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Everyone knows what time it is. Most coordination in our everyday lives relies on it. Whether in our deadlines, schedules, or even cooking timers, we can take for granted that we share a common sense of time.

In this series, I want to investigate how we've come to agree on time and connect it to my summer research with Category Labs.

Wikipedia defines the clock as "a device that measures and displays time." As we will see, time itself is defined by clocks, so this definition is rather circular. So we will go the other way and start with the clock.

We will find more use in the physics definition of a clock (from The Feynman Lectures on Physics, Volume I, Chapter 5-2):

What really matters anyway is not how we define time, but how we measure it. One way of measuring time is to utilize something which happens over and over again in a regular fashion—something which is periodic.

Let's start with the day as something periodic. Count the days from one noon to the next. We then have the clock as a sequence of ticks (the passing of days), and a count of them.

$$C = (t_0, t_1, t_2, ...)$$

Note that the clock produces only the count and not regularity in the increments. The clock makes no claim that the duration between $t_0$ and $t_1$ equals the duration between $t_1$ and $t_2$.

This is not a shortcoming of the formulation. This is the situation. The clock on its own cannot establish that its ticks are at regular intervals, because there is nothing to be regular against. The Feynman lecture gives an example of a man flipping an hourglass as soon as the last grain of sand goes from the top to the bottom. He can then count how many flips occur in each day. However, how can he rule out that the hourglass is slow at night and fast during the day? Another passage from the lecture:

All we can say is that we find that a regularity of one kind fits together with a regularity of another kind.

Regularity is not a property of a clock. It is the relation between two. Take $C': \mathbb{N} \to \mathbb{N}$ to be the function of hourglass flips with respect to the days, where the flip count starts with the day count, i.e., $C'(0) = 0$. Plotting the hourglass flips against the days gives:

Linearity indicates regularity between the two, where for every $d$ days we have $h$ hourglass flips. The dotted curve represents an irregular hourglass that varies greatly in its number of daily flips. The solid line represents our common notion of the day, where for every day, there are $24$ flips of the hourglass. The dashed line represents another possibility for an hourglass $C''$: for every day, there are $27$ flips.

In this scenario, we cannot distinguish whether $C'$ is slow or $C''$ is fast. We must agree on which one to treat as correct, and calibrate the other to it. This is how time becomes standardized. Nothing in the comparison forces the choice. Further, we could fix the day and say it is $24$ flips, or fix the hourglass and let the day be the one that varies. Both are consistent with everything we have observed.

The next post will briefly go through the history of standardizing the second.