↩ ExploRational · A Catalogue of Divergent Rationality
explorational · a divergent move
“Should I take the job?” hides salary, autonomy, tempo, and the rest as one unaskable lump. A verdict is a point; the dimensions make a space — and the space holds positions no yes-or-no could express. How many axes does the choice actually have?
The move is small and it changes everything: instead of asking whether to take the job, ask along which axes the jobs differ. Pay is one. Autonomy is another. Then commute, prestige, growth, the daily tempo, the stress you carry home. Replace the lump verdict with the three-to-seven axes that genuinely vary, and the question stops being a coin toss and becomes a location — this offer is high here, low there, and the trade-off is suddenly visible instead of swallowed. A verdict is a point on a line; dimensions open a space, and a space can hold positions — equal overall, opposite in kind — that no single yes-or-no could ever express.
But the axes lie to you. Candidate dimensions often duplicate one another: “prestige”, “title”, and “how it sounds at parties” may all measure status. A dozen listed criteria can collapse to three independent dimensions, so a long list may triple-count one concern. The practical ceiling is low: past about seven independent axes, the mind can no longer hold them apart. Count the dimensions on which your options vary. The instrument below does that on a matrix you can edit and break.
the signature instrument
Score each option (a row) on each candidate axis (a column), 0–10. Everything to follow is computed live by real linear algebra — the correlation matrix among your axes, a principal-component decomposition (Jacobi eigen-solver on that matrix), the explained variance, and the effective dimensionality: the number of axes your choice really has, which is almost always far below the number you typed. Edit a cell, add a synonym, merge a duplicate; watch the count refuse to move.
Should I take the job?
6 options · 7 candidate axes
7 axes typed → 2.9 the choice really has. The rest are duplicates in disguise.
above threshold: 3 component(s) carry the variance · participation ratio 2.90
a verdict is a point, dimensions are a space
“Should I take the job?” is unanswerable because it is underspecified. It bundles a dozen concerns into one word and then demands a single bit back — yes or no — as though the concerns pointed the same way. They never do. The offer that pays best is rarely the one with the shortest commute; the prestigious post is often the punishing one. A yes/no forces you to secretly weight and collapse all of that into a point, and the collapse happens in the dark, where you cannot inspect it or argue with it.
The divergent move is to refuse the collapse for a moment and ask for the axes instead. It opens the frame by asking: not "is this good?" but "good along which dimension, and how do the options differ there?" Each axis you name is a direction the options can spread out in, and once you have three or four real ones you are no longer choosing between offers — you are reading a map. The map's gift is the position a verdict cannot hold: two jobs that score identically overall yet sit at opposite corners of the space, one all-money-no-time, the other all-time-no-money. The average hid them. The dimensions show them, and now the choice is about which corner is yours.
what to try
Press “+ add a synonym axis.” It copies your most-varying axis with a hair of jitter — a new name for a concern already counted. The typed count climbs; the effective dimensionality barely moves. Do it three times. Ten axes, still about three real ones. Thoroughness you can fake; dimensions you cannot.
Load “The job decision” and read the space. Find two options sitting far apart yet at a similar distance from the centre — near-tied on any single verdict, opposite in kind. A yes/no would call them interchangeable. The map shows they are two different lives.
Hit “⋈ merge the most-correlated pair.” The instrument collapses the tightest duplicate — prestige and title at r ≈ 0.97 — into one axis. Seven columns become six, and the effective dimensionality hardly flinches. The duplicate was never adding a dimension; it was padding the count.
how many axes are really there
Naming axes is cheap; you can generate twenty over coffee. The question that matters is how many of them are independent — how many carry information the others do not. Two axes that rise and fall together across your options are, mathematically, one axis seen twice: their correlation is near ±1, and a single number predicts both. The instrument makes this visible in the correlation grid, where a blood-red cell between "prestige" and "title" is the signature of a duplicate in disguise. Collapse them and you lose a column but no dimension, because there was only ever one there.
Principal-component analysis is the honest count. It rotates your axes into a new set of directions — the principal components — that are uncorrelated by construction, and reports how much of the options' variation each one carries. When your seven axes are secretly three concerns, three components soak up nearly all the variance and the rest fall to the floor near zero. The headline number, the effective dimensionality, summarises this in one figure: the participation ratio (Σλ)² / Σλ², which equals the true count when the components share variance evenly and shrinks toward one when a few dominate. It is a soft, honest count of how many directions your choice genuinely runs in.
And it is usually low. There is a well-worn ceiling around seven: past roughly seven truly independent axes, people cannot hold the dials apart, and the "extra" dimensions you list are almost always recombinations of the first few. So when the finder tells you a sprawling twelve-criterion spreadsheet is really three-dimensional, it is returning the decision to a size a mind can actually think in — and pointing your attention at the three axes that were doing all the work.
model
Your scores form an options × axes matrix. Each column is standardised (mean-centred, scaled to unit variance) so that a generous 0–10 axis and a stingy one count equally. From the standardised matrix the instrument computes the correlation matrix among axes — every pairwise correlation — and then finds its eigenvalues and eigenvectors with a small Jacobi rotation solver running right in your browser. The eigenvectors are the principal components; the eigenvalues λ are the variance each one explains, and because it is a correlation matrix they sum exactly to the number of axes.
From those eigenvalues everything follows. Explained variance for a component is λ / Σλ — its share of the total. The effective dimensionality is the participation ratio (Σλ)² / Σλ²; a second reading counts how many components clear an explained-variance threshold you set with the slider. The space is a genuine projection: each option's standardised scores dotted onto the first two eigenvectors, giving its coordinates on the plane that captures the most variation possible in two directions. Two identical options land on the same dot; two that trade off land far apart. Treat it as an honest instrument for one question — how many axes, and where do my options sit on them? — not an oracle for which option to pick.
the move ↔ the model
| The move | The model |
|---|---|
| a candidate axis | a way the options might differ — a column of scores you hope is a real, separate concern. |
| the correlation matrix | which axes secretly move together; a near-±1 cell is two names for one thing. |
| a principal component | a real underlying dimension the finder recovers — a direction your options actually spread along. |
| explained variance | how much each recovered dimension matters; how much of the options' variation it accounts for. |
| effective dimensionality | the true number of axes — the participation ratio, usually far below the number you typed. |
| correlated axes | duplicates in disguise: the padding that inflates the count without adding a dimension. |
how this opening fails
risk · duplicates in disguise
The catalogue's named risk for this move, and the seductive one. A long criteria list feels rigorous, but rigour is an illusion when the columns are one concern under several labels — "prestige," "title," and "how it sounds at parties" triple-count a single worry and outvote the axes that stand alone. The finder exposes this: the effective dimensionality does not rise when you add a synonym, and the correlation grid names the culprits outright. If two axes can never separate two options, they are the same axis, and your thorough-looking spreadsheet is padded.
risk · variance is not value
This instrument counts axes of variance among the options you happened to list — not axes of value. The dimension that matters most to you might be one your current options simply don't differ on: if every offer pays about the same, "pay" carries almost no variance and PCA all but discards it, though it may be the thing you care about most. Low variance is not low importance. So the finder tells you how many directions your option set runs in, not which direction you should weight — and the fix is to add an option that stresses the axis you fear is being ignored, then look again.
Turn the lump verdict into its axes, and the choice becomes a space you can move in.
can you use it?
RECOGNITION — Which situation calls for this move? A: two job offers keep coming out tied on a twelve-criterion spreadsheet. B: you suspect your only hypothesis has unexamined rivals. C: you have named your criteria and want to decide which deserves the most weight.
A. A tied total is the lump verdict hiding a trade-off; the axes show where the offers actually differ. C is the near-miss — weighting is a value judgment, and the finder counts axes of variance, not of value.
THE NEAREST NEIGHBOR — What separates this move from writing a long criteria list?
Independence. A list counts names; this move counts axes that carry information the others do not — two criteria that rise and fall together across your options are one dimension seen twice.
PRODUCTION — Three apartments; your criteria are rent, price per square foot, monthly cost, light, and commute. How many dimensions does the choice really have? Run it before looking.
About three: rent, price per square foot, and monthly cost move together — one cost axis — leaving cost, light, commute. Yours works if at least one pair of criteria collapsed, and each survivor can still separate two options.