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John Philoponus and the conformity of mathematical proofs to Aristotelian demonstrations Orna Harari

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John Philoponus and the conformity of mathematical proofs to Aristotelian demonstrations Orna Harari
4
John Philoponus and the conformity
of mathematical proofs to Aristotelian
demonstrations
Orna Harari
One of the central issues in contemporary studies of Aristotle’s Posterior
Analytics is the conformity of mathematical proofs to Aristotle’s theory of
demonstration. The question, it seems, immediately arises when one compares Aristotle’s demonstrative proofs with the proofs in Euclid’s Elements.
According to Aristotle, demonstrative proofs are syllogistic inferences of the
form ‘All A is B, all B is C, therefore all A is C’, whereas Euclid’s mathematical proofs do not have this logical form. Although the discrepancy between
mathematical proofs and Aristotelian demonstrations seems evident, it is
only during the Renaissance that the conformity of mathematical proofs
to Aristotelian demonstrations emerges as a controversial issue.1 The
absence of explicit discussions of the conformity of mathematical proofs
to Aristotelian demonstrations in the earlier tradition seems puzzling from
the perspective of contemporary studies of Aristotle’s theory of demonstration. The formal discrepancies between Aristotelian demonstrations and
mathematical proofs seem so obvious to us that it is difficult to understand
how the conformity between mathematical proofs and Aristotelian demonstrations was ever taken for granted. In this chapter I attempt to bring to
light the presuppositions that led ancient thinkers to regard the conformity
of mathematical proofs to Aristotelian demonstrations as self-evident.
Neither an outright rejection nor an explicit approval of the conformity of mathematical proofs to Aristotelian demonstrations is found in
the extant sources from late antiquity; however, two approaches to this
issue can be detected. According to one approach, found in Proclus’
commentary on the first book of Euclid’s Elements, the conformity of
1
206
The first Renaissance thinker to reject the conformity of mathematical proofs to Aristotelian
demonstrations is Alessandro Piccolomini. His treatise Commentarium de certitudine
mathematicarum disciplinarum, published in 1547, initiated the debate known as the Quaestio
de certitudine mathematicarum, in which other Renaissance thinkers, such as Catena and
Pereyra, sided with Piccolomini in stressing the incompatibility between mathematical proofs
and Aristotelian demonstrations, whereas other thinkers, such as Barozzi, Biancani, and
Tomitano, attempted to reinstate mathematics in the Aristotelian model. I discuss this debate
and its ancient origins in the conclusions.
Philoponus and Aristotelian demonstrations
certain mathematical proofs to Aristotelian demonstrations is questioned.2
According to the other approach, found in Philoponus’ commentary on
Aristotle’s Posterior Analytics, the conformity of mathematical proofs
to Aristotelian demonstrations is taken for granted.3 Nevertheless, these
thinkers did not address the same question that Aristotle’s contemporary
interpreters discuss. Whereas contemporary studies focus on the discrepancy between the formal requirements of Aristotelian demonstrations
and mathematical proofs, the ancient thinkers focused on the non-formal
requirements of the theory of demonstration – namely, the requirements
that demonstrations should establish essential relations and ground their
conclusions in the cause.
In view of this account, I attempt to explain why the question whether
mathematical proofs meet these non-formal requirements does not arise
within the context of Philoponus’ interpretation of Aristotle’s theory of demonstration. Regarding the requirement that demonstrative proofs should
establish essential relations, I show that Philoponus considers it nonproblematic in the case of all immaterial entities including mathematical
objects. I show further that Philoponus’ assumption that mathematical
objects are immaterial renders the requirement that the middle term should
serve as a cause irrelevant for mathematical demonstrations, since according to Philoponus causes are required only to explain the realization of
form in matter. Accordingly, the dependence of mathematical proofs on
definitions is sufficient, in Philoponus’ view, to guarantee their conformity
to Aristotelian demonstrations. In substantiating this conclusion, I then
discuss Proclus’ argument to the effect that certain mathematical proofs do
not conform to Aristotelian demonstrations. I show that within the context
of Proclus’ philosophy of mathematics, in which geometrical objects are conceived of as realized in matter, consideration of the question whether mathematical proofs meet the two non-formal requirements – a question which
Philoponus ignores with regard to mathematical demonstrations – led
Proclus to argue for the non-conformity of certain mathematical proofs to
2
3
Proclus’ commentary on the first book of Euclid’s Elements was translated into Latin in 1560 by
Barozzi and it played an instrumental role in the debate over the certainty of mathematics. For
the reception of Proclus’ commentary on the Elements in the Renaissance, see Helbing 2000:
177–93.
Philoponus’ commentary on the Posterior Analytics has been hardly studied; hence it is
difficult to assess its direct or indirect influence on the later tradition. Nevertheless, it seems
that the several traits of Philoponus’ interpretation of the Posterior Analytics are found in the
medieval interpretations of Aristotle’s theory of demonstrations, such as the association of
demonstrations of the fact with demonstrations from signs which is found in Averroes (see
n. 38) and the identification of the middle term of demonstration with real causes (see n. 27).
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Aristotelian demonstrations. As a corollary to this discussion, I conclude my
chapter with an attempt to trace the origins of contemporary discussions of
the conformity of mathematical proofs to Aristotelian demonstrations to the
presuppositions underlying Philoponus’ and Proclus’ accounts of this issue.
I thereby outline a possible explanation for how concerns regarding the
ontological status of mathematical objects and the applicability of Aristotle’s
non-formal requirements to mathematical proofs evolved into concerns
regarding the logical form of mathematical and demonstrative proofs.
Philoponus on mathematical demonstrations
In the Posterior Analytics i.9, Aristotle states that if the conclusion of a demonstration ‘All A is C’ is an essential predication, it is necessary that the middle
term B from which the conclusion is derived will belong to the same family
(sungeneia) as the extreme terms A and C (76a4–9). This requirement is
tantamount to the requirement that the two propositions ‘All A is B’ and ‘All
B is C’, from which the conclusion ‘All A is C’ is derived, will also be essential
predications. The example that Aristotle presents in this passage for an essential predication is ‘The sum of the interior angles of a triangle is equal to two
right angles’. In his comments on this discussion Philoponus tries to show
that the attribute ‘having the sum of its interior angles equal to two right
angles’ is indeed an essential attribute of triangles. He does so by arguing
that Euclid’s proof meets the requirements of Aristotelian demonstrations:
For having [its angles] equal to two right angles holds for a triangle in itself (kath’
auto). And [Euclid] proves this [theorem] not from certain common principles, but
from the proper principles of the knowable subject matter. For instance, he proves
that the three angles of a triangle are equal to two right angles, by producing one
of the sides and showing that the two right angles, the interior one and its adjacent
exterior angle, are equal to the three interior angles,4 so that such a syllogism is
produced: the three angles of a triangle, given that one of its sides is produced, are
equal to the two adjacent angles. The two adjacent angles are equal to two right
angles. Therefore the angles of a triangle are equal to two right angles. And that
the two adjacent angles are equal to two right angles is proved from the [theorem]
that two adjacent angles are either equal to two right angles or are two right angles.
Whence [do we know] that adjacent angles are either equal to two right angles or
4
The proof that Philoponus describes is not identical to Euclid’s proof. Philoponus’ reference
to ‘two right angles’ implies that he envisages a right-angled triangle, whose base is extended
so as to create two adjacent right angles. Euclid’s proof refers to an arbitrary triangle. This
discrepancy does not affect Philoponus’ reasoning, as he states in the sequel that two adjacent
angles are either equal to two right angles or are two right angles.
Philoponus and Aristotelian demonstrations
are two right angles? We know it from the definition of right angles, [stating] that
when a straight line set up on a straight line makes the adjacent angles equal to each
other, the two equal angles are right. Well, having brought [the conclusion] back to
the definition and the principles of geometry, we no longer inquire further, but we
have the triangle proved from geometrical principles.5
In showing that Euclid’s proof conforms to the Aristotelian model of demonstration, Philoponus focuses on two issues: (1) he presents Euclid’s proofs
in a syllogistic form, and (2) he grounds the proved proposition in the
definition of right angle. The notion of first principles, on which Philoponus’
account is based, includes only one of the characteristics of Aristotelian first
principles – namely, their being proper to the discipline. In Philoponus’
view, the dependence of Euclid’s geometrical proof on geometrical first
principles, rather than on principles common to or proper to other disciplines, is sufficient to establish that this proof conforms to the Aristotelian
model. Two other characteristics of Aristotelian first principles are not taken
into account in this passage. First, Philoponus does not raise the question
whether the middle term employed in this proof is related essentially to the
subject of this proof; that is, he does not consider the question whether a
proposition regarding an essential attribute of adjacent angles can by any
means serve to establish the conclusion that this attribute holds essentially
for triangles.6 Nor does he express any reservations concerning the auxiliary
construction, in which the base is extended and two adjacent angles are
produced. Second, Philoponus does not mention Aristotle’s requirement
that the first principles should be explanatory or causal; he does not raise
the question whether the middle term in his syllogistic reformulation of
Euclid’s proof has a causal or explanatory relation to the conclusion. Thus
Philoponus’ account of the conformity of Euclid’s proofs to Aristotelian
demonstrations raises two questions: (1) why Philoponus ignores the question whether mathematical propositions state essential relations; and (2)
why the causal role of the principles of demonstration is not taken into
account. The following two sections answer these questions respectively.
Essential predications
Philoponus addresses the question whether mathematical proofs establish
essential predications in his comments on the Posterior Analytics i.22. He
5
6
116. 7–22, Wallies. All translations are mine.
For Philoponus’ syllogistic reformulation to be a genuine Aristotelian demonstration, one has
to assume that adjacent angles and triangles are related to each other as genera and species.
This assumption is patently false.
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formulates this question in response to Aristotle’s contention that sentences
whose subject is an attribute, such as ‘the white (to leukon) is walking’ or
‘the white is a log’ cannot feature in demonstrations, because they are not
predicative in the strict sense (Posterior Analytics 83a1–21). This contention jeopardizes, in Philoponus’ view, the status of geometrical proofs. The
subject matter of geometry, according to Philoponus, is shapes and their
attributes. Hence, Aristotle’s narrow conception of predication may imply
that proofs that establish that certain attributes belong to shapes are not
demonstrative because they prove that certain attributes, such as having the
sum of the interior angles equal to two right angles, belong to other attributes, such as triangles (239.11–14).7 Philoponus dismisses this implication
saying:
Even if these [attributes] belong to shapes accidentally, they are completive [attributes] of their being (symplērōtika tēs ousias) and like differentiae that make up the
species they are [the attributes] by which [shapes] are distinguished from other
things.8 … Just as ‘being capable of intellect and knowledge’ or ‘mortal’ or any of the
[components] in its definition do not belong to ‘man’ as one thing in another, but
[man] is completed from them, so the circle is also contemplated (theōreitai) from
all the attributes which are observed in it. Similarly, also the triangle would not be
something for which ‘having three angles equal to two right angles’ or ‘having the
sum of two sides greater than the third’ do not hold, but if one of these [attributes]
should be separated, immediately the being of a triangle would be abolished too.9
This account does not answer Philoponus’ original query; it does not tackle
the question whether proofs that establish predicative relations between
two attributes are demonstrative. Instead, Philoponus focuses here on the
question whether the attributes that geometry proves to hold for shapes
are essential, arguing that mathematical attributes like differentiae are
parts of the definitions of mathematical entities. However, the analogy
between the differentiae of man and mathematical propositions is not as
obvious as Philoponus formulates it. The attributes ‘capable of knowledge’
and ‘mortal’ distinguish men from other living creatures; the former distinguishes human beings from other animals and the latter distinguishes
7
8
9
Philoponus presupposes here Aristotle’s categorical scheme, in which terms belonging to the
nine non-substance categories are attributes of terms belonging to the category of substance.
According to Aristotle’s Categories the term ‘shape’ belongs to the category of quality. Hence,
Philoponus claims that geometry studies attributes of attributes.
The term ‘completive attributes’ (symplērōtikos) refers in the neo-Platonic tradition to
attributes without which a certain subject cannot exist. On these attributes and their relation to
differentiae, see De Haas 1997: 201 and Lloyd 1990: 86–8.
239.14–25, Wallies.
Philoponus and Aristotelian demonstrations
them from divine entities, which are also capable of knowledge but are not
mortal. By contrast, the geometrical attributes that Philoponus mentions
in this passage do not distinguish triangles or circles from other shapes.
Admittedly, the attribute ‘having the sum of the interior angles equal to
two right angles’ holds only for triangles, yet, unlike ‘having three sides’, it
is not the feature that distinguishes triangles from other shapes. It seems,
then, that in accounting for the essentiality of mathematical attributes,
Philoponus expands the notion of differentia, so as to include all the attributes of mathematical entities. He does not distinguish between attributes
that enter into the definition of an entity and necessary attributes; he concludes from the statement that a triangle will cease to be a triangle if one
of its attributes were separated from it that these attributes are essential.
Thus, rather than explaining why mathematical attributes are essential in
Philoponus’ view, this passage reflects his assumption that the essentiality
of mathematical attributes is evident. This assumption, I surmise, can be
understood in light of Philoponus’ interpretation of the principles of
demonstration.
In his comments on the Posterior Analytics ii.2,10 Philoponus accounts
for the distinction between indemonstrable premises and demonstrable
conclusions in terms of the distinction between composite and incomposite
entities. Incomposite entities, according to this discussion, are simple or
intelligible substances such as the intellect or the soul, which are considered
(theōroumenon) without matter.11 In the case of such entities, Philoponus
argues, the defining attribute is not different from the definable object
and therefore propositions concerning such entities are indemonstrable
or immediate. Another characterization of indemonstrable premises is
found in Philoponus’ interpretation of Aristotle’s discussion of the relationship between definitions and demonstrations in the Posterior Analytics
ii.2–10. In addressing the question whether it is possible to demonstrate a
definition, Philoponus draws a distinction between two types of definition:
formal and material. Formal definitions are the indemonstrable principles
of demonstration that define incomposite entities; they include, according to Philoponus, the essential attributes (ousiodōs) of the defined object.
Material definitions, by contrast, serve as demonstrative conclusions and
10
11
The editor of Philoponus’ commentary on the Posterior Analytics, M. Wallies, doubted the
attribution of the commentary on the second book of the Posterior Analytics to Philoponus
(v–vi). The authenticity of the commentary on the second book does not affect my argument,
because all the references I make here to the commentary on the second book accord with
views expressed in Philoponus’ other commentaries.
339. 6–7, Wallies.
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include the attributes that are present in matter.12 In this interpretation,
then, the ontological distinction between incomposite and composite
entities accounts for two characteristics of the principles of demonstration: their indemonstrability and their essentiality. The question whether
certain propositions meet Aristotle’s requirements is not answered by an
examination of their logical characteristics, but by the ontological status of
their subjects.
It follows from this discussion that from Philoponus’ viewpoint the
immateriality of the subject of predication is sufficient to guarantee the
essential relation between a subject and its attributes.13 This assumption
may explain Philoponus’ approach to the issue of the essentiality of mathematical propositions. Mathematical objects, according to Philoponus, are
abstractions from matter14 – that is, they belong to the class of incomposite
objects that serve as the subjects of formal definitions. Thus, in light of
Philoponus’ characterization of these definitions, it plausible to regard all
attributes of mathematical objects as essential, because the immateriality of
these objects seems to entail, in Philoponus’ view, the essentiality of their
attributes. In what follows, I show that the ontological distinction between
incomposite and composite entities also explains why the causal role of
the middle term is not taken into account in Philoponus’ discussion of the
conformity of Euclid’s proofs to Aristotelian demonstrations.
Causal demonstrations
In his commentary on Aristotle’s Physics ii.2, Philoponus examines the
tenability of Aristotle’s criticism of the theory of Forms, which involves,
according to Aristotle, separation from matter of the objects of physics,
although they are less separable than mathematical objects. In so doing,
Philoponus draws a distinction between separability in thought and separability in existence, claiming that he agrees with Aristotle that the forms
12
13
14
364.16–18, Wallies.
Two reasons may explain why Philoponus does not consider the possibility that immaterial
entities have accidental attributes. First, it is commonly held in the ancient tradition that only
individuals have accidental attributes, which belong to their matter. Second, Philoponus’
notion of essential predication is more formal than Aristotle’s. In characterizing essential
predications Philoponus appeals to extensional, rather than intensional, considerations. In his
view, attributes that belong to all members of a species and only to them are essential (e.g., In
An. Post. 63.14–20, Wallies; In DA 29.13–30.1, Hayduck; In Cat. 64.9, Busse).
For Philoponus’ conception of mathematical objects, see (e.g.) In Phys. 219.10; In DA, 3.7–11.
For a discussion of this view, see Mueller 1990: 465–7.
Philoponus and Aristotelian demonstrations
of natural things cannot be separated in existence from matter, but he disagrees with Aristotle’s view if it implies that these forms cannot be separated
by reason and in thought.15 Although Philoponus’ account of the indemonstrability of the principles of demonstration presupposes the possibility
of separating the definitions of both mathematical and physical entities,
the ontological difference between these classes of objects is nevertheless
maintained. In his commentary on Aristotle’s De anima, Philoponus draws
a distinction between physical and mathematical definitions, arguing that
physical definitions should refer to the matter of physical substance, their
form and the cause by virtue of which the form is realized in matter.16
Mathematical definitions, by contrast, refer only to the form:
The mathematician gives the definitions of abstracted forms in themselves, without
taking matter into account, but he gives these [definitions] in themselves. For this
reason he does not mention the cause in the definition; for if he defined the cause,
clearly he would also have taken the matter into account. Thus, since he does not
discuss the matter he does not mention the cause. For example, what is a triangle?
A shape contained by three lines; what is a circle? A shape contained by one line.
In these [definitions] the matter is not mentioned and hence neither is the cause
through which this form is in this matter. Unless perhaps he gives the cause of those
characteristics holding in themselves for shapes, for instance, why a triangle has its
angles equal to two right angles.17
Philoponus’ distinction between physical and mathematical definitions
has two related consequences for the methods employed in physics and
mathematics. First, although both physical and mathematical demonstrations are based on indemonstrable formal definitions, these definitions
adequately capture the nature of mathematical objects but they fail to
exhaust the nature of physical objects. In the case of physical demonstrations, the formal definition captures only one aspect of the object: its
form. Full-fledged knowledge of physical objects requires reference also
to the matter of this object and the cause of the realization of the form
in matter. Indeed, in both the commentary on Aristotle’s De anima and
the commentary on the Posterior Analytics, Philoponus considers formal
definitions of physical objects deficient. In the commentary on De anima,
Philoponus argues that definitions that do not include all the attributes
15
16
17
225.4–11, Vitelli. For the relationship between Philoponus’ discussion of separability in
thought of physical definitions and his analysis of demonstrations in the natural sciences, see
De Groot 1991: 95–111.
55.31–56.2, Hayduck.
57.35–58.6, Hayduck.
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of an object are not physical definitions, but are dialectical or empty. His
example of such an empty definition is the formal definition of anger:
‘anger is a desire for revenge’. The adequate definition of anger, according
to Philoponus, is ‘anger is boiling of the blood around the heart caused by
a desire for revenge’.18 This definition refers to the form, the matter and the
cause. Similarly, in the commentary on the Posterior Analytics, Philoponus
claims that neither the formal nor the material definition is a definition in
the strict sense; only the combination of these two yields an adequate definition.19 This conception of definition is evidently inapplicable to mathematics. Mathematical objects are defined without reference to matter or
to their cause, hence formal definitions provide an exhaustive account of
these objects.
The second consequence of Philoponus’ distinction between physical
and mathematical definitions concerns the explanatory or causal relations in demonstrative proofs. Although in the above-quoted passage
Philoponus contends that the cause is also studied in mathematics when a
relation between a mathematical object and its attributes is proved, it seems
that this cause is different from the one studied in physics. According to
the above passage, physics studies the cause of the realization of form in
matter, but since mathematics does not deal with the matter of its objects,
its explanations do not seem to be based on this type of cause. Furthermore,
Philoponus’ analysis of physical demonstrations in terms of the distinction
between formal and material definitions gives rise to a problem that has
no relevance for mathematical demonstrations. This interpretation gives
rise to the question of how the material aspect of a physical entity, which is
a composite of form and matter, can be demonstratively derived from the
formal definition, given that this definition does not exhaust the nature of
the composite entity. Stating this question differently, how, in Philoponus’
view, can a proposition regarding a substance taken with matter be
demonstratively derived from a proposition regarding its form, which is
considered in separation from matter? Evidently this question does not
arise in the mathematical context. Mathematical definitions do not refer to
matter; hence, they give an exhaustive account of mathematical objects. In
what follows, I show that Philoponus answers this question by appealing to
extra-logical considerations. More specifically, I show that the causal role
of the middle term in demonstrations provides Philoponus with the means
of bridging the gap between formal definitions and material definitions.
18
19
43.28–44.8, Hayduck.
365.1–13, Wallies.
Philoponus and Aristotelian demonstrations
In his comments on the Posterior Analytics ii.2, Philoponus presents the
following explanation for Aristotle’s remark that the questions ‘what it is’
(ti esti) and ‘why it is’ (dia ti) are the same:
For if the ‘what it is’ and the ‘why it is’ are different, it is insofar as the former is
sought with regard to simple [entities] and the latter with regard to composite [entities]. Yet these [questions] are the same in substrate, but different in their mode of
employment. Both the ‘what it is’ and the ‘why it is’ are studied in the case of the
eclipse being an affection of the moon. And we use these, the ‘what it is’ and the
‘why it is’, differently. But if we take an eclipse itself by itself, we seek what is the
cause of an eclipse, and we say that it is a privation of the moon’s light due to screening by the earth. But if we seek whether an eclipse exists (hyparkhei) in the moon,
namely why it exists, we take the ‘what it is’ as a middle term, namely privation of
the moon’s light coming about as a result of screening by the earth.20
Although this passage is presented to account for the identity between the
questions ‘what it is’ and ‘why it is’, Philoponus dissociates these two questions. The distinction he draws here is based on the ontological distinction
between simple and composite entities. The question ‘what it is’ is asked
with regard to simple entities, whereas the question ‘why it is’ is asked with
regard to composite entities. In the case of composite entities, Philoponus
argues, ‘what it is’ and ‘why it is’ are different questions. The definition
of an eclipse and the cause of its occurrence are not identical. The exact
significance of Philoponus’ distinction between these questions is not clear
from this passage. The examples presented by Philoponus seem to blur his
distinction between an eclipse considered in the moon and an eclipse considered in separation from the moon, as the accounts given for both cases
are identical – ‘privation of the moon’s light due to screening by the earth’.
This difficulty in understanding Philoponus’ distinction between ‘what
it is’ and ‘why it is’ may stem from his attempt to accommodate his view,
which dissociates these questions, with Aristotle’s claim that these questions are identical. As a result, Philoponus follows Aristotle in exemplifying the answers to these questions by one and the same account. However,
according to Philoponus’ other discussions of the definitions of entities,
which are considered in separation from matter, the account for the eclipse
taken in separation from the moon should be the formal definition ‘screening by the earth’, whereas ‘privation of the moon’s light due to screening by
the earth’ is the full definition, resulting from a demonstration that relates
the formal definition to the material definition.21 Despite the difficulty in
20
21
339.20–9, Wallies.
371.19–25, Wallies.
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understanding the distinction made in this passage, Philoponus clearly
does not follow Aristotle here in assimilating definitions with explanations.
This conclusion finds further support in Philoponus’ comments on the
Posterior Analytics i.4.
In the Posterior Analytics i.4, Aristotle presents four senses in which
one thing is said to hold for another ‘in itself ’. The first two senses are
predicative and they constitute Aristotle’s account for the predicative
relations that the premises of demonstration should express. According
to the first sense, a predicate holds for a subject in itself if it is a part of
the definition of the subject. According to the second sense, a predicate
holds for a subject in itself if the subject is a part of the definition of the
predicate. The third sense distinguishes substances that exist in themselves
from attributes, which depend on substances, by virtue of their being
said of them. The fourth sense distinguishes a causal relation between
events from an incidental relation between events. In his comments on
this fourfold distinction Philoponus argues that only the first two senses
of ‘in itself ’ contribute to the demonstrative method,22 yet he also regards
the fourth sense (i.e. the causal sense) as relevant to the theory of demonstration. According to Philoponus, the causal sense of ‘in itself ’, though it
does not contribute to the formation of the premises of demonstration,
contributes to the ‘production of the whole syllogism’.23 More precisely,
Philoponus argues that the causal sense of ‘in itself ’ expresses the relation between the cause, taken as the middle term of demonstration, and
the conclusion. The example Philoponus presents of this contention is
the following syllogism: The moon is screened by the earth. The screened
thing is eclipsed. Therefore, the moon is eclipsed. Commenting on this
syllogism, Philoponus remarks that the fact that screening by the earth
is the cause of the eclipse of the moon is not expressed in the premises
of this demonstration, but its causal force becomes evident from its role
as a middle term.24 In this discussion, then, Philoponus employs two different senses of ‘in itself ’ in accounting for the relations expressed in the
premises of demonstration and the relation between the middle term and
the conclusion. The premises of demonstration, according to Philoponus,
are ‘in itself ’ in one of the two first senses delineated by Aristotle. That
is, their predicate is either a part of the definition of the subject or their
subject is a part of the definition of the predicate. By contrast, the middle
22
23
24
65.10–11, Wallies.
65.15, Wallies.
65.16–19, 65.20–3, Wallies.
Philoponus and Aristotelian demonstrations
term and the conclusion of a demonstration are related according to the
fourth sense of ‘in itself ’ – that is, they are related as cause and effect.25 So,
according to Philoponus, the derivation of the demonstrative conclusion
is not solely based on the transitivity of the predicative relation stated in
the premises. In addition to the transitivity of the predicative relation, the
demonstrative derivation is based on causal relations between the middle
term and the conclusion. Such a distinction between logical relations and
extra-logical or causal relations is explicitly drawn at the beginning of
Philoponus’ introduction to his commentary on the second book of the
Posterior Analytics:
In the first book of the Apodeiktike (i.e. the Posterior Analytics), he showed how
there is a demonstration and what is a demonstration and through what premises it
has come about, and he showed further how a demonstrative syllogism differs from
other syllogisms and that in other syllogisms the middle term is the cause of the
conclusion and not of the thing and in demonstrative syllogism the middle term is
the cause both of the conclusion and of the thing.26
It follows from this discussion that Philoponus’ ontological distinction
between physical and mathematical entities yields different accounts for
physical and mathematical demonstrations. The distinction between the
three facets of physical entities – i.e. the form, the matter and the cause for
the realization of form in matter – is reflected in Philoponus’ interpretation of the theory of demonstration. In this interpretation, demonstrations,
like physical entities, have three components: indemonstrable premises,
regarded as formal definitions, demonstrative conclusions, which are
material definitions, and the middle term, which serves as the cause that
relates the formal definition to the material definition. Philoponus’ distinction between the form of a physical entity and the cause of the realization
of form in matter finds expression in the distinction he draws between
the formal definition considered in itself and that formal definition in its
role as the middle term in demonstration. This distinction implies that
25
26
The analysis of demonstrative derivation in causal terms is widespread in Philoponus’
commentary on the Posterior Analytics (e.g., 24.22–4; 26.9–13; 119.19–21; 173.14–20;
371. 4–19). The causal analysis of demonstrative derivation underlies Philoponus’ introduction
of a second type of demonstration, called ‘tekmeriodic demonstration’, in which causes are
deduced from effects (In An. Post. 33.11; 49.12; 169.8; 424.13, Wallies; In Phys. 9.9–10.21,
Vitelli). On Philoponus’ notion of tekmeriodic proofs and its reception in the Renaissance, see
Morrison 1997: 1–22.
334.1–8, Wallies. The distinction between the middle term as the cause of the thing and the
middle term as the cause of the conclusion is also found in the Latin medieval tradition of
interpreting the Posterior Analytics. See De Rijk 1990.
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demonstrative derivation rests on two relations: the transitivity of the predicative relation that the premises state and the causal relation between the
middle term and the conclusion. This distinction is applicable to physical
demonstrations, for which the cause of the realization of form in matter is
sought. The demonstrative derivation in these demonstrations is based not
only on logical relations but also on causal relations. Mathematical entities, by contrast, have only one facet: the form. Accordingly, Philoponus’
account of the conformity of mathematical demonstrations to Aristotelian
demonstrations focuses only on the formal requirements of the theory
of demonstration. The conformity of mathematical demonstrations to
Aristotelian demonstrations is guaranteed if the conclusions can be shown
to depend on the definitions of mathematical entities. Since mathematical
objects have no matter, mathematical demonstrations can be based only
on logical derivation; the question whether the middle term is the cause of
the conclusion does not arise in this context, as the separation from matter
renders superfluous questions concerning causes.27
The analysis of Philoponus’ interpretation of Aristotle’s theory of demonstration reveals the importance of the ontological distinction between
simple and composite entities for his account of conformity of mathematical proofs to Aristotelian demonstrations. The assumption that mathematical objects are analogous to simple entities by being separated in thought
from matter does not give rise to two questions that may undermine the
conformity of mathematical proofs to Aristotelian demonstrations. The
first question is whether mathematical predications are essential; the
second is whether the middle term in mathematical proofs is the cause of
the conclusion. The first question does not arise because the separation
from matter implies that only the essential attributes of entities are taken
into consideration. The second does not arise because causal considerations
are relevant only with regard to composite entities, as it is only in their case
that the cause of the realization of form in matter can be sought. Hence,
given the assumption that mathematical entities are separated in thought
from matter, the question whether mathematical proofs conform to the
non-formal requirements of Aristotle’s theory of demonstration does not
arise. This conclusion gains further support from Proclus’ discussion of the
conformity of mathematical proofs to Aristotelian demonstrations.
27
This conclusion may explain Proclus’ otherwise curious remark that the view in which
geometry does not investigate causes is originated in Aristotle (In Eucl. 202.11, Friedlein). If
this explanation is correct, Philoponus’ conception of mathematical demonstrations seems to
reflect a widespread view in late antiquity.
Philoponus and Aristotelian demonstrations
Proclus on the conformity between mathematical
proofs and Aristotelian demonstrations
Proclus’ philosophy of geometry is formulated as an alternative to a conception whereby mathematical objects are abstractions from material or
sensible objects.28 According to Proclus, mathematical objects do not differ
from sensible objects in their being immaterial, but in their matter. Sensible
objects, in Proclus’ view, are realized in sensible matter, whereas mathematical objects are realized in imagined matter. In Proclus’ philosophy of
geometry, then, mathematical objects are analogous to Philoponus’ physical objects; they are composites of form and matter. Proclus’ philosophy
of mathematics is at variance not only with Philoponus’ views regarding
the ontological status of geometrical objects but also with Philoponus’
views regarding the conformity of Euclid’s proofs to Aristotelian demonstrations.29 In his discussion of the first proof of Euclid’s Elements in the
commentary on the first book of Euclid’s Elements, Proclus questions the
conformity of certain mathematical proofs to the Aristotelian model:
We shall find sometimes that what is called ‘proof ’ has the properties of demonstration, in proving the sought through definitions as middle terms – and this is a
perfect demonstration – but sometimes it attempts to prove from signs. This should
not be overlooked. For, although geometrical arguments always have their necessity
through the underlying matter, they do not always draw their conclusions through
demonstrative methods. For when it is proved that the interior angles of a triangle
are equal to two right angles from the fact that the exterior angle of a triangle is
equal to the two opposite interior angles, how can this demonstration be from the
cause? How can the middle term be other than a sign? For the interior angles are
equal to two right angles even if there are no exterior angles, for there is a triangle
even if its side is not extended.30
In this passage, Proclus claims that Euclid’s proof that the sum of the interior angles of a triangle is equal to two right angles (Elements i.32) does
not conform to Aristotle’s model of demonstrative proofs. In so doing,
he focuses on the causal role of the middle term in Aristotelian demonstrations. Proclus argues that Euclid’s proof does not conform to the
Aristotelian model because it grounds the equality of the sum of the interior angles of a triangle to two right angles in a sign rather than in a cause.
28
29
30
In Eucl. 50.16–56.22, Friedlein.
A discussion of the relationship between Proclus’ philosophy of geometry and his analysis of
mathematical proofs is beyond the scope of this paper. For this issue, see Harari 2006.
206.12–26, Friedlein.
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Proclus’ reason for regarding this Euclidean proof as based on signs rather
than on causes concerns the relationship between the auxiliary construction employed in this proof and the triangle. According to Proclus, the
extension of the triangle’s base is merely a sign and not a cause of the equality of the triangle’s angles to two right angles because ‘there is a triangle
even if its side is not extended’. The exact force of this statement is clarified
in Proclus’ discussion of the employment of this auxiliary construction in
another Euclidean proof – the proof that the sum of any two interior angles
of a triangle is less than two right angles (Elements i.17). In this discussion,
Proclus claims that the extension of the triangle’s base cannot be considered
the cause of the conclusion since it is contingent: the base of a triangle may
be extended or not, whereas the conclusion that the sum of any two interior angles of a triangle is less than two right angles is necessary.31 Hence,
in questioning the conformity of certain Euclidean proofs to Aristotelian
demonstrations, Proclus raises the two questions that Philoponus ignores
in the case of mathematical demonstrations. Unlike Philoponus, Proclus
asks whether the middle term in Euclid’s proofs is the cause of the conclusion and whether it is essentially related to the triangle.
Furthermore, Proclus’ attempt to accommodate Euclid’s proofs of the
equality of the sum of the interior angle of a triangle to two right angles
with Aristotle’s requirement that demonstrations should establish essential
relations indicates that he shares with Philoponus the assumption that
demonstrations regarding material entities require an appeal to causal considerations. In concluding his lengthy discussion of Euclid’s proof that the
sum of the interior angles of a triangle is equal to two right angles, Proclus
says:
We should also say with regard to this proof that the attribute of having its interior
angles equal to two right angles holds for a triangle as such and in itself. For this
reason, Aristotle in his treatise on demonstration uses it as an example in discussing essential attributes … For if we think of a straight line and of lines standing in
right angles at its extremities, then if they incline so that they generate a triangle we
would see that in proportion to their inclination, so they reduce the right angles,
which they made with the straight line; the same amount that they subtracted from
these [angles] is added through the inclination to the angle at the vertex, so of
necessity they make the three angles equal to two right angles.32
The procedure described in the passage, in which a triangle is generated
from two perpendiculars to a straight line that rotate towards each other
31
32
311.15–21, Friedlein.
384.5–21, Friedlein.
Philoponus and Aristotelian demonstrations
up to their intersection point, is also presented by Proclus in his comments
on propositions i.16 and i.17 of the Elements. In both cases, he regards this
procedure – and not Euclid’s auxiliary construction in which the triangle’s
base is extended – as the true cause of the conclusion.33 Proclus’ appeal to
this procedure in searching for the true cause of these conclusions indicates
that in attempting to accommodate Euclid’s proofs with Aristotle’s requirement that demonstrations should establish essential relations, he grounds
mathematical conclusions in causal relations rather than in logical relations. Proclus considers the proposition that the sum of the interior angles
of a triangle is equal to two right angles essential not because it is derived
from the definition of a triangle, as Aristotle’s theory of demonstration
requires, but because the proposition is derived from the triangle’s mode
of generation. Viewed in light of Philoponus’ interpretation of Aristotle’s
theory of demonstration, Proclus’ attempt to accommodate Euclid’s proof
with Aristotelian demonstrations seems analogous to Philoponus’ account
of physical demonstrations. In both cases, causal considerations are
employed in rendering proofs concerning material objects compatible with
Aristotelian demonstrations.
This examination of the presupposition underlying Philoponus’ and
Proclus’ views regarding the conformity of mathematical proofs to
Aristotelian demonstrations has led to the following conclusions.
(1) The pre-modern formulation of the question of the conformity of
mathematical proofs to Aristotelian demonstrations concerns the
applicability of the non-formal requirements of the theory of demonstration to mathematical proofs. More specifically, this formulation
concerns the questions whether mathematical attributes are proved
to belong essentially to their subjects and whether the middle term in
mathematical proofs serves as the cause of the conclusion.
(2) The emergence or non-emergence of the question of the conformity
of mathematical proofs to Aristotelian demonstration is related to
assumptions concerning the ontological status of mathematical objects.
This question does not arise in a philosophical context in which mathematical objects are conceived of as separated in thought from matter,
whereas it does arise when mathematical objects are conceived of as
realized in matter.
(3) Demonstrations concerning composites of form and matter were
understood in late antiquity as based on causal relations, viewed as
additional to the logical necessitation of conclusions by premises.
33
310.5–8, 315.15, Friedlein.
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Causal considerations are employed with regard to mathematical demonstrations, when mathematical objects are considered material; they
are not employed when mathematical objects are considered separated
in thought from matter.
Conclusions
In concluding this chapter, I examine the relationship between the modern
formulation of the question of the conformity of mathematical proofs
to Aristotelian demonstrations and its formulation in late antiquity. The
modern discussions of the relationship between Aristotle’s theory of demonstration and mathematical proofs focus on Aristotle’s formal requirement
that demonstrations should be syllogistic inferences from two universal
predicative propositions, which relate the subject and predicate of the conclusion to a third term, called the ‘middle term’.
The disagreement among Aristotle’s modern commentators concerns
whether mathematical proofs can be cast in this logical form. For instance,
Ian Mueller, who says they cannot, argues that in a syllogistic reformulation
of Euclidean proofs the requirement that the inference should have only
three terms is not always met, because the mathematical proofs depend
on the relations between mathematical entities and not on their properties
taken in isolation from other entities.34 The possibility of expressing mathematical relations in syllogistic inferences is also central in modern attempts
to render Aristotle’s theory of demonstration compatible with mathematical proofs. Henry Mendell, for instance, shows that Aristotle’s theory of
syllogism does have the formal means that make possible syllogistic formulations of mathematical proofs. In so doing, he argues that the relation
of predication, which is formulated by Aristotle as ‘x belongs to y’, can be
read flexibly so that it also accommodates two-place predicates, such as ‘x
equals y’, or ‘x is parallel to y’.35 Mendell’s argument, like Mueller’s, focuses
on the possibility of expressing relations within the formal constraints of
the theory of syllogism. The extra-logical consequences of the expansion
of the theory of syllogism to relational terms and their compatibility with
Aristotle’s theory of demonstration are not at the centre of either Mendell’s
or Mueller’s argument. More specifically, they do not address the question
of whether relational terms or mathematical properties can be proved to
34
35
Mueller 1975: 42.
Mendell 1998.
Philoponus and Aristotelian demonstrations
be essential predicates of their subjects.36 This question, as I showed, was
central in the discussions of the conformity of mathematical proofs to
Aristotelian demonstrations in late antiquity.
The non-formal requirements of the theory of demonstration were also
central in the Renaissance debate over the certainty of mathematics.37
Piccolomini’s objective in his Commentarium de certitudine mathematicarum disciplinarum was to refute what he presents as a long-standing
conviction that mathematical proofs conform to the most perfect type
of Aristotelian demonstration, called in the Renaissance demonstratio
potissima. The classification of types of demonstrations that underlies
Piccolomini’s argument is based on Aristotle’s distinction between demonstrations of the fact (hoti) and explanatory demonstrations or demonstration of the reasoned fact (dioti). This distinction has been further
elaborated by Aristotle’s medieval commentators and it appears in the
Proemium of Averroes’ commentary on Aristotle’s Physics as a tripartite
classification of demonstrations into demonstratio simpliciter, demonstratio propter quid and demonstratio quid est. It is in this context that
Averroes claims that mathematical proofs conform to the perfect type of
demonstration, in his terminology demonstratio simpliciter.38 According to
this classification, the different types of demonstration differ in the epistemic characteristics of their premises, hence in the epistemic worth of the
knowledge attained through them. Following this tradition, Piccolomini’s
argument for the inconformity of mathematical proofs to Aristotelian
demonstrations focuses on these characteristics. According to Piccolomini
potissima demonstrations are demonstrations in which knowledge of the
cause and of its effects is attained simultaneously; the premises of such
demonstrations are prior and better known than the conclusion; their
middle term is a definition, it is unique and it serves as the proximate
cause of the conclusion. Mathematical demonstrations, so Piccolomini
and his followers argue, fail to meet these requirements. The importance
of the non-formal requirements of the theory of demonstration for the
Renaissance debate over the certainty of mathematics comes to the fore
in the following passage from Pereyra’s De communibus omnium rerum
naturalium principiis et affectionibus:
36
37
38
This question is not utterly ignored in modern interpretations of the Posterior Analytics. See
McKirahan 1992; Goldin 1996; Harari 2004.
For a general discussion of the Quaestio de certitudine mathematicarum, see Jardine 1998. For
the influence of this debate on seventeenth-century mathematics, see Mancosu 1992 and 1996.
Aristotelis opera cum Averrois commentariis, vol. iv, 4.
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Demonstration (I speak of the most perfect type of demonstration) must depend
upon those things which are per se and proper to that which is demonstrated;
indeed, those things which are accidental and in common are excluded from
perfect demonstrations … The geometer proves that the triangle has three angles
equal to two right ones on account of the fact that the external angle which results
from extending the side of that triangle is equal to two angles of the same triangle
which are opposed to it. Who does not see that this medium is not the cause of the
property which is demonstrated? . . . Besides, such a medium is related in an altogether accidental way to that property. Indeed, whether the side is produced and
the external angle is formed or not, or rather even if we imagine that the production
of the one side and the bringing about of the external angle is impossible, nonetheless that property will belong to the triangle; but what else is the definition of an
accident than what may belong or not belong to the thing without its corruption?39
Pereyra’s argument for the inconformity of mathematical proofs to
Aristotelian demonstrations is similar to Proclus’ argument. Like Proclus,
Pereyra focuses on the question whether mathematical proofs meet the
non-formal requirements of the theory of demonstration. More specifically,
he raises the two questions that were at the centre of Proclus’ discussion of
this issue: (1) Do the premises of mathematical proofs state essential or
accidental relations? (2) Are Euclid’s proofs, which are based on auxiliary
constructions, explanatory? These questions are viewed in this passage as
interrelated; real explanations are provided when the relation between a
mathematical entity and its property is proved to be essential. This requirement is met if the premises on which the mathematical proof is based state
essential relations. The only allusion to the syllogistic form of inference
made in this passage is to the middle term in syllogistic demonstrations.
However, like Proclus, Pereyra considers the middle term only in its role
as the cause of the conclusion. Its formal characteristics, such as its position, are not discussed here. Thus, pre-modern and modern discussions
of the conformity of mathematical proofs to Aristotelian demonstrations
concern different facets of the theory of demonstration. Whereas the
modern discussions focus on the formal structure of Aristotelian demonstrations, pre-modern discussions concern its non-formal requirements.
Accordingly, the questions asked in these discussions are different. The
modern question is whether syllogistic inferences can accommodate relational terms whereas the pre-modern question is whether mathematical
proofs establish essential relations.
39
The translation is based on Mancosu 1996: 13. The complete Latin text appears on p. 214, n. 12
of Mancosu’s book.
Philoponus and Aristotelian demonstrations
Nevertheless, when the pre-modern discussion of the conformity of
mathematical proofs to Aristotelian demonstrations is viewed in light of its
underlying ontological presuppositions, a conceptual development leading
to the modern formulation of this question may be traced. Discussions of
the conformity of mathematical proofs to Aristotelian demonstrations in
late antiquity were associated with discussions of whether mathematical
objects are immaterial or material;40 that is, whether they are conceptual or
real entities. This ontological distinction is reflected in different accounts
of the relation of derivation, on which demonstrations are based. Whereas
demonstrations concerning immaterial objects are based on definitions
and rules of inference alone, demonstrations concerning material objects
require the introduction of extra-logical considerations, such as the causal
relations between form and matter. Thus, the question of the ontological status of mathematical objects reflects the epistemological question:
whether extra-logical considerations have to be taken into account in
mathematics. When discussions of the conformity of mathematical proofs
to Aristotelian demonstrations in late antiquity are viewed in isolation
from ontological commitments, they seem to be conceptually related to
modern discussions of the nature of mathematical knowledge. The need to
take into account extra-logical considerations when mathematical objects
are considered material is equivalent to Kant’s statement that mathematical
propositions are synthetic a priori judgements. Developments in modern
logic led to a reformulation of Kant’s statement in terms of logical forms.
Kant’s contention that mathematical knowledge cannot be based on definitions and rules of inference alone was regarded by Bertrand Russell as
true for Kant’s time. According to Russell, had Kant known other forms
of logical inference than the syllogistic form, he would not have claimed
that mathematical propositions cannot be deduced from definitions and
rules of inference alone.41 In light of this account, the modern discussions
of the conformity of mathematical proofs to Aristotelian demonstrations,
which focus on whether syllogistic inferences can accommodate relational
terms, may be understood as evolving from the pre-modern discussions
of whether mathematical proofs establish essential relations, and to establish this conclusion, two conceptual developments have to be traced: the
process by which the question whether mathematical propositions are
40
41
This assumption seems to underlie the Renaissance discussions of this issue as well. In the
eleventh chapter of his treatise Piccolomini attempts to reinstate the status of mathematics
as a science by claiming that mathematical objects are conceptual entities, existing in the
human mind.
Russell 1992: 4–5.
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essential has become dissociated from questions concerning the ontological status of mathematical objects, and the process leading to the development of modern logic.
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