Here's a few mental models I use to keep them straight.

Let's take a simple situation: You have 4 shirts and 8 pants, how many outfits can you make?

In essence, you are picking a spot on this grid:

Shirts and Pants exist in separate dimensions, whose area represents distinct solutions. We can pick any spot *in the grid* and we have 4 x 8 = 32 options.

Now, suppose we had 4 shirts and 8 pants and had to pick a single item to sell. Here, they lie along the same "clothing item" dimension:

We can randomly pick any point *along the line* and have 4 + 8 = 12 options.

Think "different dimensions vs. same dimesion" or "grid vs. line".

Another interpretation is AND (multiplication) vs. OR (addition).

Let's say we must pick one of 4 shirts AND one of 8 pants. We need both to stay out of trouble. The scenario is:

`pick among 4 shirts AND among 8 pants = 4 * 8 = 32 choices`

What if McDonald's softens their regulations and allows a shirt OR pants? (But not both -- yikes.) Then, we have:

`pick among 4 shirts OR among 8 pants = 4 + 8 = 12 choices`

Writing out the scenario is often easier to think through, especially with numerous dimensions (shirts, pants, hats, shoes).

As you internalize the analogies, you'll quickly recognize whether multiplication or addition is needed.

Let's go meta for a minute. The permutation formula is:

How can we think about this?

The numerator (n!) is the max volume assuming each of the n choices has its own dimension. The number of rearrangements of 8 people is 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1.

But suppose we only care about the first 3 decisions -- picking a Gold, Silver and Bronze among 8 contestants. In this case, we shrink our solution space by dividing out the 5 dimensions we aren't using (which has 5! options on its own). We are left with 8! / 5! = 8 * 7 * 6 = 336 choices, with the general formula frac(n!)((n-k)!).

(If multiplication creates dimensions, then division should remove them.)

Now, let's say the medals are identical: we're giving a tin can to 3 out of 8 people. We need to further remove dimensions, because we have 3! = 3 * 2 * 1 = 6 redundancies for each permutation in our solution space. We again shrink our solution space:

(I imagine the solution space volume getting denser.)

Ah! That's what's happening with the combination and permutation formula. We create the max volume and shrink it by the dimensions we are not using. Mentally translate the scenario into a version that makes sense to you.

Here's how I think through a few sample problems.

*You flip a coin 10 times. How many ways can you get at least 7 heads?*

First off, the total number of possibilities is 2^10 = 1024. Intuitively, I see each flip as a decision along a different dimension, not the same number line. This means we have 2 * 2 * 2 *... possibilities, not 2 + 2+ 2 + ... possibilities.

Geometrically, this would be a 10-dimensional "choice space", or, written out:

`(Heads OR tails) AND (Heads OR tails) AND (Heads OR tails) AND ...`

Ok. Now, how can we get at least 7 heads? That means we had 0 tails [10 heads], 1 tails [9 heads], 2 tails [8 heads], or 3 tails [7 heads].

Switching to the written description, this becomes:

`choices we want = (0 tails OR 1 tail OR 2 tails OR 3 tails)`

Given our 10 flips, the number of outcomes are:

- 0 tails = 1 choice (all heads)
- 1 tail = 10 choices (exactly one flip was tails)
- 2 tails = C(10,2) =
`10*9/(2*1) = 45 choices`

based on the combination formula - 3 tails = C(10,3) =
`10 * 9 * 8 / (3 * 2 * 1) = 720 / 6 = 120 choices`

So, the total is

`choices we want = (1 + 10 + 45 + 120) = 176`

And for kicks, the chance of seeing this happen is:

`176 / 1024 = 17.2%`

Multiplication goes beyond "repeated addition". It's a general notion of combining for which I'm still discovering interpretations. Let's not get tied into a single meaning.

Happy math.

Turning AND/OR statements into arithmetic maps nicely to Boolean logic.

If A and B are variables with the values 1 or 0, we can write:

`A AND B = A * B`

`A OR B = A + B`

In most languages, a positive number evaluates to "true", so A + B = 2 is true. Note that this OR is an "inclusive OR" that allows both values to be true. To force an exclusive OR, we could take the remainder after dividing by two:

`A XOR B = (A + B) % 2`

Most programming languages have separate operators for AND (`&&`

), OR (`||`

) and XOR (`^`

), but it's nice seeing how logic works with regular arithmetic.

Additionally, "if/then/else" statements can be converted to arithmetic.

If `y`

is a variable (1 or 0) that determines a result, instead of:

if (y) { result = ResultIfTrue; } else { result = ResultIfFalse; }

we can use the single statement:

`result = y * ResultIfTrue + (1 - y) * ResultIfFalse`

This version avoids the needs for branching, which is expensive for a CPU, and is a formula we can optimize with Calculus (used in machine learning algorithms).

]]>I've long internalized negatives as "opposite" and subtraction as "opposite of addition" so in my head, I had a notion of "opposite of opposite of addition" which simplifies down to "addition".

But that inner verbalization was still pretty abstract. After thinking of a better intuition, here was my reply:

Great question! I had to think about it for a bit. Addition and subtraction are related, but slightly different, than positive and negative numbers.

Imagine going on a walk. You're facing forward, and take 8 steps forward. This is really:

0 + 8

0 is your starting point. The "+" means "facing forward" and "8" means "8 steps in the direction you're facing". Ok.

Now, let's say we want to keep facing forward and take 6 more steps. That'd be:

8 + 6 = 14

Which gives us 14 steps from our starting point. What if we had faced backwards and took 6 steps?

8 - 6 = 2

Which means we're pretty close to our starting point, just 2 steps away. What if we had faced backwards but *walked backwards* 6 steps?

8 - (-6) = 14

Ah! The addition/subtraction tells us which way to face, and the positive/negative tells us if our steps will be forward or backward (regardless of the way we're facing).

In a sense, the addition/subtraction acts as a verb ("face forward" or "face backward"), and the positive/negative acts as an adjective ("regular steps" or "backwards steps"). Or maybe it's an adverb, modifying how we walk (walk forwardly, walk backwardly). You get the idea.

For older students, "subtracting a negative" can be seen as "cancelling a debt". If I have a debt of $30, and someone "subtracts it", I've effectively gained $30. In general, if you remove a disadvantage, you have improved your situation -- a positive.

These explanations are a bit abstract, the walking one is more fun to try directly. I actually walked around while thinking through the intuition. (If you're adventurous, you might start thinking about taking side steps, or jumping, and how that would be represented.)

Happy math.

When doing simple arithmetic, we only track the final location, not orientation. Facing backwards and walking backwards might have us looking at 0 while we advance forward. But mathematically, our endpoint is the same: 8 - (-6) = 8 + 6 = 14.

If we care about the way we're facing, we need a more complex math object (a vector) to keep track of our orientation as well as position ("14, facing forward" vs. "14, facing backward"). Perhaps we'd use a line integral, moving along a path and tracking the direction we face as we go.

A fitting analogy leads to questions about what else is possible.

]]>It's a wonderful book, a mix of philosophy and narrative storytelling. The central thesis is that an ineffable sense of Quality (capital Q) underlies art and science/technology, and by following it, we can unlock joy, creativity, and technical excellence.

- 00:00 - Book summary
- 11:52 - Conversation with Kalid starts
- 14:10 - The distinction between "classic" and "romantic" worldviews
- 15:38 - On the historical significance of the book
- 18:45 - Zen and the art of math?
- 20:24 - Is "classical"/"romantic" wired in your brain?
- 24:15 - Ghosts, knives and analysis
- 28:20 - On "Copernican" revolutions and ghostbusting
- 29:04 - Pointers and meta-knowledge
- 30:11 - The analytical "knife"
- 32:35 - Is our brain a knife by design?
- 35:42 - Our brains are "radical simplification machines"
- 37:14 - On the failure of formalisms
- 38:59 - Godel, incompleteness and paradoxes in reason itself
- 41:04 - Is math the most "true" thing we know?
- 42:05 - Gumption traps and the practical value of the book
- 45:32 - Peace of mind and digs on Japanese manufacturing?
- 47:10 - Solutions to the monkey trap
- 48:04 - Capital "Q" Quality
- 50:13 - Is the metaphysics useful?
- 51:12 - ...but it is true?
- 52:19 - Tyler Cowen on agnosticism
- 53:34 - Is quality fundamental or emergent?
- 54:18 - Scott can't say the word "philosophizing"
- 54:47 - We need to balance intuition and measurement
- 56:34 - Final thoughts

In terms of process, I read the paperback book, kept notes/highlights in the margins, transferred them a the Google doc, then summarized a few practical takeaways:

The distinction between Science and Art is a human one, the knife we pick. (“The universe can be divided into Bananas and Non-Bananas”). This may be the source of our biases and frustrations.

There exist “gumption traps” that can drain our enthusiasm (which is the source of real understanding, enjoyment, mastery).

The power of our a priori concepts (our mental models) which is what we deal with, not our sense data directly.

Writing, experience things directly. Writing about one side of one coin, the back of your thumb. It’s your experience now.

Work through things at a speed / pace consistent with your nature (walking up the mountain). To live only for some future goal is shallow.

The multi-step process of taking notes, transferring them, and summarizing them to myself was quite enjoyable, and I got much more from the text than I would have otherwise.

Overall, the book is a well-written, thoughtful meditation on merging technical and artistic quality within the human experience. This is the balance I'm striving to hit with the lessons on the site, finding ways to unify technical understanding with artistic intuition and the wonder of discovery (that Aha! moment).

It's an endless game to find a better way to explain difficult concepts, and the book reminds me to enjoy the journey for its own sake.

Join Scott's book club for updates and discussions on upcoming books. Thanks again to Scott for having me on -- I had a blast and hope to participate in many more. If you're looking for more excellent and thoughtful analyses on the learning process, check out ScottHYoung.com.

]]>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 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."