Unit I — Learned Ignorance, Polygon Exhaustion & the Coincidence of Maximum and Minimum¶
De Docta Ignorantia · Dedicatio, Liber I, Capitula I–IV & Epistola Auctoris (Kues, February 12, 1440)
Intellectus igitur, qui non est veritas, numquam veritatem adeo praecise comprehendit, quin per infinitum praecisius comprehendi possit, habens se ad veritatem sicut polygonia ad circulum.
The intellect, therefore, which is not truth, never comprehends truth so precisely that it could not be comprehended infinitely more precisely, standing to truth as a polygon stands to a circle. — De Docta Ignorantia I.3 §2
Dedicatory Prologue to Cardinal Julian Cesarini¶
Original GEOMETOR Translation from the Latin · h I, n. 1 · Paris 1514 fol. 1r · Rotta 1913 p. 1
To his revered teacher, the father beloved of God and most reverend Lord Julian, most worthy Cardinal of the Holy Apostolic See.
Your supreme and long-tested intellect will rightly wonder what I intend when, venturing somewhat incautiously to lay bare my unpolished awkwardnesses, I choose you as judge—as though any leisure remained to you amid the gravest public affairs of your cardinalatial office at the Apostolic See, and as though, after mastering the highest knowledge of all Latin writers who have hitherto shone forth and now of the Greeks as well, you could be drawn by the novelty of a title toward this perhaps clumsy conception of mine, when you have long known full well what measure of mind I possess. Yet I hope that this very wonder—not because you suppose anything hitherto unknown is set down here, but rather out of curiosity at the boldness by which I have been led to treat of learned ignorance—will entice your mind, ever eager for knowledge, to look within. Natural philosophers report that a certain sharp pang at the mouth of the stomach precedes appetite, so that nature, striving to preserve itself, may be aroused and restored. In just this way I judge that wonder, on account of which philosophy begins, runs ahead of the desire to know, so that the intellect, whose very being is to understand, may be brought to perfection through the study of truth. Uncommon things, even when monstrous, are wont to stir us. Therefore, matchless teacher, deem in your kindness that something of worth lies hidden here, and receive from a German such manner of reasoning in divine matters as immense toil has rendered deeply welcome to me.
Book I, Chapter 1 — How To Know Is Not-To-Know¶
Original GEOMETOR Translation from the Latin · h I, nn. 2–4 · Paris 1514 fol. 1r–1v · Rotta 1913 pp. 2–4
By a divine gift we observe implanted in all things a certain natural desire to exist in the best manner that the condition of each nature permits, and toward this end to act and to possess fitting instruments, within which an innate judgment is suited to the purpose of knowing, lest impulse be in vain and fail to reach rest in the beloved poise of its own nature. Should it perchance turn out otherwise, this must happen by accident, as when illness deceives taste or mere opinion misleads reason. Wherefore we say that a sound and free intellect knows the true—which it insatiably longs to reach by surveying all things through its inborn discursive motion—when it clasps what it has apprehended in a loving embrace, doubting not at all that to be most true from which no sound mind can dissent. Now all who investigate judge what is uncertain proportionally by comparing it with something presupposed as certain; every inquiry, therefore, is comparative and employs the medium of proportion. When the matters investigated can be compared to the presupposition by a close proportional reduction, the judgment of apprehension is easy; when we require many intermediate steps, difficulty and toil arise—just as is familiar in mathematics, where earlier propositions are led back quite easily to the first self-evident principles, whereas later propositions are reduced more arduously because only through the intermediation of the earlier ones.
Every inquiry, accordingly, consists in comparative proportion, whether easy or difficult; for which reason the infinite as infinite, since it escapes all proportion, remains unknown. Yet because proportion expresses agreement in some one respect together with otherness, it cannot be understood apart from number. Number therefore encompasses everything capable of proportion. Hence number produces proportion not in quantity alone, but in all things that can in any manner agree and differ, whether substantially or accidentally. Perhaps for this reason Pythagoras judged that all things are constituted and understood through the power of numbers. Nevertheless, exactness of combinations in bodily things and the precise adaptation of the known to the unknown surpass human reason—so much so that it seemed to Socrates that he knew nothing save that he was ignorant, while wisest Solomon declared all things difficult and beyond explanation in speech, and another man of divine spirit said that wisdom and the place of understanding are hidden from the eyes of all the living. If this is so—as even most profound Aristotle affirms in First Philosophy, that in the things most manifest by nature such difficulty befalls us as befalls a night-owl trying to gaze upon the sun—surely, since the desire within us is not in vain, what we desire is to know that we do not know. If we can attain this fully, we shall attain learned ignorance. For nothing more perfect will come even to the man most zealous in learning than to be found most learned in the very ignorance that is proper to him; and anyone will be the more learned the more he knows himself to be ignorant. Toward this end I have undertaken the labor of writing a few words concerning learned ignorance itself.
Book I, Chapter 2 — Preliminary Elucidation of What Follows¶
Original GEOMETOR Translation from the Latin · h I, nn. 5–8 · Paris 1514 fol. 1v–2r · Rotta 1913 pp. 5–6
Being about to treat of the greatest doctrine of ignorance, I find it necessary to approach the very nature of maximality. Now I call ‘Maximum’ that than which nothing can be greater. Yet fullness belongs to the One. Unity, therefore—which is also beingness—coincides with maximality. And if such unity is universally absolved from every relation and contraction, it is manifest that nothing stands opposed to it, inasmuch as it is absolute maximality. Thus the Absolute Maximum is One, which is all things; in which all things are, because it is Maximum. Moreover, since nothing is opposed to it, the Minimum coincides with it at the same time; wherefore it is also in all things; and because it is absolute, it is in act all possible being, contracting nothing from things, while all things come from it. In the first little book I shall labor to investigate this Maximum—which by the indubitable faith of all nations is also believed to be God—incomprehensibly above human reason, guided by Him who alone dwells in light inaccessible.
In the second place, just as absolute maximality is absolute beingness through which all things are what they are, so too the universal unity of being derives from it—a unity called ‘maximum from the Absolute’ and hence existing contractedly as the universe, whose unity is contracted in plurality, apart from which it cannot exist. Although this maximum embraces all things within its universal unity, so that all things deriving from the Absolute are in it and it is in all of them, nevertheless it possesses no subsistence outside the plurality in which it exists, because it does not exist without contraction, from which it cannot be absolved. Concerning this maximum, namely the universe, I shall add a few remarks in the second book.
In the third place, the maximum of a third consideration will subsequently be made manifest. For since the universe has subsistence only contractedly in plurality, among those many things themselves we shall seek one maximum in which the universe subsists most greatly and most perfectly in act as in its goal. And because such a being is united with the Absolute, which is the universal bound, since it is the most perfect goal surpassing all our capacity, concerning that Maximum which is simultaneously contracted and absolute—whom we name Jesus, ever blessed—I shall subjoin certain reflections according as Jesus Himself shall inspire.
Whoever wishes to reach the meaning, however, must lift his intellect above the force of words rather than cling to the literal properties of terms, which cannot be fitted strictly to such great intellectual mysteries. Exemplary guidance too must be used transcendently, leaving sensible things behind so that the reader may swiftly ascend to simple intellectuality. To seek out this path I have striven to lay it open to ordinary minds as clearly as I could, shunning all harshness of style and revealing straightway that the root of learned ignorance lies in the ungraspable exactness of truth.
Book I, Chapter 3 — That Precise Truth Is Incomprehensible¶
Original GEOMETOR Translation from the Latin · h I, nn. 9–10 · Paris 1514 fol. 2r · Rotta 1913 pp. 7–8
Because it is self-evident that there is no proportion of the infinite to the finite, from this it is likewise crystal clear that wherever one encounters exceeding and exceeded, one never arrives at the unqualified Maximum, since things that exceed and are exceeded are finite. Such a Maximum, by contrast, is necessarily infinite. Given anything whatsoever that is not the unqualified Maximum itself, it is manifest that a greater can be given. Moreover, because we find equality to exist by degrees—so that one thing is more equal to this than to that according to generic, specific, local, influential, and temporal agreement and difference, along with like conditions—it is plain that neither two nor more things can be found so similar and equal that they could not still be infinitely more similar. Hence measure and thing measured, however equal they may be, will forever remain distinct.
A finite intellect, therefore, cannot attain the truth of things with precision by means of likeness. For truth is neither more nor less, consisting in a certain indivisible, which nothing that is not the true itself can measure with exactness —just as what is not a circle cannot measure a circle, whose being consists in a certain indivisible. Thus the intellect, which is not truth, never comprehends truth so precisely that it could not be comprehended infinitely more precisely; for it stands to truth as an inscribed polygon stands to a circle: the more angles the inscribed polygon possesses, the more similar to the circle it becomes, yet it is never made equal to the circle, even if one multiplies its angles to infinity, unless it resolves itself into identity with the circle. Clearly, then, concerning the true we know nothing other than that we know it to be incomprehensible precisely as it is—truth standing as most absolute necessity, which can be neither more nor less than it is, and our intellect standing as possibility. Accordingly the quiddity of things, which is the truth of beings, is unattainable in its purity; though sought by every philosopher, by none has it been found as it is; and the more deeply we are schooled in this ignorance, the closer we draw to truth itself.
Book I, Chapter 4 — The Absolute Maximum Is Incomprehensibly Understood, and with It the Minimum Coincides¶
Original GEOMETOR Translation from the Latin · h I, nn. 11–12 · Paris 1514 fol. 2r–2v · Rotta 1913 pp. 9–11
Since the unqualified and Absolute Maximum, than which nothing can be greater, is greater than can be comprehended by us—because it is infinite truth—we attain it in no other way than incomprehensibly. For inasmuch as it does not belong to the nature of things that admit of exceeding and exceeded, it stands above everything that can be conceived by us; indeed, whatsoever things are apprehended by sense, reason, or intellect differ within themselves and from one another in such wise that there is no precise equality among them. Maximal equality, therefore, which is neither other than nor diverse from anything, surpasses every intellect. Wherefore the Absolute Maximum, since it is everything that can be, is wholly in act; and just as it cannot be greater, by the very same reason it cannot be lesser, since it is everything that can be. Now the Minimum is that than which nothing can be lesser. And because the Maximum is of just this kind, it is manifest that the Minimum coincides with the Maximum.
This becomes clearer to you if you contract maximum and minimum to quantity. For maximal quantity is maximally large, whereas minimal quantity is maximally small. Absolve, then, maximum and minimum from quantity—subtracting ‘large’ and ‘small’ intellectually—and you clearly behold that maximum and minimum coincide; for ‘maximum’ is a superlative just as ‘minimum’ is a superlative. Absolute maximality, therefore, is no more maximal than minimal, because within it the minimum is the maximum coincidently.
Oppositions, accordingly, belong solely to things that admit of exceeding and exceeded, and to these in differing ways; to the Absolute Maximum they belong not at all, because it stands above all opposition. Since, then, the Absolute Maximum is absolutely in act all things that can be, without any opposition whatsoever so that in the Maximum the Minimum coincides, it likewise stands above every affirmation and negation. All that is conceived to be is no more than it is not; and all that is conceived not to be is not any more than it is. Rather, it is ‘this’ in such a way that it is all things, and all things in such a way that it is none of them; and maximally ‘this’ in such a way that it is minimally the same. For to say ‘God, who is absolute maximality itself, is light’ is no other than to say ‘God is maximally light in such wise that He is minimally light.’ Otherwise absolute maximality would not be all possible things in act, were it not infinite and the bound of all things and boundable by none of them, as by God’s own loving-kindness we shall explain in what follows.
Yet this transcends all our understanding, which cannot combine contradictories in their principle along the pathway of discursive reason, since we walk by way of things made manifest to us by nature; falling far short of this infinite power, reason cannot join together contradictories separated by an infinite distance. Above every discursive movement of reason, therefore, we see incomprehensibly that absolute maximality is infinite, that nothing is opposed to it, and that with it the Minimum coincides. ‘Maximum’ and ‘Minimum,’ as taken in this little book, stand as transcendent terms of absolute signification, so that above all contraction to quantity of bulk or of power they embrace all things in their absolute simplicity.
Author’s Epistolary Colophon to Cardinal Julian (Kues, February 12, 1440)¶
Original GEOMETOR Translation from the Latin · h I, nn. 263–264 · Paris 1514 fol. 25v · Rotta 1913 pp. 188–189
Receive now, revered father, the things that I long desired to attain by diverse pathways of learning yet could not reach before, until, while returning by sea from Greece, I was led—by a gift from on high, I believe, from the Father of lights, from whom comes every best gift—to embrace incomprehensible things incomprehensibly in learned ignorance, by transcending the corruptible truths knowable in a human manner. This learned ignorance I have now brought to completion, in Him who is Truth, within these little books, which from the self-same principle can be either condensed or expanded. In these deep matters the whole endeavor of our human mind ought to be to raise itself unto that simplicity where contradictories coincide: with this the conception of the first book labors. From that principle the second book draws forth a few insights concerning the universe, uncommon to many and above the ordinary road of philosophers. And now at last I have completed the third book concerning ever-blessed Jesus, advancing always from the same foundation; and the Lord Jesus has continually grown greater to me in intellect and affection through the increase of faith. For no one holding the faith of Christ can deny that along this path he is kindled higher in desire, so that after long meditations and ascents he sees most sweet Jesus alone as worthy of love and, joyfully forsaking all things, embraces Him as true life and everlasting joy. To one who thus enters into Jesus all things give way, and neither any writings whatsoever nor this world can cast up difficulty, because such a one is transformed into Jesus through the Spirit of Christ indwelling him, who is the goal of intellectual desires; whom do you, most devoted father, earnestly entreat with suppliant heart on behalf of me, a most wretched sinner, that together we may merit to enjoy Him eternally.
Completed at Kues, February 12, 1440.