Questions Sixteen and Seventeen
<QUESTIONS SIXTEEN
AND SEVENTEEN>
[QUESTION SIXTEEN
Whether Quantity is a Genus] Of quantity, one<type> is continuous, the other is discrete.1
t is asked first whether quantity is a genus.
1. It seems that it is not <a genus>:
Since "no genus is predicated denominatively of a species."2
But this is true: "a line is of a certain quantity (quanta)," and thus <it is predicated> of others which are posited as species of quantity. Therefore, none of them is its species; therefore,
<quantity> has no species; therefore, it is not a genus. Proof of the major: no univocal predicate is denominative; every genus with respect to its species is a univocal predicate, according to Aristotle earlier <in the text>;3 therefore, etc. Proof of this major: since the account of a univocal predicate is the account of a subject, which is manifestly true of a genus with respect to its species; but the account of no denominative is the account of a subject, as it is said in the chapter "On Substance,"4 since
"white" is predicated of a body; however, it is impossible for the account of white to be predicated <of a body>.
2. Second, quantity does not have species, since it is not predicated equally of continuous and discrete; and a genus is predicated equally of its species. Proof of the assumption: because continuous quantity is naturally prior to its division; since
1. Categories, Ch. 6 (4b 20–21).
2. Topics, Bk. II, Ch. 2 (109b 5–6).
3. Categories, Ch. 5 (3b 8–9).
4. Categories, Ch. 5 (2a 31–32).
QUESTIONS 16–17
insofar as a continuum is divisible, it is naturally prior to its division. Therefore, a continuum is also prior to number, since number is posterior to the division of a continuum, since it is caused from that <division>, according to Aristotle in Bk. III of the Physics.5
3. To the opposite is Aristotle.
[QUESTION SEVENTEEN
Whether Quantity is One Genus]
econdly, it is asked whether quantity is one genus.
It seems that it is not:
4. According to Aristotle in Bk. I of the Posterior Analytics and in Bk. X of the Metaphysics:6 "In every one genus there is one first and smallest that is the meter and measure of all the others." But in quantity there is not one first and smallest, but two, namely, unit and point. Therefore, quantity is two genera.
5. Second, all species of any genus are opposites, since they include either first opposite differentiae of the genus, or opposite differentiae dividing another intermediate genus, if they are contained under some intermediate genus <that is> the same; and <something> including one opposite is not predicated of another <thing> including the other <opposite>, in the way in which the differentiae divisive of a genus ought to be opposite. Therefore, no species of one genus is predicated of any species of that genus. But some species of quantity is predicated of another denominatively, for example, "a line is numbered"; therefore, etc.
6. Third, an attribute and a subject are not in the same genus as species. First, since an attribute is predicated denominatively of a subject, therefore also the genus of an attribute
<is predicated denominatively of a subject>; but the genus of 5. Physics, Bk. III, Ch. 7 (207a 32–b 2).
6. Posterior Analytics, Bk. I, Ch. 22 (84a 8–10), and Metaphysics, Bk. X, Ch. 1
(1052b 18).
JOHN DUNS SCOTUS
a subject is not predicated denominatively of a subject; therefore, the genus of a subject and <the genus> of an attribute are not the same. The first implication is evident <by arguing> from an inferior to a superior. Second, since then an attribute would not be predicated of a subject, as is evident from the preceding argument, which shows that no species of any genus can be predicated of another <species>; but longitude is an attribute of a line, which is in the genus of quantity, and longitude is in the genus of quantity; therefore, the genus of quantity is not one.
7. To the opposite is Aristotle.
[I. To The Sixteenth Question] 8. It must be said that quantity is a genus, since it is predicated essentially (in quid) of many <things> differing in species,7 for example, of continuous and discrete quantity. For when it is asked "what" (quid) each of them is, one can suitably respond
<by saying> "quantity." And it is the most general <genus>, since it does not have a higher genus.8 For nothing is predicated essentially (in quid) of it except being, which is not a genus, since it is not univocal.9
[II. To The Seventeenth Question] 9. And it is one genus, since it is said according to one account (ratio) of all its species.
[A. What is the Account of Quantity:
The Opinion of Albert] [1. Exposition of the Opinion] 10. But what that one account is, is posited <as> a twofold way.
In one way the account of measure is posited.10 First, since 7. Topics, Bk. I, Ch. 5 (102a 32).
8. Porphyry, Isagoge, Ch. "On Species" (AL I6 9; ed. Busse, 4.17–18).
9. For a fuller discussion on the nature of quantity, see Robert Andrews, "Question Commentaries," 289.
10. Albertus Magnus, Praedic. tr. 3 c. 1 (ed. Borgnet, I 194b).
QUESTIONS 16–17
Aristotle11 proves here that speech is a species of quantity through the fact that it is measured. Second, since the species of quantity are varied according to the diversity of its account.
Since some continuous measure is intrinsic to <the thing> being measured, <and> some <continuous measure> is extrinsic. The intrinsic <one> measures either according to one dimension only, and then it is a line; or <according to> two <dimensions>, and then it is a surface; or <according to> three <dimensions>, and then it is a body. The extrinsic <measure> is twofold: <the measure that is> not common with others (propria), and then it is place; or <the measure that is> common <with others>, and then it is time. And the measure of place expresses <what is> not common with others (propria), since only one <place> is of one <thing> being measured; <and> time is common, since it is the same for many <things> being measured. Time itself, however, is not the proper (propria) measure of anything temporal.
It is also manifest that number and speech are measures.
[2. To the Opinion of Albert] 11. Against that <opinion>: the account of a genus is equally found in all its species; the account of measure is not <equally found> in this way in continuous and discrete <quantity>. Since the greatest property proper to one is to be a measure, and since that <property> is essentially (per se) the principle of number, from that <property> the account of measure is first distributed to number, and from number to continuous <quantities>, as is manifestly held in Bk. X, Ch. 2, of the Metaphysics.12
12. If it is said that the discrete is prior to the continuous <quantity> in itself, not in comparison to its own genus:—this
<reply> concedes the point, since neither participates in the account of the genus through the other. But the continuous does not have the account of measuring except from the discrete, as it is taught there above.13 Therefore, the account (ratio) of measure is not the account (ratio) of quantity, as it is a genus.
11. Categories, Ch. 6 (4b 32–35).
12. Metaphysics, Bk. X, Ch. 1 (1052b 20–1053a 9). The argument is that if measure is the ratio of quantity, unity (or one) would also be a species of quantity, which it is not since it is not included in number.
13. Cf. supra, n. 2.
JOHN DUNS SCOTUS
[B. The Opinion of Scotus] 13. For this reason it is said otherwise that although measure pertains to quantity essentially (per se), nevertheless it is not the essential account of quantity insofar as the account of the genus is attributed to it; but the account of divisibility <is the essential nature of the genus>. And this <is the case> from that division which is <a division> into parts of the same sort (ratio). This is shown in three ways:
14. First, in Bk. V of the Metaphysics,14 where "quantum" is defined, insofar as it can have a definition expressing at least what is said through the name, there it is described thus: "quantum is what is divisible into those <parts> which are in it, of which <each> single <one> is by its nature this something," that is, to be divided into parts of the same nature (ratio). Therefore, quantity is essentially (per se) divisible into parts of this kind; for where there is an essential predication (per se) in the first mode, it holds from concrete <things> to abstract <things>.15
15. Second, it is shown <in this way>: since continuous and discrete quantity are not defined here through measure, but through this: "of which the parts are joined together,"16 etc.
Therefore, the essence of them is to be partible.
16. Third, since in Bk. V of the Metaphysics,17 the species of quantity are taken through the nature (ratio) of divisibility, since
<what is> divisible into non-continuous <quantity> is number or speech; <and what is divisible> into continuous <quantity> according to one dimension is a line; according to two <dimensions> is a surface; <and> according to three <dimensions> is a body. And time and place can be taken according to other differences of divisibility.
17. Against this second way:18 continuous quantity is potentially divided; discrete <quantity> is actually <divided>; but those
14. Metaphysics, Bk. V, Ch. 13 (1020a 7–14).
15. It should be noted that quantum is the concrete form of quantitas, which is the abstract form.
16. Categories, Ch. 6 (4b 25–5a 14).
17. Metaphysics, Bk. V, Ch. 3 (1020a 7–14).
18. Cf. supra, n. 15. These arguments, namely, nn. 17–19, are also alleged
QUESTIONS 16–17
things which are actually and potentially some <one thing>, are not equally that <one thing>; therefore, divisibility does not pertain equally to continuous and discrete <quantity>. The assumption is evident, since the parts of a number are actually divided from each other, <whereas> the parts of a continuum are not.
18. Second, "that of which it is essentially (per se) a potency, is also its act."19 Therefore, if continuous quantity is essentially
(per se) divisible, it can be essentially (per se) divided; therefore, continuous quantity can be essentially (per se) non-continuous, which does not seem to be given.
19. Third, Aristotle20 proves that speech is a quantity, since it is a measure (est mensura); therefore, this is the essential account of quantity.
20. To the first <argument>,21 one can say that discrete quantity is one and undivided while it remains in its species—whence it <has> its unity—although its material <elements>, which exist under this form, are divided. For example, "five" <while> remaining in its species is formally undivided, since its division destroys the form of five, for when its division has been made into "three" and "two," it no longer remains "five"; although its material <elements> as unities are divided. Therefore, it must be denied that it is taken concerning the discrete <in this way>.
21. To the second <argument>,22 one can say that the act of this potency, <namely,> "a continuum is divisible," is not this: "a continuum is divided," but this: "a continuum is being divided," that is, it is in the process of division; and this can be essentially
(per se) in a continuum.
22. To the third <argument>:23 it is not denied that to be a quantity is essentially (per se) to be a measure; but <it ought to in Duns Scotus's Questions on the Metaphysics of Aristotle in Bk. V, q. 9, nn. 2, 4, and 8.
19. On Sleep and Waking, Ch. 1 (454a 8). Etzkorn and Wolter translate this phrase as: "the subject of actuality is identical with that of potentiality" in Questions on the Metaphysics of Aristotle, V, q. 9 n. 4.
20. Categories, Ch. 6 (4b 32–35). Actually, Aristotle says that speech is measured by long and short syllables, not that it is a measure.
21. Cf. supra, n. 17.
22. Cf. supra, n. 18.
23. Cf. supra, n. 19.
JOHN DUNS SCOTUS
be said> that this aspect is not essentially of quantity as its genus, since it is not equally in its species.
[III. To The Principal Arguments Of The Sixteenth Question] 23. To the first argument of the first question,24 one can say that quantity is not said denominatively of a line <taken> absolutely, nor of some of its species taken absolutely; but <it is said of a line> in the abstract and essentially (in quid). But it is said denominatively of a line in its matter and of other species <in their matter>, and so its species is taken as a concrete
<accident>. And it is not astonishing that a concrete <accident> of a genus is predicated of a concrete <subject> of <another> species. Therefore, however common the use of speaking is to say "line is a quanta" or "divisible"—and so of other species— this is simply false about these as they are species, since then they would be the subjects of quantity or of divisibility. For such a mode of predicating pertains to all accidents with respect to their subjects.—Otherwise, it is said that the major is true in every other genus than quantity, and it is especially not <true> of this one. Since quantity is not only the principle of measure or division for other <genera>, but also for itself; and for this reason it names itself and its species, which is not true of the others.
24. To the second <argument>,25 I say that in every genus some species can be prior to other <species> in being, <when> comparing those among themselves; but none is prior to another in participating in the nature (ratio) of the genus. Thus, continuous quantity is naturally prior to discrete <quantity>, as the argument (ratio) shows, although not in receiving the predication of quantity, since divisibility exists in neither of the species through the other.
24. Cf. supra, n. 1.
25. Cf. supra, n. 2.
QUESTIONS 16–17
[IV. To The Principal Arguments Of The Seventeenth Question] 25. To the first argument of the second question,26 it is said that point and unit are not two absolute firsts, since point is reduced to unity as to what is prior to it. "For a point is a unit having position," as it is said in Bk. I of the Posterior Analytics;27 and in this way only unity is first.
26. Against this: "as principle is to principle, so also is a caused <thing> to a caused <thing>."28 Therefore, if unity is a material or a formal <element> in a point—one of which is necessary if a point is a unit having position—then a material or a formal <element> in a continuum will similarly be discrete. It is manifest that it is not a material <element>, since a continuum is not comprised from discrete <elements>; therefore, it is a formal <element>, which Aristotle denies in Bk. VII of the Metaphysics,29 <when he> reprehends Plato, who posits it thus.
27. Therefore, it must be said that just as magnitude is not multitude, neither materially nor formally, so point is not unit, neither materially nor formally; since point is the indivisibility of magnitude, and unit is <the indivisibility> of multitude.
Neither is this definition, "point is a unit," etc., from Aristotle.
But if this <position> is said anywhere about a point, it is <said> by Plato, who posits numbers to be the formal <elements> in a continuum, and so <he holds that> unity is in a point. But neither according to Aristotle nor according to Plato is this definition held in Bk. I of the Posterior Analytics, but <it is held> in this way:30 "Unity is a substance without position; however, a point is a substance having been posited," which is manifestly false, according to Aristotle, but true according to Plato, who claims that quantities are differentiae of things.
26. Cf. supra, n. 4.
27. Posterior Analytics, Bk. I, Ch. 27 (87a 35–37).
28. Latin: Sicut principium ad principium, ita principiatum ad principiatum. Cf.
Auctoritates Aristotelis (ed. J. Hamesse, p. 317).
29. Metaphysics, Bk. VII, C 11 (1036b 14–20).
30. Cf. supra, n. 25.
JOHN DUNS SCOTUS
28. Otherwise, to the argument31 one can say that the major, namely, "in every genus <there is one first and smallest that is the meter and measure of all others>," is not true of every most general genus, since scarcely in some most general genus can some one first (aliquod unum primum) be assigned as the measure of all the other <species>, or <actually it can be assigned> in none of them <other than quantity>. But this proposition is understood about a genus in the natural order; and it expresses that genus in the natural order, to which a unity of a receptive
<thing> corresponds; that is, it is only said of forms having the same receptive <nature> to a first <measure>. And Aristotle exemplifies this mode there where he states this proposition:
"such a genus is not quantity."32
29. Otherwise, one can say that although there are two first <things> and smallest <things> in the genus of quantity, nevertheless there is only one first and smallest there that is the measure of the others. For a point is not the measure of a continuum, since then taken at different times it would cause a continuum, and the continuum would be composed from indivisibles. But one is essentially the measure of number; however, one applied to some part of a continuum is the measure of the continuum, as it is said in Bk. X of the Metaphysics;33 therefore, only one is the measure in this genus.
30. To the second argument,34 I concede that no species of a genus is predicated of another <species> in the abstract, insofar as <the species> includes opposite differentiae. Nevertheless, it is possible that <a species can be predicated of another species> in the concrete, insofar as the species, which is made a subject, is taken as it is in matter, and not as it is an opposite species.
31. To the third <argument>, 35 I say that "longitude," "latitude," "magnitude," "multitude," etc., are equivocals, as they denote quantity and excess in quantity. In the first way they are species of quantity; in the second way <they are> attributes (passiones) <of quantity>. In the first way they are the same as a line, surface, etc., nor do they have opposites, <as> "short" or "nar31. Cf. supra, n. 4.
32. Categories, Ch. 6 (5b 3–4).
33. Metaphysics, Bk. X, Ch. 1 (1052b 18).
34. Cf. supra, n. 5.
35. Cf. supra, n. 6.
QUESTIONS 16–17
row," since a line, however short, is a longitude in that mode.
In the second way they are not in themselves (secundum se) in the genus of quantity; but <they are in the genus of quantity> in the first way, that is, as species; but not in the second way, that is, as attributes (passiones). This is manifest about multitude, to which, as it is a species, "unity" is opposed; and <to which> <insofar> as it is an attribute, "paucity" <is opposed>. And, according to Aristotle in Bk. I of the Topics,36 two <things> are not opposed to the same univocal <thing>.
32. To this <solution> one can argue that quantity is not one genus, since there is only one first division of one genus through two differentiae, since there are only two first species.
But quantity is divided equally at first into a twofold division, namely, through <division into> the continuous and the discrete, and through <those species> constituted (per constare) from parts having position and <through those species> not having position, etc.37 The proof of the major is according to Aristotle in Bk. I of the Topics:38 "If <those things> in question are differently diverse (propositi sint diversae differentiae), for example, of color in an utterance and in body, it is equivocal"; and according to Boethius, in the book On Division,39 every division of a genus is made through two differentiae. Proof of the minor: neither division is a subdivision of the other, since then both members of the other division would be contained under some member of the remaining division, which is not true, as is evident by deduction.
33. To this <argument>, one can say that there is not a twofold first division in the genus through formal differentiae that constitute the species. Nevertheless, it is possible that one <division> is through formal differentiae, <and> the other <division is> through material differentiae, which divisions are not under one another. Just as in the genus of substance, beyond its formal division into "animate" and "inanimate," "body" can be divided
36. Topics, Bk. I, Ch. 15 (107b 17–18).
37. Categories, Ch. 6 (5a 36–38).
38. Topics, Bk. I, Ch. 15 (107b 28–29).
39. Boethius, Liber de divisione (ed. J. Magee, p. 8; PL 64, 877C).
JOHN DUNS SCOTUS
into "corruptible" and "incorruptible," and each differentia in respect to itself is exceeded and exceeding, and so it is in the question <at hand>: the first division is formal, the second is material, according to the parts of quantity.—To the proof of the major: Boethius and Aristotle understand <this> of a formal division through differentiae that are constitutive of the species.