I always needed practical examples, which is why it was helpful to learn physics alongside calculus my senior year in high school. Knowing where the physics equations came from was easier than just blindly memorizing the formulas.

The specific example of things clicking for me was understanding where the “1/2” came from in distance = 1/2 (acceleration)(time)^2 (the simpler case of initial velocity being 0).

And then later on, complex numbers didn’t make any sense to me until phase angles in AC circuits showed me a practical application, and vector calculus didn’t make sense to me until I had to actually work out practical applications of Maxwell’s equations.

And the solutions we have for 5 or 10 appear elegant: perfect 45° angles, symmetry in the packed arrangement.

5 and 10 are interesting because they are one larger than a square number (2^2 and 3^2 respectively). So one might naively assume that the same category of solution could fit 4^2 + 1, where you just take the extra square and try to fit it in a vertical gap and a horizontal gap of exactly the right size to fit a square rotated 45°.

But no, 17 is 4^2 + 1 and this ugly abomination is proven to be more efficient.

she got a blood transfusion from Banner

The existence of blood transfusions implies the existence of blood cisfusions.

FitnessGram Pacer Tests, obviously

Cats keep rodents under control so that our stored grain isn’t destroyed or contaminated.

This study analyzes historical results of three different man versus horse races (in Wales, in Virginia, and in California). The data shows that human performance decreases with temperature, but less so than horses, so that 30°C is approximately where the best humans can start outperforming the best horses that year.

I would think that even with 15 minutes of intermittent pauses/checks, that time is still productive for cooling the animal and would add less than 15 minutes to the theoretical total if they were allowed to run the whole time.

In order to make a firearm from scratch you must first create the universe.

All of us who learned Spanish in the U.S. also know “¿Dónde está la biblioteca?”

Just a bunch of canned phrases like that kicking around in our brains.

It’s a few things that stem from bipedalism:

• We can run and breathe entirely separately. Most quadrupeds lack the ability to run and take breaths independently of the pace of each step. Watching cheetahs sprint, for example, show that they have no choice but to exhale every time their legs come together and inhale every time their legs push apart.
• Running on our hind legs only frees up our hands to be able to use tools and weapons, maybe even water containers for drinking on the go.
• We can see further by standing up, and can make tactical decisions based on terrain, while still running pretty much full speed.

Combined with our unusual ability to cool ourselves by sweating, this gives us an advantage over pretty much any animal in the heat. Wolves and horses can still outrun humans in the cold, but lack the cooling mechanisms to maintain pace in the same heat that we can.

When I learned about taxonomy in the 90’s they hadn’t really sequenced many genomes, so taxonomy was still very much phenotype driven, rather than the modern genetic/molecular approaches. I just assumed that everything I learned has become out of date.

Gray sex.

You know, sex between people of Pierce’s age.

Black, brown, then the fucking colors of the rainbow in order, gray, white.

If you need a mnemonic to memorize that, you’re gonna have some trouble actually building out your lookup table in your head of immediately knowing that red=2, yellow=4, etc.

10 digits gets the diameter of the earth to within an inch.

Put another way, 10 digits means that your error will be caused by your imprecise model of the Earth’s shape, rather than imprecision in the value of pi.

What would be the “n” in that Big O notation, though?

If you’re saying that you want accuracy out to n digits, then there are algorithms with specific complexities for calculating those. But that’s still just an approximation, so those aren’t any better than the real-world implementation method of simply looking up that constant rather than calculating it anew.

Predate rationalism? Modern rationalism and the scientific method came up in the 16th and 17th centuries, and was built on ancient foundations.

Phlogiston theory was developed in the 17th century, and took about 100 years to gather the evidence to make it infeasible, after the discovery of oxygen.

Luminiferous aether was disproved beginning in the late 19th century and the nail in the coffin happened by the early 20th, when Einstein’s theories really started taking off.

Plate tectonics was entirely a 20th century theory, and became accepted in the second half of the 20th century, by people who might still be alive today.

Science is a process for learning knowledge, not a set of known facts (or theories/conjectures/hypotheses/etc.).

Phlogiston theory was science. But ultimately it fell apart when the observations made it untenable.

A belief in luminiferous aether was also science. It was disproved over time, and it took decades from the Michelson-Morley experiment to design robust enough studies and experiments to prove that the speed of light was the same regardless of Earth’s relative velocity.

Plate tectonics wasn’t widely accepted until we had the tools to measure continental drift.

So merely believing in something not provable doesn’t make something not science. No, science has a bunch of unknowns at any given time, and testing different ideas can be difficult to actually do.

Hell, there are a lot of mathematical conjectures that are believed to be true but not proven. Might never be proven, either. But mathematics is still a rational, scientific discipline.