Yes, you should worry; it is weird!

When Reality Changes but No One Tells the Children

It's a regular experience in maths and science education: you're introduced to an idea that is, quite obviously, bizarre. The teacher, however, presents it with calm indifference, as if this conceptual landslide were no more than a change in bus timetable. You ask a question. The room shifts. They mutter something about "just notation" and move on. You’re left wondering: did reality just warp, or am I the only one paying attention?

1. Imaginary Numbers

Once denounced as a monstrous fiction - what kind of number squares to –1? - the imaginary unit i now appears in A-level maths without apology. It lives on a plane no one can see, invented solely to make polynomials behave and and casually invokes the spooky masterpiece e^iπ = –1.

Is any of this stuff real? (Pun intended.)

2. Infinity

Once the playground of mystics, now a fixture in every calculus class. We sum to infinity, we integrate to infinity; just don’t ask what infinity actually is. The teacher will wave it off as “just notation” and quietly hope you don’t bring up Zeno.

Bishop Berkeley wouldn’t have let that slide.

3. Transfinite Numbers

Cantor proved some infinities are bigger than others. He was called a heretic and died in a sanitorium. Today, ℵ₀ and ℵ₁ are chalked up like counting numbers. Whether they actually exist is left to metaphysics - or madness.

We now play with infinite sets like Lego bricks. It's best not to ask where the ceiling is.

4. Numbers as Real Entities

What is the number 2? You’ve never seen it. It’s nowhere in space or time. Yet it's treated as if it sits beside your pencil case. Ask where numbers live, and your teacher looks faintly alarmed and has no reply.

You’ve just tripped over Platonism. Don’t expect anyone to mention it.

5. Action at a Distance

Newton hated the idea that gravity could reach out across the void. So did Berkeley. Yet GCSE physics has the Sun tugging Earth like a yo-yo on a string. Fields show up later as a fix, but how exactly does a “field” live in empty space which is... empty?

This was once considered an ontological scandal. Now it’s a diagram with arrows.

6. Energy as a Property

We're taught energy is “stored” in objects like jam in a doughnut. But energy depends on your frame of reference. It’s not intrinsic, it’s relational. Try explaining that to someone sliding boxes down ramps.

You'll have to wait for graduate school before they sort that one out for you.

7. The Reality of Atoms

Atoms were controversial well into the 20th century. Today we draw them like tiny solar systems and teach them to children. That model’s false, but don’t worry: the reality is fuzzier, deeper, and no one understands it anyway.

Also: we have no idea what an electron actually is. Just smile and say “cloud.”

8. Orbitals as Clouds

“Here’s where the electron probably is,” says the textbook, handing you a pastel blob. Strictly speaking, the blob shows a probability density in 3D space, even though the wavefunction itself lives in an abstract mathematical configuration space (Hilbert space). The distinction is usually swept under the rug... until you ask about two electrons at once.

So what does that mean? Never mind; just colour it in neatly.

9. Evolution by Natural Selection

Once blasphemous, now doctrine. But the theory’s metaphysics - brutality, randomness, directionlessness - rarely get aired. “Organisms adapt” is euphemistic code for “bad stuff happens and the winners reproduce.”

Also: don’t ask about recent human evolution. That’s quietly absent from the syllabus.

10. The Species Concept

Biology likes boxes: genus, species, subspecies. The genome laughs: no species there. Hybridisation, gene flow, ring species* - they all blur the lines. Still, you’ll be tested on Linnaean clarity.

“It’s complicated,” says the teacher. Then they'll mark you as wrong.

11. The Arrow of Time

Entropy increases. That’s that. But the laws of physics don’t care about direction. So why does time seem to flow one way? “Because, for some unknown reason, the Big Bang had extremely low entropy,” the teacher 'explains'. 

'Well, glad that's been sorted out,' you say to yourself.

12. Simultaneity in Relativity

One person’s 'now' is another’s 'not yet' and someone else's 'already happened'. Simultaneity is relative. Einstein took years to accept it. The equations and diagrams purport to make it clear, but no one can visualise Minkowski spacetime. And if the future is as real as the past and present - wow, that's pretty mind-blowing!

Welcome to spacetime as it really is.

13. Wave–Particle Duality

Light is a wave. No, it’s a particle. Actually it’s both. Or neither. It depends who’s asking and how. This was once a crisis; now it’s a footnote. The cat is alive and dead. No one blinks.

Don’t think: just calculate.

14. Energy Non-Conservation in Cosmology

Energy is conserved, unless spacetime is curved and expanding. In which case, not so much. Turns out those mysterious “conservation laws” actually arise from the time and space symmetries of the laws of physics, and if you bend the stage, the rules shift.

Who's ever heard of Emmy Noether^**, anyway?

15. Calculus and Infinitesimals

We teach limits to sanitise calculus, banishing infinitesimals as ghostly nonsense. Then we write dx and treat it like a tiny thing. Don't ask.

Later, Abraham Robinson sneaks infinitesimals back in; now rigorously defined, they're legit again.

Students still don’t know what dx is. Nor, frankly, do most teachers.

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* A ring species forms when a population spreads out geographically in a ring-like fashion around a geographic barrier (like a mountain or a desert). Along the way, neighbouring populations can interbreed with each other, gradually changing as they go. But when the two ends of the ring meet again, the terminal populations are so different that they can no longer interbreed.This entirely subverts the transitive definition of 'species'.

** In general relativity, energy conservation fails because the time-translation symmetry needed for Noether’s theorem doesn’t exist globally (eg if the metric is time-varying as in the FRW case). Emmy Noether, who discovered the deep link between symmetry and conservation laws, is mostly unknown to students, despite having written one of the most important theorems in physics. There is no justice.