I'm very good at math and scored high during all my life - but I must admit that nothing I have found until now couldn't be done better by programming. So, if someone wants to understand the nature of the world, not for an university or for jobs, but for it's sake - can you provide me one single example of where advanced math could be more useful than just understanding the subject and using computers to do the calcs.

Cryptography. Say I give you an integer with around 180 digits. Now I ask you to find a prime factorization for that number, what do you do then? Brute force? It's a very tough problem in general, and you WILL need advanced math to even attempt it, let alone make a program.

I think that the problem is more one of checking the computer's accuracy. Numerical methods can be unstable for certain classes of problems, and all non-trivial programs have bugs anyway.

If you don't understand the math, you won't be in a position to properly evaluate a computer generated answer.

Just think about calculators for a second. If I ask you to divide 117.938 by 19.767, clearly a calculator will be much quicker. But if you come up with an answer of 56.556, how will you know that you accidently pressed '1' twice without knowing that 120 / 20 is about 6?

First, programs, to be of any use, must be written by someone who understands the mathematics that goes into doing any calculations.

I think you make an unwarranted dichotomy between advanced mathematics and understanding a subject. You can claim to understand physics without math, but I would argue it's through a set of vague analogies. Mathematics gives structure and definition to analogies and allows you to know in what ways the analogy is correct or incorrect.

I personally believe that everything we do in physics is model building that has no real truth in nature. Consider an artist that makes a copy of a painting. You would argue that visually it is a similar painting, but it is never the original. Going back to physics I think learning physics without math is like making a photocopy of the copy of the original.

Advanced math is needed to understand the subject. Try to understand quantum mechanics without knowing linear algebra. How will you understand http://en.wikipedia.org/wiki/Schrödinger_equation without knowledge of PDE's??
You can't just separate advanced math and physics.

Furthermore, who will write all this fancy programs for you?? People who know the math!!

TMFKAN64 you don't get the point. The computer can provide wrong answers as you. This has nothing to do with the issue. You can check it. I'm proposing a replacement of hand algebra to programming as the primordial tool to your self development of scientific knowledge. Literary: changing the paper to the keyboard.

Same to Micromass. It's about changing arcaic hand algebra to programming as the main tool for humans to work with math. Where I wrote "math skills", visualize it as youself working into your algebra on a paper, getting an integral algebrically, solving a differencial equation by hands. Couldn't this be replaced by understanding what an integral is, and using the computer to solve it for you? You don't have to get deep into every topic to use it properly. Time wasted is advancement lost.

Reading the topic again I admit it's not clear so this is bad news for the proposed discussion.

A respectable mathematician (don't know his name, sorry), went to an elementary school where they focussed on abstract math instead of doing algebra. So he went to one of the kids and he asked him: "what is 2+4". The kid thought for a while and answered "It equals 4+2 because of commutativity". The kid had no idea what 2+4 meant and that it equaled 6.

I'm sorry, but I find this very sad. People need to know how to calculate. They need not be good at it or it needs not be fast, but they need to know how it goes. You can't possibly talk about integrals and derivatives without ever having calculated one.

You clearly have never done any higher math. What you are basically saying is: "Let's get an idea of what a car is, and then let's forget about building one ourselves, let the robots do it.", except that you almost make it seem as if that's equivalent to spending the time building the car yourself.

How would you ever develop any mathematical maturity whatsoever? How would you advance mathematics if you didn't develop intuition and problem solving skills? Mathematics is not just a "read it, know it" type of monster. It's a common saying that you have to fight math to understand math.

Yes, your example is perfect. Why is being able to calculate an integral more important to knowing what it is and what it does? If anything, understanding rules as d/dx x^n = nx^(n-1) is essentially what happened to the 2+4 guy.

You clearly have never done any higher guessing. Neither reading. Quote where I said against fighting math, developing intuition and solving problems. I'm arguing in favor to using programming as a replacement of the old pen and paper. You clearly have no idea of what I'm talking about.

People are strange. I'm done here. Got my answer. Thank you.

Personally, I want to feel comfortable with the material. And to be comfortable, that means that I absolutely must see and compute actual examples. I have tried reading through a math text before without doing any computations. I remembered very little and I was not comfortable with the material at all.

Just knowing what an integral means is ok. But never calculate one makes you out of touch with what those things actually are. It's hard to explain, but it's really true.

Furthermore, what are you going to do if there are no numbers anymore, but only symbols?? How will you derive a formula from a given law?? If you don't know how to handle the numbers, then you will be hopeless if there are symbols instead.

Well I suppose you and I have different views of what doing math on a computer entails. I did not mean to offend anyone, perhaps I was unintentionally harsh.. There's a good reason pencil and paper is a tried-true-tested technique for math...

The military wants to write an algorithm to track am object falling out of the sky. You solve the differential equation numerically every timed you want to update the position of the object with your best guess, therussian programmer solves it once and just repeatedly evaluates the solution. Your program takes too long to run to successfully track the enemy nukes and the russians win the cold war in a very hot mess

Don't you think weird a technology available for thousand years is still the one used here? They didn't use ink and paper because they had a choice, this was just the only way they had to keep data and express concepts. Programming is essentially the same. There's nothing I can express on a paper I can't on notepad - except on the later I replace stuff like dividing by hand, solving an equation, and others, by an automated method. I won't lose a single bit of the knowledge. When you are solving an equation or performing a multiplication you are not thinking on your main problem anyway, it actually takes you off the focus. You know it.

Now this last answer really did it for me. I got an insight here. Thank you guys.

Oh come on. You asked a question, Div gave a fairly good answer. If you disagree with him, then bring in counterarguments. The thing you mention is very interesting, and I would love having a discussion with you about this. Don't back off because people disagree with you.

Perhaps Div misunderstood you, well: try to explain better to him what you mean.

A very related issue is whether mathematical proofs are actually necessary for something. Why should non-mathematicians (or mathematicians) even bother with proofs?? This is quite an interesting discussion.

True. But you get a feel for the math involved. You get a feel about how hard it is to calculate [itex]\int{x^2dx}[/itex]. Furthermore, you can stand in awe of the smart men who made calculating areas so easy!!

And I'll bring it up again: what if your calculations involve symbols rather than numbers?? In that case, your computer will be useless. I know computers can do symbolic manipulations but it is still not good enough for if you want to do theoretical physics.

Also (and I'm sure you will disagree with this), there is a certain beauty in the calculations. Actually being able to find the area under a parabola made me almost high when I first done it. After a while it gets boring, but calculations really can be pretty.

At what point do you think "understanding" kicks in? You say "understand what an integral is". Do you simply mean:

I understand the definition of a Riemann Integral:
[tex]\int_a^b f(x)dx=\lim_{n\rightarrow \infty}\sum_{i=1}^n f(x_i)[x_i-x_{i-1}][/tex], and that's it?

Micromass, what do you mean? This is where the computer will exceed. How familiar you are with functional and object-oriented programming? You don't just manipulate symbols, you manipulate objects with a well-defined behavior. Are you working with vectors? Great.

Well, you just defined 2 vectors, one whose position is defined by 2 functions of t, one being the derivative of the other, displaying, as result, accelerated and uniform movements for the x and y axis, and other defined by the cross product of the first vector and the unitary x vector. You can visualize it moving on the front of your eyes ready on. There are functions, symbols, objecs. Lots of concepts. No number.

Matterwave, for god's sake, that's the absolute opposite of what I'm saying. Understand it intuitivelly, play with it using programming instead of paper+ink / imagination. Seems like I'm really bad on expressing myself on english (: