've given really capture your intuition?

But I need to go back to writing the article.

Most math concepts -are- fairly easy to explain and, from the sounds of it, you don't actually want a math book. You want a book that gives you the executive summary or a run-down of specific tools for specific situations. The flaw is not in the books or the teaching, it is what you are expecting from mathematicians.

Math is, in a very general sense, about understanding and reasoning about highly structured objects. Mathematicians have developed methods and notation to achieve that end, not to make it easily digestible by the uninitiated. Rigor is an important part of this. It is what allows mathematicians to be so sure their work is correct and it is often what makes math seem so arcane. It is not always necessary for teaching, but professional mathematicians are often the teachers and the ideas are intimately tied to their rigorous formulations in their head.

For example, continuity is a fairly intuitive concept in calculus, but the rigorous epsilon-delta definition is necessary in proofs. The basic concept, while easy to explain and understand is useless, while the precise formulation, though much more opaque, is ubiquitous in analysis simply because it is an incredible tool... for mathematicians.

An engineer likely just needs the machinery built on top of the analysis: the derivative and integral. If that's all you need, then pick a book that is focused on applications and a development of general intuition, not a book designed for mathematicians-in-the-making.

Do you really think that continuity is an obvious concept?

Are you aware, for example, that there are functions that are continuous at all irrational points and discontinuous at all rational points?

Which comes back to the problems with the OPs argument. Sure, you can explain continuity in a few simple sentences but sooner or later you will start to hit corner cases and you find that those few simple sentences are really quite ambiguous. It's much like using pseudocode to describe an algorithm. I've found that the best mathematical texts provide both - a 'morally correct' definition in plain english to describe the intention and a mathematical description which defines the precise meaning.

I have my doubts about that claim. In most cases, the 'you' that thinks something 'is easily explainable with a few words of plain text' often is a different 'you' from the one that struggled grasping the concept. That struggle is what made you think the concept is 'easy'.

Next time you encounter such a 'aha erlebnis', try saying the 'few words of plain text' to someone who still has to master the subject, and _immediately_ test him or her on the subject. I predict that, oftentimes, (s)he understood nothing of what you said.

Case in point: at university, I had a prof who said that it was perfectly OK if you could not do any practice integral. Trust me, he said, if, a month from now, you do these exercises, you will not understand what is hard about them.

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