What is an INDIVIDUUM ? Part Two

An investigation into St Thomas' Principle of



Expositio super Librum Boëthii de Trinitate

quaestio 4, Articulus 2


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The conception of an "Essence" that remains constant during alterations of intrinsic and extrinsic state [ i.e. the Essence of some being remains the same while that being passes from one intrinsic state to another, or is affected by external, i.e. accidental agents, that change its extrinsic (accidental) state (See NOTE 1 )], on the one hand refers to the existence of "accidental determinations", and to "individuation" of that Essence on the other.
St Thomas devotes a complete Article to individuation in his Commentary on Boëthius' treatise on the Trinity. Because the ontological assessment of the phenomenon of individuation is so important for an evaluation of classical metaphysics, it is worthwile to give this Article special attention, in order to dive deep into the problem of individuation, hopefully resulting in important insights.
A second reason to do so is to experience the flavor of a typical medieval metaphysical treatise. By doing so we will acquire a feeling for what metaphysics is all about.
A third reason is the fact that in this article also some other important topics are treated of, for example the aristotelian categories (predicaments).
We shall first give some information about the structure of the Article (It is Article 2 of Question 4 of the Commentary), and then give the full text (in an English translation) with comments inserted between square brackets [ ]. Like many medieval treatises, St Thomas' Commentary on Boëthius' treatise on the Trinity is divided into QUESTIONS. Each Question is in turn divided into ARTICLES. Each Article is in turn divided as follows in :

  1. A number of opposing arguments (opposing the main thesis), which could possibly brought forward.
  2. Arguments that totally oppose the arguments under (1): the sed contra arguments.
  3. The main thesis (body, corpus).
  4. Replies to the arguments under (1), on the basis of the findings in the main thesis.
Individuation is central in a "thing-metaphysics", i.e. a metaphysics in which THE THING, in so far as it is a thing (a being) is considered, and for which the World consists of "self-beings", which are clearly separated from each other. These are individuals. They imply numerical diversity, i.e. diversity, not necessarily by difference in content, but by repetition (repetition of the same content). The text asks whether this numerical diversity is caused by the diversity of accidents (i.e. accidental properties of the beings in question), or by some other ontological aspect of things.

Introduction into, and structure of, the Text

Whether the diversity of accidents causes a numerical diversity.

The cause or causes of numerical diversity are being sought.
Possible causes are :

Structure of the Text

The text starts with six Ojections against the accidents as cause :
  1. Matter is the cause of numerical multiplicity.
  2. That something which is the cause of substance, and thus of the unity of a thing (that would accordingly be the Form) is also the cause of numerical multiplicity because its diversity results in numerical multiplicity.
  3. Accidents are not a cause of numerical diversity because they are -- as forms -- repeatable.
  4. First substance is the cause of numerical multiplicity.
  5. Accidents are not a cause of numerical diversity because they are not necessarily connected with the thing (substance).
  6. This objection excludes all accidents as being the cause of numerical diversity, except quantity.
Next St Thomas gives four sed contra arguments, which, with respect to the cause of numerical diversity, lay all the emphasis on the accidents as cause :
  1. The combination of accidents is unique for each individual, and by that reason constitutive for being individual.
  2. Form implies specific diversity.
  3. Matter implies generic diversity.
  4. Accidents imply numerical diversity.
  5. Matter is common (i.e. it is a common element of physical beings) and is the basis of division into species and consequently is not a cause of numerical division, because this only comes after the division into species.
  6. The form is the cause of specific diversity (2), and thus the accidents are the cause of numerical multiplicity.
  7. The individuals of the same species only differ accidentally.
In the BODY of the Article St Thomas gives his main Thesis, which, posits itself as it were somwhere in the middle between the Objections and the Sed Contra's :

Individuation, it is true, results from the accidents, specifically from the dimensive quantity, but via matter : This matter is a substrate, and becomes a dimensive substrate from (i.e. by virtue of) dimensive quantity, resulting in the possibility of delimitation of that substrate, so that we -- this he also expounds in other works, so in De Ente et Essentia -- can distinguish a "materia signata" (designated matter), and this means that the substance will enter into a -- albeit (ever) changing -- here-and-now state, and that in turn means that it is individual.

The articulation of this MAIN THESIS is as follows :

St Thomas gives his Thesis.
The whole BODY is divided into numbered sections (1 to 8) for easy reference.
Generic diversity must be reduced to (an effect of) matter.
Specific diversity must be reduced to (an effect of) form.
Numerical diversity must partially be reduced to (an effect of) matter, partially to the accidents.
Explanation of generic diversity.
Matter is the principle of this type of diversity. We cannot know matter, just like that, but only :

  1. As a relation of the substrate-aspect (= the potential) to the 'forming-over' aspect (= the actual, the form, received by that substrate, the substrate's in-formation).
  2. From the (knowledge of the) form, that confers to matter an actual state of being.
On the basis of 1. and 2. we now obtain the generic diversity.
  1. Generic diversity originates by that something that makes possible (i.e. allows for) different types of predications about the Substrate -- as is explained in Met. V, lectio 9, nr. 891/2 in St Thomas' Commentary on the Metaphysics of Aristotle (See The System of Categories in the Essay on Substance and Accident ) :

    • what the subject is.
    • what the measure of the subject is.
    • what the disposition of the subject is.
    • to what the subject is related.
    • what the subject has on it (i.e. what it 'possesses, what it has).
    • when the subject is (relating to the point in time).
    • where the subject is.
    • what the orientation of the subject is.
    • what the subject does.
    • what the subject undergoes.
And so this results in the ten highest Genera.
  1. Generic diversity originates by a diversity of participation in the Pure Act (This 'Pure Act' is a being without any substrate, and consequently without any potentiality. It is all act, it is completely actual. For St Thomas this is God). This diversity relates to the different degrees of intensity of participation and results in different degrees of similitude to the Pure Act (this is a more or less Platonic motiv). This yields the following generic diversity : corporal thing -- living (corporal) thing -- rational (living (corporal)) thing, etc. These are generic distinctions (diversity) within the genus of Substance. And because the difference, (also called) the differentia, is taken from the form (i.e. the form-over-matter), the form is the principle of specific diversity (See the text of the BODY of the Article at 4.).
    Let me explain this. When we consider, for example, a "corporal thing, that is living", then "living" is a differentia (a difference). This differentia contracts the genus "corporal thing" to the species "living corporal thing". This species is not necessarily the 'species specialissima', i.e. not necessarily the ultimate species. Indeed, here it is a sub-genus. "Living corporal thing" means "organism" and this can be further contracted to the many kinds of existing organisms.
    So, as we said, the form is the principle of specific diversity, where this "specific" includes not only the (determination to the) ultimate species, but also all the genera above this ultimate species, EXCEPT THE HIGHEST GENERA, because there are by definition no differences that contract some genus to these highest genera. And it is matter that is the principle of the diversity among these highest genera. These highest geners are the ones listed above. They are the 10 categories of Aristotle.
At 5. (in the text of the Body of the Article) St Thomas treats of the difference between a logical and a metaphysical view.
A genus in a metaphysical sense signifies a common aspect present in Reality, while a genus in a logical sense can signify a common element which only functions as a commom element in our way of gaining knowledge, and that consequently 'exists' only as such, i.e. as an ens rationis (a being of reason). So "res corruptibiles" and "res incorruptibiles" (respectively: corruptible things and incorruptible things) do not have a real common element as a (metaphysical) component in both of them.
But the logician can include them in the term "being" (and this is not a common component, but a term that transcends everything). There are no things (beings) in which "being" does not occur.

From 6. St Thomas starts to consider numerical difference and individuation, i.e. he starts to consider numerical diversity.
The specific and generic nature (of a thing) consists of matter and form. An individual consists of this matter and this form. A form is in itself not individual, but it can be individualized by matter, but the matter must be quantitatively delimitable, so that the form can be received in this matter. Matter then must be subjected to quantity (and quantity is directly next to matter -- see St Thomas in In Met.V, lectio 9, 892, quoted in NOTE 2 ), and more precisely, subjected to dimensions. As complete beings (here this means: wholly determined) the dimensions, as determined dimensions, are subsumed in the genus of Quantity. But in so far as they are determined to this or that (i.e. determined in this direction or in that), they do not garantee numerical identity, but do so only as non-delimited (not-determined, not-terminated), i.e. individuality is effected from the nature of the dimensions as incomplete beings, because these dimensions can now be delimited (determined) -- determined-while-also-varying-all-the-time, and in this way matter becomes designated matter (materia signata). Said differenly, matter, that has become dimensive by virtue of dimensive quantity, now becomes -- varying all the time -- materia quantitate signata [this is not a term used by StThomas It means: matter, that is designated by quantity.], by virtue of which numerical diversity becomes possible (This is a very preliminary sketch. I will work out things later).

The next and last section of the text (following the BODY of the Article) consists of Replies to the six Objections :

The answering takes place on the basis of the findings in the main thesis (the BODY, corpus).
So, with respect to matter distinctions are made. Among others a "materia signata" is distinguished, further a matter that is rendered dimensive (i.e. matter which has become a dimensive substrate) by the dimensiones interminatae. It always concerns the cooperation of matter and quantity in individuating.

With this I have concluded the exposition about the structure of the text.

We now start with an English rendition of the text of St Thomas. What is included between square brackets [...] does NOT belong to that text, but contains supplements, explanations, comments and interpretations (some of these are quite extensive).
The English translation is adapted from MAURER, A., Thomas Aquinas. Faith, reason and theology. Toronto, 1987.


About those items which belong to the cause of Multiplicity


Whether the diversity of accidents causes a numerical diversity.

We proceed to the second article as follows :

It seems that a difference of accidents cannot be the cause of a plurality in number.

1. For the Philosopher [With "the Philosopher" is always meant Aristotle] states that those things are one in number "whose matter is one." It follows that there is plurality in number where there is plurality of matter. Therefore it is not a difference of accidents, but rather a diversity of matter that causes numerical diversity.

2. The Philosopher says that the cause of the substance and of the unity in things is the same. But accidents are not the cause of the substance in individuals and therefore neither of their unity. So they cannot be the cause of numerical plurality.

3. Because all accidents are forms, they are by their very nature communicable [ i.e. distributable over several 'cases' ] and universal. But nothing like this can be the principle of individuation for something else. Hence accidents are not the principle of individuation. But some things are numerical diverse inasmuch as they are divided as individuals [ i.e. delimited -- separated -- from other individuals ]. Consequently accidents cannot be the source of numerical diversity.

4. Just as things differing in genus or species in the genus [ i.e. category, predicament ] of substance differ substantially and not only accidentally, so also things that differ in number.

[Thus an animal does not only differ from man in accidental characteristics, but also with respect to substance, to be precisely, with respect to Second Substance. When two items within the genus (Category) of substance differ numerically, which is already the case with two human persons, say Socrates and Plato, then they also do not differ in accidental characteristics only, but also with respect to substance, more precisely First Substance : Socrates is a substance, separated and distinguished from Plato which is another substance, and also separated from Peter, another substance again.]
But some things are said to be different in genus or species through what is in the genus of substance. Similarly, therefore, things are said to differ in number through what is in the genus of substance and not through accidents.

5. If a cause is taken away, so too is its effect. Now every accident can be removed from its subject, either in fact or in thought. Consequently, if an accident were the principle of identity and diversity in number, it could happen that in fact or in thought the same things were sometimes one and sometimes divers in number.

6. What is posterior is never the cause of what is prior. But among all accidents quantity holds the first place, as Boëthius says. Among quantities, moreover, number is by nature prior because it is more simple and abstract. Consequently no other accident can be the principle of numerical plurality.

[ Because, according to the Objection, before NUMBER no accidents appear anymore, so that the rest of the accidental domain cannot contain a principle of numerical multiplicity, because NUMBER is already that principle, and because the other accidents are posterior.]
ON THE CONTRARY, there is the statement of Porphyry, that an individual is produced by an assembling of accidents which cannot be found in any other individual. But the principle of individuation is the principle of numerical diversity. Accidents [ i.e. the collection of accidents ], therefore, are the principle of plurality in number. [ See, with respect to this first ON THE CONTRARY, the end of the BODY of the Article. ]

2. We find nothing in the individual except form, matter and accidents. Now diversity of form is not the cause of diversity in number, but rather diversity in species, as is clear in the Metaphysics. Diversity in genus is the result of the diversity of matter, for the Philosopher states that things differ "in genus if they do not have their matter in common and are not generated out of each other." Therefore nothing can bring about diversity in number except diversity of acidents. [ See for this Reply to 2. ]

3. What is found in common in several things different in species is not the cause of diversity in number, because the division of a genus into species precedes the division of the species into individuals. But matter is found in common in things diverse in species, because the same matter is the subject of contrary forms. Otherwise things having contrary forms would not be changed into each other. Therefore matter is not the principle of diversity in number, and neither is form, as has been proved. So it follows that accidents are the cause of this diversity.

[ "the same matter" which is the bearer (i.e. subject) of contrary forms, in "the same", not numerically, but with respect to content, because it always concerns here an already in-formed subject, unless we speak about a so called substantial change. But also here "the same matter" does not mean numerically the same, but both are pure potency. Thus such matter can, with respect to content (i.e. (just) an initial in-formation), or with respect to being potential, be common (i.e. be shared by several things), while it is at the same time not impossible that it is numerically many. This being-numerically-many it does not possess by itself however, but by virtue of something else, and thus is matter no principle of numerical diversity. ]
4. In the genus of substance we find nothing but genus and difference, but the individuals of one species differ neither in genus nor by reason of substantial differences. Therefore they differ only by reason of accidental differences. [ See for this, among others, the end of the BODY of the Article, and Reply to 2. ]

REPLY     [ Main Thesis ]:

1. In order to clarify this question and those raised in the text of Boëthius we must see what causes the three kinds of diversity mentioned in the text. There are only three items [ See NOTE 3] in a composite individual in the genus of substance : matter, form and the composite of the two. It is from among these, then, that we must find the causes of each of these diversities. We should know that diversity in genus is reducible to diversity of matter, while diversity in species is reducible to diversity of form, but diversity in number is reducible partly to diversity of matter and partly to diversity of accidents.

[This is in fact St Thomas' reply to the Question (with respect to the numerical diversity), and he is going to get into it in more detail : First he expounds in what way (i.e. by virtue of what) a generic diversity comes into being, next by which a specific diversity is generated. After this he will distinguish between a logical and metaphysical view of genus, species and difference, and finally he will explain individuation more in detail by means of continuous quantity which is constituted by its having dimensions. ]
Now because a genus is the beginning of knowledge, being the first part of a definition, while matter in itself is unknown, it follows that diversity in genus cannot be taken from matter as it is considered in itself, but only in the way in which it is knowable. Now it is in fact knowable in two ways. In one way by analogy or proportion, as the Physics says [ The "Physics" is a work of Aristotle on Natural Philosophy ]. For example, we may say that matter is that which has the same relation to natural things as wood has to a bed [ Accordingly this is the relationship of the potential -- in this case a substrate-item -- to the actual -- in this case an in-formational-item. ]. In another way matter is known through form [ Thus in this case not through a substrate-item, but through an in-formational-item ], through which it has actual existence, for everything is known inasmuch as it is actual and not inasmuch as it is potential, as is said in the Metaphysics.

2. Accordingly diversity in genus is derived from matter in two ways. First, because of a different relation to matter. In this way the primary genera of things are distinguished with reference to matter. What is in the genus of substance is related to matter as to one of its parts [ Matter here is an ontological part of substance. And as such we get the (supreme) genus SUBSTANCE ]. What is in the genus of quantity does not have matter as one of its components, but it is related to matter as its measure [ And as such we get the (supreme) genus QUANTITY ]. Quality in turn is related to it as its disposition [ And as such we get the (supreme) genus QUALITY ].

[ Thus the genera of Substance and Quantity are characterized by their relation to matter, and in this way we get generic diversity][Quality however immediately follows upon form, and in this way dispositions' (the) matter, while quantity immediately follows upon matter. In In Met.V, lectio 9, nr 892 we read this as follows : "(...) and this predicate either inheres in it [ i.e. in the subject ] per se and in an absolute way, as following upon matter, and as such it is quantity, or it follows upon form, and as such it is quality". ].
Through these two genera all the other genera receive different relationships to matter. Matter itself is a part of substance, and it gives to substance its nature as a subject, because of which it bears a relation to accidents [ In this way we get all the other (supreme) genera. ]. In a second way diversity in genus is derived from matter inasmuch as matter is perfected through form. Matter is pure potentiality and God is pure actuality, and for matter to be brought into the actuality that is form is nothing but its sharing, though imperfectly, a likeness of the primary actuality. Thus what is composed of matter and form is in between pure potentiality and pure actuality.
[ Here St Thomas is following a neoplatonic motiv of down-flow (and at the same time in-flow) of beingness from the First Act into matter. But matter participates in God in the sense that it also is created by God, by conferring being upon it.]
3. Matter, however, does not uniformly receive an equal resemblance to the primary actuality. By some things it is received imperfectly, by others more perfectly. Some things share the divine likeness [ In fact : possess a divine likeness in virtue of which they participate in God in this modest way ] only to the extent that they subsist [ from 'below' : by themselves, 'upward' : as received (namely by their participating in God)], some to the extent that they live, some to the extent that they know, and still others to the extent that they understand. Now the likeness itself of the primary actuality existing in any matter is its form. But a form of this kind causes only existence in some things, in others existence and life, and so with other forms, each of which remains one and the same, for a more perfect likeness possesses everything a less perfect likeness does, and more besides. There is something common, accordingly, in both likenesses, which in one is the ground of imperfection and in another the ground of perfection, as matter is the basis of both actuality and privation. Thus matter, along with this common factor, still functions as matter with respect to the perfection and imperfection mentioned above.
[ I.e. it still has the character of a substrate, despite its possessing already a certain degree of in-formation. In this context matter should possess already a certain degree of in-formation, otherwise it could not exist as a substrate for a privation. ]
[ In this way we get the (sub)genera within the (supreme) genus of substance, like 'non-living thing', 'living thing', 'non-sensitive living thing', 'sensitive living thing', 'non-rational sensitive living thing', 'rational sensitive living thing', and also all (sub)genera subsumed under them.
4. A genus, accordingly, is derived from this material factor, whereas differences are taken from the above-mentioned perfection and imperfection. For example, we take the genus "living body" from the common material condition of possessing life, while we derive the difference "sentient" from an added perfection, and the difference "insentient" from an imperfection. In this way the diversity of these material factors results in a diversity in genus, for example, the diversity between animal and plant.
[ A higher genus is contracted to a lower genus by a difference, and this lower genus is again contracted (by a difference) until we arrive at the lowest genus, and this will finally be contracted to a species, again by a difference. ]
For this reason matter is called the principle of diversity in genus, and for the same reason form is the principle of diversity in species, for the differences that determine species are taken from the above-mentioned formal principles, which are related to the aforesaid material factors, from which genera are derived, as form to matter.
[ "are taken" here always means that the content of a concept derives from something in Reality. ]

[ Before I will proceed further with the text of St Thomas I will give a detailed explanation of : Generic Diversity (diversitas secundum genus) ]


Recall from the Section Difference between "First Substance" and "Second Substance", in the the Essay on the Criteria for something to be a Substance, that "First Substance" refers to a subsistent being (as opposed to Accidents).
All ten Predicaments (Categories) are about First Substance.
Second Substance is about what a given first Substance essentially is.
The nine other Predicaments are about what a given first Substance accidentally is.
And because a first Substance, is not itself seated in any substrate, it represents that substrate, and that means it represents matter, and ultimately prime matter, because prime matter is the ultimate substrate and as such confers a substrate aspect to First Substance. So the "about First Substance" of the Predicaments boils down to "about (prime) matter".

A genus, like ANIMAL, is, as genus, a so-called logical intention (intentio logica, De Ente et Essentia, Prologus, line 9), i.e. a mental item that functions as a sign (In the Thomistic Image-theory it even is a likeness of the thing signified). The sign signifies something in the extramental world :   it intends a segment of this world. But before that, the sign was already generated by a causal interaction of that world with the knower. It is caused by an active aspect of that world, that has been -- via the senses -- passed onto the intellect, in which it appears as an intention, after abstraction (i.e. subtraction) of the here-and-now aspect. It is the mentioned active aspect FROM WHICH, St Thomas says, the intention "is taken". But this intention refers, in its function of signifying, back to the extramental Reality, right to this mentioned aspect. However this reference does not relate exclusively to this aspect, but includes implicitly a possible substrate, and also a possible (further) in-formation (i.e. a reception of some form).
So a genus is an intention, it is "taken" from a substrate aspect (See NOTE 4) (of an extramental thing) and accordingly intends this substrate aspect, however not exclusively so (not "cum praecisione") but implicitly also refers to all possible in-formations of this substrate aspect. However this must be further specified, i.e. restricted :
If the genus belongs to the supreme genus of SUBSTANCE, i.e. if that genus is a subgenus of the supreme genus of Substance, for example the genus ANIMAL, then it is taken from that substrate aspect that can be identified with the 'determinable Essence' (i.e. an essence -- of a thing -- that could still be further specified), and the genus in turn intends this determinable Essence, in which all possible determinations of this Essence however are not excluded (as this would be possible in the cognitive process), they are just left out of consideration (See De Ente et Essentia, Cap.2, line 140).
If, on the other hand, the genus is a subgenus of one of the supreme accidental genera, for example the genus COLOR (as a subgenus of the supreme genus of QUALITY), then only a (determinable) essence in a qualified sense (i.e. in a certain restricted sense) is intended, not the (determinable) Essence of the given thing.
In the present discussion, however, we restrict ourselves for the time being to the (status of the) subgenera of the supreme genus of SUBSTANCE.
Now because in this way certain differences are left out of consideration we 'find' identical aspects (or 'layers') in Reality. The unity of the genus is accomplished by an aspect of indetermination (See De Ente et Essentia, Cap.2, line 225). Thus we can have, for example, the genus ANIMAL, and leave out of consideration whether it is a rational animal or not. So a possible generic diversity corresponds to the diversity of substrate aspects (while a specific diversity corresponds to their respective possible in-formation). And this means :   different degrees of in-formation (actualization, specification) of those substrates (because non-informed substrates cannot differ).
Let me clarify this further :
The genus ANIMAL, although in classical texts interpreted as the lowest organic genus, is in fact a higher genus. And as such we will employ it in the discussion to follow. The genus PLANT is also such a high genus. They both reside at the same level of generality. Their different contents reflect and exemplify a generic diversity that extends horizontally through the organic world.
But these (higher) genera, ANIMAL and PLANT, are pointing to substrate aspects (in the extramental world) that can still be further in-formed by (still) essential determinations. Such determinations are called differences (differentiae).
In this way substrate aspects appear that have a higher degree of in-formation than did the original substrate aspects, and now we have a diversity that extends vertically. Such substrates with a higher degree of in-formation are called lower genera. By again adding differences (differentiae) to these lower genera, still lower genera appear. When we continue this process (of adding differences) we finally end up with a species (And from now on only accidental determinations can be added, leading to (the specification of) individuals (of that species)).
So indeed we see, when going from higher genera to lower genera (and finally to a species), an intensification of in-formation (actualization) of substrate aspects. For example the genus VERTEBRATE (i.e. vertebrate animal) refers to a substrate aspect (in extramental reality), while the genus MAMMAL also refers to a substrate aspect, but the latter is much more in-formed than the former : A mammal is a vertebrate-that-breastfeeds-its-young.
These in-formation intensities determine a scale, spanning between pure potency (prime matter) and pure act (God, or whatever equivalent to Him), and so a certain position on this scale corresponds to a certain genus (as a subgenus of the supreme genus of SUBSTANCE). This means that the status of in-formation (its intensity or degree) of the (still further) determinable Essence is that something from which the genus is "taken", and, while taken from a part, it -- as a sign -- refers back to the whole :   Determinable Essence + Determination. With respect to the Determination however the reference is implicit. When we descend to a lower and lower intensity of substrate in-formation, we finally arrive at prime matter. The genus that is taken from this prime matter is the genus of SUBSTANCE, that, in its function of intention, refers back to this prime matter again, but, again, not by exclusion, but by making implicit reference to ALL possible (cases of) in-formation of this prime matter. So here we have let the degree of indeterminateness become maximal. Prime matter, that is as such without any content, but only functioning as a substrate, provides the explicit meaning of SUBSTANCE, the (ultimate) subject (i.e. something that "stands under").
Something is intended with the genus of SUBSTANCE (In the text it says : Id enim, quod est in genere substantia (See NOTE 4a ) if matter is a(n) (ontological) part thereof, namely the substrate. If we totally generalize on this relation of substrate-information (the reception of some (further) form by the substrate), then it becomes a potency-actuality relation, and then also the immaterial beings belong to the genus of SUBSTANCE, because then potency is not necessarily matter (We then can also have to do with the relation Form (potency) -- Existence (act)).

What we have done above is the derivation (in a general sense) of the subgenera, like ANIMAL, PLANT, etc., of the supreme genus of SUBSTANCE and with it the derivation of that supreme genus itself. It concerned a "relation to matter" which here is a relation to the many possible substrate aspects that appear when they are elevated as it were by the addition of further forms resulting in (still) more in-formed substrate aspects, and that St Thomas calls the progressively perfecting of substrates by adding further forms (that integrate fully with the original formal content).
But this is not the whole story concerning the generic diversity, because besides the subgenera of the supreme genus of SUBSTANCE we still have other genera that contribute to generic diversity, namely, besides the supreme genus of SUBSTANCE, the supreme accidental genera, QUANTITY, QUALITY, RELATION, etc. This generic diversity thus is a diversity that obtains between the ten supreme genera, SUBSTANCE, QUANTITY, QUALITY, RELATION, etc. Also this diversity can be seen as "relations to matter", but in a different way as the one we discussed above :   St Thomas says in 2. of the body (REPLY) of the Article :

"What is in the genus of substance is related to matter as to one of its parts.
What is in the genus of quantity does not have matter as one of its components, but it is related to matter as its measure. Quality in turn is related to it as its disposition.
Through these two genera [quantity, quality] all the other genera receive different relationships to matter. Matter itself is a part of substance, and it gives to substance its nature as a subject, because of which it bears a relation to accidents."

(After this text St Thomas discusses the other way by which diversity in genus is derived from matter, namely inasmuch as matter is perfected through form, and this we have discussed above.)

In what way should we consider an Accident (as a subgenus of one of the supreme accidental genera) so that it can indeed legitimately be placed under such an accidental supreme genus (accidental category)?

If we consider those aspects in Reality which do not enjoy (ontologically) independent existence (towards the ontological bottom), and if we moreover consider those aspects (Accidents) in an abstract way (NOTE 4b), then we intend those aspects by means of concepts, which, when we let, in the process of intending, the indefiniteness (indeterminateness) become maximal, end up with (one of the) nine accidental categories, i.e. the accidental supreme genera. Expressed differently, not until we consider an Accident in an abstract way, it cannot be placed under a category, at least this is what St Thomas posits in De Ente et Essentia, Cap.6., line 123 -127. So the abstract term primarily (with respect to an Accident) refers to the accidental form and secondarily to the substrate :   snubness is the curvature of the nose, where "nose" is, for a while, considered as a Substance.

The different Accidents can be reduced to nine basic types, and this means nine "relations to matter" (analogiae ad materiam). These nine genera (categories), plus the genus of Substance, thus correspond to ten relations to matter.
Being is an analogical concept, which means that it comes in several grades, like for example potential-being actual-being. Also the Predicaments (Categories) reflect a grading in Being, especially in the distinction between Substance (logically corresponding to the Subject) and the Accidents (that are in a logical context -- 'said' (accidentally) of that subject, i.e. are the (accidental) predicates).
The analogous nature of Being is often expressed by the term "analogia entis" which means :   "the analogy of Being".

[ Now we will continue with the text of St Thomas.]

5. It should be borne in mind, however, that because the material factor which is the basis of the genus includes both matter and form,

[It concerned after all matter which had already received a certain (amound of) in-formation, but which can still function as a substrate for further in-formation ("perfection") ],
the logician considers the genus only on its formal side, with the consequence that his definitions are called formal [ This probably relates to nominal definitions versus real definitions, defining not names but things. ] The natural philosopher, on the other hand, takes both aspects of the genus into consideration. Thus it can happen that something is in a genus from the logician's point of view which is not in a genus from the perspective of the natural philosopher. For it is possible that the resemblance to the primary actuality that something receives in matter of a particular kind something else might receive without matter, and something else again might receive it in matter of an entirely different kind. For example, it is evident that stone comes to subsist in matter potential to existence. The sun comes to subsist through matter potential to place and not to existence.
[Because the matter of celestial bodies was considered as eternal : The matter of the heavens does not have potential being, it IS already, and cannot NOT be. ]
And an angel subsists without any matter whatsoever. So the logician, finding in all of these the source from which he derived a genus, places all of them in the same genus of substance. The natural philosopher and metaphysician, however, who take into account all [ Thus not only ONE (common) aspect. ] the principles of a thing, assert that they are in different genera, for they do not find them sharing the same matter. This agrees with the statements of the Metaphysics, that the perishable and the imperishable differ in genus, and that things are in the same genus which have their matter in common and are generated out of ech other.
[I will try to make clearer the difference between the logical view and the metaphysical view, about which St Thomas here speaks, without too much pretention, and at the same time note that according to me it does not relate to a crucial text-section in the present context, certainly not, if we limit the individuation-problem for the time being first of all to material beings, and as yet not consider celestial bodies (in the medieval view), and also not (consider) the separate substances (with "separate substances" is meant : substances without matter).
The contraries RATIONALE (= rational, possessing reason) and IRRATIONALE (= not possessing reason, like the (lower) animals) belong to one genus, namely ANIMAL (= sensitive living being). They, as differences (differentiae) further determine this genus, either to the species ANIMAL RATIONALE, or to the species ANIMAL IRRATIONALE, and because of that RATIONALE does not belong to the essence of ANIMAL (St Thomas, in In Met. X, lectio 12, nr.2142, expresses himself as follows : "does not belong to the substance of that genus"). ANIMAL is only in potency with respect to RATIONALE (i.e. ANIMAL could become ANIMAL RATIONALE. In the same way IRRATIONALE does not belong to the essence of ANIMAL. That there are genera, like ANIMAL, which relate only potentially to RATIONALE and IRRATIONALE, is caused by the fact that not every genus of things is either RATIONALE (rational) or IRRATIONALE (irrational): only the genus ANIMAL is relevant in this respect.
However we have a totally different case with CORRUPTIBLE and INCORRUPTIBLE, for a genus is a genus of either corruptible things, or of incorruptible things, because either CORRUPTIBLITY or INCORRUPTIBILITY always belongs to the Essence of something, and thus implying an essential difference, and consequently a generic difference, resulting in the fact that CORRUPTIBLE THINGS and INCORRUPTIBLE THINGS always belong to different genera : CORRUPTIBLE and INCORRUPTIBLE never share matter of the same kind, while the genus should be taken from matter.
But they could be similar to each other by virtue of one or another common aspect like having (ontologically) independent being, being a being, or they could feature a "distinctio realis" (show a real distinction between Essence and Existence). And based on this the logician can indeed subsume these items (certain CORRUPTIBLE THINGS and INCORRUPTIBLE THINGS) under one and the same genus. In this case the genus only refers to 'the-possibility-of-consideration-under-one-and-the-same-sense', and consequently only refers to our way of knowing and considering, and not to the (considered) extramental Reality.
[Next St Thomas will treat of numerical difference and individuation ]

6. It is clear, then, how matter causes diversity in genus and form diversity in species. Diversity in individuals of the same species should be understood as follows. As the Philosopher says, just as the parts of a genus and species are matter and form, so the parts of an individual are this matter and this form. It follows that just as diversity of matter or form taken absolutely causes diversity in genus and species, so this form and this matter bring about diversity in number. Now no form as such is individual of itself. I say "as such" because of the rational soul, which in a sense is of itself an individual substance, but not insofar as it is a form. For any form that can be received in something as in matter or in a subject can be predicated by the mind of many things, which is opposed to the nature of an individual substance.

[ A "this something" (i.e. an individual something) cannot be instantiated anymore, it is already totally determined (NOTE 5). A form is in itself not this form, and thus it is of itself not a "this something". With respect to the soul, which is a form (See S.T. Ia 75. 1), this seems problematic because the soul also is a "this something" (in a sense in itself). The soul is however not a "this something" as a form (i.e. in its function of form), because a form can be attributed to more than one thing, and this does not -- as he (St Thomas) says -- comply with a "this something". Being a "this something" of the (rational) soul is accomplished because it is the soul of this body. When the soul disattaches itself from the body it remains individual, implying that it is nonetheless 'in a certain sense in itself' a "this something". With these matters we must realize that the human soul has a two-fold character : It is (1) the form of a body, and (2) a subsisting form. ]
Consequently form is rendered individual through being received in matter. But because matter in itself lacks all differentiation, it can individuate the received form only insofar as it itself bears some distinguishable mark. So form is individuated by being received in matter, but only as it is received in this particular matter, determined to this place and this time. Matter, however, is divisible only through quantity. Thus the Philosopher says that if quantity were taken away, substance would remain indivisible. Accordingly matter is made to be this and designated owing to the fact that it is subject to dimensions.

7. Now dimensions

[In De Ente et Essentia St thomas speaks of "three dimensions", implying that the three dimensions of the spatial continuum are meant.]
can be understood in two ways. In one way inasmuch as they are determinate, and by this I mean that they have a definite measurement and shape. In this sense, as complete beings, they are located in the genus of quantity. Now when dimensions are understood in this way they cannot be the principle of individuation, because there is often a variation in such determination of dimensions in the same individual, and thus it would follow that the individual would not always remain the same in number.
[St Thomas departs from (i.e. starts his considerations with) the kind of individuality that we encounter in the case of human persons : Such a type of individual is an individual, which is not exclusively distinguishable by space-time coordinates, but also by other characteristics, albeit that they are not fundamental for being-an-individual, but only diagnostic for it -- See the end of the Body of the Article -- such an individual is the 'historical individual' (versus the here-and-now individual). Thus St Thomas models his view of being-an-individual (concerning all material substances) after the (paradigm of the) human person, and this implies that the kind of individuality visible in human persons must fit into a broader view of individuation, in other words, this general view of individuation must be able to include the accepted view of the individuality of a human person. ]
In another way dimensions can be taken as indeterminate, simply as having the nature of dimensions, though they can never exist without some determination, any more than the nature of color can exist without being definitely black or white. Taken in this way dimensions are located in the genus of quantity as something incomplete. It is through these indeterminate dimensions that matter is made to be this designated matter [ haec materia signata ], thus rendering the form individual. In this way matter causes diversity of number in the same species [ Comment by Maurer : "St Thomas took the notion of indeterminate dimensions from Averroes. ... After the period of this commentary he spoke only of determinate dimensions, or simply of dimensive quantity, as the principle of individuation. ..."].
[In other works St Thomas speaks about "terminated dimensions" instead of "interminated dimensions", which could be rationalized as follows :
The dimensions, for example the size of a thing, must necessarily be terminated (i.e. determined), because interminated dimensions, for example an indetermined size of something, cannot imply a spatial setting of that something, while an individual must have such a setting. But although these dimensions should be determined (terminated), it is not necessary that the limits, the terminations, should have constant values. These limits may vary, as long as there are limits (delimitations) at all. And in this way we obtain the "terminated dimensions" (dimensiones terminatae). These generally determined dimensions are thus the "terminated dimensions", featuring in other works of St Thomas, and are identical to the "interminated dimensions" (dimensiones interminatae) in the present text. So a human individual is indeed all the time quantitatively determined (i.e. delimited, and thus having a finite width, length and thickness), while his width, length and thickness constanly change, so the dimensions are not undetermined in an absolute way, they are 'generally' determined, just as 'generally' as '(an) individual' is, when we are not looking specifically at Socrates, but looking at the 'general' individual : Socrates or Plato or Peter etc.

Before proceeding further with the text of St Thomas it seems useful to me to ponder for a while about QUANTITY :
Quantity is a highest genus (a supreme genus) with respect to a certain way of being, and consequently is NOT definable, because the degree of indetermination in (the proces of) signification is maximal. There is not a still higher genus beyond Quantity (under which quantity could be subsumed), so we don't have a difference that together with this higher genus could constitute a definition (of quantity). We, of course could 'define' quantity by stating that it confers MEASURE, but this is identical to QUANTITY. It is just another word for it.
Addition of differences like discrete / continuous to QUANTITY reveals definable species (subgenera) of Quantity.
Quantity is an 'accidental form' (for a body -- i.e. a corporal thing -- AS body it is a property (proprium accidens), because ALL bodies are quantified), which confers measure onto matter. By virtue of continuous quantity (quantitas continua) we get a coherent measure, while this measure will not be coherent by virtue of discrete quantity (quantitas discreta).
The continuum is constituted by dimensions (the "three dimensions" from De Ente et Essentia). Out of this continuum the discretum is generated by division, and consequently a numerical diversity (diversitas numero, See Reply to 6.). According to St Thomas matter (the substrate-aspect of a thing) becomes divisible by virtue of quantity, and that means first of all the continuous quantity (constituted by dimensions), because continuous quantity is the most fundamental subgenus of quantity (See Reply to 6.). This implies that because of that the form (following the dimensional substrate) is dispersed and can then be individuated by division into several individuals (individual cases) of that form. Individuation does not occur until after a process of division has taken place, and so not already by the 'interminated dimensions', but by 'terminated divisions' (divisiones terminatae).

   A more precise view concerning the 'dimensiones interminatae and terminatae' could perhaps be as follows, based on BOBIK, J., 1954 (See the references after the end of the text) : All the time it concerns the problem about the difference between the individual nature and the specific nature :
This difference is caused by several items (aspects) in which both natures (the individual and the specific) differ.
Three aspects play a role in the individual nature, and they together contribute to the phenomenon of being-an-individual :

  1. Numerical unity.
  2. Last (i.e. ultimate) subject.
  3. Determination to the here-and-now.
1. The individual nature has numerical unity, undivided in itself (first part of the definition of INDIVIDUAL (individuum)) and because of that numerical identity and numerical distinction with respect to other individual natures of the same species. This is first of all garanteed by dimensive quantity. This is dimension in so far not terminated (dimensiones interminatae). These convert matter into dimensive matter (treated of in the Commentary on Boëthius' treatise on the Trinity). The dimensiones interminatae take care of numerical unity.

2. An individual is a last subject. It is separated from other individuals (second part of the definition of INDIVIDUAL (individuum)). It cannot again be received into yet another subject anymore. Said logically : it cannot be predicated of another subject : in a downward direction it is not further communicable anymore (in contast with the specific nature, like HUMAN BEING that CAN be predicated of a subject -- like "Socrates (subject) is a HUMAN BEING (predicate) -- and so be communicable to, say, Socrates). Prime matter renders substance into first substance, i.e. into a last subject, still containing content. This subject is accordingly not instantiable anymore and thus unique, and in this way prime matter takes care of this aspect of being-an-individual. This is being discussed in De principio individuationis (On the principle of individuation) and in De natura materiae et dimensionibus interminatis (On the nature of matter and the interminated dimensions)(See NOTE 6). Prime matter takes care for being-a-last-subject.

3. Quantitative and qualitative (constantly changing in time) determination of the individual to the here-and-now (the Semaphoront). This is garanteed by determined (terminated) dimensions (Total determination in a quantitative sense -- materia sub quantitate determinata, = materia signata (designated matter)), and also in a qualitative sense) : As such it concers the "accidentia designantia hanc materiam individualem" (= the accidents referring to this individual matter) (S.T. Ia q.3, a.3, c.). This "hanc materiam", i.e. materia sub quantitate determinata (De principio individuationis) is the materia signata of the De Ente et Essentia. Quantitative and qualitative determinations take care of the 'being-here-and-now'. These determinations constanly change, as has been said, but are here not considered in so far as changing.

That this here-and-now determination can freely vary (for instance as we see this in the succession of the stadia of metamorphosis in insects, or in stadia connected with growing older) without loss of individuality, is garanteed by the 'dimensiones interminatae', because variation can proceed along these dimensions (See NOTE 7), so that the nature (of something) can exist continuously under here-and-now conditions, despite changes. Said differently : the dimensiones interminatae result in an individual, that numerically differs from other individuals, but keep the possibility of variation within an individual open, while the dimensiones terminatae again and again determine a here-and-now condition. In this way numerical unity is conserved, despite the changes mentioned (Treated of in De Ente et Essentia, De principio individuationis and in De natura materiae et dimensionibus interminatis).

The explanation just given with respect to 'dimensiones terminatae and interminatae' is still preliminary. It nonetheless is -- according to me -- a suitable basis for further historical and thematical discussion (See the coming discussion concerning this text of St Thomas). ]

[We will now proceed further with the present text of St Thomas.]
8. It is clear, then, that matter taken in itself is the principle of neither specific nor numerical diversity. But it is the principle of generic diversity inasmuch as it underlies a common form, and so likewise it is the principle of numerical diversity as underlying indeterminate dimensions. Because these dimensions belong to the genus of accidents, diversity in number is sometimes reduced to the diversity of matter and sometimes to diversity of accidents, and this because of the dimensions mentioned above. But other accidents are not the principle of individuation, though they are the cause of our knowing the distinction between individuals. In this sense individuation can also be ascribed to the other accidents.
[I.e. in this way also these other accidents individuate, at any rate they let us distinguish between individuals : we can in this way recognize certain individuals. We do this on the basis of accidental differences, or perhaps more so on the basis of an every time unique combination (collection) of accidents.]

Reply to 1. The Philosopher's statement that those things are one in number whose matter is one is to be understood as referring to designated matter, which is the subject [i.e. it is the substrate ] of dimensions. Otherwise we would have to say that all generable and corruptable things are one in number because their matter is one.

Reply to 2. Because dimensions are accidents, they cannot by themselves be the source of the unity of an individual substance. But matter, as the subject of these dimensions, is understood to be the principle of such unity and plurality.

[ See ON THE CONTRARY 2. So determination (delimitation) towards numerical unity and plurality must proceed through matter.]
Reply to 3. By definition the individual is undivided in itself and divided from other things by the last of all divisions.
[ After this, a further division will make it specifically different (i.e. the thing will become something else). Because when we are going to divide an individual (cut it into chunks), then we will obtain different substances (I will address this point in the ensuing discussion (Part Three))]
Now among accidents quantity alone has of itself the special characteristic of division. So dimensions of themselves have a certain character of being individual with reference to a definite position, position being a quantitative difference.
[ With respect to "with reference to a definite position" the Latin text reads : secundum determinatum situm. This could be more appropriately translated as : "with reference to ordered dispersion". I think that SITUS (accusative: situm) should be distinguished from POSITIO (position), because situs connotes an ordering or succession, and this is compatible with the concept of dimension. Dimension, namely, is something ALONG WHICH a measurement can take place, thanks to a certain ordering (of, say, points, or possible positions). See In Met.V, lectio 9, nr.892, where St Thomas treats of the Categories (Predicaments). Within the ten Categories he distinguishes a subgroup in which, it is true, the predicate is taken from (something) outside the subject, but nonetheless signifies a measure of that subject (it is a measure for that subject) :
"If it [the predicate] is a measure of it [i.e. of the subject], the category is -- because an external measure either is [point in] time, or [the] place [of it] -- taken either from time, and as such it will be [the category] WHEN, or from place, and then it will -- if the ordering of parts at a [that] place is not considered -- be [the category] WHERE. And if this ordering is taken into consideration, then it will be [the category] SITUATION (situs)."
We could perhaps interprete these matters as follows :
If situs means ordered dispersion along a dimension, and in so far as this is a quantitative feature, then situs belongs to the category of Quantity.
If we have a case where situs is not a quantitative feature, then it belongs to another category, namely Situs (Situation), or it belongs to the category of Quality, namely when this ordered dispersion is an intrinsical figure (configuration)(i.e. an intrinsical ordering -- situation -- of the parts of the thing of which a situs is predicated.
Thus a dimension is individual on two scores : because of its subject, just like any other accident. And also because of itself, insofar as it has position [ thematically perhaps better : ordered dispersion ]. Owing to its position, even if we abstract from sensible matter [ St Thomas treats of this in S.T.Ia, 85, 1 ] we can imagine a particular line and a particular circle.
[ Because even after abstraction from sensible matter, intelligible matter remains (in these mathematical objects). This matter is the mathematical continuum, i.e. the dimensive (aspect).]
So it rightly belongs to matter to individuate all other forms because it is the subject of that form which of itself has the trait of being individual. Indeed even determinate dimensions themselves, which are grounded in the already completed subject, are in a sense individuated by matter that has been rendered individual by the indeterminate dimensions that we conceive beforehand in matter.
[ A dimension communicates unity, continuity, from the subject, satisfying the first part of the definition of INDIVIDUUM, namely the "in se indivisum" (undivided in itself). In itself a dimension an ordered dispersion (situs), in virtue of which things can take different positions resulting in their being separated from each other, also satisfying the second part of the definition of INDIVIDUUM, namely the "ab aliis divisum" (separated from others). And this is accomplished by a last possible division (ultima divisione), because that implies the separating (spatial) positioning of things (the next division would destroy the whatness of the things, because then the threshold of the minimum of the form (of the thing with that nature) is crossed. ]

   O  R  D  E  R  E  D       D  I  S  P  E  R  S  I  O  N

Situs of some individuals :  ordered dispersion along a dimension ]

Reply to 4. Things differing in number in the genus of substance differ not only in their accidents but also in their form and matter.
[ So this is the "substantial differences" (differunt secundum substantiam) from the Objection ]
[ Like for instance Socrates, as (a) first substance, differs from Plato.
But if you ask why their form is different [ And here it concerns numerical diversity ] , no other reason can be given except that it exists in a different designated matter [ The form differs numerically, because it is received into another designated matter ]. Neither can any other explanation be found why this matter is divided from that except quantity. Consequently matter, as subject to dimension, is understood to be the principle of this diversity.
[ Thus he says : Yes, indeed they also differ substantially, namely with respect to matter, but this comes from quantity. So in the accomplishment of individuation (i.e. in the 'process' of individuating (something)) also an accident is involved. ]
Reply to 5. That argument is based on complete accidents, which are consequent upon the existence of form in matter [ I.e. which are ontologically consequent upon the reception of the form in matter ] , not on indeterminate dimensions [ these are incomplete accidents ], which are conceived of before the reception of form in matter [ Those (indeterminate dimensions) consequently cannot be taken away or thought away ], for without these the individual cannot be understood [ Because variation within the individual must be possible ], just as it is unintelligible without form [ Because the form precedes, according to its nature, all accidents, also quantity as (being an) accident. The form sets certain limits with respect to the determination.
Summarizing, this all means : If we think away all accidents -- appearing after the composite (of matter and form), the dimensionality, which is already in matter before the form, nevertheless remains. In S.T.III 77, 2, c. St Thomas says that "dimensive quantity is the first determination of matter", and this is compatible with the answer to the fifth Objection, in which however St Thomas relies more on the intelligibility of an individual with or without dimensions : We cannot think of an individual without dimensionality. We can, with respect to an individual, say, Socrates, think away all accidents ( In a modern conception, laid down in the thematic part of this website, we could question this ), except dimensionality (as a subgenus of the supreme genus of Quantity). And this is why this accident can also not be thought away -- as long as we hold on the individuum -- and therefore he in fact says : Sticking to the individual we cannot think away the individual (or (cannot think away) that something which causes individuality). Because within the domain of thinking the individual (aspect) is prior (namely directly observable) to dimensionality. Thus if we think away dimensionality, we in fact think away individuality. This dimensionality remains after thinking away of the accidents because this dimensionality is IN matter.
   Of course we CAN think away dimensionality, but then we automatically do not have a 'hoc aliquid' (a 'this something') anymore, but then it is also NOT true anymore -- as is posited in the Objection -- that 'the same (thing)'(See NOTE 8) (
eadem, neutr. accusative plural, because in the sentence of the (Latin) text we find the word 'diversa' (this is plural)), either in reality or in thought, is at some time numerically ONE (in the case of not thinking away), and at another time (numerically) diverse (in the case of thinking away)(See NOTE 9). In the Objection a distinction is being made between something that is (in) itself, namely substance, and those items that can be predicated of this substance. This is correct. But it is INcorrect to conclude from that this one substance would have some accident that causes this substance to be many substances (making it numerically divers): This is only possible in thought, with respect to second substance (substantia secunda). This second substance then can occur in several individuals, instead of in only one. With regard to first substance we never can say that that ONE first substance (i.e. the one, pointed to with the finger) is many substances after thinking away of the accidents.
Indeed St Thomas says that numerical unity and diversity is accomplished by incomplete accidents. In the present case these are the interminate dimensions (dimensiones interminatae) which are 'before the form already in matter. The accidents alone (i.e. all that appears after the composite of matter and form, and by implication after the form) are not sufficient for garanteeing numerical unity and diversity. If this were so then indeed the same thing (namely that one substance) would sometimes be ONE, sometimes MANY, dependent on which accident was being thought away. The conclusion of the Objection would then read as follows : The accidents alone are not sufficient for numerical unity and diversity. Dimensioned matter is necessary to make this possible. That multiplicity, here numerical diversity, cannot be an accident (accidens) of ONE substance is in fact immediately clear. It can be an accident of second substance : In "TWO PEOPLE", A and B, "TWO" is an accident, namely belonging to Discrete Quantity, but it does neither belong to A nor to B. It belongs, as has been said, exclusively to second substance : MAN can, accidentally, be two (where MAN is considered to be second substance).
First substance does not have accidents that -- by exclusively their own activity -- make this substance ONE substance. This always proceeds through matter (thus through first substance itself). First substance after all does not have any accident that -- through matter or not through mater -- makes it into many substances. This because "many substances" is NOT (a) substance. "Many substances" (for example an aggregate of substances) IS ontologically independent, but misses the character of (being a) Totality.
First substance is numerically ONE because of -- as we now have found out -- matter which stands under (the influence of) dimensive quantity. And because of that it is an ens extensum, i.e. a (spatially) extended being.
We can however legitimately say that that one substance becomes MANY (other) substances, by division, and in that case we only can say : These MANY substances are MANY, implying that there never can arise any contradiction, as suggested in the Objection (The contradiction was : Socrates is ONE substance, but, when the accidents are thought away, Socrates can be MANY substances). So we can never say: That one substance IS (now) many. We can only say : Either that one substance is ONE, or those many substances are MANY.
Reply to 6. Number, formally speaking, is prior to continuous quantity, but materially speaking continuous quantity is prior,
[ And all the time it concerns material division. Further (according to the Objection) we should -- with respect to individuation -- focus on the most fundamental (thus first) accident.
"Formally prior" here means logically prior.
"Materially prior" is : in fact prior.
And because it concerns a material division, we must look for the materially prior, and that is in this context continuous quantity. The Objection thus was, so to say, 'almost correct'.
for number is the result of the division of the continuum, as the Physics says. In this way the division of matter according to dimensions [ continuous quantity ] causes diversity in number.
[ Thus it turns out that plurality (number) CAN still be caused by another accident, because there is still another relevant accident before it, namely continuous quantity. ]
As for arguments ON THE CONTRARY, it is clear from what has been said to what extent they are to be granted and to what extent they arrive at erroneous conclusions.

******** END OF TEXT ********

See for important discussions about the Thomistic view of individuation, and the several accompanying aspects, among others :

BOBIK, J., 1965 (1970), Aquinas on Being and Essence, p.75--80. pp.268.
BOBIK, J., 1954, Dimensions in the individuation of bodily substances, Philosophical Studies 4.
BOBIK, J., 1953, La doctrine de saint Thomas sur l'individuation des substances corporelles, Revue Philosophique de Louvain, 51, p.5--41.

For a general thematical contemporary discussion with regard to individuation,
see, among others, GRACIA, J., 1988, Individuality.

This brings us at the end of the Text-Study.
In the NEXT PART we will summarize the Thomistic view of Individuation, followed by a more extensive thematic discussion of the same topic.

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