As far as I know, Immanuel Kant (1724-1804) believed that space and time do not exist objectively, independent of us, and that they are added by our minds to our perceptions. I am eager to know what his argument/justification was for this claim. Later, Arthur Schopenhauer (1788-1860) accepted this, and argued that multiplicity and plurality are only apparent, the reality is one single entity, because, he said, to distinguish things, time and space are required – two things can be differentiated only if they occur at different times or places.
There's a lot going on in the body of your question, but I'm only going to address the Kant part.
Kant's position is subtly different than what you are claiming. For a thorough discussion, see the SEP article on Kant's view on time and space.
I'm primarily just going to work from that and try to simplify it with reference to precisely what you are claiming and asking. Let's start, however, by distinguishing two things:
- The claim that time and space do not exist objectively.
- The claim that I have no objective unfiltered access to time and space.
On my reading, Kant is committed to something close to 2 but not to 1. 1 is a claim about what isn't "out there." 2 is a claim about the way minds work.
Thus, Kant states
Space is not an empirical concept which has been derived from outer experiences. For in order that certain sensations be referred to something outside me (that is, to something in another region of space from that in which I find myself), and similarly in order that I may be able to represent them as outside and alongside one another, and accordingly as not only different but as in different places, the representation of space must already underlie them [dazu muß die Vorstellung des Raumes schon zum Grunde liegen]. Therefore, the representation of space cannot be obtained through experience from the relations of outer appearance; this outer experience is itself possible at all only through that representation (Critique of Pure Reason A23/B38 as quoted at SEP).
The idea here is that space for the subject is not something we experienced and then abstracted as an idea. Instead, it 's something that we have to bring to our experience to experience anything in space:
- Space is a necessary a priori representation that underlies all outer intuitions (A24/B38-9 also from SEP).
For Kant, "space" is a concept we bring when we experience. This is a major part of Kant's philosophy -- the idea that we do not have unmediated access to an "out there." For some people, this sounds like crazy talk; for others, this seems obvious. But the basic picture for Kant is that we bring framing devices that structure our experience of things and what we encounter is structured according to them (to get technical we bring "manifolds of sensibility" in time and space and "categories of the understanding" (12 exactly for Kant) which we use to understand objects).
This highlights that Kant believes 2. To see that Kant does not believe 1, we go to the refutation of idealism which occurs only in the B version. Again, there's a helpful SEP article. The basic thrust of the refutation of idealism is that the time and space we experience really are linked to an out there and are not products of our imagination. (See this question for more discussion), but we experience them through the space and time we have as part of our knowing apparatus.
To give an analogy, Kant's view is that we have an operating system that loads in "time" and "space" before we can experience anything. And then if we want to understand an object, when we try to understand something we automatically bring concepts with us. For Kant, (on most sympathetic interpretations) this is not a denial of the underlying thing behind what we try to understand but only a statement of what we bring to the task of knowing.
Schopenhauer is not merely following Kant but removing the underlying realism about what is "out there." Maybe that makes Schopenhauer more consistent considering this is one of the stickiest problems in Kant scholarship.