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I would disagree. There is an entire conceptual apparatus that developed with
17th-19th century mathematical physics - generator functions, matrix algebra,
vector calculus - etc. I don't see how you can make meaningful predictions in
QM outside that framework.
Electrostatics is actually a bad example to use, precisely because the theory is mathematically identical to
Newtonian gravity. Even relativity would have been no problem to these guys. Einstein's contribution, while crucial,
is technically really very small, as it amounts to doing hyperbolic geometry instead of the Euclidean one. Newton
knew more about conics than most mathematicians probably do today.
As to making useful predictions in QM without these methods, remember that matrix mechanics is only Heisenberg's
picture. The Schroedinger picture is about wave equations, which had already been worked out in the middle of the
17th century by Euler and the Bernoulli gang.
$E(X_t|F_s) = X_s,\quad t > s$
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