Backporch

The pillars · the fabric in between

The pillars

The units everything else builds from. Each one answers a single question — what is a ___? — and together they run one arc: from what could happen to what you do about it. Chance, number, information, certainty, change, representation, action. Get the atoms right and the formulas assemble themselves.

How this is built. Eleven units — each an answer to “what is a ___?” — grouped in five acts, on one arc from what could happen to what you do about it. Then eleven deep dives that take a single thread all the way down, and the mind castles for keeping it in memory. Start anywhere; the arc is just one path through.

The five acts · chance → action

I · UncertaintyChance · Measurement · Distribution · Quantity · Information II · TruthCertainty — Binomial & Taylor as theorems III · ChangeChange (e) · Rotation (Euler)IV · RepresentationBasis · Frame · Covariance · Eigen · PCA · SVD V · ActionWhat you do about it

↓ each thread, taken all the way down

Eleven deep dives · hover a node to trace it, click to open

Hover a node to trace its threads · click to dive in. L² geometry is the hub the other threads hang from.

The units

Each pillar answers “what is a ___?”

A unit is an irreducible idea — the atom, not the molecule. They run an arc from what could happen to what you do about it; the numbers are one path, not the only one. (Next door, Paradigms is the companion: if these are the atoms, those are the frames you view them through.)

I · UncertaintyChance · Measurement · Distribution · Quantity · Information5

Chance — outcomes, events, and a measure

Everything begins with one triple, $(Ω, F, P)$ — a probability space. $Ω$ is the sample space, the set of every outcome $ω$ that could happen. $F$ is a σ-algebra of events: subsets of $Ω$ , closed under complement and countable union. An event is a yes/no question about the outcome — “did it land in this set?”

$P$ is a probability measure, $P : F \to [0, 1]$ , with $P (Ω) = 1$ and countable additivity for disjoint events:

P (i = 1 ⋃ \infty A_{i}) = i = 1 \sum \infty P (A_{i}), A_{i} \cap A_{j} = \emptyset .

The quiet point that trips everyone: probability is assigned to events — sets of outcomes — not to outcomes one at a time. On a continuous space every single outcome has probability zero; only sets carry mass.

Measurement — a random variable is a function

Here is the reveal. A random variable is neither random nor a variable. It is a function — a deterministic, measurable map from outcomes to numbers:

X : Ω \to R, {ω : X (ω) \leq x} \in F for every x \in R .

Not random: $X$ is a fixed rule. Hand it an outcome $ω$ and it returns the same number every time. The randomness lives upstream, in which $ω$ nature draws from $Ω$ ; $X$ merely reports a number about whatever was drawn.

Not a variable: a variable holds a value. $X$ holds nothing — it is the whole mapping. That measurability condition, ${ω : X (ω) \leq x} \in F$ , is exactly what makes a sentence like “ $P (X \leq x)$ ” mean anything: it must point back to an event the measure can weigh.

P (X \leq x) = P ({ω \in Ω : X (ω) \leq x}) .

Seen whole as a derivation: why correlation is a cosine — a random variable read as a vector.

Distribution — from events to a law

The “distribution” of $X$ is just the measure $P$ pushed through the function — the pushforward. For any set $B$ on the line,

P_{X} (B) = P (X^{- 1} (B)), F_{X} (x) = P (X \leq x) .

The CDF $F_{X}$ collects it; the PMF or PDF is how $P_{X}$ lays mass along $R$ . And here is why the catalog next door even works: the distribution forgets $Ω$ entirely. It remembers only what $X$ reported. Two unrelated experiments — coin flips and decaying atoms — can share one distribution because the underlying functions push their measures to the same place.

Seen whole as a derivation: Student's t from a normal and a χ² — a distribution built by pushing measures through functions.

Quantity — what a number is allowed to mean

Before any of this, a measurement has to mean something. The scale you measured on is a contract: it dictates the operations — and therefore the statistics — you're allowed to use. Get this wrong and the math is a lie before it begins (the average of ZIP codes is not a location).

Scale	Structure	Legit operations	Example · safe stats
Nominal	labels, no order	$=, \neq =$	blood type, city · mode, counts
Ordinal	ordered, unequal gaps	$<, >$	Likert, rankings · median, rank tests
Interval	equal gaps, no true zero	$+, -$	°C, calendar dates · mean, sd
Ratio	equal gaps, true zero	$+, -, \times, \div$	length, count, $ · geometric mean, CV

So an uploaded column is never “just numbers.” It lives on one of these scales — or it's qualitativeand needs a transformation before any model touches it: one-hot for nominal categories, ordinal/integer encoding when order is real, target or frequency encoding for high-cardinality labels, learned embeddings for the very high-cardinality. Skewed ratio data often wants a $lo g$ or a Box–Cox first; comparisons across columns want standardization, $z = \frac{x - μ}{σ}$ . The discipline is the same every time: classify the scale, then offer only the mathematics that scale will honestly support.

Seen whole as a derivation: what is a bucket? — choosing which differences a measurement is allowed to ignore.

Information — what is a bit?

A bit is the unit of surprise. Ask one yes/no question whose answer you genuinely don't know, with both answers equally likely, and the answer carries exactly one bit. Information is resolved uncertainty — which loops straight back to the first pillar: a bit is the answer to one optimally-posed event.

Make it precise. The surprise of an outcome of probability $p$ is $- lo g_{2} p$ bits — rarer means more surprising. The average surprise of a whole distribution is its entropy:

H (X) = - i \sum p_{i} lo g_{2} p_{i} bits .

A fair coin gives $H = 1$ bit; a two-headed coin gives $H = 0$ — no surprise, no information. Entropy is largest for the uniform distribution (maximum ignorance) and is the floor on how few bits can encode the source (Shannon). The gap between the model you coded for and the truth is the relative entropy (KL divergence) — literally the extra bits you pay for being wrong:

D_{KL} (p ∥ q) = i \sum p_{i} lo g_{2} \frac{p _{i}}{q _{i}} \geq 0.

Three threads run out of this one atom. Physics: erasing a bit costs at least $k T ln 2$ of energy (Landauer) — information is not abstract, it has a thermodynamic price. Quantum: a qubit holds a superposition until a measurement resolves it — the QBist reading, a bit as the outcome of an act. Inference: maximizing entropy subject to what you know is the most honest prior, and minimizing KL is what fitting a model actually is. The currency under all of it is the bit.

II · TruthCertainty — what a proof guarantees1

Certainty — theorems are true statements

The unit of guarantee. A theorem is not a guess or a pattern that has held so far; it is a statement proven true from axioms — and that is a different kind of object from everything above it, which described the world. Two theorems are pillars under nearly everything in this library, and each is the “simple is dense” idea made literal: one line, an ocean inside.

The Binomial theorem.

(x + y)^{n} = k = 0 \sum n (k n) x^{k} y^{n - k} .

Those coefficients $(k n)$ are the weights of the Binomial distribution. Set $x = p, y = 1 - p$ and the theorem says the probabilities sum to one: $k \sum (k n) p^{k} (1 - p)^{n - k} = (p + (1 - p))^{n} = 1$ It's also the spine of generating functions and the moment-generating function.

Taylor's theorem.

f (x) = k = 0 \sum n \frac{f ^{(k)} ( a )}{k !} (x - a)^{k} + R_{n} (x) .

Local truth: near a point, a smooth function is a polynomial, plus a remainder you can bound. It powers the delta method (how variance propagates through a transform), asymptotic expansions, and the characteristic-function proof of the Central Limit Theorem. And notice the remainder term $R_{n}$ — that's the nuance you don't drop. Keep it and the statement stays true; throw it away for a cleaner-looking line and you've started to lie.

III · ChangeChange (e) · Rotation (Euler)2

Change — the exponential, its own rate of change

Some objects are pillars because everything leans on them. The exponential is the first. Among all functions, $e^{x}$ is the one that equals its own derivative — change exactly proportional to size.

\frac{d}{d x} e^{x} = e^{x}, e^{0} = 1.

That single property — $f^{'} = f$ — pins it down uniquely, and it is why the exponential is the native language of anything whose growth or decay is proportional to how much is already there: populations, compound interest, radioactive decay, the heat equation, the Gaussian's $e^{- x^{2} /2}$ . The number $e$ itself is just compounding pushed to its continuous limit:

e = n \to \infty lim (1 + \frac{1}{n})^{n} \approx 2.71828.

It isn't a base anyone chose; it's the base that makes the calculus vanish. Differentiate or integrate $e^{x}$ and nothing changes. That invariance is the whole reason it is everywhere.

Rotation — Euler's formula, growth turned sideways

Feed the exponential an imaginary number and something astonishing happens: it stops growing and starts turning.

e^{i θ} = cos θ + i sin θ .

On the real line $e^{x}$ stretches; on the imaginary axis it rotates — one function wearing two faces. Set $θ = π$ and you get the most compressed true sentence in mathematics, five constants in one equation:

e^{iπ} + 1 = 0.

Complex numbers are not a trick for the square roots of negatives; they are the plane's natural arithmetic of rotation and scaling. A complex number is a small instruction — stretch by this much, turn by this angle. Euler's formula says the exponential is the bridge between the additive world of angles and rates and the multiplicative world of rotations and scalings — which is exactly why it runs through Fourier analysis, signal processing, and quantum mechanics.

Seen whole as derivations: what is a holomorphic function? (rotation-and-scale made into calculus, the runway to modular forms) and how a series draws a curve (Euler's rotation, stacked into shape).

IV · RepresentationRepresentation (basis) · Frame (change of basis) · Covariance · Eigen · PCA · SVD6

Representation — basis, logs, and rootsA vector is just a list of amounts — how much of each chosen direction. Logs and roots are the same move on the multiplicative line.

To name a point you need a frame. A basis is a set of independent directions that span a space; every vector is then just a list of amounts — how much of each direction.

v = c_{1} e_{1} + c_{2} e_{2} + \dots + c_{n} e_{n} .

The coordinates $c_{i}$ are not inside the vector; they live in your choice of basis. Pick different axes and the same arrow gets new numbers. Logarithms are the same move on the multiplicative line: a log expresses a number in units of how many times you multiply — turning multiplication into addition, exponentials into straight lines, and inverting the exponential exactly:

lo g_{b} (b^{x}) = x, ln x = lo g_{e} x .

Roots are fractional exponents — $b^{1/ n}$ undoes the $n$ -th power — so roots, logs, and powers are three readings of one operation in different directions. Each expresses a quantity in a chosen unit.

The frame — change of basisAlmost every powerful idea is one move: re-express the object in a basis chosen to make it simple. The reality doesn't move; the coordinates do.

Here is the pillar the others were building toward. Almost every powerful idea in analysis is one move: take an object and re-express it in a basis chosen to make it simple.

Taylor's theorem expresses a function in units of powers of $(x - a)$ . Fourier expresses it in units of pure frequencies. A change of variable re-expresses an integral in coordinates that fit its symmetry. Standardization expresses a measurement in units of its own standard deviation, $z = \frac{x - μ}{σ}$ . A logarithm expresses a number in units of multiplication. Every one is a change of basis:

f (x) = k \sum c_{k} ϕ_{k} (x) — the same f, different building blocks ϕ_{k} .

The reality doesn't move; the coordinates do. There is no privileged basis handed down by nature — only bases that are useful for a purpose. Mastery is fluency in switching: seeing a Gaussian as easy in the frequency domain, a rotation as easy in the complex exponential, a skewed variable as easy in log units.

This is where the pillars close into a perimeter — and where QBism reads as more than physics. A QBist takes a probability to be an agent's own commitment, relative to the questions that agent chooses to ask, not a property sitting out in the world. A basis is the same kind of object: a frame an agent adopts to make a problem legible, never the thing itself. The deepest move in mathematics and in inference turns out to be one move — choose the representation that turns your question into one you can answer, and stay honest that the choice was yours.

Seen whole as a derivation: Taylor's theorem — change of basis into the powers $(x - a)^{k}$ .

Covariance — a pillar, studied deeply

The template for the deeper cards to come: collapsed it's one line; expand it for the explicit formulas, the live geometry, and the same idea seen from other fields.

Eigenvectors — the directions a map only stretches

Before you can keep the top of a spectrum, you need the spectrum. Drag a vector and find the two directions a symmetric map leaves pointing where they started — the eigenvectors. This is the engine inside both cards below.

PCA — selecting the top of the covariance

The same template, one level up: covariance hands you the matrix Σ; PCA is the move of diagonalizing it and keeping only the directions that carry the most variance.

SVD — one map, two frames

Covariance was a self-map; SVD is the rectangular cousin — one matrix carrying you from a domain to a different co-domain, rotate–stretch–rotate. The picture is two panels; the bridge back to PCA is squaring the spectrum.

V · ActionAction — what you do about it1

Action — what you do about it

Change of basis was the last move about the world as given — how to see it. An action is the first about the world as chosen — what to do. It is the unit the whole arc builds toward, and the reason the studio exists: to help people act well under uncertainty.

Decision theory gives the skeleton. Hold beliefs $P (ω)$ over states and a utility $u (a, ω)$ over act-and-outcome; the right action maximizes expected value:

a^{⋆} = ar g a \in A max E_{P} [u (a, ω)] = ar g a max ω \sum P (ω) u (a, ω) .

But the deeper point is what an action is. To observe is to read $P (Y ∣ X)$ ; to act is to intervene, $P (Y ∣ do (X))$ — you reach in and set $X$ , cutting the arrows that used to point into it (causal inference). In category theory an action is a morphism — an arrow, a process — and the process-relational bet is that arrows, not objects, are what reality is made of. In QBism a measurement simply is an action an agent takes, with the Born rule the discipline for betting on what returns.

So the arc closes where it should: from what could happen to what you do about it. Every other unit — the number, the bit, the frame — is in service of choosing the next arrow well, and owning that the choice was yours.

Seen whole as a derivation: what is a process? — a map as a doing, its fibers the buckets it forgets.

Eleven deep dives — the threads from the map above, in wordsexpand ▾

The same eleven the SpineGraph maps — each takes one unit's thread all the way down, with worked notation and a deck of reconstruction cards. The graph stays scannable; the long derivations live one click away.

The geometry of L² →

One inner product, and covariance becomes a cosine: correlation, the law of cosines, whitening, Mahalanobis distance, PCA & SVD, and why the curse of dimensionality is that same geometry breaking down.

Markov chains & Monte Carlo →

Trade proof for samples: estimate π, e and γ by dice (with a live lab), why simulation can't prove √2 irrational, Markov chains and MCMC, variance reduction & quasi-Monte Carlo, and the honest state of the q- and p-adic analogs.

Change of basis & variable →

Re-express the object until the question is easy: the Jacobian law, a catalog of moves, deriving t and F from χ² by symmetry (live forge), Fourier series as a change of basis (live builder), and how you know which leap to make.

What is a number? →

Size is meaning: Ostrowski (only ℝ and the p-adics), the p-adic world where high powers of a prime are small, q-deformed integers behind Ramanujan's q-series, and continuous growth

A = P e^{r t}

with annuity formulas — each one visualized.

Generating functions & moments →

One function carries a whole distribution: the MGF and moments-by-differentiation (live), sums → products, cumulants, and the characteristic function as the Fourier transform of the density — wired to Proofs, the Library, and the Judgement Lab.

The distribution foundry →

Where the Γ's in the catalog come from — no crystal ball. Three bricks built live from raw draws: convolved waits (a simplex volume becomes

(k - 1)!

), the Gaussian in polar coordinates (Γ is the sphere-area constant), and integrated-out latent scales (the Γ-ratios of Beta and Student's t) — plus the brick they can't build alone: dependence, isolated as a copula.

Scaling & regularization →

Reshape the data or constrain the model: normalization vs standardization vs regularization from every angle (when/why/how/where/what), the ridge-vs-lasso geometry and shrinkage paths (live), and the Bayesian reading where a penalty is a prior.

Modular forms — the shape of |f(τ)| →

A modular form as a landscape:

∣ f (τ) ∣

over the upper half-plane as an interactive 3D surface and filled contour for

Δ = η^{24}

E_{4}

, and

E_{6}

— the zeros at

ρ

and

i

, the fundamental domain, and an honest account of what the picture shows and what it doesn't yet.

The Eigenbook →

One word climbed all the way up: eigenvectors become Hecke eigenforms, then the Coleman–Mazur eigencurve, then Galois representations where the eigenvalues are traces of Frobenius, then p-adic L-functions. Following Bellaïche — scaffolded, then filled rung by rung.

Affine space & varieties →

Forget the origin and the maps become affine (

x \mapsto a x + b

); polynomial zero-sets become varieties paired with ideals (Cox); complete to projective space and a cubic

y^{2} = x^{3} + a x + b

becomes an elliptic curve whose chord-and-tangent law is a group (live) — counting its points mod p hands straight to the Eigenbook.

Nonlinear dimensionality reduction & manifold learning →

Data on a curved surface: PCA can only flatten it, but Isomap bends the ruler to geodesic distance and unrolls the swiss roll, while Laplacian Eigenmaps reads the graph's spectrum (graph analytics). Every method is eigenvectors of a matrix built from the data — the spectral thread, running straight into TDA. Live: PCA vs Isomap vs Laplacian Eigenmaps on the roll.

Mind castles

Keeping it in memory

A derivation, a proof, and a memory walk are the same ordered chain seen three ways. This section is the third — how to make a chain stick once it makes sense. The other two live next door; a chain like Taylor, from the Mean Value Theorem is proved and made draggable on Proofs, then explored step by step on Derivations, and only then walked to memory here.

Provedrag it until it won't break, then read why it can't → Proofs Exploreone chain, step by step, in both views → DerivationsKeepbind it to a place and a color → here

The method of loci — memorize a derivation

A derivation is an ordered chain, and order is exactly what a memory palace (Kevin Horsley's method of loci) is built to hold. Pin each step to a spot on a familiar walk, tie it to one vivid image, and the sequence becomes a stroll you can't lose. Pick a chain from Explore and walk it to memory — the full memory walk, with a test mode, lives on Derivations.

The palette — color, image, and beauty

Loci hold order; this holds identity. Dual-coding theory says a concept bound to a vivid image and a color is recalled far better than words alone — the picture-superiority effect, with color and beauty as extra retrieval hooks. Give each unit a color, a glyph, and one striking image, place them in a room you know, then close the page and rebuild it. Click a tile to bind its image, or hit Test me to recall each unit from color and image alone.

Click a tile to bind a vivid image to its color and place.