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Could you refer to a tutorial or a bit more description about what you mean by 'manipulate abstraction over the mathematical concepts', especially when doing hi

Could you refer to a tutorial or a bit more description about what you mean by 'manipulate abstraction over the mathematical concepts', especially when doing higher math? Simple arithmetic, or even computer-algebra simplifications are too simplistic. They help with high-school stuff but are useless for higher math where the bread and butter are theorem-proof-theorem-proof based learning/understanding/research. Geometric plotting/vis, again, good for high-school and freshman stuff mainly. (Plotting is useful for research, data science, numerics, what not, but that's a whole other story. And computer graphics is a rabbit hole). Same with Analysis. Good to have an automatic integrator for Calculus, useless for Real Analysis. Numerical calculations are too laborious and you have to have a good understanding of a wide body of numerical literature, plus have experience. (again a rabbit hole) That leaves us with proof-assistants, model-checkers, and small-scale automatic theorem-provers. And experience research mathematicians are still learning those. Am I missing something?

gnramires 6y ago
Not a mathematician. Used Sage for engineering purposes. But I think one example would be: you're investigating something and get a large linear system, a polynomial, etc. Sage can solve equations with symbols. It saves a lot of manual labor. Various expressions can be manipulated, simplified, etc. It can readily verify equality of huge expressions (which can be error prone by hand), and so on. More specific fields have more specific tools that could be used to assist, so it would depend on the field I guess. See here: https://ask.sagemath.org/question/24866/how-to-simplify-with-sage/ https://ask.sagemath.org/question/24866/how-to-simplify-with... Edit: I also think it should be fairly common to need to perform an actual numerical calculation in mathematical work? I think it's quite better than a simple calculator for that too, since it gives exact e.g. rational results, and approximations would be unfit for a proof (unless extensive precautions are made). I remember I got an expression for an eigenvalue of a matrix (as a function of some parameters) that would be very laborious to get by hand.

gnramires 6y ago

Not a mathematician. Used Sage for engineering purposes. But I think one example would be: you're investigating something and get a large linear system, a polynomial, etc. Sage can solve equations with symbols. It saves a lot of manual labor. Various expressions can be manipulated, simplified, etc. It can readily verify equality of huge expressions (which can be error prone by hand), and so on. More specific fields have more specific tools that could be used to assist, so it would depend on the field I guess. See here: https://ask.sagemath.org/question/24866/how-to-simplify-with-sage/ https://ask.sagemath.org/question/24866/how-to-simplify-with... Edit: I also think it should be fairly common to need to perform an actual numerical calculation in mathematical work? I think it's quite better than a simple calculator for that too, since it gives exact e.g. rational results, and approximations would be unfit for a proof (unless extensive precautions are made). I remember I got an expression for an eigenvalue of a matrix (as a function of some parameters) that would be very laborious to get by hand.