I profoundly disagree. I believe that mathematics can be learned and appreciated by many, because mathematics is a language for articulating inner urges and perceptions that are common to a wide range of creatures, not only humans. This idea that mathematics is only for the few is self fulfilling, and perpetuated by narrow ideas of what it constitutes.

I agree. I've noticed this issue where your anticipated difficulty of something ends up determining your experience of its difficulty. With math—most of the time—there's nothing intrinsic in what you're learning that would make it more difficult than say, learning the rules to a complex board game, or the structure of one's government—and yet, people go into learning math with knowledge of its reputation as only being suited to certain special kinds of minds. If you suspect you're one of the chosen you can go into it with relish; if not, you respectfully put in your time, knowing you're in someone else's territory, and sort of meekly do what you can.

I think most of the difficulty is 1) determined by your expectation of it 2) initial foreignness and the invisibility of the gap between the concepts you need to understand something, and the ones you already have (i.e. when an outsider just looks at some mathematical statement they have no comprehension of, it's not clear that there are a few layers of concepts underlying it which could be smoothly traversed, as long as one puts the time into it and is given some direction.)

> I've noticed this issue where your anticipated difficulty of something ends up determining your experience of its difficulty.

That assessment didn't come from nowhere.

Given that there seem to be aptitude differences for most physical things, I would find it hard to believe that there are no aptitude differences for most mental things, especially for something so unnatural.

Not to mention that all the things you speak of are very real difficulties to many people that they're not going to get any visible pathway through.

> I believe that mathematics can be learned and appreciated by many, because mathematics is a language for articulating inner urges and perceptions that are common to a wide range of creatures, not only humans

I'm not sure where you're getting the 'wide range of creatures, not humans' part, or even why that matters, but I agree, mathematics is a formalization/projection of our own thinking process, which pretty much every human has.

I agree. I think that mathematics is being teached the wrong way. Especially in the USA, the state of math education is absolutely terrible.

I suspect a more visual approach to be more useful than the classical textbook approach. It is true that a lot of textbooks do use pictures, but video's would help a lot, I think.

There is a wonderful book, "Visual Group Theory", that demonstrates this approach [1].

However, there are people who learn better algebraically than visually. So, combining different approaches would probably be optimal -- the problem currently is that often a very dry (and anti-historical) endless litany of definition, theorem, proof is taught.

[1] http://web.bentley.edu/empl/c/ncarter/vgt/

Yet another highly recommended book is Visual Complex Analysis by Needham: http://usf.usfca.edu/vca/.

There is a huge difference between the pursuit of new mathematics and the appreciation and understanding of existing work. I think OP is talking about the former and it certainly isn't for everyone, at least not the way it is currently implemented. To understand this you need to look at the career track of newly minted math PhDs and what one needs to do to secure tenure.

I do agree that math can and should be widely appreciated.

> To understand this you need to look at the career track of newly minted math PhDs and what one needs to do to secure tenure.

You can pursue new mathematics without having a PhD. I think the most difficult/frustrating part about academic/institutional mathematics is the politics and bureaucratic bullshit.

It's a nice sentiment but it isn't true. Most people don't have the necessary intellectual chops to understand even Euclidean geometry.

You don't know that.

This reminds me of when people say the best career is the one they happen to have, and everyone should do what they do. But... people are different.

It's the fallacy that has led to the extreme oversupply of post-graduate academic job seekers: far too many people are in college, getting advice from professors who think, "well, it worked for me, you should do it too!"