I'm on the Transcendental Aesthetic (and so am well aware that Kant's reasoning here may become clearer later on) -- what confuses me is how it seems to follow, from the discovery that time and space, being pure forms of the sensible intuition, bear only on objects of the senses, that these objects of the senses have nothing to do with objects in themselves; that the thing in itself is wholly unknown, and that we cannot assume at all that its features resemble those which appear to us. What am I missing here? What is it about our perception that renders us completely unable to say anything with certainty about things in themselves; unable even to posit the unique qualities of a thing such that it appears to us as rooted in the thing in itself? I mean we know there are unique objects, e.g. a rose which is of a particular height and number of petals and which is thus distinct from other roses -- why is it impossible for us to say that these differences (between different roses) have their basis in the things in themselves? Or does Kant indeed allow that we can assume this is so, but forbids us from any pretension to knowing through experience what a rose in itself actually is? But, in that case, what is it in particular about the Transcendental Aesthetic that leads to this conclusion? The mere fact that we have a priori intuitions, which structure our experience, and which we cannot do without?
"What is it about our perception that renders us completely unable to say anything with certainty about things in themselves; unable even to posit the unique qualities of a thing such that it appears to us as rooted in the thing in itself?"
Perception can give the same sensory experience for many different things-as-they-are. There is some process of transformation, some loss of data, some insertion of noise, that can change the experience nearly arbitrarily. So given a sensory experience, you can't tell which of the possible things-as-they-are gave rise to it.
Consider virtual reality, as in a computer game. You run round a maze - turn left at the bridge, go up the stairs on the right, run past the zombie, and hit the switch on the wall to your left. Your sensory experience is of a 3D maze with objects arranged in space. Things-as-they-are is a box containing electrons whizzing in and out of transistors embedded in a slab of silicon, wired to a flat panel of tiny lights. There is software/data that is, in some sense, the zombie you see. But the characteristics of the zombie we perceive have very little to do with its software representation. The electrons are not in the same spatial arrangement as the parts of the zombie, they're not the same colour. The zombie and the walls and the maze and the space they all live in are nowhere, except inside our own heads.
"I mean we know there are unique objects, e.g. a rose which is of a particular height and number of petals and which is thus distinct from other roses -- why is it impossible for us to say that these differences (between different roses) have their basis in the things in themselves?"
Because those properties might not be the result of the thing itself, but our perspective on it. The number of petals a rose has depends on what time you are looking at it. First there are no petals, then the buds open into flowers and there are lots of petals, then the flowers die and drop off, then next year there are more flowers. The number of petals it will have over its entire history in space and time is something you can't see directly. You also can't see the petals on the other side of the rose bush, or if you are short-sighted and have blurry vision, or all sorts of reasons. Indeed, the problem might not even be with the rose, but with your ability to count.
And it might be a rose in a 3D computer game. The height and the number of petals might be included in its software representation, or they might be added as part of the transformation process from software representation to pixels on a screen to image in the eye to model in the brain. Maybe the rendering software glitches, causing part of the rose bush to disappear when you stand too close to it. Maybe it's an optical illusion, and the glitch is in your brain. Or maybe it's not a computer model of a rose, but a computer model of a painting of a rose.
There is no evident end to the possible layers of misperception.
We know that misperception happens, because we often experience that sudden realisation that something is not what we thought it was, and we often experience objects we know are not actually there in pictures and paintings. We know that there is a lot about the world we can't perceive, like the whole new world we discovered when we invented microscopes. We ask the question: where does the thing-as-I-perceive-it actually exist? Out there? Or in here?
As with many of the perplexing questions one encounters when studying philosophy, the answer depends on how you interpret the words being used. Take a stone. You can know a lot about the stone by looking at it, feeling it, weighing it, scanning it to create a 3d model in a computer, placing a sample of it in a machine that determines its chemical components, comparing it with other rocks and so on. Clearly it is nonsense to say it is impossible to know anything about the stone. However, there are things you cannot know about the stone from the outside. You cannot know the history of the stone- you cannot know exactly how the fundamental particles of which it is comprised are arranged. You cannot see the inside of a stone, except by breaking it apart, in which case it is no longer 'the stone'.
You might take the view that Kant uses the term 'the object in itself' to mean all of the things you can't know about the object simply through the senses, in which case his position makes perfect sense, albeit being somewhat empty, since he is saying that you cannot perceive with the senses things you cannot perceive with the senses.
You should also bear in mind that some of the properties we attribute to the stone are purely imagined. For example, take the colour of the stone. Colour, like beauty, is entirely in the mind. There is no such thing in physics as yellow, for example- yellow is a mental sensation humans experience when light of a given range of wavelengths impinges on the retina.
Scientific experiences (see the book The Man Who Mistook his Wife for a Hat) show that blind-since-born people that get sudden vision can't visually recognise objects they know perfectly by touch (such is called the Molyneux Problem).
Imagine seeing a puzzle over a colored rug; how would someone, who has never learned how to discriminate visually one object from another, distinguish two objects among such colored mess? However, such task is elementary for a kid with a pinch of understanding.
So, empirically, existence shows to emerge at the level of understanding, not sensing (yes, that's the very same emergence those followers of the complex systems pseudo discipline talk about: objects and features do not emerge physically, they do so only subjectively, and more precisely, in the kantian Transcendental Analytic stage).
In addition, theoretically (metaphysically), it seems quite logical: boundaries are subjective constructs; outside of our senses, nature has no boundaries; everything interacts with everything, all time; therefore, no objects exist in nature; out there, there are only atoms, which are almost radically and absolutely unknown to us; we can't access such huge amounts of information, not only due to our rational potentials, but also due to our limited physical capabilities (sensitivity included); so, that's a second argument in the sense that objects exist only in our understanding.
However, you will notice in the Transcendental Logic that objects seem to raise as part of the understanding, but the object as itself seems not to be associated with a pure knowledge. Instead, the categories depend on it, which from my personal perspective, is clearly a contradiction: a pure concept which structure is nothing but empirical. The kantian notion of categories was taken from Aristotle, but does not fit precisely to Kant's goals.
The essential idea behind Kant's demarcation of noumena/phenomena is that he views "things in themselves" as meaning things as they are without mediation through a means of perception/ consciousness of an observer. Therefore, according to Kant, any attempt to "know" a thing by learning about it through perception and pure reason is inherently impossible --- the very act of observing and thnking about the thing means that its properties are mediated through the perception/consciousness of the observer, so what is being observed is phenomena, not noumena. Note that Kant believed that you could not know about the noumenal world through pure reason so you had to use a non-sensory method (faith) to conceive the noumenal world. As Kant put it, "I had to deny knowledge in order to make room for faith".
One can understand Kant's argument by positing the contrapositive question: How can I learn about a thing without using my perception or consciousness? Obviously, the answer is that we can't --- any attempt to learn about a thing will necessarily be mediated through our perception/consciousness. Even Kant's critics would agree with this, so the subsequent question then becomes: If we learn about a thing using our perception/consciousness (which is the only way to do it), are we learning about that thing, or are we only learning about our own perception/consciousness and its relationship to the world? Kant's view is the latter, but many philosophers have disputed his view that you cannot know "things in themselves". Ayn Rand famously characterised Kant's argument as saying that "...man is blind, because he has eyes — deaf, because he has ears — deluded, because he has a mind — and the things he perceives do not exist, because he perceives them."
Of the noumenon Kant says
the concept of a noumenon is necessary, to prevent sensible intuition from being extended to things in themselves, and thus to limit the objective validity of sensible knowledge.
For Nietzsche that sensible knowledge has perspectivity — different ideas of what a rose is:
To the extent to which knowledge has any sense at all, the world is knowable: but it may be interpreted differently, it has not one sense behind it, but hundreds of senses.—"Perspectivity."
Heidegger distinguishes the noumenon i.e. truth from the disclosed:
aletheia, disclosure regarded as the opening of presence, is not yet truth. Is aletheia then less than truth? Or is it more because it first grants truth as adequatio and certitudo, because there can be no presence and presenting outside of the realm of the opening?
He also brings time out of purity and into subjectivity, as Derrida elucidates:
To clarify what he calls Kant’s “obscure assertion” that “time affects a concept, in particular, the concept of the representations of objects” (Kant and the Problem of Metaphysics, 133), Heidegger shows what time as pure intuition must signify: originarily, it can in no way signify affection of something by something, affection of a being by another being, affection of an existing subject by something outside it: because time is nothing, as such it cannot affect anything. It is affection of self by self. ... This auto-affection as temporality is not a characteristic affecting transcendental subjectivity, one of its attributes; it is, on the contrary, that starting from which the self, the Selbst, the I think constitutes itself and announces itself to itself.