User:IssaRice/List of mathematical difficulties

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

the following types of frustration are excluded:

• an exercise/problem i couldn't solve, but as soon as i read a solution, i immediately understood it (and slapped myself on the forehead)
• things i keep mysteriously forgetting for some reason, but each time i look it up again it makes sense.

months/years of confusion

• material implication (introductory sources don't even mention the deduction theorem...)
• epsilon-delta definition of a limit (part of the problem here was that i was going through this stuff in high school with absolutely no guidance from anyone who knew math, and this was back when i had no idea how to learn math)
• 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
• the idea of a "distribution" -- what was confusing here was that people use it to mean pmf, pdf, cdf, or the borel map thingy, and it took me a long time to realize they are all interchangeable in a way
• all the equivalent properties of injective/surjective linear maps
• 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.
• all the different views of solomonoff induction (and the proof that if you look at proportion of valid programs, considering length=n in the limit is equivalent to considering length<=n in the limit.)
• the idea that different classes of probability distributions aren't mysterious -- that you can always describe a simple causal/mechanical model and formally derive the distribution that way. The problem here is that every statistics/probability book i have looked at just says "here are twelve common distributions you should be familiar with" and then goes on to just list the pdf/cdf, compute the mean/variance/etc without any explanation. i thought i would never understand why some things have one distribution while other things have another... until i actually tried. (a similar lesson applies to all of math in general: things seem super mysterious until you tinker around with a simple example and see why it works, after which it seems obvious.)

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
• MCMC/metropolis-hastings