I cannot help but think that much of (what I take to be) the confusion with regard to analytical and synthetical judgments stems from a general deficiency in grasping what Kant means by the object. We must, in my opinion and in this regard, go back to a different understanding of Anschauung* as the most critical term in the Kantian lexicon. Its traditional and almost universal rendering by "intuition" is simply fraught with distracting connotations.
* which is transliterated as "at-look" or "on-look". The Germans use this term a great deal to express such sentiments as "in my view of things, I see . . ." or "according to my view," etc.
Take the simple case of seeing someone point with a finger: in order to view that finger as pointing it is not sufficient merely to look at an outstretched finger as though that were what a pointing finger were (although, of course, it is that also!). For I can just as easily see that finger as a representation of the number one; or as showing me a certain hue of some (skin) color; or a dirty or broken finger nail, and so on infinitum. Each of these sightings requires a certain envisagement (Anschauung); for none of these sightings is the object on its own such that it might be intuited. For our purposes here we must see this finger as the first of two parts of a straight line segment, the far end point of the second of which touches a far object (the object of the pointing). Now that is sheer envisagement/viewing, for there is obviously no line present to be intuited (unless we mean by intuition that we are reading someone's mind, like what I sometimes call an" intellectual sighting" perhaps!?).* **
* It is interesting, I think, that in order to know whether another person is pointing with, or only showing, his*** finger, we must take his two eyes also into account and see if they "converge" on the finger or make a triangulation with the finger at some distant object. It is all quite artificial, and yet quite real--it is synthetic.
** If intuition were the right term for Anschauung, then the born blind should be able to intuit or sense a pointing finger by simply touching it. But this I greatly doubt. For they have no conception of space, but merely the feelings of muscular exertions, and so hence no sense of remote, spatial referencing.
In a similar way of putting an imaginary line into space such that the finger is a part of that line, our minds more generally put objects and pictures into things in that space (and also in time). Clouds and foliage, for example, present stuff (textures, shadings) to our eyes, and our minds order and orient that material in a certain way, automatically (it seems), and we find ourselves actually sighting faces and other shapes in, and by means of, that stuff;* and not as though we were imagining that, in the way that I look at my table and imagine Lilliputians walking across it, or a battleship sailing through its wood; no! we actually see the face in the clouds,** even though it is the play of our imagination with the textures, and once we see the face, it is difficult then not to see it, and to revert back to seeing the cloud or foliage as such without a face--so powerful is the effect of the envisagement.
* And often other people simply cannot see this figure, until we finally have to make a drawing on tracing paper and they look through that at the cloud (but which sighting of a face in the tracing is also an envisagement/viewing, but often easier to grasp and sight than the face in the cloud).
** It comes across to us as though the face were actually there in the cloud on its own, just like the human face seems to actually be on the front*** of a human head.
*** And where "front" is also an interesting envisagement, for there are no fronts on their own to be intuited in some mysterious way. We simply see them, and they are the way that we look at things and not in the things themselves.
In order to make this otherwise merely subjectively valid sighting objective, we must add an object. And this we do by dreaming up something called a face and which we can describe via a rule, e.g., a forehead, two eyes ... and chin, which we can then point out in an sighting for an identification. This then constitutes the objective aspects, i.e., we can point it out to others.* And it is also then this object which constitutes the basis of analytical judgments concerning the face, for it is the concept of this object which makes possible the original sighting (reality, or objective sighting) of the object in the first place. And so, of course, it must hold of the object in all cases. For without it there is no more of an object per se on its own than there is a face (actually) in the cloud, i.e., the face that is sighted in the cloud is, without the concept, merely a sensitive or empirical manifold which happens to be configured by our envisagemental/viewing capacity as a unity, much as I happen to notice that the top of my coffee cup on my table at this moment splits the edge of a book a few inches away into two equal parts. That object, that configuration of cup and book, will not remain with me unless I come up with a rule to describe it and then (for utilitarian purposes [only]) give it a name. That face on a human head or in a cloud will also have no more endurance for me subjectively without such a rule and name** *** than that coffee cup/split-book configuration.
* Whether we actually ever do or not; for it is the capacity to do so that gives us the objectivity; for some people cannot see it even when pointed out.
** From an epistemological standpoint, the initial sighting and retention of a face will be as a thing on its own, e.g., Momma or Daddy. It is when other "Mommas" are heard talked about, I suspect, that we begin to look for the rule to join the two or more "things" (Mommas) into a single consciousness as two or more faces, and whereby Momma becomes one of several mommas.
*** I often speculate as to whether all animals see faces. I suspect that dogs do and that cats do not, the latter seeing rather little animals where we see eyes (snails?), noses (frogs?), lips (worms?) and tongues (serpents?). But even if dogs do spy what I call a face, still, without a concept of face, they are caught in the Never-Never Land of mistaken identity, for if to them my face is a thing on its own, then my happy face and my sad face are two diverse things, and my happy face and that of someone who has a similar face are one and the same face. And so it is a different kind of world indeed!
The first object that we contribute (via our envisagement/Anschauung) to the world is objective time and space. The subjective, (what I call) split-finger space (of my finger touching by nose*) we account for by dreaming up an objective space in which we and all things (which can be sighted) are found, and by means of which I can ascribe the split finger phenomenon to a brain unification of the respective and diverse sightings of my two eyes (and which I attain to by experimentation and observation in conjunction with that conception of this objective space).** ***
* and which is meaningful, of course, only to two-eyed people.
** This is an excellent means of Kantian reflection, it seems to me. I notice the split in the finger as it touches my nose, and how the finger differs as I alternatively close each eye in tandem, and how the finger unifies according to a rule or pattern as I slowly focus on it at a distance from my nose, etc. Thomas Reid was taken by this "doubling" phenomenon in the late 18th century; and Captain Hook (of Peter Pan fame) even devised an alternative explanation for this sort of phenomenon in his celebrated explanation of the rainbow).
*** And we do the same thing for time, i.e., we dream up a time in which our subjectively valid time and that of others are placed in a coherent juxtaposition, i.e., my "earlier" and "later" are coordinated with your "earlier" and "later".
Now once we have recognized an object (and can make analytical statements about it), then it makes sense to speak of discovering something about that object, e.g., that balls fall and that balls bounce, while bean bags only fall. And here predicates are added synthetically.*
* But we are sometimes confused by forgetting that the original object is a synthetic provision of the mind itself, but the result of which, since the object is ultimately (subjectively) a concept/rule, is the capacity to make analytical statements, although always only of an empirical necessity; and occasionally we find a table which is upside down, for example, even though the analysis of the concept indicates that a table is used to support things at a level suitable for humans to write on, among other things; and so the violation is merely a call for an explanation within the context of the larger object, i.e., nature and the world, e.g., the table is being painted or repaired or stored away, etc. Given the necessity of the empirical concept, I speak of deviations (which call for an explanation) as opposed to merely variations (which means that there is nothing ever given which is fixed, so to speak, and so a bear with only three legs would not be deformed, for example, but rather that would be simply the way that bears sometimes appear).
The diversity of the concept will depend upon the particular viewing/Anschauung, and this will vary according to our own make up and, more importantly, according to our exposure. The ancient Hebrews (per Genesis 1) thought of the sky as a metalic dome (and somewhere Kant also discussed such an envisagement of the dome/sky, i.e., a low dome covering the earth, perhaps also where he spoke about the further part of the sea seeming to rise higher than the portion closer to his viewing position on the shore).
There are (at least) two important ramifications of Kant's idealism with regard to the object. In the first place it is so far removed from Berkeley's idealism (for whom there simply was no object, and thus what we called the object was merely a set of sighting/perceptions which we coordinated to be one and the same by means of space*) that Kant, far rather, provides an object such that what are otherwise merely appearances/Erscheinungen like the split-finger touching our nose, instead of being treated as things on their own, as the animals likely do, and where (with the animals) the finger actually does in fact split (if the animals every notice this), are reduced to mere images of things (but still which are all that is ever given to us, the object remaining always a figment of our imagination--a rule for unifying specific appearances), and whereby then our assertion as to the reality of the object becomes justified, and we are thereby saved from Berkeleyism.
* Which means, of course, that space and time had to precede, subjectively, in order for this arrangement and arraying to be produced. And this we do, as indicated above where the objective (so-called real) space is dreamed up in order that the subjective (split-finger) space might be incorporate in that with the (now understood) subjective space of others.
In the second place (and the subject of a growing interest on my part), this idealism opens the door for us to think differently of the object when we come to the consideration of freedom and necessity, for now, by means of this concept of the object as ideally provided, we can distinguish between the empirical thing on its own (provided for the purpose of experience) and the transcendental thing on its own (which we can think for the purposes of legal and moral thinking).*
* The concept of the thing on its own (or in itself) is difficult to articulate and yet is perhaps the most fundamental of any human concept. It is totally nondescript with regard to theoretical knowledge of objects, but it is only by means of this concept that we are able to distinguish the rainbow and the split fingers from the rain and from solid fingers. Without the concept of the thing on its own as a contrast, there would be no way of expressing the curious character of the rainbow and the split fingers, and we would be left with no reason to think that the rain could not suddenly go out of existence (as the rainbow actually does) and the chair in the corner could not suddenly split into two (or more!) ghost chairs. Now the interesting thing is that we don't just suddenly assume the rainbow and the split fingers to be specifically odd, as though we could somehow intuite that in some mysterious way, but rather because we come to think of everything we see and sense about us as an appearance or specter of things, and for which reason we require the concept of the thing on its own. By virtue of this a priori conception of things on their own we are able to fathom that what we see and sense is a function of our own viewing and so not a thing on its own. And it is in this framework and attitude that we undertake our investigations of the world revealed to us through the senses, and then actually distinguish the appearance from the thing on its own, in that we then conceive of a thing on its own which is empirical, e.g., the rain or the solid finger, and thus use that recognition in order to discern a difference with the rainbow and the split finger so that we come to call the latter an appearance or a specter. But this physical distinction between thing on its own and appearance (rain and rainbow) is itself based on a more profound transcendental distinction between thing on its own and appearance (all that strikes our senses). And it is this earlier and profound distinction between the thing on its own and appearance which gives us a reference for the distinction between transcendental freedom and the the pervasive realm of nature (which has to do merely with appearances, and never with things on their own.
Personal comment: my approach to a study of Kant is to imagine that he is a highly disciplined and highly imaginative and highly observant man; and then I try to inculcate these same characteristics in myself. For example: I notice (as Hume reported concerning his table*) that objects do in fact (seem to)** change shape and size as I approach them. Kant will have noticed this, too, of course; and whereas Hume was unable to account for his knowledge of this, namely that the change was only apparent and real only in his eye, Kant conceived of a means of accounting for that which is what we all must do, namely we dream up a space in which our two eyes are also located along with the physical objects we look at, and are able thereby to put our own, local space into that objective space as merely a viewpoint within that space. When I look at the change in objects as I move about in space, I realized that I do the same thing. Kant then developed that into a full blown theory call Transcendental Idealism.***
* An Enquiry Concerning Human Understanding, Section 118.
** We might say, with only a bit of exaggeration, that Kant's Critique of Pure Reason gives the (only?) justification and meaning for the bracketed expression here, i.e., "seems to." For there is simply no impression whatsoever anywhere in all of existence that could ever suggest to us that what we see is not what is, for there is nothing in addition to what we see/sense that could be given to us (through some sort of intuition) such that we could tell that what we see/sense is only a subjectively based sighting, and not what things really are. For there is absolutely no reason whatsoever that things could not get larger and smaller on their own, and it is a far reach indeed to dream up space (and time) and put these things and ourselves into them in order to reach that so-called simple answer, namely that we see things in space and time. It is far more simple to say that things are as they appear to us. Thus the world in which we live and that we recognize is a massive concoction of the human mind whereby we are able to tame what otherwise would be true wildness, i.e., taking specters/Erscheinung as things on their own, and make sense and order of it all. Hume, of course, knew this, but he was unable to explain it due to his faulty grasp of mathematics (at least according to Kant's estimation of Hume's plight. See Hume's Two Errors on my home page and the note below).
*** Now Hume, although a fellow Scot with myself (I am proud to say, and by descent), is far sharper than I am. But I have the advantage of Kant's thinking, and so have already the suggestion of the solution which I can then seize upon and seek to work out myself. Otherwise it would have been impossible for me to have spied Hume's error. For it never would have occurred to me to think that arithmetic were synthetic. And if Hume had made that jump in his thinking, according to Kant, then he would have realized that there were an object provided to arithmetic whereby (alone) its advances would have been possible. That object was the wispy pantomimic of (or rather in) the objective space that Hume then would have had to see that we ourselves dream up (as I have tried to indicate above) and into which then, of course, we would have inserted our, likewise, dreamed up objects (like solid fingers which only seem to split) so that the appearances would be relegated to images of objects instead of being unmanageable things on their own--Kant's ceative genius is so massive as to be staggering.