By this I mean that they exist indepent of experience of an agent. I am afraid I cannot make it more clearer than this. Perhaps you can define it if you understood what I mean and then we discuss your definition.
It is pretty obvious that “[does not] have form independent of a thinker” should at least mean that the object ceases to exist when no one is thinking of it (there is no thinker), and that the thinker gets to decide what the form is going to be. Otherwise an object clearly has form somewhat independently of the thinker. Yet, apparently, that was not what you meant, since you do not think any “Little Johnny” can decide that 2+2=5, while claiming that numbers have form “not independently of thinker”.
Do you agree with the definition of objects which was provided in the first comment in post #50?
No, I do not agree with that definition. But that’s beside the point. The point is that you also do not really agree with that definition.
Roman Empire as a political power which used to exists in the past is an abstract object.
That completely misses the point. The point is that you cannot explain what changed when the Holy Roman Empire was disbanded. In your view, it was an abstract object and it stayed an abstract object. Thus, according to you, nothing has changed. And yet, it is clear that something has changed.
Think of mathematics as underlying truth of any reality. By this I mean that you cannot find any reality that 2+2=5. This however doesn’t mean that mathematical objects exist without a thinker.
If reality exists without a thinker, then so does any “underlying truth”. And thus you cannot keep all those claims - something has to go.
Actually, there is one argument for existence of God that points out that the only available place for abstract objects (like numbers) is God’s mind. But, since you deny God’s existence, abstract objects become mysterious to you…
Because it has to become a part of our experience.
That does not answer the question. In other places you have claimed that hearing and thinking also count as experiencing, so visibility shouldn’t be special here.
I said that they have form when they are part of our experience.
That does not answer the question. You have also claimed that angels cannot be imagined and thus cannot be experienced. Are you taking that claim back?
Can you distinguish between aleph-zero, imaginary unit and quaternions? Yes. This means that each of them have a specific form inside your thoughts.
That does not answer the question. The question was not “Can you assert that something has form?”. We know that you can. In words of Hamlet, it is “as easy as lying”…

No, the question was what those forms of aleph-zero, imaginary unit and quaternions actually are. You should give actual geometrical shapes, since that’s how you understand forms.
Also, that begs the question - you haven’t shown that distinguishing objects requires difference of form yet.
I meant that two examples occupy different amount of space in a thinker thoughts.
That does not answer the question. (Are you beginning to see a pattern here?) You still are not giving meters, thus we only have bits and characters. And if something can be measured in bits and not in meters, then it has no definite geometrical shape - and all kinds of problems for your philosophy follow.
Well, vision is a part of our experience. Objects have clear forms inside our visions. Hearing is similar to vision but it is less obvious that objects have form when we hear them. Thinking is similar to hearing. Is that clear now?
That does not answer the question. The question asked where do you get a premise like “Form is necessary for us to distinguish objects.”. Your answer does not address it in any way.
I discuss them in the previous comment.
That does not answer the question (yes, neither here nor above). You haven’t explained how to imagine aleph-zero, imaginary unit or quaternions. Neither have you taken back the claims that would seem to imply that imagining them is necessary.
The obvious reason for that is that none of those objects can be imagined - and that’s fine. However, your philosophy demands that imagining them would be possible…
Any experience is related to a conscious mental state. Any mental state is related to a specific process inside the brain. Any process is related to specific function of a set of neurons. There is a one to one corespondence between the experience of abstract object and the configuration of a specific set of neurons inside our brain.
Proof, please.
I’m afraid that such a bold claim (unfortunately, pseudoscientific) can only be sustained if you will avoid learning anything about even the most simplified of artificial neural networks (that is, perceptrons)…
I hope things are clear by now.
Yes, they are, but in another way.
In your profile you write: “Religion: Seeking the Truth”. If you really are seeking the truth, you should try to find out the weakest points of your current “best guess”. Instead, you keep trying to evade the challenges directed at them. That’s a terrible strategy for seeking. In fact, it is not a very good strategy of defence either (a couple of “I don’t know.” would have looked much better).
So, if you are still going to evade difficult questions and challenges, I’m afraid that there isn’t much we can expect to get from this discussion…