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Interesting train of thought. Hard to imagine a world with methods beyond observation and experimentation. The advent of controlled experiments is a good example of a new method.

Though, are computer simulations not just applying mathematics and controlled experiments to the extreme? So, I don't see that as a new method, yet. Really wonder what it will look like.

I was reading the Wikipedia article on Oliver Wendell Holmes Jr. since you mentioned him as someone who rebuked Plato. He has an interesting quote:

"the best test of truth is the power of the thought to get itself accepted in the competition of the market. ... That, at any rate, is the theory of our Constitution. It is an experiment, as all life is an experiment."

I see the subtlety here. On one hand, Plato and Holmes both seem to agree on the endless struggle to find the truth. It's like a tree continuously growing towards the sun, branching off in whatever direction necessary. There is no fixed tree. However, markets select for what spreads, and spreading isn't the same as being true. How can we be sure that what survives is the truth?