See: Math Big Picture
Collect and organize examples of figuring things out in mathematics.
Extending the domain
- Lie algebra (infinitesimal) vs. Lie group (broader domain).
- Kan extension - extending the domain - every concept is a Kan extension
- Apibrėžti "gebėjimus" ir kaip matematinis mąstymas suveda skirtingus gebėjimus suvokti kelis, keliolika, keliasdešimts, tūkstančius ir t.t. daiktų
Analysis - arithmetic hierarchy
- Calculus (delta-epsilon)
- Differentiable manifolds
- Difference between set and vector space (or list?) Set has empty set, but vector space has zero instead of the empty set. So there are no functions into the empty set, but there are functions into zero. Vector space is not distributive. Can't just take the union, need to take the span. Thus R and (S or T) is not equal to (R and S) or (R and T). A line and a plane is not a line and line or a line and line.
- (x-a) x is unknown (conscious question); a is known (unconscious answer); and the difference x-a gives the discrepancy. So a polynomial can be thought of as constructed as the product of the discrepancies - or it can be constructed as coefficients for powers. Note that EVERY polynomial with complex coefficients can be constructed as a product of such discrepancies. So it's a type of completeness. What makes that true?
Solutions to the problem of the brachistochrone