Hi, just a question that I've been wondering : Can the brain understand every mathematical idea or you must take some things for granted and trust in them ? Because sometimes I try to understand everything and I find it not that easy ? For example, I can understand simple things like 2+2 =4 but I don't think every concept in math is that clear as adding, substracting, etc.
So my final question in general is, should I try to understand everything about the mecanism of something ( in math) ? What do you people do in these situations ? Do you understand everything like it was your own creation? Thanks
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.
I have to agree with muxdemux. Learn what you can and use it but use what you do not fully understand when you can.
Sort of like driving your car or using your computer. Few people understand the nuts and bolts aspect of the things we have around us but we do learn how to use them.
My signature was a quote from John Von Neumann, one of the great mathematicians of all time. If he did not understand it all, what chance does everyone else have? Did not stop him from applying it. Actually, you are measured in the world more easily with your applications than with your theoretical understanding.
If it ain't broke, fix it until it is.
Thinking is cheating.
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!
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Ok thanks everyone. Just another question, when people say understand things intuitively, what do they exactly mean ? If I can give an example, sometimes, when I learn something, I understand in it's general aspect (Intuitively?) but I can't explain it in words, I just ''understand'' it but can't explain it to myself in words. Is it this, understand things intuitively ? Understand without going in too much details ?
And a last question, if maths are created (discovered?) by man, how come we can't understand everything about them ? Do they exist independently of man and have their own way of working ?
Last edited by Al-Allo (2012-12-25 08:28:19)
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.
If it ain't broke, fix it until it is.
Thinking is cheating.
Understanding something intuitively means that you understand it because it seems logical and natural. For example, you intuitively know that the equation 2+2=4 is true, because you know that if you have two apples and a friend gives you another two apples you will have four apples (and a good friend ).
Last edited by anonimnystefy (2012-12-25 09:15:56)
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.