Karl Weierstrass had a very interesting critique of Riemann's work. Supporters of Riemann, claim that a pure logician would never have been able to see the things that the "geometric imagination" of Riemann's could. While the Weierstrassians valued "algebraic truth".
In the current state of math, I think most working mathematicians would agree that Riemann probably saw further ahead than Weierstrass and for the majority of his life did not have the logical tools to prove what he knew was right. This was a task for those after him. Unlike Weierstrass who mostly just continued on with the existing math of the time.
However, this claim that the "geometric imagination" is what allowed him to see further and that it is of a divine character is somewhat flaky. I see no reason why Weierstrass couldn't have done the things Riemann did you, and what more with the added benefit of being rigorous. So then, I ask all the philosophers and pure mathematicians, why do we need geometry?
Have pure mathematicians been wasting their time on something which provides no rigorous truth when they should be investing in algebraic truths? Have we been deceiving ourselves with the elegance of geometry into thinking this signifies some sort of divinity? Does this mean a blind man could only be a logician and never see ahead into math beyond his time? I very much doubt it. However, I am no Riemann, nor am I a qualified philosophers or pure mathematician.
P.S. for the applied mathematicians this question is not for you. Geometry is plenty useful for the physical world.
