Basics of Mathematics

[Amr Morsi]

Are Mathematics (excluding Geometry) depends only on addition and subtraction together with positive and negative quantities (away from whether imaginary is included or not)? Or, more or less?

[DrRocket]

Are Mathematics (excluding Geometry) depends only on addition and subtraction together with positive and negative quantities (away from whether imaginary is included or not)? Or, more or less?

 

no

[Amr Morsi]

Wow! You are so confident!

[DrRocket]

Wow! You are so confident!

 

It comes from understanding the subject. Any other mathematician would tell you the same thing.

Edited August 29, 2011 by DrRocket

[ajb]

Are Mathematics (excluding Geometry) depends only on addition and subtraction together with positive and negative quantities (away from whether imaginary is included or not)? Or, more or less?

 

 

The answer is no, though of course the ideas of adding objects and inverses is very important.

[Shadow]

While I'm not sure it directly answers your question, you should think about reading Where Mathematics Comes From by Lakoff and Núñez. While controversial, it does shed interesting light on mathematics as a whole.

[DrRocket]

The answer is no, though of course the ideas of adding objects and inverses is very important.

 

That makes two actual mathematicians providing the same answer.

[Amr Morsi]

O.K. Thanks, I was just asking and I, myself do have the answer very clearly and convinced.

 

Thanks guys.

 

It really does.

 

This is what I think.

[Dekan]

Are Mathematics (excluding Geometry) depends only on addition and subtraction together with positive and negative quantities (away from whether imaginary is included or not)? Or, more or less?

 

You seem to be right. Otherwise, how could Maths be done by computers? Computers only do addition and subtraction - and subtraction is just negative addition.

 

So computers are really just adding-up machines.

 

Yet these adding-up machines can solve complex problems in Maths. Doesn't that prove that Maths, ultimately, must boil down to nothing more than adding-up?

Edited August 29, 2011 by Dekan

[ajb]

That makes two actual mathematicians providing the same answer.

 

See the power of adding objects in action in the real world right in front of us.

 

Yet these adding-up machines can solve complex problems in Maths. Doesn't that prove that Maths, ultimately, must boil down to nothing more than adding-up?

 

It proves that a lot of interesting problems can be solved on the computer via direct computation. Large parts of applied mathematics and related topics like industrial physics or engineering say, can be tackled using numerical methods which ultimately do rely on adding real numbers.

 

However there are large parts of mathematics that cannot really be understood in terms of numbers. Sometimes you may be able to employ matrices and finite dimensional real vector spaces etc. in order to build representations. For example with Lie algebras we have Ado's theorem: every finite dimensional Lie algebra over a field of characteristic zero has a faithful representation as square matrices under the commutator.

 

(The generalisation of Ado's theorem is the universal enveloping algebra, but lets not get too distracted!)

 

So for finite dimensional Lie algebras over the real or complex numbers things are at worse multiplication of square matrices, which does boil down to a collection of numbers multiplied and added together.

 

But this will not work in absolute generality. Nor is picking a representation always the most convenient thing to do. Philosophically the modern attitude is that mathematical objects exist independent of any representation.

 

There are plenty of abstract objects that have nothing to do with real or complex numbers. An example would be infinity.

 

I will think on for better accessible examples...

Edited August 30, 2011 by ajb