PAOLO
MANCOSV
ARISTOTELIAN LOGIC AND
EUCLIDEAN MATHEMATICS:
SEVENTEENTH-CENTURY
DEVELOPMENTS
OF THE QUAESTZO DE CERTZTUDZNE
A4ATHEA4ATZCARUA4
Introduction
AMONG the factors which played a role in the birth of Galilean science is the
process of critical revision of Aristotelian
philosophy which took place during
the fifteenth and the sixteenth centuries.
As part of this process must be
included the sixteenth-century
reflection on the ‘epistemology of mathematics’.
Within the background
of Aristotelian
philosophy
a number of issues were
raised about the nature of mathematics
which led some authors (e.g. Piccolomini, Catena, Pereyra) to the paradoxical
thesis that mathematics
is not a
science. These positions,
understandably,
generated
the reactions of other
authors (e.g. Barozzi, Biancani, Tomitano)
who tried to reinstate mathematics
into the framework of ‘Aristotelian
science’.’ This debate is often mentioned as
*Department
of Philosophy,
Yale University,
P.O. Box 3650 Yale Station,
06520, U.S.A.
Received 21 February 1991; in revised form 14 June 1991,
New Haven,
CT
‘By ‘Aristotelian science’ is meant here the conditions which a body of knowledge must satisfy in
order to meet Aristotle’s definition of scientific knowledge. For Aristotle, to know scientifically is,
among other things, to know the cause on which the fact depends and scientific demonstrations
arc
those which produce scientific knowledge. See for example, Posterior An&tics.
Book 1. Section 2.
J. Barnes in Aristotle’s Poslerior Annlyrics (Oxford, 1975, p. 96), uses ‘explanation’
to’render the
and its cognates render aifia and its
Greek aitia. He explains his choice as follows. ‘ “Explanation”
cognates; the traditional
translation
is “cause”. Ar&totle’s synonyms for aitia are ro dioti and to
diu fi (literally, “the wherefore” and “the because of what” - I translate “the reason why”); thus
to give the aitia of something is to say why it is the case, and X is aiton of Y just in case Y is
because of X (cf. [H.] Bonitz, Index Aristofelicus, Berlin, 1870, 177’50-2). Hence “cause”, as it is
used in colloquial
English, is a fairly good translation
of aifia (cf. the conjunction
“because”).
Philosophical
usage, however, seems generally to base itself on a Humean analysis of causation;
and an aitia is not a Humean cause. For this reason it is probably advisable to adopt a different
translation;
“explanation”
seems better than “reason”.’
Stud. Hist. Phil. Sci., Vol 23, No. 2, pp. 241-265,
Printed in Great Britain.
241
1992.
0039-3681/92
S5.00 +O.OO
@ 1992. Pergamon Press Ltd.
242
the Quaestio de certitudine
Studies in History and Philosophy of Science
mathematicarum.2
The fundamental
issues raised by
this debate were essentially the following two:
(a) What is the relationship
between Aristotelian
logic and Euclidean mathematics? In other words, can mathematics
be considered, as was often thought,
the paradigm exemplification
of the idea1 of ‘Aristotelian
science’, described in
the Posterior Analytics, or does it fall short of it? This led to a careful analysis,
at least by Renaissance
standards,
of the nature
of mathematical
demonstrations.
(b) If mathematics
does not derive its certainty by the form of its demonstrations, how are we to justify its certainty and evidence?
Scholarly work, especially by Giacobbe,
has shown that the Quaestio de
certitudine mathematicarum
crossed the Italian boundaries
to reach as far as
Portugal
and France.
In conclusion
to his article on Pereyra, Giacobbe
conjectured
that the Quaestio might have had a diffusion and importance
that
went beyond the part of the debate he had uncovered.
The problem of the
fortune of the Quaestio has also been raised by Nicholas Jardine in connection
with the problem of continuity
“between the ‘new’ sciences and epistemologies
of the seventeenth
century and earlier developments”.
In particular
Jardine
gives the following appraisal of the situation for mathematics:
The sources and fortunes of the sixteenth-century
Italian discussions of the status of
mathematical
demonstrations
and the grounds of certainty in mathematics
have
been little studied. These debates, are, however, reflected in the treatments
of the
status of mathematics
by Christophorus
Clavius, Giuseppe Biancani and Galileo’s
*For a first summary introduction
to the debate the reader is referred to N. Jardine, ‘The
Epistemology
of the Sciences’, in The Cambridge History of Renaissance Philosophy, eds C. B.
Schmitt, Q. R. D. Skinner, E. Kessler (Cambridge,
1988), pp. 685-711,
especially pp. 693-697.
For bibliographical
information
and a detailed analysis of the authors just mentioned
see the
following works by G. C. Giacobbe:
‘II commentarium
de certitudine mathematicarum
discipii-,
narum di Alessandro Piccolomini’, Physis 14 (1972), 162-193; ‘Francesco Barozzi e la Quaestio de
certitudine mathematicarum’
Physis 14, (1972), 357-374;
‘La riflessione metamatematica
di Pietro
Catena’, Physis 15 (1973). 178-196; ‘Alcune cinquecentine
riguardanti
il process0 di rivalutazione
epistemologica
della matematica
nell’ambito
della rivoluzione
scientifica rinascimentale’,
in La
Eerio, 13 (1973), 7-44; ‘La Quaestio
de certitudine
mathematicarum
all’interno
della scuola
padovana’,
in Atti del convegno di storia della logica (Padua,
1974), pp. 95-l 12; ‘Epigoni nel
seicento della “Questio de certitudine
mathematicarum”:
Giuseppe Biancani’, Physis 18 (1976),
540;
‘Un gesuita progress&a
nella “Quaestio
de certitudine
mathematicarum”
rinascimentale:
Benito Pereyra’, Physis 19 (1977), 5 l-86; Alle radici della rivoluzione scientifica rinascimentale: le
opere di Pietro Catena sui rapporti rra matematica e Iogica (Pisa, 1981). See also M. Pedrazzi, ‘Sul
tentativo di Alessandro
Piccolomini
di ridurre a sillogismo la I dimostrazione
degli Elementi di
Euclide’, in Culmra e Scuola 13, 1974, fast. 52, 221-230; on Moletti and Tomitano see the articles
in L. Olivieni (ed.), Aristotelismo Veneto e scienza moderna, 2 vols, (Padua, 1983), respectively by
Carugo, pp. 509-517, and Davi Daniele, pp. 607-621. The following are some of the primary
sources:
A. Piccolomini,
Commenrarium
de cerritudine
mathematicarum,
1547; F. Barozzi,
Opusculum, in quo una Oralio. & duae Questiones: altera de certitudine, & altera de medielare
Marhematicarum
conlinentur, 1560; P. Catena, Universa loca in logicam Aristorelis in marhematicas
disciplinas, 1556; Super loca malhemalica
contenta in Topicis et Elenchis Aristotelis, 1561; Oratio
pro idea merhodi,
1563; B. Pereyra,
De communibus
omnium rerum naluralium principiis et
affeclionibus
libri quindecim,
1576; Collegium
Conimbricense,
Commenlarii
In oclo libros
Physicorum Arisroteli, 1594; G. Biancani, De mathemalicarum
nafura dissertatio, 1615.
Aristotelian
Logic and Euclidean Mathematics
243
friend and mentor Jacopo Mazzoni, treatments which interestingly combine insistence on the certainty and excellency of mathematics and mathematical demonstration with emphasis on the substantial role of mathematics in the study of nature.’
The aim of this paper
is to show that the Quaestio
de certitudine
mathemati-
carum had a diffusion which reached, geographically,
as far as England and
Poland and chronologically
as far as the 1670s. I will analyse the contributions
to the Quaestio by Smiglecius, Wallis, Hobbes, Barrow and Gassendi.4 It is my
claim that the material I present gives an unequivocally
positive answer to the
problem of continuity
raised in Jardine’s article: great portions of the epistemological, and more generally philosophical,
reflection on mathematics
in the
seventeenth
century cannot
be understood
without
referring
back to the
Renaissance
debates on the Quaestio.
The structure of the paper is as follows. The first section will provide some
background
to the Renaissance
contributions
to the Quaestio. This will set the
stage for the contributions
of the authors under consideration.
Section 2 will
analyse the Quaestio 14, sectio 14 of the Logica by Smiglecius.’ Section 3 will
describe Wallis’ reactions to the theses held by Smiglecius and some of the
objections raised by Hobbes against Wallis’ answers to Smiglecius. Section 4
will analyse the contributions
of Isaac Barrow against the theses held many
years before by Pereyra; I will also argue that the real target of Barrow’s
attacks was not Pereyra but Gassendi.
1. The Quaestio de certitudine mathematicarum
The Quaestio de certitudine mathematicarum
originated with the publication
in 1547 of a treatise by Alessandro Piccolomini (15081578)
entitled Commentarium de certitudine mathematicarum
disciplinarum which can be rightly considered one of the most important
Renaissance
contributions
to the study of
the nature of mathematics.
Piccolomini’s
project can be characterized
as an
attempt
to refute a widespread
argument
which aimed at showing
the certitude
‘See Jardine, op. cit., note 2, p. 709. The previous quotation
is on p, 708.
‘Bibliographical
information
on Wallis, Hobbes, Barrow and Gassendi is easily available. For
Smiglecius see Universal Lexicon. ad vocem; and Sommervogel,
Biblioth?que de la Compagnie de
J&us, ad vocem. Smiglecius died in 1618 and he is important
in the history of Catholicism
for his
relentless campaign against the Socinians.
SThe Loaica’s first edition was published at Ingolstadt
in 1618. The other three editions were
published in Oxford in 1634, 1638,. 1658 respectivGy. I have used the 1658 edition whose title page
reads: LOGICA/MARTINI/SMIGLECII
SO-ICIETATIS
1ESU.S.
THEOLOGLE/Doctoris./
Selectis Disputationibus & qurestionibus illusfrata./Et in duos Tomos bistributa:/
In qua/&icquid
in
Aristotelico
organ0 vel cognitu necessarium,/vel
obscuritate
perplexum,
tam clare & perspicue,
quam so-/lide ac nervose pertractatur./Cum
Indice Rerum copioso/AD/Perillustrem
ac Magnificum Dominum/Dm.
THOMAM
ZAMOYSCIUM,
&c./OXONII,/Excudebat
A. LichJeld, Acad.
Typogr. Impensis H. CRIPPS,/J.
GODWIN
& R. BLAGRAVE,
An. Dom. 1658./Gum Privilegie.
244
Studies in History and Philosophy of Science
of mathematics
(asserted by Aristotle and reiterated by Averroes and a long
list of Aristotelian
commentators’)
arguing from the assumption
that mathematics makes use of the highest type of syllogistic demonstrations,
which in
Renaissance
terminology
were called demonstrationes potissimae (see below).
Piccolomini’s
contribution
was twofold.
First he argued that the demonstra-
tions of mathematics
are not potissimae and that therefore the argument
outlined above is bogus since the premiss is false. He achieved this by a careful
definition
of what counts as a demonstratio potissima and by showing why
mathematical
demonstrations
do not, and cannot possibly, fit into such a
category. Piccolomini’s
position was innovative:
by separating
the syntactic
features of Euclidean mathematics
from those of Aristotelian
logic, he can be
seen as an important
moment in the series of events that led to the emergence
“of mathematics
as the linguistic tool of the new science”.’ However, Piccolomini believed in the certainty of mathematics.
Thus in the last part of the
treatise he argued for the certainty
of mathematics
by emphasizing
the
conceptual
nature of the objects of mathematics
which, being created by the
human mind, have the highest level of clarity and certainty.
In order to proceed it is essential to introduce the terminological
distinctions
concerning
different types of demonstrations
introduced
by Aristotle in the
Posterior Analytics and further elaborated
by Averroes in his Proemium to his
commentary
on Aristotle’s Physics. Aristotle (Posterior Analytics, Book 1,
Section 13) had distinguished
two types of demonstrations:
roO &i and ro8
6~5~ (henceforth hoti and dioti), or in the Latin terminology
quia and propter
quid, i.e. of the ‘fact’ and of the ‘reasoned fact’. The first type of demonstration
proceeds from effects to causes whereas the second type proceeds from causes
to effects.8 In Averroes’ Proemium the distinction
becomes threefold; demonstrations are classified under the genera of quia, propter quid and potissima.
by Piccolomini
who characterized
the demonwhich gives both the cause and the effect
of an event (simul et quia et propter quid). He seems to identify it with a
syllogism of the first figure with universal premiss. More specifically, Piccolomini required the middle to have the form of a definition and to determine the
proximate cause of the effect in a unique way. Chapter 11 of his treatise was
This is the distinction
proposed
strati0 potissima as a demonstration
hPiccolomini quotes, among others, Albert, Thomas, Marsilius, Zimarra, Nifo, Acciaiolo.
Giacobbe,
‘La riflessione metamatematica’,
op. cit., note 2, p. 358.
XThese were sometimes identified, for example in Zabarella, with the cornpositive and resolutive
methods used by the mathematicians.
For an analysis of this problem see E. Berti, ‘Differenza tra il
metodo risolutivo degli aristotelici e la “resolutio”
dei matematici’,
in Aris~ofelismo Veneto, op.
cit., note 2, pp. 435-457. The Aristotelian classification
has been influential even in recent work in
philosophy
of science on explanation.
See for example B. A. Brody, ‘Towards an Aristotelian
Theory of Scientific Explanation’,
Philosophy of Science 39 (1972). 20-31; and B. van Fraassen,
‘The Pragmatics
of Explanation’,
American Philosophical Quarterly 14 (1977). 143-151.
Aristotelian Logic and Euclidean Mathematics
devoted
to showing
that demonstrations
245
in mathematics
do not conform
to
any of the above restrictions.9
There were several reactions
with him that demonstrations
to Piccolomini’s
in mathematics
work. Several scholars
did not conform
agreed
to the strictest
Aristotelian
standards for demonstrationes potissimae; this group included, for
example, Catena and Pereyra. However, their motivation
for denying that
mathematical
demonstrations
were potissimae were quite different. Whereas
Pereyra claims this to be the case in order to denigrate them, Piccolomini and
Catena do so in order to emphasize their autonomy
and certainty (indeed
Catena claims that mathematical
demonstrations
serve as the model for
methods in all disciplines).
By contrast,
people like Barozzi, Biancani and
Tomitano
argued that at least some demonstrations
in mathematics
did
conform to the requirements
for demonstratio potissima, although not all of
them (for example Barozzi and Biancani explicitly excluded proofs by contradiction from the realm of demonstrationes potissimae).”
We will see that the
main issues raised by the Quaestio were still alive in the second part of the
seventeenth
century. The positions held by Piccolomini,
Catena and Pereyra
called into question
cherished beliefs in mathematics
as the paradigm
of
science; or worse, excluded mathematics
from the realm of science altogether.”
Mathematicians
of the calibre of Wallis and Barrow felt this claim could not
go unchallenged.
‘More details can be found in Jardine, op. cif., note 2. Jardine summarizes the distinction made
by Aristotle, between dioti and hofi demonstrations
thus: ‘One of his examples of the former
(slightly expanded) is: heavenly bodies which are near the earth do not twinkle; the planets are
near the earth; hence the planets do not twinkle. This syllogism demonstrates
the presence of an
observed effect, not twinkling, in a subject, the planets; and it does so by means of a middle term,
being near the earth, which constitutes
the proximate cause of that effect. By rearrangement
of
terms a ‘demonstration
of the fact’ [hot11 is obtained in which the middle term specifies the effect
rather than the cause. Thus we have: heavenly bodies which do not twinkle are near the earth; the
planets do not twinkle; hence the planets are near the earth. Part of Aristotle’s intention in this
passage [Posterior
Analytics, Book I, Section 131 is, it seems, to distinguish demonstrative
syllogisms from related syllogisms whose premisses, whilst true, fail to explain the conclusion.’
Ibid., p. 686.
‘OOn the issue of proofs by contradiction
in the seventeenth century see P. Mancosu, ‘On the
Status of Proofs by Contradiction
in the Seventeenth Century’, Synrhese 88 (1991) 1541. This
paper also deals with other ramifications
of the Quaestio as for example in Rivaltus and Guldin.
“The most forceful statement
of such positions is provided
by Pereyra. “Mea opinio est.
Mathematicas
disciplinas non esse proprie scientias: in quam opinionem adducor turn alijs turn
hoc uno maxim6 argumento.
Scire est rem per caussam cognoscere propter quam res est; & scientia
est demonstrationis
effectus; demonstratio
autem (loquor de perfectissimo demonstrationis
genere)
constare debet ex his quae sunt per se & propria eius quod demonstratur;
quae verb sunt per
accidens, & communia,
excluduntur
a perfectis demonstrationibus,
sed Mathematicus,
neque
considerat essentiam quantitatis,
neque affectiones eius tractat prout manant ex tali essentia, neque
declarat eas per proprias caussas, propter quam insunt quantitati,
neque conficit demonstrationes
suas ex praedicatis
proprijs
& per se, sed ex communibus,
8t per accidens, ergo doctrina
Mathematics
non est proprie scientia: Maior huius syllogismi non eget probatione,
etenim apertt
elicitur ex his quae scripta sunt ab Arist. I. Post. Confirmatio
Minoris ducitur ex his, quae scribit
Plato in 7. lib. de Republ. dicens Mathematicos
somniare circa quantitatem,
& in tractandis suis
demonstrationibus
non scientific6 sed ex quibusdam
suppositionibus
procedere. Quamobrem
non
vult doctrinam
eorum appellare intelligentiam
aut scientiam, sed tantum cogitationem:
in quam
WIPSZ%Z-D
246
Studies in History and Philosophy of Science
2. Smiglecius
It should be evident from what has been said so far that disagreement
about
the Quaestio de certitudine mathematicarum centred around the issue whether
mathematical
demonstrations
were demonstrationes potissimae. Indeed Piccolomini explicitly acknowledged
that this problem
had provided
the main
reason for writing the Commentarium.
To this very issue the Polish logician
Martin Smiglecius devoted the Quaestio 14, sectio 14, of his Logica (1618):
“Whether mathematical
demonstrations
are most perfect and have the features
of potissimae demonstrations”.‘2
Smiglecius’
scholastic
style of exposition
presented in an orderly and schematic fashion the several positions which had
been held with respect to the nature of mathematical
demonstrations.
It should
be remarked
that Smiglecius did not associate the different positions with
specific names.
Smiglecius began by expounding
the ‘first proposition’
asserting that potissimae demonstrationes, if they exist at all, can only appear in mathematics.
The
argument hinged upon a sceptical position concerning our ability to know the
essences of natural things. He then proceeded to mention a similar argument
used to exclude the possibility that any other science than mathematics
could
possibly be about necessary
things (de re necessaria) and proceed from
necessary principles. I3
sententiam multa s&bit Pro&s in I. lib. suorum Commentariorum
in Euclidem. Verum, tametsi
neque Platonem
neque Proclum
neque alias Philosophos
graves, haberemus
auctores
huius
sententiae, tamen id per se manifestum
fit cuivis qui vel leviter modo attigerit eruditum illum
Mathematicorum
pulverem. Nam si quis secum reputet atque diligenter consideret demonstrationes geometricas,
quae continentur
libris Elementorum
Euclid. plant intelliget eas sic esse
affectas ut ante diximus: ac ut de multis unum aut alterum proferam
exemplum,
Geometer
demonstrat
triangulum
habere tres angulos aequales duobus
rectis, propterea
quod angulus
externus, qui efficitur ex latere illius trianguli
producto,
sit aequalis duobus angulis eiusdem
trianguli
sibi oppositis:
Quis non videt hoc medium non esse caussam illius passionis quae
demonstratur?
cum prius natura sit triangulum esse, & habere tres angulos aequales duobus rectis,
quPm vel produci latus illius, vel ab eo latere fieri angulum aequalem duobus rectis, quam vel
produci latus illius vel ab eo latere fieri angulum aequalem duobus internis? Praeterea, tale medium
habet se omnino per accidens ad illam passionem;
nam sive latus producatur,
& fiat angulus
externus, sive non, immo tametsi fingamus productionem
illius lateris; effectionemq, anguli externi
esse impossibilem,
nihilominus
tamen illa passio inesset triangulo;
at, quid aliud definitur esse
accidens quam quod potest adesse & abesse rei praeter eius corruptionem?
Ad haec, illas
propositiones,
Totum est maius sua parte, aequales esse lineas quae ducuntur
a centro ad
circumferentiam,
illud latus esse maius, quod opponitur
maiori angulo, & id genus alia, quam
crebro usurpat in demonstrando?
in quam multis demonstrationibus
eas pro medio adhibe.t &
inculcat Mathematicus?
ut necesse sit ex his demonstrationibus
quae constant praedicatis communibus, non gigne perfectam scientiam”. De communibus
, op. cit., note 2, pp. 24-25. The above
passage contains
many of the themes that characterized
the Quaestio: (a) the scientificity
of
mathematics;
(b) the causal nature of the syllogism; (c) the use of Proposition
I.32 from Euclid’s
Elements.
‘Smiglecius,
Logica, op. cit., note 5, pp. 580-583.
“Ibid., p. 580. It is to be remarked that such sceptical positions concerning our ability to know
the essence of natural things were already widespread in the Renaissance.
Aristotelian
Logic and Euclidean Mathematics
247
Fig. I
Having presented the argument
which excluded the possibility that other
sciences, in particular physics, can have demonstrationes potissimae, Smiglecius
proceeded to discuss the arguments in favour of the thesis that mathematical
demonstrations
do indeed possess the features of demonstrationes potissimae.
The first one relies on the authority
of Aristotle
who, according
to this
interpretation,
in the Posterior Analytics had defined the potissima demonstration as a syllogism of the first figure because the mathematical
sciences use that
figure in their proofs. And this argument would not hold without an implicit
assumption,
on Aristotle’s part, that mathematical
demonstrations
are potissimae. The second argument
appeals to the fact that demonstrationes potissimae should be causalI and about necessary objects; and both conditions
can
be found in mathematical
demonstrations.
However, whereas there does not
seem to be any problem about the necessity of mathematical
demonstrations,
the problem about causality is more delicate, since some claim that mathematical demonstrations
are not based on real causes but only causes relative to
our knowledge (“causas . . . cognoscendi”). I5 In any case, they are causal as is
illustrated by using a problem from Euclid which had been a locus classicus in
the previous discussions on the causality of mathematical
demonstrations.
It is
problem I.1 from Euclid’s Elements where it is shown how to construct an
equilateral triangle over a given segment. The construction
uses two auxiliary
circles which have their centres
equal to the segment (see Fig.
sides are equal to the radius of
The causal nature of the proof
at the endpoints of the given segment and radii
1). Now the triangle ABC is equilateral since its
the same circle and thus are equal to each other.
is argued as follows. Either this property of the
‘%r its primary meaning the appeal to causality
relies on the idea that the essence of a
geometrical figure (often identified with its definition), say a triangle, causally determines its other
(non essential) properties. However, the several meanings of causality in relation to mathematical
demonstrations
will emerge in the course of the paper.
‘>“De causis vero etsi quidam dubitent eas non habere veras causas essendi, sed cognoscendi,
tamen revera, tales habent causas, quibus positis sequitur talis proprietas”.
Op. cit., note 5, p. 581.
This position was held by the Archimedean
commentator
David Rivaltus in his Archimedis opera
quae extant novis demonstrationibus commentariisque illustrata per Davidem Rivalturn a Flwantia
(1615).
248
Studies in History and Philosophy of Science
triangle can be shown to depend causally on the nature of the triangle or not.
If so then one can prove the sought property from the essential nature of the
triangle even if no such demonstration
is yet available. If there is no such cause
then we have a contradiction
since no true effect in the world can be without
some true cause.16 One should note that this position tends to concede much to
those who contested that mathematical
demonstrations
were potissimae. The
argument aims at establishing
the possibility for some mathematical
demonstrations
to be potissimae even if de f&to there may be none with such
property.
Smiglecius then proceeded to present the ‘second proposition’:
mathematical
demonstrations
are not potissimae, because they do not argue from true and
necessary causes. This position can be argued from two different types of
assumptions.
The first one proceeds from a number of claims about the
ontological
status of mathematical
objects:
And indeed some argue this point from the fact that mathematical entities like
quantities and figures, as they are considered in mathematics, are not to be found in
nature [in rerum natura] .
Moreover,
it is required of a true and perfect
demonstration
to be about a real entity and not about an imaginary one. Otherwise,
it will not have truly and really any properties but only through imagination.”
However,
Smiglecius
finds this latter position untenable.
Indeed, asserts Smiglecius, all that is required is the possibility of the existence of the subject of a
mathematical
demonstration,
and this is guaranteed
by God’s potentiam.
It is not required of a demonstration
that its subject exist in actual reality; otherwise,
in winter there could be no science of the rose; and there could be no science of a
future eclipse. It suffices in fact that it could exist in reality. Indeed it is not in doubt
that those exact figures, as they are defined by mathematicians,
can be given by
God’s power.‘*
For example, nothing prevents God creating a line independently
of a plane,
i.e. a mere length without breadth.
The true argumentation
in favour of the ‘second proposition’
is that
mathematical
demonstrations
do not possess the real causes of being (non
continent in se veras causas essendi) and consequently
lack the necessity that
can only originate from true causes. In other words, mathematical
demonstrations are deficient in both features of a potissima demonstration,
i.e. causality
161bid.
““Et quidem nonnulli probant id ex eo, quod entia mathematics,
a Mathematics
considerantur,
non dentur in rerum natura
ut quantitates
& figurae, prom
Porro ad veram & perfectam
demonstrationem
requiritur; ut sit de ente reali, non de ente imaginario; alioquin non habebit vere
& reamer ullas proprietates,
sed tantum per imaginationem.”
Ibid.
‘““Neque vero ad demonstrationem
requiritur ut subjectum actu reali existat, (alioqui de rosa in
hyeme, & de ecclypsi futura non posset esse scientia) sed satis est, ut realiter existere possit. Non
est autem dubium exactas illas figuras, quales definiunt Mathematici
per Dei potentiam
dari
posse.” Ibid., p. 582.
249
Aristotelian Logic and Euclidean Mathematics
and necessity.
There are two types of arguments
One proceeds
a posteriori
for the proof of this assertion.
(inductio) and the other one a priori. Let us consider
the argument
a posteriori. If one analyses
Euclid’s problem I.1 mentioned
above, it is clear that the reason why the triangle is equilateral is not because of
the two circles used in the construction:
For in the first demonstration in Euclid, the triangle is shown to be equilateral from
the fact that it is constructed between two circles and all its sides are drawn from the
centre to the circumference.
Nobody can fail to see that this does
true cause of being [veram causam essendi]. In fact the triangle is
account of its being constructed between two circles. For it would
even if it were not constructed between two circles. From whence
not determine the
not equilateral on
still be equilateral
it follows that this
cause is accidental to that property.19
Another example which is considered in this connection,
and which is another
locus classicus of the Quaestio, is Euclid’s proposition
I.32 which shows that
the sum of the internal
angles of a triangle equals two right angles by
exploiting the external angle:
Similarly in proposition
32 of the first book of Euclid it is shown that a triangle has
the three angles equal to two right angles. For, producing one side, the external
angle is equal to the two internal angles. But this is not the true cause of being. For
the triangle would still have the three angles equal to two right ones even if the
external angle was not there [non esset].‘O
The second argument
proceeds a priori. One argues that in the potissima
demonstration
the cause of the property is the essence of the subject from
which the property originates;
however, in mathematics
one does not argue
from the essence of the subject but from the subject’s relationship
[habitudinem]
to other figures. Thus mathematical
demonstrations,
concludes the argument,
do not argue by veram causam essendi. The argument for the second premiss
appeals to the claim that figures and quantities
are physical properties [accidentia] and the science
that deals with them is physics. Thus whereas physics
proves by true causes, mathematics
argues from the relationship
that one
figure has to other figures. However, this way of arguing proceeds from
extrinsic causes since the figure about which we are proving something does
not depend for its being on the auxiliary figures used to show its properties.
The argument
against
necessity
is simply a variation
of the previous
one.
“+‘Nam in prima demonstratione
Euclidis, demonstratur
triangulum
esse aequilaterum,
ex eo,
quod sit constructum
intra duos circulos, habetq; omnia sua latera a centro ad circumferentiam:
Ubi nemo non videt, non assignari
veram causam essendi, non enim triangulum
idcirco est
aequilaterum,
quia est constructum
intra duos circulos: Nam etiamsi non esset intra duos circulos
constructum,
adhuc esset aequilaterum,
unde talis causa, est accidentalis illi proprietati.”
Ibid.
‘O‘YSimiliter in 32. propositione
primi libri Euclidis, demonstratur,
triangulum
habere tres
angulos aequales duobis rectis: quia product0 uno latere, angulus extrinsecus est aequalis duobus
angulis internis: at haec non est causa vera essendi: quia etiamsi non esset ullus angulus
extrinsecus, haberet nihilominus
triangulus tres aequales duobus rectis.” Ibid.
Studies in History and Fhilosophy of Science
250
At this point, after having surveyed the different positions,
Smiglecius is
ready to give his opinion
on the whole issue. Against the argument
by
authority, while conceding that Aristotle gave mathematical
examples for the
properties of the potissima demonstration,
he argues that Aristotle never gave
an example where all these properties held at once. Moreover, Aristotle never
asserted that mathematical
proofs should originate ex veras causas essendi.
Concerning the causality of demonstrations
Smiglecius denies that in mathematics demonstrations
proceed by using the true causes of the subject:
Concerning the second argument, it must be denied that true causes of being are in
mathematics.
For even if necessary properties have true causes of being, namely the
essence of the subject, yet such causes are not considered by the mathematician,
since he knows that they belong to Physics. He considers his business only to
demonstrate
[properties of] the figure through the figure, or through something
extrinsic to the figure.*’
Moreover, there is an important difference between mathematics
and the other
sciences. In the other sciences there might not be de facto potissimae demonstrationes but they are possible. In mathematics
such possibility is not given.
For in other sciences, even if de facto there are not potissimae demonstrations,
there
could be, as far as the nature of the objects and of the science allows. For the objects
have the true causes of being of their properties and to demonstrate
through such
causes does not go beyond the formal nature of the object. But in mathematics,
neither are the true causes of being of several properties given, nor is it the business
of the mathematician
to demonstrate
through them but that of the physicist. For
example, being equilateral is a mere accidental property of the triangle, which can
belong or not belong to it. Consequently,
it has no cause of being other than an
accidental one, that is, the exact construction
of the triangle.22
Smiglecius concluded the Quaestio by posing the problem whether Aristotle’s
work in laying down conditions
for the demonstratio potissima had been in
vain. Not at all; Aristotle’s important
contribution,
concludes our logician,
consists in having given a perfect ideal toward which demonstrations
should
strive.
>‘“Ad secundum, Negandum
est in Mathematicis
esse “eras causas essendi: Nam etsi proprietates necessariae habeant “eras suas causas essendi, nempe essentiam subjecti, tamen tales causas
non curat Mathematicus,
cum sciat eas ad Physicam pertinere: sui vero officii censet esse solum,
figuram per figuram demonstrare,
seu per aliquid figurae extrinsecum.”
Ibid., p. 583.
‘2“Nam in aliis scientiis, etsi de facto non sint demonstrationes
potissimae, possunt tamen esse,
quantum est ex natura objectorum & scientiae: Nam & objecta habent “eras causas essendi suarum
proprietatum,
& per tales causas demonstrare,
non excedit rationem
formalem
objecti; at in
Mathematicis,
neque dantur
verse causae essendi plurimarum
proprietatum,
neque per eas
demon&rare
ad Mathematicum
spectat, sed ad Physicum. Nam triangulum,
verbi gratia, esse
aequilaterum,
est proprietas me& accidentalis. quae potest adesse, & abesse, & ex consequenti non
habet aliam causam essendi, nisi accidentalem,
hoc est, ipsam constructionem
exactam trianguli
.
”
Ibid.
Aristotelian Logic and Euclidean Mathematics
251
Let us now recapitulate.
Smiglecius asserts that demonstrations
in mathematics in principle cannot satisfy the conditions
for the demonstrationes potissimae. This is not the case for physics
demonstrationes
potissimae.
This position
which at least in principle could have
is very close to that held forty years
before by Pereyra in his De communibus omnium rerum naturalium principiis et
afictionibus.
However, Smiglecius never drew, as Pereyra did, the paradoxical
conclusion
that mathematics
is not a science. Smiglecius’ theses found an
opponent in the mathematician
Wallis.
3. Wallis and Hobbes
of the Quaestio occurs in his Mathesis universalis: sive
(1657).“’ In his book he felt it necessary to deal
with the concept of demonstration
in order to show that Arithmetic
and
Geometry are true sciences. This is in fact the title of the third chapter of the
book: “Of mathematical
demonstrations.
Where it is shown that the mathematics are truly sciences.”
Wallis began by distinguishing
the dioti demonstration,
“which teaches the
proper affections of the subject by proper causes”, from the hoti demonstration, where one only needs an “argument
from the effect”. Wallis claimed that
nobody had put in doubt the certainty or evidence of mathematics.
However,
some, and here Wallis referred to Smiglecius’
exposition,
have doubted
whether mathematical
demonstrations
can reach the standard
of potissimae
demonstrationes
(which Wallis implicitly identified with the dioti demonstraWallis’
discussion
arithmeticurn
opus integrum
tions). Wallis then proceeded
to argue against the two main arguments
exposed by Smiglecius in defence of that thesis. Recall that the first one hinged
upon the imaginary nature of mathematical
entities, while the second one was
based on the exclusion of the possibility
that mathematical
demonstrations
proceed by true causes. Wallis contested both arguments. The first one claimed
that “mathematical entities as considered commonly by mathematicians, do not
exist wholly in nature”; moreover,
there cannot be a science of such objects
because imaginary
entities cannot have real properties. Wallis did not think
this position
represented
a serious threat. He claimed that it is the same
whether the object of a science exists or not; what matters is its potential
existence, like that of the rose in winter. By putting more emphasis on the
abstraction
rather than on the separation from physical objects, Wallis aimed
at bringing mathematical
entities closer to the realm of physical objects.
=‘Mathesis universalis: sive arithmeticurn opus integrum, in Wallis, Opera Marhematica
~01s. 1693-99), vol 1.
(Oxford,
3
Studies in History and Philosophy of Science
252
For it is one thing to abstract and another to deny. Indeed the mathematician
abstracts his magnitudes from the physical body; however, he does not deny it them.
The mathematician
no more asserts the existence of quantity without physical body
than the physicist maintains the existence of corporeal substance without quantity.
Each considers the one abstracted from the other. Therefore mathematical
entities
exist, or rather they can exist, not only in the imagination but in reality; truly not in
themselves but in the physical
body, although they are considered in abstraction.24
Having given his position on the first objection, Wallis tackled the difficult
problem of the causality of mathematical
demonstrations.
Against those who
argued that “mathematical
demonstrations
do not contain in them the true
causes of being, and therefore nor do they contain the eminent necessity which
originates from true and proper causes,” he replied by discussing proposition
I.1 from Euclid’s Elements. Recall that for those who denied mathematical
demonstrations
to be potissimae (for example, Pereyra) the construction
of the
equilateral triangle by use of two circles gave only an accidental cause for the
triangle being equilateral
but not the essential cause. Against this position
Wallis makes eight different points.
The first one is an attempt at redefining what must count as a scientific
demonstration:
First, I say that although it be true what is stated, namely that mathematical
demonstrations
do not proceed by a true and proximate cause, nevertheless they are
sufficiently scientific if they proceed either by a more distant cause or through an
effect, or by some other middle term taken from another part of the demonstration,
and with certainty conclude the thing to be as it is affirmed to be, and not to be able
to be otherwise. For the demonstration
to be scientific it suffices that it proceed from
the nature of the thing through a necessary medium
neither he [Smiglecius], nor
any other that I know, will deny pure mathematics to be a science, although they do
not concede to it the greatest perfection of demonstration.25
Thus Wallis proposed an alternative
definition of a scientific demonstration;
this would include demonstrations
which do not proceed by true and proximate causes, as for example proofs that make use of remote causes or argue
from effects to causes
(per efictum),
or by use of some other
middle
term
““Aliud enim est abstrahere,
aliud negare: Mathematicus
suas quidem magnitudines
abstrahit a
corpore Physico, non tamen de illo negat: Net magis asserit Mathematicus
quantitatem
exsisteze
sine corpore Physico; quam Physicus, substantiam
corpoream
exsistere sine quantitate:
uterque
tamen alteram ab altera abstractam
contemplatur.
Exsistunt igitur, saltem exsistere possunt, non
tantum imaginarie sed realiter, entia Mathematics;
non quidem per se, sed in corpore Physico; licet
abstracte considerentur.”
Ibid., p. 21.
25”1. Dice, quod, utut illud verum esset quod affirmatur, nempe demonstrationes
Mathematicas
non per veram & proximam causam procedere; sunt tamen satis scientificae, si vel per causam
remotiorem
vel per effectum, vel per aliud aliquod medium ab alio argumentandi
loco deductunt
procedant,
modo certo concludant
rem ita esse prout affirmatur,
atque aliter esse non posse.
Sufficit enim ad hoc, ut demonstratio
sit scientifica, si proceduf per medium necessarium ex nufura
rei: [ut & ipsi quidem, qui hoc objiciunt, non negant, & Smiglecius etiam directe affirmat, Disp. 15.
qu. I & &hi. Quare] net ipse, net quos scio, alii, negabunt, Disciplinas pure Mathematicas
scientias
esse, utut perfectissimum
illis demonstrandi
modum non concedant.”
Ibid., p. 22.
Aristotelian Logic and Euclidean Mathematics
253
already deduced somewhere else. One more remark concerning
the above
quotation.
We know that Piccolomini and Pereyra explicitly denied mathematics to be a science.
Thus
we must
conclude
that
Wallis
had
no direct
knowledge of the primary sources of the debate and that Smiglecius constitutes
his main source on the Quaestio de certitudine mathematicarum.
In his second remark Wallis simply mentions that even those who object
against mathematical
demonstrations
their lack of features characterizing
the
potissimae demonstrations,
have doubts whether such perfect demonstrations
can be found anywhere.
In the third point Wallis gives a reinterpretation
of the example concerning
the equilateral triangle (Euclid I. 1). In his opinion Euclid’s demonstration
does
not assert that the triangle is equilateral because it is constructed
between two
circles, but that the sides of the triangle are equal to each other. This, he
concludes, is to prove from true causes.
In his fourth point Wallis distinguishes
between the demonstration
of the
Euclidean theorem and the demonstration
of the construction
of the problem.
He asserts that the latter is done by use of the true causes whereas the former
follows from the latter by use of remote or accidental causes. Indeed, he argues
in his fifth point, one should not expect in any science that all the proofs enjoy
the same degree of perfection:
5. It is however to be added that in no science is it to be expected that all the
demonstrations
belonging to it proceed by an equal degree of perfection. It is more
than enough, if some demonstrations
proceed by true and intimate causes, and
therefore
are roi, 616~1, although
those demonstrations
roi, 616~1 are mingled
everywhere with other roi, &I.~”
The
last
remark
concedes
much
to those
who
claimed
the inferiority
of
mathematical
demonstrations
on the ground that they are not causal. This is
admitted explicitly by Wallis in his sixth point. However, as he points out in
the seventh remark, there are some propositions
which are causal (and hence
dioti) in mathematics.
In his eighth and last remark he distinguished
three types of demonstrations
(sive modus, sive gradus, sive species): proofs by contradictions,
ostensive hoti
and dioti. It is interesting at this point to quote a passage from a different work
of Wallis, the Znstitutio logicae (I 687), where, while making the same distinction, he asserted that mathematical
demonstrations
“do not all possess the
same degree of excellency.
Although
they have the same certainty
they
26“5.Etiam addendum est. in nulla quidem scientia expectandum
esse, ut omnes ibidem
demonstrationes
aequali perfectionis gradu procedant.
Abunde sufficit, si demonstrationes
aliquae
sint per “eras et proximas causas, ideoque roi, 616r1, quanquam
demonstrationibus
illis ro6 616~1,
passim immisceantur
aliae soi, 6~1. Ibid., p. 23.
254
Studies
in
History and Philosophy of Science
however do not possess the same evidence.“”
Thus different types of proofs
have different degrees of evidence although all of them have the same degree of
certainty. The lowest type of demonstration
is the proof by contradiction.
As
an example in the Mathesis universalis Wallis gives a proof of the proposition
“Qf all the straight lines inscribed in a circle, the longest is that which passes
through the centre”. In his Institutio logicae he mentions
the Archimedean
proof of the quadrature
of the circle. The second type of demonstration
is
ostensive hoti:
The second type of demonstration is ostensive ZOB &I. As when [see Fig. l] the
segment AC is demonstrated to be equal to segment BC seeing that both can be
shown equal to AB. And those which are equal to the same are equal among
themselves. And truly this demonstration is direct [ostensiva] but only zoi, &land
not also ZOO 81i)~l. For the common equality of both to the same third certainly
indicates the equality between them, but it is not the cause of that equality. Indeed,
AC and BC would be equal to each other even if AB were not drawn.‘K
Finally
Wallis gives us an example
of a dioti demonstration:
Indeed, the third type of demonstration, which is the most perfect of all, is ostensive
zoi, 616~ which demonstrates what it is and why it is. It is this kind of demonstration
if someone demonstrates
all the radii of the same circle to be equal from the
definition of a circle (and this is a possible definition) as a plane figure delimited by a
single curve which is everywhere on it equidistant from the middle point of the space
enclosed. For if the fact that its periphery is equidistant from the center gives the
essence of the circle then it follows immediately, just as from a true and a proximate
cause, that all radii, each of which measures that distance, are also equal. And this
demonstration
is ostensive TOO &&I, inferring from the proximate and immediate
cause.2s
Thus Wallis’ position with respect to the Quaestio is easily summarized
in the
statement that some mathematical
proofs are causal. However, it is clear that
the importance
of Wallis’ text lies not so much in the specific details of his
argument
concerning
the causality
of mathematical
demonstrations
but in the
*‘Inslitutio Logicue, in Opera, op. cit., note 23, vol. 3. “Cumque Demonstrafiones omnes (quae
hanc merentur appellationem)
tales sint; non tamen eodem omnes dignitatis gradu haberi solent.
Utut enim eadem cerfitudo, non tamen eadem est omnium evidentiu.” Ihid., p.180.
*YS.ecunda demonstrandi
ratio, est, Ostensiva ~06 &I. Ut, si recta AC demonstretur aequalis
esse recfae BC; quoniam utraque fuerat aequalis ipis AB: quae autem eidem sunt aequalia, sunt &
aequalia inter se. Estque haec demonstratio
quidem ostensiva, sed tantum rob &I, non utem sod
616r1. Communis enim utriusque aequatio eidem tertiae, indicat quidem earum aequalitatem
inter
se, non autem illius causa est. Essent enim ipsae AC, BC, sibi invicem aequales, etiamsi ipsa AB
non fuisset ducta.” Marhesis, op cit., note 23, p. 23.
29“Tertia vero demonstrandi
ratio, quae & omnium perfectissima,
est Ostensiva soi, 616m Quae
demonstrat & quod sit, & quare sit. Ejusmodi est demonstratio,
siquis Radios onme.~ejusdem circuli
aequales esse inde demonstret,
quod definiatur circulus (saltem definiri possit) Figura plum, mica
linea curva contenta. quae a medio comprehensi spatii aequaliter uhique d&at. Si enim circuli
essentia postulet, ut ipsius peripheria aequaliter B centro distet; immediate sequitur, tanquam B
causa vera & proxima, radios omnes, quibus illa distantia mensuratur,
etiam aequales esse. Estque
demonstratio
haec Ostensiva rob ~516~1,
A causa proxima & immediata desumpta.”
Ibid., pp. 23-24.
Aristotelian
Logic and Euclidean Mathematics
overall intention
guiding
his discourse.
When we compare
255
Wallis’ text with the
Renaissance contributions
on the Quaestio we are struck by a complete change
of perspective. In Piccolomini’s
treatise, for example, Aristotelian
logic was the
language of science and its scientific syllogism the ideal form at which every
scientific demonstration
had to aim; Aristotle had proposed,
as Smiglecius
said, a “. . . perfect ideal to which all demonstrations
must aim, so that they
will be considered
the more perfect the closer they approach the ideal itself
. .” In Wallis’ text, as well as ,in the ones by Hobbes and Barrow to be
analysed
later, it is mathematics
that is the paradigm
of science and its
reasonings the paradigm of scientific demonstrations.
This is in fact the essence
of Wallis’ first remark.
From the technical point of view I find the most interesting point of Wallis’
contribution
to consist in his recognition
that proofs by contradiction
should
be classified in a different category from the hoti demonstrations.
While being
prepared to discuss the issue within the framework of Aristotelian
terminology Wallis brings to bear his experience as a mathematician
in distinguishing
as separate types of proofs the proofs by contradiction
and the direct ones.
This distinction
is much closer to the phenomenology
of mathematical
discourse,
and to the experience
of the mathematician,
than any of the
distinctions
based on the causal criterion. This is a sign of the fact that the
actual practice of the mathematician,
and not the abstract attempts of the
Aristotelian
philosophers,
were beginning to shape the essential distinctions
in
philosophy
of mathematics. 3o However,
Wallis’ position
left open several
problems as his obstinate adversary Hobbes was ready to point out.
Hobbes’ objections
to Wallis’ position on this issue are contained
in his
Examinatio
et emendatio mathematicae
odiernae, qua&s explicatur in libris
Johannis Wallisii distributa in sex dialogos (1660).” The Examinatio is made up
of six dialogues between two speakers A and B who comment on passages
from Wallis’ Mathesis universalis. The section begins by discussing Wallis’
definition
of demonstration
(“a demonstration
is a syllogism which demonstrates the properties of the subject by proper causes”) which is found wanting
on two counts. First, the definition is circular since the verb demonstrat is used
in the definition
of demonstration.
Second, one should speak of a series of
syllogisms as opposed to a syllogism in order to have an accurate definition.
Hobbes’ revised definition reads: “A demonstration
is a syllogism or a series of
syllogisms starting from the definitions of the names and ending with a derived
conclusion.”
Soon after, Hobbes makes a point also of including axioms as
starting points of demonstrations,
“as are the axioms assumed by Euclid”.
‘OFor further reflections on this issue see op. cif., note 10.
j’The section concerned with Wallis’ theory of demonstrations
goes from p. 35 to p. 43 of vol. 4
of the Opera Philosophica quae latine scripsit omnia, ed W. Molesworth
(London,
1845).
Studies in History and Philosophy of Science
256
Speaker
A continues
by reading
the passage
where Wallis
distinguished
proofs from dioti proofs. Hobbes’ point is that only the dioti reasonings
really demonstrations
whereas the hoti arguments are not:
hoti
are
For, the demonstration is rot5 616~r when one shows on account of what cause the
subject has a certain property. And therefore seeing that every demonstration is
scientific, and that to know that such a property is in the subject comes of the
knowledge of the cause which necessarily produces that property, it follows that
nothing can be a demonstration if not zoi, 616r1.~~
This is emphasized a few lines later by remarking that only reasonings from the
causes to the effects are demonstrations
(he also thinks of them, as Wallis did,
as equivalent
to the potissimae) whereas reasoning from effects to causes are
not:
Clearly this is what Aristotle intended, and Wallis does not say otherwise, when he
calls it an argument from effect. It must be said that to the twofold investigation of
the philosophers, namely of the effects from causes and of the causes from effects,
correspond two types of ratiocination. Namely, demonstration a priori, that is
ratiocination from definitions, which is scientific; and a posteriori, ratiocination from
possible hypotheses: which, even if it is not scientific, if over a long period no effect
appears to confute the hypothesis, the mind finally accepts it not less than in science.
We seek
in vain a definition of demonstration
rot? 6~1, which is not a
demonstration.”
Speaker A continues
by mentioning
the exchange between Smiglecius and
Wallis. On the issue of the ontological status of mathematical
entities Hobbes
expressed doubts as to whether mathematical
objects could be defined and
properly distinguished
from other abstract objects. He then went on to criticize
Wallis’ thesis that there are different degrees of perfection in the demonstrations belonging to a science. Hobbes’ argument
reduces to the fact that all
demonstrations
are scientific and that knowledge does not admit of degrees: we
either know or we do not. To Wallis’ point that hoti demonstrations
are
intermingled
remarked:
with
dioti
demonstrations
in
a
science
Hobbes
scathingly
‘*“Nam demonstratio
TOOb16rt est, quando quis ostendit propter quam causam subjectum talem
habet affectionem.
ltaque quoniam
demonstratio
omnis est scientifica, et scire talem esse in
subject0 affectionem
est a cognitione
causae quae illam necessario producit.
nulla potest esse
demonstratio
praeterquam
roi, 61ort.” Ibid., p. 38.
““Videtur id voluisse Aristoteles, neque dissentiente Wallisio, qui earn appellat argumentum
ab
effectu. Dicendum ergo est, duplici philosophorum
inquisitioni,
nimirum effectuum ex causis et
causarum ex effectibus, duplex respondere ratiocinationis
genus, nempe priori, demonstrationem,
id est, ratiocinationem
ex definitionibus,
quae est scientifica;
posteriori,
ratiocinationem
ex
hypothesibus
possibilibus;
quae etsi scientifica non sit, si tamen nullus appareat effectus, ne in
longissimo quidem tempore, quae hypothesin redarguat, facit ut animus in earn tandem acquiescat,
non minus quam in scientia. Frustra autem demonstrationis
706 &I quaerimus
definitionem, quae
demonstratio
non est”. Ibid., p. 39.
Aristotelian Logic and Euclidean
Mathematics
257
It is then enough if not all theorems in Euclid’s Elements are demonstrated
by
demonstrations
708 616t1, i.e. it suffices if we know some of its theorems to be true,
while we do not know of others whether they are true or not. Is this man qualified as
professor of geometry, when, as we have shown, he neither knows what are the
principles of that science nor, as it appears here, what it is to demonstrate, i.e. to
teach? I
wonder who allowed him to get the Savilian Chair.”
Given Hobbes’ contempt
of Wallis’ adherence
to the classical distinction
between hoti and dioti demonstrations
one should expect that Wallis’ tripartition of proofs would also be challenged by Hobbes. Indeed, Hobbes tried to
argue that all the kinds of proofs mentioned by Wallis are in truth dioti proofs.
For example, with respect to the proof used’ by Wallis as an example of
ostensive hoti, Hobbes claims that the use of the two auxiliary circles corresponds to an appeal to efficient causes. Hobbes then returns to the meaningfulness of a definition for hoti demonstrations:
is stated what is true or what is false, how
is one demonstration
quod and the other propter quid? For we do not know that a
thing is so unless we know by which cause it is so; according to what we Aristotelians
usually say, to know is to know by causes.3s
Tell me then, since in every demonstration
Thus mathematics
can claim its status as science whereas physics, which uses
hoti argumentations,
has no certainty whatsoever. In vain therefore, concludes
Hobbes, does Wallis look for hoti demonstrations
in Euclid’s Elements.
In conclusion
one must ,admit, according
to Hobbes, only one type of
demonstration:
the dioti, to which direct proofs and proofs by contradiction
alike belong. Thus Hobbes holds the following two main theses: all demonstrations in mathematics
are causal, since to know is to know by causes; and
moreover
they are all dioti. How strongly was Hobbes committed
to this
position? It seems to me that as in the case of Wallis we should reach for the
overall intention underlying
Hobbes’ text rather than for the specific solutions
which may depend on the context of the specific polemic. For Hobbes
geometry is “the only Science it hath pleased God hitherto to bestow on
mankind”.36 Euclid’s Elements represented
for him a storehouse of scientific
demonstrations
all of equal value and dignity. In order to make his point he
did not hold back in the above passages from claiming things that he was to
deny in other works, as, for example, in the case of proofs by contradiction:
““Abunde
ergo sufficit, si in Elementis Euclidis non omnia theoremata
demonstrentur
demonstrationibus
to8 616r1; id est. sufficit si alia ejus theoremata
sciamus esse Vera, alia an vera sint
nccne, nesciamus. An geometriae professor idoneus est, qui neque, ut ante ostendimus,
scit quae
sint illius scientiae principia,
neque, ut apparet hoc loco, quid sit demonsrrare, id est, quid sit
docere? Miror qui factum sit, ut cathedram
nactus sit Savilianam.”
Ibid., p. 40.
““Die ergo, cum in omni demonstratione
dicant quod verum est, vel quod falsum, quomodo una
demonstratio
est quod, alia propter quid? Nescimus enim quod res ita est, nisi sciamus propter quid
ita est: juxta id quod solemus dicere Aristotelici, scire est per causarn scire.” Ibid., p. 42.
‘bLeviathan, in Englbh Works of Thomas Hobbes, ed. W. Molesworth
(London:
1845). vol. 3,
pp. 23-24.
258
Studies in History and Philosophy of Science
Therefore in demonstrations
that tend to absurdity
along the operation of the cause.”
it is not good logic to require all
We witness in Hobbes a strong impatience
toward the attempt to reduce
geometrical
demonstrations
to the foreign categories of Aristotelian
logic.
With Hobbes it is geometry, and all the forms of reasonings used in it, which
represents
the highest achievement
of the human mind. However, we can
honestly say that he did not argue his position. His arguments for showing that
all mathematical
demonstrations
are dioti are completely insensitive to all the
previous distinctions
the Aristotelian
tradition had elaborated.
Moreover, his
appeal to the Aristotelian
notions only constitutes a formal move on his part.
For Hobbes it is physics, once thought the science which in theory could
exhibit demonstrations
embodying
the conditions
for demonstrationes potissimae, which is shown to depend on hoti reasonings,
and thus unscientific
reasonings.
When Hobbes was writing the Galilean revolution had been achieved. The
new language of science was mathematics
and not Aristotelian
logic any more.
A new ideal of science had taken over the philosophical
world. Hobbes’ text
simply reflects this deep change in the organization
of knowledge.
4. Barrow and Gassendi
Barrow’s contribution
to the Quaestio occurred in his Lectiones,” especially
the fifth and the sixth which were entitled respectively “Containing
answers to
the objections
which are usually brought against mathematical
demonstrations”, and “Of the causality of mathematical
demonstrations”.
As in Wallis’
case, Barrow was motivated
by the aim to show that mathematics
is a real
science against those who “both have been, and still are so subtle as to deny
that the Mathematics
are truly Sciences, and that they afford true Demonstrations”.39 However, whereas Wallis seems to have been unaware of the literature
on the Quaestio except for Smiglecius’ scholastic exposition,
Barrow quotes
directly from two of the primary sources: Biancani and Pereyra. Barrow’s
discussion is very extensive. Consequently,
I will emphasize only the points
most closely related to our issue.
The evidence and truth of mathematical
axioms, said Barrow at the beginning of his fifth lecture, were already questioned
in Greek times. One of the
main sceptical objections which were usually raised in this connection
is that
“English works, ibid., vol I, p. 62.
‘8Mathemnticis Professoris Lectiones, in The Mathematical
Works (Cambridge, 1860). The
quotations
are from the English translation
by J. Kirby, The Usefulness of Mathematical Learning
(London: 1734); reprint by Cass Publishing Company,
London, 1970.
‘VThe Usefulness
., Ibid., p. 66.
Aristotelian
universal
Logic and Euclidean Mathematics
axioms
are obtained
opens up the problem
science.
It suffices
259
by induction
of certainty
and
therefore
are fallible.
and the role that sense and intellect
This
play in
here to say that Barrow rejected the theory that all principles
of mathematics
depend only upon induction
from the senses. However, he is
ready to add, sensation plays a role in showing the possibility of a mathematical hypothesis.
It is in connection
with the problem
of existence of
mathematical
entities that Barrow argues a few pages later against Biancani
and Vossius who held that mathematical
figures have no real existence outside
the mind. By contrast, similarly to Wallis, Barrow wants to put emphasis on
the potential actualizability
of mathematical
entities. In Barrow’s organization
of the lecture this was simply a digression from the main aim, i.e. to show the
certainty of mathematical
axioms. More relevant for our topic is Barrow’s
discussion of the claims of “those who study to detract not from the certitude
and evidence, but from the dignity and excellence of mathematics”.
Amongst
these detractors Pereyra is mentioned:
For they attempt to prove that Mathematical
Ratiocinations
are not Scient$c,
Causal and Per-feet, because the Science of a Thing signifies to know it by its Cause;
according to that Saying of Aristotle; We are supposed to know by Science, when we
know the Cause. And to use the Words of Pererius, who was no mean Peripatetic, A
Mathematician neither considers the Essence qf Quantity, nor treats of its Affections,
as they j?oM*,frotn such Essence, nor declares them by the proper Causes by which they
are in Quantity,
common
nor &rms
and accidental
their Demonstrations
from
proper
and essential,
but from
Predicates.40
Barrow must have thought
Pereyra’s objections
represented
a challenge.
Indeed he spent part of his fifth lecture and the whole sixth lecture arguing
against them. He has no doubts that mathematical
Aristotelian
strictures for scientific ratiocinations:
To which I answer, that those scientific
Demonstration,
who was most observant
Mathematical
Ratiocinations.4’
ratiocinations
satisfy the
Conditions,
which Aristotle prefixes to
of its Laws, do most fitly agree with
He goes on to assert that mathematical
ratiocinations
in fact use premisses
which are universal, necessary, primary and immediate.
Moreover they are
“More Known and More Evident than the Conclusions
inferred”. Finally, they
are also causal. Barrow’s comments on this point are important
because he
“Ibid., p. 80. See also the quotation in note I I.
4’lbid.
Studies in History and Philosophy of Science
260
states
that
at least
some
mathematical
demonstrations
are diotL4* This
argued by appealing, first of all, to Aristotle’s examples who, Barrow
took his only examples of causal demonstrations
from mathematics.
sixth lecture he proposed to show that
is
claims,
In his
Mathematical Demonstrations are eminently Causal, from whence, because they only
fetch their Conclusions from Axioms which exhibit the principal and most universal
Affections of all Quantities, and from Definitions which declare the constitutive
Generations and essential Passions of particular Magnitudes. From whence the
Propositions that arise from such Principles supposed,, must needs flow from the
intimate
Essences and Causes of the Things.4’
The main elements to be analysed in a demonstrative
science are the notions
of subject, affection of the subject, and common axioms. In demonstrations
we
usually want to show how certain affections belong to a subject; common
axioms are instrumental
in allowing us to do so. Barrow does not have much
to say about subjects of demonstrations;
by contrast, he spends a great deal of
time explaining
the nature of affections and that of axioms. There are two
types of affections: common and proper. Common affections are those “which
agree with their Subject necessarily, but not solely, as being also capable of
being truly attributed
to other Subjects”.
For example, it is a common
property of an isosceles triangle to have the internal angles equal to two right
ones but this property is also enjoyed by the scalene triangle. By contrast,
“proper AfSections, are such as agree with their Subject both necessarily and
solely, i.e. they do so reciprocate with their Subject, that if they be supposed, it
is also supposed of Necessity.”
Barrow gives the example of the circle. It
@“Last/y, he [Aristotle] demands them to be the Causes of their Conclusions;
which last
Condition
may be accepted two Ways: Either first only as they contain the Reason which
necessarily causes the Conclusions
to be believed as true, and produces a certain Assent, i.e. as the
mean Term assumed obtains a necessary Connection
with the Terms entring the Conclusion;
whence arises that which is called a Demonstrationroi, &I thatCIThing is: or secondly more strictly,
as this mean Term applied is more than a necessary Effect and a certain Sign, i.e. as it is a proper
Cause of the Attribute or Property, which is predicated of the Subject in the Conclusion; and hence
is that called a Causal Demonstration,. or a Demonstration SOB616~1why a Thing is. But there is no
Reason to doubt, but the last Condition understood
in the former Sense agrees with the Premisses
of every Mathematical
Syllogism, since there are no such Syllogisms, which do not most strongly
compel the Assent; nor does this follow because the Premisses are necessarily true (for otherwise
they are not admitted by Mathematicians),
but this Necessity argues that there is an essential
Connection
and Causal Dependance
of the Terms between themselves in which it is founded,
because the Accidents may be separated,
and consequently
the accidental
Predicates are only
attributed
to the Subject contingently.
[I. An. Post. c. 61 Things Essential (says Aristotle) are
necessarily in every Genus. but Things Accidental are not necessary: And every such kind of
Argumentation
begetting
a lesser Degree of Science is reckoned
a more low and ignoble
Demonstration, because it shews a Thing to be so only from its Effect or Sign, not from its Cause;
but yet this most clearly convinces the Mind, and most validly confirms the Truth. There is
therefore no Mathematical
Discursus which proceeds not thus far. But that the foresaid Condition
taken in the latter Sense does also agree with many Mathematical
Ratiocinations,
i.e. that the
mean Term assumed in them has the Force of a Cause in Respect of the Property attributed to the
Subject in the Conclusion,
Aristotle is our first Author
.” Ibid., pp. 81-82.
“Ibid., p. 83.
Aristotelian
Logic and Euclidean Mathematics
261
belongs only to the circle among geometrical figures to have equal radii and
every figure which has “equal Radii from the Center to the Perimeter or
Circumference is a Circle”. However, proper affections may not be unique. For
example it is a proper affection of the circle “that every two Right Lines that
can be drawn from the Extremities
of the Diameter to any Point in its
Circumference
will make a
disagreeing
with Aristotle
Barrow’s opinion it is only
affection as a starting point
feature, adds Barrow, that
consists:
Right Angle”. On this issue Barrow admits to
who thought
definitions
should be unique.
In
a matter of convenience
how to choose a proper
for a demonstrative
chain.@ It is exactly in this
the causality
of mathematical
demonstrations
Such in Reality and no other is the mutual Causality and Dependence of the Terms
of a Mathematical Demonstration, viz. a most close and intimate Connection of them
one with another; which yet may be called a formal Causality, because the remaining
Affections do result from that one Property, which is first assumed, as from a Form.
Nor do I think there is any other Causality in the Nature of Things, wherein a
necessary Consequence
can be founded.4s
Thus we must add Barrow’s name to the list of those who claimed in the
debate surrounding
the Quaestio that mathematical
demonstrations
are causal,
since they make use of syllogisms proceeding from forma1 causes. However, we
must notice how much less stringent
are Barrow’s requirements
for such
demonstrations.
For example, the issue of the unicity of the middle had played
a major role in the Quaestio; nonetheless,
Barrow does not worry very much
about this.
Are there any other types of mathematical
causal demonstrations
beside the
ones which rely on formal causes? Barrow argues at length that geometry does
not admit demonstrations
which argue from efficient or final causes.46 The
argument
depends on a form of theological voluntarism.
God can alter the
normal causal course of nature:
For every Action of an eficient Cause, as well as its consequent Effect, depends upon
the Free- WilI and Power of Almighty God, who can hinder the Influx and Efficacy of
any Cause at his Pleasure; neither is there any Effect so confined to one Cause, but it
may be produced by perhaps innumerable others. Hence it is possible that there may
be such a Cause without a subsequent Effect, or such an Effect and no peculiar Cause
to afford any Thing to its Existence. There can therefore be no Argumentation
from
an efficient Cause to the Effect, or contrarily from an Efict to the Cause, which is
lawfully necessary.47
@Ibid., pp. 84-85. “It is all one, as to the Nature of the Thing, from which the Discursus takes its
Link of the Chain you take hold of, the Whole will follow.” Ibid., p. 88.
451bid.,p. 88.
Rise, for whichsoever
“He does not consider
“Ibid., pp. 88-89.
SHIPS
232-E
material
causes which had however
played a certain
role in the Quaestio.
Studies in History and Philosophy of Science
262
However,
God cannot
modify
necessary
truths:
For necessary Propositions have an universal, immutable and eternal Truth, subject
to nothing, nor to be hindered by any Power.48
The two claims combined
seem to exclude the hoti reasonings
from mathema-
tics; for, they proceed from effects to causes. Thus Barrow agrees with Hobbes
also on the claim that all mathematical
demonstrations
are dioti. The same
arguments were also applied to final causes. Barrow then proceeded to discuss
the nature of axioms. His main conclusion is that their use in a demonstration
preserves the causal nature of the demonstration.
This concludes Barrow’s
general discussion of the nature of causality in demonstrations.
The last part of the lecture provided
an articulate
analysis of Euclid’s
proposition
I.1 and I.32 which had played the role of paradigm
examples
throughout
the debate. After discussing I.1 at length Barrow proceeded to
discuss 1.32:
But as I remember Pererius, and others, do produce another Instance, also blaming
that celebrated Proposition
which is the thirty-second
of the first Element, as not
scientifically demonstrated.49
He then went on to summarize the main criticisms by Pereyra and made four
replies. In the first reply he invoked the authority of Aristotle who, claimed
Barrow, quoted this proposition
as an example of causal demonstration;
in his
second remark he argued that since a triangle is constituted
by straight lines
then what is essential to lines also pertains to the triangle. “But it is the
Property of a Right Line that it may be produced; therefore this Production
is
not altogether accidental or extrinsical to a Triangle.” The third point argued
that division of the external angle is the most natural means to obtain the
sought result. His fourth and last point was that one can give a step by step
analysis showing that Euclid’s proof conforms to his schema for mathematical
demonstrations
which, as he has already argued, embodies the form of causal,
and hence scientific, demonstrations.
At this point Barrow could finally
conclude by boasting the superiority of mathematical
demonstrations:
it seems to me . . . that Demonstrations,
though some do outdo others in Brevity,
Elegance, Proximity to their first Principles, and the like Excellencies, yet are all
alike in Evidence, Certitude, Necessity, and the essential Connection
and mutual
Dependence of the Terms one with another. Lastly, that Mathematical
Ratiocinations are the most perfect Demonstrations.5o
*Ibid., p. 90.
‘vlbid., p. 91.
wlbid., p. 98, p. 99. This is inconsistent with a number of statements made by Barrow
XXI and XXIII on the lower dignity of proofs by reductio ad absurdurn.
in lectures
Aristotelian Logic and Euclidean Mathematics
As in the case of Wallis and Hobbes
still being
willing
to frame
263
one should
his arguments
remark
in the context
that Barrow,
while
of the Aristotelian
logical terminology,
begins with the basic presupposition
that mathematics
is
the science par excellence. Nothing can be more remote from his perspective
than the subtle scholastic distinctions
that had characterized
the Renaissance
contribution
to the Quaestio and of which Smiglecius’ Logica is only a dim
reflection. Barrow too is writing when the battle for Aristotelian
logic as the
universal language of science had already been lost.
However, this seems to leave an open problem. If the reorganization
of
knowledge brought about by the Galilean revolution
in physics had already
taken place, why was it necessary to reiterate the scientific nature of mathematics? Wallis’ contribution
could be explained by his willingness to address a
text in Logic, i.e. Smiglecius’, that was much read at Oxford during the period
in which he was writing and teaching there. Hobbes’ intervention
was only a
consequence
of Wallis’ statements. But what can we say about Barrow? There
does not seem to be any reason why he would want to spend so many pages
arguing against an author like Pereyra who was far from being an authority.
However, Barrow himself gives us a clue in the right direction when he says
that “some both have been, and still are so subtle as to deny that the
mathematics
are truly sciences, and that they afford true demonstrations.”
I
believe that the real person Barrow is addressing is not Pereyra but Gassendi.
It was in 1665 that Barrow delivered the lectures I analysed. Only a few
years before, in 1658, Gassendi’s Opera Omniu had been published. The third
volume contained a work which Gassendi had written in 1624 but had never
published: the second part of the Exercitationes paradoxicae adversus AristoteZeos. The sixth Exercitatio had the title: “That no science exists, and especially
no Aristotelian
science”. Gassendi
argued there that none of the so-called
sciences could be said to provide Aristotelian
knowledge, i.e. causal knowledge
from the essences of the subjects. Of course he could not leave unanswered the
challenge
that mathematical
sciences represented
for such a position.
He
himself acknowledged
that it was general opinion that nobody, nisi is sit
furiosus,
could deny the certainty and evidence of mathematics.
Gassendi’s
only weapon was to quote the opinion of Pereyra who, he claimed, was a
Peripatetic and nevertheless denied that mathematics
was a science. Gassendi
quoted at length the passage from Pereyra which I reported in note 11. This is
not the place to give a complete account of what Gassendi was trying to
achieve in the more general context of his Work.5’ It suffices to say that
5ee B. Rochot, ‘Gassendi et les mathkmatiques’,
Revue d’Histoire des Sciences 10 (1957),
69-78. See also P. Mancosu and E. Vailati, ‘Torricelli’s Infinitely Long Solid and its Philosophical
Reception in the Seventeenth Century’, Isis, 82 (1991), 50-70.
Studies in History and Philosophy of Science
264
Pereyra’s position was used to support
knowledge is from appearances:
his general
attempt
to show that all our
Therefore, I conclude that whatever certainty
related to appearances,
natures of things.52
and evidence there is in mathematics is
and in no way related to the genuine causes and inmost
Thus, Barrow had in mind not an obscure Jesuit from the previous century but
an adversary of the calibre of Gassendi, whose influence on the philosophical
world had already proved to be immense and which consequently
deserved an
extensive confutation.53
Gassendi’s appeal to the Quaestio to support his sceptical position as to the
nature of our knowledge raises the further issue of the relationship
between
scepticism in the seventeenth
century and the Quaestio. There is a book by
Wilhelm Langius, De veritatibus geometricis (1656), which contains references
to the Quaestio in the context of a defence of geometry against the sceptical
attacks on the certitude
of mathematics.
Langius shows awareness of the
primary literature concerning
the Quaestio and, against a sceptical use of the
debates on the nature of mathematical
demonstrations,
appeals to Barozzi who
“with various arguments,
derived both from authority
and from certain
reason, very learnedly and firmly established
that mathematical
demonstrations not only are to be called true and proper demonstrations,
differently from
what some thought, but also that they are the highest of all and the most
certain.“54 It is thus evident from Gassendi’s and Langius’ assertions that there
52“Concludo
ergo, quaecumque
est certitude
& evidentia
in disciplinis
Mathematicis
eas
pertinere
ad apparentiam;
nullo autem modo ad causas germanas
vel naturas etiam rerum
intimas.” Exercitationes,in Opera omnia in six tomes divisa, (Leyden: 1658) vol. 3, p, 209. English
translation
in The Selecred Works of Pierre Gassendi, ed. Brush, (New York, Johnson Reprint
Corporation,
1972) p. 107.
*)It should be remarked that such attacks by proxy were a common strategy in the period.
‘*Popkin called attention to Langius’ work in The History of Scepticism from Erasmus to Spinoza
(Berkeley, California,
1979) where he states that “with regard to mathematics
the sceptical
atmosphere of the seventeenth century was apparently strong enough to require that some defence
be given for this ‘queen of the sciences’. There is a work by Wilhelm Lang&,
of 1656, on the truth
of geometry, against sceptics and Sextus Empiricus”
(p. 85). The full quote by Langius is the
following. “Equidem non me fugit, multa a Petro Ramo viro doctissimo atq; insigni Geometra
contra methodum
Euclideam
fuisse proposita:
quae tamen ideo tacitus praetereo,
quod illum
vitium, non ipsas veritates concernat, net tam Geometricum
sit, quam Logicum. Fuere alii qui de
natura demonstrationurn
Mathematicarum,
varia disputarunt;
quibus egregie satisfecit Franciscus
Barocius patritius Venetus in celeberrima Pataviensi Academia Mathematum
Professor Publicus,
qui in illa Oratione, quam publice habuit. turn cum primum Mathemata
protiteri inciperet, variis
argumentis
tam ex auctoritate,
quam ex solida ratione petitis, pererudite ac solide comprobavit
demonstrationes.
Mathematicas
non modo vere & proprie demonstrationes
appellari, secus quam
aliqui sentirent, sed & omnium primas esse ac certissimas. Qui ergo haec plenius cognita habere
cupit illum adeat. Neq; enim his diutius immorari
libet, cum ipsa prima principia
totaq;
Wilhelmi Langii De Veritatibus
Geometrica
materia a cavillis malevolorum
satis sit vindicata.”
Geometricis Libri II. Prior contra Scepticos & Sex&m Empiricum & c. Posterior. contra Marcum
Meibomium, Copenhagen,
1656, pp. 156157.
265
Aristotelian Logic and Euclidean Mathematics
is an important
the Quaestio.
connection
between
It may be worthwhile
scepticism
in the seventeenth
to investigate
this problem
century
and
further.
Added in pro05 Two more texts related to the Quaestio that have recently come
to my attention
are Paulus Vallius’ Logica (1622) and Hugh Simple’s De
Disciplinis
Mathematicis
(1635).
Vallius’
text
is dependent
on the
Pereyra-Smiglecius
tradition
whereas
Simple
follows,
often
verbatim,
Biancani’s De Nature Mathematicarum.
Attention
to Simple was drawn by
Peter Dean in his “Jesuit Mathematical
Science and the Reconstitution
of
Experience
in the Early Seventeenth
Century”,
Studies in History and
Philosophy of Science 18, (1987) 133-175.
Acknowledgements -
I would like to thank Dr P. Cramer, Prof. W. Knorr, Prof. E.
Vailati and two anonymous referees for their useful comments on previous drafts of this
paper.