> Without an understanding of probability, linear algebra and differential equations, there would have been no quantum mechanics. Somebody observing the photoelectric effect could not have developed those tools for their "application".
I disagree strongly, and history might too. For example, Heisenberg invented matrix multiplication for quantum mechanics. In general, physicists have been very happy to invent entire fields of mathematics (eg, calculus!) for their direct application. At the very least, I think you're overselling your point.
Taking nothing away from Heisenberg, he reformulated a particular quantum system in the new language. But in order to be able to apply QM to all kinds of systems, one needs to also reformulate it in a universal language. At that point, Max Born interpreted Heisenberg's example using his knowledge of math -- Here's the story I found: https://en.wikipedia.org/wiki/Heisenberg%27s_entryway_to_mat...
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Max Born had two seminal contributions:
1. Interpreting the noncommutative structure in Heisenberg's observation as linear operators
2. The "Born rule" that relates amplitudes involving the wavefunction (quantum state) to probabilities.
For both of those, he used existing math knowledge. He did not re-invent linear algebra or probability theory. And differential equations were part of the standard toolkit of physicists by then (given successes of the theory of heat and the wave theory of light)
I disagree strongly, and history might too. For example, Heisenberg invented matrix multiplication for quantum mechanics. In general, physicists have been very happy to invent entire fields of mathematics (eg, calculus!) for their direct application. At the very least, I think you're overselling your point.
http://mathoverflow.net/questions/185954/when-exactly-and-wh...