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Both circles and sets are considered abstract objects. I can visualise a circle in my mind (can 'see it through my mind's eye') but can't visualise a set or a number. I have no picture of a set in my mind. This makes me wonder if there are two types of abstract objects, those that can be visualised and those that can't be.

Are there two types of abstract objects, those that can't be visualised and those that can be?

If yes, are there terms to mark that distinction?

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    To be specific, you can only visualize an example of a circle - not a general circle. You can also visualize an example of a set. A circle, being a set of points, is an example of a set.
    – causative
    Jul 15 at 20:25
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    Kant arguably distinguished between arithmetic which relies on pure reason and geometry which relies on pure intuition. Intuition, for Kant, is the ability to process sensory data and pure intuition is the ability to imagine sensory data in an a priori way. For example when you imagine a circle, you are using pure intuition.
    – David Gudeman
    Jul 15 at 22:05
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    But the fundamental reason you find it easy to visualize a (geometric) circle but not a set is that there is in a certain sense only one circle, but in no sense is there only one set. There are many different sets, even if you identify any two setsX, Y if there is a bijection between them.
    – Daniel Asimov
    Jul 16 at 2:52
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    An important difference is that while almost all mathematicians (and laypersons) 'believe' in circles, there is a significant bunch of mathematicians that don't believe in arbitrary sets. They usually call themselves intuitionists, constructivist, sometimes finitist. A good answer laying out the spectrum from believable to not believable
    – Rushi
    Jul 17 at 2:54
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    Why just two types while not a set of more plural types? And you're starting to think in terms of ZFC sets now, and after enough learning you may easily visualize that which is a set and that which is not as hinted in Russell's paradox...
    – Double Knot
    Jul 20 at 20:28

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Imagine Harder

I don't know about you, but I can visualize both circles and sets. Finite sets are easiest. I can even visualize an abstract set with five elements. Think of five different things, with some fog covering them so you cannot tell exactly what each of them it. Just that there are five of them.

Of course this picture does not describe all sets simultaneously. But the same for your circle. We can only imagine a particular circle, and not the generic circle. Another problem is the circle one imagines might have a very slight kink in it when you zoom up really close. This is analogous to the fog in the visualization of sets.

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    What color is the mathematical object that you visualize?
    – g s
    Jul 17 at 8:23
  • @gs Black and white
    – Daron
    Jul 17 at 9:52
  • @Daron I see. I think of it this way: What I can visualize are the things from which I have formed a set, much like how I can only visualize two people hanging out not their friendship in itself.
    – Harshit Rajput
    Jul 17 at 13:02
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    @HarshitRajput I might imagine a friendship as a montage of all the things they did together. Something like this: youtube.com/watch?v=9NPykPA7Mgw
    – Daron
    Jul 17 at 13:53
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You can't visualize a circle, nor an instantiation of a circle. You can visualize only an object with the characteristic of circularity. Calling such an object a circle is to use a homonym, with no greater significance than any other homonym.

Similarly, (noting that the boundaries of an object are arbitrary) you can visualize an object with the characteristic of quantity, e.g. a dozen eggs, or an object with the characteristic of containing separately identified members (which one might call setness), e.g. "the things on your desk".

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