This happened hundreds of times in physics throughout the twentieth century, because physicists were specifically trained to do mathematics from scratch. The main reason is that it was too time consuming in pre-internet times to learn the specialized jargon of each subfield, so it was easier just to rederive the stuff.
The most significant early success of this sort of willful ignorance is probably the development of special relativity from essentially nothing. The Minkowski geometry of relativity is remarkable, because if you interpret the words "point" and "line" as usual, and the word "circle" as a unit hyperbola with 45-degree angle asymptotes (the unit circle of relativity), it satisfies all the explicit axioms of Euclid's geometry, as set out in the elements, including the axiom of parallels, but is not Euclidean. The essential difference is that circles are not closed curves, so that certain implicit betweenness properties fail. There are distinct points which are at a zero "distance" from one another, the hypotenuse of a right triangle is always shorter than one of the sides, etc. This is amazing to me, because of the number of people who have considered models of geometry before Einstein (including all the heavy focus on non-Euclidean geometry for the previous century). All the bigwigs missed Minkowski geometry.
Aside from Einstein's work, there are the following mathematical developments from physics, all of which came out of nowhere mathematically:
- Quantum mechanics, in particular, the theory of the canonical commutation relation [x,p]=i and its relationship with wave operators and random walks.
- Dirac's distribution theory (delta-functions): this completed the notion of Eigenvalue of a linear operator to include Eigenvalues and Eigenfunctions for the x operator in quantum mechanics.
- Majorana spinors--- these were due to the discovery of the Dirac equation. The representation theory of SO(p,q) is now entirely dependent on dirac matrices and the Majorana and Weyl conditions.
- Wigner's random matrix theory. This was completely ab-initio, and is now very active mathematics.
- Anderson localization: this is also a mathematical surprise--- the eigenfunctions of randomized potentials are localized in space. The full resulting theory has still not been made part of mathematics, but Anderson's paper is an ab-initio (although not rigorous) argument.
- Metropolis algorithm--- this essentially inaugurated monte-carlo methods, and I do not know any previous work it builds on.
- Feynman's path integral--- this was developed within mathematics as the Weiner integral at about the same time, but the physics work is completely ab-initio. Needless to say, the results are not going into mathematics easily (in my opinion, this is mostly due to the reluctance of mathematicians to make every subset of R measurable).
- Candlin's fermionic path integral (Berezin integrals)--- Candlin in 1956 develops the whole theory of path integrals for fermionic fields from scratch in a Neuvo Cimento article with next to no citations (in either direction). The theory was ignored for a decade for no apparent reason.
- Mandelstam's double dispersion relations (and dispersion relations in general).
- Kraichnan's inverse cascade--- generally the statistical theory of nonlinear classical equations is developed from scratch by Kraichnan and others. The biggest shocker is the inverse cascade--- in two dimensions, eddies go up from small scales to big scales.
- Zimmermann's forest formula--- this is now part of mathematics, due to Kreimer and Connes, but Zimmermann did it from scratch in physics.
- The theory of second order phase transitions and modern renormalization by Widom/Wilson.
- Wilson's theory of operator product expansions, (which is not a part of mathematics yet)
- Supersymmetry is developed from scratch by several groups with no previous motivation in mathematics (not much in physics). The original germ of an idea is in Golfond and Likhtman, but the person who does most of the early theory work is Pierre Ramond. Wess and Zumino's work also comes out of nowhere.
- Virasoro algebra/Kac-Moody algebra-- Virasoro algbera is the theory of infinitesimal conformal maps under composition, so it should have been classical mathematics, but as far as I know, it wasn't. The theory started (as far as I know) with the study of string theory in the early 1970s.
- Mirror symmetry--- this owes to previous work in T-duality in string theory, not in mathematics.
- Witten's global anomalies--- these are not yet part of rigorous mathematics, but they are ab-initio, and were a complete surprise.
I got tired, but there are hundreds, maybe thousands of examples, because all the results in the physics literature were generally ab-initio. It is a standard practice for some mathematicians to scan the physics literature for original ideas and incorporate them into mathematics.