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First, I came across the Penrose triangle and then a variety of impossible objects. I understand that an object might not exist in a specific domain. For example, Penrose triangle does not exist in 3 dimensional Euclidean space. However, I have difficulty to understand how we can infer something cannot exist at all in whatever imaginable domain. If something cannot exist, even as a mental state, then how could somebody study its properties and conclude that object cannot exist? Thus, whatever, we are able to imagine must exist even though only as an abstract object or as a mental state. Am I right or am I missing something?

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    Because those objects are defined as belonging to a specified domain if they are to exist at all. Namely, three-dimensional objects, "whatever imaginable" is moot. "Whatever we are able to imagine must exist" is overly optimistic about the powers of our imagination, it is quite adept at producing much nonsense. Or, if you prefer, at extracting nonsensical descriptions from what is supposedly imagined.
    – Conifold
    Commented Jul 5, 2020 at 11:16
  • I'm certain you can come up with certain axioms of geometry that allow you to talk about that triangle without running into contradictions. I think the article is slightly misleading in saying that it is "impossible". Commented Jul 5, 2020 at 11:49

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Descriptions of objects can be just sets of individual properties. Like it has 4 legs, it is made of wood, it fits through a door. Such properties can freely be combined to sets. They just remain sets of individual properties. Some such sets then describe real objects, some describe possible objects, some do not describe any possible object. But they can still be thought about as individual properties.

It is possible to reason about the "raw" set of properties, to decide if it is consistent within itself, or consistent with other facts of nature that we believe in.

As an image, you can imagine pieces of a puzzle that do not match, so you can never finish the puzzle. But you can still have all those pieces, carry them around, check which ones fit to each other. There is no need to first complete that puzzle to decide that the puzzle cannot be completed. The individual pieces alone are sufficient for that.

Thus while it might be said what we "reason about impossible objects", that's just short for saying instead that we "reason about inconsistent sets of properties that are labelled to describe objects".

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