Francesco d’Appignano

on the Eternity of the World and the Actual Infinite

 Russell L. Friedman*

In her groundbreaking Diskussionen über das aktuell Unendliche in der ersten Hälfte des 14. Jahrhunderts, Anneliese Maier showed that Francesco d’Appignano was among the first of the scholastics to break with accepted Aristotelian ideas and hold that an actually infinite magnitude, multitude, or intensive form could exist.[1] Aristotle had strictly denied the possibility of an actual infinite, but had allowed for the existence of a potential infinite, e.g. in the continuum, where the number of parts could be increased without end; in general, as Aristotle himself says, the infinite is “what always has something outside it”.[2] Most medieval thinkers accepted Aristotle’s rejection of an actual infinite, but Maier isolated several thinkers from the first half of the fourteenth century, who thought that an actual infinite was possible. Among these thinkers Francesco d’Appignano stood out, stating unequivocally that “God could make an actually infinite effect”, and specifying that he understood this actual infinite to be a true transfinite that “exceeds any finite whatsoever by every determinate proportion given or able to be given.”[3] Moreover, Maier linked Francesco’s thought with that of the “radical infinitist”, Gregory of Rimini (OESA; read Sentences at Paris 1343-44; d. 1358), whose ideas on infinity have been called “by far the most sophisticated ... to emerge from scholasticism.”[4]

In part because of Maier, Francesco’s view of infinity is one of the very few areas of his thought that has received further attention from modern scholars. In particular, since Maier wrote, Francesco’s ideas on the infinite, and particularly on our knowledge of the infinite, have been discussed by Notker Schneider, who made use of Francesco’s long Metaphysics commentary, editing a substantial excerpt from it.[5] However, Francesco dealt at length with the subject of the infinite in at least six places throughout both versions of his commentary on the first and second books of the Sentences, and of these discussions, only one, on the infinite intensity of God’s power (= I Scriptum, d. 2, q. 3), has been edited, in this case relatively recently by Father Mariani.[6] Thus, because there is a great deal of little studied material that appears to contain interesting and significant ideas, further discussion of Francesco’s thought on this topic is warranted. I want to contribute to this discussion by focusing here on Francesco’s treatment of the possibility of the world’s eternity: q. 12 of the main version of his II Sentences (hereafter II Sent. (A)).[7] The question asked here is “whether the creation of any creatable thing was possible from eternity, and whether this was indeed the case” (Utrum creatio cuiuscumque rei creabilis fuerit possibilis ab aeterno, et utrum fuerit de facto).

                Francesco’s answer to whether any creatable thing was in fact created from eternity, i.e. whether the created world really did have no beginning, is, as we might expect, short and to the point. He says: “nothing besides God in fact existed from eternity; rather all things have existed from time.” But he continues: “nevertheless all things through <God’s> power could have existed from eternity”.[8] The remainder of his extensive (ca. 6700 word) treatment of the issue is devoted to showing that all attempts to prove that the created world must have had a beginning of its existence fail. It is Francesco’s view that there is absolutely nothing that prevented God from having created a beginningless world; God simply chose to do it that way, and we know this only on the basis of Scripture.

                I want to show in the following that Francesco’s treatment of the eternity of the world was composed nearly entirely as a reaction to the presentation of Peter Auriol on the same subject. This in itself should not be surprising: Peter, a French Franciscan, read the Sentences at Paris in 1316-18, just a year or so before Francesco himself lectured on the Sentences there in 1319-20, and Peter was Franciscan regent master in theology from 1318-20, so he was a figure who must have loomed large on Francesco’s horizon. In the process of showing the way that Francesco reacts to Peter, I hope to give an impression both of the way Francesco argues for his position, and of the way he understands the infinite. But first I must give a brief and simplified description of Peter Auriol’s discussion of the possibility of an eternal world.

                As Lauge Nielsen has recently shown,[9] Peter’s discussion of the eternity of the world is embedded in a complex overarching strategy to show the inadequacy of human cognitive apparatus to decide questions such as whether the world is eternal. Peter’s discussion of this issue, then, is structured in the following way. He argues on the one hand that there are no successful a priori arguments proving that God created the universe in time, i.e. there is no a priori proof that the universe is not eternal. On the other hand, he maintains that attempts to prove the eternity of the world fail, because the notion of ‘infinite past time’ involves a logical contradiction. Thus, one can prove neither that the world is eternal nor that it is not eternal. Again: this is a simplification of Peter’s view; nevertheless, it is true to say that he holds that proof that the world is eternal and proof that it is not eternal are equally problematic. Peter uses this apparent aporia to make a basic epistemological point: he claims that the reason why we cannot decide this issue one way or the other has to do with the limitations of our cognitive apparatus.[10] There is, of course, only one correct answer for Peter — we know from Scripture that the world had a beginning — but our limited intellects are incapable unaided of arriving at this correct answer in a convincing fashion.

                If I were to describe the use that Francesco makes of Peter’s discussion of the eternity of the world, I would say that he “hijacks” it. Francesco ignores entirely the intricate epistemological considerations that prompt Peter Auriol to argue that we can prove neither that the world had a beginning in time nor that it is eternal. Francesco instead uses Peter’s presentation as a framework for his own. This becomes clear when Francesco gives his justification for structuring his discussion the way he does. Under the major topic of whether any creatable thing could have been created from eternity, Francesco isolates two subtopics. The first subtopic is whether creation from eternity is possible in the case of “durable beings” (entia permanentia), that is to say beings whose existence is at least potentially a lasting phenomenon, not necessarily coming and going at an instant. The second subtopic is whether eternal creation is possible in the case of successive things (successiva), things like movement and time which have an existence characterized by flowing and continual development. Francesco claims:

 

First it will be seen about durable beings, second about successives, for concerning the latter there are special difficulties, on account of which some great doctors, granting that God could make durable entities from eternity, nevertheless deny this about all successives.[11]

 

Now, this in a nutshell sums up Peter Auriol’s discussion of the issue — if one ignores his overarching epistemological goal, and reads Peter’s treatment extremely literally. Thus, Peter discusses the question “is it logically inconsistent for a product to be produced from eternity just because it is produced by choice (elective) and contingently.”[12] Here Peter discusses exclusively durable beings or, as he calls them, ‘products’, arguing, as already mentioned, that no a priori argument succeeds in proving that God could not have created such beings from eternity, if God had so chosen. On the other hand, Peter also discusses whether the term ‘infinite past time’ is internally inconsistent. Here Peter’s answer was, as we have seen, a resounding ‘yes’ (we will return to his arguments for this position later). Infinite past time — i.e., an eternal successive thing — for Peter is a contradiction in terms.[13] So, reading Peter literally as Francesco did, one can say that Peter argued that durable beings could exist from eternity, while successive things, like time, could not, and this is precisely the position Francesco described in the quotation above. Now it should be stressed that Peter’s point was that, if we discuss the eternal duration of the universe from the one point of view, it is possible, while from the other point of view it is impossible; from this Peter concluded that there was an obvious problem with human cognitive apparatus. Francesco, on the other hand, is not in the slightest interested in epistemological conundrums; rather he adopts Peter’s structure for his own treatment because, as he says: “the whole universe is made up of these two kinds of beings, namely successives and durables.”[14] In Francesco’s view, Peter Auriol’s mode of proceeding gave the possibility of an exhaustive treatment of the issue. It also gave Francesco something to build upon and to respond to precisely and, as we shall see, Francesco uses Peter’s specific arguments in his own discussion. This is what I mean when I say that Francesco hijacked Peter’s presentation: he read it literally and reacted to it only at its face value. Let us turn to Francesco’s own discussion of the possibility of the world’s eternity.

 

Could durable beings have existed from eternity?

The first issue that Francesco addresses, then, is whether durable entities, these entia permanentia, could have existed from all eternity. A comparison of his discussion with that of Peter Auriol shows a great deal of overlap — not surprising when one considers that they both maintain that arguments trying to prove that durable beings could not have existed from eternity fail — although Francesco’s discussion is definitely more orderly and perhaps slightly more extensive than Peter’s is. Francesco’s conclusion is easily summarized:

 

I grant that God was able to produce some creature from eternity. For creation is nothing other than the total production of a thing — by its nature it is nothing more than this; thus that creation is from nothing (de nihilo), i.e. that nothing precedes it, this is something accidental (accidit) to creation, for the total production or creation of a thing is compatible with each of these, namely both with the fact that nothing precedes it and with the fact that nothing does not precede it, and consequently it is compatible with eternity, that is with not having a beginning to its duration, and with its opposite.[15]

 

For Francesco, creation, that is to say the total production of some thing, in itself is fully compatible with either having a beginning or not having a beginning. Probably the major motivation that lies behind his position is Francesco’s conviction that God is fundamentally free, and contingently wills both to create and the manner in which he creates. For Francesco, it would unduly limit God’s freedom to claim that he was not able to  have created from eternity, that he had to have created a world with a beginning along with time. Thus, Francesco argues as follows: “the exclusive efficient cause why a creature has being is because God gives it being; then the exclusive cause why a creature does not have being is because God does not give it being” (because “in exclusive causes, if affirmation is the cause of affirmation, also negation is the cause of negation”); Francesco continues:

 

Then God was either a necessary or a contingent cause of this non-entity of a creature, speaking about cause in the way in which a negation has a cause, namely [as] privative [and] not [as] positive. If [God was a] contingent [cause], then he was able to eliminate (expellare) this negation, i.e. the non-entity, through the affirmation opposed to it, namely by giving being to [the creature]. If [, on the other hand, he was a] necessary [cause of the non-entity of creatures], since God is the contingent cause of the entity of a creature, it then follows that he causes non-being more necessarily than he causes being; but this is false with respect to the same type of cause, and affirmation and negation are reduced to the same mode of causing.[16]

 

Thus, God was certainly able to create creatures whose non-being did not precede their being, i.e. whose being did not have a beginning. If God were unable to do this, then in creating, God would not act totally contingently, an unacceptable consequence for Francesco. God must be just as free to create from eternity as he is to have created in time, otherwise he would not have created contingently.

                It had been argued in the thirteenth century that, if God in eternity created an eternal world, then God would have created of necessity, because eternal things are necessary. Francesco denies this, comparing creation and predestination: God contingently predestines those who are to be saved from eternity; therefore he could also contingently create the universe from eternity, because if eternity is compatible with the contingency of predestination, then it is also compatible with the contingency of creation. “And if you claim” continues Francesco “that there is no similarity between predestination and creation, because through predestination a thing does not have any being, but through production God posits a thing in real being outside (extra), this does not stand up to scrutiny, because just as predestination contingently posits an object in willed being (esse volito), so production [posits an object] in real being.”[17] Again we see the stress on God’s utter freedom to have created however he liked, with or without a temporal beginning to the world.

                Francesco ends his presentation by confronting what was perhaps the most potent argument against the eternity of the world: that if the world were eternal there would now be an actually infinite number of immortal souls, which was ruled out by the traditional Aristotelian denial of the possibility of an actual infinite. Francesco does not think that this is necessarily the case, however: God could certainly have made the world eternal, yet have found a way to keep finite the number of immortal souls. Moreover, even if one were to concede that an infinite number of immortal souls resulted necessarily from the world’s eternity, Francesco does not believe that this would rule out an eternal world, because “I say that [an actual infinite] is not impossible for God.” As mentioned above, Francesco believed that an actually infinite magnitude, multitude, or intensive form was possible, and this is reflected clearly in his discussion of the eternity of the world.[18]

 

Could successive beings have existed from eternity?

Next Francesco turns to the question of the eternal existence of successive beings, that is to say of time and movement. If there were any doubt remaining that Francesco had structured his entire discussion in reaction to Peter Auriol’s treatment of the same issue, here this doubt is immediately dispelled. First, Francesco repeats that “some, granting that God could make durable beings from eternity, nevertheless say that no successive being, for example movement or time, can be made from eternity.”[19] This, as we have seen, is Francesco’s literal reading of Peter’s text, not taking into account Peter’s epistemological goal. A great deal of Francesco’s discussion of this topic is centered around Peter’s claim that ‘infinite past time’ involves a contradiction in terms. In support of Peter’s claim, Francesco first presents what he calls “five specific arguments”; thereafter he offers some “common” or “general” arguments to the same end. Francesco goes on to rebutt all of these arguments.

                The “five specific arguments” are taken directly from Peter Auriol’s II Sentences, and are presented by Francesco in the same order in which they are found there; thus, there can be absolutely no doubt that Francesco had Peter in mind when he wrote this text.[20] The reason why Francesco calls these five arguments ‘specific’ is that in them Peter Auriol employed his own understanding of the infinite in order to show that the phrase ‘infinite past time’ involves a contradiction. Peter thinks that infinity is always “act mixed with potency”; for Peter, by definition something infinite has a successive type of existence, constantly changing and developing, and never reaching an end point or completion.[21] Now, clearly, because any infinite is “act mixed with potency”, no infinite admits of fully actualized real existence. This is the basis of all of Peter’s arguments against infinite past time. Thus, according to Peter, because what is past is gone and hence completely actualized, so that no potentiality is left in it at all, the past is inconsistent with infinity, which is “act mixed with potency”. ‘Infinite past time’, then, is a contradiction in terms.[22] Further, any attempt to claim that we can, one day at a time, move back from today to yesterday to the day before yesterday and so on into infinity, Peter derides as confusing the past with the future: time is unidirectional, therefore this thought experiment merely deceives us into believing that an infinite past is possible.[23]

                Francesco’s response to Peter Auriol’s line of argument makes it immediately clear why Peter was an obvious choice as an opponent for Francesco to argue against: the differences between the way Francesco thinks about the infinite and the way Peter does are so marked that they allow Francesco to explain his position very clearly through contrast with Peter. Francesco maintains:

 

In reply to the evidence of the solution of all their arguments, it must be known that all of these arguments or their motives imagine that, if there were infinite past time, it would only be potentially infinite. But with this assumed these arguments are indeed unassailable. But I imagine the direct opposite. For I maintain that, if time past were infinite, it would not be potentially infinite or an infinite in act mixed with potency, rather it would be actually infinite (infinitum in actu) — note: not actually remaining or existing, but actually passing by (in actu praetereunti), for we ought to attribute such infinity to the thing as befits its entity. And so, just as the infinity of a durable thing (res permanens) is all at once, just as its entity is, so the infinity of a past successive thing would be, certainly not actually so that all its parts actually existed, but successively actually passing by (in actu praetereunti), just as its entity is.[24]

 

Francesco denies on two planes Peter Auriol’s contention that all infinites are successives which always involve act mixed with potency. First, it is not true that all infinites are successives: whereas for Peter an infinite only has a successive type of existence, always changing and developing and never being fully actualized, as we have seen, Francesco holds that God could create an actually infinite durable being, like a magnitude, multitude, or intensive form. Second, while it is certainly true that time and movement are successive beings, this in itself, for Francesco, does not preclude their actual infinity, as Peter claimed it did. In fact, Francesco gives a kind of taxonomy of infinites depending upon the type of infinite entities we are dealing with:[25]

                1) actually infinite durable being, that is the infinite magnitude, multitude or intensive form that God could create.   

                2) actually infinite successive being: if past time were infinite, this would describe it.

                3) potentially infinite successive being: this is the future.

What Peter Auriol said was true of all infinites by definition, according to Francesco really applies only to a truly potential successive infinite like future time (nr. 3), where act is in fact mixed together with potency. Past time, on the other hand, is called by Francesco an actually infinite successive being, i.e. actually infinite and not merely potentially so, and yet also continually developing as time passes by, as we would expect from a successive thing, like time.

                Francesco makes still more clear his conception of infinite time, and thereby of the infinite, when he confronts several of the most difficult arguments that had been raised against an infinite past — arguments that he calls ‘common’ since they had been common in the scholastic discussion since at least the middle of the thirteenth century.[26]

One of the most forceful of these arguments had been based on the claim that an infinite, since it is endless, cannot be transversed; therefore infinite past time is impossible since all past time has in fact been traversed.[27] While Francesco grants that no infinite can be traversed, nevertheless he denies that in the case of infinite past time an infinite would be traversed at all, if traversed is understood as requiring both a starting point and a finishing point, since the world’s being eternal would mean precisely that it had no temporal beginning. Thus, Francesco claims that the eternity of the world would indeed require that all past time, even infinite past time, had passed by or had existed, but not that an infinite had been traversed from start to finish.[28] This is in fact a major difference between past and future time, according to Francesco: while the past, as we have seen, could be an actually infinite successive being, and hence an infinite amount of time can be actually passed by, nevertheless the future, since it is a potentially infinite successive being, can never be totally passed by.[29] Once again we see that different types of infinites have different characteristics depending on their natures.

Another argument Francesco addresses took its point of departure from the revolutions of the sun and the moon: the moon revolves around the earth twelve times for every once that the sun revolves around the earth; therefore if the past were infinite, one infinite would be twelve times larger than another, an unacceptable solution.[30] Francesco replies first that this argument would just as well argue against an everlasting future as it would against an everlasting past; so it cannot be demonstrative.[31] Moreover, even though for every single revolution of the sun there are twelve revolutions of the moon, if all of these revolutions were added together over the course of infinite time, all of the revolutions of the moon would be infinite as would all of the revolutions of the sun. Infinites simply do not work the same way as finites do for Francesco — an infinite is not a finite just bigger, it obeys fundamentally different rules. Francesco says:

 

Although every revolution — past or future, of the sun or of the moon — is finite, nevertheless taken all together they are infinite, both those of the sun and of the moon. And so all [the revolutions] of the moon cannot be greater than all [the revolutions] of the sun.[32]

 

That infinites are governed by different rules than finites for Francesco is confirmed by his response to another of the common arguments against infinite past time. This argument insists that, if time could be from eternity, then the part could be equal to the whole, for if I took all of infinite time up to today and removed one day from it, what remained would still be infinite; therefore the part — infinite time minus one day — is equal to the whole.[33]

                Francesco simply denies that infinites work like this:

 

Take away some part from infinite time or from any other infinite quantity and what remains is infinite. And when you say in the minor that [the remaining infinite] was part of the preceding quantity, I say that that is false, for it is inconsistent with the infinite as it is infinite to have the nature of the part, or even of the whole.[34]

 

Again we see that infinites simply work differently than finites do: there is no part and whole, no greater and lesser with the infinite; normal intuitive laws of finite mathematics do not apply. As Anneliese Maier remarked,[35] Francesco looks at the infinite as a true transfinite, not simply as a maximum finite number. The infinite obeys laws of its own, for Francesco, and even comes in different varieties depending on the type of entity the infinite in question possesses.

 

Conclusion

Summing up, there are three prominent features of Francesco’s discussion of the eternity of the world as I have presented it. First, Francesco’s entire discussion is built up around a confrontation with Peter Auriol’s discussion of the same subject. Francesco, however, in a sense “hijacks” Peter’s discussion, by disregarding Peter’s intricate argumentation for the inadequacy of human cognitive apparatus for determining the question of the world’s perpetual or finite duration. Francesco instead reads Peter’s treatment literally: on Francesco’s presentation, Peter was correct to maintain that no a priori argument can prove that creatures were not able to have existed from eternity; on the other hand, Peter’s insistence that infinite past time is logically inconsistent is simply incorrect. Thus, here we have a fine example of the important role that Peter played in the formation of Francesco’s thought.[36] It can be suggested on this basis that, rather than as some type of “scotist”, Francesco is most fruitfully looked at as a highly independent and innovative thinker who in many contexts reacted strongly to Peter Auriol’s system.

                Second, Francesco’s proofs for the proposition that durable things (entia permanentia) could be created with no temporal beginning rest to a large degree on his understanding of divine freedom and power. According to Francesco, we would be unduly limiting God’s freedom, by holding that he were bound in some way to create the universe along with time.

                Third and finally, should be mentioned the view of the infinite that is revealed in Francesco’s proof that time and movement could be infinite. Francesco, in line with his position that God can produce actually infinite magnitudes, multitudes, and intensive forms, holds that infinite time or movement are possible: since time is a successive entity, infinite past time would be an actual successive infinite, while infinite future time is a potential successive infinite. Francesco proposes, then, a type of taxonomy of infinite things based upon the type of entity they possess. Moreover, Francesco clearly thinks that the infinite is more than merely a maximum finite, it is a transfinite, which obeys its own special rules: thus, for infinites, the whole is not necessarily greater than the sum of its parts and one infinite cannot be greater than another. We see here that Francesco may well represent an important step on the road to Gregory of Rimini’s extreme infinitism.

 

 

 

APPENDIX

 

Francesco d’Appignano, II Sent. (A), q. 12, art. 1 (B 24rb; C 121ra; Y 107rb; Z 110rb-va)   Arguo enim primo sic: infinitum est actus permixtus potentiae, ex hoc sic illud cui repugnat actus permixtus potentiae non potest esse infinitum. Sed motui sive tempori praeterito repugnat quod sit actus permixtus potentiae, tempus enim praeteritum totum fuit in actu et non est aliquid eius in potentia ad aliquid, cum totum transierit et in actu completo fuerit. Ergo, etc. Et si forte dicas quod accipiendo aliquam partem temporis praeteriti, puta diem hesternam, et sic post illam accipiendo aliam priorem, et sic quasi retrocedendo accipiendo unam post aliam, erit processus in infinitum, quia numquam devenietur ad primum instans temporis, contra hoc arguitur et bene, quia tunc per istum modum non accipitur tempus praeteritum per modum praeteriti sed magis secundum modum futuri.                             Praeterea, secundo arguitur sic ad principale: ratio infiniti in quocumque accipitur ex ista parte ad quam est processus, non ex illa parte a qua non est processus; et ratio huius est quia ex parte a qua est status non itur in infinitum, sed ex parte ad quam proceditur; hoc patet in numeris in quibus non proceditur in infinitum ex parte unitatis, sed tantum eundo versus multitudinem; hoc idem patet de divisione continui quae quidem vadit in infinitum ex parte magnitudinis ad quam fit processus divisionis. Sic ergo patet ista, videlicet quod ratio infiniti accipitur tantum ex ista parte versus quam est processus non ex illa ex qua non est processus. Sed processus in successivis est a praeterito in futurum. Ergo impossibile est tempus praeteritum esse infinitum. Sed si esset aeternum, esset necessario infinitum. Ergo etc.                   Praeterea, tertio arguitur sic: de ratione infiniti est quod non sit totum in actu, sed quod semper sit in potentia ad aliquid. Sed id quod est praeteritum totum extitit in actu ita quod non restat aliquid eius. Ergo impossibile est quod praeteritum fuerit infinitum. Et si dicas ut prius quod est infinitum procedendo versus partem temporis praeteriti accipiendo unam post aliam in infinitum, iam commutas praeteritum in futurum.                       Quarto arguitur sic: de ratione infiniti est quod habeat esse incompletum et imperfectum, quia semper habet annexam potentiam. Sed praeterito repugnat habere esse incompletum et imperfectum, cum totum quod pertinet ad rationem eius sit completum et nihil restet accipiendum. Ergo etc.         Praeterea, quinto: secundum Philosophum III Physicorum infinitum est extra quod semper est aliquid accipere. Sed extra tempus praeteritum non est aliquid accipere sive non restat /Z 110va/ eius aliquid accipiendum, cum totum sit acceptum. Ergo impossibile est tempus praeteritum esse infinitum.

Peter Auriol, II Sent., Quaestio prohemialis, art. 2 (ed. Rome, 1605, pp. 3b-5b)   Primo sic: de ratione infiniti est quod sit actus permixtus potentiae; rationi autem praeteriti repugnat contradictorie quod sit actus permixtus potentiae; ergo ratio infiniti repugnat contradictorie rationi praeteriti [...] Respondetur ab aliquibus ad minorem, quod in praeterito semper est potentia coniuncta cum actu, sicut in infinito, quia quando vis accipere praeteritum, quando aliquid acceperis, adhuc restat aliud sumendum. Ut accipe diem hesternam, restat tertia dies accipienda, et sic semper in infinitum, nam illud quod acceptum est, est in actu; quod restat accipiendum, est in potentia. Contra istam solutionem ostendo quod confirmat propositum, et reduco rationem sic. Ex parte illa inest tempori infinitas, ex qua inest ei potentia, et non ex illa, ex qua nulla inest potentia sibi, sed ab eo excluditur omnis potentia. Non enim habet infinitatem ex illa parte ex qua tantum est in actu per Commentatorem, qui dicit ubi supra quod infinitum non est actus entis, quod est in actu puro non admixto potentiae et quod infinitum non est in re, nisi secundum potentiam exhibentem in actu et ex continuatione potentiae cum actu non ex corruptione eius, alias corrumperetur ratio infiniti. Sed tempori, ex parte qua praeteritum est, nulla inest potentia, et hoc non ex parte qua accipitur in ante, sed ex parte futuri, qua accipitur in post. Ergo infinitas solum inest tempori a parte post et in tempore futuro et non a parte ante et in tempore praeterito [...] Secunda ratio principalis accipitur ex processu, et arguo sic. Ratio infiniti est ratio, quae necessario in processu est, ita quod est accidens entis, quod est in processu, nec invenitur nisi in processu[m] et ex parte illa ad quam proceditur, non autem ex parte illa a qua proceditur. Sed tempus non procedit in praeteritum, sed a praeterito. Ergo omnis infinitas est in tempore praecise a parte futuri. Maiorem declaro primo exemplo [...] Exemplum pono de infinito reperto in numero et in magnitudine, quod non est nisi in fluxu. Infinitas enim utrobique est processus versus partem ad quam, non versus partem a qua, non enim processus numeri in infinitum est procedendo ad unitatem sed ab unitate. Consimiliter processus divisionis in infinitum in continuo non est ad totum, sed a toto. Ergo patet propositum in exemplo. Probo autem idem per rationem, quia ex quo infinitas in processu non est per rationem termini a quo, quia ille non est in potentia, sed in actu perfecto. Ergo est per rationem termini ad quam. Minor autem rationis patet, quia tempus procedit a praeterito in futurum, sicut numerus ab unitate, ergo infinitas est ex parte temporis futuri, et non ex parte temporis praeteriti [...] Tertia ratio principalis sumitur ex non cessare, et est ista. Illud de cuius ratione est quod eius potentia non exeat ad actum nec possit cessare, illud repugnat rationi praeteriti. Sed de ratione infiniti est quod numquam cesset et quod eius potentia numquam exeat ad actum. Ergo [...] Respondetur quod in praeterito remanet adhuc potentia, quoniam quantumcumque intelligam praeteritum in infinitum, semper accipio aliquid in actu, et remanet mihi accipiendum aliquid in potentia. Concedo quod intellectus possit hoc facere et intelligere in praeterito aliud et aliud in infinitum procedendo; et similiter quod Deus posset illud facere, sed hoc esset procedendo in post et non in praeteritum, sed quasi in futurum; sed accipiendo praeteritum in ratione praeteriti, de ratione eius est quod totum exiverit in actu, et ideo dicere quod praeteritum sit infinitum, est dicere quod infinitum sit in actu exclusa ratione potentiae. Quarta ratio principalis est ista secundum Philosophum et Commentatorem. Ratio infiniti est ratio partis et causa huius est quia ratio infiniti est ratio imperfecti et incompleti. Tunc sic: ex parte illa inest tempori infinitas, ex qua parte habet rationem imperfecti et incompleti et rationem partis, sed quod sit imperfectum et incompletum et pars non inest tempori sub ratione qua praeteritum, sed sub ratione qua est futurum; immo sub ratione qua est praeteritum habet quod sit completum et perfectum. Quare etc. [...] Ultima ratio sumitur ex dictis Philosophi [...] et ideo [Philosophus] concludit quod infinitum est extra quod est aliud semper. Tunc sic. Illi repugnat infinitas cui repugnat esse extra quod aliud est. Sed tempori praeterito, ratione qua praeteritum, repugnat esse extra quod est aliud. Ergo etc.