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Discussion 4 to Talk Back 12
On Squaring Saint Anselm's Circle.

by Bob McCallum

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Ian, in your essay on Anselm’s argument (hardly a ‘proof’), you said:

‘Consider this also: Can you conceive of a square circle? Yes, you can: "An object with four equal sides with all points equidistant from the centre". Could you draw this? Of course not. Does it exist in re? Of course not, not as far as we know. But the concept clearly exists, in intellectu.’

I checked the 3 discussion responses to your essay, assuming someone would point out the obvious problem associated with your description of a ‘square circle,’ but didn’t see one, so I’m responding.

A square is not defined simply by having 4 equal sides. Those sides must also be straight lines. And not only that, the angles between adjacent sides must be 90 degrees. Thus, you’re asking for us to ‘conceive of’ a curved straight line. Now, by definition, one can’t have such a line, unless perhaps one considers an infinitely large circle, any arc of which would approach a straight line in the limit; however, the angles between the ‘sides’ certainly would not be 90 degrees. In addition, one of the properties of a circle, as opposed to a square, may be that its area is a transcendental number (because of the occurrence of π in its computation / definition), whereas the area of a square may not have to be. However, it’s been such a long time since I’ve thought about this that I’m not sure about any restrictions on the numerical characteristics of the numerical values of the areas of these two figures.

In any event, my conclusion on whether one can ‘conceive of’ a square circle, i.e. of’ a curved straight line, is exactly the opposite of your ‘Yes, you can,‘ i.e. No, I can't. Thus, it appears to me that such a concept clearly does not exist in intellectu.