Reproducibility is one of the main principles of a scientific method, that is, the entire process of a given experiment should be reproduced independent of the experimenter. Now, imagine a given scientific discovery, for example quantum mechanics. It starts with some observations, then some ideas and finally formulation based on some mathematical proofs. Now, the question is: if this process, from idea to formulation, is reproducible? More precisely, are the results independent of the people who contributed in formulating quantum theory, such as Planck, Einstein, Heisenberg, Bohr, and many more?
My short answer to this question is “No”. For example, it is not clear that how Planck came up with his role of quantization in his 1900 paper, and if it was not suggested by him, probably this could still remain as a mystery for many more years to come. Or, it is not clear that how Heisenberg came up with his matrix formulation of quantum mechanics. See this reference, in which possible derivation of the 1925 “magical” paper of Heisenberg is discussed. The same is true for Born probabilistic interpretation of quantum states. So, as it is clear, all these steps depend highly on the genius of the people who contributed to the field. These derivations are not reproducible, hence, the process of scientific discoveries itself is outside of the realm of science, surprisingly.
In the ancient eastern science there were two methods for obtaining knowledge. The first method, which matches with the definition of the modern science, is by extrinsic observation and speculation and is called “obtainable” science. I call this method an algorithmic method. It means that, there is an algorithm to obtain that knowledge. The second method is derived by a deep intellectual intuition and it depends on the individual intellect and its level of awareness. I call this method an intuitive method.
Almost all of the mathematical and geometrical proofs belong to the second method of obtaining knowledge, that is by deep intuitions. Actually, this is the part which distinguishes human from machine, and this is the part which I believe can not be simulated by AI (artificial intelligence). Solving an equation, or winning games like chess, all belong to the algorithmic method, so the higher the power of computation, the easier to solve or win. But proving a mathematical or geometrical problem does not require a computational power; it requires ingenuity or a deep intellectual intuition.
The second method is strongly criticized by the modern science, however, ironically, the process of scientific discoveries belong to the category of the intuitive method. In the eastern science, there is a long history of how to improve awareness and intellectual intuition accordingly . This is, in fact, what is missing in the modern science.
 The procedure of improving awareness, actually, is the source of different religions in the east. Each religion came up with an answer to this question, as this is also the only way one can go beyond the material world, if there is any. That’s why whence the modern world started to criticize the religions, it also denies the second method totally without paying much attention to aforementioned subtleties.