In the philosophical concept of presentism, it is posited that only the present exists. The past and future do not exist. This leads me to the thought : if everything is in the present, then does time, as the fourth dimension, cease to exist ?
Does presentism say that time does not exist?
There's a slight misconception here. Presentism is the theory that only present things exist.
This means that things in the past do not exist and things in the future do not exist. So, there is no real sense in which we can say that Winston Churchill exists and no real sense in which we can say that the 100th President exists.
This is more about what entities we are ontologically committed to than the existence of time itself.
If you want to make the claim that time does not exist then you will have to justify it by providing an alternative theory to General Relativity to explain the existence of gravity. According to GR, if you step off a cliff, you experience what seems to be the force of gravity attracting you to the Earth because you are coasting along a geodesic in spacetime which is curved in such a way that as your time coordinate increases, your spatial coordinates change in the direction of the Earth's surface.
One way to reconcile presentism with GR is to relax the claim that future and past times do not exist, but recognise that although they exist they are empty, since all of matter exists in the current present. You exist in a given region of spacetime now, and have immediately vacated it by passing on into later regions. You might find some claims to the effects that presentism, of the sort I have outlined, is incompatible with Special Relativity, but those claims are based on a naive interpretation of the meaning of simultaneity, so you can safely ignore them.
Presentism is an A-theory of time
Finally, the presentist, in virtue of being an A-theorist, must deal with the arguments against the A-theory that were mentioned above, including especially the worry that the A-theory is incompatible with special relativity
Galileo compared the periodic motion of a pendulum, or the diminishing level of water in a chamber, to the motion of some other object, such as a rolling ball on an incline plane or horizontal surface. Technically motion is the change in position of an object in space. This change of position is called displacement. The displacement of the clock is compared with the displacement of some other object. Galileo puts the object displacement in the numerator and the clock displacement in the denominator to define speed. He defines accleration as the change of speed.
By counting the periodic displacement changes in the clock we have a means to specify the elapsed time or change in time on a discrete or continuous basis. Time is the inverse of frequency of a resonant or rotating system. The resonant displacement has cycles or oscillations that map to the trigonometric functions on the unit circle. The resonance property of nature, the capacity to compare displacement, the ability to make ratios, and human memories give us the concept of elapsed time.
I would argue that resonance is a property of eternal nature and time is a human invention. We age and die. Many, but not all, natural events seem to be irreversible from our perspective. We imagine the arrow of time from birth to death or from the Big Bang to some ultimate state. We also imagine the Bang Bang Bang theory of an expanding and collapsing universe or the invisible multiverse, etc. But eternal nature exists independent of our human perceptions of time! The observer and the observed both have a property of eternity which transcends the perception of time!
@Marco Ocram, indeed targeted a fundamental part of the topic, but relativity is in fact considered as deficient (by presentism) in describing reality and so openly challenged. Besides that, it's not that presentism considers time not to exist, but that things do not exist except in the present. This view (presentism/A-theory) seems more compatible with a teleological understanding of reality in which the ontology of things is enclosed in their potentiality (in it's Aristotelian meaning), rather than existing independently in a 4d space time. So, the general idea is that the evolution of something, ex. a seed becoming a tree, is the expression of the potentiality that is actualized in the present; that potentiality always exist within the seed, in every present...
From a mathematical point of view we would consider as follows. Let us suppose that we live in two dimensional world of Flatland. Our motion and our perception would be limited to planar frame. This does not mean that a third dimension (say height) would not exist. In fact a being of three dimensional world would perceive it (think approximately human and ant). This would mean that we -as "Flatlanders"- would be at a constant height (a constant point of the third dimension), hence the third dimension would not vary for us. The non variation of the third dimension for us is the reason of non perception or the "absence" for us but not for its inexistence. If presentism supposes that there is absolutely only presence, then yes, it is necessary to accept the absence of time.
However I cannot get away from mentioning that there is a statement about God's view: God can see the world as a whole, in an extratemporal way, with past, present and future coexisting. This is an example of relative presentism, for we, human beings, seem that we are immersed in temporality.
