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Day or night: perspective taking and the cutoff under a dichotomy

Level: 101 · for anyone who has argued about whether something is "really" one thing or the other

One line: "Day or night?" sits on a number that changes smoothly, the sun's height, and a yes/no answer needs a cutoff on that number. Once the cutoff is named, every moment is decided exactly; different people name different cutoffs, and most arguments over a dichotomy are an argument over a cutoff nobody said out loud.

The passage

Chapter 1 of Todd Zakrajsek's The New Science of Learning (third edition, 2022), a guide for college students, is called "Learning from Multiple Perspectives", and it has a section on perspective taking that opens with a claim: dichotomous thinking is weak thinking. A dichotomy splits the world in two, black or white, day or night, and the people who lean on one are often very sure of themselves without seeing what it costs them (the author cites Dunning, 2011). The example is day and night. Midnight and noon are easy. But dusk? When exactly does day stop? And does it stop at the same moment for someone in Times Square, where billboards make midnight bright, and for someone in Starbuck, Washington, population 137, where a blazing sunset eases into full dark?

The passage stops at the question. Mathematics can go one step further, because the sun's position is something a program can compute. The answer it gives is neither "there is no line" nor "the line is obvious". It is: the line is exactly where you put it, and there are several standard places to put it.

A yes/no word on a smooth quantity

Section 1 of the program prints the sun's height above the horizon at Starbuck on the summer solstice, 21 June 2026. It falls from +64.5° at noon, through 0° a little before 21:00, to −20° around one in the morning. Nothing in that column says "day stops here". The quantity is continuous; the word is binary. To go from one to the other you need a cutoff c, and then

it is night ⇔ the sun is lower than c

That is a definition, so it is an "if and only if" by agreement, the kind if A then B ↗ says every definition is.

Four cutoffs, four answers

Astronomers did not pick one cutoff. They picked four, each for a job, and section 2 asks when night begins under each:

Verified output of day_or_night.py — regenerated by tools/run_examples.py, never hand-typed.

1. The quantity underneath: the sun's height, Starbuck, 21 June 2026

   local time   sun's height
     12:00       +64.51 deg
     18:00       +25.85 deg
     20:00        +6.28 deg
     20:30        +1.83 deg
     21:00        -2.36 deg
     21:30        -6.23 deg
     22:00        -9.74 deg
     22:30       -12.82 deg
     23:00       -15.42 deg
     00:00       -18.97 deg
     01:00       -20.01 deg

   It falls smoothly through zero. Nothing in the column says where
   day stops; a yes/no answer needs a cutoff, and someone has to pick it.

2. Four cutoffs, four answers to 'when does night begin?'

   night begins at        cutoff    Starbuck    Times Sq
   sunset                 -0.833       20:49       20:30
   civil dusk             -6.000       21:28       21:04
   nautical dusk         -12.000       22:22       21:47
   astronomical dusk     -18.000       23:39       22:37

   In Starbuck the four answers span 170 minutes.
   Every one of them is exact to the minute and anyone can recompute it.
   They disagree because they answer four different questions.

3. Same sky, different verdicts: Starbuck at 22:00

   The sun is at -9.74 deg. Under each cutoff:
     sunset               night    (top edge of the sun leaves the horizon)
     civil dusk           night    (too dark to read outdoors)
     nautical dusk        not yet  (sea horizon no longer visible)
     astronomical dusk    not yet  (sky as dark as it gets)

   Two people who call 22:00 'night' and 'not night yet' agree about the sky.
   They disagree about the cutoff, and the argument ends once they say it.

4. No minute is the one: the sorites

   From sunset to astronomical dusk: 170 one-minute steps.
   The largest drop in one minute:  0.138 deg
   The smallest drop in one minute: 0.054 deg

   No single minute changes the sky by even a quarter of a degree,
   so 'if it is day now, it is day a minute later' feels safe. Applied
   170 times it carries you from sunset to full dark. The premise is
   false for every cutoff: at exactly one minute it fails, and the
   cutoff is what says which minute.

5. Where a cutoff is never reached: the lowest sun on the solstice

   The sun's declination today is +23.45 deg, so at local midnight
   it sits at  -(90 - latitude - declination)  degrees.

    latitude   lowest sun   last cutoff the sun sinks past
      40.76N      -25.79    astronomical dusk
      46.52N      -20.03    astronomical dusk
      48.00N      -18.55    astronomical dusk
      50.00N      -16.55    nautical dusk
      55.00N      -11.55    civil dusk
      60.00N       -6.55    civil dusk
      66.00N       -0.55    none: the sun never sets
      67.00N       +0.45    none: the sun never sets

   North of 48.55N there is no astronomical night in late June.
   Ask 'when does night begin?' there with the astronomers' cutoff and
   the honest answer is 'it doesn't', not a time.

6. A third way out: a degree of day instead of a yes or no

   Let  day(e) = 0 below -18 deg, 1 above -0.833 deg, a straight line
   between. Then 'day and not day' has the value min(day, 1 - day).

   local time   height      day   day and not-day
     20:30      +1.83     1.00        0.00
     21:00      -2.36     0.91        0.09
     21:15      -4.33     0.80        0.20
     21:30      -6.23     0.69        0.31
     22:00      -9.74     0.48        0.48
     22:30     -12.82     0.30        0.30
     23:00     -15.42     0.15        0.15

   At noon and midnight 'day and not day' is 0, as a two-valued logic
   insists. At dusk it rises to one half. Fuzzy logic keeps both halves
   of the dichotomy and reports how much of each; a cutoff throws that
   number away to get a yes or no.

In Starbuck the four answers run from 20:49 to 23:39, almost three hours apart. Each is exact to the minute, and anyone who reruns the program gets the same times. They are not four guesses at one answer. They are the answers to four different questions: when is the sun gone, when can you no longer read a newspaper outside, when can a sailor no longer see the horizon, when is the sky as dark as it will get.

Section 3 makes the same point at one moment. At 22:00 the sun is 9.74° below the horizon. By the sunset and civil cutoffs it is night; by the nautical and astronomical ones it is not night yet. Two people who say "it's night" and "it isn't" at 22:00 in Starbuck are not disagreeing about the sky. They agree about the sky to two decimal places. They are using different cutoffs, and the argument is over as soon as each says which one.

That is the mathematical content of perspective taking, and a study technique in its own right: before deciding who is right, in a class discussion or in your own head, find the quantity underneath and the cutoff each side is using. It is also the answer to the overconfidence the passage warns about. The dichotomous thinker is sure because they never noticed there was a cutoff to choose; metacognition measures the same gap between feeling sure and being right.

The sorites: no minute is the one

The old argument called the sorites (from the Greek for heap) goes: one grain of sand is not a heap; adding one grain to something that is not a heap never makes a heap; so there are no heaps. For dusk: it is day at sunset; one minute never turns day into night; so it is never night.

Section 4 measures the premise. Between sunset and astronomical dusk there are 170 one-minute steps, and no step lowers the sun by more than 0.14°. "If it is day now, it is day a minute later" feels safe every single time. But once a cutoff is named, that sentence is simply false at exactly one minute, the one where the sun crosses c. The sorites does not show that night is an illusion. It shows that a vague word has no cutoff until someone supplies one, and that small steps can add up to a large change without any one step looking large.

Where a cutoff is never reached

The dichotomy can fail in a second way: the quantity may never get to the cutoff at all. At local midnight on the solstice the sun is at −(90° − latitude − 23.45°). Section 5 tabulates it. At Times Square (40.8° N) it reaches −25.8°, well past astronomical dusk. At Starbuck (46.5° N) it just makes it, −20.0°. North of 48.55° N, which is less than 150 km north of Starbuck, it never gets to −18°, so on the astronomers' cutoff night does not begin at all in late June; at 55° N not even nautical dusk comes, and near the Arctic Circle the sun does not set. Asked "when does night begin?", the honest answer there is "it doesn't", which a dichotomous thinker who knows day and night can't give.

A third way out: degrees instead of yes or no

A cutoff throws information away. Section 6 keeps it: let day(e) be 1 when the sun is above −0.833°, 0 when it is below −18°, and a straight line in between. Then "day and not day" can be scored as min(day, 1 − day). At noon and at 01:00 it is 0, as two-valued logic insists. At dusk it climbs to one half: it is as much day as night. This is fuzzy logic (Zadeh, 1965), and it is what the passage's "nearly everything is open to interpretation" looks like as arithmetic. It is not vaguer than the cutoffs. It is a different, equally precise, choice: a number in [0, 1] instead of a bit.

What the program cannot do

The passage's Times Square point is not about the sun. Billboards make the street bright; they do not make the sun higher. "Is it dark?" there is a question about a different quantity, the light at street level, and a program cannot compute a street's billboards without making up data. So this page makes the smaller claim it can back: even for the one quantity everybody agrees on, the sun's height, "day or night" has several right answers, one per cutoff. A second quantity only adds more.

Try it

  1. Before arguing, name the number. "Is this course hard?", "is this a good grade?", "is this person tall?": say which quantity is underneath and what cutoff you are using.
  2. Find a cutoff someone else uses. A pass mark, a speed limit, a fever of 38 °C, a blood test "above 50" (see a share above a cutoff ↗). Who chose it, and for what job?
  3. Change the date. Set DAY_OF_YEAR to 355, the winter solstice, and rerun. Which cutoffs move, and why does no latitude in the table miss astronomical night now?
  4. The cards.

Flashcards

The page as a deck of Anki cards: day_or_night.txt. Import with File → Import. Tags: definition, twilight, sorites, fuzzy, trap.

Po polsku, w skrócie

Pytanie „dzień czy noc?" ma dwie odpowiedzi, ale pod nim leży wielkość, która zmienia się płynnie: wysokość Słońca nad horyzontem. Żeby z liczby zrobić „tak" albo „nie", trzeba wybrać próg. Dopiero wtedy każda chwila jest rozstrzygnięta, i to dokładnie.

Astronomowie mają cztery takie progi: zachód Słońca (−0,833°), zmierzch cywilny (−6°), żeglarski (−12°) i astronomiczny (−18°). W Starbuck w stanie Waszyngton, w dniu przesilenia letniego, dają one cztery godziny początku nocy, od 20:49 do 23:39. Każda jest poprawna, bo każda odpowiada na inne pytanie. Dwie osoby, z których jedna mówi o 22:00 „już noc", a druga „jeszcze nie", nie spierają się o niebo, tylko o próg. Spór kończy się, gdy każda powie, którego używa.

Paradoks stosu (sorites) mówi: jedna minuta nie zmienia dnia w noc, więc noc nigdy nie nadchodzi. Program pokazuje, że żadna minuta nie obniża Słońca o więcej niż 0,14°, a jednak 170 takich kroków prowadzi od zachodu do pełnej ciemności. Przy ustalonym progu zdanie „minutę później wciąż jest dzień" jest fałszywe dokładnie raz. Na północ od 48,55° szerokości Słońce w czerwcu w ogóle nie schodzi poniżej −18°, więc w sensie astronomicznym noc się nie zaczyna. Trzecie wyjście to logika rozmyta: zamiast „tak/nie" liczba od 0 do 1, która o zmierzchu mówi „w połowie dzień, w połowie noc".

See also