I don't see how removing the very nice ability to document source code with 
Markdown and Latex Math mode will help to improve notebooks. I'd certainly 
not like to read mathematical formulas inside Python comments. That would 
be very unreadable. There is also HTML output of cells, which is persisted 
in the notebook's JSON afaik, so it's not suddenly all only Python or R or 
whatever source code if you remove Markdown. It is also the visualizations 
of already computed things, which is persisted. That in itself is quite 
important if you have lengthy calculations and simply want to show a 
notebook without rerunning the computations.

With the other parts I can agree. Running selected code could be a great 
feature. However, I like the ability to separate "computation ideas" / 
"steps" by using separate code cells and putting Markdown documentation of 
what I am doing in between the code cells.

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