Is there any philosophical theory that postulates that time can only arise out of a three or plus dimensional space? I think I heard someone say that time is the fourth dimension implying that time can only exist in a three dimensional or fourth dimensional space. Is there any theory that claims this and what does it say specifically about the other conditions necessary for time to exist?
Nicolas Oresme, a 14th C French theologian held that time could be modelled as a line going from an observation of Aristotle and also of Campanus of Novarra, a 13th century commentor on Euclid. From the former, he understood that time and motion were continuums and continuously related and from the latter, he understood that anything that is understood as a continuum could be imagined as a line, surface or a solid. Thus is his Questions on the Geometry of Euclid, he says:
Aristotle, who in the fourth [book] of the Physics imagines time by means of a line
Now if we are to imagine space and time together as both Newton and Einstein did, and if we also imagine them as continuums as Campanus suggested we could do, then the minimum dimension for spacetime is 2d: a single dimension of time and a single dimension of space. For as Aristotle said, time and space are continuums that are continously related and not identified. Since the time can be imagined as a line, it is 1d. Since space must exist for anything to change, and it is a continuum, and the minimal dimension of a continuum is 1d, we derive the result above: the dimension of spacetime is minimally 2d.
This is implicit in Oresme, but of course he didn't make this explicit as he would have granted the dimensionality of space as that given by empirical experience and would have held it as neccessarily so, that being set as 3d.
It should go without saying that simply that because time can be 'imagined' as a line, that it should not be identified with a line. This is a common mistake, and even more so after Einstein imagined spacetime as a 4d continuum. In QM, the nature of time is shown to be other than this and its probably because of the two very different conceptions of time in quantum and gravitational theiry that had made cobstructing a unified theory of both so difficult to attain.
According to Kant our mind organizes the data from the senses using a priori categories like time and (three dimensional) space. Time and space are not outside in the universe but in our own minds. This strange idea was the only way that Kant could find to overcome the criticisms of Hume to an empirical notion of space and time.
Today we expect science to have the last word on everything related to knowledge. Maybe we are asking too much of science. If we look at Newton, for instance, he assumes the concepts of time and space as given, without examining them. Einstein's General Relativity relates space and time, but we can't say that he really defines or examines them. Allan Connes Non Commutative Geometry constructs mathematical models for Quantum Mechanics where space is something different from the usual thing. Our notion of three dimensional space is not very useful at the subatomic level. Long story.
There are some problems with Kant's vision of time and space but I can't find any new framework that allows us to ignore him.
1.) I do not know about philosophical ideas which claim: Time needs at least 3 spatial dimensions in order to exist.
In the opposite direction, there exist speculations that time is 2-dimensional. For a first orientation in the internet see https://en.wikipedia.org/wiki/Multiple_time_dimensions
2.) From Special and also General Relativity we learn that time and space are united as spacetime. Spacetime has 4 dimension (3+1). The 4-dimensional spacetime is a physical object: Gravitational waves are waves of spacetime - not in(!) spacetime. Hence spacetime and a posteriori also its time component depend on the physical objects in the neighbourhood. E.g., masses like the sun, neutron stars or black holes change spacetime, and this effect has been observed.
One can split spacetime into 3-dimensional space and 1-dimensional time. But this splitting has a certain arbitrariness and can be done in different ways by different observers. Hence the spltting has no independent meaning.
3.) Physicists study in the context of quantum field theory models with 2-dimensional spacetime (1+1), but also 5-dimensional and even higher-dimensional models.
4.) A radical new view on the question of spacetime is taken by Loop Quantum Theory. Here spacetime is quantized. The quanta of spacetime are generated during the interaction of two systems. In general, it is not possible to arrange the quanta of time into a linear order as we are used from our everyday experience. These phenomena arise at the level of the Planck scale, which is much smaller than any scale of today‘s experimental physics.
5.) IMO questions of space, time, spacetime etc. should first be investigated by physics. The concepts and their relations should obtain a mathematical formulation and become embodied into a physical theory. Afterwards it may help to start a philosophical reasoning about the fundamental concepts of the theory.