On Proofs (500-1599)

"Geometry enlightens the intellect and sets one's mind right. All of its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence." (Ibn Khaldun, cca. 14th century)

"Among the various branches of philosophy, logic has two prerogatives: it has both the honor of coming first and the distinction of serving as an efficacious instrument throughout the whole body [of philosophy]. Natural and moral philosophers can construct their principles only by the forms of proof supplied by logicians. Also, in order to define and divide correctly, they must borrow and employ the art of the logicians. And if, perchance, they succeed in this without logic, their success is due to luck, rather than to science. Logic is 'rational' [philosophy], and we may readily see from the very name, what progress in philosophy can be expected from one who [since he lacks logic] lacks reason." (John of Salisbury, "Metalogicon", 1159)

"But every art has its own special methods, which we may figuratively characterize as its 'approaches' or 'keys'. Seeking is a necessary preliminary to finding, and one who cannot endure the hardship of inquiry cannot expect to harvest the fruit of knowledge. Demonstrative logic, however, seeks methods [of proof] involving necessity, and arguments which establish the essential identification of terms that cannot be thrust asunder. Only that which cannot possibly be otherwise is necessary." (John of Salisbury, "Metalogicon", 1159)

"The method of demonstration is therefore generally feeble and ineffective with regard to facts of nature (I refer to corporeal and changeable things). But it quickly recovers its strength when applied to the field of mathematics. For whatever it concludes in regard to such things as numbers, proportions and figures is indubitably true, and cannot be otherwise. One who wishes to become a master of the science of demonstration should first obtain a good grasp of probabilities. Whereas the principles of demonstrative logic are necessary; those of dialectic are probable." (John of Salisbury, "Metalogicon", 1159)

"Reason may be employed in two ways to establish a point: first for the purpose of furnishing sufficient proof of some principle, as in natural science, where sufficient proof can be brought to show that the movement of the heavens is always of uniform velocity. Reason is employed in another way, not as furnishing a sufficient proof of a principle, but as confirming an already established principle, by showing the congruity of its results […]" (St. Thomas Aquinas, "Summa Theologica", cca. 1266-1273)

"Geometry enlightens the intellect and sets one’s mind right. All of its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence." (Ibn Khaldun, "The Muqaddimah: An Introduction to History", 1377)

"Reason may be employed in two ways to establish a point: first for the purpose of furnishing sufficient proof of some principle, as in natural science, where sufficient proof can be brought to show that the movement of the heavens is always of uniform velocity. Reason is employed in another way, not as furnishing a sufficient proof of a principle, but as confirming an already established principle, by showing the congruity of its results [...]" (Saint Thomas Aquinas, "Summa Theologica", 1485)

"At first, the thing seemed to me to be based more on sophism than on truth, but I searched until I found a proof." (Rafael Bombelli. "L'algebra", 1550 [1572]) [comment on the casus irreducibilis]