[ Question ]

What areas of maths are useful across disciplines?

by tcelferact 25d17th Nov 20191 min read11 comments

6


As a humanities graduate who just read Superforecasting by Tetlock and Gardner, I was impressed and excited by Bayes' Theorem. I am also disappointed that I had never previously studied it, as it is a useful tool that I could have applied to many decisions involving prediction/uncertainty, i.e. almost regardless of my profession.

My question: what else am I missing? What areas of mathematics are useful to pretty much anyone trying to affect high-impact social change? I want to study them! I suspect there are other areas of probability I haven't encountered that can improve my decision-making, but I am also open to completely different topics.

I don't mind a bit of a slog, e.g. I am a computer programmer and started learning multivariable calculus when I became interested in neural networks. In other words, I'm not asking for easy maths (though I wouldn't mind it), I'm asking for widely applicable maths. Thank you!

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7 Answers

I never studied maths or any math-heavy discipline formally (my background is in philosophy), but recently I completed the entire Khan Academy math curriculum. Speaking purely from personal experience, the most valuable math I learned was just basic algebra I had studied in high school but never really mastered. Besides that, I'd say statistics, linear algebra, and parts of calculus (especially series) have been the most useful so far.

Brian Tomasik's great article on education matters for altruism has a section listing useful disciplines and areas. Within maths, it mentions "probability, real analysis, abstract algebra, and general 'mathematical sophistication'" (statistics is also listed, but as a separate discipline).

Statistics is certainly valuable. I don't have any particular recommendations on books/sources for general stats.

I do think Pearl's graphical causal models are a beautiful and useful statistical tool for thinking about interventions. (For a shorter introduction, there's Thinking Clearly About Correlations and Causation: Graphical Causal Models for Observational Data and my own series.)

I would say probability and statistics.

If you want to do evidence-based things, you will want to be able to read an academic paper, including the maths. So you will want to be able to not just understand what is meant by a p-value, but also be able to have thoughts like:

  • this paper used a normal distribution, but really a students t / logit / whatever distribution would have been better, I wonder how big a difference that makes?
  • they used a normal approximation for a binomial here -- and I do/don't think that seems like a reasonable approximation
  • The paper claims this looks like a fairly good fit -- they could have used a chi squared test here; I wonder why they didn't?

This sort of ability wouldn't be useful if the existing body of research consistently used statistics well, but I don't think that's the case.

Final caveat: answering this question is hard because it's so broad. I'm extrapolating from my own experience, but what's useful for you might be different.

The Model Thinker sounds like it may be a good fit for you (There's also a Coursera course by the author). It explicitly aims to be interdisciplinary and provides a broad sampling of different mathematical tools. I think it also hits a nice balance on the rigor-fluff scale. Here's the table of contents:

  • The Many-Model Thinker 1
  • Why Model? 13
  • The Science of Many Models 27
  • Modeling Human Actors 43
  • Normal Distributions: The Bell Curve 59
  • Power-Law Distributions: Long Tails 69
  • Linear Models 83
  • Concavity and Convexity 95
  • Models of Value and Power 107
  • Network Models 117
  • Broadcast, Diffusion, and Contagion 131
  • Entropy: Modeling Uncertainty 143
  • Random Walks 153
  • Path Dependence 163
  • Local Interaction Models 171
  • Lyapunov Functions and Equilibria 181
  • Markov Models 189
  • Systems Dynamics Models 201
  • Threshold Models with Feedbacks 213
  • Spatial and Hedonic Choice 227
  • Game Theory Models Times Three 243
  • Models of Cooperation 253
  • Collective Action Problems 269
  • Mechanism Design 283
  • Signaling Models 297
  • Models of Learning 305
  • Multi-Armed Bandit Problems 319
  • Rugged-Landscape Models 327
  • Opioids, Inequality, and Humility 339

I haven't actually read it, but I've heard good things about More Precisely: The Math You Need to Do Philosophy and the table of contents looks good to me:

  1. SETS
  2. RELATIONS
  3. MACHINES
  4. SEMANTICS
  5. PROBABILITY
  6. INFORMATION THEORY
  7. DECISIONS AND GAMES
  8. FROM THE FINITE TO THE INFINITE
  9. BIGGER INFINITIES

Sets, relations, probability, information theory and decisions and games are all especially valuable and widely applicable IME. It also claims to not require anything more than a high school math background.

If you liked Superforecasting, you'd probably also like How to Measure Anything--it's quite good IMO. It has some nice math like the expected value of perfect information.

I'm biased towards some versions of graph / network theory , dynamical systems and multiobjective optimization theory. Since you are into neural nets and multivariable calculus it sounds like you are already doing a version of these. (I was in an interdisplinary field and took a fair amount of applied math and physics, many of the details of which i never used or really remember--i can look them up--my applied interests were in between very technical and 'fermi' (back of the envelope) problems and i usually tried to phrase them both ways---one solvable. and the other intractable.

I never had a class in statistics but i studied it a bit on my own (partly because one area i did use a bit was statistical physics, though alot of that does not like what you see in a statistics text thoughn they overlap, and also newer texts sometimes sort of have both fields---neural nets to an extent can be viewed as analogous or closely related to statistical physics (sometimes almost the same formalism). .

since i was into applications (and usually not the ones i was assigned to do which were more in biochemsitry and biotech--fields that don't really interest me even though formally they can be phrased in analogy to ones i was interested in, i never could really get into the research (felt they were not problems of high priority to solve, or at least were 'aesthetically' inintersting---just alot of tecnique. its like music--i'm more into forms of modern pop/underground 'Fermi' music, rather than (tecnical) classical, thogh they can overlap. ).