By 1900s formulas and transformations were viewed as things, not process, and the objects of algebra were much more abstract and far more general.
The most succesfull and most systematic work along these lines carried out by Lagrange, he reinterpreted the classical formulas in terms of the solutions that were being sought. What mattered, he said, was how certain special algebraic expressions in those solutions behaved when the solutions themselves were permutted- rearranged. One that could remain exactly the same no matter how the solutions were shuffled- could be expressed in terms of the coefficients of the equation, making it a known quantity.
Galois realized that the solutions for algebraic equations had to do with symmetry.
Group theory: Frame work for studying symmetry. This led to a more abstract view of algebra, and mathematics. Abstract methods are more powerful than concrete ones. It made clear that negative results may still be important, and that an insistence on proof can sometimes lead to major discoveries.
The most succesfull and most systematic work along these lines carried out by Lagrange, he reinterpreted the classical formulas in terms of the solutions that were being sought. What mattered, he said, was how certain special algebraic expressions in those solutions behaved when the solutions themselves were permutted- rearranged. One that could remain exactly the same no matter how the solutions were shuffled- could be expressed in terms of the coefficients of the equation, making it a known quantity.
Galois realized that the solutions for algebraic equations had to do with symmetry.
Group theory: Frame work for studying symmetry. This led to a more abstract view of algebra, and mathematics. Abstract methods are more powerful than concrete ones. It made clear that negative results may still be important, and that an insistence on proof can sometimes lead to major discoveries.