Just because the heavens don't tick doesn't mean Earthly stuff won't tick as well.

People are going to count their heart beats, breaths and bowel movements.

Humanity may have used the sky as its sole time piece for millenia, but once we started to pay people by the hour we got creative. The sundial was the last time piece to depend on astronomy. The chinese would count time by seeing how long standardized ropes would take to burn. On this side of the world people would use hourglasses and water clocks. Sand and water take a fixed amount of time to clear a reservoir of a fixed size with a hole at the bottom.

Your people may not have the concept of a day, but they will still have some measure which they can use to make the life of office workers miserable. "How dare you say that the new feature for our app isn't ready yet, Rob? You told us it would take three turns of the hourglass during last scrum meeting!"

"I'll be back before you can walk to the lake."

If not an exact sense or measurement for time, maybe they'd have an approximate one based on distance.

Just as we say "an hours' drive" or used to say "about a weeks' journey," though obviously people drive, cart, and walk at different speeds, a civilization may generally agree on a common-sense "average."

So imagine they reverse distance and time to say things like, "She'll return in about the lake's distance" (meaning the time it would take to walk back from the lake), or "It would take a round-trip to Symmett to prepare such a huge feast!"

If eventually a more precise measure of time is needed, some other standards might be the speed of sound (echoes) or the time it takes for something to fall from a certain height, and then candles, hourglasses, waterclocks, etc. can be developed from them. I do think any advancing civilization will need to specify time (i.e. I'm not just saying it out of human-centrism) because for two people to meet at a place, you need both spatial and temporal coordinates. If they wish to communicate and commerce efficiently, they'll have to carry some agreed upon measure of time.

First off, the term that you are looking for is "mensurable" or "quantifiable". "Quantized" means the same thing in mathematics, signal processing, physics, music, computer science and everywhere else I've heard it used.
It means that some property can only have values that are integer multiples of some minimum unit. (I'm not guessing here - I have degrees in mathematics and physics).

Now to the answer.

At some point, I would expect that the utility of measuring the passage of time would occur to someone on the world you describe. Someone will notice that breathing rates, heart rate, or some other chemical or biological process occurs at different rates for different people or objects. It's trivial to think of an economic (or selfish) reason to measure these differences. With such a motive, people will try to figure out how to measure time. And there are plenty of ways to do it without astronomical or seasonal references. Someone might experiment with an hourglass, a water clock, candle clock, and eventually, a pendulum clock.

It will doubtless be harder to measure long time periods accurately, but eventually, someone will discover that you can get an extremely accurate clock by observing the oscillations of something that they can measure (we do this with cesium in an atomic clock).

So, as long as your people can make tools, I think measuring the passage of time will be inevitable, just because it is so useful (and interesting).

Let's take the situation you posit regarding time and consider an analogous situation regarding distance:

Imagine a world in which there is no clear, objective standard for measuring distance. Every tree is a different height and every branch, a different length. No two pairs of rocks are the same distance from each other. People are all different heights, with no universal length of their limbs or digits. Even something as simple as the length of a single person's hair could be radically different from one moment to the next.

In a world where distances are so chaotic, would people even form the concept of quantifying distance, much less find an agreed standard for doing so?

Of course, we don't have to imagine that world. We live in it. And, therefore, we know that, yes, quantified distance is a universally-accepted concept, which has been standardized multiple times, in multiple ways. ("That's the great thing about standards - there are so many to choose from!")

Not only that, but we also know that, despite the variability in the sizes of body parts, they formed the basis of some of the earliest widespread distance-measurement systems. To this very day, one of the major units of those systems even shares the name of the body part it was based on - the foot.

"But," I can imagine you saying, "my foot is the same size from one day, or month, or year to the next. It's utterly consistent!"

Yes, that's true, but how about another widely-used "archaic" measure of distance: What good treasure map doesn't measure distance in "paces", the distance covered in a single step? How consistent is that distance? I know that the length of my pace varies from day to day, or even moment to moment, based on my mood, my energy level, the terrain, and several other factors. And, of course, one person's pace can vary greatly from another's, due to those same factors, plus differences in height, flexibility, fitness level...

The pace is far from a consistent measurement, and yet it was (and is!) still widely used to measure distance when more accurate measures are not readily available.

So, yes, given that humanity has quantified distance at least since the beginning of history (and I suspect well into pre-history, probably even before we became "human" in the modern sense) despite the lack of universal repeating distances to use as an objective base distance measurement, it is exceedingly likely that any similarly-sapient species would quantify time, even without having universal repeating intervals to use as an objective base time measurement.

Moreover, the utter necessity of having some way to specify both times and locations if one wishes to coordinate the activities of multiple entities makes me quite confident that it would be essentially impossible to develop anything remotely approximating civilization or technology without having the ability to specify both of them. You can't say "meet me at this place at this time so we can hunt together/research how to make fire/bring in the harvest/mate/launch our exploratory expedition/whatever" unless you have the concepts of both "at this place" and "at this time", as well as a means to express those concepts.

Biological rhythms might be irregular, but chemistry remains the same.

Chemical reactions aren't based on solar cycles, and they're the same everywhere in the universe. As soon as your species starts doing chemistry they're going to need to measure time, and also will have a means to do so.

And what's the granddaddy of all chemical reactions? Fire. As soon as one of your creatures tries to search a cave using a torch or cook an egg, they'll start to figure out this "time" thing.

Even in a tidally locked world has a changing sky.

The world you describe has a constant sun and no moon, but it still has stars. And those stars will change as the planet orbits through the year. When your creatures venture onto the dark side of your world, they will discover the yearly cycle that can't be measured by the non-existant seasons.

What they will start to notice is ratios of the duration of many different events or processes in the world around them. If they learn these ratios they are much more likely to survive.

For one pregnancy, the fruits grow back on the local trees about twice.
The grains are harvested about four times for every three times the fruit grows back.
Water has to be brought to the grain fields between 18 and 23 times for every grain harvest. The chickens lay three to five eggs between each watering of the grains. Traders can make a trip to the neighboring village about twice between grain harvests. These sorts of ratios are going to remain pretty constant over time and would show them that each of these processes each take about the same amount of time every time. By comparing many events and processes they will notice that they take a roughly consistent amount of time. Their uncertainty will be in the range of days.

At the beginning the people are not going to be too bothered by accuracy. What they are going to want is to not waste time doing the wrong kind of work. Accuracy will come once they have to coordinate with a lot of people. Each region will come up with definitions for duration of each event. Once they have to coordinate over long distances will they start to formalize the them with greater accuracy. The equivalents of minutes and seconds are not going to show up until they need to measure something with that sort of accuracy.

Yes.

Time is an inextricable part of the way the world works.

1. Motion is a change in position over time. How do you catch prey? You have to go faster than them. You have to be there before they are.




2. Everything has a resonant frequency. Go outside for a walk. Notice how you swing your arms back and forth? Try swinging them faster. It's hard, huh? You generally have to swing them in phase with the movement of your legs. You can go twice as fast, or 4 times if you really try hard, but good luck holding a rhythm that goes out of sync with your walking pace.




3. Pendulums always swing at regular intervals. On your tidally locked planet things will definitely grow to face the sun. And then things will grow tall, so that their sunlight isn't blocked by stuff that falls on them. An arms race for height will cause there to be plenty of tall things. You have a high-up place, and something to swing from it: bam, pendulum!




4. Things fall at a constant acceleration. See previous bullet for explanation for things in high places. There will be things that fall. These things will always take the same time to fall. If your sentient species discovers acceleration, they will need time to comprehend it.




5. Heat takes time to be conducted. These life forms most likely live in the twilight zone of their planet: the unmoving ring in between night and day that could achieve viable conditions. They definitely know about heat gradients, and the difference between hot and cold. If they go to a warm area, and then back to the cold area, they will notice that it takes time to cool down.

The lowest common denominator of the above phenomena is time. If this species develops any scientific knowledge at all, they will have to come up with a numerical system for comprehending time.

Now, there's a difference between time, the circular clock face, and time, in seconds. There's no need for a cyclical method of measuring time if you don't live in a day/night cycle. But you need a unit of measurement for time to advance beyond a certain point. Humans invented cyclical time before we thought of time as something independent of perception. Time was dusk, dawn, day, and night. Later, we invented the sundial, and realized that time ticks on at the same speed, regardless of how much fun you're having. No species could progress without this.

I am going to assume that your creatures have a social system. I'm also going to call them humans sense that's easier to say.

There are a lot of different reasons why social systems form, but sense your people are capable of language I am going to assume that it was for trade and to cooperate in hunting, both of which have TIGHT connections to time.

How much do you pay the Sheppard for watching your livestock? By the hour.
How long do you cook bread for? How long do you wait before reaping your fields? When someone says they'll send a shipment to you, how long do you wait before you tell them they're late and in breach of contract?

If your people don't care about answering these questions, they're not intelligent enough to form an understanding of time even in a world with cycles.

The comments by AlexP, which you dismissed as “argue, irrespective of the validity of the argument”, are actually spot on: quantification/measurement does not require quantisation. Let me elaborate.

In physics, making a measurement does not mean counting the repetitions of a base unit. A measurement generally does not give an exact integer and claims that this is the correct value; instead it gives uncertainty bounds and claims that the true value lies somewhere between.

For instance, when you measure a length with a ruler, you do not count the ticks from zero until the one that exactly aligns with the end of the object. That actually never happens: if you look closely enough, the ticks will always be a bit off from the edge.

What you do instead is, you look for the tick that is closest to the edge, say $157:mathrm{mm}$, and then announce that the length is $(157pm1):mathrm{mm}$. Likewise, if you measure a time span, you may indeed go about it by counting the beats of an oscillating clock until the end, but that will never give you the exact time you meant to measure but only an approximation thereof. That approximation is indeed quantised, but that doesn't mean your physical model itself is quantised – it's just an implementation detail.

In many cases, even the expected-value part of the result of the measurement will not be an integer of any base unit at all. For instance, to measure an electrical capacitance $C$, one option is to charge the capacitor up to a given voltage (say, $1:mathrm{V}$), then discharge it through a resistor of known value (say, $1:Omega$) and measure to what voltage it has dropped after a given time (say, $1:mathrm{s}$). The physical model of how a capacitor works suggests that the time-dependent voltage is
$$
U(t) = U_0 cdot e^{-tfrac{R}{C}},
$$
thus
$$
t_mathrm{e}cdotfrac{R}{C} = -log left(frac{U(t_mathrm{e})}{U_0}right)
$$
$$
C = frac{t_mathrm{e}cdot R}{log(tfrac{U_0}{U(t_mathrm{e})})}
$$
Here, even if the individual measurements all give integer multiples of some small unit-resistance/time/voltage, then the result for the capacitance will involve the logarithm of such an integer, and that is in general an irrational number which can therefore not be expressed as integer multiple of any given unit.

Of course the result will still have a measurement uncertainty, meaning there will be such an integer that is consistent with the measured value. But that does by no means imply that this quantisation is conceptually the right interpretation of the capacitance – quite the opposite: most of physics is completely based on the assumption that the underlying principles are continuous rather than quantised.

Even in the absence of periodic natural astronomical phenomena (years, seasons, days etc), biological events would presumably still happen over particular time-scales. Life-time; gestation time; the time before dying of thirst or starvation; the time to learn to speak etc. So for sure it would still be useful to measure and quantify time.

As to how time would be measured: even primitive time-keeping devices on earth (hour glasses, pendulums etc) would work and so would sophisticated ones (quartz clocks, atomic clocks etc)! What's missing is something very basic for measuring long time scales, like counting the days and nights, or years.

So for a really primitive society, perhaps they would could count the number
of cycles of a biological event (rather than an astronomical one, like the years or days). E.g., count the menstrual cycles, or count the generations of a simple creature that lived for a reasonable amount of time. E.g, count the generations of mice that passed between two events.

Yes

Music.

If you can think, you can create your own rhythms, your own paradise of periodicity in an inconstant world.

Sound doesn't exist?

Dance.

Nothing ever moves?

You should be able to communicate somehow. Time is still required to express ideas, to consume them. Eventually you'll think... man, it really takes John a long... time? just to tell me he ate an apple.

Sapients can be very impatient, and I'm sure they'll take a good crack at inventing some kind of clock, just to complain about exactly how long you made them wait.
(To be pedantic, I assume you mean sapient - cats are sentient, but I doubt they care a lick about time in general).

If you're just a brain in a jar, with no discernible inputs or outputs... well. There's this thing in my head asking me to let it kick it like it's 1986, for some reason. I'd put good odds on some kind of internal rhythm to develop eventually, even in a brain in a jar.

Would they create concept of time, or especially effective way of measuring it a bit later? Possible. Would it be so shocking for them? No. Reason? Biology. We can't escape it.

• heart beat (escape that)


• circadian rhythm


• menstrual period


• pregnancy length (escape that, also for all major species)


• reproduction seasons of other species (in case of many species there is clear edge in doing it in one big go to overwhelm predators or find mate, example cicadas that perfectly aim their mating season every 13-17 years, yes prime number to make predators live harder)


• time necessary for their crops to mature from seedlings

EDIT: Clarification
How would animals manage? Not sure. Solar (spot) cycles? Star location on twilight zone for migratory animals? There is NO natural cycle taking 13 or 17 years, but because of evolutionary pressure animals managed to get it right. I would consider as highly unlikely if no animal managed anything like that.

Plants do NOT have to be synchronised, even on Earth. However, there are plenty of short lived plants, that have a few months cycle from seed to crops that are harvested. It does not have to be related to any special cycles like seasons. On Earth, in very good climate rice can be harvested 3 times per year.

If they are civilised, they are likely to have agriculture. If they have agriculture, then yield they should have some staple crops, with a few specially productive dominating. Sooner or later they start some big scale monocroping. Whatever is the time it takes from them to produce edible seeds, that period is roughly constant for any specific breed.

EDIT2:

"There would be no need for migration" Escaping from some weather? No. Going for a mating season? Exactly.

Evolved or were created by some deity or high tech aliens? If evolved then getting some biological cycles / processes in roughly fixed amount of time is highly beneficial and such adaptation should evolve independently multiple times. If you say that actually your sentient specie didn't get it - fine, blind spots or damaged vitamin-C coding gene happen. If you say that nothing in your environment got it - then honestly that world indeed needs some new kind of physics or mathematics, as at least normal probability seems to be no longer working.

Oh... by occasion... the specie is curious enough to develop physics but not curious enough to venture on the dark side of the planet and observe how cyclically visible stars change? ;)

OK, I think I understand your question. You want to know why would these people want to measure time; and how would they figure out that it even CAN be measured? Surprising how many people miss the point. Anyway.

When answering this question, I look at the real world today. Today, we have figured out how to measure a myriad of things we didn't even know about a few hundred years ago. Pretty much all of electrical engineering falls under that category, as well as quantum mechanics, etc. So why did we do it? Why and how did we figure out that "here is a thing and here is a way we can measure this thing"?

Well, because we like to understand, predict and control the world around us. And we're not even done with that yet. There's still awfully many things that we don't understand, can't predict and can't control yet.

But why do we want to do this? Well, to have better lives, of course. There are plenty of things out there that threaten us, can harm us and can kill us - natural phenomena (winds, floods, droughts, lightning, earthquakes; etc); diseases and traumas; wild animals; other humans; and probably more that I haven't thought of.

And we also have wants and desires. We want an easy life. A happy life. A secure life. We like to socialize and gain social status. We like to indulge in physical and mental pleasures. Etc.

All in all there are many, many things that we want - but can't readily get. And this is where all the desire to control and predict comes from. And so we start to puzzle out how the world works.

And from this starting point on you can take a look at our own history. It starts simple, of course, with the transition from hunter-gathers to farmers. We learned to control the plants and animals around us, to get a stable source of food. But the yields were low at first, and the fruits small, so we started to figure out how to get larger yields and better fruits. Thus we discovered the method of selective breeding (both for plants and animals). And we also learned how to make better shelters from wood and later stone. How to craft tools. Tame the fire. Make the wheel. Every step along the way solved some immediate problem and made the life just a little bit better for everyone involved.

Eventually we arrived at the scientific method and realized the value of simply trying to better understand our world without a specific goal in mind. Scientists today work on problems both immediately practical, and purely theoretical, because history has shown that even the most obscure bit of theory might find an unexpected use in the future.

Lasers are one famous example of that - their inventors had no idea what they could do with them. It was a pretty cool invention, but without any immediately visible use. And look at them today. They're practically a cornerstone of modern engineering.

So, what would be the turning point for your civilization? At what point would they become interested in the passage of time? I think this is a pretty open-ended question. There are a lot of things that could have triggered it, and you as the author of the story have a pretty free hand in choosing the one that suits you best.

Certainly modern technology is not possible without precise tracking of time. Chemistry also involves it a lot - as soon as your alchemists got serious, they would need to keep track of time.

But even before that, I think human societies would benefit much from even a rough measure of time. OK, so maybe you don't need to get 3 square meals every 24 hours, but you there is some upper limit of how long you can go without food. Once you want a stable source of food, to make sure that you won't need to go hungry (or thirsty), you need to plan ahead. Remember - in the early days there were no refrigerators, so food had to be eaten pretty soon after acquiring it. (That or cure it, but even cured foods don't last forever) And that already involves some rough measure of time. It probably won't be quantified, but they'll come up with something.

As societies grow they will inevitably clash over some resources or whatever, and war will arise. That too depends a lot on time. The questions of "can we get our forces to point X before the enemy does?" and "Will we last here long enough until reinforcements arrive?" become pretty important.

Coordination, of course, will be important even before that. Unless your sentient beings lead solitary lives (in which case their sentience will not get very far), they will need to cooperate. We have only come as far as we have because from the earliest days on we have realized that a group of people can do things that no single person can do alone. But to do big projects, you'll need to coordinate time, even if just roughly. You need to know when all the people need to come together to do something. And if it's a large project that takes a lot of time, you'll also need to coordinate rest schedules.

Remember - even though the lives of your people might not have a natural rhythm to them, it doesn't mean that they aren't filled with things to do. Everyone has a hundred tasks they need to do soon. They can't just go to the project site and idle around waiting for others to show up. That's wasted time, and even if your people won't precisely be able to measure how much they've wasted, they'll certainly understand that they could have been doing better things instead of just sitting on their hands.

So, all in all, I think it's more than likely that your civilization will develop some sort of rough way of tracking time early on. And a rough way is completely enough for a long time, by the way. Precise time tracking in our world is actually a surprisingly recent innovation. Clocks in homes have been around for maybe a little more than a hundred years. Before that people did use quite a rough scale of time (date + morning/midday/evening or some such). So that's the kind of accuracy your people would be satisfied with.

In our world precise time tracking first became necessary because of ship navigation. However that also depends on the position of stars and the rotation of the planet, which, in your case, doesn't happen. So what could be the trigger in your world? Well, I don't really know. I'd like to suggest advances in chemistry - those themselves being spurned on by the desire to cure diseases and create riches. But you could choose another route. I would suggest looking up history of clocks and time keeping. See what was the next thing that made good use of precise clocks after ship captains.

If this sentient species had no experience that the passage of time was measurable and periodic, would they ever develop a method to quantize it?

I guess this fits as a frame challenge. If your aliens never experience the passage of time as measurable then no, they would never quantize it because they couldn't experience it.

BUT, everything we know of exhibits the flow of time and all species seem to 'experience' time. Mountains build and erode. Stars begin and end. The universe changes. Animals migrate due to some evolutionary seasonal 'sensor' that tells them Winter or Spring is coming.

This is all due to time flowing.

Aside: Time is not periodic. As far as we know, the arrow of time is quite unidirectional - especially at macroscopic scales (we haven't seen a star reverse its aging.)

Further trying to define things, according to Wikipedia and its associated sources, "Sentience is the capacity to feel, perceive or experience subjectively."

Animals obviously are sentient according to that definition and to my knowledge at least, they have not found a way to quantize time. I doubt the animals have any actual 'thoughts' about it becoming dark in some discrete period of time.

That being said, I'll go forward assuming you mean beings such as humans who possess such character traits as "creativity, intelligence, sapience, self-awareness, and intentionality" (again from Wikipedia) as well as sentience.

So, here are some examples a 'conscious' being might eventually notice where time could be measurable.

Time elapses between meals and hunger appears and disappears as time goes by. Perhaps hunger is random - but your being would still experience the passage of time and sense it happens again and again.

Another example, is moving from one place to another - say from one village to another. Although the sun never moves in the sky it still takes some average amount of time to walk there, or ride a horse there, or sail a boat there. If we were to communicate to another that they should take extra food for this journey we have acknowledged it takes a period of time to get there. And perhaps that's their first measure of time - hunger periods. It takes 4 hunger periods to walk to the next village.

Of course the experience of time is readily apparent as we grow older and our bodies change. Most animals on this planet have a period before being able to reproduce and a period after which reproduction is possible. Maybe they begin to define an epoch of life according to these three phases.

As others have said, harvest periods are quite appropriate for defining and measuring an epoch of time. Plants grow at a rate proportional to incident energy (and other factors but in this particular world the weather would seem to be fairly regular.) This gives a rather predictable time period for the harvest of particular plant species. Perhaps that is their first time measurement. Harvest periods for 'corn' could be a 'year'. Harvest periods for 'rice' could be a 'month'.

Animals also have fairly regular lifetimes the aliens could use to 'measure' time. How long between when a grasshopper is born and dies while in captivity for example.

In summary, it seems inevitable that your aliens would eventually define and measure time in whatever epochs they discovered and needed.

The root of your question is really this: without a solar cycle, would an observer within that system have any way to develop an objective sense of time?

Given that, of course they would, though it might take longer. And therefore yes, for the objective observer, time would indeed be quantized as you use the term here.

An observer (or group of observers) would first note natural periodicities that trend to constants - the steady flow of water into a vessel, a sound heard moments after seeing the clap in the distance in calm air, the time it takes objects little-affected by air resistance to fall from a height.

You mention the cycles of plant life and described it as essentially random on this planet, but that's different than "things drop dead at random points in their lifecycles". In fact, plants would live some average amount of time before being ripe or otherwise best to enjoy or use... as there would be no seasons and presumably little weather whatever process that maintains good growing conditions would be mostly constant and acyclic, so if anything their lifecycles would be more clockwork in nature than what we experience here on Earth. Once agriculture begins it would be far easier to see that a crop planted now can be harvested in, say, three hundred buckets "time".

A heartbeat would still be a heartbeat, and this too would perhaps be less cyclic (but would such a creature shelter from its sun for slumber?)

The dark side of this world would be cold and probably incompatible with plant life... but perhaps explorers there would witness the stars and return with tales of their slow, steady paths across the heavens (though none of what I've described requires that).

So despite not having a diurnal cycle there would be a multitude of other cycles by which to suss out the foundations for observation, just as we ourselves have, and possibly more quickly thanks to fewer confounding factors. As for observers not trusting their shared perceptions of time, I think while that's possible in pre-agricultural times, once you are, to wit, on a cycle of farming and other shared occupations requiring an investment of energy metered out over time, you learn about consensus. After all, at some point in the distant past we finally agreed on our observations enough to call them truths... though some were falsified with more evidence, others were reinforced, and so here we are.

Very interesting question. It's hard to even think about an answer without involuntarily relying on our own concept of time, but I'll try.

TL;DR: Time derived quantities (like speed or efficiency) are important in the world. A measure of time would be created to accurately describe them.

I think an accurate notion of time would not emerge immediately. It would be a process that starts off very inaccurately but gradually improves over time.

Initially, people would start noticing differences in magnitude between certain events.

• It takes multiple breaths to walk a mile.


• You can walk a mile multiple times after eating before getting hungry again.


• You have to eat more than once before crops become fully grown.


• People can generally outlive multiple pets.

Even though none of these things have a single definitive value, people would notice there is some scale to these things. Let's call this the Thyme scale (named after Justin Thyme the Second who made the general public aware of this).

The examples I gave were obviously of different magnitudes, but as soon as people are aware there is such a thing as the Thyme scale they would start comparing/ordering things according to it that are of more similar magnitude.

On a practical level, Thyme derived quantities would probably be discovered before discovering Thyme itself. Speed or efficiency for example:

• Suppose something happens in Town A for which they require help from a blacksmith. There are 2 competing blacksmiths living in Town B (both at approximately the same distance from Town A). A couple of messengers leave at once from Town A to each fetch a blacksmith, but they each move on their own pace.
  
  Messenger 1 brings back his blacksmith first, who then fixes the problem.
  
  Messenger 2 is slower and thus brings back his blacksmith only after the problem was already fixed by the first one.
  
  Being faster is important for business. Being slow costs money.




• Suppose Isaac and John start working together (at the same time) and suppose that Isaac finishes his work while John has only done half. This means Isaac worked more efficiently than John by a factor of 2. More efficiently means that he is able to do more work in the same Thyme distance, or, that he manages to do the same amount of work in a smaller Thyme distance.
  
  Employers prefer Isaac because he gives them the competitive advantage.

People would quickly find out that most of these properties are inversely related to the Thyme scale. In order to more accurately describe these they would create units of Thyme.

Since there is no periodicity to be found in the world, these units would probably initially be based on derived quantities. E.g.:

• The distance between A and B divided by Hermes's speed between A to B.


• 100 divided by the efficiency of Isaac to pick 100 apples.

There would be efforts to normalise these different units. In time they would come to a standardised unit, e.g.:

• 1 Thyme = how long it takes for 100 grams of sand to fall through an hourglass with an aperture of 1 mm.


• 1 Thyme = some constant divided by the frequency of the microwave spectral line emitted by atoms of the metallic element cesium.

With such a unit, people would start building devices to keep track of universal Thyme. Those clocks would not be cyclical but work more like the internal representation of time in our own computer systems, indicating Thyme "since Epoch". A random event would be chosen and agreed upon to represent the start of Thyme-counting. Thyme would then be counted as the number of units away from that instant.

The only reason for clocks to be cyclical in your world may be the infeasibility of creating a device able to indicate an infinite amount of Thyme. Assuming your civilisations start off without advanced technology, this infeasibility "might" introduce a notion of cyclicality anyway. (An hourglass needs to be turned every X Thyme, starting a new cycle)

Thyme would not be thought of as multiples of 12 or 60, but rather follow the most used numerical base.

Practical "day to day" usage would lead to the same thing because people are generally interested in things that happen in the close Thyme vicinity of the present. We wouldn't say it is 1553420000 seconds since Epoch. The first digits change so rarely that they would not be mentioned every time, out of practicality. Instead, your people would speak about "when the next kiloThyme marker is passed". It may not be related to a cycle in nature, but it is practical.

In the purest form of "no cycles" and therefore no measure of passage of time in a repeatable way I don't find your question interesting. I think it's asking to depart too much from my context of the universe to reject an understanding of time entirely. Sorry if I can't go with you on that journey, but I don't believe that life would exist in such a world. Nothing about reality as I know it works in your cycle-less timeless sandbox. So I have a hard time connecting with the idea at all. I feel compelled to try to imagine, instead, a radically different scale of cycles, from immeasurably long to infinitesimally short.

So if you'll indulge me in a slightly off-topic journey, here we go!

Separately from the idea of being tidally locked, to try to get a cycle-less existence, I think it might be more interesting to consider a "rogue planet" ejected from its solar system and galaxy, travelling at extreme speed throughout space due to a close encounter with a super-massive black hole or something like that. No cycles, no orbits -- always above escape velocity of anything it encounters. The planet is on a one way trip, never to repeat its journey. It would endure millions of years of icy coldness, interrupted by periods of excitement occurring as it passes a star or a black hole, the duration of which could be a day or could be thousands of years. Life would be irrevocably changed on this planet, but sadly it would be thrust into a low energy frozen state unless it brought its own energy with it from within. But I can tell you that these chaotic one-off astronomical close encounter events are not significant sources of regular change that life needs to evolve. So that won't work because I believe...

Life needs cycles.

The cycle of day/night might be the most important way the Earth has established the entropy conditions needed for evolution. Each time a part of the earth turns to face the sun, it is warmed, and then it cools as the Earth turns away. The tides come in and out. The tilt of the earth provides a yearly cycle of temperature conditions that involve the formation and destruction of massive amounts of ice. These are powerful mechanical forces. Water expands when it freezes and jackhammers rock apart then thaws to fill the cracks and repeat the process. Evaporation and condensation produce powerful water cycles that carve out valleys and move liquid water to high altitudes. Heating and cooling of the atmosphere drives powerful wind forces that shape the earth. I'm no expert in tectonics and magnetic dynamos, but I'd confidently presume that the cyclic forces at play there cause other powerful acts of change such as continents to drift and volcanoes to erupt that play a major role in the changes the Earth undergoes.

Without these powerful forces at play, the perfectly motionless planet essentially reaches a high entropy state very quickly. The planet is hot on the sun side and cool on the dark side with a steady state of a calm temperature gradient reached very quickly. There may be some excitement in regions near transition states (the "sunset ring" around the planet where temperature is moderate near the freezing point of water) but it would be a chaotic subtle twitching of change, not the powerful cycles that drive real change of any significant magnitude. No powerful cycles mean no powerful reactions. Nothing major changes. Cycles of local entropy drive change and evolution.

So what would be the cycle that would drive evolutionary change on your imagined world? Are you supposing that there is none? I predict it would would be a very dead and dull world on a physical level.

To liven things up but still stay true to your "tidally locked" supposition, perhaps we could suppose that it is instead the mechanics within the the atoms themselves that are now the most powerful cycles on the planet to drive change. Evolutionary clocks now cycle at rates of billions of cycles per second with the rhythms of the vibrations of fundamental particles, but with forces too small to move anything heavier than electrons. Perhaps what this planet has going for it is something electrical where it is the electricity or magnetic energy within the planet that has evolved into something sentient. Perhaps the world is highly conductive where electrons move freely and it has evolved a world of electromagnetic beings. Imagine a solid state motionless world of electrical thought energy! I could imagine the world is alive and thriving on extremely small scales, but nearly devoid of chemistry and physical mechanics at the scales we know them - all the action is entirely in the electromagnetic realm.

Anyway, I hope you didn't mind me exploring this from a few different angles. I think in the strictest sense of answering your direct question though, it doesn't seem possible to me that life as we know it would evolve without the celestial motions we are familiar with, and I think the cycles and rhythm of time is too fundamental to dispose of entirely, or else life would have to take some very radically different and unfamiliar form. And from a standpoint of fiction, I would feel very disconnected from a world without measurable time.

Since they are born and die they know that time is, however they call it. They have difficulty to standardize it, since everything seems unrepeatable, unique. But maybe they don't, because it us who needs things to repeat themselves in order to begin to think that something is going on. Maybe the absence of similarity between the things is a better basis for time perception and a cycling is our, rather cumbersome, from their points of views way to understand time. Maybe on a planet where there is no way to figure out what is time using the Earth's consideration there are its own methods to do this and the hour and minute hands of their watches are chaotically chasing each other in an magnetic field of the planet which is perfectly fine with them. But I think time can be understand based solely on gravity and they have it.

Even on Earth, time-keeping is a lower priority in some cultures than others. Extrapolating from there, it’s easy to see how the natives of your planet might have a very loose sense of time indeed, or none at all, at least when we consider their innate mental furniture. Time and its measurement might be a concept only really understood by intellectuals. But I do believe that there will be plenty of opportunities and incentives for the intellectuals to develop that concept. Many aspects of technology will make it inevitable that sooner or later, somebody notices the advantages of being able to measure and standardize time:

• Working out the amount of resources needed for a task. The local candle maker will tend to have moulds of a fixed size and use a certain consistency of wax. Customers will inevitably ask themselves: do we have enough candles to complete that underground task we have to do? Similar concerns might apply to planning the amount of oxygen needed for diving or mountaineering. Or even the amount of food you need to pack for a large group of beings traveling a fixed distance (estimated based on some sort of metabolic average).




• Regulating speed. Do they drive? Sooner or later they will notice that when a vehicle is driven more noisily, and with more wind rushing through one’s scalp tentacles, then more accidents happen and they tend to be more serious. Somebody will start measuring speed, and others will start drafting legislation.




• Do they use radioactive material? Is the sample safe to handle yet? How much less powerful will it be by the (what did you call it? thyme?) the construction crew have finished building the reactor?

And on and on.

One way: They have quantized distances already. Then they use distances to quantize time. They will build a very high tower and drop stones from them and use the duration of the fall to quantize time. Of cours to use this they need to have discovered that the weight of things they drop doesn't matter as long as the things are dense enough, exactly like Galileo with his Pisa Tower experiment: they dropped two different things and see that they crash exactly at the same time (note that the story is a bit more complicated, please read the Wikipedia for details, a thought experiment is involved).

What matters in my opinion are two things:

First, they see that the drop times are always the same and are able to abstract from this experience and conclude that they can use this to quantize time. They say their "second" is when they drop things from a ten "meter" high structure.

Secondly and more vaguely, their physics derivation process is different: they derive time from distance, this is backwards from what Galileo might have done. Galileo knows about time and uses this knowledge in his thought experiment of dropping things from a tower. The sentient beings however are in the process of learning how to quantize time and therefore need to use things they know well, like distance and later the law of the free fall to quantize time. To make a good story I would invent a very successful experiment showing the law of the free fall without using time.

EDIT

It seems that I didn't sufficienlty address the question why the sentient beings would want to quantify time at all.

This is subtle. And we tend to forget that we ourselves had a completely different attitude before a scientific breakthrough. The law of the free fall says that we only need to accelerate at the start and decelerate before the destination. This is not intuitive for people because of the damned friction. If we abstract away friction we just speed up a short time then we can coast for a long time and finally break just before we arrive. To understand this is a breakthrough. I suggest that quantization of time could be similarly difficult for the sentient beings. They might however understand the law of free fall a lot better than people at the start of the Renaissance and use this understanding to quantize time. Exactly like we people used our understanding of time to arrive at the law of free fall. That's what I meant that the physics derivation process is backwards.

And now for the spark which initiates the search for the insight. What exactly did make Galileo investigating the law of free fall? I don't know. Perhaps you should try to look for a story about this spark of Galileo as an inspiration to tell the story you want to tell.

I'd say that they notice that things change around them. If they don't notice even this then what would they need a time measurement for? They know what is time. It is what happens when a thing changes. How to measure it? Take a million inhabitants, measure the lengths of their noses and divide by the million. You are going to get one standard nose length. Accidentally notice that various insects spend more or less of something, crawling from its base to its tip. If you don't know what insect is crawling the nose now what would be your best guess, how much of something most of the insects take to cross the nose? Ask this a million of inhabitants what they think. Let them sing an interval, add all of them together and divide by a million. Transfer the interval orally from generation to generation, paying high salary to the keepers. Keep them in the unknown locations.

The plants and animals will adopt cycles depending on other species and the signals they get from them. For example one plant will grow when another's blossom is over, etc. To some extend, this happens on earth, too. This would eventually stabilize in some equilibrium that is only evolving slowly (evolution, too slow for any single being to notice).

Humans would then measure time by those fluctuations, i.e. "the season of the cherry blossom" and so on.

Speech

If they are intelligent and communicate, they will have some sort of language. Unless they communicate telepathically, of course. Speaking takes time if you need to articulate, and if you communicate with others you will have a common basis, some kind of language with words that are equal for any speaker.

Prayers/Songs

This is a follow up to "Speech", and it might not be religious prayers but part of rituals. All the more if there is no outside way of "counting", if they have developed a communication system and a mathematical system to count, and they are using fire for cooking, then counting as the basic way to know how long something needs to cook is the start. And from there, it could be something like "3 XY prayers" for cooking an egg, for example. (Unless cooking an egg is a random thing and they would never know when the egg is done. In such a world, cooking would be a mess and probably noone would bother with it.)

Someone else mentioned music. I would just call it songs. There is no need to bother with rythm or anything. If they like to talk and use their voices, they will also sing, especially if there are also animals that sing (like our birds). If they sing together and they have good memories, then whenever they sing separately the same song, they will more or less take the same time from start to finish.

You seem to be asking for some naturally-occurring phenomenon in which certain events repeat with a reasonably steady period, so as to naturally suggest the idea of something repeating with a reasonably steady period - but with no ties to astronomical phenomena.

Consider: Dripping water. Water does tend to collect in pockets at heights above ground, and sometimes these pockets are steadily replenished so as to overflow. When the amount of replenishment is sufficiently small, the overflow is in the form of repeated drops. With the fall being of fixed height, and replenishment steady, the amount of time between drops will be constant (to within the limits of unaided perception). It wouldn't take much for someone to notice this.

[Is this plausible? Look up the definition of 'clepsydra']

If they were a hierarchical species, it's likely that time would be measured relative to the length of a founder-equivalent reign. This is how time was traditionally marked in monarchical societies. I'd guess that the most likely to evolve subunit would be either a base related to digits/appendage count (base ten for humans, base 8 for an arachnid species) or based on subunits of 60 - which is divisible by 1,2,3,4,5 & 6 and a natural subunit.

Cell/Life cycle can be used a clock.

In your planet, I think zones of temperature would be in a shape of circles and rings with no seasonal variations. This is an ideal condition for life cycle of cells to be at a constant rate - for both plants that depend on light and animals that eat always-available plants. So cell cycle (i.e. time needed for mitosis) can be used as a clock. Similar logic can be used with time needed for one species visible to naked eye and with a much shorter life cycle than the sentient one to develop from newborn (or 1-cell embryo) to mature organism capable of making new embryos. Biological clock is probably not as good as something based in chemistry or physics, but it could be a nice start.

Life span varies by considerably among species. Even if your planet has a very long period of rotation, I don't see why a sentient species that evolve in that planet cannot have life span that spans MANY rotations.

The stars in the sky can still be used to measure discrete periods of time even if nobody ever lives long enough to witness a full cycle. Since all the stars would be moving at a constant speed across the sky, you could simply set up two parallel poles, and measure how long it takes any given star to cross the gap between those two poles when viewed from a fixed point. This would be a constant for every star that crosses this gap. You could then use this to calibrate sand or water timers. To ensure consistency in multiple locations, you could manufacture constant-width measuring devices to enforce a standard. This is similar to how a sextant or astrolabe works.

As far as how many cycles of "star-crossing-gap" constitutes a full "day", that would be fairly arbitrary, but if the race that evolves on this world requires a semi-regular sleep cycle, you could simply take the average of a large population and define it that way for consistency, as indeed you would have to do if you wanted anything like an organized society. Everything else is just multiplication. A week or a month or a year don't need to be anything special, just a fixed number of "days" based on whatever mathematics or holy numbers or whatnot you choose.

Just want to add another perspective by using computers and our understanding of space-time as an analogy.

First, let me state that computers do not recognize time by itself. It requires an external clock that is subsequently mapped to human-understandable time (1 time step -> 0.1 ns, for example).
But barring that external clock to quantize (discretize) the executions of computers instructions (note that in computers we artificially synchronize executions based on the clock, otherwise, it will "just happen"), the amount of "time" (from we as external observer that already understand time) required to accomplish tasks (or a sequence of instructions) are not quantifiable.

So for the computers without external clock, time would be something inherent to their fabric of existence.

Another perspective before getting to the final point is how we understand the concept of fabric of space-time. (image below from the book "We Have No Idea")

We can't really measure the expansion of our very fabric of space-time directly, since any measure we use would also be expanding, unless we can find something that doesn't expand with the fabric of space-time.

Now, back to the computer analogy, I guess it won't be able to see the concept of time, unless there are things in the universe that stays constant with respect to time (something that does not "expand" with the chaos of time in your universe).

However, we now do learn about the speed of light and the fabric of space-time (but much much later, definitely not the first thing that we would come up with). So I believe there would be some cases that caused your creatures to try to understand the concept of time, with much difficulty, just as now we try to understand the concept of space-time.

A final remark would be to note that this might have some similarity if we assume speed of light is much lower than the current speed, that we can move very close to the speed of light. That can cause things to happen differently for different people (think about the different frame of reference in our current universe that might cause ordering of events to be different for different observers). So it might be hard for the creatures in your universe to understand that there are absolute ordering (passage of time) of events. So you might want to start from there.