index 0118fe8..f059e88 100644
@@ -15,11 +15,11 @@ but most real-world ordering isn't total. it's **partial**. consider prioritizin
- "finish essay" is more important than "do laundry"
- but is "study for exam" more important than "finish essay"? depends on deadlines, weight, your energy level
-you can't always compare. some things are just... incomparable. and that's fine — partial orders are a legitimate mathematical structure, not a failure of decision-making. partial orders are sets with a specific relation — which makes them a [[structural/set-theory-as-thinking|set theory]] concept at heart.
+you can't always compare. some things are just... incomparable. and that's fine — partial orders are a legitimate mathematical structure, not a failure of decision-making. partial orders are sets with a specific relation — which makes them a [[set-theory-as-thinking|set theory]] concept at heart.
## the hidden math in sorting
-sorting algorithms are one of the most studied topics in computer science (see [computer science](/wiki/stem/computer-science)). but the key mathematical insight is about *comparisons*: to sort n items, you need at least n log(n) comparisons. this is a proven lower bound — no cleverness can beat it.
+sorting algorithms are one of the most studied topics in computer science (see [[computer-science|computer science]]). but the key mathematical insight is about *comparisons*: to sort n items, you need at least n log(n) comparisons. this is a proven lower bound — no cleverness can beat it.
this has a practical consequence: if you're trying to rank 100 job applicants by doing pairwise comparisons, you need at minimum about 665 comparisons. ranking is inherently expensive. that's why we use heuristics, filters, and thresholds instead of trying to create a perfect total ordering of everything.
@@ -41,9 +41,9 @@ comparison requires a notion of "how much better." this leads to **metrics** —
these seem obvious, but many real-world "distances" violate them. travel time isn't symmetric (one-way streets). perceived similarity isn't symmetric (people say "north korea is like china" more than "china is like north korea"). whenever you're comparing things, it's worth asking: does my comparison method actually behave like a real metric?
-metrics are deeply connected to [[structural/topology-as-thinking|topology]] — every metric defines a topology, and the topological properties (connectedness, compactness) determine what kinds of ordering and comparison are possible in a space.
+metrics are deeply connected to [[topology-as-thinking|topology]] — every metric defines a topology, and the topological properties (connectedness, compactness) determine what kinds of ordering and comparison are possible in a space.
-this connects to [linear algebra as thinking](/wiki/structural/linear-algebra-as-thinking) — when you embed things as vectors, the distance between them (cosine similarity, euclidean distance) gives you a metric that enables meaningful comparison in high-dimensional spaces.
+this connects to [[linear-algebra-as-thinking|linear algebra as thinking]] — when you embed things as vectors, the distance between them (cosine similarity, euclidean distance) gives you a metric that enables meaningful comparison in high-dimensional spaces.
## ordering in practice
@@ -51,7 +51,7 @@ this connects to [linear algebra as thinking](/wiki/structural/linear-algebra-as
chess ratings, competitive gaming, and matchmaking systems all use elo or elo-like systems. the core idea: after each match, update both players' ratings based on the *expected* vs *actual* outcome. if a 2000-rated player beats a 1500-rated player, not much changes. if the 1500 beats the 2000, big update.
-elo is mathematically elegant because it converts a partial order (we only observe some matchups) into a total order (a single number per player). it's also a beautiful example of [bayesian updating](/wiki/immediate/probability-in-daily-life) — each game is new evidence that shifts our estimate.
+elo is mathematically elegant because it converts a partial order (we only observe some matchups) into a total order (a single number per player). it's also a beautiful example of [[probability-in-daily-life|bayesian updating]] — each game is new evidence that shifts our estimate.
### pareto optimality
@@ -61,4 +61,4 @@ this is the mathematical way of saying "it depends on what you value." ordering
## the deep point
-ordering feels like common sense, not math. but the math of orders reveals why ranking is so hard: transitivity failures, incomparable options, multiple dimensions, and the fundamental impossibility results of social choice theory. understanding this doesn't make decisions easier, but it does make you stop expecting a "correct" ranking where none exists. and comparison always rests on [[immediate/counting-and-measurement|measurement]] — what you can order depends on what you can measure.
\ No newline at end of file
+ordering feels like common sense, not math. but the math of orders reveals why ranking is so hard: transitivity failures, incomparable options, multiple dimensions, and the fundamental impossibility results of social choice theory. understanding this doesn't make decisions easier, but it does make you stop expecting a "correct" ranking where none exists. and comparison always rests on [[counting-and-measurement|measurement]] — what you can order depends on what you can measure.
\ No newline at end of file