My advice would be to intuitively understand as much as you can. That said, I certainly wouldn't expect anyone to understand everything that way.

Bobbym's signature is probably relevant here.

hi Al-Allo

Some people imagine that learning and understanding math is like building with bricks.  You start with a firm foundation (that's your 2 + 2 = 4) and gradually build up a taller and taller structure with the later levels depending on what has been placed below.

I don't find that model works for me.  My model is more like a jigsaw puzzle.  From a distance, there are big pieces labelled number, algebra, geometry and so on, but when you look more closely you can then see that each piece is actually made up of lots of smaller pieces ... algebra for example has pieces for equations, quadratics, re-arranging formulas and so on.  Topics like trigonometry are on the border between algebra and geometry.

There are lots of pieces that I know exist but I haven't managed to place them (understand) in the puzzle yet.  Some areas I'd like to master one day when I get time and enthusiasm; others I'm happy to take on trust, knowing that somebody understands them and can work with the topic when necessary.

I find this model helps when I'm trying to understand a new area of math.  Let's say I've bought a new math book with 10 chapters.  And that I get stuck at chapter 3.  The building block model requires that I re-read chapters 1 and 2 so that I have a better foundation before tackling chapter 3 again.  Whenever I've tried that I just get stuck at the same place.  But with a jigsaw you don't have to fit the bits in order.  You do the edge, and the bits with writing on, or maybe a telegraph pole because those bits are easier to fit and the area not yet done shrinks and so becomes easier too.  So instead of re-reading chapters 1 and 2, I take its results on trust and read on the remaining chapters.  Some of that sticks and makes it easier to sort out why ch. 3 is the way it is.  Sometimes the way is clearer when I know what's around the corner and I may get to the end understanding maybe 40%.  But now I know what I don't know and, helpfully that makes it easier to re-read the mystery parts and try to improve my understanding of those.

General relativity is like that for me.  I've got special R. and I know I've got to learn about tensors, (I hope that's right) so, one day, when I've got some spare time, I'll try 'chipping away' at the edges of that part of the puzzle.

Will the puzzle ever be complete.  Eeekkk! No!  For one thing the pieces are fractal-like    http://www.mathisfunforum.com/viewtopic.php?id=18601  so there's always more structure to understand, and also, mathematicians keep adding more areas.  But that's what makes it fun! 

Bob

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Hi;

The first part of that question I can not answer. What exactly does intuitive mean applied to math?

The second part I grasp better. I am formalist. Mathematics is a game invented by man. He built his world and constructed his science with it. He put imaginary lines on his world and out into space. He imagined circles and triangles and that they had deep significance. It is very possible that to an alien race our mathematics is incomprehensible or ridiculous.

Why can't we get all of it down in our heads? Ever played any chess, poker, pool, basketball...? All invented by man and still largely unexplored. Only a finite number of brain cells can stuffed into a human skull. It is not surprising that there are fields that are larger than man's capacity.