I largely agree that this is one type of mathematics.

Just an open thought on whether there can be 2 types of mathematics.

1. One (like you said) where the initial experiences commence the journey, but then the rest of it progresses through identifying and proving pre-existing relations between pure ideas purely using reason/logic alone.

2. And another mathematics where the attempt is to keep refining calculations to make agree with observations in the real world. For example a method of calculation for measuring area of land or some astronomical phenomenon.

Or even square root of 2 or value of Pi but using various series expansions to continuously improve the accuracy to help in better measurement of land or navigation of ships by sea.

One can say that the second is just another branch of science. Or is it another kind of mathematics, of course with some relations to the axiomatic mathematics of pure ideas…

Roddam Narasimha wrote about this in EPW many years back in this article:

https://www.epw.in/journal/2003/35/review-science-policy-review-issues-specials/axiomatism-and-computational-positivism

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