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"Second, instead of mathematical equations, we use binary computer programs. "
That substitution is extremely domain-narrowing. The modern mathematics has happily
reclused itself into the borders it had itself drawn for itself - Godel's incompleteness, etc...
After that it is even more happily narrowed these borders through mentioned above substitution.
The purpose of the borders is to be transcended. Leibniz didn't happily recluse himself
into counting tortoise steps ahead of Achilles steps. No. Instead he transcended the borders of the sequential counting process. The same way the modern mathematics should strive to transcend the borders of the sequentiality imposed by the natural numbers (ie. Godel's incompleteness), Turing machine and the likes...
Withe best regards,
an MS in Mathematics with high GPA.