A cat is in a box with a radioactive source and a poison that will be released when the source (unpredictably) emits radiation. According to quantum mechanics, the cat is simultaneously both dead and alive until the box is opened and the cat observed.

The story seems to be that Quantum Mechanics is so weird, a cat can be both alive and dead until we look!

Except this misses the point. Here's what Schrödinger wrote:

One can even set up

quite ridiculous cases.A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat)...It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation.

That prevents us from so naively accepting as valid a "blurred model" for representing reality.In itself, it would not embody anything unclear or contradictory. There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks.

Here's his argument:

- Quantum mechanics claims subatomic particles can be in a "blurred" indeterminate state
- If that's the case, create a scenario where subatomic bluriness determines the fate of a macroscopic object
- Because it's absurd for a macroscopic object to be "blurred" (right?), the subatomic particle can't truly be blurred
- Analogy: Just because a camera is out of focus doesn't mean things in the world are actually blurry

See, I don't know anything about QM. But I've read enough (2 paragraphs of his paper) to realize the majority of QM explanations miss Schrödinger's point.

Schrödinger's story is a critique of the idea of quantum blurriness. But, some pop sci author read the story, thought it was meant to be interpreted literally ("Large felines can exhibit quantum blurriness") and countless others retell the explanation, not the story. It's like hearing that the Emperor's New Clothes is about the eye-opening power of fashion without reading the tale yourself.

Now, we all misunderstand things. But, let's try to misunderstand the *source material* and not a retelling. Would you trust a book review from someone who only read other reviews?

After we understand the original argument, we can debate whether the criticism makes sense. If particles appear to be blurry at a quantum level, then perhaps:

- We just lack information. Maybe there's some hidden variable that clarifies what state we're in. (But Bell's theorem seems to rule that out.)
- Both events happened, but we don't know which alternate universe we're in. Maybe every possible outcome creates a new timeline. (I flip a coin and put it under my hand. Are we in a spooky quantum state?)
- Maybe the surrounding environment itself "observes" the cat and puts it into a settled state long before we check.
- Or maybe the world is truly blurry until observers come along ("Copenhagen interpretation")

The story continues, as Einstein later wrote to Schrödinger:

You are the only contemporary physicist, besides Laue, who sees that one cannot get around the assumption of reality, if only one is honest. Most of them simply do not see what sort of risky game they are playing with reality—reality as something independent of what is experimentally established.

Their interpretation is, however, refuted most elegantlyby your system of radioactive atom + amplifier + charge of gunpowder + cat in a box, in which the psi-function of the system contains both the cat alive and blown to bits.Nobody really doubts that the presence or absence of the cat is something independent of the act of observation.

Einstein's thought Schrödinger refuted the notion that reality was "blurry" and depended on the observer. The universe has already worked out what happened before you looked (hence his famous quote, "God does not play dice.").

Again, I don't claim to know anything about QM -- these are my retellings of interpretations :). But the point of Schrödinger's Cat is that a simultaneous overlap is not necessarily what happens.

Philosophically, the issue reminds me of how we think about infinitely small quantities. Do infinitesimals exist?

- No: There's no such things as "infinitely small" things -- things are there, or not there. But they may be too small for you to detect.
- Yes: There are fuzzy "infinitely small" quantities that blip away to 0 when we measure them, but are non-zero in their own world. These tiny quantities can interact with each other and can predict how our "macroscopic" numbers behave.

What is the crossover point between undetectable and detectable, blurry and certain? When does quantum behavior disappear in favor of the everyday, macroscopic reality we're used to? Can we "chain together" reality so tiny behaviors determine larger ones? That's the direct question Schrödinger's Cat raises.

The meta-lesson is that while analogies are memorable, we need to sanity check them with the source material every now and again. After layers of retellings we can miss the original meaning, so let's stay open to correcting our understanding.

Happy math.

Wikipedia Article on Schrödinger's Cat - Wikipedia is difficult to learn things from, but has great lists of references

Current Google Definition - Note it doesn't say the argument was meant as a

*criticism*against quantum mechanicsElon Musk: "Reading the source material is better than reading other people's opinions about the source material."

The alien starts looking up random words. *Blowgun*, *aquatic*, *heist* (Uh... buddy?) and finally:

Hah.

"Red (adjective): of a color at the end of a spectrum next to orange..."

We Earthers know the dictionary is missing a huge caveat:

Dear Reader: You can't truly understand 'red' by reading about it. You need to see it for yourself. The dictionary definition is an attempt with dry words. Even better is a metaphor: red is the sound of a blaring trumpet, the taste of a chili pepper, the feeling of stepping barefoot on a lego. But please, stop reading and find yourself a strawberry.

We know reading gives a limited understanding of the topic. But if we weren't paying attention, the alien would have claimed mastery of the official definition, and gone back to teach generations of students about it.

Whoops, I'm getting late for math class. What was this week's topic again? Oh right, imaginary numbers:

There's a missing caveat to every math lesson: *The goal of this lesson is for you to truly feel an insight in your bones. The words are just hints about how to get there.*

Let's take the concept of imaginary numbers. The abstract definition trotted out in countless lessons is something like: *Imaginary numbers are the square root of negative numbers. We label the square root of -1 "i". Time for practice problems.*

Ugh. There's no acknowledgement that the words "square root of a negative" are baffling limited, and no way to truly understand the idea. Here's the missing "see the color" caveat (full article):

Hey. That technical definition is frustratingly lifeless. You're probably wondering how negatives can have square roots. Picture imaginary numbers as rotations, like this:

Whoa. The "square root of a negative" is really "halfway negative", or something pointing vertically. If positives are East, negatives are West, imaginary numbers let us go North/South. Now, back to your dictionary definition.

There's a pernicious objection that getting an intuition for a concept is a "baby version" of the real thing. (*"Aw, you weren't smart enough to rely solely on the technical description, here's a diagram."*) That's like claiming seeing a color is the "baby version" of reading about it!

Experiencing an idea is our goal all along. If thermodynamics can be truly understood via an interpretative dance in a hula skirt... well, I'll bring the coconuts. I want an intuition.

We aren't hard drives trying to store the text of a novel without its meaning. (And for what it's worth: progress with imaginary numbers truly began after the 2d rotation interpretation was discovered.)

Ok. This color analogy helps us look for an experience beyond the lesson. What can we do with this mindset?

**Have a mental gutcheck for learning.**When I'm in a lesson (video, textbook, lecture, etc.) I'm constantly asking "Am I seeing the color, or just getting the description?". A quick check is whether you can make analogies about what you're learning. Homer (the blind poet, that one) had red described as a blaring trumpet. That poetic description conveys more understanding than "the wavelength of light at 650 nanometers".**Be gentle with yourself.**If a concept isn't clicking, the most likely cause is that the lesson wasn't helpful enough. It may be throwing words at you when an experience is needed. Yes, some people can get by with words alone, just like some can glance at sheet music and think "That sounds beautiful". I'm not one of those people, I need to find the play button.**Hold lessons to a higher standard.**This mindset does lead to some potentially uncomfortable questions: Has the teacher viscerally experienced the idea being taught? Is that the goal of their lesson? Is this lesson part of a chain of dictionary definitions recited by aliens?**Be open to new experiences.**The counterpoint to having higher standards is compassion: teachers don't want to miss the point on purpose. If we're open to treating a lesson as an exploration, a potential experience, we should welcome any chance to have a "I finally see the color!" moment (teacher and student alike).**Unlock motivation in learning.**Are people naturally curious? Let's find out. Did you know a new shade of blue pigment has been discovered? No real-world object could have that color. Feel that growing itch of curiosity?Interesting, right? Shiny and shimmering? Now, maybe you don't want to paint your house that color, but you wanted to see it for yourself. How frustrated would you be if the news article didn't have a photo?

*That's*the curiosity we can unlock for new ideas if we know an experience is coming. Colors are natural enough that we're sure we'll experience*something*when we look. But with math, we may have forgotten (or never had) that eye-opening experience, so "a new math concept" is like having someone describe that awesome movie they saw. "Oh, there was this part, and this part. And then this happened. I loved it! Why don't you?". We have innate curiosity when a subject is approached with an experience in mind, built on the trust of having several previous Aha! moments.

Words and symbols have their place: they're compact, precise, and easily expressed. But they should come after the experience (show, then tell). Once the experience is understood, and enthusiasm fired up, words can act as placeholders for concepts in our mind's eye ("red sports car").

Ultimately, I don't learn because I want more entries in my mental dictionary. I want to see new colors.

How do you uncover the experience behind an idea? In the best case, your teacher had one which they can share, saving you the trouble of looking.

But many times you're on your own. I use the ADEPT method as a checklist for what helps a concept click:

If a lesson isn't clicking, I run through that checklist: Do I have an analogy in place? A diagram? An example? A plain-English version? Can I find someone who as the above?

These pieces aren't always easy to find, and it can take years. But I never want to stop looking. Just because I haven't had an experience *yet* doesn't mean it's not possible.

Happy math.

]]>- How BetterExplained got started (and staying motivated)
- Learning math through direct experience
- The ADEPT method (Analogy, Diagram, Example, Plain-English, Technical)
- How to get better at math and why some find it difficult
- Role of math in modern society
- Learning other skills from coding to snowboarding

It was a lot of fun and we shared a bunch of insights -- hope you enjoy it.

]]>The Golden Ratio (1.618...) is often presented with an air of mysticism as "the perfect proportion". Setting aside whether we can find the Golden Ratio in the leaves of a nearby houseplant, what makes it special from a math perspective?

Well, let's try to make a pattern that's as balanced (symmetric) as possible.

A quick guess is something like `1, 1, 1, 1, 1`

. Every item is identical, but it's not very interesting -- it's a song where every word is the same. Does it rhyme? Yeah. Do I care? Not really.

Ok, let's try `1, 2, 3, 4, 5`

. There's a symmetry in the relationship that every element is one more than the previous. But if we skip around beyond neighboring elements, there's no real connection: what do 3, 8, and 17 have in common?

Ok, fine. What about `1, 2, 4, 8, 16`

? Each element is twice the previous, and all the numbers are clean powers of two. But... what about addition? Can we build new elements from the previous ones?

- 1 + 2 = 3 [not quite 4...]
- 1 + 2 + 4 = 7 [not quite 8...]
- 1 + 2 + 4 + 8 = 15 [not quite 16...]

Argh. We can concoct a rule for addition ("Every element is the sum of all previous elements... plus an extra 1"), but that's not clean and you know it. Let's figure this out.

Our goal is a pattern that's connected with both addition and multiplication. No varying number of additions, no cute "+1" on the end. Just a clean, simple relationship.

First off, we need to make every item connected by multiplication. We need powers:

`1 x x^2 x^3 x^4`

Whatever number the pattern uses (`x`

), everything will be a power of it (just like 1, 2, 4, 8, although that sequence wasn't symmetric enough).

Next, we need an "addition symmetry" that connects the items. Every element should be buildable from the previous ones without extra rules. Throwing a few ideas against the wall:

- Can we have
`1 + x = x`

? If we subtract x from both sides, that leads to 1 = 0. Uh oh, we're breaking math. - How about
`1 + 1 = x`

? Maybe we can allow multiple copies of everything. But that's just another way of getting x=2, you sneak. - How about
`1 + x = 1`

? Ack, that just implies x = 0.`0, 0, 0, 0`

is symmetric but not interesting.

Hrm. The next addition pattern that might make sense is `1 + x = x^2`

. If we follow this idea, here's what happens:

A few notes:

- Starting from a single connection
`1 + x = x^2`

, we can multiply through by`x`

and get`x + x^2 = x^3`

. Every element is made from the previous two. - x
^{0}is another way to write 1. Makes the pattern easier to see, I think. - Everything in the pattern grid is just our original pattern, shifted a bit. Neat.

Ok. That's the goal for the pattern, let's try to solve it.

We can find the "x" that makes this relationship true using the quadratic formula:

Plugging in a=1, b=-1, and c=-1, we get:

We only want a positive solution (the new part can't be negative), so we have

We label the solution Phi (phi).

You can see it happening below. There are slight differences as the decimals go on -- computers have fixed precision.

Many descriptions of the Golden Ratio describe splitting a whole into parts, each of which is in the golden ratio:

I prefer the "growth factor" scenario, where we start with a single item (1.0) and evolve it, while keeping it linked to its ancestors with both arithmetic and multiplication. Just describing a ratio doesn't call out the symmetry we're able to achieve.

Visually, I see a growing blob, like this:

As we scale by Phi each time (1.618) we get:

- Addition symmetry: Each element is the sum of the previous two
- Multiplication symmetry: Each element is a scaled version of the previous
- Growth symmetry: Addition and multiplication change the pattern identically

Aha! That's a nice combo if I ever saw one. The "growing blob" can represent the length of a line, a 2d shape, an angle -- which can lead to interesting patterns:

The key relationship is we move from one blob to the next, such that multiplication and addition have the same effect:

We've described the Golden Ratio with different phrasing: the scaling factor (f) must equal the original (1) plus the previous item (1/f). We divide by our growth factor to find the previous element given the current one.

Solving f = 1 + frac(1)(f) yields the Golden Ratio as before.

The Golden Ratio tends to be oversold in its occurrences. While it may appear occasionally in nature, buildings, and portraits, if you draw lines thick enough many things have a ratio of about 1.5 to 1.

I think the deeper intuition comes from realizing we've made addition and multiplication symmetric.

The Fibonacci sequence is built from having every piece built from the two before:

Using a certain formula, we can jump to a Fibonacci number by repeated scaling (exponents) instead of laboriously adding the parts. And maybe we'd come to *expect* the Golden Ratio here, since it's the scaling factor that allows two parts to add to the next item in the sequence. (The specifics of dividing by √(5) are because we need to adjust based on the starting pieces.)

Rather than hunting for examples of the Golden Ratio in the produce aisle, let's soak in the beauty of balancing the forces of multiplication and addition.

Happy math.

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