1The importance of mathematics, and particularly of a geometry of the infinite, in the theology of Nicholas of Cusa, no longer requires any demonstration.  It is equally well known that fourteenth-century scholasticism saw a spectacular development of mathematical reasoning in philosophy but also in theology,  one of its most celebrated representatives being Duns Scotus (1265 or 1266–1308), who is said to have introduced rationes mathematice into theology; it is also he who introduces (in reference to God) the concept of an infinite entity.  It is therefore tempting to place Nicholas of Cusa among the descendents of Duns Scotus and scotism; and a number of critics have indeed done so. 
2In the following study, we shall question the pertinence of this rapprochement, and we shall endeavor to situate the infinite in Nicholas in relation to the use made of this concept by the scholasticism of the preceding century. Of course, in order to do so we must make certain clarifications as to the sense(s) in which Nicholas uses the word “infinite,” something that is far from being obvious, as can be seen from the following passage from chapter 3 of the Complementary Theological Considerations, to which we shall return:
If we suppose an infinite circle, it is then necessary that the center, the half-diameter and the circumference, should be completely equal. The center of the infinite circle is also infinite, because one cannot hold that the infinite should be greater than the center; for one cannot hold that that which cannot be smaller than the infinite and unlimited should be larger than the center. For the center is the end of the semi-diametric line, and the end of the infinite is itself also infinite. The center of the infinite circle is therefore infinite. Just as its half-diameter and its circumference likewise. The equality between the center, the half-diameter and the circumference of the infinite circle is thus complete. And since there cannot exist many infinites, for if so none of them would be infinite, the existence of many infinites implying a contradiction, the center, the half-diameter and the circumference must therefore constitute one sole infinite. 
4In this study we shall pursue the following plan: we shall begin by presenting the divine infinite in Scotus, on the basis of questions 2 and 3 of the Quodlibet V; then we shall give an overview of the Cusanian doctrine of learned ignorance, or the Not-other, which we shall follow with a brief comparison between the Cusanian infinite and the Scotist infinite; then we shall tackle the symbolism of mathematical figures in Nicholas, upon whose basis, through a transposition to the infinite, something of the divine form is conjectured; finally we shall conclude in opposing the Cusanian approach to the infinite to that of the scholastics of the thirteenth to fourteenth centuries.
The Divine Infinite in Duns Scotus
5Let us begin by recalling, and insisting upon, the fact that Scotist theology rests upon the well-known foundation of the univocity of beings, whether they are divine, created, or creatable (which designates entities whose existence implies no contradiction). This implies in particular that although, for example, the wisdom of God is beyond all assignable relation to that of man, they are both cases of the same concept of wisdom. Scotus insists upon this point, saying that if our concepts had two analogous reasons, one for the mundane and one for God—that is to say, two different reasons but expressed in the same way—then “there would be absolutely no evidence in our theology” and “unless being implies one unique, univocal intention, theology would simply perish […].”  The result is that being, separate from the natural real since it also refers to God or to creatables, is naturally knowable. We see immediately that the agreement with the doctrine of learned ignorance, one of the foundations of Cusanian theology, is problematic; we shall come back to this point.
6Scotus exposes his doctrine of the infinitude of God many times, but his clearest explanation is in the quodlibetal question V, which we shall paraphrase.  His starting point is the quantitative infinite as defined by Aristotle in the Physics: “[The infinite is] such that we can always take a part outside what has been already taken.”  Scotus’s argument is as follows: since “as great as it might be, what has been already taken is infinite,” Aristotle can deduce “that the infinite in quantity, having only a potential being, does not correspond to the notion we have of a whole, for a whole is that of which there is nothing outside of itself;  this potential infinity is thus not perfect, for that which is perfect is that which lacks nothing that would perfect it. [whereas] this infinity does lack something.” Scotus thus has recourse to an argument very frequent in the fourteenth century, reasoning secundum ymaginationem. He imagines “that all the parts susceptible to being taken are so taken simultaneously, or subsist simultaneously.” One would then have, he says, “an actual infinite quantity, for it would now be as large in actuality as it was in potential. And all the parts which, in the potential infinity, would have been reduced in actuality to an infinite succession, and would have received being one after another, would be simultaneously conceived as being actual, so that this imagined actual quantitative infinite would be a perfect whole.”
7With another argument secundum ymaginationem, Scotus then defines an infinite in act in entitate—that is to say, in an entity, or rather in “beingness [étantité]” (a neologism on the model of “humanity” that is preferable here because of the ambiguity of the word “entity”). This argument rests upon an analogy, founded on the existence of intensive magnitudes, with what takes place in quantity. Scotus sums up his approach as follows:
Beginning with the notion of infinite posited in book 3 of the Physics, if we move from there in imagination to that of an actual infinitude in quantity, reasoning as if such an infinitude were possible; and then if, going further, we move to that of an actual infinitude in beingness, which is possible since there is no contradiction between the fact of being and infinitude, we arrive to a certain extent at understanding in what sense we must admit that there is a being that is infinite intensively, in other words infinite in perfection or in excellence—that is to say, a being lacking nothing in the order of beingness. 
9This, for Scotus, is what characterizes God: he is the infinite being, in other words he is “formally infinite” whereas all other beings are finite. This infinite perfection is explained in a formulation that is, as we shall see, close to that of Nicholas of Cusa:
Take for example this entity: whiteness. It is exceeded by another entity, for example by science, in the relation of one to three: and then it is exceeded by the intellective soul in the relation of one to ten; and then, by the highest of angels, let us say, in the relation of one to a hundred. In whatever way one progresses in the scale of beings, there is always a determinate relation in which the highest exceeds the lowest. This is not because there is a relation in the proper sense, such as that used by mathematicians, for an angel is not constituted of something inferior to it along with something else added to it, since it is more simple than that which is inferior to it; this must be understood in the sense of a relation of excellence or perfection, like the relation in which one species exceeds another. It is in this sense that one can say that the infinite being exceeds in beingness a finite being, not in some determinate relation (like that which may hold between finite beings) but beyond any assignable relation. 
11What consequence does this infinitude in beingness have for the divine attributes of wisdom, kindness, etc.? Scotus explains that “intensive infinitude expresses an intrinsic mode of its beingness [that of the infinite being]”  in the sense that “if we abstract from everything that is a property or quasi-property [attributes] of this beingness, even in this case its infinitude is not removed from it.” And he demonstrates in the quodlibetal question VII  that, from the fact that in any being, properties and quasi-properties are subordinated to beingness, their infinitude follows from that of beingness.
The Cusanian Infinite and Its Proximity to the Scotist Infinite
12As was remarked above, it is tempting to compare the Scotist formula that “God is formally infinite” whereas all other beings are finite, with the statement that opens the Learned Ignorance, “God is the absolute maximum.” But since Nicholas shows immediately that he is also the absolute minimum,  we arrive at a characterization that cannot at all be attributed to the Scotist infinite in beingness.
13To see this more clearly, we must return to the much-disputed interpretation of the famous doctrine of the coincidence of opposites. The critic today considers that “in Nicholas one cannot isolate the ‘internal’ possibilities of God from his possibilities in relation to the world, and that God is never envisaged by Nicholas of Cusa as a reality in itself, independently of the relations that other things entertain with it. […] The first chapters of the Learned Ignorance, moreover, show clearly that the doctrine of the maximum is a part of a fundamental non-knowledge of truth [i.e. learned ignorance].”  In other words, when Nicholas says that God is the absolute maximum, he envisages “God as he is for the human mind.” So that one can only approach the Cusanian conception of God by starting out from the relations that he entertains with creatures, and it is only on this basis that one can try to think his infinity and his transcendence. This is what is explained in one of the last treatises of Nicholas, the De Non-aliud, where God is defined as the Not-other, a concept which, according to Nicholas himself, corresponds to what he previously sought to express with his “coincidence of opposites.” This proximity is very clearly explained by Jean-Michel Counet. 
14According to Nicholas, finite things are characterized by a modality of being that is their “alterity.” In the finite world, things are what they are, each other than the others; they are therefore defined by this alterity. Inversely, to say that God is the Not-other signifies that God is not posited as different from all that is not him [he is not other than anything whatsoever]—in fact he is in each thing and gives each thing its being by identifying with it; Nicholas’s formula is that God is “all in all,” which Jasper Hopkins, in his translation of De Non-Aliud, annotates as follows: “‘In X, God is not other than X,’ i.e., ‘In X, God is X’”.  In the finite world, that of alterity, things are other than each other because they lack the being of others, whereas God, who is Being in its plenitude, lacks nothing. Let us remark incidentally that, since all things of the finite world are “other”, the negation Not-other can be identified with “infinitum” as negation of “finitum,” which justifies our considering infinitas as the essential attribute of the divine nature.
Does Not Such a Thesis Lead to Pantheism?
15Nicholas’s response would be that, for him, there are two modes of existence: when things exist in the mode of the other, they are finite, distinct, and changing in space and time; and when they are in the mode of Not-other they are eternal, immutable, and divine. Counet emphasizes that this point of view, for which God is not a being comparable in relation to creatures, but an ontological mode of things themselves, is borrowed, notably, from Meister Eckhart.  In fact, there are not two worlds—that of sensible reality on one hand and that of divine Ideas on the other—but one and the same world, at once eternal, infinite in God, and manifesting itself as spatio-temporality delimited and contracted in the finite.
16Let us come back to things. They are not illusions since they possess a quiddity which results from the fact that the Not-other gives them their being through its identification with them; as we have said, in a thing God is not other than that thing. And since they are not other than themselves, Nicholas can say that their participation in the Not-other is their essence. In other words, the Not-other founds the identity of the thing and constitutes its essence:
For how would the sky be not other than the sky if in it Not-other were other than sky? Now, since the sky is other than not-sky, it is an other. But God, who is Not-other, is not the sky, which is an other; nonetheless, in the sky God is not an other; nor is He other than sky. (Similarly, light is not color, even though in color light is not an other and even though light is not other than color.) 
18The coincidence of opposites in God is thus explained as follows: all the formal reasons which are distinct in the finite are identical in God, where they are in act; so that in God, all things, and in particular opposites, are actual and coincident. However, to say of God, as Scotus does, that he has a perfection [a beingness] that is determined at a supreme degree, would oblige us to concede that this perfection has a different, and thus other, being than that which the same perfection has in creatures; God then could not be the Not-other. For Nicholas, God does not only possess perfection to a preeminent degree, he is identified with all things, he is in act everything that could be; and not all that he could be—a formula that would return to the possibles of God in himself. It is in this sense that God is the absolute maximum and infinite. But we have seen that it he is also contained in each creature [he is identified with its essence], and this is why he is absolute minimum. And as this identification with things applies even to the most insignificant among them, he possesses formally in act all degrees of being. Ultimately, the maximum is also the minimum; and also all the intermediaries between maximum and minimum. This is what is meant by the coincidence of opposites. It is clear that equality in God, and more generally, theological concepts, no longer obey the usual logic; quite on the contrary: it is on the basis of theology that Nicholas tries to create a new, metaphysical, logic.
19An important objection, raised by Counet,  to a “Scotist” interpretation of the Cusanian infinite, would be that, according to Nicholas himself, this infinity must be a measure of things, and that it is hard to see how an infinity could be a measure of the finite. The objection falls with the thesis of the Not-other God, since the Not-other is anterior to everything that is not him, given the fact that every definition includes him (the definition of X can be reduced to: X is not-other than X), it is therefore legitimate to say that he is the reason and the measure of all things.
20Still, we must insist on the fact that the coincidence of opposites sees God such as he is present in the human mind and for the human mind, and not as pure Act. The sight of God, the love of God, etc., are identified with a relation of ontological identity between God and the creature. And (this is the meaning of learned ignorance) there is no place for man’s seeking to know something of God outside of this sight, since this sight is our very being, and we cannot know him except through the categories of our proper nature.
21Let us come back to the initial question of a comparison between the Scotist infinite and the Cusanian infinite. We have seen that the latter, given the fact that it contains formally in act everything that can be, was firstly called absolute maximum. It is therefore close to the Scotist infinite that contains in act all the degrees of beingness of created or creatable beings. But it differs from it, since the metaphysical logic of Nicholas, characterized by the coincidence of opposites, implies that, since he is the maximum (containing all things in act), he is the minimum (contained qua essence in all things). Ultimately, the fundamental difference is that Scotus tries to define the divine essence within the framework of a positive theology (founded on the univocity of ens), whereas Nicholas, given the principle of learned ignorance, considers this to be impossible. Certainly, he cannot but recuse this univocity, which is meaningless for him, since, for him, God is Being, and not a being which can only be approached via its relation to the creature. This said, he is not in complete disagreement with Scotus, since the latter would affirm that, if one renounces the univocity of ens, theology would perish. Nicholas could concede this, by saying: a certain theology would indeed perish, and this is why we must invent another.
The Symbolism of Mathematical Figures
22Another aspect of Nicholas’s intervention on the infinite is the method of so-called “theological figures,” demonstrated most notably in the Complementary Theological Considerations: a method through which he considers that the examination of mathematical figures allows us to conjecture something of the divine form.
23Before discussing this, we must begin by recalling the status that he accords to mathematics (a status that, it seems, has no counterpart in Scotus).
24In a passage in the Complementary Theological Considerations, Nicholas describes the difference between the contemplation of things by intelligence “in truth,” and their perception by the senses.
[…] the mind—which itself is not free of all otherness (not free, at least, of mental otherness)—sees [geometrical] figures as free of all otherness [when it contemplates the drawing of a triangle it abstracts from the thickness of the mark, its dimensions, etc.]. Therefore, it views them in their truth, but it does not view them beyond itself. For it views them, and this viewing cannot occur beyond itself. For the mind views [them] mentally and not beyond the mind—just as the senses, in attaining [them] perceptibly, do not attain [them] beyond the senses but [only] within the scope of the senses. 
26In what follows, he explains that God has endowed man with an indispensible light with which to view material realities. All of this is coherent with scholastic tradition.
[…] in the mind a light-of-truth is present; through this light the mind exists [the action of God, his light, gives being to things, in particular to the mind] and in it the mind views itself and all other things [however this vision is only imperfect, as in a mirror—see the following citation]. By way of illustration: in a wolf’s sight there is a light through which the seeing occurs; and in this light the wolf sees whatever it sees. God concreated with the wolf such a light for its eyes, in order that the wolf would be able to hunt, for the sake of sustaining its life; without this light the wolf could not seek its prey at nighttime. If so, then God did not fail to concreate with the intellectual nature (which is nourished from the pursuit of truth) the light that is necessary for it. 
28Note the slippage of the meaning of the word “light,” from the divine light that gives being, to the light created by God which enlightens the mind and is indispensible in its search for truth.
29It is thus in a divine light that the mind grasps material realities. And Nicholas explains that, because mathematical figures cannot exist in the sensible world (where absolute equality or an absolute rectilinearity are impossible), but exist in God and in our mind, they allow the mind to grasp certain objects: mathematical figures are constructed by our reason and physical realities are but similar to them, so that the mind that possesses within it the ideal figures can measure the empirical figures against them.
30This is what is explained in the following quotation from De Possest, which ends by emphasizing the importance of mathematical science to the understanding.
CARDINAL: […] For regarding mathematical [entities], which proceed from our reason and which we experience to be in us as in their source [principium]: they are known by us as our entities and as rational entities; [and they are known] precisely, by our reason’s precision, from which they proceed. (In a similar way, real things (realia) [that is, empirical realities] are known precisely [by God but not by us], by the divine [intellect’s] precision, from which they proceed into being.) These mathematical [entities] are neither an essence (quid) nor a quality (quale); rather, they are notional entities elicited from our reason. Without these notional entities [reason] could not proceed with its work, e.g., with building, measuring, and so on. [This should be understood in relation to the light created by God to enlighten the mind in its search for truth, in the preceding quotation.] But the divine works, which proceed from the divine intellect, remain unknown to us precisely as they are. If we know something about them, we surmise it by likening a figure to a form. [We conjecture that the form in God is the mathematical figure.] Hence, there is no precise knowledge of any of God’s works, except on the part of God, who does all these works. If we have any knowledge of them, we derive it from the symbolism and the mirror of [our] mathematical knowledge [ex enigmate et speculo cognite mathematice]. E.g., from figure, which gives being in mathematics, [we make an inference about] form, which gives being [in fact it is God who gives being to the thing in identifying himself with its essence]: just as the figure of a triangle gives being to the triangle, so the human form, or species, gives being to a man [in the divine light the man is perceived in his essence]. We are acquainted with the figure of a triangle since it is imaginable; but we are not acquainted with the human form, since it is not imaginable and does not have quantity (whether the quantity be discrete quantity or a combination of quantities). Now, anything which does not admit of multitude or magnitude cannot be either conceived or imagined, and no image of it can be fashioned. Hence, it cannot be understood precisely. (For everyone who understands must behold images.) And so, with regard to [any of] these [divine works] we apprehend that it is, rather than apprehending what it is.
BERNARD: So if we rightly consider [the matter, we recognize that] we have no certain knowledge except mathematical knowledge. And this latter is a symbolism for searching into the works of God. Thus, if great men said anything important, they based it upon a mathematical likeness—for example, that species are related to one another as are numbers, that the sensitive is in the rational as a triangle is in a quadrangle, and many other such [comparisons]. 
32In conclusion, it is via the analogy figure/form in God, an analogy in fact justified only by the ideal character of the unique mathematical figure that corresponds to a variety of empirical forms, that the mind, which knows mathematical figures perfectly, can know, or at least postulate that it knows—since in fact it has no such knowledge—certain forms.
33But Nicholas goes beyond this. For him, not only do mathematical figures allow the mind to perceive objects in the divine light through their essence, but in addition he accords to these figures a symbolic status that allows the mind, in spite of the principle of learned ignorance, to conjecture something of the divine form itself. This is the famous double transposition (transumptio) described in Learned Ignorance:
But since from the preceding [points] it is evident that the unqualifiedly Maximum cannot be any of the things which we either know or conceive [learned ignorance]: when we set out to investigate the Maximum symbolically [rather than through images], we must leap beyond simple likeness. For since all mathematicals are finite and otherwise could not even be imagined, if we want to use finite things as a way for ascending to the unqualifiedly Maximum, we must first consider finite mathematical figures together with their characteristics and relations. Next, [we must] apply these relations, in a transformed way, to corresponding infinite mathematical figures. Thirdly, [we must] thereafter in a still more highly transformed way [transumptio, Lagarrigue: analogical extrapolation], apply the relations of these infinite figures to the simple Infinite, which is altogether independent even of all figure. At this point our ignorance will be taught incomprehensibly how we are to think more correctly and truly about the Most High as we grope in the midst of enigma [in enigmate] by means of a symbolism. 
36— A circle whose radius is growing indefinitely and which remains tangential at a point to a straight line, will coincide at infinity with the straight line. The circumference is then curved at minimum and straight at maximum, and curvature and rectilinearity are not opposed, but coincide.
37— In an isosceles triangle ABC, the sum of angles adds up to two right angles; if the top A is brought near to BC, and the angle at A tends toward two right angles, the triangle tends towards a straight line (segment) that is at the same time one and three; there is identity, not for reason but for the intellect, of the straight line and the triangle.
38The suggestion is that when distinct geometrical figures, like circles, triangles, straight lines, are taken to infinity, they coincide, and that in some way a coincidence of opposites is realized: the curve becomes straight, the triangle becomes rectilinear, etc. In reality finite figures are destroyed, and in the finite world, that of quantity, the coincidence is unrealizable; but it is through the use of quantity that we gain a presentiment of what happens at the level of the absolute maximum. After the first transposition, certain coincidences of opposites become apparent; coincidences which, in the second transposition, will be related to divine traits.
39But one could also say that this coincidence is potential at the level of the finite and actual at the level of the infinite, for an infinite figure is in act everything that the corresponding finite figure is potentially. Nicholas thus develops this latter property—with a very extensive use of the couplet potential/act—and draws paradoxical consequences from it, to the point where one sometimes has the impression that he is undertaking an exploration of all possible consequences. Here is an example: a finite straight line turning around one of its ends engenders a half-circle (it is potentially a half-circle), which in turn engenders a sphere by turning around its diameter; so, ultimately, the finite line is potentially a sphere; but the infinite line is identified with the essence of every (finite) line, and is in act everything that the finite line is in potential—which makes it possible to say that, since the finite line is indefinitely divisible into distinct parts (which are “other”), the infinite line is indefinitely divisible into infinite parts. And as there cannot be anything greater than an actual infinity, the line coincides with the infinite sphere. Here we are obviously far away from actual infinity as depicted in the fourteenth century—we shall speak of this again later.
40Following this double transumptio, Nicholas accords symbolic values to infinite geometric figures—circle, sphere, triangle—which he considers to refer to distinct aspects of the absolute maximum united in the divine reality:
41— The infinite triangle, as we said, is at once one and three, and symbolizes the trinity.
42— The infinite circle, with its identically infinite circumference and diameter and its centre (the midpoint of the diameter) that is situated everywhere; it will be called infinite in its turn and thus identical with the circle as a whole. This is the meaning of the passage from the Complementary Theological Considerations cited at the beginning of our article: the infinite circle is the maximum space, but, since it is identified with an infinite point, it is the centre that is everywhere; it thus symbolizes the maximum absolute, which coincides with the minimum absolute, and thus, with the divine unity.
43— The infinite sphere is, like the circle, at once identical to the diameter and to the surface; it is at once length, width, and depth. All that exists in act being integrated into the infinite sphere, the latter is the figure that represents the complete actuality of the divine principle, which is all that can be. 
44Finally, in having recourse to geometrical situations of this type, Nicholas symbolizes, through the relation of finite to infinite, the relation between God and creatures.
45— Thus, in the case of circles with a growing radius tangent to a given point, the curvatures vary and have neither maximum nor minimum, just as among beings there is neither maximum nor minimum. In addition, the curve is relative to the straight line, for that which is more curved is that which is further from the straight; which means that we can say that the straight line measures the curve. The inverse is not true, for the straight line has no need of the curve for its definition. This proximity of the curved line to the straight line can be considered as a participation of the curve in the straight line qua straight, and thus in the infinite straight line.  One can then put these relations of participation in correspondence with the appropriate relations between different types of entity: finite substances, symbolized by curves, participate in the entity of maximum absolute symbolized by the infinite straight line.
46— As to the relation between finite straight lines and infinite straight lines, it furnishes a model of the relation between God and finite beings, namely that God is the absolute maximum and that each real thing participates in him through its essence.
47What justification does Nicholas give for this symbolization through mathematics? In truth hardly any, unless that the method brings to light non-rational properties formally similar to the divine characteristics that his negative theology has allowed him to conjecture. This said, the absence of rational justification is not an objection to his method, since his infinite, unlike that of Scotus, is not attainable through rational argument; and, of course, because of learned ignorance, we know that we can have no perfect knowledge of the divine essence.
48We have seen that Nicholas makes constant use of the concept of the “infinite.” We could say that this places him in a lineage of the scholastic tradition of the thirteenth and above all the fourteenth century, particularly illustrated by Scotus and the Scotists, and among them particularly Jean de Ripa, and the philosophy of several English masters known as the Mertonians, and a Parisian, Nicolas Oresme.
49Jean de Ripa, extending Scotus’s work, establishes the existence of an infinite perfection of creatable beings, a perfection that must be distinguished from the divine perfection and must be inferior to it; he thus admits, in a very un-Aristotelian fashion, the possibility of two unequal infinities, whereas we have seen that the unicity of the infinite was one of the foundations of Nicholas’s reasoning.
50What is more, the infinite is one of the preferred themes of fourteenth-century natural philosophy. Although actual infinity is still considered impossible, it is studied at length in the framework of an imaginary physics (i.e. with the aid of arguments secundum ymaginationem: if God were to make an actual infinity, or if we were to imagine an actual infinity, etc.). Among the most refined studies, we must cite the monumental study of the comparability of infinities in question 3.12 of Oresme’s Physics,  which, after a meticulous study of the notion of comparability between magnitudes, establishes that two infinities are never comparable—a thesis that becomes, after Oresme, a widespread opinion. We could also cite Oresme’s Questions on Euclid’s Geometry about mathematical properties, like the divergence of the harmonic series, or the sums of series derived from geometric series.  These latter are taken up again in the Treatise on the Configuration of Qualities, where they are systematically calculated using geometric representations,  something which necessitates the introduction of the limits of figures, as in Nicholas. Nevertheless, the similarity stops there, because, whereas the Oresmian geometrical constructions are elements of a true geometric algebra allowing the demonstration of infinitary mathematical properties, Nicholas aims to use the symbolization of certain divine characteristics by “limit cases” as a basis for the elaboration of a new, metaphysical, logic. The latter seems, what is more, to have a largely exploratory character, as seems to be demonstrated by the variety of situations proposed, but above all the polysemy, often noted, of certain terms—that of the word “infinite” among others. With Nicholas, we are very far from scholastic precision, with its definitions, distinctions, objections, counter-objections, etc. So the similarity with Scotus remains very much a partial one, for if the Cusanian maximum absolute has certain points in common with the Scotist infinity in perfection, the approaches of the two philosophers are opposed in their presuppositions as in their objectives. Fundamentally, Nicholas’s approach in his utilization of mathematics appears profoundly original, or in any case unprecedented in the scholasticism of the thirteenth to fourteenth centuries.
51However his posterity, in particular his Leibnizian inheritance, hardly seems contestable. The simplest way to bring this to light is to return to the process of the passage of finite figures to the infinite, corresponding to the first part of the double transumptio described above—a process at the end of which is realized the coincidence of curve and straight line, of triangle and rectilinear, etc. Such coincidences are unacceptable in mathematics, where 1 is always different from 3, where the curve and the straight line are of different genres, etc. Moreover, this is why, for Nicholas, the squaring of the circle cannot be envisaged in the domain of rational mathematics, the method of Archimedes saying only that a curvilinear surface is neither larger nor smaller than the rectilinear surface. To pass from this observation to an equality, one must exit from this domain for a passage to the limit that is accessible only to the intellectual vision. It is this lesson that Leibniz will seem to have retained, when he undertakes to resolve the question of geometry through recourse to the infinitely small. For there it is a matter of mathematical entities between which equality is no longer mathematical formal equality, since Leibniz considers as equal two infinitely small magnitudes that differ only by infinitely small magnitudes of a higher order. Even if Leibniz makes no explicit reference to Nicholas of Cusa, here we find ourselves very close indeed to the intellectual vision of the latter.
See Jean-Michel Counet, Mathématiques et dialectique chez Nicolas de Cuse (Paris: Vrin, 2000), especially chapter 7. The mathematical writings of Nicholas have been translated into French in Nicholas de Cuse, Les Êcrits mathématiques, trans., introduction and notes Jean-Marie Nicolle (Paris: Honoré Champion, 2007). [The major works, in Jasper Hopkins’s English translation, are available at http://www.jasper-hopkins.info. Accessed May 2, 2013.]
The reference publication is: John Emery Murdoch, “From Social into Intellectual Factors: An Aspect of the Unitary Character of Late Medieval Learning,” in The Cultural Context of Medieval Learning (Boston Studies in the Philosophy of Science vol. 26), ed. John Emery Murdoch and Edith Dudley Sylla (Dordrecht, Netherlands and Boston, MA: Kluwer, 1975), 271–348.
For a clear presentation of Duns Scotus and his metaphysics, the reader is referred to Putallaz’s preface in Duns Scotus, Traité du premier principle (Tractatus de primo principio), Latin text edited by Wolfgang Kluxen, trans. Jean-Daniel Caviglioli, Jean-Marie Meilland, and François-Xavier Putallaz, ed. Ruedi Imbach, introduction by François-Xavier Putallaz (Paris: Vrin, 2001), 9–70.
Cf. Counet, Mathématiques et dialectique, 167–169.
Nicholas of Cusa, “Complementary Theological Considerations,” in Nicholas of Cusa: Metaphysical Speculations, ed., trans. J. Hopkins (Minneapolis, MN: Arthur J. Banning Press, 1998), 751, accessed May 2, 2013, http://www.jasper-hopkins.info. [“Complément théologique 2,” in Nicholas of Cusa, Trois traités sur la docte ignorance et la coïncidence des opposés, trans., introduction, notes and commentary Francis Bertin (Paris: Cerf, 1991), 95–96. As the difficulty of Nicholas’s text is well-known and translations diverge noticeably, most of the quotations drawn from one of the translations cited in the bibliography have been compared with the Latin text: Nicholas of Cusa, Opera omnia, Basiliae ex officina Henric Petrina, 1565.]
Duns Scotus, Lectura in IV Sententiarum, I, 3, §113, cited in Duns Scotus, Sur la connaissance de Dieu et l’univocité de l’étant (trans., introduction, and commentary Olivier Boulnois) (Paris: Presses universitaires de France, 1988), 12.
The translation used [in the original French article] is drawn from: Joël Biard and Jean Celeyrette, De la théologie aux mathématiques. L’infini au XIVe siècle (Paris: Les Belles Lettres, 2005), 41–50.
Aristotle, Physics III, 6, 207a 7–8.
Aristotle, Physics III, 6, 207a 8–9.
Biard and Celeyrette, De la théologie, 43.
Biard and Celeyrette, De la théologie, 44.
In all the quotations, the italics indicate a phrase emphasized by myself; the parenthesized passages being commentaries or personal explanations.
Biard and Celeyrette, De la théologie, 50–55.
Nicholas of Cusa, On Learned Ignorance, chapter 4, trans. J. Hopkins (Minneapolis, MN: Arthur J. Banning Press, 1990), available at http://www.jasper-hopkins.info. Accessed May 2, 2013, 8 [De la docte ignorance, trans., introduction and notes Jean-Claude Lagarrigue (Paris: Cerf, 2010), 84–86].
J.-M. Counet, Mathématiques et dialectique, 56, summarizing Kurt Flasch, Die metaphysik des Einen bei Nikolaus von Kues. Problemgeschichtliche Stellung und Systematische Bedeutung (Leiden: Brill, 1973), 158–174.
J.-M. Counet, Mathématiques et dialectique, 57–61.
Jasper Hopkins, Nicholas of Cusa on God as Non-Aliud: A Translation and Appraisal of De Non Aliud (Minneapolis, MN: Arthur J. Banning Press) available at http://www.jasper-hopkins.info. Accessed May 2, 2013, 1167.
Meister Eckhart, Œuvres, Traités et sermons, trans. P. Petit (Paris: Gallimard, 1987), 268–273; cited by Counet, Mathématiques et dialectique, 69–70.
Nicholas of Cusa, “De Non-Aliud,” in Hopkins, ed., Cusa on God as Non-Aliud, 1118 [cited in J.-M. Counet, Mathématiques et dialectique, 59–60].
Counet, Mathématiques et dialectique, 55.
Nicholas of Cusa, “Complementary Theological Considerations,” 748 [“Complément théologique 2,” 91; cited by Counet, Mathématiques et dialectique, 293].
Nicholas of Cusa, “Complementary Theological Considerations,” 748 [“Complément théologique 2,” 91–92], reviewed in Counet, Mathématiques et dialectique, 127–128.
Nicholas of Cusa, “De Possest,” in Hopkins, ed., A Concise Introduction, 936, accessed May 2, 2013, http://www.jasper-hopkins.info. [Trialogus de possest, trans. with notes by Pierre Caye et al. (Paris: Vrin, 2006), 73–75.]
Nicholas of Cusa, On Learned Ignorance, I.12: 20 [translation modified] [De la docte ignorance, 106].
Nicholas of Cusa, Complementary Theological Considerations, V: 755 [“Complément théologique 2,” 103].
Nicholas of Cusa, On Learned Ignorance, I.13: 20–22 [De la docte ignorance, 108–110].
Nicholas of Cusa, On Learned Ignorance, I.23; 38–39 [De la docte ignorance, 139–140].
Nicholas of Cusa, On Learned Ignorance, I.23; 38–39 [De la docte ignorance, 141].
Stefan Kirschner, Nicolaus Oresmes. Kommentar zur Physik des Aristoteles (Stuttgart: Franz Steiner, 1997), 256–270.
Edmond Mazet, “La Théorie des series de Nicole Oresme dans sa perspective aristotélicienne, ‘Questions 1 et 2 sur la Géométrie d’Euclide,” Revue d’Histoire des Mathématiques 9 (2003): 33–80.
Marshall Claggett, ed., trans., Nicole Oresme and the Medieval Geometry of Qualities and Motions (Madison, WI: University of Wisconsin Press), tertia pars, 413–435.