The "best" mathematically-informed topics?

+1 for Structure and Interpretation of Computer Programs (aka SICP, aka "the wizard book") - this is a legendary programming book. Here is an interactive version: https://xuanji.appspot.com/isicp/.

I also agree on the important algorithmic ideas, with one addition: algorithmic analysis. Just as you can describe the movement of the planets with a few simple equations, and that's beautiful, you can describe any sequence of steps to finish a task as an algorithm. And you can mathematically analyze the efficiency of that sequence: as the task gets larger, do the number of steps required to finish it grow linearly, quadratically, logarithmically (we hope)? This is a broadly applicable and powerful idea, since pretty much everything (even learning) involves a sequence of steps or process.

Thank you!

Be careful here. Statistical intuition does not come naturally to humans - Kahneman and others have written extensively about this. Learning some mathematical facts (relatively simple to do) without learning the correct statistical intuitions (hard to do) may well have negative utility. Unjustified self confidence is an obvious outcome.

Something to consider: what's a good field in which to learn basic statistics (sticking with the "learn by doing, when possible" theme) ?

Definitely involves math (since you asked).

What do you mean by decision theory?

One of the options I would call 'decision analysis,' and doesn't require much more than algebra and a basic understanding of probabilities. I wrote an introduction to it, and my current book recommendation on the subject is Decisive by the Heath brothers, and I remember it being much more about the psychology of decision-making (and practical heuristics you can apply) than math.

A second option is "what's the difference between CDT and EDT?", but that's something you shouldn't really approach until you understand causal graphs, which you shouldn't really touch until you understand probabilities and graphical networks (like Bayes nets).

A third option is "what are those exotic things they talk about on LW like TDT and UDT?", and I don't feel qualified to tell you the right way to approach that.

As the token epidemiologist in the Less Wrong community, I should probably comment on this.

The utility of learning epidemiology will depend critically on what you mean by the word:

If you interpret "epidemiology" as the modern theory of causal inference and causal reasoning applied to health and medicine, then learning epidemiology is very useful, so much so that I believe that a course on causal reasoning should be required in high school. If you are interested in learning this material, my advisor is writing a book on Causal Inference in Epidemiology, part of which is freely available at http://www.hsph.harvard.edu/miguel-hernan/causal-inference-book/ . For more mathematically oriented readers, Pearl's book is also great.

If you interpret "epidemiology" to mean the material you will learn when taking a course called "Epidemiology", or to mean the methods used in most papers published in epidemiologic journals (ie endless Cox models, p-hacking, model selection algorithms and incoherent reasoning about confounding), then what you will get is a broken epistemology with negative utility. Stay far away from this - people who don't have the time to learn proper causal reasoning are better off with the heuristic "if it is not randomized, don't trust it" . This happens to be the mindset of most clinicians, and appropriately so.

I personally found that in general it is quite hard to figure out which math you need to fully understand a certain result if you don't already know the math. If you start with an overly ambitious goal (for example if you start with the Einstein equations of General Relativity and say to yourself: 'Lets backtrack to see which math I need') I suspect that you will have trouble figuring out which math to learn

Noted, but I do have the advantage of being able to ask. The next post will ask, "for each topic, formula, or problem on this list, what math do I need to know to understand and solve it?"

Microeconomics contains several useful tools for thinking about the world, but those tools are only somewhat driven by math. For example, let's consider "supply and demand."

In a basic econ course, you'd focus on the math (which is basically just algebra). You'd have a increasing linear function for 'supply,' and a decreasing linear function for 'demand.' (Both represent the quantity supplied or demanded at a particular price.) You would then find the intersection of those two lines, and this is the "market price." You would then consider increasing or decreasing the supply or demand curve, and notice how this changes the intersection.

In a basic econ philosophy book, you'd focus on the feedback mechanism that generates the lines discussed above. "Suppose there are more people who want to buy the good than there are sellers of the good at a particular price," the armchair philosopher would say, "and one of the buyers will craftily offer to pay more than the listed price in order to secure a spot in line." This would give you a sense of why the supply line is increasing and the demand line is decreasing.

The idea of what an equilibrium means, and the math underpinning their study, seems useful. But for the math you're potentially better off looking at control theory / controls engineering / feedback systems / signal processing / there might be more names I'm forgetting.

I don't think macroeconomic tools are personally relevant to you, but you can learn about the models in any standard undergraduate text, and it probably won't even require calculus.

Numerical optimization is only kind of in economics. This is the class of problems where you have an exact statement of what you want ('this is my objective function, these are my constraints') and it just takes a lot of pushing numbers around to figure out what the best option is. Once you know linear algebra, you can figure out the basic approaches here, but I don't think this is particularly useful unless you're employed in a context where you need to solve these sorts of logistical problems.

Depends whether you want to go genetics, or a more theoretical approach. But complete expert-level knowledge of the subject would probably include fitness landscapes, network theory, a basic understanding of what's fashionably called 'big data' (for bioinformatics), linear algebra (including a familiarity with the dynamics of formally unsolvable systems and chaos theory), and a LOT of probability theory.

Price's Equation

I don't think that trying to solve the Schrödinger equation itself is particularly useful. The SE is a partial differential equation, and there's a whole logic of differential equations and boundary conditions, etc. that provides context for the SE. If you're serious about trying to understand quantum mechanics, I think the concept of Hilbert space/abstract vector spaces/linear algebra in general is a bigger conceptual shift than just being able to solve the particle in a box in function space. It's also just a really useful set of concepts that makes learning things like optimization, coordinate/fourier transforms, etc. easier/more intuitive.

Until I had the wave function explained to me as some vector in a high dimensional space that we could map into x-space or p-space or Lz-space I don't think I really had a good grasp on quantum mechanics. This is anecdote not data, your mileage may vary.

Try the wave equation first? If you want to think of particles like waves it might be useful to know what a wave is. Note that you already need to have heard of a respectable chunk of calculus to solve this equation.

The Schrodinger equation itself can be understood as simply a particular instance of integral calculus. If I recall my undergrad days correctly, you didn't even need linear algebra. Once you know the calculus, quantum waveforms don't require a whole lot of additional mathematical insight.

Where did you get the Khan Academy image? Could I see the full version?

A dependencies graph would be of great philosophical and practical interest.

After compiling a list of topics, the next step is to figure our what math underpins each one, and order them according to increasing assumed mathematical knowledge.

Thanks for the suggestion.

Can this be my karma sink post?

The book Music and Mathematics is a bit of a jumble of essays from various people. It's not a coherent whole or anything, BUT, if you are the least bit interested in this subject I would strongly you recommend reading the first essay (i.e. chapter). A lot of "music theory" mumbles around circles of fifths and chord progressing and things with vague pretenses of being math, but that chapter includes the only fundamental mathematical idea in the entire book.

Conveniently that chapter, and some others, are here.

I am not sure exactly what you mean, but if you mean the math of its article rankings then it's important to note that the amount of very early upvotes, the subreddit a post is posted to, the name and preview thumbnail, and the time of day it is posted all have tremendous impact on its ikelihood to hit the front page.

http://cseweb.ucsd.edu/~jmcauley/pdfs/icwsm13.pdf

Have you read Margaret Boden's The creative mind: myths and mechanisms? It might make studying relevant math more rewarding.

The Bayes' theorem, of course.

"If a field involves math, and you cannot do the math, you are not qualified to comment on that field."

The "politics of elections" are explicable mathematically using formal methods like game theory. That said, it's social science, which is naturally inexact, which makes it difficult. If you want to criticize it, how about learning the math yourself?

Isn't that like saying psychology is useless since humans have "free will"? It may not be perfectly predictive, but it's still interesting and useful to know what the underlying math and incentives tend to.

In any case, if there are major exceptions that deviate form the mathematical political optima, I want to know why that is.

I'm surprised that this is the first thing to get a point. [My surprise is flagged for consideration with regards to what it means about my model] Who else is it that even knows what sacred geometry is?

Median Voter Theorem is a good one.

If, in a two-person race, you will vote for whomever is is closest to your position, even if they're only a tiny deviation from he median towards you and their opponent is a tiny deviation away from you, then candidates are incentivized to cater as closely as possible to the median voter.

Many factors complicate this in real life elections, but it's still a good concrete start.

This is mostly just applied Game Theory for theoretical political science and Basic Statistics for empirical political science, so I upvoted those instead.