Why mathematics has to be too much formal?

Question

This is very general question I am asking here. But I think it really needs to be addressed. Whenever I come across a new concept in mathematics, I try to understand it by searching on the internet. It is always the case for me that when I do find some stuff about those concepts, many times they are elaborated heavily with complex notations and highly formal language.I think a person with some/no mathematical background can never get it. Thankfully there are blogs/sites which explain it in more natural language help me not only understand those concepts but also show the beauty behind those mathematical concepts. Don't you think that extra formality and over usage of symbols really drives away people to go deep into those concepts and hinders the progress of mathematics as less people get involved. No branch of science has progressed with just few peoples getting involved. It is always seen that as more and more people get involved, more progress is done. We have seen that open source softwares evolve way better than proprietary softwares.

Answer 1

To steal a good alliteration from Gauss: the notions of mathematics require intuition; the notations of mathematics require precision. If someone isn't able to understand (much less formulate their own) precise logical statements and arguments, either in symbols or in "jargon", they aren't going to be contributing anything to mathematics research. There's no reason that just getting "more people" involved will help mathematics (or any other field for that matter).

Answer 2

It's true that we can talk about math informally in order to gain intuition and understanding. We can discuss examples. We can explain how we think about the math.

But when it comes down to it, everything still has to be stated precisely, and for this, formal language is necessary. That's the beauty of mathematics - everything is rigorously defined, to infinite precision. The exposition by itself is not enough. How can you prove that two lines intersect only at exactly one point if we don't define what a line is? We can say intuitively that it should be that way - but we can't prove it until we say what exactly a line is and what exactly it means for two of them to intersect.

You are probably frustrated because the internet is not, in general, a very good place for motivated mathematical discussion, especially in lower level disciplines. When people google about math, they're generally searching for homework solutions, so the solutions, without much explanation, tend to be what are most commonly available.

Books are much, much better for this. Many people who write math books try to motivate the subject and provide helpful exposition on how to understand the complicated ideas they present.

It takes more patience to read a math book than a short article, but that's just the name of the game. Math is hard, and it's silly to think that there should be some presentation which allows anyone to immediately understand it without careful thought. For that reason, math will probably never be as popular as watching TV or listening to music. But, that's just how it has to be. I still like it.