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Thanks for contributing! I would recommend starting with a careful look at formal contradiction and then working your way up to incompleteness and undecideability; the linked textbook is quite good.
– Corbin
Commented Jun 25, 2023 at 15:30
"[A square circle] cannot be imagined or conceived within one's mind" I've tried to understand the full implications of Einstein's general relativity. I don't get it — it doesn't really fit into my brain. This is famously common. Yet, I freely accept that someone smarter than me can conceive this within their mind. Therefore, I have to allow the possibility that someone can perfectly conceive of a square circle.
– mattdm
Commented Jun 25, 2023 at 15:55
Or: I can understand the concept of a tesseract (the 4D analog of a cube). I can even understand the visualization of that theoretical object as it interacts with 3D space. However, I am unable to truly conceive of its entire 4-dimensional form, as I have no actual way to visualize anything beyond the 3 dimensions by which I understand existence. The entire concept of a 4th spatial dimension does not, as far as I can sense, have any real existence, but it's plausible that this is a limitation of my faculties. This could be true of square circles and other apparent contradictions as well.
– mattdm
Commented Jun 25, 2023 at 16:01
@mattdm In the answer, it was not stated that I cannot imagine a square circle so therefore no one can, but rather that no one can imagine a square circle simpliciter. I would hold that someone smarter than me cannot conceive of a square circle, as this has nothing to do with the degree of intelligence possed by someone. Once the "circle" component of the "squared circle" is imagined, the imagined object can no longer be "squared" and vice versa.
– Max Maxman
Commented Jun 25, 2023 at 18:15
@mattdm As for the second comment, it seems that you have misunderstood my claim. I simply stated that contradictions have no referents, and for something to be a thing, it requires a referent. In anticipation of the most common objection (that the referent, in this case, is the concept of the square circle) I pointed out that it cannot be so, as a contradiction is inconceivable in the mind. The example you brought of the 4th dimension simply shows that something can be "inconceivable" in the mind and possible at the same time, but that is not the point of contention here.
– Max Maxman
Commented Jun 25, 2023 at 18:35

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