Aristotle's famous and fateful definition of time reads:
"Time is the number of movement with regard to before and after."
(ὀ χρόνος ἀριθμὸς κινήσεως κατὰ τὸ πρότερον καὶ ὕστερον ἐστιν. Phys. IV xi 219b2; cf. also De Caelo I ix. 279a15)
This hermeneutic cast of time remained in force throughout Western history without question until Heidegger's thinking put it into question in 1927 with the publication of Being and Time. His treatise shows that this "arithmetic" clock-time is a "vulgar" derivative of an originary time which provides the answer to the question concerning the "meaning of being" (die Frage nach dem Sinn des Seins). This gives us something to chew on philosophically that has so far been neglected, not only in analytic philosophy, not only throughout modern science, but even in Heideggerian scholarship. I won't go into the details of this neglect here (cf. however A Question of Time).
Instead, here is a simple observation that apparently has not been made throughout the two-and-a-half millennia separating us from Aristotle: Aristotle's definition is viciously circular. The simplest things are hardest to see. To define time as the number of movement/change with regard to "before and after" presupposes that "before and after" — which themselves are temporal determinations — are already understood. The tradition has invariably focused on the number of movement, that is, on the clock, no matter whether this clock is taken to be a sun dial measuring the sun's regular return to the same longitude each day or the frequency of electronic transitions in a caesium 133 atom. The temporal "before and after" are tacitly presupposed.
As any philosopher worth his or her salt knows, philosophy's job is to unearth the tacit presuppositions on which any (philosophical) thinking is implicitly based, thus making them explicit, visible. This is how philosophy progresses, namely, by moving backwards into the tacit presuppositions. With Heidegger, philosophical thinking has moved backwards into the most elementary phenomena possible: being and time — phenomena that are taken for granted by everyone and every philosophy and every science, thus in their questionability invariably overlooked.
Without always already (a priori) understanding before and after, and the now in between them, we would not be human beings. Aristotle also famously said, and modern science agrees, that only that which is now exists all. All that is before was and is no longer. All that is after is not yet. For modern science what is 'now' are the sensuous data it collects with its detection instruments of all kinds and of all degrees of sophistication, right through to the Large Hadron Collider in CERN and the planned LISA laser interferometer in outer space for detecting the long-sought gravitational waves.
Without presupposing before and after, it would be senseless, for instance, to talk of a Big Bang, of an expanding universe, or any other competing variant thereof. After all, ever since Aristotle, physics has been the science of ta\ kinou/mena , i.e. of that which can move/change. Mathematized physics has long since laid claim to the title of the fundamental science on which all other sciences are based, from chemistry, biochemistry right through to neuroscience and the scientific quest for cracking the phenomenon of so-called consciousness, which seems — misguidedly — to be holy grail for today's science.
One of the two most promising approaches to the long-sought-for theories of quantum gravity, along with string theory, is loop quantum gravity associated with the names Carlo Rovelli, Lee Smolin and Abhay Ashtekar . In his most enlightening 2004 book, Quantum Gravity Rovelli commendably provides many philosophical side-reflections, including on meanings of time. Apart from the first meaning, the "time of natural language" with its "existence of memory and expectations", all the other meanings of time listed up to the penultimate one — including "time-with-a present, ... thermodynamical.... Newtonian ...special relativistic ... cosmological... proper... clock... parameter time" (p.60) — are one-dimensionally linear.
With the identification of 4D space-time with the gravitational field in loop quantum gravity, the former vanishes and along with it parametric time with respect to which all motion in the electromagnetic and gravitational fields is infinitesimally or discretely differentiated — in favour of a covariance in changed measurements in measuring instruments, including clocks. Time becomes just another measurement read off an experimental apparatus and thus surreptitiously remains counted clock-time.
Rovelli calls this final meaning "no time", "the idea about time underlying every theory in which there is no fundamental notion of time at all" (ibid.). What this "underlying" "idea about time" is or could be Rovelli does not say. In truth, he merely hides the traditional conception of counted clock-time in the mathematics.
Since Rovelli does not tell us anything about a deeper conception of time, let's provide the answer, which can come from a serious engagement with and critique of Heidegger's thinking. This "fundamental notion of time" is hidden in the "before and after" in Aristotle's definition, which he nowhere explicates Before and after name two dimensions of ecstatic (Latin meaning literally out-standing) time to which, along with the now of the third dimension, the present, into which human existence stands out. Only through belonging to this three-dimensionally stretched time are we humans human beings who understand the world with our minds. Time and mind are identical in a hermeneutic cast as time-mind. 'Was' and 'will be' no longer name dimensions in which entities do not exist, but in which they presence and 'are' in two distinctive modes of absence. Absence is itself a mode of presence in which entities that purportedly 'are not' exist, that is. are — for the human mind.
This 3D time is genuinely three-dimensional, that is, not one-dimensionally linear and thus confined to a notion of succession. Hence events occur to the human mind with maximum degrees of freedom in which the three temporal dimensions remain continually and 'simultaneously' open to mental leaps. Being mental and thus non-physical, such leaps are not subject to the relativistic absolute limit of the speed of light.
Moreover, the mind is not on the 'subjective' side of time-mind, and time is not on the 'objective' side. If time were solely objective, it would be 'in itself', an sich, separate from mind like a Kantian Ding an sich. If time were merely subjective, as it is in Kant's transcendental ego, it would be merely 'for us', für uns. In truth, time-mind is an-und-für-sich, neither inside (in so-called consciousness) nor outside (in the so-called external world), but pre-spatially no-where in an identity of time and mind, of Zeit und Geist, in a way consonant with Hegel's identity of subject-object in the Idea, which is likewise an-und-für-sich.
Since all modern, mathematized physics, including the most advanced theories of quantum gravity, is duty-bound to make predictions, thus calculating motion, it must remain unquestionably committed to a notion of linear, successive, 1D time, even when it deals in quantum-mechanical "transition probabilities". Hence the three-dimensionality of time-mind must be truncated for the sake of mastering movement calculatingly. Modern science remains in denial regarding its deep-seated tunnel vision. Why? Because its essence (Wesensbestimmung) is the will to calculating, effective power over movement and change of every conceivable kind.
Hence it comes as no surprise that all the modern sciences, starting with physics, and through to the social sciences of economics, psychology, sociology, etc. are all positivist, i.e. based on empirically observable 'facts' to the exclusion of any purported 'transcendent' dimension. Aided and abetted by analytic philosophy, scientific thinking firmly keeps the lid on the ontological difference and denounces any attempt to re-open it. However, modern science and analytic philosophy come too late in closing the lid, because all human being implicitly understands three-dimensional time by virtue of always already understanding 'before' and 'after' and 'now'. None of these dimensions is a being, but is to be found by descending into the ontological difference between being and beings first discovered by Greek philosophy, and then even further into this 3D-time itself AS the clearing that enables beings of all kinds to presence and absence AS what and who they are for our hermeneutically imbued human minds. This explicit uncovering of the open clearing of 3D-time is the task of thinking for our age. Its fourth dimension resides in human being's standing-out (ex-sisting) into this temporal clearing and therefore can be called mind. Mind and time belong intimately together.
On the other side of the divide, so-called Continental philosophy is impotent in the face of the positivist onslaught because — with the rare exceptions of those few who to date have taken Heidegger's world-shattering message of Being AND Time seriously — it is just as clueless about the ontological difference as are science and analytic philosophy. Continental philosophy has a penchant for merely literary gestures and an aversion against mathematics and mathematized science, and also against conceptual thinking. Thus it is merely an opposed rhetorical position, appealing to those with similar prejudiced proclivities, without providing true insight into the originary time underlying the ontological difference. It is this insight into the temporal clearing that poses the genuine historical challenge to positivist science and its allied, subservient analytic philosophy.
Further reading: A Question of Time and 'The End of Science and the Beginning of Wisdom'.