User:IssaRice/List of mathematical difficulties

This pages lists some of the concepts in math that I had the most difficulty with.

months of confusion

• material implication (introductory sources don't even mention the deduction theorem...)
• the idea of a random variable, and how it relates to the sample space
  • expansion of sample space
  • the fact that we often only care about properties of random variables that are shared among all random variables with the same distribution
• singular value decomposition, and classification of linear operators (once i did everything in a 2d real vector space, and could picture the geometry of each type of operator, everything made sense)
• determinant of a matrix (watching https://www.youtube.com/watch?v=xX7qBVa9cQU cured me once and for all!)
• chain rule of differentiation (the problem here was that i needed to study linear algebra beforehand, but literally no calculus book will tell you this very important fact!)
• lots of the basic concepts in computability theory: recursive set, recursively enumerable set, partial recursive function, etc. etc. I think the boolos/jeffrey/burgess book is great in a way, but also really sucks in a way (it just doesn't emphasize the stuff i want emphasized). eventually everything started to make sense, and now it seems so intuitive that i think "how did i not understand this?" I swear there is a much faster way to learn all of this without all the suffering.
• lots of stuff in mathematical logic: the different uses of $\models$, the difference between a logic, language, theory, axioms, interpretation, model, sentence, wff, formula, etc. etc.

days of confusion

• why the godel completeness theorem and incompleteness theorem don't contradict each other

ongoing confusion

• recursion theorem/diagonalization lemma (i still don't understand this, but i think i'm getting there... i probably need to know more category theory first, maybe lambda calculus)
• why i should study topology when i have metric spaces already