Part 1, which begins our examination of the question 'what is time?' can be viewed here.
Part 2 Some travels through time.

This is a slightly whimsical interlude, presented as a break from the 'hard' stuff.

The first part of what follows is a speculation about a journey back through time. 
At some point in this fiction-science journey, fact begins to creep in.

"the end of our exploring,
Will be to arrive where we started,
And know the place for the first time."

TS Eliot, 'Little Gidding'

A Brief Journey Through Time

Through the ages, people find ways to know what part of the day it is. 
Nature knows, and the facts can be readily observed.  The sunflower tracks
the path of the sun.  Birds sing their dawn chorus.  Insects chirp and
fireflies flash as the last light fades.  Frogs croak.

But none of that helps when you have to pay your taxes, right?
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The Beginnings of intelligence.

Let us journey together, back through time's mists of cliche to the dawn of man.  Try to imagine what life would feel like to a hominid with some human-like qualities, but lacking the gift of speech.  Like any animal, this hominid would be heavily influenced by the natural cycles of the year, season, lunar month, and day.  Let's assume that our ancestor had more than an animal awareness, a spark of what we would recognise as intelligence, and just a little curiosity - perhaps more than a little.  As we journey through time, we will encounter a small child performing typical tasks, but with a child's free time to wonder at nature.  I can never resist a pun, so let us call her Jean.  She has an older brother, Andrew.

Jean is sitting by a small stream.  She picks a flower and casts its petals into the water, watching them drift away.  From time to time she looks upstream, seeming to wonder why the petals don't come back.

Left with just a stalk, she casts it, too, into the water.  She is puzzled, but has no words with which to manipulate, master and control that puzzlement.  The sun is going down.  A tall sapling casts its shadow nearly to the stream.  Jean idly picks up a small stone and places it at the end of the shadow.  Stones stop things moving.  Jean plucks another flower and plays the same game.  She turns to the stone.  The shadow has moved.  She places another stone, a bigger one.  She looks at the low, red sun.  She knows that darkness will bring dangers. She goes home to the cave.

The social groupings  of the cave allow for little interchange of ideas or goods.  There is no money. There is no meaningful exchange of grunts.  The exchange of blows may be observed from time to time.  The group we are observing is very fortunate.  There is plenty of game, fruits and vegetables.  The stream can be fished by hand.  Accordingly, there is much free time.  In this society, free time is the only thing of real value.  It allows the group to flourish.

The next day, after Jean, and her brother, Andrew have done the chores assigned them they are free to play.  Jean tugs at Andrew and gestures towards the stream.  At the stream, Andrew tries his luck at catching fish.  Jean plucks a flower and begins her play.  She remembers the stones and casually looks at them.  For some reason she is not at all aware of she looks at the tree, its shadow, and then the sun.  She picks up a small stone and places it on the end of the shadow.

Jean and Andrew start playing a game of tag, running in and out of the trees, jumping over the stream, laughing in a gutteral kind of fashion.  These kids sure know how to have fun.  The sun is high and hot.  Some instinct in Jean makes her aware that about half of the day is gone.  She feels within herself a sense of power.  She can do things like pinning a fleece with rocks in the stream to wash it.  Jean and Andrew are having so much fun, but they know that the day will wear on.  Jean has an idea.  They must pin down the sun-shadow.  If it doesn't move ...

Jean tugs and gestures and grunts at Andrew, making him aware of what she wants to do.  He doesn't understand at all, but loves to show off his strength.  And so, just to please his sister, he places the largest rock he can on the sun-shadow.  Jean makes signs to show her intention that the rock should keep the shadow from moving.  They go off to play again.  A long time later they return.  The sun-shadow is between the large rock and the small pebble.  Jean has just learned that, whether or not you know what time 'is', you sure as shootin' can't stop it.

Over the years, Andrew puts more stone around the tree.  He comes to know when to expect winter and summer, when this plant or that plant will bear its fruits.  Andrew becomes the tribe's shaman.  Jean gets no credit.  Even before the invention of glass, that damned ceiling is there for women.

The group, and especially Andrew, gains fame as being able to know the 'when' of nature.  They grow rich in free time.  They learn that by banging two stones together you can make sharp stones.  By banging two stones together, you can also make fire.  Life goes on ...
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A Few Words About Weights and Measures

It may well be that the precursor to money was the same thing as the precursor to math.  The pebble.  Pebbles are so incredibly fascinating that pebbleology should be a part of every civilised nation's national curriculum.  Perhaps not.

Things you can do with a few pebbles.

If you have a good supply of  pebbles, and a piece of wood with two holes in it, you have the foundation of a global standard system of weights and measures.  I kid you not.  The piece of wood can be as rough as you like.  The two holes must be nearly, but not quite, the same size.  This is called a go no-go gauge. It is based on the idea that a thing can be too big to go through the bigger hole, or too small to not go through the smaller hole.  A pebble which will not go through the smaller hole, but will go through the bigger hole is neither too small, nor too big, but 'just right'.  You can now select your pebbles to a close standard size by using the Goldilocks test.  If they all came from the same source then they should all be about the same weight.  We can test that with a beam balance.  Just place  equal numbers of pebbles in each bag or pan.  By careful selection and substitution you can find all the pebbles that are too heavy or too light.  Perhaps you never knew that Goldilocks was a scientist?  She was doing research into sleep therapy and the don't think a about a bear test.

Now that you have your pebbles, you can:

1 - Put them in a line, touching each other, against some other object so as to measure it.
2 - Put them on a scale and balance them against some other object.
3 -Put them in a container at a higher level and let them drop one at a time into a lower container with a very satisfying click-clonk-clink sound.  You can sell this one as a quack cure for soothing bad nerves.
4 - use them as tokens in barter and exchange.  You may need a lot of pebbles for this.  People will hoard your pebbles as a hedge against future needs.

Now you are all ready for business.  You can open a school of mathematics. 

Q: If W sells 3 pebbles of matchwood to Y at 2 pebbles per pebble, how many pebbles must Y pay.
You boy!  I did not say "Why pay?"  Now go and wait outside in the queue for ritual disembowelments.

Q: a chariot leaves Ur at 21 pebbles.  It arrives at the army barracks at 47 pebbles.  How long did the journey take?
Supplementary question:  If the distance between Ur and the barracks is 37,329 pebbles, what was the average speed of the chariot in pebbles per pebble?

Comments and End of Interlude:

Whatever we measure, the units are entirely arbitrary.  We are so used to using our weights and measures that we treat them as a part of the greater reality of the cosmos.  Velocity is just so many pebbles per pebble.  If  the ancients had measured acceleration, they just might have seen how crazy it is to measure events in time in terms of pebbles per pebble per pebble.  They might have done as we do today, and used a real person's name, an honorific.  They would probably have started measuring velocity in terms of something like  Bean units.
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From using clocks as a means to mark the hours, tides and seasons we have moved to the belief that a clock is a measuring tool.  It isn't.  It is, however complex, just a different version of the pebble counter.

But What is a clock?

All clocks relate changes of energy state to changes of physical state.  Hourglasses and water clocks relate fluid levels to potential energy states.  A candle uses chemical energy and fuel levels. A mechanical clock relates rotary orientations in pointers to potential energy states in springs or weights.  From another perspective, a weight-driven clock is an amplifier whose pointers show more accurately the vertical motion of a falling weight retarded by friction. Even a sundial conforms to the energy-physical state idea: the sun's shadow depends on the relative orientations of sun and earth's local surface.  Gnomon shadow angle and length can be viewed as pointers to the physical state of the earth's rotation, orientation and orbital location.  There have been many designs of clock, from the water-clock, or clepsydra to the caesium fountain clock.
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In the absence of inertia, there can be no clocks.   Absent inertia and the whole history of the cosmos unrolls in an unmeasurable instant.  That instant may be viewed as a dimensionless point.  It cannot be a point in time  since, without inertia, there is no possible way to derive the concept of time.  Imagine a world in which a mechanical clock has no inertia. Winding and unwinding would occupy the same unmeasurable instant. A cat would be a superimposition of all possible physical and evolutionary states, past, present and future, of all cats dead or alive.  It would beat Schrodinger's cat hands down in a wierdness competition.

Without inertia, we could quite literally throw a non-stop party.  The guests would all be instantly greeted, seated and sated.  The whole world could arrive, but the booze could never run out.  One bottle would be more than enough.  The booze could never run out because the guests all depart at the instant they arrive.  Also, since time doesn't exist, there is no time for them to drink before they leave. 

An absence of inertia is the direct equivalent of a superposition of all possible states and locations in the cosmos. A superposition of locations is implicit because a fundamental particle can occupy any new location instantly. A cosmos without inertia is a cosmos without time - the ultimate singularity.  That singularity cannot exist if inertia is added to the mix.

A clock, then, is a device for indicating the physical state of a system which is using or cycling energy.  Clearly, the concept of inertia is necessary to our understanding of the universe, as is the concept of energy.  Given that a change of physical state requires a change of energy state, it follows that if clocks measure anything at all, it is a flow of energy that is measured.  Any ordinary alarm clock could readily be adapted to show on its dial the amount of energy remaining in the spring.  In the case of a battery-operated clock, we could at least in principle dispense with the clock mechanism and use an electricity meter to more directly measure the chemical energy changes in the battery.

Remarks:
I cannot conceive of any experiment which could demonstrate the measurement of time without an inertial mechanism.  Thus, either we do not need the concept 'time', or we do not need the concept of 'inertia'.  Each is but a shadow cast by the other.

Continued in part 3

Edit: I will be making mention of Foucault's Pendulum in a later discussion of inertia.