Aristotelian Logic and Euclidean Mathematics: Seventeenth-Century ...

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. Piccolo-

mini, 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, New Haven, CT 06520, U.S.A.

Received 21 February 1991; in revised form 14 June 1991,

‘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 Greek aitia. He explains his choice as follows. ‘ “Explanation” and its cognates render aifia and its 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, 1992. Printed in Great Britain.

0039-3681/92 S5.00 +O.OO @ 1992. Pergamon Press Ltd.

241

242 Studies in History and Philosophy of Science

the Quaestio de certitudine mathematicarum.2 The fundamental issues raised by

this debate were essentially the following two:

(a) What is the relationship between Aristotelian logic and Euclidean mathe-

matics? 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 demonstra-

tions, 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 insis- tence 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 episte-

mological, 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 Commen-

tarium de certitudine mathematicarum disciplinarum which can be rightly con-

sidered 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 Magni- ficum Dominum/Dm. THOMAM ZAMOYSCIUM, &c./OXONII,/Excudebat A. LichJeld, Acad. Typogr. Impensis H. CRIPPS,/J. GODWIN & R. BLAGRAVE, An. Dom. 1658./Gum Privilegie.


of mathematics (asserted by Aristotle and reiterated by Averroes and a long

list of Aristotelian commentators’) arguing from the assumption that mathe-

matics 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, Piccolo-

mini 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; demon-

strations are classified under the genera of quia, propter quid and potissima.

This is the distinction proposed by Piccolomini who characterized the demon-

strati0 potissima as a demonstration which 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, Piccolo-

mini 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

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.


devoted to showing that demonstrations in mathematics do not conform to

any of the above restrictions.9

There were several reactions to Piccolomini’s work. Several scholars agreed

with him that demonstrations in mathematics did not conform 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 contra-

diction 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

WIPS Z%Z-D


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 Picco-

lomini 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 potis-

simae 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.


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 demonstra-

tion 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 potis-

simae. The second argument appeals to the fact that demonstrationes potis-

simae 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 mathe-

matical 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 at the endpoints of the given segment and radii

equal to the segment (see Fig. 1). Now the triangle ABC is equilateral since its

sides are equal to the radius of the same circle and thus are equal to each other.

The causal nature of the proof 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).


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 demon-

strations 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 Smig-

lecius, 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 demonstra-

tions are deficient in both features of a potissima demonstration, i.e. causality

161bid. ““Et quidem nonnulli probant id ex eo, quod entia mathematics, ut quantitates & figurae, prom

a Mathematics considerantur, non dentur in rerum natura 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.


and necessity. There are two types of arguments for the proof of this assertion.

One proceeds a posteriori (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 not determine the true cause of being [veram causam essendi]. In fact the triangle is not equilateral on account of its being constructed between two circles. For it would still be equilateral even if it were not constructed between two circles. From whence 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 [acci-

dentia] 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.

250 Studies in History and Fhilosophy of Science

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 mathe-

matics 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 demonstra-

tiones 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.


Let us now recapitulate. Smiglecius asserts that demonstrations in mathema-

tics in principle cannot satisfy the conditions for the demonstrationes potis-

simae. This is not the case for physics which at least in principle could have

demonstrationes potissimae. This position 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

Wallis’ discussion of the Quaestio occurs in his Mathesis universalis: sive

arithmeticurn opus integrum (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 mathema-

tics are truly sciences.”

Wallis began by distinguishing the dioti demonstration, “which teaches the

proper affections of the subject by proper causes”, from the hoti demonstra-

tion, 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 demonstra-

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 (Oxford, 3 ~01s. 1693-99), vol 1.


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 proxi-

mate 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.


already deduced somewhere else. One more remark concerning the above

quotation. We know that Piccolomini and Pereyra explicitly denied mathema-

tics 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 distinc-

tion, 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.


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.


overall intention guiding his discourse. When we compare 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 termino-

logy 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 demon-

strates 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).


Speaker A continues by reading the passage where Wallis distinguished hoti

proofs from dioti proofs. Hobbes’ point is that only the dioti reasonings are

really demonstrations whereas the hoti arguments are not:

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 demonstra-

tions 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 with dioti demonstrations in a science Hobbes scathingly

remarked:

‘*“Nam demonstratio TOO b16rt 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.


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’ triparti-

tion 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 corre-

sponds to an appeal to efficient causes. Hobbes then returns to the meaningful-

ness of a definition for hoti demonstrations:

Tell me then, since in every demonstration 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

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 demonstra-

tions 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.


Therefore in demonstrations that tend to absurdity it is not good logic to require all along the operation of the cause.”

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 potis-

simae, 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 demonstra-

tions”, 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 Demonstra-

tions”.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 begin-

ning 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.


universal axioms are obtained by induction and therefore are fallible. This

opens up the problem of certainty and the role that sense and intellect play in

science. It suffices 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 mathe-

matical 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, nor &rms their Demonstrations from proper and essential, but from

common and accidental 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 ratiocinations satisfy the

Aristotelian strictures for scientific ratiocinations:

To which I answer, that those scientific Conditions, which Aristotle prefixes to Demonstration, who was most observant of its Laws, do most fitly agree with Mathematical Ratiocinations.4’

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.


states that at least some mathematical demonstrations are diotL4* This is

argued by appealing, first of all, to Aristotle’s examples who, Barrow claims,

took his only examples of causal demonstrations from mathematics. In his

sixth lecture he proposed to show that

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 Demonstration roi, &I that CI Thing 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 SOB 616~1 why 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.


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 Right Angle”. On this issue Barrow admits to

disagreeing with Aristotle who thought definitions should be unique. In

Barrow’s opinion it is only a matter of convenience how to choose a proper

affection as a starting point for a demonstrative chain.@ It is exactly in this

feature, adds Barrow, that the causality of mathematical demonstrations

consists:

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 Rise, for whichsoever Link of the Chain you take hold of, the Whole will follow.” Ibid., p. 88.

451bid., p. 88. “He does not consider material causes which had however played a certain role in the Quaestio. “Ibid., pp. 88-89.

SHIPS 232-E


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 Ratiocina- tions 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 in lectures

XXI and XXIII on the lower dignity of proofs by reductio ad absurdurn.


As in the case of Wallis and Hobbes one should remark that Barrow, while

still being willing to frame his arguments in the context 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 mathema-

tics? 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 Aristote-

Zeos. 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.


Pereyra’s position was used to support his general attempt to show that all our

knowledge is from appearances:

Therefore, I conclude that whatever certainty and evidence there is in mathematics is related to appearances, and in no way related to the genuine causes and inmost natures of things.52

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 demonstra-

tions 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; Geometrica materia a cavillis malevolorum satis sit vindicata.” Wilhelmi Langii De Veritatibus Geometricis Libri II. Prior contra Scepticos & Sex&m Empiricum & c. Posterior. contra Marcum Meibomium, Copenhagen, 1656, pp. 156157.


is an important connection between scepticism in the seventeenth century and

the Quaestio. It may be worthwhile to investigate this problem 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.

Aristotelian Logic and Euclidean Mathematics: Seventeenth-Century ...

Documents

Transcript of Aristotelian Logic and Euclidean Mathematics: Seventeenth-Century ...