Everything Is Logarithms
5 hours ago
- #logarithms
- #dimension
- #vectors
- Introduces the concept of a 'baseless logarithm' as an abstract object without a specified base, allowing change-of-base formulas to be seen as unit conversions, akin to vector coordinate changes.
- Logarithms behave like vectors, with baseless logarithms analogous to geometric vectors and based logarithms like coordinate projections, using operations resembling vector division.
- Connections exist between logarithms and various mathematical concepts: p-adic valuations νp(n) project logarithms onto prime components, and order of vanishing in complex analysis is similar.
- Vectors in differential geometry can be viewed as logarithms of translation operators, with expressions like ln(T^v) = v_x ∂x + v_y ∂y.
- Dimension operator dim in linear algebra acts like a logarithm, with dim_K V = log_|K| |V|, relating cardinalities and bases, and suggests fractional dimensions.
- Functions can be interpreted as logarithms when 'setified' using combinatorial expansions, hinting at deeper connections between arithmetic and set operations.
- The overarching theme is that many mathematical operations (logarithms, vectors, dimensions, functions) are instances of the same fundamental structure, akin to general covariance in physics, suggesting unified theories.