## 1. Life and Works



Johannes Kepler was born on December 27, 1571 in Weil der Stadt, a
little town near Stuttgart in Wurttemberg in southwestern
Germany. Unlike his father Heinrich, who was a soldier and mercenary,
his mother Katharina was able to foster Kepler's intellectual
interests. He was educated in Swabia; firstly, at the schools Leonberg
(1576), Adelberg (1584) and Maulbronn (1586); later, thanks to support
for a place in the famous *Tubinger Stift*, at the
University of Tubingen. Here, Kepler became *Magister
Artium* (1591) before he began his studies in the Theological
Faculty. At Tubingen, where he received a solid education in
languages and in science, he met Michael Maestlin, who introduced him
to the new world system of Copernicus (see *Mysterium
Cosmographicum*, trans. Duncan, p. 63, and KGW 20.1, VI, pp.
144-180).



Before concluding his theology studies at Tubingen, in
March/April 1594 Kepler accepted an offer to teach mathematics as the
successor to Georg Stadius at the Protestant school in Graz (in
Styria, Austria). During this period (1594-1600), he composed
many official calendars and prognostications and published his first
significant work, the *Mysterium Cosmographicum* (= MC), which
catapulted him to fame overnight. On April 27, 1597 Kepler married his
first wife, Barbara Muller von Muhleck. As a consequence of
the anti-Protestant atmosphere in Graz and thanks also to the positive
impact of his MC on the scientific community, he abandoned Graz and
moved to Prague in 1600, to work under the supervision of the great
Danish astronomer Tycho Brahe (1546-1601). His first contact
with Tycho was, however, extremely traumatic, particularly because of
the Ursus affair (see below Section 4.1). After Tycho's
unexpected death on October 1601 Kepler succeeded him as Imperial
Mathematician. During his time in Prague, Kepler was particularly
productive. He completed his most important optical works,
*Astronomiae pars* *optica* (=APO) and
*Dioptrice* (=D), published several treatises on astrology
(*De fundamentis astrologiae certioribus*, *Antwort auf
Roeslini Diskurs; Tertius interveniens*), discussed
Galileo's telescopic discoveries (*Dissertatio cum nuncio
sidereo*), and composed his most significant astronomical work,
the *Astronomia nova* (=AN), which contains his first two laws
of planetary motion.



On August 3, 1611 Kepler's wife, Barbara Muller, died. In
1612 he moved to Linz, in Upper Austria, and became a professor at the
*Landschaftsschule*. There, he served as Mathematician of the
Upper Austrian Estates from 1612 to 1628. In 1613, he married Susanne
Reuttinger, with whom he had six children. In 1615, he completed the
mathematical works *Stereometria doliorum* and *Messekunst
Archimedis*. At the end of 1617 Kepler successfully defended his
mother, who had been accused of witchcraft. In 1619 he published his
principal philosophical work, the *Harmonice mundi* (=HM), and
wrote, partially at the same time, the *Epitome astronomiae
copernicanae* (=EAC). In 1624 Kepler continued his investigations
on mathematics, publishing his work on logarithms (*Chilias
logarithmorum*...).



In pursuit of an accurate printer for the *Tabulae
Rudolphinae*, he moved to Ulm near the end of 1626 and remained
there until the end of 1627. In July 1628 he went to Sagan to enter
the service of Albrecht von Wallenstein (1583-1634). He died on
November 15, 1630 at Regensburg, where he was to present his financial
claims before the imperial authorities (for Kepler's life,
Caspar's biography (1993) is still the best work. KGW 19
contains biographically relevant documents).


## 2. Philosophy, theology, cosmology



There is probably no such thing as "Kepler's
philosophy" in any pure form. Nevertheless, many attempts to
deal with the "philosophy of Kepler" have been made, all
of which are very valuable in their own way. Some studies have
concentrated on a particular text (see, for instance, Jardine 1988,
for the *Defense of Tycho against Ursus*), or have followed
some particular ideas of Kepler over a longer period of his life and
scientific career (see, for instance, Martens 2000, on Kepler's
theory of the archetypes). Others have tried to determine from a
philosophical point of view his place in the development of the
astronomical revolution from the 15th to the
17th centuries (Koyre 1957 and 1961) or in the more
general context of the scientific movement of the 17th
century (Hall, 1963 and especially Burtt, 1924). Still others have
discussed a long list of philosophical principles operating in
Kepler's scientific world, and have claimed to have found, by
means of such an analysis, compelling evidence for the interaction
between science, philosophy, and religion (Kozhamthadam, 1994). If, in
the particular case of Kepler, philosophy is immediately related to
astronomy, mathematics and, finally, "cosmology" (a notion
which arises much later), the core of these speculations is to be
sought in the spectrum of problems with which he dealt in his
*Mysterium Cosmographicum* and *Harmonice Mundi* (on
this topic, Field 1988 is one of the most representative works on
Kepler). In addition, because of the particular circumstances of his
life and his fascinating personality and genius, the literature on
Kepler is extremely wide-ranging, covering a spectrum from literary
pieces like Max Brod's *Tycho Brahes Weg zu Gott*
(1915)--though still not free from mistakes concerning
Kepler--to general introductions in the genre of historical
novels, and even fictional stories and charlatanry on astrology or,
running for some years now, portraying him as the assassin of Tycho
Brahe. However, according to recent reports, it is still a matter of
controversy whether Tycho was assassinated at all (see the
 report at phys.org).



Kepler mastered, like the best scientists, the most complicated
technical issues, especially in astronomy, but he always emphasized
his philosophical, even theological, approach to the questions he
dealt with: God manifests himself not only in the words of the
Scriptures but also in the wonderful arrangement of the universe and
in its conformity with the human intellect. Thus, astronomy represents
for Kepler, if done philosophically, the best path to God (see
Hubner 1975; Methuen 1998 and 2009; Jardine 2009, Kirby 2019). As
Kepler at the core of his greatest astronomical work confesses (AN,
Part II, chap. 7, KGW 6, p. 108, Engl. trans., p. 183), at the
beginning of his career he "was able to taste the sweetness of
philosophy ... with no special interest whatsoever in
astronomy." And, even in his later work, after having calculated
many ephemerides and different astronomical data, Kepler writes in a
letter of February 17, 1619 to V. Bianchi: "I also ask you, my
friends, that you do not condemn me to the treadmill of mathematical
calculations; allow me time for philosophical speculation, my only
delight!" (KGW 17, let. Ndeg 827, p. 327, lin. 249-
51).



Especially where Kepler deals with the geometrical structure of the
cosmos, he always returns to his Platonic and Neoplatonic framework of
thought. Thus, the polyhedral hypothesis (see Section 3 below) he
postulated for the first time in his MC represents a kind of
"formal cause" constituting the foundational structure of
the universe. In addition, an "efficient cause," which
realizes this structure in the corporeal world, is also needed. This
is, of course, God the Creator, who accomplished His work according to
the model of the five regular polyhedra. Kepler reinterprets the
traditional statements about the Creation as an image of the Creator
giving to the ancient ideas a more systematic and a quantitative
character. Even the doctrine of the Trinity could be geometrically
represented, taking the center for the Father, the spherical surface
for the Son, and the intermediate space, which is mathematically
expressed in the regularity of the relationship between the point and
the surface, for the Holy Spirit. In Kepler's model, we have to
be able to reduce all appearances to straightness and curvature as
providing the foundation for the geometrical structure of the
world's creation. The very first category, through which God
produced a fundamental similitude of the created World to Himself, is
that of quantity (see MC, chapter 2, KGW pp. 23-26).
Furthermore, quantity was also introduced into the human soul for the
specific purpose that this fundamental symmetry could be apprehended
and known scientifically.



This kind of speculation also belongs to the basic principles of
Kepler's philosophical optics. In Chapter 1 of APO ("On
the Nature of Light"), Kepler gives a new account of this
"Trinitarian Cosmogony." As he admits in a letter to
Thomas Harriot (1560-1621), his approach here is more
theological than optical (KGW 15, let. 394, p. 348, lin. 18). Similar
to his speculations in MC, Kepler explains again the symmetry between
God and the Creation, but now he goes slightly beyond the limits of a
theologico-geometrical reflection. Firstly, he seems to assume that
the bodies of the world were provided in the Creation with some powers
which enable them to exceed their geometrical limits and to act on
other bodies (magnetic power is a good example of this). Secondly, the
principle of symmetry introduced into matter constitutes "the
most excellent thing in the whole corporeal world, the matrix of the
animate faculties, and the chain linking the corporeal and spiritual
world" (APO, Engl. trans., p. 19). Thirdly, as expressed by
Kepler in a wonderful, long Latin sentence, with multiple
subordinations, this principle "has passed over into the same
laws (*in leges easdem*) by which the world was to be
furnished" (ibid., p. 20; the original passage is in KGW 2, p.
19: the marginal note in the edition is "*lucis
encomium*"). Finally, these reflections are concluded with a
remark, in which--as with Copernicus, Marsilio Ficino, and
others--the central position of the Sun is legitimated because of
its function in spreading light and, indirectly, life. Similar
speculations are still present in EAC (KGW 7, pp. 47-48 and
267). It is also worth noting that these speculations are of vital
importance to the special way in which Kepler conceived of astrology
(see, for instance, *De fundamentis astrologiae certioribus*
with Engl. trans. and commentary in Field 1984).


## 3. The five regular solids



Philosophical, geometrical and even theological speculations related
to the five regular polyhedra, the cube or hexahedron, the
tetrahedron, the octahedron, the icosahedron and the dodecahedron,
were known at least from the time of the ancient Pythagoreans. Since
Plato's *Timaeus*, these five geometrical solids played a
leading role, and for the later tradition they became known as the
"five Platonic solids".



![figure 1](figure1.jpg)
Figure 1.
Table 3 in *Mysterium Cosmographicum*, with Kepler's
model illustrating the intercalation of the five regular solids
between the imaginary spheres of the planets (cf. KGW 1,
pp. 26-27).




Plato establishes at the physical and chemical level a correspondence
between them and the five elements--earth, water, air, fire and
ether--and tries to provide this correspondence with geometrical
grounds. A further source of historically decisive importance is the
fact that the five regular polyhedra are treated in Euclid's
*Elements of Geometry*, a work that for Kepler, especially in
the Platonic approach of Proclus, has a central position. At the very
beginning of HM Kepler complains about the fact that the modern
philosophical and mathematical school of Peter Ramus (1515-1572)
had not been able to understand the architectonic structure of the
*Elements*, which are crowned with the treatment of the five
regular polyhedra. In addition, a revival of Platonic philosophy was
taking place in Kepler's time and inspired not only philosophers
and mathematicians but also architects, artists and illustrators (see
Field 1997).



Amidst this general interest in the regular polyhedra during the
Renaissance, Kepler was specifically concerned with their application
in the resolution of a cosmological problem, namely the reality of the
Copernican system (see Section 5 below). To achieve this goal, he
introduced his polyhedral hypothesis already in MC, where he looked
for an "a priori" foundation of the Copernican system (see
Aiton 1977, Di Liscia 2009). The background for such an approach seems
to be that the "a posteriori" way, which according to
Kepler was taken over by Copernicus himself, cannot lead to a
necessary affirmation of the reality of the new world system, but only
to a probable, and hence to an "instrumental",
representation of it as a computational device. This is the
"Osiander" or "Wittenberg Interpretation" of
Copernicus which Kepler directly attacked not only in his MC but also
later in his AN (see Westman 1972, 1975 and 2011). In MC, he claimed
to have found an answer to the following three main questions: 1) the
number of the planets; 2) the size of the orbits, i.e., the distances;
3) the velocities of the planets in their orbits. By referring to the
polyhedral hypothesis (see Figure 1), Kepler found a definitive and
simple answer to the first question. By intercalating the polyhedra
between the spheres which carry the planets, one must inevitably
finish with the sphere of Saturn surrounding the cube--there are
no more polyhedra to be intercalated and, as remains the standard case
for Renaissance and modern astronomy, there are no more planets to be
carried by the spheres. It is absolutely decisive for the consistency
of the argument that the necessity of the hypothesis is guaranteed by
the fact that it already exists as a mathematical demonstration (by
Euclid, *Elements XIII*, prop. 18, schol.), according to which
there are only five regular ("Platonic" for the tradition)
polyhedra. For the second and third questions the answer is, of
course, not as evident as for the first one. However, Kepler was able
to show that distances which are derived from the geometrical model of
the five regular bodies fit much better with the Copernican system
than with that of Ptolemy. The answer to the third question needs, in
addition, the introduction of a notion of power that emanates from the
Sun and extends to the outer limit of the universe (see Stephenson
1987, pp. 9-20).



In HM, Kepler continues his investigations of the polyhedra at a
cosmological as well as a mathematical level. In the second book,
dealing with "congruence" (which here does not mean, as it
does today, "the same size and shape", but the property of
figures filling the surface together with other regular polygons
- on the plane--or, to build closed geometric solids
- hence, in space), Kepler made new mathematical discoveries
working with tessellations. He discovered two new solids, the
so-called "small and great stellated dodecahedrons".



The treatment of the regular polyhedra constitutes one of the two
principal pillars of HM, Book 5 (chapters 1-2), where the third
law is formulated (see Caspar in KGW 6, Nachbericht, p. 497, and Engl.
trans. p. xxxiii). Following his approach from MC and complementing it
with occasional references to his EAC, Kepler makes use again of the
Platonic solids to determine the number of the planets and their
distances from the Sun. Meanwhile, he has learned that the application
of his old polyhedral hypothesis has limits. As he tells us in a
footnote in the second edition of his MC from 1621, he was earlier
convinced of the possibility of explaining the eccentricities of the
planetary orbits by values derived "a priori" from this
hypothesis (MC, Engl. trans., p. 189). Now, with access to the
observational data of Tycho, Kepler had to exclude this explanation
and look for another. And this is one of the most significant
achievements of his basic harmonies (which in turn are derived from
the regular polygons), constituting the second great pillar of Book
5.


## 4. Epistemology and philosophy of sciences



Almost all of Kepler's scientific investigations reflect a
philosophical background, and many of his philosophical questions find
their final answer, even if they are of scientific interest, in the
realm of theology. From a very modern point of view, one could
highlight Kepler's epistemological thought in terms of four
different items: realism; causality; his philosophy of mathematics;
and his--own particular--empiricism.


### 4.1 Realism



Realism is a constant and integral part of Kepler's thought, and
one which appears in sophisticated form from the outset. The reason
for this is that his realism always runs parallel to his defense of
the Copernican worldview, which appeared from his first public
pronouncements and publications.



Many of Kepler's thoughts about epistemology can be found in his
*Defense of Tycho against Ursus* or *Contra Ursum*
(=CU), a work which emerged from a polemical framework, the plagiarism
conflict between Nicolaus Raimarus Ursus (1551-1600) and Tycho
Brahe: causality and physicalization of astronomical theories, the
concept and status of astronomical hypotheses, the polemic
"realism-instrumentalism", his criticism of skepticism in
general, the epistemological role of history, etc. It is one of the
most significant works ever written on this subject and is sometimes
compared with Bacon's *Novum organum* and
Descartes' *Discourse on Method* (Jardine 1988, p. 5; for
an excellent new edition and complete study of this work see Jardine /
Segonds 2008).



The focus of the epistemological issues could be ranked *mutatis
mutandi* with modern discussion surrounding the scientific status
of astronomical theories (however, as Jardine has pointed out, it
would be sounder to read Kepler's CU more as a work against
skepticism than in the context of the modern realism/instrumentalism
polemic). For Pierre Duhem (1861-1916), for instance, the
position of Andreas Osiander, which was adopted by Ursus and which
was, according to Duhem, naively criticized by Kepler in his MC,
represents the modern approach known as "instrumentalism".
According to this epistemological position, held by Duhem himself,
scientific theories are not to be closely linked to the concepts of
truth and falsehood. Hypotheses and scientific laws are nothing more
than "instruments" for describing and predicting phenomena
(seldom for explaining them). The aim of physical theories is not to
offer a causal explanation or to study the causes of phenomena, but
simply to represent them. In the best-case scenario, theories are able
to order and classify what is decisive for their predictive capacity
(Duhem 1908, 1914).



Contrary to Tycho and Kepler, Ursus held a fictionalist position in
astronomy. Yet in the very beginning of his work *De
hypothesibus*, Ursus makes a clear declaration about the nature of
astronomical theories, which is very similar to the approach suggested
by Osiander in his forward to Copernicus' *De
revolutionibus*: a hypothesis is a "fictitious
supposition", introduced just for the sake of "saving the
motions of the heavenly bodies" and to "calculate
them" (trans. Jardine 1988, p. 41)



Following his approach in MC and anticipating the opening pages of his
later AN (see particularly AN, II.21: "Why, and to what extent,
may a false hypothesis yield the truth?" Engl. trans., pp.
294-301), Kepler addresses the question of Copernicanism and its
reception by thinkers such as Osiander, who emphasized that the truth
of astronomical hypotheses cannot necessarily be deduced from the
correct prediction of astronomical facts. According to this
interpretation, Copernican hypotheses are not necessarily true even if
they are able to save the phenomena, otherwise one would commit a
*fallacia affirmationis consequentis*. However, according to
Kepler, "this happens only by chance and not always, but only
when the error in the one proposition meets another proposition,
whether true or false, appropriate for eliciting the truth"
(trans. Jardine, p. 140). To be noted is that, as Jardine (2005, p.
137) has pointed out, the modern scientific realist departs from a
real independent world, while Kepler's notion of truth
presupposes that neither nature nor the human mind are independent of
God's mind (Jardine 2005, p. 137).


### 4.2 Causality



The reality of astronomical hypotheses--and hence the superiority
of the Copernican world system--implied a physicalization of
astronomical theories and, in turn, an accentuation of causality.
Despite Kepler's criticism of Aristotle, this aspect can
actually be considered the realization in the field of astronomy of
the old Aristotelian ideal of knowledge: "knowledge" means
to grasp the causes of the phenomena.



Thus, on the one hand, "causality" is a notion implying
the most general idea of "actual scientific knowledge"
which guides and stimulates each investigation. In this sense, Kepler
already embarked in his MC on a causal investigation by asking for the
*cause* of the number, the sizes and the "motions"
(= the speeds) of the heavenly spheres (see Section 3 above).



On the other hand, "causality" implies in Kepler,
according to the Aristotelian conception of physical science, the
concrete "physical cause", the efficient cause which
produces a motion or is responsible for keeping the body in motion.
Original to Kepler, however, and typical of his approach is the
resoluteness with which he was convinced that the problem of
equipollence of the astronomical hypotheses can be resolved and the
consequent introduction of the concept of causality into astronomy
- traditionally a mathematical science. This approach is already
present in his MC, where he, for instance, relates for the first time
the distances of the planets to a *power* *which emerges
from the Sun* and decreases in proportion to the distance of each
planet, up to the sphere of the fixed stars (see Stephenson 1987, pp.
9-10).



One of Kepler's decisive innovations in his MC is that he
replaced the "mean Sun" of Copernicus with the real Sun,
which was no longer merely a geometrical point but a body capable of
physically influencing the surrounding planets. In addition, in notes
to the 1621 edition of MC Kepler strongly criticizes the notion of
"soul" (*anima*) as a dynamical factor in planetary
motion and proposes to substitute "force" (*vis*)
for it (see KGW 8, p. 113, Engl. trans. p. 203, note 3).



One of the most important philosophical aspects of Kepler's
*Astronomia Nova* from 1609 (=AN) is its methodological
approach and its causal foundation (see Mittelstrass 1972). Kepler was
sufficiently conscious of the change of perspective he was introducing
into astronomy. Hence, he decided to announce this in the full title
of the work: *Astronomia Nova, Aitiologetos, seu physica coelestis,
tradita commentariis de motibus stellae Martis. Ex observationibus G.
V. Tychonis Brahe*: *New Astronomy Based upon Causes or
Celestial Physics Treated by Means of Commentaries on the Motions of
the Star Mars from the observations of Tycho Brahe* ...
(trans. Donahue). In the introduction to AN Kepler insists on his
radical change of view: his work is about physics, not pure
kinematical or geometrical astronomy. "Physics", as in the
traditional, Aristotelian understanding of the discipline, deals with
the causes of phenomena, and for Kepler that constitutes his ultimate
approach to deciding between rival hypotheses (AN, Engl. trans., p.
48; see Krafft 1991). On the other hand, since his celestial physics
uses not only geometrical axioms but also other, non-mathematical
axioms, the knowledge obtained often has a kernel of guesswork.



In the third part of AN, chapters 22-40, Kepler deals with the
path of the Earth and intends to offer a physical account of the
Copernican theory. By so doing he includes the idea that a certain
notion of power should be made responsible for the regulation of the
differences in velocities of the planets, which in turn have to be
established in relation to the planets' distances. Now, the
Copernican planetary theory departs from the general principle that
the Earth moves regularly on an eccentric circle. For Kepler, on the
contrary, the planets are moved irregularly, and the slower they are
moved, the greater their distance is from the center of power, the
Sun. Addressing the physical aspects of his new astronomy, he deals in
chapters 32-40, perhaps the most idiosyncratic of the work, with
his notion of motive power. Here, he combines different approaches and
sources, sometimes producing--for the purpose of simplifying the
whole geometrical construction of geometrical astronomy by introducing
a power causing motions--a new confusion at the dynamical level.
To begin with, it is not always absolutely clear what kind of power
Kepler has in mind. He inclines, above all, to the idea of a magnetic
power residing in the Sun, but he also mentions light and, at least
indirectly, gravity (which he does not bring into operation in the
central chapters of the AN but which is to a certain extent implied in
his explanations using the model of the balance and which he surely
accepts as true for the Sun-Moon system, as he explains in the general
introduction). Secondly, it is not always clear what this power is and
how it acts, especially when he is speaking merely analogically,
"as if" (particularly in the case of light). Essentially,
Kepler breaks down the motions of the planets into two components. On
the one hand, the planets move around the Sun--at this state of
the discussion--circularly. On the other hand, they exhibit a
libration on the Sun-planet vector. The rotation of the Sun is
responsible for the motion of the planets. Irradiating from the
rotating Sun is a power which spreads at the ecliptic plain. This
power diminishes with distance to the source of the power, that is, to
the Sun. A decisive work for Kepler's development in his
physical astronomy is William Gilbert's (1544-1603) *De
magnete* (London, 1600), a work which also intends to offer a new
physics for the new Copernican cosmology and which surely influenced
Kepler's thoughts about this power. One of the main problems
was, of course, how to apply the general principles of magnetism to
planetary motion, first to explain the difference in velocity on a
circular path, and later to give an account of the motion on an
ellipse. Kepler conceives of a model with parallel magnetic fibers
which links the Sun with the planets in such a way that the rotation
of the Sun causes the motion of the planets around it. The fibers are
born in the planets parallel and perpendicular to the lines of apsides
by a kind of "animal power". The planets themselves are
polarized, that is, with one pole they are attracted to the Sun, with
the other pole they are pushed away from it. This explains very well
the direction of planetary motion: the planets all move in one
direction *because* the Sun rotates in that direction.
Nevertheless, a further problem still seems to remain unresolved:
according to Kepler's explication, the planets should move
around the Sun as fast as the Sun itself rotates, which is not the
case. This phenomenon can be explained by referring to a property of
matter, which for Kepler has an axiomatic character: the
*inclinatio ad quietem*, that is, the tendency to rest (see
especially AN, chap. 39; KGW 3, p. 256). As a consequence, the planets
are moved around the Sun slower than they would be if the power of the
Sun were at work alone.



Kepler's causal approach is above all present in his
*Epitome*, a voluminous work which exercised a considerable
influence on the later development of astronomy. In the second part of
Book 4, he deals with the motion of the world's parts. Not the
two first laws but rather the third law, which he had recently
announced in his HM, is Kepler's starting point; for this law,
rather than a calculational device for the path of one planet,
represents a general cosmological statement, and thus it is more
convenient for his approach here. At the same time, it should be
pointed out that the third law is not necessarily the best point of
departure for a dynamical, causal approach to motion, as Kepler
intends here; for, in comparison with the previous causal approaches,
the question of the location of the cause of power responsible for the
production of motion remains relevant. The spheres, which in the
traditional view transported the planets, had been abolished since the
time of Tycho. Furthermore, Kepler is clearly against the
"moving intelligences" of the Aristotelian tradition. The
fact that the orbits are elliptical and not circular, shows that the
motions are not caused by a spiritual power but rather by a natural
one, which is internal to the composition of matter. The planets
themselves are provided with "*inertia*", a
property, as Kepler understood it, that inhibits motion and represents
an impediment to it. The motive power (*vix motrix*) comes
indeed from the Sun, which sends its rays of light and power in all
directions. These rays are captured by the planets. Kepler, however,
tries to explain this behavior of the planets less through astrology
and much more through magnetism (a physical phenomenon which was by no
means clearly understood in his time). Firstly, the Sun rotates and,
by so doing, sets in motion the planets around it. Secondly, since the
planets are poles of magnets and the Sun itself acts with magnetic
power, the planets are, at different parts of their orbits, either
attracted or repelled; in this way the elliptical path is causally
produced. Kepler partially gives up the mechanical approach by
postulating a soul in the Sun which is responsible for its regular
motion of rotation, a motion on which, finally, the entire system
depends. In fact, the planets are also supposed by Kepler to rotate
and are therefore provided with "a sort of soul" or some
such principle which produces the rotation.



In addition to astronomy and cosmology Kepler expanded his causal
approach to include the fields of optics (see Section 6 below) and
harmonics (Section 7 below).


### 4.3 Philosophy of mathematics



Beyond his own original talent, it is clear that Kepler was trained in
mathematics from his earliest studies at Tubingen. At least
officially, his positions at Graz, Prague, Linz, Ulm and Sagan can be
characterized as the typical professional occupations of a
mathematician in the broadest sense, i.e., including astrology and
astronomy, theoretical mechanics and pneumatics, metrology, and every
topic that could in some way be related to mathematics. Besides the
field of astronomy and optics, where mathematics is ordinarily applied
in different ways, Kepler produced original contributions to the
theory of logarithms and above all within his favorite field, geometry
(especially with his stereometrical investigations. For the medieval
proportions theory as a background for Kepler's logarithms, see
Rommevaux-Tani, 2018). Thus, on account of his natural predilection
and talent and the importance of mathematics, particularly of
geometry, for his thought, it is not surprising to find many different
passages in his works where he articulated his philosophy of
mathematics. However, Kepler's principal exposition on this
topic is to be found in his HM, a work in which the first two books
are purely mathematical in content. As he himself declares, in HM he
played the role "not of a geometer in philosophy but of a
philosopher in this part of geometry" (KGW 6, p. 20, Engl.
trans., p. 14).



While in philosophical questions related to mathematics, Proclus and
Plato were Kepler's most important inspirational sources, he did
not always see Plato and Aristotle as completely opposed, for the
latter--in Kepler's interpretation--also accepted
"a certain existence of the mathematical entities" (KGW
14, let. Ndeg 226, p. 265; see Peters, p. 130). To a great extent
Kepler understood his mathematical investigations of HM as a
continuation of Euclid's *Elements*, especially of the
analysis of irrationalities in Book 10. The central notion that he
works out here is that of "constructability". According to
Kepler, each branch of knowledge must, in principle, be reducible to
geometry if it is to be accepted as knowledge in the strong sense
(although, in the case of the physics, this condition is, as the AN
emphasizes, only a necessary and not a sufficient condition). Thus,
the new principles he was elaborating over the years in astrology were
geometrical ones. A similar case occurs with the basic notions of
harmony, which, after Kepler, could be reduced to geometry. Of course,
not every geometrical statement is equally relevant and equally
fundamental. For Kepler, the geometrical entities, principles and
propositions which are especially fundamental are those that can be
constructed in the classical sense, i.e., using only ruler (without
measurement units) and compass. On this are based further notions
according to different degrees of "knowability"
(*scibilitas*), which begins with the circle and its diameter.
Once again, Kepler understood this within the framework of his
cosmological and theological philosophy: geometry, and especially
geometrically constructible entities, have a higher meaning than other
kinds of knowledge because God has used them to delineate and to
create this perfect harmonic world. From this point of view, it is
clear that Kepler defends a Platonist conception of mathematics, that
he cannot assume the Aristotelian theory of abstraction and that he is
not able to accept algebra, at least in the way he understood it. So,
for instance, there are figures that cannot be constructed
"geometrically", although they are often assumed as safe
geometrical knowledge. The best example of this is perhaps the
heptagon. This figure cannot be described outside of the circle, and
in the circle its sides have, of course, a determinate magnitude, but
this is not knowable. Kepler himself says that this is important
because here he finds the explanation for why God did not use such
figures to structure the world. Consequently, he devotes many pages to
discussing the issue (KGW 6, Prop. 45, pp. 47-56, see also KGW
9, p. 147). Certainly for a geometer like Kepler, approximations
constitute - as *mathematical* theory--a painful and
precarious way to progress. The philosophical background for his
rejection of algebra seems to be, at least partially, Aristotelian in
some of its basic suppositions: geometrical quantities are continuous
quantities which therefore cannot be treated with numbers that are, in
the inverse, discrete quantities. But the difference from the
Aristotelian ideal of science remains an important one: for Aristotle,
a crossover between arithmetic and geometry is allowed only in the
case of the "middle sciences", while for Kepler all
knowledge must be reduced to its geometrical foundations. Recent
research has tried to show that Kepler has followed the criteria of a
"Pythagorean methodology" when studying the phenomena of
light refraction. This would be evident in Kepler's application
of the principle of "sameness is knowable by sameness"
and, additionally, in his tendency to avoid the infinite or, rather,
to set limits to infinite magnitudes (Cardona Suarez 2016).


### 4.4 Empiricism



A general presentation of Kepler's philosophical attitude and
principles is not complete without reference to his link to the world
of experience. For, despite his mainly theoretical approach in the
natural sciences, Kepler often emphasized the significance of
experience and, in general, of empirical data. In his correspondence
there are many remarks about the significance of observation and
experience, as for instance in a letter to Herwart von Hohenburg from
1598 (KGW 13, let. Ndeg 91, lines 150-152) or from 1603 to
Fabricius (KGW 14, let. Ndeg 262, p. 191, lines 129-130), to
mention only two of his most important correspondents. Looking for
empirical support for the Copernican system, Kepler compares different
astronomical tables in his MC, and in AN he makes extensive use of
Tycho's observational treasure trove. In MC (chapter 18) he
quotes a long passage from Rheticus for the sake of rhetorical support
when, as was the case here, the data of the tables he used did not fit
perfectly with the calculated values from the polyhedral hypothesis.
In this passage, the reader learns that the great Copernicus, whose
world system Kepler defends in MC, said one day to Rheticus that it
made no sense to insist on absolute agreement with the data, because
these themselves were surely not perfect. After all, it is
questionable whether Kepler, using for instance the *Prutenic
Tables* (1551) of Erasmus Reinhold (1511-1553), had access
to complete and correct empirical information to confirm the
Copernican hypothesis in grand style, as he claimed (for an analysis
of Reihold's tables and their influence see Gingerich 1993, pp.
205-255).



The situation changed completely when Kepler came into contact at
Prague with Tycho's observations (which, as Kepler often
reports, were seldom at his disposal). However, a change of attitude
is evident in AN, where he used Tycho's observations without
restriction (which is something he makes clear in the work's
title). In part 2 (chap. 7-21), he presents the "vicarious
hypothesis", which in the end he refutes. This hypothesis
represents the best result which can be reached within the limits of
traditional astronomy. This works with circular orbits and with the
supposition that the motion of a planet appears regular from a point
on the lines of apsides. Against the traditional method, here, Kepler
does not cut the eccentricity into equal parts but leaves the
partition open. To check his hypothesis, he needs observations of Mars
in opposition, where Mars, the Earth, and the Sun are at midnight on
the same line. From Tycho, he "inherited" ten such
observations between the years 1580 and 1600, and to them he added
another two for 1602 and 1604. In chapters 17-21, Kepler carries
out an observational and computational check of his vicarious
hypothesis. On the one hand, he points out that this hypothesis is
good enough, since the variations of the calculated positions from the
observed positions fall within the limits of acceptability (2 minutes
of arc). In fact, Kepler presents this hypothesis as the best
hypothesis which can be proposed within the framework of a
"traditional astronomy", as opposed to his new astronomy,
which he will offer in the following parts of the work. On the other
hand, this hypothesis can be falsified if one takes the observations
of the latitudes into consideration. Further calculations with these
observations produce a difference of eight minutes, something that
cannot be assumed because the observations of Tycho are reliable
enough. Kepler's famous sentence runs: "these eight
minutes alone will have led the way to the reformation of all of
astronomy" (AN, KGW 3, p. 286; Engl. trans., p. 286). There
seems to be agreement that Kepler's AN contains the first
explicit consideration of the problem of observational error (for this
question see Hon 1987 and Field 2005).



Kepler also gave an important place to experience in the field of
optics. As a matter of fact, he began his research on optics because
of a disagreement between theory and observation, and he made use of
scientific instruments he had designed himself (see, for instance, KGW
21.1, p. 244). Recent research on the problem of the *camera
obscura* and the "images in the air" shows, however,
the limits of a traditional approach to Kepler's optics
following the main current of the history of physics. Rather, his
notion of *experimentum* needs to be contextualized within the
social practices and epistemological commitments of his time (see
Dupre 2008).



Finally, it should be mentioned that a similar significance is
assigned to experience and empirical data in Kepler's
harmonic-musical and astrological theories, two fields which are
subordinated to his greater cosmological project of HM. For astrology,
he uses meteorological data, which he recorded for many years, as
confirmation material. This material shows that the Earth, as a whole
living being, reacts to the aspects which occur regularly in the
heavens (see Boner 2013, pp. 69-99). In his musical theory Kepler was
a modern thinker, especially because of the role he gave to
experience. As has been noted (Walker, 1978, p. 48), Kepler made
acoustic experiments with a monochord long before he wrote his HM. In
a letter to Herwart von Hohenburg (KGW 15, ep. 424, p. 450), he
describes how he checked the sound of a string at different lengths,
establishing in which cases the ear judges the sound to be
pleasurable. Kepler does not accept that this limitation is founded on
arithmetical speculations, even if this was already assumed by Plato,
whom he often follows, and by the Pythagoreans. On the basis of his
experiments, Kepler found that there are other divisions of the string
that the ear perceives as consonant, i.e., thirds and sixths.



If cosmology is the main framework of Kepler's interest, there
is no doubt that, as Field has pointed out, he "felt the need to
seek observational support for his model of the Universe" (Field
1988, p. 28; see also Field 1982).


## 5. Copernicanism reformed and the three planetary laws



Today Kepler is remembered in the history of sciences above all for
his three planetary laws, which he produced in very specific contexts
and at different times. While it is questionable whether he would have
understood these scientific statements as "laws"--and
it is even arguable that he used this term with a different meaning
than we do today--it seems to be clear that all three laws (as a
linguistic convention, we may continue to use the term) suppose some
fundamentals of Kepler's philosophy: (a) realism, (b) causality,
(c) the geometric structure of the cosmos. Besides this, it should be
remarked that the common denominator of all three laws is
Kepler's defense of the Copernican worldview, a cosmological
system which he was not able to defend without reforming it radically.
It is noteworthy that already at the very beginning of his career
Kepler vehemently defended the reality of the Copernican worldview in
a way that he characterized, taking over the terminology from the
standard Aristotelian epistemology, as "a priori" (see
above Section 3 above and Di Liscia 2009).



![figure 2](figure2.jpg)
Figure 2. 
Kepler's first law of ellipse and second law of areas (modern
representation with greatly exaggerated eccentricity).




The first two laws were published initially in AN (1609), although it
is known that Kepler had arrived at these results much earlier. His
first law establishes that the orbit of a planet is an ellipse with
the Sun in one of the foci (see Figure 2). According to the second
law, the radius vector from the Sun to a planet P sweeps out equal
areas, for instance \(SP\_1P\_2\) and \(SP\_3P\_4\), in equal times. The
planet *P* is therefore faster at perihelion, where it is
closer to the Sun, and slower at aphelion, where it is farther from
the Sun. In accordance with his dynamical approach, Kepler first found
the second law and, then, as a further result because of the effect
produced by the supposed force, the elliptical path of the planets
(for the two first planetary laws see especially Aiton 1973, 1975a,
Davis 1992a-e, and 1998a; Donahue 1994; Wilson 1968 and 1972; for
Kepler's system of conics in connection with his optics see the
recent study by Del Centina, 2016).



Perhaps the most significant impact of Kepler's two laws can be
found by considering their cosmological consequences. The first law
abolishes the old axiom of the circular orbits of the planets, an
axiom which was still valid not only for pre-Copernican astronomy and
cosmology but also for Copernicus himself, and for Tycho and Galileo.
The second law breaks with another axiom of traditional astronomy,
according to which the motion of the planets is uniform in swiftness.
The Ptolemaic tradition in astronomy was, of course, aware of this
difficulty and applied a particularly effective device for saving the
"appearance" of acceleration: the equant. Copernicus, for
his own part, insisted on the necessity of the axiom of uniform
circular motion. Ptolemy's equant was understood by Copernicus
as a technical device based on the violation of this axiom. Kepler, on
the contrary, affirms the reality of changes in the velocities of the
planetary motions and provides a physical account for them. After
struggling strenuously with established ideas which were located not
only in the tradition before him but also in his own thinking, Kepler
abandoned the circular path of planetary motion and in this way
initiated a more empirical approach to cosmology (though see
Brackenridge 1982).



Kepler published the third law, the so-called "harmonic
law", for the first time in his *Harmonice mundi* (1619),
i.e., ten years later. In his *Epitome*, he provided a more
systematic approach to all three laws, their grounds and implications
(see Davis 2003; Stephenson 1987). In Book 5, chapter 3, as point 8 of
13 (KGW 6, p. 302; Engl. trans., pp. 411-12), Kepler expresses,
almost accidentally, his fundamental relationship connecting elapsed
times with distances, which in modern notation could be expressed
as:



\[
\left(\frac{T\_1}{T\_2}\right)^2 = \left(\frac{a\_1}{a\_2}\right)^3
\]


with \(T\_1\) and \(T\_2\) representing the periodic times of two
planets and \(a\_1\) and \(a\_2\) the length of their semi-major axes. A
further formulation of this relationship, which is often found in the
literature, is: \(a^3/T^2 = K\), which expresses with \(K\) that
the relationship between the third power of the distances and the
square of the times is a constant (however, see Davis 2005,
pp. 171-172; for the third planetary law see especially
Stephenson 1987). As a consequence of the third law, the time a planet
takes to travel around the Sun will significantly increase the farther
away it is or the longer the radius of its orbit. Thus, for instance,
Saturn's sidereal period is almost 30 years, while Mercury needs
fewer than 88 days to go around the Sun. For the history of cosmology,
it is important to make clear that the third law fulfils
Kepler's search for a
*systematic* representation and defense of the Copernican
worldview, in which planets are not absolutely independent of each
other but integrated in a harmonic world system.


## 6. Optics and metaphysics of light



Kepler contributed to the special field of optics with two seminal
works, the *Ad* *Vitellionem paralipomena* (=APO) and
the *Dioptrice* (=DI), the latter motivated in large part by
the publication in 1610 of Galileo's *Sidereal Messenger*
(*Sidereus Nuncius*). In his *Conversation with the Sidereal
Messenger* (*Dissertatio cum Nuncio Sidereo* ... *a
Galillaeo Galilaeo*, KGW 4, pp. 281-311), he supported the
factual information given by Galileo, indicating at the same time the
necessity of giving an account of the causes of the observed
phenomena. The background for his investigation into optics was
undoubtedly the different particular questions of astronomical optics
(see Straker 1971). In this context he concentrated his efforts on an
explanation of the phenomena of eclipses, of the apparent size of the
Moon and of atmospheric refraction. Kepler investigated the theory of
the *camera obscura* very early and recorded its general
principles (see commentary by M. Hammer in KGW 2, pp. 400-1 and
Straker 1981). In addition, he worked intensively on the theory of the
telescope and invented the refracting astronomical or
'Keplerian' telescope, which involved a considerable
improvement over the Galilean telescope (see especially DI, Problem
86, KGW 4, pp. 387-88). Besides these impressive contributions,
Kepler expanded his research program to embrace mathematics as well as
anatomy, discussing for instance conic sections and explaining the
process of vision (see Crombie 1991 and especially Lindberg
1976b).



In Chapter 1 of APO ("On the Nature of Light"), Kepler
expounds 38 propositions concerning different properties of light:
light flows in all directions from every point of a body's
surface; it has no matter, weight, or resistance. Following--but
also inverting--the Aristotelian argument for the temporality of
motion, he affirms that the motion of light takes place not in time
but in an instant (*in momento*). Light is propagated by
straight lines (rays), which are not light itself but its motion. It
is important to note that although light travels from one body to
another, it is not a body but a two-dimensional entity which tends to
expand to a curved surface. The two-dimensionality of light is
probably the main reason why it is incorporeal. Motion in general
plays a significant role in Kepler's philosophy of light. For
Straker, the supposed link between optics and physics (especially in
Prop. 20, where the mechanical analyses are introduced) "reveals
the full extent of his commitment to a mechanical physics of
light" (Straker 1971, p. 509).



Two questions are intensively discussed by modern specialists.
Firstly, to what extent is the attribution of a mechanistic approach
to Kepler justified? Secondly, how should one determine his place in
the history of sciences, especially in the field of optics: do the
main lines of thought in Kepler's optics indicate a
*continuity* or rather a *rupture* with tradition? There
are well-grounded arguments for different positions on both
questions. For Crombie (1967, 1991) and Straker, Kepler develops a
mechanical approach, which can be particularly appreciated in his
explanation of vision using the model of the *camera obscura*.
Besides this, Straker stresses that Kepler's basic mechanicism
is also powerfully assisted by his conception of light as a
non-active, passive entity. In addition, the concept of motion and the
explanations using the model of the balance are indicative of a
commitment to mechanicism (Straker 1970, pp. 502-3). On the
contrary, Lindberg (1976a), who supports the "continuity
side" of the dispute, has quite convincingly showed that, for
Kepler, light has a constructive and active function in the universe,
not only in optics but also in astrology, astronomy, and natural
philosophy (for Kepler's criticism of the medieval tradition see
also Chen-Morris / Unguru, 2001).


## 7. Harmony and Soul



From a philosophical point of view, Kepler considered the HM to be his
main work and the one he most cherished. Containing his third
planetary law, this work represents definitively a seminal
contribution to the history of astronomy. But he did not reduce his
long prepared project to an astronomical investigation--his first
thoughts on the notion of "harmony" arose already in 1599,
although he did not publish his work until 1619--but instead
extensively discussed its mathematical foundations and its
philosophical implications, including astrology, natural philosophy
and psychology. Thus, Kepler's third planetary law appears in a
context which goes far beyond astronomy and to a great extent takes up
again the perspective of his youthful MC. When talking about
"harmony" in Kepler, one needs to keep in mind that the
background giving its full significance to this notion goes far beyond
music theory. Of course, music is involved and plays a determining
role - and along with it the corresponding mathematical concepts
stemming from the Pythagorean tradition. However, in Kepler
"harmony" represents the most general idea, the very core
of the world as an ordered whole and, therefore, the ultimate goal to
be pursued in his philosophical investigations. Social and political
aspects are also included. It is surely not a minor detail that
Kepler, on the threshold of the Thirty Years' War, dedicated his
HM to King James I of England "the man whom Kepler believed was
best suited to apply its lessons to the most pressing harmony of all:
the harmony of church and state" (Rothman 2017, p. 6).



According to Kepler, it is necessary to distinguish
"sensible" from "pure" harmony. The first is
to be found among natural, sensible entities, like sounds in music or
rays of light; both could be in proportion to one another and hence in
harmony. He resolves this matter by combining three of the
Aristotelian categories: quantity, relation and, finally, quality.
Through the function of the category of relation Kepler passes over to
the active function of the mind (or soul). It turns out that two
things can be characterized as harmonic if they can be compared
according to the category of quantity. But the fact that at least two
things are needed shows that the property of "being
harmonic" is not a property of an isolated thing. Furthermore,
the relationship between the things cannot be found in the things
themselves either; rather, it is produced by the mind: "in
general every relation is nothing without mind apart from the things
which it relates, because they do not have the relation which they are
said to have unless the presence of some mind is assumed, to relate
one to another" (KGW 6, p. 212; Engl. trans., p. 291). This
process takes place through the comparison of different sensible
things with an archetype (*archetypus*) present in the
mind.



The next central question directly concerns gnoseology, for Kepler
gives a psychological account of the path followed by sensible things
into the mind. He resumes the scholastic species theory: immaterial
species radiate from the sensible things and affect the sense organs
by acting firstly on the "forecourts" and then on the
internal functions (for the notion of 'species' in Kepler
see Rabin 2005, which has found both favorable and critical reception
in the literature). They arrive at the imagination and from there go
over to the *sensus communis*, so that, according to the
traditional teaching, the sensible information received is now able to
be processed and used in statements. From here onwards, the sensible
things are "preserved in the memory, brought forth by
recollection, [and] distinguished by the higher faculty of the
soul" (Caspar 1993, p. 269, cf. KGW 6, p. 214, lines
18-23; Engl. trans., p. 293). While harmony arises as an
activity of the soul/mind consisting in relating quantitatively,
Kepler adds, taking over the Aristotelian doctrine of categories, that
harmony is a "qualitative relation" as well, involving the
"quality of shape, being formed from the regular figures",
which provides the grounds for comparison (KGW 6, p. 216, lines
37-41; Engl. trans., p. 296). If this is how
"things", i.e., sensible entities, find their way into the
soul in order to be compared, it by no means represents--as
Kepler admitted--a sufficient explanation of how non-sensible
things, i.e., mathematical entities, find their way into the mind. How
do they come into the soul? Kepler accepted Aristotle's
criticism of Pythagorean philosophy concerning numbers: both Kepler
and Aristotle are convinced that numbers constitute ontologically a
lower class within mathematical entities (for Kepler, they are derived
from geometrical entities). Nevertheless, Aristotle's philosophy
is insufficient to grasp the essence of mathematics. By aligning
himself with Proclus, from whom he quotes a long passage of his
*Commentary on Euclides*, Kepler defends Plato's theory
of *anamnesis* against Aristotle's doctrine of the
*tabula rasa*. His discussion lies at the origin of the
classical debate between empiricism and rationalism which was to
dominate the philosophical scene for generations to come. A connection
with idealism is, of course, apparent (see, for instance, Caspar 1993,
Engl. trans., p. 269), and it is a fact that Kepler was positively
received within German Idealism of the 19th century
(Schwaetzer 2016, esp. 217-19). Historically, however, it seems
to be more accurate to link his position with the philosophical
tradition of St. Augustine.



Besides psychology and gnoseology, the other main spectrum of
questions Kepler deals with in Book 4 is his theory of
"aspects", i.e., astrology (HM, IV, Cap. 4-7), a
further field of application of his psychology and further evidence of
the role of geometry in his philosophy. The "aspects",
i.e., the angles between the planets, Moon and Sun, are all he wishes
to save from the old astrology, which he harshly criticizes; for the
aspects are or can be reduced to geometrical structures, the
archetypes, which can be recognized by the soul. According to Kepler,
there is no "mechanical influence" of the heavens (stars
and constellations are not relevant in his astrology) which exerts a
determining effect on the Earth and on human life. Rather, both the
Earth and human beings, ultimately, like all other living entities,
are provided with a soul in which the geometrical archetypes are
present. By the formation of an aspect in the heavens, symmetry arises
and stimulates the soul of the Earth or of human beings. "The
Earth," Kepler writes, responds to "what the aspects
whistle" (KGW 11.2, p. 48; for his astrology see especially
Field 1984 and Rabin 1997, Boner 2005 and 2006).