## 1. Life and Works



In the later Middle Ages Albert of Saxony (*Albertus de
Saxonia*) was sometimes called *Albertucius* (Little
Albert), to distinguish him from the thirteenth-century theologian
Albert the Great. He is however a master of great importance in his
own right. He was born at Rickensdorf, in the region of Helmstedt
(Lower Saxony) in present-day Germany, in the beginning of the 1320s.
After initial schooling in his native area, and possibly a sojourn at
Erfurt, he made his way to Prague and then on to Paris. He was member
of the English-German Nation and became a master of arts in 1351. He
was Rector of the University of Paris in 1353. He remained in Paris
until 1362, during which time he taught arts and studied theology at
the Sorbonne, apparently without obtaining a degree in the latter
discipline. His logical and philosophical works were composed during
this period. After two years of carrying out diplomatic missions
between the Pope and the Duke of Austria, he was charged with founding
the University of Vienna, of which he became the first Rector in 1365.
Appointed canon of Hildesheim in 1366, he was also named Bishop of
Halberstadt the same year, serving in that office until his death on
July 8, 1390.



Not having left any theological writings or a commentary on
Aristotle's *Metaphysics* (at least none that we know
of), Albert is primarily known for his works on logic and natural
philosophy. Albert's masterwork in logic is the *Logica*
(later designed as *Perutilis logica*, Very Useful Logic).
Albert also composed a voluminous collection of *Sophismata*,
which examines numerous sentences that raise difficulties of
interpretation due to the presence of syncategorematic words --
i.e., terms such as quantifiers and certain prepositions, which,
according to medieval logicians, do not have a proper and determinate
signification but rather modify the signification of the other terms
in the propositions in which they occur. He also wrote several
question commentaries: *Quaestiones* on the *Ars Vetus*
or Old Logic (i.e., the *Isagoge* of Porphyry and
Aristotle's *Categories* and *De
Interpretatione*), *Quaestiones* on the *Posterior
Analytics*, and a series of 25 *Quaestiones logicales*
(Logical Questions), addressed to semantic problems and the status of
logic. Albert of Saxony also commented the main texts of the
Aristotelian natural philosophy (*Physica, De caelo, De generatione
et corruptione, De sensu et sensato, Meteorologica*) together with
the *De sphaera* of John of Sacrobosco. By contrast, the
*Quaestiones de anima* that were attributed to him are of
dubious authenticity. He also wrote commentaries on *Ethics*
and *Economics*, and texts of mathematics: a treatise *De
proportionibus*, a *Quaestio de quadratura circuli*, a
*Tractatus de maximo et minimo*.



The most renowned philosopher when Albert studied and taught in the
Faculty of Arts at Paris was John Buridan. Albert was one of several
masters contemporary with or immediately following Buridan whose work
transformed logic and natural philosophy in the later Middle Ages. For
a long time he was thought to have been a pupil or follower of
Buridan, but this idea is now widely questioned. Some of his works on
logic and physics were composed before Buridan had lectured on these
subjects for the last time, and Buridan clearly takes notice of them,
whether for criticizing or for adopting Albert's views. In
logic, he seems to have been influenced by certain ideas and methods
imported from England. His logic depends very much on Ockham's,
but the influence of William Heytesbury is also evident in his
*Sophismata*. Walter Burley was another important influence on
Albert, though this is somewhat puzzling in view of the fact that they
had opposing views on the nature of universals. In any case, Burley
seems to have been on Albert's mind when he wrote his commentary
on the *Nicomachean Ethics* as well as when he was developing
his theory of consequences.



These different influences have sometimes made Albert seem no more
than an eclectic compiler of the views of others. But, in addition to
providing the context for some of his own contributions,
Albert's fluency with the views of his contemporaries gives him
a unique place in the development of logic and philosophy at the
University of Paris in the fourteenth century.


## 2. Logic



On most topics the *Logica* (a text based on lectures first
given before 1356 and finally revised for publication before 1360) is
influenced by Ockham's *Summa logicae*, though it offers
an independent approach in the treatises on obligations, insolubles,
and consequences, which had assumed greater importance during this
period. As has been known for some time, this work is a remarkable
handbook organized into six treatises: the first defines the elements
of propositions; the second treats of the properties of terms; the
third of the truth conditions of different types of proposition; the
fourth of consequences (including syllogisms, and in fact adding to it
the theory of topics); the fifth of fallacies; and the sixth of
insolubles and obligations.



In the first part of the *Logica*, which sets out the
terminology of the entire text, Albert returns to the Ockhamist
conception of the sign and in so doing distances himself from the
position defended by Buridan. After clearly including the term (an
element of the proposition) in the genus of signs -- by which he
provides, in the tradition of Ockham, a semiotic approach to
logico-linguistic analysis -- he establishes signification
through a referential relation to a singular thing, defining the
relation of spoken to conceptual signs as a relation of subordination.
He is also an Ockhamist in his conception of universals, which he
regards as spoken or conceptual signs, and in his theory of
supposition, which essentially restates the Ockhamist divisions of
supposition, despite of some refinements as *descensus
copulatim*, perhaps influenced by Heytesbury. In particular, he
restores the notion of simple supposition -- i.e., the reference
of a term to the concept to which it is subordinated, when it
signifies an extra-mental thing -- which is criticized and
rejected by Buridan. Finally, Albert is close to the *Venerabilis
Inceptor* in his theory of the categories, where he refuses to
consider quantity as something absolutely real, reducing it instead to
a disposition of substance and quality. Albert in fact contributed as
much as Ockham to the spread of this conception of the relation
between substance and quantity in natural philosophy in Paris and
Italy.



Albert's treatment of relations is, on the other hand, highly
original. Although (like Ockham) he refuses to make relations into
things distinct from absolute entities, he clearly ascribes them to an
act of the soul by which absolute entities are compared and placed in
relation to each other, an "act of the referring soul [*actus
animae referentis*]". This leads him to reject completely
certain propositions Ockham had admitted as reasonable, even if he did
not construe them in quite the same way, e.g., 'Socrates is a
relation'. Both Ockham and Buridan had allowed that the term
'relation' could refer to the things signified or connoted
by concrete relative terms (whether collectively or not).



So Albert was not content with merely repeating Ockhamist arguments.
More often than not, he developed and deepened them, e.g., in
connection with the notion of the appellation of form. This property
of predicates, which had previously been used by the *Venerabilis
Inceptor*, was employed by Albert in an original manner when he
adopted it instead of Buridan's appellation of reason
(*appellatio rationis*) to analyze verbs expressing
propositional attitudes. Every proposition following a verb such as
'believe' or 'know' appellates its form. In
other words, it must be possible to designate the object of the belief
via the expression understood as identical to itself in its material
signification and without reformulation. Another area in which Albert
deviates from Ockham is his rejection of the idea that any distinction
with multiple senses must have an equivocal proposition as its object.
According to Albert, equivocal propositions can only be conceded,
rejected, or left in doubt.



He adopts the same position as Burley about the issue of the complex
subject-term (including an oblique case) in a categorical proposition,
as in 'Any man's donkey is running [*Cuiuslibet hominis
asinus currit*]'. He supports a logico-semantical model in
which 'man [*homo*]' is the logical subject while
'a man's donkey [*hominis asinus*]' is the
grammatical one. This theory is criticized by Buridan, for whom the
logical subject is always the grammatical subject.



Albert's semantics becomes innovative when he admits that
propositions have their own proper significate, which is not identical
to that of their terms (see especially his *Questions on the
Posterior Analytics* I, qq. 2, 7, 33). Like syncategorematic terms
(see his *Questions on the Categories*, qu. 1 'On
Names'), propositions signify the "mode of a thing
[*modus rei*]". This position is not repeated in the
*Logical Questions*. In any case, Albert avoids hypostatizing
these modes by explaining them as relations between the things to
which the terms refer. It cannot be said here that Albert is moving
towards the "complexly signifiable [*complexe
significabile*]" of Gregory of Rimini, although his remarks
are reminiscent of the latter theory. Still, he uses the idea of the
signification of a proposition to define truth and to explain
'insolubles', i.e., propositions expressing paradoxes of
self-reference. On Albert's view, every proposition signifies
that it is true by virtue of its form. Thus, an insoluble proposition
is always false because it signifies at the same time that it is true
and that it is false.



The *Questiones circa logicam* (*Questions on Logic*)
were written at roughly the same time as the *Logica* and the
*Questiones circa artem veterem*, that is to say about 1356.
They explore in a series of disputed questions the status of logic and
semantics on topics such as the relation of words to concepts, the
difference between natural and conventional signification, etc., as
well as the theory of reference and truth. Albert defines
signification by *representation*. He distinguishes two ways of
understanding *suppositio*, the first as the act of the mind
itself; the second as an operation constituting one of the properties
of terms.



In his *Sophismata,* Albert usually follows Heytesbury. The
distinction between compounded and divided senses, which is presented
in a highly systematic way in Heytesbury's *Tractatus de
sensu composito et diviso*, is the primary instrument (besides the
appellation of form) for resolving difficulties connected with
epistemic verbs and with propositional attitudes more generally. This
is abundantly clear in his discussion of infinity. Rather than
appealing to the increasingly common distinction between the
categorematic and syncategorematic uses of the term
'infinite' and then indicating the different senses it can
have depending on where it occurs in a proposition, he treats the
infinite itself as a term. Albert's approach involves analyzing
the logical and linguistic conditions of every proposition involving
the term 'infinite' that is significant and capable of
being true. This leads him to sketch a certain number of possible
definitions (where he appears to take into account the teachings of
Gregory of Rimini), as well as to raise other questions, e.g., on the
relation between finite and infinite beings (in propositions such as
'Infinite things are finite [*infinita sunt
finita*]'), on the divisibility of the continuum, and on
qualitative infinity. There are echoes in Albert not only of the
approach Buridan had systematically implemented in his
*Physics*, but also of the analyses of English authors --
again, especially Heytesbury. As is often the case, the treatment
proposed by Albert in the *Sophismata* provides good evidence
of the extent to which philosophers were gripped by questions about
infinity at that time.



Finally, one of the fields in which Albert is considered a major
contributor is the theory of consequences. In the treatise of the
*Perutilis Logica* devoted to consequences, Albert often seems
to follow Buridan. But whereas Buridan maintained the central role of
Aristotelian syllogistic, Albert, like Burley, integrated syllogistic
and the study of conversions into the theory of consequences.
Consequence is defined as the impossibility of the antecedent's
being true without the consequent's also being true --
truth itself being such that howsoever the proposition signifies
things to be, so they are. The primary division is between formal and
material consequences, the latter being subdivided into consequences
*simpliciter* and *ut nunc*. A syllogistic consequence
is a formal consequence whose antecedent is a conjunction of two
quantified propositions and whose consequent is a third quantified
proposition. Albert is thus led to present a highly systematic theory
of the forms of inference, which represents a major step forward in
the medieval theory of logical deduction.


## 3. Natural Philosophy and Mathematics



It is this analysis of language together with a particularist ontology
that places Albert in the tradition of nominalism. This is combined
with an epistemological realism that emerges, e.g., in his analysis of
the vacuum. In certain respects, Albert's work is an extension
of physical analysis to imaginary cases. Distinguishing, as Buridan
did, between what is absolutely impossible or contradictory and what
is impossible "in the common course of nature"
(*Questions on De Caelo* I, qu. 15), he considers hypotheses
under circumstances that are not naturally possible but imaginable
given God's absolute power (e.g., the existence of a vacuum and
the plurality of worlds). However, even if we can imagine a vacuum
existing by divine omnipotence, no vacuum can occur naturally
(*Questions on the Physics* IV, qu. 8). Albert refuses to
extend the reference of physical terms to supernatural, purely
imaginary possibilities. In the same way, one can certainly use the
concept of a point, although this would only be an abbreviation of a
connotative and negative expression. There is no simple concept of a
point, a vacuum, or the infinite, and although imaginary hypotheses
provide an interesting detour, physics must in the end provide an
account of the natural order of things.



Historically, Albert's reputation in natural philosophy is at
least as high as the one he had in logic. His commentaries on the
*Physics*, on *De caelo* or on *De generatione et
corruptione* are close to Oresme's and
Buridan's. He appeals to the authority of his "revered
masters from the Faculty of Arts at Paris" at the beginning of
his questions on *De caelo*. Even so, it should be noted that
his *Physics* was composed soon after 1351, before the final
version of Buridan's *Questions on the Physics* (between
1355 and 1358). On some points we know that Buridan modifies his
positions between the third and the final versions of his commentary,
and we can presume that this is in response sometimes to Oresme's
ideas, but sometimes also to Albert's.



On some other issues, the oppositions remain strong between the two
masters. We have already seen that on the question of the status of
the category of quantity, then at the border of logic and physics,
Albert followed Ockham and distanced himself from Buridan by reducing
quantity to a disposition of substance or quality. This move becomes
evident in certain physical questions, e.g., in the study of
condensation and rarefaction, where Albert openly disagrees with
Buridan by arguing that condensation and rarefaction are possible only
through the local motion of the parts of a body, and without needing
to assume some quantity that would have a distinct reality on its own.
Nevertheless, he defines the concept of a "lump of matter
[*materie massa*]" without giving it any autonomous
reality, although it does help fill out the idea of a 'quantity
of matter', which Giles of Rome had already distinguished from
simple extension.



Similarly, Albert is sometimes seen as standing alongside Ockham on
the nature of motion, rejecting the idea of motion as a flux
(*fluxus*), which is the position Buridan had adopted. In
contrast to Buridan, Albert treats locomotion in the same way as
alteration (movement according to quality): in neither case is it
necessary to imagine local motion as a *res successiva*
distinct from permanent things, at least if the common course of
nature holds and one does not take into account the possibility of
divine intervention.



Concerning the motion of projectiles, gravitational acceleration, and
the motion of celestial bodies, Albert's position is similar to
Buridan's major innovation, i.e., the theory of
*impetus*, a quality acquired by a moving body (see
Buridan's *Questions on the Physics* VIII, qu. 13, on
projectile motion). Like Buridan, he extends this approach to
celestial bodies in his commentary on *De caelo*, clearly
following its consequences in rejecting intelligences as agents of
motion and in treating celestial and terrestrial bodies using the same
principles. Nevertheless, he formulates the idea of *impetus*
in more classical terms as a *virtus impressa* (impressed
force) and *virtus motiva* (motive force). Albert makes no
pronouncements about the nature of this force, claiming that this is a
question for the metaphysician. His work also mentions the mean speed
theorem, a method of finding the total velocity of a uniformly
accelerated (or decelerated) body, which had been stated (though
without being demonstrated) in Heytesbury's *Tractatus de
motu*, and also adopted by Nicole Oresme. Albert was part of a
general scientific trend which sought the first formulations of the
principles of dynamics.



He shares with Buridan the theory according to which, to explain the
fact that a part of the earth is above the water, we have to
distinguish, for the earth, a centre of magnitude and a centre of
gravity, because of the evaporation of water, but also because of the
rarefaction of a part of the earth under the influence of the solar
heath (*Quaestiones in De caelo*, II, q. 25 and 28). He
explained also a number of curious natural phenomena, taking
particular interest in earthquakes, tidal phenomena, and geology.



Albert wrote a treatise on proportions dedicated to the analysis of
motion, highly influenced by the treatise *De proportionibus
velocitatum in motibus* of Thomas Bradwardine. He explains in a
synthetic way the elements of the theory of proportions, applying this
theory to different motions (local motion, alteration, augmentation,
and diminution). Motion is to be studied "from the point of view
of the cause" and "from the point of view of the
effect". Like Oresme, Albert adopts the idea that motion varies
according to a geometrical progression when the relation of motive
force to resistance varies arithmetically. His treatise is less
innovative than Oresme's, but it is a clear exposition that was
very widely read.



Albert was interested in certain mathematical problems. In addition to
authoritative arguments and purely empirical justifications, his
question on the squaring of the circle uses properly mathematical
arguments appealing to both Euclid (in the version of Campanus of
Novarra) and Archimedes (translated by Gerard of Cremona). His most
original contribution is a proposal to dispense with Euclid's
proposition X.1, replacing it with a postulate stating that if A is
less than B, then there exists a quantity C such that A<C<B.



Finally, a treatise *De maximo et minimo*, considering the
limits of active and passive potencies, written in the tradition of
the Oxford Calculators, at the frontiers of logic and natural
philosophy.


## 4. Impact and Influence



Albert of Saxony is not a compiler. Even if he shares many ideas with
the contemporary masters of arts, he is often original. His teachings
on logic and metaphysics were extremely influential. Although Buridan
remained the predominant figure in logic, Albert's
*Logica* was destined to serve as a popular text because of its
systematic nature and also because it takes up and develops essential
aspects of the Ockhamist position. But it was his commentary on
Aristotle's *Physics* that was especially widely read.
Many manuscripts of it can be found in France and Italy, but also in
Erfurt, Vienna and Prague. Thanks to Albert of Saxony, many new ideas
raised in Parisian physics and cosmology in the later Middle Ages
became widespread in Central Europe. Albert's *Physics*,
much more than Oresme's and even Buridan's, basically
guaranteed the transmission of the Parisian tradition also in Italy,
where it was authoritative along with the works of Heytesbury and John
Dumbleton. His commentary on Aristotle's *De caelo* was
also influential, eclipsing Buridan's commentary on this text.
Blasius of Parma read it in Bologna between 1379 and 1382. A little
later, it enjoyed a wide audience at Vienna. His *Treatise on
Proportions* was often quoted in Italy where, in addition to the
texts of Bradwardine and Oresme, it influenced the application of the
theory of proportions to motion.



Albert played an essential role in the diffusion throughout Italy and
central Europe of new ideas discussed in the Mid-Fourteenth century by
Parisian masters, but which were also clearly shaped by Albert's
own grasp of English innovations. He manifests an undeniable
originality on many topics in logic and physics.