Time has been central in Mathematics, from its inception, however time itself, has remained mysteries.
Early civilization began to study the phenomenon, in order to handle situations such as agriculture, religion, etc.
All religions give primacy to time, via astrology, creation myths, and definitions of eternity.
Philosophers have tried to make progress, arguing that time is a fundamental property, that it is an illusion, or a property of mind, and not reality.
We've grown increasingly skilled at measuring time more and more precisely, without actually understanding what it is we're measuring.
Quantum Mechanics and Relativity theories, have made obvious how peculiar the nature of time, leaving nothing but paradoxes left in some cases.
Opposite of this, the entirety of Calculus has built upon an assumed understanding of time, where an objects position at time t, is given by f(t), and it's velocity is dtdx, the derivative of f(t) with respect to time, and the acceleration is the second derivative.
This forces us to treat time as continuous, with any given interval of time being divisible, whereas QM implies that time is quantized (or quantizable), creating conflict between the time of mathematics and physical theory (which relies on mathematics to get it's point across).
One of, if not the earliest drivers for the development of Mathematics, was time, as the passage of time governed the important cycles of nature, such as the passing of the seasons, harvesting times, and the flooding of the nile, for the Egyptians.
The process of creating temporal measurements was tricky, dividing the year into months, and the months into days.
The Sumerians, divided the day into 12 periods, and each of these periods, into 30 parts.
It was the Babylonians, however, that gave us our 24/60/60 measurement of the days, breaking them down into hours, minutes and seconds.
Early devices for measuring time, were largely based around the movement/position of the sun, such as the well known sundial, and lesser known gnomon, however, the path of the sun is not uniform or consistent (at least, not via knowledge of the movement of celestial bodies at the time), and there were other, somewhat ad-hoc devices for measuring time at night, such as clepsydras, or water clocks, or the hourglass.
The measurement of time possessed great religious significance, being tied to astronomical events that dictated religious observations, and crop cycles (which ~dictate the survival of a civilization).
More sophisticated representations of time were put forth by Pythagoras and the Buddha, both of which argued that time was cyclical in nature, whereas Judaism and Christianity placed time as a result of the act of creation, where the creator exists outside of time.
Rustin Cole also felt that time, was a flat circle.
Early contributions to our understanding of time:
Zeno of Elea presented a number of paradoxes that made very plain some of the counterintuitive aspects of time that we still struggle with, such as the arrow paradox, which states that an object in motion, in any duration-less instant, is neither moving away, or towards anything, but since these duration-less instants compose the time an object spends in motion, there is no motion.
The base of this argument is that if an object moved in an indivisible unit of time, then that unit of time therefore would be divisible.
Aristotle disagreed, stating that "...time is not composed of indivisible 'nows'...".
Plato felt that time began when the creator fashioned the world, and that time is a "moving image of eternity", as the creator couldn't bestow eternality to a creation in it's fullness.
Aristotle however, rejected the idea of created time, and chose to relate time to motion: in the measurements of time via water, celestial bodies, or hourglasses, motion was a key property of the process, so for Aristotle, time was motion, that could be enumerated.
St. Augustine, ported a lot of Platonic beliefs to Christianity, agreeing that time started with creation, but still not feeling as if he deeply understood the phenomenon: "What then is time? If no one asks of me, I know; if I wish to explain to him who asks, I know not."