I missed a doctor’s appointment the other day. I made a pretty typical calendar mistake. My doctor emailed me a week before the appointment asking if I could reschedule from our previously-agreed time to 10:45 AM. And I agreed—except I accidentally entered the appointment time into my calender as 11:45 AM instead of 10:45 AM.

In the end, it was just a doctor’s appointment. No big deal. But it got me thinking: is it a coincidence that I made that particular mistake? What I mean is that, to me at least, 10:45 AM and 11:45 AM don’t “feel” all that different. They are both safely after what I consider to be a normal time to be eating breakfast, but also before what I consider to normal time to be eating lunch. The whole two-hour block from 10 AM to noon feels like one big homogenous lump of morning-time. So it makes sense to me how—in the twenty or so seconds that I had to keep the appointment time in my working memory as I navigated from my email to my Google Calendar—how I could have misremembered the appointment time. (Consider the fact that there is a zero percent chance that I would have misremembered the time as 10:45 PM!)

Now, you could argue that an 11:45 AM appointment time bleeds into lunch which should make it feel texturally different from an earlier appointment time. But whether it’s logical or not, I tend to associate appointments more strongly with their starting times than their midpoints or ending times. (I also tend to eat lunch on the later side.)

What times of day are the most distinguishable? The least distinguishable? We think of daily time as a circle (represented commonly by a clock). On a clock, a second is a second, and the second hand sweeps through the same angle regardless of when that second occurs. But that wouldn’t be an accurate representation of how time flows inside our heads. Sure, topologically, our internal representation of the daily cycle is still circular. But in terms of geometry? Time moves faster or slower at different times of day. There is a big difference between, say, 8 AM and 9 AM—the transition from early morning to proper morning—but a relatively smaller difference between 2 AM and 3 AM, when most of us are fast asleep.

Would there be discontinuities? You could argue, for example, that midnight is a hard, discontinuous threshold. If you are a student with a homework assignment due at midnight, then there is a qualitative difference between 11:58 PM and 12:02 AM.

But I would argue in terms of “internal experience”, there is no discontinuity. I associate 11:58 PM with the panicked last-minute editing of my essay so that I can submit it just before the Canvas portal closes. And I associate 12:02 AM with wiping my brow, having just in the nick of time managed to get the assignment in (or the vague shame of having to email the teacher asking if pretty please, you wouldn’t believe what happened…).

What about non-monotonicity? For example, is 2 AM more similar to 3 AM than it is to 2:12 AM? If that were case, it would mean that daily time wouldn’t be a metric space (as the semantic distance between time points wouldn’t obey the triangle inequality). My instincts lean towards “no” for most people, but different people will have different maps of associations between times of day.

Consider someone working on a nuclear submarine who has to enter a passcode at the top of every hour, or the entire arsenal will be launched. For them, 2 AM might feel very similar to 3 AM and to 4 AM—all just another hourly check.