Unit II — Incorruptible Certitude & the Three-Step Method of Mathematical Transsumption¶
De Docta Ignorantia · Liber I, Capitula XI–XII (1440)
Primo necesse est figuras mathematicas finitas considerare cum suis passionibus et rationibus, et ipsas rationes correspondenter ad infinitas tales figuras transferre, post haec tertio adhuc altius ipsas rationes infinitarum figurarum transumere ad infinitum simplex absolutissimum etiam ab omni figura.
First we must consider finite mathematical figures with their properties and ratios, next transfer those ratios correspondingly to infinite figures of the same kind, and third transsume the ratios of those infinite figures still higher unto the Simple Infinite wholly absolved from every figure. — De Docta Ignorantia I.12 §1
Book I, Chapter 11 — That Mathematics Aids Us Most Greatly in Apprehending Diverse Divine Truths¶
Original GEOMETOR Translation from the Latin · h I, nn. 30–32 · Paris 1514 fol. 4v–5r · Rotta 1913 pp. 23–25
All our wisest and most divine teachers have agreed that visible things are truly images of invisible things and that the Creator can thus be seen recognizably from creatures as though in a mirror and in an enigma. Now the fact that spiritual realities—unattainable by us in themselves—may be investigated symbolically takes its root from what was said earlier: namely, that all things bear to one another a certain proportion (albeit hidden from us and incomprehensible), so that out of all things one universe arises and in the one Maximum all things are that very One. And although every image seems to approach likeness to its exemplar, nevertheless apart from the Maximal Image—which is the very thing the Exemplar is in unity of nature—no image is so similar or equal to its exemplar that it could not be infinitely more similar and equal, as has already become clear from the preceding chapters.
Whenever an inquiry proceeds from an image, however, nothing doubtful may remain within the image through whose transsumptive proportion the unknown is investigated, since the pathway to uncertain things can lead only through things presupposed and certain. Now all sensible things exist in a kind of continual instability owing to the material possibility abounding within them. Yet when objects more abstract than these are considered—not as wholly stripped of the material accompaniments without which they cannot be imagined, nor yet as wholly subject to fluctuating possibility—we see them to be most firm and most certain to us; such are mathematical entities themselves. Wherefore in them the wise skilfully sought examples of the realities to be searched out by the intellect, and none of the ancients esteemed as great approached difficult matters by any likeness other than the mathematical; so much so that Boethius, the most learned of the Romans, declared that no one wholly lacking practice in mathematics could attain knowledge of divine things.
Did not Pythagoras, the first philosopher both in name and in deed, place every investigation of truth in numbers? Him the Platonists and our own foremost thinkers followed so closely that our Augustine, and after him Boethius, affirmed beyond doubt that number was in the Creator’s mind the principal exemplar of things to be created. How else could Aristotle—who wished to appear singular by refuting his predecessors—convey to us in the Metaphysics the difference of species save by comparing them to numbers? And when the same Aristotle wished to impart knowledge of natural forms, showing how one is contained in another, he necessarily fled to mathematical forms, saying: ‘Just as the triangle is in the quadrangle, so the lower form is in the higher.’ I pass over in silence countless similar examples of his. The Platonist Aurelius Augustine likewise, whenever he inquired into the magnitude of the soul, its immortality, and other loftiest themes, turned to mathematics for aid. This pathway so pleased our Boethius that he steadfastly asserted that every doctrine of truth is comprehended in multitude and magnitude. And if you wish me to speak still more concisely: did not the doctrine of the Epicureans concerning atoms and the void—which denies God and shatters all truth—perish at the hands of the Pythagoreans and Peripatetics solely by mathematical demonstration, namely that one cannot arrive at indivisible and simple atoms, which Epicurus had posited as his starting-point?
Walking along this road of the ancients and concurring with them, we say that, since no pathway for approaching divine realities lies open to us except through symbols, we shall be able to make all the more fitting use of mathematical signs on account of their incorruptible certitude.
Book I, Chapter 12 — How Mathematical Signs Ought To Be Used for Our Purpose¶
Original GEOMETOR Translation from the Latin · h I, nn. 33–34 · Paris 1514 fol. 5r · Rotta 1913 pp. 25–27
Yet since it is established from what has gone before that the unqualified Maximum can be none of the things known or conceived by us, when we propose to investigate it symbolically we must leap beyond simple likeness. For inasmuch as all mathematical objects are finite and cannot even be imagined otherwise, if we wish to use finite things as an example for ascending to the unqualified Maximum, first we must consider finite mathematical figures together with their properties and intelligible ratios; second, we must transfer those very ratios correspondingly to infinite figures of the same kind; and third, after this, we must transsume the ratios of those infinite figures still higher unto the Simple Infinite that is wholly absolved even from every figure. Then at last our ignorance will be taught incomprehensibly how we, who labor in an enigma, ought to think more rightly and more truly concerning the Most High.
Acting in this way, therefore, and setting out under the guidance of Maximal Truth, we note that holy men of most exalted mind who applied themselves to figures spoke in varied ways: most devout Anselm compared Maximal Truth to infinite rectitude; following him, let us turn to the figure of rectitude, which I picture as a straight line. Other very skilled thinkers compared to the superblessed Trinity a triangle of three equal and right angles; and because such a triangle necessarily consists of infinite sides, as will be shown, it may be called an infinite triangle—and these too we shall follow. Others, striving to figure infinite unity, called God an infinite circle. Those, finally, who contemplated the most actual existence of God affirmed God to be as it were an infinite sphere. We, however, shall demonstrate that all of these simultaneously conceived rightly concerning the Maximum, and that the judgment of them all is one and the same.