Mathematical notation outdated?

[Gerenuk]

Now as we have computers, why don't we introduce a nice graphical and colourful notation that would make formulas much more understandable? And I mean revamp all of it - not just fancy writing. Tensor contraction for example are ideal candidates for visualisation. It would be much easier to do algebra in the head. Of course some clever hotkeys for typing in this notation would be needed, but at the same time a computer algebra system could check transformations.

One drawback is of course that one always need a computer, but they are everywhere.

I have some ideas for such a system.
Are some ideas out there already?

 

[cristo]

How would you write down such notation? Whilst computers are used to type things up, mathematics is still done by pen and paper! What if you wanted to print off and read a paper but only had a black and white printer?

 

[Hurkyl]

You mean like Penrose's notation for tensors?

 

[Crosson]

It is inevitable that the standard curriculum will be modernized to account for the ability of computers to do tedious symbolic manipulation, so that students have more time to focus on the mathematical subtleties that require human undersanding.

On the otherhand I can't think of anyway that computers could significantly improve notation.

 

[jimvoit]

You might be interested in the paper:
Mathematical Notation: Past and Future (2000). Stephen Wolfram October 20, 2000.

 

[Gerenuk]

How would you write down such notation? Whilst computers are used to type things up, mathematics is still done by pen and paper! What if you wanted to print off and read a paper but only had a black and white printer?

Yes, mathematics is still done by pen and paper since there is no easy program to handle notation. Colour or not is another question. But I would sacrifice backwards-compability for new-age clearness.

 

[Gerenuk]

You mean like Penrose's notation for tensors?

Yes, that was actually one example I was thinking of. But also I thought of some more cartoonish drawing of basic school algebra to would be more visual to pupils. With colours or blobs or so :)

 

[Hurkyl]

There's some deep sense in which tensors are "two-dimensional" arithmetic (John Baez talks about it a lot -- look for stuff he writes on monoidal categories, adjunctions, and on category theory in general). Roughly speaking, it involves arithmetic on vector spaces which is compatable with the arithmetic on vectors in a certain way. Anyways, that's the reason why string diagrams work so well for dealing with tensors. In general, I think it would be unlikely to have such a natural higher-dimensional notation for doing arithmetic in various algebraic strctures.