4 ms·

Not a mathematician but every use of the term algebra I have seen has some way related back to what I learned in grade school about sets of real numbers, operat

by tmerr 5y ago

Not a mathematician but every use of the term algebra I have seen has some way related back to what I learned in grade school about sets of real numbers, operations, properties, and how to solve equations. It's the same thing with linear algebra, they just put restrictions on the kinds of equations to solve. Then more ... abstract there is abstract algebra that goes beyond real numbers and looks at other kinds of sets & operations and classifies them as fields, rings, and more. I wonder if there are examples of the term "algebra" that are unrelated to this. If not that might be one difference between algebra and "a calculus" which as you said seems to refer to any system for formal calculation/reasoning.

rsj_hn 5y ago
When most mathematicians use the world "algebra", they mean abstract algebra. It's not so much a research area in itself but there are many interesting subfields. For example, algebraic geometry is in some sense just the study of solutions of polynomials, but there are algebraic analogues of many interesting geometric notions you encounter in plain old Riemmannian geometry. Then there is algebraic number theory, the studies of modules (generalizations of linear algebra), and then you have questions about representation theory, you can ask questions about moduli spaces of curves defined over a field, of vector bundles (or rather "coherent sheaves") defined over a field, classification of m-dimensional algebraic structure up to birational equivalence, resolution of singularities, etc. All of these have purely algebraic analogues and are topics of interest without needing to resort to the real numbers or any analytical techniques. Talking about "algebra" is a bit like talking about "analysis". It's more of a dividing line of techniques than a specific field.

4 ms·

Not a mathematician but every use of the term algebra I have seen has some way related back to what I learned in grade school about sets of real numbers, operat

by tmerr 5y ago

rsj_hn 5y ago
When most mathematicians use the world "algebra", they mean abstract algebra. It's not so much a research area in itself but there are many interesting subfields. For example, algebraic geometry is in some sense just the study of solutions of polynomials, but there are algebraic analogues of many interesting geometric notions you encounter in plain old Riemmannian geometry. Then there is algebraic number theory, the studies of modules (generalizations of linear algebra), and then you have questions about representation theory, you can ask questions about moduli spaces of curves defined over a field, of vector bundles (or rather "coherent sheaves") defined over a field, classification of m-dimensional algebraic structure up to birational equivalence, resolution of singularities, etc. All of these have purely algebraic analogues and are topics of interest without needing to resort to the real numbers or any analytical techniques. Talking about "algebra" is a bit like talking about "analysis". It's more of a dividing line of techniques than a specific field.