Cusa: On informed ignorance
Nicholas of Cusa wrote De docta ignorantia (On informed ignorance) in which he looked as philospohical notions which he compared with the problem of squaring the circle. The quotations below come from Nicholas of Cusa, Die Kunst der Vermutung (Bremen, 1957) and J E Hofman (ed.), Schriften des Nikolaus von Cues Vol 2 Die Mathematischen Schriften (Hamburg, 1950). In De docta ignorantia, Nicholas of Cusa advocated 'informed ignorance':

If we achieve this, we shall have attained to a state of informed ignorance. For even he who is most greedy for knowledge can achieve no greater perfection than to be thoroughly aware of his own ignorance in his particular field. The more be known, the more aware he will be of his ignorance. It is for that reason that I have taken the trouble to write a little about informed ignorance. ...
Thus wise men have been right in taking examples of things which can be investigated with the mind from the field of mathematics, and not one of the Ancients who is considered of real importance approached a difficult problem except by way of the mathematical analogy. That is why Boethius, the greatest scholar among the Romans, said that for a man entirely unversed in mathematics, knowledge of the Divine was unattainable. ...
The finite mind can therefore not attain to the full truth about things through similarity. For the truth is neither more nor less, but rather indivisible. What is itself not true can no more measure the truth than what is not a circle can measure a circle; whose being is indivisible. Hence reason, which is not the truth, can never grasp the truth so exactly that it could not be grasped infinitely more accurately. Reason stands in the same relation to truth as the polygon to the circle; the more vertices a polygon has, the more it resembles a circle, yet even when the number of vertices grows infinite, the polygon never becomes equal to a circle, unless it becomes a circle in its true nature.
The real nature of what exists, which constitutes its truth, is therefore never entirely attainable. It has been sought by all the philosophers, but never really found. The further we penetrate into informed ignorance, the closer we come to the truth itself. ...
Nicholas of Cusa is, of course, absolutely correct when he states:

The circle cannot be squared since: the area of a circle is incommensurable with that of any noncircle.
However his arguments are somewhat muddled. Let us quote his arguments and then make a few comments: 
Those who hold firmly to the first view [that the circle can be squared] seem to be satisfied with the fact that given a circle, there exists a square which is neither larger nor smaller than the circle. ... If, however, this square is neither smaller nor larger than the circle, by even the smallest assignable fraction, they call it equal. For this is how they understand equality  one thing is equal to another if it neither exceeds it nor falls short of it by any rational fraction, even the smallest. If one understands the notion of equality in this way, then, I believe, one can correctly say that, given the circumference of a certain polygon, there exists a circle with the same circumference. If, however, one interprets the idea of equality, insofar as it applies to a quantity, absolutely and without regard to rational fractions, then the statement of the others is right: there is no noncircular area which is precisely equal to a circular area.
We have to applaud Nicholas of Cusa for studying the question of equality in the case of squaring the circle. His writings add important impetus to such a debate. However, what he seems to be saying is incorrect. Does he have a confused understanding of a correct argument? Perhaps he comes close to arguing that we cannot say that a circle is equal to a polygon simply because, given a rational number, no matter how small, we can find a polygon greater than a given circle and one smaller than that circle which differ by less than the chosen rational.

JOC/EFR August 2007
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