Not for every scientific advancement alyhough it's probably true that somewhere along the line mathematics has played an important part. When it comes to areas like quantum mechanics and particle physics, mathematics is absolutely fundamental.
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No, only for some kinds of scientific advancement.
An example of a scientific advancement that doesn't have a mathematical formula involved is discovering a new kind of fossil.
An example of a scientific advancement that doesn't have a mathematical formula involved is discovering a new kind of fossil.