By dkl9, written 2024-289, revised 2024-289 (0 revisions)

I like studying calculus, physics, and English etymology.

Calculus includes problems like ∫ `dx` arctan(`x`).
Arctan is an elementary function, so you may be expected to have its integral memorised.
I'd have to find it otherwise.
By parts: ∫ `dx` arctan(`x`) = `x` arctan(`x`) - ∫ `dx` `x` arctan'(`x`).
I remember arctan's derivative: 1 / (`x`^{2} + 1).
Short of that, I could rederive it in terms of trigonometric identities and the chain rule.
Then ∫ `dx` arctan(`x`) = `x` arctan(`x`) - ∫ `dx` (`x` / (`x`² + 1)).

I can easily enough express 1 / (`x`² + 1) as a power series, based on that of 1 / (1 - `x`), and the numerator `x` just multiplies each term.
Integrate term-by-term and get a new power series reminiscent of that for the logarithm.
From that, I could guess-and-check to get the exact logarithm — or I do it precisely from the knowledge that (ln(`f`))' = `f`' / `f`, recognising `x` as roughly the derivative of `x`² + 1.

In physics, you can derive orbital velocity in terms of centripetal acceleration (`a` = `v`² / `r`), and hesitantly get sqrt(`G``M` / `r`).
You can also derive escape velocity in terms of kinetic (1/2 `m``v`²) and potential (`G``M``m` / `r`) energies, and hesitantly get sqrt(2`G``M` / `r`).

If you recall approximate values for each for any one planet, like Earth's 8 km/s and 11 km/s, respectively, you can notice that they differ by a factor around 1.4. The small factor suggests the similar formulae are valid. If you recall sqrt(2) ≈ 1.414, it's clear that those formulae line up: both right or both wrong.

With enough English etymology knowledge, you can look at "acromegaly", recognise "acro-" to mean "tip", "-mega-" to mean "huge", and "-y" to mean "state", and infer that "acromegaly" is "huge extremities condition", with vague or absent context.

Somewhat less so, I can enjoy organic chemistry and 汉字. They repeat each other to enable easy memory, in ways described earlier. From the properties of one fatty acid, we can infer many properties of others, as their difference is the length of an alkane carbon chain, which only react by combustion. From the component 青, learned from the right of the common 情, we can infer a pronunciation close to "qing" of 晴, 清, 請, and more.

At least by comparison to all that above, I dislike learning arts and history.

My subject preferences are well-established as empirical facts. The examples above serve to make apparent a hypothesis behind the pattern: subjects are fun insofar as knowing some things helps you figure out more on your own.

Such an internal connectivity changes the experience of learning anything in the subject. "Learning" a fact I already inferred lets me listen or read with smug boredom. Learning a truly new-to-me fact lets me explore a whole new set of implications. That set of implications can in turn include new-to-me facts, triggering a possibly-exponential chain of engaging reasoning, sans extra inputs. E.g. integrating arctan, as above, could prompt expressing it in complex-logarithmic form.

Those patterns play out well in maths and hard sciences, and somewhat less so in languages. History, by contrast — under the limits of my common, poorly-educated understanding — is largely just one thing after another. Inference and prediction within it do little, mostly resulting in incidental mnemonics when events overlap or connect.