## 1. Brief Biography



Galileo was born in Pisa on February 15, 1564. By the time he died on
January 8, 1642 (but for problems with the date, see Machamer 1998b,
24-25), he was as famous as any person in Europe. Moreover, when
he was born there was no such thing as 'science'; yet by
the time he died, science was well on its way to becoming a
discipline, and its concepts and method a complete philosophical
system.



Galileo's father Vincenzo, though of noble heritage, was a
semi-itinerant court musician and composer of modest means, who also
authored treatises on music theory; his mother, Giulia Ammannati,
descended from Pisan cloth merchants. In 1572, they resettled the
family in Florence. As a boy, Galileo was tutored privately and, for a
time, by the monks at Vallombrosa, where he considered a religious
vocation and may have started a novitiate. He returned home, however,
and then enrolled for a medical degree at the University of Pisa in
1580. He never completed this degree, but instead studied mathematics,
notably with Ostilio Ricci, a mathematics teacher attached to the
Tuscan court and the Florentine Accademia del Disegno.



After leaving university, Galileo worked as a private mathematics
tutor around Florence and Siena and cultivated the support of leading
mathematicians. He visited Christoph Clavius, professor at the Jesuit
Collegio Romano, and corresponded with the engineer Guildobaldo del
Monte, Marchese of Urbino. In 1588, he applied and was turned down for
a professorship in Bologna, but a year later, with the help of Clavius
and del Monte, he was appointed lecturer in mathematics at Pisa. In
1592, he obtained, at a much higher salary, a chair of mathematics at
the University of Padua, in the Venetian Republic. Galileo also
supplemented his income by producing a calculating instrument of his
own design (see Galilei 1606) and other devices in a household
workshop, and by private tutoring and consulting on practical
mathematics and engineering. During this period, he began a
relationship with Marina Gamba, and their daughter Virginia was born
in 1600. In 1601, they had another daughter, Livia, and a son,
Vincenzo, in 1606.



In Padua, Galileo worked out much of the mechanics he would publish
later in life, and which constitute his primary lasting
contribution to physical science. However, these projects were
interrupted in 1609, when Galileo heard about the recently invented
spyglass, invented an improved telescope, and used it to make
astounding celestial discoveries. He rushed these into print in
*Sidereus Nuncius* (*Starry Messenger*), which appeared
in March 1610 and launched Galileo onto the world stage. Among others,
Johannes Kepler, Imperial Mathematician at Prague, lauded the work
(Kepler 1610). Clavius and his colleagues at the Collegio Romano
confirmed its results and threw a celebratory banquet when Galileo
visited in 1611. During the same Roman sojourn, Galileo was admitted
to what was perhaps the first scientific society, the Accademia dei
Lincei; he would style himself "Lincean Academician" for
the rest of his life. Some fascinating treatments of this period
of Galileo's life and motivations have recently appeared
(Biagioli 2006; Reeves 2008; Wilding 2014).



Galileo also used the *Starry Messenger* to solicit patronage
in his native Tuscany, naming the moons of Jupiter he had found the
"Medicean" stars, in honor of the ruling Medici family.
His negotiations were ultimately successful, and Galileo moved to
Florence as "Chief Mathematician and Philosopher to the Grand
Duke" and holder of a sinecure professorship at Pisa. His
daughters moved with him and were shortly placed in the convent of
Saint Matthew at Arcetri, near Florence. Vincenzo and his mother,
Marina, were left behind in Venice.



Once a courtier, Galileo entered into several debates on scientific
topics. In 1612, he published a *Discourse on Floating Bodies*,
and in 1613, *Letters on Sunspots*, where he first openly
expressed support for Copernican heliocentrism. In 1613-14,
Galileo entered into discussions of Copernicanism through his student
Benedetto Castelli, and wrote a *Letter to Castelli*defending the doctrine from theological objections. Meanwhile, it
had become known that Copernicanism was under scrutiny by Church
authorities. Galileo lectured and lobbied against its condemnation,
expanding his *Letter to Castelli* into the widely
circulated *Letter to the Grand Duchess Christina* in 1615
and travelling to Rome late that year. Nevertheless, in March 1616,
Copernicus's *On the Revolutions of the Heavenly Orbs*was suspended (i.e., temporarily censored), pending correction,
by the Congregation of the Index of Prohibited Books. Galileo himself
was called to an audience with Cardinal Robert Bellarmine, a leading
theologian and member of the Roman Inquisition, who admonished him not
to teach or defend Copernican theory. (The details of this episode are
far from straightforward, and remain disputed even today. See Shea and
Artigas 2003; Fantoli 2005.)



In 1623, Galileo published *The Assayer*, which deals with
the nature of comets and argues they are sublunary phenomena. This
book includes some of Galileo's most famous methodological
pronouncements, including the claim that the book of nature is written
in the language of mathematics. It also contains passages suggestive
of atomism, a heretical doctrine, for which the book was referred to
the Inquisition, which dismissed the charge.



Also in 1623, Maffeo Barberini, Galileo's supporter and friend,
was elected Pope Urban VIII. Galileo felt empowered to begin work on
his *Dialogue Concerning the Two Chief World Systems*. The
"two systems" are the Ptolemaic and Copernican, and the
text clearly, though not explicitly, favors the latter. Printing was
completed in Florence by February 1632. Shortly afterwards, the
Inquisition banned its sale, and Galileo was ordered to Rome for
trial. In June 1633, Galileo was convicted of "vehement
suspicion of heresy," and a sentence of imprisonment was
immediately commuted to perpetual house arrest. (There is more about
these events and their implications in the final section of this
 article, Galileo and the Church.)



In 1634, while Galileo was confined to his villa in Arcetri, his
beloved eldest daughter died (Sobel 1999). Around this time, he began
work on his final book, *Discourses and Mathematical
Demonstrations Concerning Two New Sciences*, based on the
mechanics he had developed early in his career. The manuscript was
smuggled out of Italy and published in Holland by the Elzeviers in
1638. Galileo died early in 1642, and due to his condemnation, his
burial place was obscure until he was re-interred in 1737.



For detailed biographical material, the best and classic work dealing
with Galileo's scientific achievements is Stillman
Drake's *Galileo at Work* (1978). More
recently, J. L. Heilbron has written a magnificent
biography, *Galileo* (2010), that touches on all the
multiple facets of his life. 


## 2. Introduction and Background



From the seventeenth century onward, Galileo has been seen by many as
the "hero" of modern science. He is renowned for his
discoveries: he was the first to report telescopic observations of the
mountains on the moon, the moons of Jupiter, the phases of Venus, and
the rings of Saturn. He invented an early microscope and a predecessor
to the thermometer. In mathematical physics--a discipline he
helped create--he calculated the law of free fall, conceived of
an inertial principle, determined the parabolic trajectory of
projectiles, and advocated the relativity of motion. He is thought to
be the first "real" experimental scientist, who dropped
stones from towers and ships' masts, and played with magnets,
clocks, and pendulums (noting the isochrony of the latter). Much of
his cultural stature also arises from his advocacy and popularization
of Copernicanism and the resulting condemnation by the Catholic
Inquisition, which has made him a purported "martyr" to
the cause of rationality and enlightened modernity in the subsequent
history of a supposed "warfare" between science and
religion. This is no small set of accomplishments for one
seventeenth-century Italian, who was the son of a court musician and
who left the University of Pisa without a degree.



Momentous figures living in momentous times are full of interpretive
fecundity, and Galileo has been the subject of manifold
interpretations and much controversy. The use of Galileo's work
and the invocations of his name make a fascinating history
(Segre 1991; Palmerino and Thijssen 2004; Finocchiaro 2005; Shea
and Artigas 2006), but this is not our topic, which are the
philosophical implications of his work.



Philosophically, Galileo has been used to exemplify many different
themes, usually as a personification of whatever the writer wished to
make the hallmark of the Scientific Revolution or of the nature of
good science--whatever was good about the new science or science
in general, it was Galileo who started it. One tradition of Galileo
scholarship has divided Galileo's work into three or four parts:
(1) his physics, (2) his astronomy, and (3) his methodology, which
might include his method of Biblical interpretation and/or his
thoughts about the nature of proof or demonstration. In this
tradition, typical treatments deal with his physical and astronomical
discoveries and their background and/or who were Galileo's
predecessors. More philosophically, many ask how his mathematical
practice relates to his natural philosophy. Was he a mathematical
Platonist (Jardine 1976; Koyre 1978), an experimentalist
(Settle 1967; Settle 1983; Settle 1992; Palmieri 2008), an
Aristotelian emphasizing experience (Geymonat 1954), a precursor of
modern positivist science (Drake 1978), or maybe an Archimedean
(Machamer 1998a), who might have used a revised Scholastic method of
proof (Wallace 1992; Miller 2018)? Or did he have no method and just
fly like an eagle in the way that geniuses do (Feyerabend 1975)?
Alongside these claims there have been attempts to place Galileo in an
intellectual context that brings out the background to his
achievements. Some have emphasized his debt to the artisan/engineer
practical tradition (Rossi 1962; Valleriani 2010), others his
mathematics (Giusti 1993; Feldhay 1998; Renn, et al. 2000; Palmieri
2001; Palmieri 2003; Peterson 2011; Palmerino 2016), his mixed (or
subalternate) mathematics (Machamer 1978; Lennox 1986; Wallace 1992;
Dear 1995; Machamer 1998a), his debt to atomism (Shea 1972; Redondi
1983), his use of Hellenistic and Medieval impetus theory (Moody 1951;
Duhem 1954; Clagett 1959; Shapere 1974), or the idea that discoveries
bring new data into science (Wootton 2015).



Still, almost everyone working in this tradition seems to think the
three areas--physics, astronomy, and methodology--are
somewhat distinct and represent different Galilean endeavors. More
recent historical research has followed contemporary intellectual
fashion and shifted foci, bringing new dimensions to our understanding
of Galileo by studying his rhetoric (Finocchiaro 1980; Moss 1993;
Feldhay 1998; Spranzi 2004), the power structures of his social milieu
(Biagioli 1993; Biagioli 2006), his personal quest for acknowledgment
(Shea and Artigas 2003), and more generally emphasizing the larger
social and cultural history (Reeves 2008; Bucciantini, et al. 2015),
in particular the court and papal culture in which Galileo
functioned (Redondi 1983; Heilbron 2010).



In an intellectualist recidivist mode, this entry will outline his
investigations in physics and astronomy and exhibit, in a new way, how
these all cohered in a unified inquiry. In setting out this path, we
shall show why, at the end of his life, Galileo felt compelled (in
some sense of necessity) to write the *Two New Sciences*,
which stands as a true completion of his overall project and is not
just a reworking of his earlier research that he reverted to after his
trial, when he was under house arrest and going blind. Particularly,
we shall try to show why both of the two new sciences,
especially the first, were so important--a topic not much
treated except recently (Biener 2004; Raphael 2011). In
passing, we shall touch on his methodology and his mathematics, and
here refer you to some of the recent work by Palmieri (2001; 2003). At
the end, we shall add some words about Galileo, the Catholic Church,
and his trial.


## 3. Galileo's Scientific Story



The philosophical thread that runs through Galileo's
intellectual life is a strong and increasing desire to find a new
conception of what constitutes natural philosophy and how natural
philosophy ought to be pursued. Galileo signaled this goal clearly
when he left Padua in 1610 to return to Florence and the court of the
Medici. He asked for and received the designation
'Philosopher', in addition to 'Mathematician'.
This was not just a status-affirming request, but also a reflection of
his programmatic aims. What Galileo accomplished by the end of his
life was a reasonably articulated replacement for the traditional set
of analytical concepts connected with the Aristotelian tradition of
natural philosophy. He offered, in place of the Aristotelian
categories, a set of mechanical archetypes that were accepted by most
everyone who afterwards developed the "new sciences," and
which, in some form or another, became the hallmark of the new
philosophy. His way of thinking became the way of the Scientific
Revolution (and yes, there was such a revolution, *pace* Shapin
1996 and others; see the selections in Lindberg and Westman 1990;
Osler 2000).



Some scholars might wish to describe what Galileo achieved in
psychological terms, as an introduction of new mental models (Palmieri
2003) or a new model of intelligibility (Machamer 1994; Machamer,
1998a; Adams, et al. 2017). However phrased, Galileo's main move
was to dethrone the Aristotelian physical categories; namely, the one
celestial element (*aether*, or quintessence--i.e.,
"fifth element") and the four terrestrial ones (fire, air,
water, and earth), along with their respective motive natures
(circular, and up and down). In their place, he left only one
element, corporeal matter, whose properties and motions he described
using the mathematics of proportional relations typified by the
Archimedian simple machines--the balance, the inclined plane, and
the lever--to which Galileo added the pendulum (Machamer 1998a;
Machamer and Hepburn 2004; Palmieri 2008). In doing so, Galileo
changed the acceptable way of talking about matter and its motion, and
so ushered in the mechanical tradition that characterizes so much of
modern science, even today. See Dijksterhuis 1961; Machamer, et al.
2000; Gaukroger 2006; Roux and Garber 2013.



As a way of understanding Galileo's accomplishments, it is
useful to see him as being interested in finding a unified theory of
matter--a mathematical theory of the material stuff that
constitutes the whole of the cosmos. Perhaps he did not realize that
this was his grand project until the time he actually wrote the
*Two New Sciences* in the mid-1630s. Despite working on
problems of the nature of matter from 1590 onwards, he could not have
written his final work much earlier than 1638; certainly not
before the *Starry Messenger* of 1610, and probably
not before the *Dialogue Concerning the Two Chief World
Systems* of 1632. He had thought deeply about the nature of
matter before 1610 and had tried to work out how best to describe
matter, but before 1632, he did not have the theory and evidence he
needed to support his claims about a unified, singular matter. The
idea of unified matter theory had to wait for the
establishment of principles of matter's motion on a moving
Earth. And this he did not accomplish until
the *Dialogue*.



Galileo began his critique of Aristotle in a treatise he drafted
around 1590, titled *De Motu* (*On Motion*). The
first part of this manuscript deals with terrestrial matter and argues
that Aristotle's theory has it wrong. For Aristotle, the matter
of the terrestrial realm within the sphere of the moon is of four
elemental kinds--earth, water, air, and fire. These possess two
formal principles that give rise to their natural motion: heaviness
(*gravitas*; in earth and water) and lightness
(*levitas*; in air and fire). Galileo, using an Archimedean
model of floating bodies, and later the balance, argues that there is
only one principle of motion--heaviness. Bodies move upward not
because they have a natural lightness, he says, but because they are
displaced or extruded by other heavier bodies moving downward. So on
his view, heaviness is the cause of all natural terrestrial
motion.



This move left Galileo with a problem: what is heaviness and how is it
to be described? In *De Motu*, he argued that the moving
arms of a balance could be used as a model for treating all problems
of natural motion. In this model, heaviness is the proportionality of
the weight of an object on one arm of the balance to the weight of
another body on the other arm. In the context of floating bodies,
heaviness is the weight of one body minus the weight of the medium.
Galileo quickly realized these characterizations were insufficient,
and so began to explore how heaviness might be related to specific
gravities; i.e., the comparative weights of bodies having equal
volume. He was trying to figure out the concept of heaviness that is
characteristic of all matter. What he failed to work out--and
this was probably the reason why he never published *De
Motu*--was this positive characterization of heaviness. There
seemed to be no way to find a standard measure of heaviness that would
work across different substances. At this point, he did not have a
useful replacement for Aristotelian *gravitas*.



A while later, in his 1600 manuscript version of *Le
Meccaniche* (*On Mechanics*), Galileo introduced the
concept of *momento*, a *quasi* force
that applies to a body at a moment, and which is somehow proportional
to weight or specific gravity (Galluzzi 1979). Still, he had no good
way to measure or compare specific gravities of bodies of different
kinds, and his notebooks during this early seventeenth-century period
reflect his trying again and again to find a way to bring all matter
under a single proportional measuring scale. He tried to study
acceleration along an inclined plane and to find a way to think of
what changes acceleration brings to *momento*. Yet the details
and categories of how to properly treat weight and movement eluded
him.



We see from this period that Galileo's law of free fall arises
out of this struggle to find the proper categories for his new science
of matter and motion. Galileo accepted, probably as early as the 1594
draft of *Le Meccaniche*, that natural motions might be
accelerated. Particularly in the cases of the pendulum, the inclined
plane, free fall, and projectile motion, Galileo must have observed
that the speeds of bodies increase as they move downwards and,
perhaps, do so naturally. But that accelerated motion is properly
measured against time is an idea he realized only later, chiefly
through his failure to find any satisfactory dependence on place and
specific gravity. Also at this time, he began to think about
percussive force. For many years, he thought that the correct science
of these phenomena should describe how bodies change according to
where they are on their paths. Specifically, it seemed that height is
crucial. Percussive force is directly related to height and the motion
of the pendulum seems to involve equilibrium with respect to the
height of the bob (and time also, but isochrony did not lead directly
to a recognition of time's importance).



One of Galileo's problems was that the Archimedian simple
machines he was using as his models of intelligibility, especially the
balance, are not easily conceived of in a dynamic way (but see
Machamer and Woody 1994). Since they generally work by establishing
static equilibrium, time is not a feature of their action one would
normally attend to. In discussing a balance, for instance, one does
not normally think about how fast an arm of the balance descends, nor
how fast a body on the opposite arm is rising (though Galileo does in
his *Postils to Rocco* circa 1634-45; see
Palmieri 2005). The converse is also true. It is difficult to model
dynamic phenomena that involve rates of change as balance arms moving
upwards or downwards because of differential weights. So it was that
Galileo's puzzle about how to describe time and the force of
percussion (the force of body's impact) would remain unsolved.
Throughout his life, he could not find systematic relations among
specific gravities, heights of fall, and percussive forces. Even while
the *Two New Sciences* was going to press in 1638, Galileo was
laboring on an additional "Fifth Day" (not published until
1718) that presciently explored the concept of the force of
percussion, which would become, after his death, one of the most
fecund ways to think about matter and its motion.



In the period 1603-9, Galileo experimented with inclined planes
and, most importantly, pendulums. These studies again exhibited to
Galileo that acceleration and, therefore, time is a crucial variable.
Moreover, the isochrony of the pendulum--the period depends only
on the length of the cord, regardless of the weight of the
bob--went some way towards showing that time is a possible term
in the equilibrium (or ratio) that needs to be made explicit to
represent motion. It also shows that, in at least one case, time can
displace weight as a crucial variable.



The law of free fall--i.e., a body in free fall from rest
traverses a distance proportional to the square of the time
elapsed--was discovered by Galileo through the inclined plane
experiments (Drake 1999, v. 2). At first, Galileo attempted to
represent this phenomenon with a velocity-distance relation, and the
equivalent mean proportional relation. His later and correct
definition of natural acceleration dependent on time was an insight
gained through recognizing the physical significance of that mean
proportional relation (Machamer and Hepburn 2004; for a different
analysis of Galileo's discovery of free fall see Renn, et al.
2000). Yet Galileo would not publish anything making time central to
his analysis of motion until 1638, in the *Two New
Sciences*.



In 1609, Galileo began his work with the telescope. There are many
ways to describe Galileo's findings, and many interpreters have
taken this to be an interlude irrelevant to his physics. However,
they are remarkable insofar as they are his start at dismantling the
celestial-terrestrial distinction entrenched in Aristotelian cosmology
(Feyerabend 1975). Perhaps the most unequivocal case of this is when
he analogizes the mountains on the moon to mountains in Bohemia in the
*Starry Messenger*. Also crucial was his discovery of the four
moons circling Jupiter, which lent credence to the Copernican system
since it meant that a planet-moon arrangement was not unique to the
Earth. The abandonment of the dichotomy between heavens and earth
implied that all matter, whether celestial or terrestrial, is of the
same kind. Further, if there is only one kind of matter, there can be
only one kind of natural motion--one kind of motion that this
matter has by nature. So it has to be that one law of motion will hold
throughout the terrestrial and celestial realms. This is a far
stronger claim than he had made in 1590, which concerned only the
terrestrial elements.



A few years later, in his *Letters on
Sunspots* (1613), Galileo offered new telescopic evidence
that supported the Copernican theory. But these observations also
served as additional reasons for dissolving the celestial-terrestrial
distinction. One was that the sun is not an immutable aetherial
sphere, but has changing spots (*maculae*) on its surface.
Another was that the sun rotates circularly around its axis, like the
Earth. A third was the discovery that Venus undergoes a full sequence
of phases (like the moon), which entails that Venus revolves around
the sun, and suggests that the Earth is likewise a celestial body
moving around the sun. Certainly the phases of Venus contradicted the
Ptolemaic ordering of the planets.



Later, in 1623, Galileo argued for a quite mistaken material thesis.
In *The Assayer*, he tried to show that comets are
sublunary phenomena and that their properties could be explained by
optical refraction. While this work stands as a masterpiece of
scientific rhetoric, it is somewhat strange that Galileo should have
argued against the superlunary nature of comets, which the great
Danish astronomer Tycho Brahe had demonstrated earlier.



Yet even with all these developments, Galileo still needed to work out
general principles concerning the nature of motion for this newly
unified matter. In this respect, Galileo differed from Ptolemy (at
least of the *Almagest*), Copernicus, or even Tycho Brahe, who
treated their planetary systems--be they Earth- or
sun-centered--merely as models of the planets' observed
motions; that is, as mathematical conceits for calculating observable
positions. For Galileo, by contrast, Copernicanism was also a
commitment to a physically realizable cosmography. Consequently, he
needed to work out, at least qualitatively, a way of thinking about
the actual motions of matter. He had to devise (or shall we say,
discover) principles of local motion that would fit a central sun,
planets moving around that sun, a whirling Earth, and everything on
it.



This he did by introducing two new principles. In Day One of
his *Dialogue Concerning the Two Chief World
Systems* (1632), Galileo argues that matter will move
naturally along circular trajectories, neither speeding up nor slowing
down. Then, in Day Two, he introduces his version of the famous
principle of the relativity of observed motion. This latter holds that
observers cannot detect uniform motions they share with objects under
observation; only differential motion can be seen. Of course, neither
of these principles was entirely original with Galileo. They had
predecessors. But no one needed them for the reasons that he did,
namely that they were necessitated by a unified cosmological
matter.



One key effect of these principles is that the diurnal terrestrial
rotation asserted by the Copernican system is unobservable. The Earth
and all the objects on it naturally move in circles around the
Earth's axis once a day, but since human observers share this
motion, it cannot be detected. We only notice departures from shared
rotation, such as bodies falling or rising. Consequently, "all
experiments practicable upon the Earth are insufficient
measures" for proving its stability or its mobility,
"since they are indifferently adaptable to an Earth in motion or
at rest" (Galilei 1967, 6). This blunted standard objections to
Copernicanism on the grounds that there is no evidence of terrestrial
motion.



Having dispelled these arguments against the Copernican system,
Galileo then dramatically argues in its favor. In Day Three of the
*Dialogue*. Salviati, Galileo's avatar, has Simplicio,
the ever-astounded Aristotelian, make use of astronomical
observations, especially the facts that Venus has phases and that
Venus and Mercury are never far from the sun, to construct a diagram
of the planetary positions. The resulting diagram neatly corresponds
to the Copernican model. Then in Day Four, Galileo offers a supposed
proof of Copernican theory on the basis of the tides, asserting that
they result from the combination of the Earth's diurnal rotation
and its annual motion around the sun.



In the *Dialogue*, things are more complicated than we
have just sketched. Galileo, as noted, argues for circular natural
motion. Yet he also introduces, in places, an intrinsic tendency for
rectilinear motion. For example, Galileo recognizes that a stone
whirled circularly in a sling would fly off along the rectilinear
tangent if released (Galilei 1967, 189-94; see Hooper 1998).
Further, in Day Four, when he is giving his mechanical explanation of
the tides, he nuances his matter theory by attributing to water an
additional power of retaining an impetus for motion such that it can
generate a reciprocal movement once it is sloshed against a side of a
basin. This was not Galileo's first dealing with water. We saw
it first in the *De Motu* around 1590, where Galileo
discusses submerged and floating bodies, but he learned much more in
his dispute over floating bodies (which produced the *Discourse on
Floating Bodies* in 1612). In fact, a large part of that debate
turned on the exact nature of water as matter, and what kind of
mathematical proportionality could be used to correctly describe it
and bodies moving in it (see Palmieri 1998; 2004).



The final chapter of Galileo's scientific story came in 1638,
with the publication of the *Two New Sciences*. The second
science, discussed in the last two Days, deals with the principles of
local motion and has been much commented upon in the Galilean
literature. But the first science, discussed in the first two Days,
has been misunderstood and infrequently discussed. It has misleadingly
been called the science of the strength of materials, and so seems to
have found a place in history of engineering, since such a course is
still taught today. However, this science is not about the strength of
materials *per se*. It is Galileo's attempt to
provide a mathematical science of his unified matter (see Machamer
1998a; Biener 2004; Machamer and Hepburn 2004). Galileo realizes that,
before he can work out a science of the motion of matter, he must have
some way of showing that the nature of matter may be mathematically
characterized. Both the mathematical nature of matter and the
mathematical principles of motion he believes belong to the science of
"mechanics," which is the name he gives for this new way
of philosophizing.



So it is in Day One that Galileo begins to discuss how to describe
mathematically (or geometrically) the causes of the breaking of beams.
But this requires a way to reconcile mathematical description with the
physical constitution of material bodies. In this vein, Galileo
rejects using finite atoms as a basis for physical discussion, since
they are not representable by continuously divisible mathematical
magnitudes. Instead, he treats matter as composed of infinitely many
indivisible--which is to say, mathematical--points. This
allows him to give mathematical accounts for various properties of
matter. Among these are the density of matter, its coherence in
material bodies, and the properties of the resisting media in which
bodies move. The Second Day lays out the mathematical principles
concerning how bodies break. Galileo does all this by reducing the
problems of matter to problems of how a lever and a balance function,
which renders them mathematically tractable via the law of the lever.
He had begun this back in 1590, though this time he believes he is
getting it right, showing mathematically how bits of matter solidify
and stick together, and how they break into bits.



The First Day also contains Galileo's account of the
acceleration of falling bodies, and the argument that they fall
equally fast in a vacuum, whatever their weight. This discussion
contains the famous thought experiment refuting the Aristotelian
theory of fall, according to which the speed of a body's fall is
proportional to its weight. In this "short and conclusive"
argument, Galileo supposes that two bodies, one heavier than the
other, are suddenly conjoined in the midst of falling. On the one
hand, if Aristotle is correct, the faster fall of the heavier body
will be retarded by the slower motion of the lighter body, so that the
conjoined body will fall slower than the original heavy body. And yet,
the conjoined body is heavier than either original body, so it should
also fall faster. Hence, there is a contradiction in the Aristotelian
position (Gendler 1998; Palmieri 2005; Brown and Fehige 2019).



Galileo's second new science, in Days Three and Four of the Two
New Sciences, deals with the mathematical description of local motion
and the laws governing it. This is now the motion of all matter, not
just sublunary stuff, and the treatment takes the categories of time
and acceleration as basic. Here is where Galileo enunciates his law of
free fall, the parabolic path of projectiles, and other physical
"discoveries" that would lay the foundation for modern
physics (Drake 1999, v. 2).



In the projected Fifth Day, Galileo would have treated the power of
moving matter to act by impact, which he calls the force of
percussion. Ultimately, Galileo was unable to give mathematical
principles of this kind of interaction, but this problem subsequently
became an important locus of interest. Rene Descartes, probably
following Isaac Beeckman, would eventually convert the problem into
the task of finding equilibrium between the forces conserved by
colliding bodies. Descartes's own mathematical treatment was
mistaken, but correct principles were given in 1668-9 by
Christiaan Huygens, John Wallis, and Christopher Wren.



The sketch above provides the basis for understanding Galileo's
career. He offered a new science of matter, a new physical
cosmography, and a new science of local motion. In all these, he used
a mathematical mode of description based upon, though somewhat changed
from, the proportional geometry of Book VI of Euclid's
*Elements* and of Archimedes (for details on the changes, see
Palmieri 2001).



It is in this way that Galileo developed the categories of the
mechanical new science, the science of matter and motion. His new
categories utilized some of the basic principles of traditional
mechanics, to which he added the category of time and so emphasized
acceleration. But throughout, he was working out the details about the
nature of matter so that it could be understood as uniform and
universal, and treated in a way that allowed for coherent discussion
of the principles of motion. It was due to Galileo that a unified
matter became accepted and its nature became one of the problems for
the new science that followed. After him, matter really mattered.


## 4. Galileo and the Church



No account of Galileo's importance to philosophy can be complete
if it does not discuss the Galileo Affair--the sequence of
interactions with the Church that resulted in Galileo's
condemnation. The end of the affair is simply stated. In late 1632, in
the aftermath of the publication of the *Dialogue Concerning the
Two Chief World Systems*, Galileo was ordered to appear in Rome to
be examined by the Congregation of the Holy Office; i.e., the
Inquisition. In January 1633, a very ill Galileo made an arduous
journey to Rome. From April, Galileo was called four times to
hearings; the last was on June 21. The next day, June 22, 1633,
Galileo was taken to the church of Santa Maria sopra Minerva, and
ordered to kneel while his condemnation was read. He was declared
guilty of "vehement suspicion of heresy," and made to
recite and sign a formal abjuration:



> 
> 
> I have been judged vehemently suspect of heresy, that is, of having
> held and believed that the sun in the center of the universe and
> immoveable, and that the Earth is not at the center of same, and that
> it does move. Wishing however, to remove from the minds of your
> Eminences and all faithful Christians this vehement suspicion
> reasonably conceived against me, I abjure with a sincere heart and
> unfeigned faith, I curse and detest the said errors and heresies, and
> generally all and every error, heresy, and sect contrary to the Holy
> Catholic Church. (Quoted in Shea and Artigas 2003, 194)
> 
> 
> 



Tradition, but not historical fact, holds that, after abjuring,
Galileo mumbled, "*Eppur si muove* (and yet it
moves)." He was sentenced to "formal imprisonment at
the pleasure of the Inquisition," but this was commuted to house
arrest, first in the residence of the Archbishop of Siena, and then,
from December 1633, at his villa in Arcetri. When he later finished
his last book, the *Two New Sciences* (which does not mention
Copernicanism at all), it had to be printed in Holland, and Galileo
professed amazement at how it could have been published.



The details and interpretations of these proceedings have long been
debated, and it seems that each year we learn more about what actually
happened. One point of controversy is the legitimacy of the charges
against Galileo, both in terms of their content and the judicial
procedure. Galileo was charged with teaching and defending the
Copernican doctrine that holds the sun is at the center of the
universe and the Earth moves. The status of this doctrine was cloudy.
In 1616, an internal commission of the Inquisition had determined that
it was heretical, but this was not publicly proclaimed. Instead,
Copernicus's book had been placed on the *Index of Prohibited
Books*--the list of books Catholics were forbidden to read
without special permission--with the status "suspended
until corrected." Even more confusingly, the
requisite corrections had appeared in 1620, but the book
nevertheless remained on the *Index* until 1835. In fact, the
Church's first *public* pronouncement that Copernicanism
is a heresy appears in Galileo's condemnation.



Galileo's own status was also problematic. In 1616, at the same
time that the Inquisition was evaluating Copernicanism, they were also
investigating Galileo personally--a separate proceeding of which
Galileo himself was not likely aware. The outcome was
Bellarmine's admonition not to "defend or hold" the
Copernican doctrine. This "charitable admonition" may (or
may not) have been followed by a "formal injunction"
"not to hold, teach, or defend it in any way whatever, either
orally or in writing." When the records of this disposition of
the 1616 case were discovered in 1633, it made Galileo appear guilty
of recidivism, having violated the Inquisition's injunction by
publishing the *Dialogue*.



To confound issues further, the case against Galileo transpired in a
fraught political context. Galileo was a creature of the powerful
Medici and a personal friend of Pope Urban VIII, connections that
significantly modulated developments (Biagioli 1993). There were also
pressures stemming from the Counter Reformation, the Thirty Years War,
and resulting tensions within Urban's papacy (McMullin 2005;
Miller 2008). It has even been argued (Redondi 1983), that the charge
of Copernicanism was the effect of a plea bargain meant to cover up
Galileo's genuinely heretical atomism, though this latter
hypothesis has not found much support.



The legitimacy of the underlying condemnation of Copernicus on
theological and rational grounds is even more problematic. Galileo had
addressed this problem in 1615, when he wrote his *Letter to
Castelli* and then the *Letter to the Grand Duchess
Christina*. In these texts, Galileo argues that there are two
truths: one derived from Scripture, the other from the created natural
world. Since both are expressions of the divine will, they cannot
contradict one another. However, Scripture and Creation both require
interpretation in order to glean the truths they
contain--Scripture because it is a historical document written
for common people, and thus accommodated to their understanding so as
to lead them towards true religion; Creation because the divine act
must be distilled from sense experience through scientific enquiry.
While the truths are necessarily compatible, biblical and natural
interpretations can go awry, and are subject to correction.



Much philosophical controversy, before and after Galileo's time,
revolves around this doctrine of the two truths and their seeming
incompatibility. Which of course, leads us immediately to such
questions as: "What is truth?" and "How is truth
known or shown?"



Cardinal Bellarmine was willing to countenance scientific truth if it
could be proven or demonstrated (McMullin 1998). But Bellarmine held
that the planetary theories of Ptolemy and Copernicus (and presumably
Tycho Brahe) are only mathematical hypotheses; since they are just
calculating devices, they are not susceptible to physical proof. This
is a sort of instrumentalist, anti-realist position (Machamer 1976;
Duhem 1985). There are any number of ways to argue for some sort of
instrumentalism. Duhem (1985) himself argued that science is not
metaphysics, and so only deals with useful conjectures that enable us
to systematize phenomena. Subtler versions of this position, without
an Aquinian metaphysical bias, have been argued subsequently and more
fully by Van Fraassen (1980) and others. Less sweepingly, it can
reasonably be argued that both Ptolemy and Copernicus's theories
were primarily mathematical, and that Galileo was defending not
Copernicus's theory *per se*, but the physical
realization of it. In fact, it might be better to say the Copernican
theory that Galileo was constructing was a physical realization of a
simplified version of Copernicus's theory, which dispensed with
many of the technical details (eccentrics, epicycles, Tusi couples and
the like). Galileo would be led to such a view by his concern with
matter theory, which minimized the kinds of motion ascribed uniformly
to all bodies. Of course, when put this way, we are faced with the
question of what constitutes identity conditions for a theory. Still,
there is clearly a way in which Galileo's Copernicanism is not
Copernicus's.



The other aspect of all this that has been hotly debated is what
constitutes proof or demonstration of a scientific claim. Galileo
believed he had a proof of terrestrial motion. This argument
concerning the cause of the tides is contained in *On the Ebb and
Flow of the Tides*, a manuscript he composed in 1616 while
Copernicanism was under the Inquisition's scrutiny, and the main
thrust of which appears in Day Four of the *Dialogue Concerning the
Two Chief World Systems*.



In the first place, Galileo restricts the possible class of causes of
the tides to mechanical interactions, and so rules out Kepler's
attribution of the cause to the moon. How could the moon cause the
tides to ebb and flow without any connection to the seas? Such an
explanation would be an invocation of magic or occult powers. Thus,
for Galileo, the only conceivable (or maybe plausible) physical cause
for the regular reciprocation of the tides is the combination of the
diurnal and annual motions of the Earth. Briefly, as the Earth rotates
around its axis, some parts of its surface are moving along with the
annual revolution around the sun and some parts are moving in the
contrary direction. (In the same way that a point near the top of a
car's wheel is rotating in the same direction as the car is
moving, while a point near the ground is rotating toward the rear.) In
the frame of the fixed stars, this creates accelerations and
retardations of the Earth's surface, and since the terrestrial
waters are not attached to the surface, they slosh back and forth as
their basins speed up and slow down. Hence the tides. Moreover, since
the Earth's diurnal and annual rotations are regular, so are the
tidal periods. Local differences in tidal flows are due to the
differences in the physical conformations of the basins in which they
occur (for background and more detail, see Palmieri 1998). Albeit
mistaken, Galileo's commitment to mechanically intelligible
causation makes this is a plausible argument. One can see why Galileo
thinks he has some sort of proof for the motion of the Earth, and
therefore for Copernicanism. 



Yet one can also see why Bellarmine and the instrumentalists would not
have been impressed. First, they did not accept Galileo's
restriction of possible causes to mechanically intelligible causes.
Second, Galileo's explanation is not precise; it does not
account for many details of tidal motion. Most significantly, the
motion of the Earth's surface varies over twelve hours, not the
six-hour cycle of the tides. Third, the argument does not touch upon
the central position of the sun or arrangement of the planets as
calculated by Copernicus. So at best, Galileo's argument is an
inference to the best partial explanation from a limited set of
features of Copernicus's theory. Meanwhile, there were
compelling considerations about the size of celestial bodies that
weighed against the Copernican cosmology, stemming from a lack of
understanding of the telescope's optics (Graney 2015).



Nevertheless, when the tidal argument is added to the earlier
telescopic observations that show the improbabilities of the older
celestial picture--the fact that Venus has phases like the moon
and so must revolve around the sun; the principle of the relativity of
perceived motion which neutralizes the physical arguments against a
moving Earth; and so on--it was enough for Galileo to believe
that he had the necessary proof to convince the doubters.
Unfortunately, it was not until after Galileo's death and the
acceptance of a unified material cosmology, utilizing the
presuppositions about matter and motion that were published in
the *Two New Sciences*, that people were ready for
such proofs. But this could occur only after Galileo had changed the
acceptable parameters for gaining knowledge and theorizing about the
world.



To read many of the documents of Galileo's trial, see
Finocchiaro 1989; Mayer 2012. To understand the long, tortuous, and
fascinating aftermath of the Galileo affair see Finocchiaro
2005; and for Pope John Paul II's 1992 rehabilitation of
Galileo, see Coyne 2005.