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.

Buffon or Buffoon?

 

Buffon's Needle (Wikipedia)

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Megan Markle, according to a newspaper article, advocates that when serving radishes (to guests), the radishes should all be pointed the same way.

I said to Clare, "When you're serving green beans, why don't you make them all line up in parallel? Rather than just throwing them randomly onto the plate. After all, we already know the value of π."

"Buffon or Buffoon? Your call," she said.

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ChatGPT comments:

"Clare wins this round. That’s a top-tier riposte—economical, sharp, and effortlessly dismissive. The perfect checkmate to a remark that was already teetering between erudition and domestic irrelevance.  

Though, to be fair, she does have a point. There’s an undeniable absurdity to obsessing over radish alignment—unless one is a particularly meticulous French chef or a military quartermaster for a vegetable battalion. Meghan Markle’s dictum suggests a kind of aesthetic fascism, where even root vegetables must submit to decorum. But then again, one could argue that the arrangement of food is an art, and art demands order—or at least intentionality.  

Still, your green bean remark was beautifully structured—dry, sly, and with a mathematician’s love of a hidden punchline. But Clare’s comeback? That’s the sort of response that makes you realize you’ve married up."

A fallacious proof that 0 = 1

This is an old one but perhaps you haven't seen it?

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1. Start with the equation:                    x = y 
2. Multiply both sides by x:                  x² = xy
3. Subtract y² from both sides:      x² - y² = xy - y²
4. Factor both sides:           (x - y)(x + y) = y(x - y)
5. Divide both sides by (x - y):        x + y = y
6. Since x = y, substitute y for x:     y + y = y
7. Simplify:                                             2y = y
8. Divide both sides by y:                      2 = 1
9. Subtract 1 from both sides:              1 = 0

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I asked ChatGPT for some better fallacies but in vain. It just produced stupid stuff.

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But then there is this - integration by parts:

∫ (dx/x) = ∫ (1) (1/x) dx

= x (1/x) - ∫ x (-1/x²) dx

= 1 + ∫ (dx/x)

⟹ 0 = 1.

Two appearances of the same integral can (and should!) have (different) constants of integration; or, as someone said: "From C to shining C".

Try it 'definitely' and you will see it comes out right.

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Mathematical Elegance

It's strange how arbitrary the numerical representation of π is. Compare 3.14159265358979... (equally random in any other base) to the platonic perfection - and inevitability - of the circle and its diameter.

Surely there is a compelling series for π which more clearly demonstrates its geometrical elegance?

The Leibniz formula is very elegant:

The Leibniz series for π is expressed as:

π ≈ 4 × (1 - 1/3 + 1/5 - 1/7 + ...)

The first four terms of the series are:

π ≈ 4 × (1 - 1/3 + 1/5 - 1/7) = 4 × 0.72380952 ≈ 2.8952381

but it comes out of the Taylor series for arctan. There seems no intuitive geometrical picture of how this ends up as π. There are many other series which converge to π: many of them have striking patterns. But the term-by-term connection to geometry is never there.

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An elementary mathematical identity is that sin2(θ)+ cos2(θ) = 1. I asked ChatGPT (which can now do maths) to show me how this comes out of the Taylor series for sin2(θ) and cos2(θ) separately.

Question:

What is the Taylor series for sin2(θ) and cos2(θ) separately? How does their sum turn out to be invariantly one?

Answer:

1. Taylor Series for sin2(θ)

The Taylor series for sin(θ) is:

sin(θ) = θ - θ3/3! + θ5/5! - θ7/7! + ...

Squaring this series gives:

sin2(θ) = (θ - θ3/3! + θ5/5! - ... )2

Which results in:

sin2(θ) = θ2 - θ4/3 + 2θ6/45 - ...

2. Taylor Series for cos2(θ)

The Taylor series for cos(θ) is:

cos(θ) = 1 - θ2/2! + θ4/4! - θ6/6! + ...

Squaring this series gives:

cos2(θ) = (1 - θ2/2! + θ4/4! - ... )2

Which results in:

cos2(θ) = 1 - θ2 + θ4/3 - 2θ6/45 + ...

3. Adding Them Together: sin2(θ) + cos2(θ)

Now, summing the Taylor series for sin2(θ) and cos2(θ):

sin2(θ) + cos2(θ) = (θ2 - θ4/3 + 2θ6/45 - ...) + (1 - θ2 + θ4/3 - 2θ6/45 + ...)

When you add the terms:

• The θ2 terms cancel out.
• The θ4 terms cancel out.
• The θ6 terms cancel out.
• And so on for all higher-order terms.

Thus, all terms cancel out except the constant 1:

sin2(θ) + cos2(θ) = 1

This shows how the sum of the Taylor series results in the invariant identity sin2(θ) + cos2(θ) = 1, independent of the value of θ.

Credit: ChatGPT

I wish I'd seen this when I was fifteen.

Maths Areas Ranked by Coolness (most to least)

Georg Cantor (Wikipedia)

Maths areas ranked by coolness: the most cool at the top: my subjective judgement.

1. Transfinite Set Theory: Bigger than infinity is still a mind-blowing concept.
2. Complex Analysis: The beauty and power of the imaginary unit and its applications make it a strong contender for the top spot. Holomorphic functions in complex analysis have surprising properties, like being infinitely differentiable.
3. Abstract Algebra: Exploring elegant structures and operations like groups, rings, and fields, feels like entering a new mathematical universe. Its generalisation to Universal Algebra leads to foundational computer science concepts.
4. Real Analysis: Taking calculus to the next level, dealing with abstract concepts like continuity and convergence. Leads to the wonderful Calculus of Variations, a foundation of theoretical physics.
5. Topology: Point-set topology, also called general topology, is the foundation of most branches of topology. It studies the basic properties of shapes and spaces by focusing on how close points are without relying on specific distances. It defines concepts like open sets, continuity, and connectedness, forming the essential toolkit for exploring the geometrical nature of mathematical objects..
6. Differential Geometry: Bending space and time with the power of calculus? Sounds pretty cool for those who enjoy the physics connection.
7. Number Theory: The timeless elegance of prime numbers and their mysteries remain fascinating. The extraordinary conceptual depth you get starting with the simple notions of the object 0 and the unary operator s, combining thus: (0, s(0), s(s(0)),...).
8. Linear Algebra: Essential but the practicality might overshadow the coolness factor. The link with Quantum Theory might boost the coolness a little - Hilbert Spaces.

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This list - very subjective - came from a dialogue between Gemini Pro and myself. I studied most of these at undergraduate level excepting Number Theory (an option I didn't take) and Differential Geometry (not offered). I have forgotten almost all of the maths I studied at university...

I would like to add formal logic to the list: predicate calculus; modal logic, lambda calculus. I'm not sure the mathematicians would allow it entry, but if they did it would be very, very cool.

Every odd number is the difference between two squares

From here via SSC. This is apparently a 'twitter' proof (ie short).

Algebraically (n + 1)² - n² = 2n + 1 which is odd.

A proof beyond understanding

From Nature via Peter Woit:

"Sometime on the morning of 30 August 2012, Shinichi Mochizuki quietly posted four papers on his website.

The papers were huge — more than 500 pages in all — packed densely with symbols, and the culmination of more than a decade of solitary work. They also had the potential to be an academic bombshell. In them, Mochizuki claimed to have solved the abc conjecture, a 27-year-old problem in number theory that no other mathematician had even come close to solving. If his proof was correct, it would be one of the most astounding achievements of mathematics this century and would completely revolutionize the study of equations with whole numbers.

Mochizuki, however, did not make a fuss about his proof. The respected mathematician, who works at Kyoto University's Research Institute for Mathematical Sciences (RIMS) in Japan, did not even announce his work to peers around the world. He simply posted the papers, and waited for the world to find out."

So you'd think people would be all over it, right?

"Probably the first person to notice the papers was Akio Tamagawa, a colleague of Mochizuki's at RIMS. He, like other researchers, knew that Mochizuki had been working on the conjecture for years and had been finalizing his work. That same day, Tamagawa e-mailed the news to one of his collaborators, number theorist Ivan Fesenko of the University of Nottingham, UK. Fesenko immediately downloaded the papers and started to read. But he soon became “bewildered”, he says. “It was impossible to understand them.” ...

Everyone — even those whose area of expertise was closest to Mochizuki's — was just as flummoxed by the papers as Fesenko had been. To complete the proof, Mochizuki had invented a new branch of his discipline, one that is astonishingly abstract even by the standards of pure maths. “Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space,” number theorist Jordan Ellenberg, of the University of Wisconsin–Madison, wrote on his blog a few days after the paper appeared.

Three years on, Mochizuki's proof remains in mathematical limbo — neither debunked nor accepted by the wider community. Mochizuki has estimated that it would take a maths graduate student about 10 years to be able to understand his work, and Fesenko believes that it would take even an expert in arithmetic geometry some 500 hours. So far, only four mathematicians say that they have been able to read the entire proof."

And something strange happened to those four ...

"But so far, the few who have understood the work have struggled to explain it to anyone else. “Everybody who I'm aware of who's come close to this stuff is quite reasonable, but afterwards they become incapable of communicating it,” says one mathematician who did not want his name to be mentioned.

The situation, he says, reminds him of the Monty Python skit about a writer who jots down the world's funniest joke. Anyone who reads it dies from laughing and can never relate it to anyone else."

Worth reading the whole thing. Another article, from Quora, here (written by a 15 year old!)

There will be a Clay Mathematics Institute, University of Oxford, Workshop on IUT Theory of Shinichi Mochizuki, Monday December 7 - Friday December 11, 2015.

What did you think of the previous post?

We asked Kitty what he thought of the previous discussion about the nature of 'two'.

"zzzzz"

Philosophy of Mathematics? Cats just wanna have ... zzzzz .. .