Schmitt C.B. (ed.) The Cambridge History of Renaissance Philosophy
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Epistemology
of the
sciences
687
The
connections
of
regressus with earlier discussions
of the
nature
of
scientific
demonstrations
are
little understood,
and the
following
remarks
are tentative.
4
Latin commentary
on
the
Posterior Analytics tended from
the
outset
to see it as
largely concerned with proximate causes
of
natural
phenomena, showing little appreciation
of
what many present-day
commentators regard
as
Aristotle's attempt
to
model
on
geometry
a
conception
of
a science
as a
deductive hierarchy
and to
show
how
diverse
phenomena
may be
understood
in the
light
of
general principles.
5
Given
this
view
of
the work,
it
is hardly surprising
that
a
passage
that
gives
explicit
examples
of
syllogisms
dealing with proximate causes should have become
a major focus
of
interest.
More
specific fuel
for the
elaboration
of
regressus
was
provided
by
passages
in
which Aristotle contrasts 'absolute' (airXcos)
and
'accidental'
(Kara
GVfjLfieftrjKos) knowledge
and by
those
in
which
he
distinguishes
things 'better known
to
us'
(TJ/JLLV
yvcopL/jLoorepov)
from things 'better known
in nature'
(rrj
<j>vo€i
yvcopipbajrepov)
.
6
To
obtain absolute knowledge
of
an
effect
we
cannot,
the
regressus theorists argued, proceed directly
to the
demonstration
of
that
effect
through its proximate cause;
for
though causes
are better known in the natural order, effects are better known
to
us, because
evident
to our
senses. Rather,
the
two types of demonstration, demonstra-
tion
of the
fact, which proceeds from
effect
to
proximate cause,
and
demonstration
of
the reasoned fact, which proceeds from proximate cause
to
effect,
must
be
combined. The precise mode
of
combination
of
the
two
types
of
demonstration needed
to
bring about
the
absolute demonstration
(demonstratio potissima)
of the
effect
which
is
productive
of
absolute
knowledge
was widely debated. The
following
scheme is, however, found
in
a
considerable number
of
accounts.
(1) By
observation
we
obtain
accidental
or
'confused' knowledge
of
an
effect.
(2) By induction combined
with
demonstration
of
the fact we obtain accidental knowledge
of
its cause.
(3) Through
a
noetic process ofnegotiatio (meditatio, consideratio)
we
obtain
absolute
or
distinct knowledge
of the
proximate cause, grasping
the
necessity
of
the link between
it and its
effect.
(4) By
demonstration
of
the
reasoned fact
we
obtain absolute knowledge
of
the
effect,
knowledge of it
through
the
cause which necessitates
it.
A
further important factor
in the
development
of
regressus
was the
conflation
of
species
of
demonstration
of
the Posterior Analytics with other
methods and procedures. For example,
in
Physics
1.1
Aristotle mentions two
4.
On the
sources
of
regressus
see
Randall
1961;
Crombie
1953;
Bottin
1972; Cambridge History 1982,
pp.
496-517
(Serene).
5. Cf.
Kosman
1964,
chs. 1, 2.
6. For
example,
Posterior Analytics 1.2
(7i
b
9-i2);
Prior Analytics
11.23
(68
b
35
ff.);
Physicsi.i
(i84
a
io
ft".).
Cambridge Histories Online © Cambridge University Press, 2008
688
Problems
of
knowledge
and
action
procedures,
one
going from
the
universal
to the
particular,
the
other from
the particular
to the
universal.
In
Latin commentary these were widely
identified (often
on the
authority
of
Themistius
and
Averroes) with
demonstration
of the
reasoned fact
and
demonstration
of the
fact
respectively.
7
Further conflations arose
in
commentary
and
exegesis
of
Galen's
Ars
medica,
the
analytic
and
synthetic procedures mentioned
at the
beginning
of the
treatise being identified with
the two
supposed species
of
demonstration.
8
It
seems
that it
was partly as
a
result
of
these conflations
that
the
nature and
validity
of
regressus came
to be
regarded
as a key to an
understanding
of the way in
which
the
masters
of
antiquity
had
attained
their knowledge
in
natural philosophy.
By
way of
example,
let us
consider
the
account
of
regressus offered
by
Agostino
Nifo
in his
Physics commentary
of
1506.
He
first raises
a
basic
difficulty
which faces those
who
regard regressus
as
productive
of
knowledge,
its
apparent circularity.
But
on
this issue
I
customarily raise
the
question whether
there
are
two procedures
in natural [sciences],
one
from
an
effect
to the
discovery
of the
cause,
the
other
from
the
cause discovered
to the
effect.
It
seems not, because then
there
would
be a
circle
in
demonstrations
. . .
9
Nifo
goes
on to
retail
a
defence
of
regressus against
the
objection:
Recent
authors
are of the
opinion
that
four types
of
knowledge occur.
The
first
is
[knowledge]
of the
effect through sense
or
observation.
The
second
is the
discovery
of
the cause through
the
effect, which
is
obtained
by
demonstratio
signi.
10
The
third
is
the
negotiatio
of
that
same cause by the intellect, by which, together with
the first, knowledge
of
the cause
is so
much increased
that it is
worthy
to be
made
the middle term
of
a
demonstratio
simpliciter. The last is knowledge
of
the reason
for
the same effect through
a
cause
so
established
that it may be the
middle term.
11
Later Nifo modified
his
views
on
regressus. Where
in his De
anima
commentary
of
1498
he had
given
an
account close
to
that
just outlined,
in
the 1520 edition
he
denies
that a
process
of
negotiatio
is
always involved;
12
7.
Themistius 1900, p. 1; Aristotle and Averroes 1550-2, iv, f. 4
va
. 8. See below p. 704.
9. Nifo 1552, f. 5
V
: 'Sed circa haec dubitare
solitus
sum, utrum in naturalibus
sint
duo processus, unus
ab
effectu ad causae inventionem, alter a causa inventa ad effectum. Videtur quod non, quia tunc
esset
circulus in demonstrationibus . . .'
10.
Nifo's terminology reflects the widespread identification of the 'demonstration of the fact' of
Posterior
Analytics
1.13 with the syllogism 'from probabilities or signs' (c£
€IKOTO>V
77 orjfieiojv)
of
Prior
Analytics 11.27.
11.
Nifo 1552, f. 8
r
: 'recentiores volunt
esse
quatuor notitias. Prima est efFectus per sensum, aut
observationem. Secunda est inventio causae per effectum, quae quidem demonstratione signi
habetur.
Tertia est eiusdem causae per intellectus negotiationem, ex qua cum prima crescit notitia
causae in tantum, ut digna sit effici medium demonstrationis simpliciter. Ultima est eiusdem
efFectus notitia propter quid per talem causam sic certam, ut sit medium.' Cf. Randall 1961,
pp.
42-3. 12. See Poppi 1969.
Cambridge Histories Online © Cambridge University Press, 2008
Epistemology
of
the
sciences
689
and
in
the
posthumous
1540
edition
of his
Physics commentary there
is
added
the
following
much-quoted recognitio:
But
when
I
more
carefully
ponder
the
words
of
Aristotle
and the
commentaries
of
Alexander,
Philoponus
and
Simplicius,
it
appears
to me
that
in the
regress
which
occurs
in the
demonstrations
of
natural
sciences,
the
first
process,
by
which
the
dis-
covery
of
the
cause
is
syllogised,
is a
mere
syllogism,
because
merely
conjectural,
13
since
through
it the
discovery
of the
cause
is
syllogised
only
conjecturally.
The
second
process,
moreover,
by
which
the
reason
for the
effect
is
syllogised
through
the
cause,
is demonstratio propter quid, not
that
which
makes
us
know
absolutely,
but
rather
ex conditione,
given
that that
is the
cause,
or
given
that
the
propositions
which
exhibit
the
cause
are
true
and
that
there
can be no
other
cause
. . . But you
will
object,
because
then
science
concerning
nature
would
not be
science.
It
should
be
said
that
science
concerning
nature
is not
science
absolutely
as is
mathematical
science;
but it is
science
propter quid,
because
the
discovery
of the
cause,
which
is
obtained
through
conjectural
syllogism,
constitutes
the
reason
for the
effect.
14
It
is
unclear whether
Nifo's
reservations about
our
ability
to
apprehend
with
certainty
the
causes
of
observed effects
are
general
or
apply only
to
certain difficult questions
in
natural philosophy. There
are,
however, other
signs
of a
moderate scepticism
in his
later commentaries. Thus
he
argues
against
the
view
that
the
definitions
on
which demonstrations
are
based
are
self-evident,
claiming
that
they
are
nota
per
se only
in the
sense
that
they
are
not demonstrable from something else.
15
And he is
ready
to
admit
that
the
non-demonstrative methods
of
Aristotle's
Topics have
a
place
in
enquiry
in
natural philosophy.
16
However,
a
caveat
is in
order: though
Nifo
is a
shrewd
and
scholarly interpreter,
it
is
hard
to
detect
a
systematic body
of
philosophical
doctrine
in his
eclectic
and
often eccentric pronounce-
ments.
17
When
we turn to the
account offegressus offered by Jacopo Zabarella
in
13.
'Conjectural'
is
potentially misleading here. Elsewhere Nifo characterises 'demonstration
of
the
fact'
as
coniectura necessaria:
Nifo
1553b
(ist edn, 1523),f.
I3
r
.
In the passage of Philoponus to which
Nifo is probably referring, the term
T€Kfxr)pLœ8r)s
is applied to such demonstrations, and the term
is
rendered by
coniecturalis
in the only Latin translation then published of Philoponus' commentary,
that
of Theodosius: Philoponus 1559 (ist edn, 1539),
p. 44 .
'By
a
reliable criterion'
is
perhaps an
appropriate
gloss.
14.
Nifo 1552,
f. 6
V
:
'Sed cum Aristotelis verba
et
Alexandri,
ac
Themistii, Philoponi,
ac
Simplicii
commentaria diligentius revolverim, videtur mihi
in
regressu,
qui fit in
naturalibus
demonstrationibus, primum processum,
quo
syllogizatur causae inventio,
esse
syllogismum
tantum,
quoniam coniecturalis tantum, cum per ipsum solum coniecturabiliter syllogizetur causae
inventio: secundum vero, quo syllogizatur per inventionem causae propter quid effectus,
esse
demonstrationem propter quid, non qui faciat scire simpliciter, sed
ex
conditione, dato quod ilia
causa sit, vel dato quod propositiones verae sint, quae représentant causam
et
nulla alia causa
esse
possit
. . .
Sed occurres, quia tunc scientia
de
natura non
esset
scientia. Dicendum, scientiam
de
natura
non
esse
scientiam simpliciter, qualis
est
scientia mathematica,
est
tamen scientia propter
quid:
quia inventio causae, quae habetur
per
syllogismum coniecturalem,
est
propter quid
effectus.' Cf. Randall 1961, pp. 43-5.
15.
Nifo
1553b,
f. 33
e
.
16.
Nifo 1540
(ist
edn,
1535),
ff.
9
r
-io
r
.
17.
Cf. Vasoli 1974,
p.
437.
Cambridge Histories Online © Cambridge University Press, 2008
690
Problems of knowledge and action
his influential Opera logica of 1578, we find ourselves in very different
territory. Here regressus is at the heart
of
an elaborate and systematic account
of
the processes of cognition and the functions of
logic,
an account
that
shows
no trace
of
scepticism about our ability to attain knowledge
of
causes
through demonstrative methods.
18
The
work opens with a treatise on the
nature
of
logic
in which Zabarella
argues
that
it is to be considered not as a practical art or theoretical science in
its own right, but
rather
as a midwife to the
arts
and sciences, being the
source of the 'instruments, by which
truth
in every matter is known and
discerned from falsity'.
19
In the four books De methodis he distinguishes
ordines, procedures for communicating existing knowledge, from methods
sensu stricto, procedures for the acquisition of new knowledge.
20
Zabarella claims
that
'demonstration of the reasoned fact' (demonstratio
propter quid or methodus compositiva) and 'demonstration of the fact'
(demonstratio quod or methodus resolutiva) are the sole methods in this strict
sense. He goes to great lengths to show
that
other procedures, for example,
the 'Platonic' composition and division used in the
hunt
for definitions and
the resolutio
(avdXvais)
whereby mathematicians work back from theorems
to principles, are not genuinely productive
of
knowledge.
21
He goes on to
argue
that
the two demonstrative methods provide all
that
is needed to
attain the goal of natural philosophy. That goal is knowledge of the
definitions
of
ajfectiones
—
observable states
of
bodies,
such as being eclipsed,
not twinkling and thundering. Such a definition must specify the genus,
subject and proximate cause
of
the state in question:
thus thunder
is defined
as a noise produced in a cloud by the extinction of fire. The genera and
subjects
of
ajfectiones zrepraecognita, 'things known in advance', and the two
demonstrative methods provide all
that
is needed, Zabarella claims, to
achieve
distinct knowledge of proximate causes.
22
This
last claim is apparently open to the Aristotelian objection
that
it
overlooks
the process of induction, described in the last chapter of the
Posterior Analytics, which leads from individual sense experiences to the
grasp
of
universals. It
turns
out
that
Zabarella, whilst
fully
acknowledging
18. On
Zabarella's
account
of the
nature
and
functions
of
logic
see W. F.
Edwards
i960;
Corsano
1958-63,
41:
507-17;
Poppi
1972a;
Aristotelismo veneto
1983,1,
pp.
155-72
(Risse).
On
Girolamo
Balduino's
account,
to
which
Zabarella
is
much
indebted,
see
Papuli
1967, pp.
125-200.
On the
fortuna
of
Zabarella's
logical
doctrines
see, for
example,
Petersen
1921; J.
Zabarella
1578, pp. xi-
xxviii
(Vasoli's
introduction
to the 1985
reprint).
19. J.
Zabarella
1607b,
cols.
51-2:
'instrumenta
quibus
in
omni
re
verum
cognoscatur,
et a
falso
discernatur'.
20. Ibid.,
cols.
138-9.
21. Ibid.,
cols.
23iff.,
266—7.
22. Ibid.,
cols.
264-6,
294-318.
Zabarella's
argument
depends
on his
interpretation
of
Aristotle's
obscure
treatment
in Posterior Analytics 11 of the
relations
between
definition
and
demonstration.
Cambridge Histories Online © Cambridge University Press, 2008
Epistemology of the sciences
691
the role in the acquisition of scientia of this type of induction, does not
consider it as an independent form of inference. With respect to its role in
the discovery
of
causes it can, he suggests, be regarded as an inferior form of
demonstration of the fact, inferior because whereas demonstrations of the
fact
can reveal to us recondite causes inaccessible to the senses, for example,
the prime mover as the cause of the diurnal rotation of the heavens,
induction can make known only causes accessible to the senses, for example,
fire as the cause of smoke.
23
Induction is not, Zabarella claims, strictly speaking a form
of
inference.
24
To
understand this assertion we have to
turn
to the theory of cognition
expounded in his De rebus naturalibus of 1590 and in his massive
uncompleted De anima commentary, published posthumously in 1605.
There Zabarella claims
that
our knowledge
of
universals is not acquired by
inference from observations. Rather sense experiences
give
rise to images
which
render the 'possible intellect' (intellectus possibilis) or rational soul
receptive to representations of the universals present in the mind of God.
The
vehicle of this inspiration is the 'active intellect' (intellectus agens).
Induction is not therefore a form of inference, but should
rather
be
considered a process 'from the same to the same', from the representations
of
universals
that
constitute the individuals of the external world to
representations of those same universals inspired into the human mind by
the transcendent active intellect. Perhaps it is his confidence in divine
revelation of universals, triggered by the images whose formation in the
imagination
follows
sense experience,
that
makes Zabarella so confident of
the ability of the natural philosopher to apprehend the premises on which
demonstrative syllogisms depend.
25
To
see this confidence in action let us
turn
to the most elaborate of his
examples of the application of regressus.
26
The
effect
to be explained is the
generation and corruption of substances, the observed fact
that
physical
objects
of
each natural kind come into existence and ultimately cease to be.
Let
us take as an example the demonstration
of
Aristotle
in book
1
of
the Physics, by
which
from the generation
of
substances he shows
that
prime matter occurs: from a
known
effect
an unknown cause. For generation is known to us by sense but the
underlying matter is in the highest degree unknown. So after the proper subject,
that
is a perishable natural body, in which each is originally present, has been
considered, it is demonstrated
that
there
is present in it a cause, on account
of
which
the
effect
is present in the same, and it is demonstratio quod which is
thus
formed:
where
there
is generation
there
is underlying
matter;
but in a natural body
there
is
23. Ibid.,
cols.
268-71.
24. Ibid.,
cols.
270, 1277. 25. Cf.
Skulsky,
1968, p. 356.
26. J.
Zabarella
1607b,
cols.
484-9;
cf.
also
1607a,
cols.
137-42.
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692
Problems
of
knowledge
and
action
generation;
so in a
natural body there
is
matter.
In
this demonstration
the
minor
premise
is
known
to
us confusedly, because we do
in
fact observe
that
natural bodies
are generated
and
perish,
but we do not
know
the
cause.
The
major premise,
although
it is not
known
by
sense, easily becomes known when some mental
consideration
has
been applied.
27
Now
that
we
have completed
the
first phase
of the
regressus,
we
have
discovered
that
prime matter
is
present
in all
natural bodies:
we
have
'confused'
knowledge
of
the cause
of
generation
and
corruption.
But we do
not yet, Zabarella claims, have 'distinct' knowledge
of
the cause,
for we do
not
yet
know
that
it is the
proximate cause
of
generation
and
corruption
rather than
a
mere constant concomitant.
To
discover this
we
must
undertake
a
consideratio
or
negotiatio about
the
essential
nature
of
prime
matter. Zabarella
gives
as
the
negotiatio
an
exposition
of
Aristotle's
theory
of
prime matter.
Matter therefore having been discovered through generation
is not yet
known
to
be
the
cause
of
generation, because
it
is
not
yet known what
that
matter
is.
For
this
reason Aristotle,
who
wished
to
teach
us a
distinct,
not
merely
a
confused,
knowledge
of
principles,
as
much
as a
natural philosopher
can
have, began
to
investigate
the
nature
and
conditions
of
the matter which
he
has discovered
. . . He
taught
us
first
of
all
how it
differs from privation.
For the
function
of
matter
is to
underlie contraries and
to
receive them. The function
of
contraries, moreover, is
for
each
to
drive
the
other
out of the
same matter. Hence matter persists under each
contrary
and
never perishes
. . .
Therefore
in its own
nature
matter must lack
all
forms
and
have
the
potentiality
to
receive all. This
is
without doubt
the
nature
of
matter,
that
it is
nothing
in
actuality
but
everything
in
potentiality.
28
27.
J. Zabarella 1607a, col. 485: 'sumamus demonstrationem Aristotelis in libro
1
Physicorum, qua ex
generatione, quae substantiarum est, ostendit materiam primam dari ex effectu noto causam
ignotam:
generatio enim
sensu
nobis cognita est, subiecta vero materia maxime incognita. Accepto
igitur
subiecto proprio, scilicet corpore naturali caduco, cui primo utraque
inest,
in eo
demonstratur causam
inesse,
propterea quod in eodem
inest
effectus, et est demonstratio quod,
quae ita formatur: ubi est generatio, ibi est subiecta materia: at in corpore naturali est generatio,
ergo in corpore naturali est materia: in hac demonstratione minor propositio est nobis nota
confuse: quoniam generari quidem, et interire naturalia corpora cernimus, sed causam ignoramus:
maior vero propositio quanquam
sensu
non cognoscitur, aliqua tamen adhibita mentali
consideratione facile innotescit.'
28.
Ibid.,
col. 488: 'Materia igitur per generationem inventa, nondum cognoscitur
esse
causa
generationis: quia quid sit ipsa materia, adhuc ignoratur: propterea Aristoteles, qui principiorum
cognitionem nobis tradere voluit non modum confusam, verum etiam distinctam, quantam
naturalis philosophus habere potest, coepit materiae inventae conditiones, et naturam investigare
.
. . Docuit in primis quomodo a privatione differat: materiae namque officium est substare
contrariis, et ea recipere: contrariorum vero officium est se ab eadem materia mutuo pellere: ideo
materia manet sub utroque contrario, et nunquam interit. . . debet igitur materia secundum suam
naturam
carere omnibus formis, et omnium recipiendarum potestatem habere: haec absque dubio
est natura materiae ut nihil sit actu, sed potestate omnia.' The qualification 'quantam naturalis
philosophus habere potest' may well imply a contrast with the knowledge of first principles, which
is the province of the metaphysician: cf.
ibid.,
cols. 529, 780, 1267.
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Epistemology of the sciences
693
As
a result of this negotiatio:
It readily becomes apparent to us
that
such matter is the cause of generation. For
because it has the capacity to receive all forms and prescribes for itself no particular
form,
but is equally suited to receive a form and its privation, it comes about
accordingly
that
nothing possessing matter can be eternal, but of necessity it
eventually
perishes, and from it something else is generated.
29
Zabarella
concludes
that
now
that
we have distinct knowledge
of
the cause
we
are in a position to set out a demonstration
of
the reasoned fact: wherever
prime matter occurs there is generation; but prime matter occurs in a
natural body; so generation occurs in a natural body. We have achieved the
desired distinct knowledge of the generation and corruption of substances,
for
we now have certain knowledge of it through the cause which
necessitates it.
30
THE
CERTAINTY
OF
MATHEMATICS
As
we have seen, in the Aristotelian tradition achievement of 'absolute
demonstration' (a^oSei^s*
a7rAa>s%
usually rendered as demonstrate potissima
or simpliciter), was generally regarded as problematic in natural philosophy,
because of the lack of evidence to us of the hidden causes on which such
demonstrations often depend. In mathematics, however, it was widely held
that
the two types of priority, priority 'for us' and priority 'for nature',
coincide.
31
So no process of working back from what is evident to our
senses to what is naturally prior but hidden from us is required;
rather
the
necessary principles on which demonstratio potissima depends are immedi-
ately
evident to us.
In his Commentarium de certitudine
mathematicarum
of 1547 the Siennese
philosopher Alessandro Piccolomini challenges the
view
that
mathematics
owes
its certainty to the status of its demonstrations as demonstrationes
potissimae.
First we intend to show both by reasons and by testimonies
that
mathematical
demonstrations are not those
demonstrationes
potissimae
that
Aristotle devises in
29. Ibid., col. 488:
'facile
nobis
innotescit,
talem
materiam
generationis
causam
esse:
quoniam
enim
potestatem
habet
recipiendi
omnes
formas,
et
nullam
certam
formam
sibi
praescribit,
sed
aeque
apta
est
recipere
formam,
et
eius
privationem:
ideo
facit
ut
nullum
materiam
habens
possit
esse
perpetuum,
sed ex
necessitate
aliquando
intereat,
et ex eo
aliud
generetur'.
30. Ibid., col. 489.
31.
Averroes
was
often
cited
in
support
of
this
view:
see, e.g.,
Aristotle
and
Averroes
1550-2,1, f.
374
r
;
Cambridge Histories Online © Cambridge University Press, 2008
694
Problems
of
knowledge
and
action
Posterior
Analytics.
In the
second place.
.
.
it
is
our
intention
to
set
out the
reason
for
which
mathematical disciplines
can be
said
to be in the
first grade
of
certainty.
32
In
the
opening chapters
he
offers
an
account
of
the
nature
of
logic
and
of
the
demonstrative methods
it
provides.
33
The
conditions
for
demonstratio
potissima are,
on
this account, stringent.
The
premises must
be
necessary,
the
major premise having
the
form
of a
definition,
and the
middle term must
specify
the
immediate cause
of the
effect
demonstrated.
34
There
follow
chapters
on the
aims, objects
and
methods
of
mathematics,
the
latter
including
a
discussion
of
the resolutive method
of
the classical geometers.
35
The
crux
of the
argument comes
in
chapter
11,
in
which Piccolomini sets
out
a
series
of
ways
in
which mathematical demonstrations
fail
to
satisfy
the
conditions
for
demonstratio potissima: they
do not
specify causes
in any
of
the
four
genera
of
causes;
the
middle term
is not
always immediate, since
there
may
be
many mathematical demonstrations
of the
same property
of a
subject
through different middle terms;
and the
major premise does
not
always
have
the
form
of a
definition.
36
Piccolomini
rests
his
case
for the
certainty
of
mathematics
on an
account
of
the
objects
of
mathematical knowledge
that
he
attributes
to
Proclus:
So
Proclus derives from Plato
the
view
that
the
mathematical things themselves,
about which demonstrations
are
made,
are
neither sensible things altogether
in a
subject,
nor
entirely freed from it,
but
that
these mathematical figures are formed
in
the imagination,
the
occasion being afforded
by
quantities found
in
sensible
matter.
Moreover,
the
intellect derives those universal principles from these quantities
that
are
in the
imagination.
37
Separation
of
the objects
of
mathematics from sensible matter
is a
standard
ground
for the
certainty
of
mathematics
in the
period,
but
Piccolomini's
position
is
less usual
in
relating
the
certainty
of
mathematics
to the
constructive role
of the
imagination.
The
issues raised
by
Piccolomini attracted considerable attention
in the
32.
A. Piccolomini 1565, f. 6o
v
: 'Primum et rationibus et authoritatibus, demonstrare intendimus
mathematicas demonstrationes non
esse
illas potissimas, quas in [Posterioribus Analyticis] construit
Aristoteles. Secundo vero loco . . . causam assignare est animus, qua mathematicae disciplinae in
primo
esse
gradu certitudinis dici possunt.' For a detailed exegesis of this work see
Giacobbe
1972a;
see also Schilling 1969, pp. 4$f[.
34.
Ibid.,
ff.
88
v
-94
r
.
35.
Ibid.,
ff. 94
v
-ioo
r
. 36.
Ibid.,
ff. ioo
r
-5
v
.
37. Ibid.,
f. 95
r
: 'Concludit ergo Proclus ex Platone, quod res ipsae mathematicae, de quibus fiunt
demonstrationes, nec omnino in subiecto,
sensibiles
sunt, nec penitus ab ipso liberatae, sed in ipsa
phantasia reperiuntur figurae illae mathematicae, habita tamen occasione a quantitatibus in materia
sensibili repertis. Intellectus autem, ex iis, quae in phantasia sunt in quantitatibus, rationes illas
universales colligit.'
Cambridge Histories Online © Cambridge University Press, 2008
Epistemology of the sciences
695
following
decades. Pietro Catena, Francesco
Barozzi
and Giuseppe Moletti,
all
teachers
of
mathematics at Padua, concerned themselves with the
nature
of
mathematical demonstrations and the sources of certainty in mathemat-
ics,
as did the logician Bernardino Tomitano, and the Jesuit theologian and
natural philosopher Benito Pereira.
38
Let us consider briefly the responses
of
Francesco
Barozzi,
who sets out to refute Piccolomini, and of Pietro
Catena,
whose position is closer to Piccolomini's.
Barozzi's
Quaestio de certitudine mathematicarum of 1560 challenges
Piccolomini's
distinctions between mathematical demonstration and
demonstrate potissima. To Piccolomini's claim
that
mathematical demon-
strations never appeal to causes,
Barozzi
responds by insisting
that
in dealing
with
materia intelligibilis they specify both formal and material causes.
39
However,
in trying to counter others of the distinctions alleged by
Piccolomini,
Barozzi
makes substantial concessions, allowing
that
not all
mathematical demonstrations are demonstrationespotissimae.
40
He then
turns
to the issue of certainty. From Aristotle he extracts three criteria for the
relative certainty
of
sciences:
a science which demonstrates both a fact (quia)
and its reason (proper quid) is more certain than one
that
demonstrates only
the fact; a science
that
deals with immaterial objects is more certain than one
that
deals with material objects; and one
that
deals with simple objects is
more certain than one
that
deals with what is complex.
41
Since
demonstrationes potissimae reveal both facts and their reasons, mathematics
possesses the first grade of certainty by the first criterion. By the second
criterion it has a degree of certainty intermediate between those of scientia
divina and scientia naturalis,
given
that
its objects are quantities partly
conjoined
with material bodies and partly separate from them.
42
Barozzi
concludes:
'It is abundantly clear
that
[mathematics] is to be put in the first
grade of certainty not only because of the matter subject to it (as a most
erudite modern author held), but also on account of its potissimae
demonstrationes. '
43
38. On
Catena
see
Giacobbe
1973, 1981;
Crapulli
1969, pp.
45-55.
On
Barozzi
see
Giacobbe
1972b;
Schilling
1969, pp. 52-3;
Rose
1977. On
Moletti
see Aristotelismo veneto 1983, 1, pp.
509-17
(Carugo);
on
Tomitano,
Aristotelismo veneto, 11, pp.
607-21
(Davi
Daniele).
On
Pereira
see
Giacobbe
1977;
Crapulli
1969, pp. 93-9;
Schilling
1969, pp.
47—9.
39.
Barozzi
1560, fF.
2i
r
fF.
A
detailed
exegesis
is
given
by
Giacobbe
1972b.
40.
Proof
by reductio ad
absurdum,
for
example,
is not
demonstrate
potissima: ibid., 2j
T
~
v
.
41. Ibid., ff.
32
v
-3
r
-
42. Ibid., f. 33
r
.
43. Ibid., f. 33
v
: 'non
solum
propter
subiectam
sibi
materiam
(ut
voluit
eruditissimus
recentior)
verum
etiam
propter
suas
potissimas
demonstrationes
in
primo
certitudinis
ordine
ponendam
esse,
abunde
patet'.
Cambridge Histories Online © Cambridge University Press, 2008
6Q6
Problems
of
knowledge
and
action
In
his
treatises
of
1556
and
1561
on
mathematical passages
in
Aristotle,
Pietro Catena emphasises
the
autonomy
of the
methods
of
mathematics.
44
He
follows
Piccolomini
in
denying
the
occurrence
of
demonstrate potissima:
no mathematical demonstration
is
potissima,
and
because
of
that
mathemat-
ical
disciplines
are not
sciences,
if
Aristotle's teaching
is
adhered to'.
45
Catena's brief Oratio
pro
idea methodi
of 1563
presents
a
case
for the
certainty
of
mathematics
that
represents
a
substantial elaboration
of
Piccolomini's
claims.
He
first considers
the
certainty
that
accrues
to
mathematics from
its
objects. There
are, he
maintains, three kinds
of
things
with
which
our
speculations
may be
concerned.
For some things
that are
bound
up in
matter come under consideration together
with
it; while others
that
are
joined
to
matter
are
conceived without
it by the
mind.
A
third kind
of
things, however, is remote from all matter and is completely known
in every respect without it. The first
are
most obscure, because
not
only
are
they
in
matter,
but
they are
not
even separated from
it by our
thought. For this reason they
are never represented
by our
intellect genuinely
and
faithfully. Those things which
lack
all matter.
.
. are surely
the
ones most evident
and
perspicuous
by
nature
to the
eyes
of
the mind;
but
since
we are
left
to the
help
of
the senses, from which
all our
knowledge
originates so long as we are confined
in
this body [lit. pleasant prison],
it
comes
about
that
things
in
themselves most clear are,
on
account
of
the feebleness of
our mind, quite hidden
in
Cimmerian darkness. There remains
the
third kind
of
things, which inhere
in the
material thing itself,
but are
separated from
it by our
thought;
and
these
are the
only things
that are
known
to us
exactly
and do not by
their obscurity engulf
the eye of
the mind
in
darkness
or
injure
it
with
too
much
light
and
brilliance. So wherever we
are
we see
the
triangle,
the
rectangle
and
other
things
of
this kind without any
coverings,
because they are
the
sort
of
things
that are
grasped
by the
senses
and
because we contemplate
the
form
itself
apart from matter
and
do not
consider wood, iron
or
anything
of that
kind.
And
this evidence
and
certainty
of the
mathematical disciplines
is
evident from [their] subjects
and the
things themselves.
46
44. Catena 1556, 1561. The former work is striking in its recognition of the centrality of geometrical
science in Aristotle's
Organon.
(Catena assumes Aristotle's familiarity with Euclid's
Elements.
As
was usual in the period he indentifies the geometer with Socrates' disciple Euclid of
Megara:
see
Giacobbe
1981, pp.
48-9.)
See Giacobbe 1981 for editions of these works and of Catena 1563.
45. Catena
15
56, p. 90: 'Dico secundo. . . quod nulla demonstratio mathematica est potissima, et ob id
mathematicae nullae sunt scientiae si stetur in doctrina Aristotelis.'
46. Catena 1563, f. 4
r_v
: 'Quaedam nam materiae implicitae, cum ea pariter in considerationem
veniunt. Aliae autem materiae adiunctae, absque ea mente concipiuntur. Tertium vero genus
rerum semotum est ab omni materia, et absque ipsa omnifariam intellectu pernoscitur. Primae sunt
obscurissimae, quia non solum sunt in materia, sed ne cogitatione quidem nostra ab ilia
seiungantur: et ob id intellectui nostro, neque pure, neque sincere unquam representantur. Quae
vero carent omni materia, . . . sunt quidem illae, mentis oculis natura illustrissimae, atque
evidentissimae. Verum cum in illarum speculatione destituamur auxilio sensuum, a quibus
proficiscitur omnis nostra cognitio, quandiu in hoc laeto carcere detinemur, fit, ut res per se
lucidissimae, propter imbecillitatem mentis nostrae, cymeriis reddantur tenebris obscuriores.
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