I think it's pretty clear, in my reply, I'm talking about the masses need for Mathematics. Which is for 'solving real-world problems'.

I agree with Terence Tao's sentiments.

Math, for the masses, is a great way to abstractly teach the masses how to critically think about things. Math, for the masses, shouldn't get bogged down in the rigour. But if one were to go on to Adv Math, then yes, rigour is needed and demanded of the mathematician.