Was physics ever deterministic? The historical basis of determinism and the image of classical physics

Abstract

Determinism is generally regarded as one of the main characteristics of classical physics, that is, the physics of the eighteenth and nineteenth century. However, an inquiry into eighteenth and nineteenth century physics shows that the aim of accounting for all phenomena on the basis of deterministic equations of motion remained far out of reach. Famous statements of universal determinism, such as those of Laplace and Du Bois-Reymond, were made within a specific context and research program and did not represent a majority view. I argue that in this period, determinism was often an expectation rather than an established result, and that especially toward the late nineteenth and early twentieth century, it was often thought of as a presupposition of physics: physicists such as Mach, Poincaré and Boltzmann regarded determinism as a feature of scientific research, rather than as a claim about the world. It is only retrospectively that an image was created according to which classical physics was uniformly deterministic.

Full text

Eur. Phys. J. H (2021) 46:8
https://doi.org/10.1140/epjh/s13129-021-00012-x THE EUROPEAN
PHYSICAL JOURNAL H
Regular Article
Was physics ever deterministic? The historical basis of
determinism and the image of classical physics
Marij van Striena
IZWT, Bergische Universit¨at Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany
Received 9 December 2020 / Accepted 9 March 2021
©The Author(s) 2021
Abstract Determinism is generally regarded as one of the main characteristics of classical physics, that is,
the physics of the eighteenth and nineteenth century. However, an inquiry into eighteenth and nineteenth
century physics shows that the aim of accounting for all phenomena on the basis of deterministic equations
of motion remained far out of reach. Famous statements of universal determinism, such as those of Laplace
and Du Bois-Reymond, were made within a specific context and research program and did not represent a
majority view. I argue that in this period, determinism was often an expectation rather than an established
result, and that especially toward the late nineteenth and early twentieth century, it was often thought of
as a presupposition of physics: physicists such as Mach, Poincar´e and Boltzmann regarded determinism as
a feature of scientific research, rather than as a claim about the world. It is only retrospectively that an
image was created according to which classical physics was uniformly deterministic.
1 Introduction
The early twentieth century brought two revolutions,
which ended the age of classical physics and formed the
beginning of the age of modern physics. The first rev-
olution was brought about by the introduction of Ein-
stein’s relativity theory in 1905, which fundamentally
changed the notions of space and time in physics. The
second revolution, the development of quantum physics
in the first decades of the twentieth century, is generally
seen as having brought an end to classical determinism.
But was there already a clear notion of ‘classical
physics’ at this point? Staley (2005,2008a) has pointed
out the obvious by noting that the classical only exists
in contrast to the non-classical, and has shown that it
was only in the context of these two revolutions that
1Staley has looked for the first uses of the terms ‘clas-
sical physics’ and ‘classical mechanics’ and has found a
few occasional uses of these terms from 1899 onward, but
argues that only after the development of the special the-
ory of relativity in 1905, these terms started to be used
to denote the physics and mechanics of the eighteenth and
nineteenth century as a whole. Whereas Staley has focused
on identifying the first uses of the terms ‘classical mechan-
ics’ and ‘classical physics,’ Gooday and Mitchell (2013)have
argued that it took somewhat longer before the term ‘classi-
cal physics’ really became established as a general term for
non-relativistic, non-quantum mechanical physics, until the
1920s–1930s.
ae-mail: vanstrien@uni-wuppertal.de (corresponding
author)
the concept of ‘classical physics’ was developed.1In the
early twentieth century, an image was created of classi-
cal physics, which formed a contrast with the modern
physics of relativity theory and quantum mechanics;
Staley therefore speaks of the ‘co-creation’ of classical
and modern physics. Central aspects of this image of
classical physics are (1) the use of Euclidean space and
time and (2) determinism, of which the ultimate expres-
sion was found in the work of Laplace (1814). In addi-
tion, it has often been thought that the physics of the
late nineteenth century can be characterized by a sense
of complacency and by the idea that the basic features
of physics had been figured out and that the remain-
ing task was to fill in the details—however, this feature
of ‘classical physics’ has been contested, for it has also
often been noted that in fact, the last decade of the
nineteenth century saw lively debate and disagreement
about foundational issues in physics (on this topic, see
Kragh 2014; Seth 2007).
We can ask to what extent the conception of clas-
sical physics, as it was formed in the first decades of
the twentieth century, provides a correct image of the
physics of the eighteenth and nineteenth century. In this
paper, I focus on the aspect of determinism. Determin-
ism forms an essential part of the image of classical
physics: however, there are a few reasons to question
whether physics was indeed deterministic up until the
introduction of quantum mechanics.
First, it has been pointed out that during the 1910s
and early 1920s, when quantum physics was still in its
early stages and its implications for causality and deter-
minism were not yet clear, several physicists already
doubted or even abandoned determinism. Moreover,
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they did so for reasons that were at least partly inde-
pendent of the developing quantum theories. Where
Forman (1971) has argued that the cause for this turn
against determinism can be found in the cultural milieu
of the Weimar republic, others (including Brush 1976b;
Ben-Menahem 1989;St¨oltzner 1999)havelookedespe-
cially at the context of statistical mechanics: within sta-
tistical mechanics, the conception of statistical laws of
nature arose, which gave rise to the idea that all laws
of nature may be statistical: the laws of physics may
arise as statistical averages, with processes at the fun-
damental level taking place by pure chance. This idea
was most notably expressed by Franz Serafin Exner
and Erwin Schr¨odinger (Ben-Menahem 1989;St¨oltzner
1999). These early acceptances of indeterminism in the
context of statistical mechanics, however, do not nec-
essarily have to be interpreted as implying that inde-
terminism already arose within classical physics: one
might argue that statistical mechanics itself became
non-classical as soon as it allowed for fundamental
chance, and that we therefore have to draw the clas-
sical/modern boundary already at this point.
Secondly, in the last decades, philosophers of physics
including Earman (1986,2007) and Norton (2008)have
claimed that it is possible to construct systems within
classical mechanics in which determinism fails. This
would mean that the idea that classical physics is deter-
ministic has been a mistake, based on the fact that
these possibilities were overlooked. One may suppose
that these subtleties were unknown during the eigh-
teenth and nineteenth century, and that therefore at
least historically, it is correct to state that physics was
deterministic during this period. However, the failure
of indeterminism described by Norton (2008), known
as the ‘Norton’s dome,’ was known and discussed by at
least a few physicists in nineteenth century France (see
Sect. 2.2).
In this paper, I aim to answer the question to what
extent mechanics, and physics as a whole, was regarded
as deterministic before the twentieth century. Within
physics, determinism is usually defined in terms of laws
and initial conditions: determinism is essentially the
claim that all processes can be fully described through
a set of fundamental laws of nature, which always have
a unique solution for given initial conditions. I argue
that during the eighteenth and nineteenth century, this
claim could not easily be established, and it seems that
only a limited number of physicists explicitly adhered
to it. This, however, does not mean that there was a
widespread acceptance of indeterminism. It is possi-
ble to take for granted that everything that happens
is uniquely determined, without thinking that this has
been established by physics. It is also possible to think
that ultimately, physics should aim at describing all
processes through deterministic laws, without think-
ing that this aim has already been accomplished. My
main claim is that during the period which we now
describe as classical, determinism was not so much an
established result of physics, but rather an expectation,
and that during the late nineteenth and early twentieth
century, it more and more took the form of a method-
ological principle or necessary presupposition of science,
rather than an ontological claim.
Section 2of this paper deals with the question to
what extent physics was in fact deterministic during
the eighteenth and nineteenth century. Claims that in
this period, physics was deterministic, are usually based
on the idea that the laws of mechanics uniquely deter-
mine all processes within classical physics. Thus, we
first have to consider the question whether the mechan-
ics of the eighteenth and nineteenth century was in
fact deterministic (Sects. 2.1–2.3) and then the question
whether mechanics sufficed to account for all phenom-
ena within physics (or alternatively, whether another
unifying framework was possible) (Sect. 2.4).
Section 3deals with reflections of physicists on the
issue of determinism. In Sect. 3.1, it is shown that in
the late nineteenth century, many physicists adopted
a position of modesty toward the ontological implica-
tions of their theories: this position undermined the idea
that physics could decide on the issue whether nature
is deterministic. Sect. 3.2 shows how during the late
nineteenth and early twentieth century, physicists could
argue for determinism as a presupposition of science,
while remaining agnostic about whether nature itself is
deterministic.
2 Foundations of determinism in classical
physics
2.1 Varieties of mechanics
One might think that determinism in classical mechan-
ics is rather straightforward: mechanics describes how
matter moves according to laws of motion. In classical
mechanics, these laws are for example Newton’s laws
of motion, according to which for each particle, the
motion is described through a second-order differential
equation (F=md2x
dt2). These equations always have a
unique solution for given initial conditions, thus ensur-
ing that for a given initial configuration of matter, there
is only one possible way the system can evolve in time.
(In fact, uniqueness of solutions only follows under an
additional assumption of continuity which is often over-
looked, namely that the force function F(x) is Lipschitz
continuous; see below).
Thus, if all change takes place through motion of
matter, as is generally presupposed in mechanics, and if
all motion takes place according to these laws of motion,
and if these indeed have unique solutions for given ini-
tial conditions, then determinism holds. The usual his-
torical reference is Laplace, who in 1814 gave the most
famous expression of determinism in physics:
An intelligence which, for one given instant, would
know all the forces by which nature is animated and
the respective locations of the entities which compose
it, if besides it were sufficiently vast to submit all these
data to mathematical analysis, would encompass in the
same formula the movements of the largest bodies in the
universe and those of the lightest atom; for it, nothing
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Eur. Phys. J. H (2021) 46:8 Page 3 of 20 8
would be uncertain and the future, as the past, would
be present to its eyes (Laplace 1814, 3–4).2
However, there are good reasons to think that
mechanics was never that simple. Generally, it is a sim-
plification to think of classical mechanics as being con-
stituted completely by Newton’s laws of motion. In fact,
the conception of the laws of mechanics as dynamical
laws, expressed by means of differential equations, was
developed only during the eighteenth century. Further-
more, from the eighteenth century until the present,
there have been various formulations of mechanics and
various proposals for the basic mechanical principles,
and attempts to reduce all of this to one well-defined
and consistent theory with one basic set of laws have
been only partially successful (Truesdell 1968; Stan
2017; Stan forthcoming-a; Stan forthcoming-b). Stan
describes how during the eighteenth century, there was
no general agreement on what exactly the laws of
motion were, and various mechanical principles, laws
of motion, and formulations of mechanics were devel-
oped: ‘this diversity of foundational perspectives defies
any attempt to show that post-Newtonian mechanics is
a unified theory’ (Stan forthcoming-a).
The most pressing issue was that mechanics dealt
with several distinct conceptions of matter (Stan 2017;
see also Wilson 2013). The main conceptions of matter
used in mechanics are mass points, rigid bodies and
deformable continua. These are fundamentally different
and are subjected to different dynamics:
•Mass points are unextended and can only undergo
translational motion. Collisions between mass points
are rare enough that they can plausibly be ignored,
so that the mass points only interact through ‘action
at a distance’; they interact through forces which act
between pairs of particles, are directed along the line
which connects them, and depend on their distance.
•Rigid bodies are extended and generally have well-
defined geometrical shapes which cannot be deformed.
They undergo both translational and rotational
motion and can exert both contact forces (one body
pushing against another) and body forces (e.g.,
gravity and electromagnetic force).
•Deformable continua are also extended and can be
divided into volume elements with an infinitesimal
extension. They can undergo translation, rotation
and deformation and can undergo internal stresses.
All three of these conceptions of matter have turned
out to be needed to describe the range of phenomena
that fall under the domain of mechanics. This frag-
mented ontology of mechanics persisted throughout the
nineteenth century and still persists today: textbook
examples in mechanics often use a mixture of the above
2For the question to what extent Laplace’s statement can
be taken to express the idea that all processes in physics
are determined through (differential) equations of motion
which have unique solutions for given initial conditions, see
Van Strien (2014a,b,c).
conceptions of matter. Most physicists content them-
selves with the idea that in classical mechanics, matter
can be modeled in different ways, depending on the sit-
uation and context.
Those who seek to base mechanics on solid founda-
tions and on a single ontological picture can argue that
only one of these conceptions of matter is foundational,
and that the others are only used as approximations.
For example, one could argue that what there really
is are mass points, but that certain problems within
mechanics can be solved much more easily if you model
certain configurations of mass points as rigid bodies or
as deformable continua. Many such proposals for onto-
logical unification have been made—more on this in the
next section—but up until the late nineteenth century,
the issue remained debated, and there was no general
consensus about the constitution of matter (Wilholt
2008). In the twentieth century, it became clear that
to answer the question what matter is like at a foun-
dational level, one has to look at quantum mechanics,
and that classical mechanics can only offer descriptions
of matter on a higher scale level.
Here, it may be objected that none of this has any
implications for the issue of determinism: the fact that
there are various formulations of mechanics and that
mechanics works with various conceptions of matter in
itself does not give us any reason to think that there
could be indeterministic processes in mechanics. This
is fair enough, but at least it shows that the issue of
determinism in classical mechanics is not that straight-
forward after all: the conception according to which
mechanics deals with a single type of matter of which
the motion is uniquely determined through a single set
of laws is too simple. To properly demonstrate that
(classical) mechanics is deterministic, one would need to
engage with mechanics in its full complexity. One possi-
bility would be to establish that there are abstract prin-
ciples which can be applied to all conceptions of matter,
such as the principle of virtual work and d’Alembert’s
principle, and establish that these principles determine
the course of any process in full detail.3Another pos-
sibility would be to establish that at bottom there is a
single type of matter and one set of laws of motion and
that this ‘bottom level’ is deterministic; however, one
then has to make plausible that the rest of mechanics
can (at least in principle) ultimately be reduced to this
bottom level.
It is no coincidence that the most well-known state-
ments of determinism in physics have been made by sci-
entists who argued that mechanics should be based on a
single conception of matter; and it is also no coincidence
that the conception of matter they adhered to was that
of the mass point, which is the conception for which
the chances of establishing determinism look best. In
the next section, we will look at the status of determin-
ism within point particle mechanics; in Sect. 2.3,we
will look at the situation in other areas of mechanics.
3On the development of such general principles, see, e.g.,
Truesdell (1968) and Darrigol (2014).
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2.2 Point particle mechanics
As was mentioned in the previous section, by far the
most well-known statement of determinism in physics
was given by Laplace (1814). However, similar state-
ments of determinism can already be found earlier, in
the work of a number of contemporaries of Laplace
in eighteenth century France, notably Maupertuis,
Condorcet and d’Holbach; moreover, Laplace himself
already expressed the idea of determinism in a lec-
ture in 1773 (Wolfe 2007; Van Strien 2014a). These
early expressions of determinism can be understood
in the context of the French enlightenment and its
optimism about science. Perhaps the most notable
precursor of Laplace’s determinism, however, is the
Jesuit mathematician and natural philosopher Roger
Boscovich: Koˇznjak (2015) has argued that in the work
of Boscovich, one finds a statement of determinism
that is both earlier (1758) and stronger than that of
Laplace, yet has remained largely unknown.4Like those
of his predecessors, also Laplace’s statement of deter-
minism did not have an immediate impact: in fact,
Laplace’s determinism only became well known and
much debated after being popularized in a famous lec-
ture by Emil Du Bois-Reymond in 1872 (Du Bois-
Reymond 1898 [1872]; see Cassirer 1956 [1936]; Hacking
1983).
Boscovich, Laplace, and Du Bois-Reymond are thus
perhaps the three most significant sources of the idea of
determinism in classical physics (even though the his-
torical influence of Boscovich’s determinism may have
been limited). All three of them strictly adhered to a
conception of matter as constituted by point particles,
or by very small particles of which the internal structure
need not be studied, with central forces acting between
these particles; and according to all three of them, the
task of physics consists in explaining all natural pro-
cesses in terms of the motions of these particles. In
fact, the conception of mass points finds its origin in
Boscovich’s work. In this section, I show how the state-
ments of determinism of Boscovich, Laplace and Du
Bois-Reymond were each embedded in a research pro-
gram which sought to reduce physics to point particle
mechanics, and then look at in how far it was possible
for them to establish determinism within point particle
mechanics.
Boscovich has become most well-known for his idea
that matter is constituted by point particles, with
central forces acting between them; the force acting
4Koˇznjak (2015) argues that Boscovich’s determinism is
more carefully formulated than that of Laplace and has a
stronger basis in physics: unlike Laplace, Boscovich does
not forget to mention that not only initial positions but
also initial velocities need to be given, and Boscovich stip-
ulates that forces need to be continuous, which is essential
in order for the equations to yield unique solutions. More
generally, Van Strien (2014a) argues that Laplace primarily
argues for determinism on the basis of metaphysical princi-
ples, whereas for Boscovich, it is apparent that his argument
for determinism is based on mathematics and mechanics.
between point particles is repulsive at short distances,
represents gravity at large distances, and in between it
oscillates a few times between repulsive and attractive.
The exact form of the force function should account
for various properties of matter, including electricity
and magnetism; Boscovich’s theory thus represents an
ambitious attempt at unification. Boscovich argues for
determinism as follows:
Any point of matter, setting aside free motions
that arise from the action of arbitrary will, must
describe some continuous curved line, the determi-
nation of which can be reduced to the following
general problem. Given a number of points of mat-
ter, & given, for each of them, the point of space
that it occupies at any given instant of time; also,
given the direction & velocity of the initial motion
if they were projected, or the tangential velocity
if they are already in motion; & given the law of
forces expressed by some continuous curve, such as
that of Fig. 1, which contains this Theory of mine;
it is required to find the path of each of the points,
that is to say, the line along which each of them
moves.
(...) Now, if the law of forces were known, & the
position, velocity & direction of all the points at
any given instant, it would be possible [for a mind
that is brilliant enough] to foresee all the necessary
subsequent motions & states, & to predict all the
phenomena that necessarily followed from them.
(Boscovich 1922 [1763]).
This statement of determinism is carefully formu-
lated. It explicitly applies to point particles and holds
as long as the forces between them are continuous; this
holds for the specific force function Boscovich proposed
to act between particles. (Also, note that he argues that
physical systems are only deterministic as long as no
‘arbitrary will’ intervenes; he thus argues for determin-
ism within the domain of physics, but is not an absolute
determinist).
Also Laplace adhered to a particle conception of mat-
ter; in fact, he was the leading figure of a research pro-
gram which Fox (1974) has termed ‘Laplacian physics.’
The main feature of Laplacian physics was that it
sought to account for all phenomena in terms of
molecules, of which the extension could be neglected
for most practical purposes and which could thus be
treated as point particles; and central forces, which
could be attractive or repulsive. On this basis, all prop-
erties and interactions of matter, including chemical
reactions, were to be accounted for. Heat, light, electric-
ity and magnetism were conceived of as imponderable
fluids, consisting of molecules. Thus, physics and chem-
istry were to be based on strong, unitary foundations
provided by molecules and central forces. According to
Fox, Laplacian physics constituted a highly unified and
ambitious research program: ‘In the years of its great-
est success, from 1805 to 1815, the program both raised
problems and laid down the general principles for solu-
tions; and, by doing so, it gave French physical science a
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Eur. Phys. J. H (2021) 46:8 Page 5 of 20 8
most uncommon unity of style and purpose’ (Fox 1974,
91).
Point particles play a central role in celestial mechan-
ics, in which planets, moons and the sun are usually
modeled as point masses in calculations of planetary
orbits. This approach was extraordinarily successful
in the eighteenth century and provided a model for
Laplace’s physics. In celestial mechanics, it had become
possible to make impressively accurate long term pre-
dictions; Laplace (1814) refers to Halley’s prediction of
the return of a comet in 1759 as an impressive feat
of prediction. After writing that an intelligence which
would have exact knowledge of the present state of the
universe, would, if this intelligence were ‘sufficiently
vast to submit all these data to mathematical analysis,’
be able to predict anything that would happen in the
future, Laplace continues: ‘The human mind offers, in
the perfection which it has been able to give to astron-
omy, a feeble idea of this intelligence’ (Laplace 1814, 4).
If all of physics can indeed be reduced to point particle
mechanics, then the problem of predicting any process
within physics is merely a (much) more complex varia-
tion on the problem of predicting planetary orbits and
orbits of comets; although predicting these motions by
far surpasses our calculation skills, there is nevertheless
good reason to assume that these motions are determin-
istic. Laplace concludes his reflections on determinism
by stating: ‘The curve described by a simple molecule of
air or vapor is regulated in a manner just as certain as
the planetary orbits; the only difference between them
is that which stems from our ignorance’ (Laplace 1814,
6).
The fact that Laplace was the leading figure of a
research program which sought to reduce physics to the
mechanics of particles raises the question in how far his
determinism was bound to this specific program. This
question becomes all the more pressing if we consider
that, according to Fox (1974), the program of Lapla-
cian physics collapsed quite suddenly between 1815
and 1825 and was abandoned by almost all of its for-
mer adherents. Fox explains this collapse both through
institutional factors and through new challenges within
physics: most scientists came to the agreement that the
scheme provided by Laplacian physics was too rigid and
too limited to account for all phenomena within physics.
He mentions a number of scientific developments of the
early nineteenth century which could not easily be inte-
grated into the program of Laplacian physics, including
the wave theory of light, the vibrational theory of heat,
Dalton’s atomic theory of chemistry and the rational
mechanics of Fourier, as well as phenomena of elastic-
ity and electrodynamics.
Although the research program of Laplacian physics
lost many of its adherents, its basic concepts and
approach continued to be influential. Laplace’s deter-
minism was popularized in Du Bois-Reymond’s well-
known lecture ‘On the limits of our knowledge of
nature’ in 1872. In this lecture, in which Du Bois-
Reymond sought to determine the potential domain
and the limits of natural science, he argued that to
understand something scientifically means to reduce it
to the motion of atoms:
Natural science—or, more definitely, knowledge of
the physical world with the aid of and in the sense
of theoretical natural science—means the reduc-
tion of all change in the physical world to move-
ments of atoms produced independently of time
by their central forces; or, in other words, natural
science is the resolution of natural processes into
the mechanics of atoms. (Du Bois-Reymond 1898
[1872], 18).
For Du Bois-Reymond, understanding natural pro-
cesses in terms of motion of atoms is a definition
of scientific knowledge. A similar definition of scien-
tific knowledge can be found in von Helmholtz (1847);
Helmholtz similarly argued that the only way to com-
pletely understand nature is to reduce natural phenom-
ena to the motion of material particles subjected to cen-
tral forces. According to Du Bois-Reymond, this then
sets the boundaries of science: everything which can
in principle be reduced to the motion of atoms falls
within the domain of what can be known scientifically,
and anything that cannot be understood in terms of
motions of atoms is fundamentally unknowable. If all
natural processes can be reduced to motion of atoms,
Laplacian determinism follows:
If we were to suppose all changes in the physical
world resolved into atomic motions, produced by
constant central forces, then we should know the
universe scientifically. The condition of the world
at any given moment would then appear to be
the direct result of its condition in the preceding
moment and the direct cause of its condition in
the subsequent moment. Law and chance would
be only different names for mechanical necessity.
Nay, we may conceive of a degree of natural science
wherein the whole process of the universe might be
represented by one mathematical formula, by one
infinite system of simultaneous differential equa-
tions, which should give the location, the direction
of movement, and the velocity, of each atom in the
universe at each instant. (Du Bois-Reymond 1898
[1872], 18).
Du Bois-Reymond uses the term ‘astronomical knowl-
edge’ for knowledge of the positions and motions of
atoms; this shows that he, like Laplace, models science
on celestial mechanics (Du Bois-Reymond Du Bois-
Reymond 1898 [1872], 26).
As in the cases of Boscovich and Laplace, also
Du Bois-Reymond’s statement of determinism thus
appeared within the context of a broader program of
reduction to point particle mechanics. It is relevant to
note here that Du Bois-Reymond himself was not a
physicist, but rather a physiologist, although one for
whom physics was central to physiology; with Johannes
M¨uller and Hermann von Helmholtz, he developed an
approach to physiology which was heavily based on the
methods and theories of physics and chemistry, and
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which aimed at mechanistic explanation of organic pro-
cesses (Finkelstein 2013). Du Bois-Reymond was par-
ticularly concerned with arguing against any vitalist
notions within physiology; he argued that physiological
processes are determined by material conditions, with-
out intervention of the mind or any type of vital force
(Van Strien 2014c). In this context, it was important for
him to argue that all natural processes, including phys-
iological ones, could be understood in terms of mere
matter and motion. It is also relevant to note that the
lecture in which Du Bois-Reymond argued for deter-
minism was a popular lecture; Romizi (2019) has argued
that Du Bois-Reymond’s determinism should be under-
stood in the context of science popularization, and that
the public demand for an all-encompassing scientific
world view contributed to the extension of determin-
ism to all of nature. Thus, also Du Bois-Reymond’s
determinism was embedded within a specific context
and within an ambitious research program, in which
the idea that all natural processes can be reduced to
atoms and central forces played an essential role.
If one assumes that all phenomena within physics
can indeed be reduced to the motion of point particles,
determinism seems to follow quite straightforwardly. Of
course, it is not possible for us to calculate exactly what
the future will hold: this would require exact knowledge
of instantaneous values of the position and momentum
of all particles—either all particles in the universe, or
within a perfectly isolated system. Nevertheless, it is
in principle possible to describe the motion of each
particle through differential equations which, although
extremely complicated and practically unsolvable for
systems of more than a handful of particles, are nev-
ertheless of a relatively straightforward form and can
be expected to have a unique solution for given initial
conditions.
However, there are a few caveats. First, it remains
to be specified which forces act between atoms, besides
gravity. In celestial mechanics, the motion of planets,
moons and comets is determined purely by gravita-
tional force; but this does not suffice to explain all
interactions of matter on smaller scales. If all processes
in physics are to be reduced to point particle mechan-
ics, additional force functions have to be introduced to
account for, e.g., properties of different materials and
electric and magnetic phenomena.
During the nineteenth century, it was established
that in order for the differential equation describing the
motion of a point particle to yield a unique solution
for given initial conditions, the force function needs to
fulfill a continuity condition. The exact condition was
formulated by Lipschitz in 1876: a force function F(x)
needs to fulfils the condition that there is a constant
K>0 such that for all x1and x2in the domain of F,
|F(x1)−F(x2)|≤K|x1−x2|.
(see Van Strien 2014b). This condition has come to be
known as ‘Lipschitz continuity.’ This mathematical the-
orem was not yet available to Boscovich or Laplace;
therefore, strictly speaking, they would not have been
able to prove mathematically that their laws of motion
always yield a unique solution for given initial con-
ditions. This is not to say that they had no insight
in whether equations of motion in mechanics could
be expected to have a unique solution: they can be
expected to have had reliable mathematical intuitions
on this matter, and in fact both seem to have realized
that determinism depends on a continuity condition.
Boscovich explicitly requires that the force function has
to be continuous in order for his theory to be determin-
istic (Boscovich 1922 [1763], 281), and Laplace appeals
to version of Leibniz’ law of continuity in his argument
for determinism (Van Strien 2014a;Israel1992).
The possibility that a point particle may be sub-
jected to a force for which the equation of motion fails
to have a unique solution was already raised by Pois-
son (1806), as well as by Duhamel (1845), Boussinesq
(1879) and Joseph Bertrand (1878). In their examples
of non-uniqueness of solutions to mechanical equations
of motion, the Lipschitz condition is in fact violated;
but they did not explicitly rely on the Lipschitz con-
dition, or on earlier results by Cauchy on conditions
for the uniqueness of solutions to differential equations.
Poisson and Duhamel consider the possibility that point
particles attract each other with a force which allows
for non-unique solutions to the equation of motion;
Boussinesq also considers this possibility and in addi-
tion designs a situation in which a point particle is
placed on top of a dome-shaped surface and subjected
to gravitational force; because of the particular shape of
the surface, the equation of motion fails to determine if
and when the point particle will roll down the surface.
The latter way of constructing an indeterministic sys-
tem in classical mechanics was rediscovered much later
by John Norton (Norton 2008) and is now known as
‘the Norton dome.’ It is an interesting case because in
this system, indeterminism arises through gravitational
force.5(Note that this is not a system within point par-
ticle mechanics, as it involves both a point particle and
a rigid surface).
Interestingly, most of the above authors did not con-
clude that there could be indeterminism in nature. Pois-
son and Duhamel argued that when an equation of
motion has more than one solution, only one of these
solutions can be correct, and the problem is how to
identify the correct solution. Bertrand sought the prob-
lem in the relation between theory and reality and
argued that when the theory allows for multiple pos-
sible future evolutions of a system, the theory must not
be entirely accurate (see Sect. 3.2).6
5This happens because the component of gravitational
force tangent to the dome is non-Lipschitz-continuous, and
this can happen because there is a well-hidden singularity
in the second derivative of the surface of the dome, at its
summit (see Malament 2008;Fletcher2012).
6For Boussinesq, things are a bit more complicated. He did
argue for indeterminism within physics, but did not allow for
occurrences to be altogether uncaused: when the equations
of mechanics fail to have a unique solution, either a vital
principle or human free will has to step in (where a vital
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Eur. Phys. J. H (2021) 46:8 Page 7 of 20 8
Collisions are a further caveat to the claim that deter-
minism is relatively straightforward within point par-
ticle mechanics: when two or three particles collide,
there is a singularity in the equations of motion, and
these equations do not say what will happen; one has
to figure out a way to continue the equations past the
collision. One way to avoid this problem is by argu-
ing that for particles without extension, the probabil-
ity of collision is infinitely small and can therefore be
ignored. Boscovich in fact postulated that at very short
distances between particles, there is a repulsive force
which goes to infinity as the distance between particles
goes to zero, so that collisions never take place.7
We thus arrive at the following conclusions. First,
the statements of determinism of Boscovich, Laplace
and Du Bois-Reymond, three of the most significant
proponents of determinism in physics in the eighteenth
and nineteenth century, were each embedded within a
specific research program, aimed at reducing all nat-
ural phenomena to the motion of atoms which can
be regarded as mass points. These research programs
were very influential, but they did not represent a gen-
Footnote 6 continued
principle will act in a way which is determined by the laws
of physiology, which are irreducible to physics, whereas the
human will is genuinely undetermined). In this way, he used
this case of possible indeterminism in physics as the basis
for an elaborate theory about organic life and free will (see
Van Strien 2014c).
7A further caveat to point particle mechanics being deter-
ministic is the possibility of ‘space invaders.’ In Newtonian
mechanics, there is no limit to the velocity of particles, and
that means that the equations allow for a particle to come
in from spatial infinity within a finite time (Earman 1986).
This means that given the current state of any system, the
future is not completely determined, since it may or may
not happen that a particle pops in from infinity. Histori-
cally, the development of the idea of space invaders started
with Painlev´e, who in 1895, while lecturing on the prob-
lem of the stability of the solar system, conjectured that
for systems of more than three particles, attracting each
other with a gravitational force, there can be non-collision
singularities (Diacu and Holmes 1999). In 1908, Von Zeipel
proved that such singularities can only take place if ‘the
motion of the system becomes unbounded in finite time’—
which at the time was generally thought to be impossible
(Diacu and Holmes 1999, 89). After some progress on the
issue was made in the 1960s and 1970s, it was finally proven
by Xia in 1992 that within a system of five point masses
subjected to Newtonian gravity, it is indeed possible that
one of the particles accelerates in such a way that it moves
off to infinity within finite time (Diacu and Holmes 1999).
Since the dynamics of point particles subjected to Newto-
nian gravity is time reversible, also the reverse process is
possible: thus, it follows that a particle may appear from
infinity within finite time. Thus, this indeed seems to be a
way in which point particle mechanics may fail to be deter-
ministic. However, it seems that during the eighteenth and
nineteenth century, this possibility remained largely unex-
plored, with the exception of Painlev´e’s conjecture in 1895
which can be seen as a first suspicion that something like
this may be possible.
eral consensus in physics, and they were ambitious and
ongoing: the endpoint of being able to account for all
natural phenomena in terms of the motion of atoms was
never reached. Furthermore, although within a system
of point particle mechanics, determinism seems rela-
tively straightforward, it is not at all trivial to rigor-
ously establish that a system of point particles is deter-
ministic, and there may be failures of determinism even
within point particle mechanics—and one can in fact
find a few examples of physicists in the nineteenth cen-
tury who were aware of this.
The point particle conception of matter was very
influential throughout the nineteenth century, but always
existed alongside other conceptions of matter. In the
second half of the nineteenth century, an increasing
number of physicists doubted whether point particles
could be regarded as foundational. Besides a general
increasing resistance to reductionism in physics (about
which more in Sect. 2.4), point particles raised a num-
ber of conceptual problems. Already Du Bois-Reymond,
while arguing for a program of reduction of all natu-
ral phenomena to the motion of atoms, argued that we
cannot know the nature of atoms; in fact, he argued
that our conception of atoms is ultimately contradic-
tory. One problem lies in the question whether atoms
are extended: whereas Du Bois-Reymond thought that
atoms can be considered as point particles for purposes
of calculation, he argued that they did in fact have
to occupy at least a small space, for something can-
not exist without being in space. However, at the same
time, he argued that for an atom to be a foundational
element in science, it would have to be indivisible, but
an atom can only be truly indivisible when it is unex-
tended.8Because of such puzzles, Du Bois-Reymond
argued that the nature of matter will always remain
unknown to us.
Other physicists also regarded the point particle con-
ception of matter as problematic. James Clerk Maxwell
argued that results from spectroscopy showed that
atoms had to be elastic, but if atoms are foundational,
this means that the property of elasticity cannot be fur-
ther explained. Moreover, this result from spectroscopy
was hard to reconcile with information about the inner
structure of atoms obtained within the kinetic theory
of gases, as well as with the point particle conception
of atoms according to which they do not have an inter-
nal structure (Maxwell 1875, 471).9Duhem argued in
8Moreover, Du Bois-Reymond argued that in order to be
foundational, atoms must be perfectly hard, for if they
aren’t, this means that they have a property which requires
further explanation. But if atoms are perfectly hard, col-
lisions between atoms involve an infinite force (Du Bois-
Reymond 1898 [1872], 21; and see Wilholt 2008).
9Specifically, spectroscopy indicated that atoms had an
internal structure and were capable of internal vibrations.
This was also hard to reconcile with the fact that within
the kinetic theory of gases, it could be derived (on the basis
of the theorem of equipartition of energy) that the ratio
between the specific heats of gases at constant pressure and
at constant volume depends on the number of degrees of
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8 Page 8 of 20 Eur. Phys. J. H (2021) 46:8
1905 that attempts to reduce all of physics to point
particle mechanics had led to increasingly complicated
and artificial theories, and that it had ultimately turned
out that it is not possible to account for, e.g., elastic-
ity in terms of point particles (Duhem 1905, 88). Henri
Poincar´e similarly argued that whereas the conception
of mass points and central forces had been very useful in
the historical development of physics, at some point, it
seemed no longer methodologically adequate to attempt
to base all of physics on this foundation (Poincar´e1921,
297–299; Liston 2017).
By the late nineteenth century, the question of the
inner structure of atoms was an open problem, and the
very existence of atoms was debated. The point par-
ticle conception of matter was confronted with both
conceptual problems and empirical challenges, and it
seemed increasingly likely that it would not be possible
to account for all natural processes in terms of point
particles and central forces.
2.3 Mechanics of rigid bodies, continua and fluids
Without the assumption that all of physics is reducible
to point particle mechanics, the picture becomes more
complicated. The two main alternatives to the point
particle conception of matter were rigid bodies and
deformable continua.
In order to develop a deterministic mechanics based
on rigid bodies, one has to formulate rules for colli-
sions of rigid bodies, or determine that no collisions take
place. The formulation of rules for collisions turned out
to be a problematic issue: in order for motion not to be
lost, collisions would have to be elastic, but this seemed
hard to reconcile with the property of rigidity. In par-
ticular, an elastic collision between hard bodies would
involve an instantaneous change of motion, which would
require an infinite force. This problem of collisions of
hard bodies was much debated during the eighteenth
and nineteenth century (Scott 1970; Darrigol 2001).10
If matter is allowed to deform during collisions, we
arrive at the conception of matter as deformable con-
tinua, which is described by continuum mechanics. But
by taking this step, the possibility of giving a com-
plete and exact description of natural phenomena gets
much further out or reach. Whereas in point parti-
cle mechanics, processes are (usually) determined by a
finite number of initial conditions, namely the position
and velocity of each mass point, plus the force laws act-
ing between particles, in continuum mechanics you need
the values of physical quantities (such as pressure and
density) over entire surfaces and volumes. Mathemat-
Footnote 9 continued
freedom of the atoms constituting the gas. Measured values
for the ratio of specific heats indicated that the number of
degrees of freedom of the atoms had to be limited (Brush
1976a, 353ff).
10 And it is not easy to establish that all rigid body systems
would be deterministic: Wilson (2009) has given examples of
systems of rigid bodies for which a deterministic description
is lacking.
ically speaking, in continuum mechanics you need to
work with partial differential equations, which require
boundary surfaces as initial and boundary conditions.
In order to deal with properties of different substances,
one needs to specify internal stresses, elasticity, friction
and viscosity. In order to make problems mathemati-
cally tractable, it is unavoidable to work with idealiza-
tions, to take surfaces to be smooth and volumes to be
homogeneous.
Physicists in the nineteenth century certainly real-
ized the necessity of simplifying assumptions and ideal-
izations in continuum mechanics. Maxwell, when com-
paring the atomic and the continuum conceptions of
matter, noted that the continuity view can be used as
long as bodies can be assumed to be homogeneous:
[A] theory that some particular substance, say
water, is homogeneous and continuous may be a
good working theory up to a certain point, but
may fail when we come to deal with quantities so
minute or so attenuated that their heterogeneity of
structure comes into prominence. (Maxwell 1875,
450).
It was not obvious that equations in continuum
mechanics would always yield exact solutions for given
initial conditions. Existence and uniqueness theorems
for solutions to partial differential equations were
largely unavailable during the eighteenth and nine-
teenth century; therefore, if physicists at the time were
concerned with establishing whether equations in con-
tinuum mechanics have unique solutions for given initial
conditions and thus yield deterministic descriptions, it
would have been difficult, if not impossible, to rigor-
ously establish this. Fluid mechanics brought particu-
lar challenges, especially regarding turbulent flow, and
physicists in the nineteenth century extensively debated
the possibility of discontinuity and unstable solutions in
fluid mechanics (Darrigol 2002).11 While the existence
of unstable solutions would not demonstrate a failure
of determinism, it would be a further obstacle to estab-
lishing a strict determinism.
Other areas of mechanics required yet different types
of mathematical equations. In order to deal with stress
on physical bodies, Ludwig Boltzmann, Vito Volterra
and E. Picard developed an approach called hereditary
mechanics, in the late nineteenth century; the idea is
that for a body subjected to stress, its deformation
11 A particular type of the continuum conception of mat-
ter was the vortex theory of atoms, developed especially by
William Thomson (Lord Kelvin) from the 1860s on, accord-
ing to which atoms are constituted by vortex rings (like
smoke rings) in a fluid, usually identified with the ether. The
motions in this fluid could be described by the equations of
hydrodynamics; thus, the ambition was to derive the prop-
erties of matter from hydrodynamics. This research program
was very popular among physicists in Victorian Britain, but
ultimately failed: the development of the theory was math-
ematically very complicated and led to few results, and by
the 1890s, it seemed impossible to explain all properties of
matter on the basis of vortex atoms (Kragh 2002a).
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Eur. Phys. J. H (2021) 46:8 Page 9 of 20 8
may not only depend on the stress applied to it at
that particular moment, but also on the stress that has
been applied to it at earlier times (D¨orries 1991;Ian-
niello and Israel 1993). In order to describe this mathe-
matically, they used integro-differential equations. This
means that in these cases, in order to make predictions
about future behavior, it does not suffice to know the
current state of the system; one also needs knowledge
about past states. In 1910, Paul Painlev´e, noting that
this went against Laplacian determinism, drew a dra-
matic conclusion: ‘The conception according to which,
in order to predict the future of a material system,
one has to know its entire past, is the very negation
of science’ (Painlev´e, quoted in Israel 1992, 269). Most
other physicists who concerned themselves with heredi-
tary mechanics, however, were not particularly both-
ered with its implications for determinism (Ianniello
and Israel 1993).
A more recent perspective on rigid body mechanics
and continuum mechanics is given by Mark Wilson, who
has summarized the situation by noting that classical
mechanics ‘must inevitably compress swatches of very
complicated physical behavior into simplified rules of
thumb’ (Wilson 2009, 175). In a paper titled ‘What is
classical mechanics anyway?’, Wilson notes:
As matters are commonly represented within mod-
ern college primers, ‘classical physics’ appears to be
a transparent subject matter firmly founded upon
Newton’s venerable laws of motion. But this placid
appearance is deceptive. Any purchaser of an old
home is familiar with parlor walls that seem sound
except for a few imperfections that ‘only require
a little spackle and paint.’ When those innocent
dimples are opened up, the ancient gerry-rigged
structure comes tumbling down and our hapless
fix-it man finds himself confronted with months
of dusty reconstruction. So it is with our subject,
whose basic concepts can seem so ‘clear and dis-
tinct’ on first acquaintance that unwary thinkers
have mistaken them for a priori verities. But the
true lesson of ‘classical mechanics’ for philosophy
should be exactly the opposite: the conceptual
matters that initially strike us as simple and pel-
lucid often unwind into hidden complexities when
probed more adequately (Wilson 2013, 43).
To sum up: in order to deal with the full range of
problems in mechanics, physicists in the nineteenth
century needed to work with various mathematical
techniques and often needed to work with simplifying
assumptions and idealizations in order to make prob-
lems mathematically tractable. Determinism was often
not a point of special concern, but would in fact be
far from trivial to establish rigorously: a demonstra-
tion that the equations of motion that were used would
always yield unique solutions for given initial condi-
tions was far beyond what was mathematically feasible.
This does not mean that there was reason to think that
mechanics allowed for indeterministic processes; but it
does show that the idea that all mechanical processes
can be fully described through a single set of (second-
order differential) equations was far removed from sci-
entific practice.
2.4 From mechanics to physics
So far, we have examined the foundations of determin-
ism within mechanics; in order to argue on the basis
of determinism in mechanics that physics as a whole
is deterministic, one would need to assume that all of
physics can be reduced to mechanics. That this can be
done was part of the research programs of Laplace and
Roger Joseph Boscovich, and also Du Bois-Reymond
adhered to this idea.
The nineteenth century saw enormous development
in different domains of physics, including electrodynam-
ics and thermodynamics—in fact, physics as a disci-
pline was only established during the nineteenth cen-
tury. There were extensive attempts to reduce elec-
trodynamics and thermodynamics to mechanics, e.g.,
by finding a mechanical model for the electromagnetic
ether, and by giving a mechanical derivation of the sec-
ond law of thermodynamics—there was a prevalent idea
that only through reduction to matter and motion, nat-
ural phenomena could really be understood. Thus, J.
J. Thomson argued in 1888 that the belief in the pos-
sibility of mechanical explanation of natural phenom-
ena was ‘the axiom on which all Modern Physics is
founded’ (Thomson 1888, 1). Some important results
were reached: probably the main success was the devel-
opment of the kinetic theory of gases, which enabled to
define heat in terms of the motion of atoms. However,
by the end of the century, there was also a movement
against the idea that physicists should actively try to
account for all phenomena through mechanical mod-
els. This movement was led by physicists such as Ernst
Mach, Pierre Duhem and Wilhelm Ostwald, who argued
that the attempts to give mechanical explanations of
all physical phenomena were often cumbersome and
unfruitful, leading physicists to devise mechanical mod-
els that were speculative and needlessly complicated,
and that it was often methodologically more fruitful to
examine relations between observable phenomena than
to attempt to reduce them to configurations of matter
and motion. Mach, Duhem and Ostwald furthermore
argued that there may well be natural phenomena and
natural laws which were fundamentally irreducible to
mechanics; an important case was the second law of
thermodynamics, which described irreversible behavior
which could not be derived from mechanics (Klein 1973;
Van Strien 2013).
Mach argued already in 1872 that the foundational
status often given to mechanics could be explained by
the contingent historical fact that mechanics was the
first area of science to develop into an exact science:
It is the result of a misconception, to believe, as
people do at the present time, that mechanical
facts are more intelligible than others, and that
they can provide the foundation for other physical
facts. This belief arises from the fact that the his-
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tory of mechanics is older and richer than that of
physics, so that we have been on terms of intimacy
with mechanical facts for a longer time. Who can
say that, at some future time, electrical and ther-
mal phenomena will not appear to us like that,
when we have come to know and to be familiar
with their simplest rules? (Mach 1911 [1872], 56–
57)
In fact, in the late nineteenth and early twenti-
eth century, there were proposals to take another
area of physics as foundational: specifically, there were
attempts at a unification of physics based on electrody-
namics or on energy and thermodynamics. In the ‘elec-
trodynamic world view,’ proposed in 1900 by Wilhelm
Wien, matter was conceived of as structures in an elec-
tromagnetic ether, and the laws of mechanics were to be
explained electrodynamically—this attempt of reduc-
tion of physics to electrodynamics looked promising
for a while, but ultimately its success remained lim-
ited (Kragh 2002b,2014). Wilhelm Ostwald proposed
to take energy as fundamental and to unify physics
by reducing it to the science of energy. This approach,
which went under the name ‘energeticism,’ drew much
attention but also harsh criticism, among others by Max
Planck and Mach; in the end, it was not very successful
(Hiebert 1971; Kragh 2014).
Boltzmann positioned himself as going against the
trend of anti-mechanism and persevered in his work
on the kinetic theory of gases, with the aim to give
a mechanical account of thermodynamic phenomena.
The kinetic theory of gases, developed from the mid-
nineteenth century by Rudolf Clausius, Maxwell and
Boltzmann among others, explained the behavior of
gases through the motions of the atoms composing
them. It started out as a fully mechanical theory; how-
ever, in the course of time, Boltzmann found it needed
to give a more central role to probabilities in the the-
ory. Whereas in Boltzmann’s earlier work, probabili-
ties used within the kinetic theory of gases were deter-
mined by the mechanical properties of the gas, at
some point he introduced probabilities which relied on
assumptions about the state of the gas as a whole, and
could not directly be reduced to its mechanical prop-
erties. According to Uffink (2007, 928), this marks the
transition from the kinetic theory of gases to statis-
tical mechanics. Even though Boltzmann’s statistical
mechanics is mechanical in the sense that it describes
systems composed of moving particles, the macroscopic
evolution of these systems cannot be fully derived from
the equations of motion of these particles without prob-
abilistic assumptions (Uffink 2007, 972).
Thus, during the nineteenth century, physics became
more complex and plural. By the late nineteenth cen-
tury, the range of phenomena studied and the types of
mathematical equations and methods used to describe
these phenomena had become very diverse, and the idea
that all of physics could be reduced to mechanics and
all of mechanics could in turn be reduced to point par-
ticle mechanics was increasingly criticized. This does
not necessarily indicate a move away from determin-
ism: of course, the fact that there are theories in physics
which cannot be reduced to mechanics does not imply
that there are cases in which determinism fails. But the
only feasible way to demonstrate that all processes in
physics are deterministic would be to find a relatively
small set of entities and laws of nature, to which every-
thing else reduces. The electrodynamic and energetic
world views were attempts at such a unification; how-
ever, they remained ambitious research programs with a
limited number of adherents, there was no general con-
sensus about how promising they were and ultimately
their goals of unification were never achieved. Gener-
ally, by the end of the nineteenth century, it had become
increasingly unfeasible to reduce all of physics to a basic
set of equations and to establish that these equations
would always have a unique solution for given initial
conditions.
In how far determinism was generally accepted in
nineteenth century physics is not easy to judge. As
Brush (1976b) notes, in the nineteenth and early twenti-
eth century, there was often a certain ambiguity in the
use of terms like randomness, indeterminism, chance,
spontaneous, and probabilistic, and it is often not clear
whether physicists argue that there is genuine indeter-
minism, or whether they are merely expressing igno-
rance of the exact paths of particles or of causes of
movement. Moreover, there were many physicists who
were simply not concerned with the question whether
or not physics is deterministic. But at least some cases
can be found of physicists who doubted or explic-
itly rejected determinism. Notable examples include
Maxwell, who argued that unstable mechanical systems
could allow for the intervention of an undetermined
free will (Maxwell 1995 [1873]) and Boussinesq, who
used the possibility of failures of determinism in point
particle mechanics as the basis for an elaborate theory
of free will and organic life (see Sect. 2.2). Generally,
those who rejected determinism in physics often did so
in the context of religion, and often argued for the pos-
sibility of free will and/or for a vitalist conception of
organic life (on indeterminism in the nineteenth cen-
tury, see Nye 1976;Hacking1983; Van Strien 2014b,c;
Romizi 2019). At the same time, many physicists seem
to have taken for granted that all physical processes
are uniquely determined, often without explicitly argu-
ing for this, and without necessarily being committed
to the idea that current theories of physics could yield
a deterministic description of all natural processes.
3 Determinism around 1900
3.1 Descriptionism and determinism
In the late nineteenth century, many physicists argued,
in different ways, that natural science cannot answer
ontological questions and cannot provide us with a true
representation of nature. It is possible to describe and
predict phenomena by means of scientific theories, but
one should not assume that these theories correspond
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with nature in every respect. Heilbron (1982) has pro-
posed the term ‘descriptionism’ to describe this trend.
This is a broad term describing a general tendency,
which can according to Heilbron be found in the work
of diverse physicists such as Mach, Poincar´e, Duhem
and Boltzmann: despite significant differences in their
philosophies of science, they all share a certain ontolog-
ical humility and emphasize that science cannot yield
true representations of nature.
This trend of descriptionism was not a break with
the past, but was continuous with earlier developments.
Already 1876, Gustav Kirchhoff argued in the intro-
duction to his Vorlesungen ¨uber Mechanik (Kirchhoff
1876), that the aim of mechanics is merely to describe
the motions of matter, and not to inquire into the causes
of motion. Mach argued in the 1880s that science can
merely describe the relations between observable phe-
nomena, and that the aim of science is to merely arrive
at efficient ways to report experiences.12 Whereas Mach
and Kirchhoff stressed that we should stick with the
phenomena and not speculate about underlying causes,
a different strand of descriptionism was based on the
liberal use of mechanical models. Maxwell and Kelvin
devised detailed mechanical models, but used these
models freely, switching between different models and
not being too concerned about inconsistencies between
them. This free attitude stemmed from the fact that
they did not demand that these models gave a true and
exact representation of nature. Maxwell thought that it
was generally fruitful to work with concrete mechanical
models, and argued for a method of analogies: analo-
gies enable us to work with a ‘clear physical concep-
tion,’ without being committed to specific hypothe-
ses about, e.g., the inner structure of matter (Maxwell
1856). Maxwell’s use of models was praised by Boltz-
mann, who argued for the heuristic value of visualizable
models, or ‘pictures,’ of reality, which do not have to
be taken as true representations of nature (Boltzmann
1902). In a lecture in 1899, Boltzmann argued that
[N]o theory can be objective, actually coinciding
with nature, but rather (...) each theory is only a
mental picture of phenomena, related to them as
sign is to designatum.
From this it follows that it cannot be our task to
find an absolutely correct theory but rather a pic-
ture that is as simple as possible and represents
phenomena as accurately as possible. One might
even conceive of two quite different theories both
equally simple and equally congruent with phe-
nomena, which therefore in spite of their difference
are equally correct.
Many questions that used to appear unfathomable
thus fall away of themselves. How, it used to be
said, can a material point which is only a men-
tal construct, emit a force, how can points come
together and furnish extension, and so on? Now
we know that both material points and forces are
12 See,e.g.,hislecture‘Die¨okonomische Natur der
physikalischen Forschung’ from 1882 (Mach 1903, 215–242).
mere mental pictures. The former cannot be identi-
cal with something extended, but can approximate
as closely as we please to a picture of it. The ques-
tion whether matter consists of atoms or is con-
tinuous reduces to the much clearer one, whether
the continuum is able to furnish a better picture of
phenomena (Boltzmann 1974, 91; and see De Regt
1999; De Courtenay 2002).
More origins of descriptionism can be found, and
they can be traced further back in time. Schiemann
(2008) and Pulte (2000,2009) have described how dur-
ing the nineteenth century, the ideal of absolute truth
in science was abandoned.13 Pulte (2009) argues that
this abandonment of absolute truth was partly brought
about by internal problems within mechanics, which
led to an abandonment of the idea that mechanics is
built up deductively from absolutely certain axioms.
Romizi (2019) uses the term ‘ent-ontologisierung’ (de-
ontologization) to describe the shift in the attitude of
scientists toward scientific theories in the late nine-
teenth century and draws the origin of this development
all the way back to Kant, who already argued that sci-
ence cannot describe the world as it really is, and that
for example causality is not a feature of the world in
itself but rather a category of understanding.
The development which Heilbron describes as ‘descrip-
tionism’ was thus a broad movement, encompassing var-
ious other—isms, such as conventionalism, positivism,
and instrumentalism. There were significant differences
between the positions of, e.g., Maxwell, Mach, Poincar´e,
Duhem, and Boltzmann: they had different attitudes to
atomism, made different judgments about the heuristic
value of hypotheses and of detailed mechanical models,
and attached different values to unification and con-
sistency in science. How broadly the idea that scien-
tific theories cannot be taken as true representations of
nature was shared can be seen from the fact that also
Du Bois-Reymond, despite his strong mechanical reduc-
tionism, argued that matter and force cannot be taken
to exist the way we conceive of them. We have already
seen in Sect. 2.2 that Du Bois-Reymond thought that
the nature of matter and force remain unknown to us; in
fact, he argued already in 1848 that both our concepts
of matter and of force are abstractions from reality
(Du Bois-Reymond 1848, xlii; Finkelstein 2013, 283). A
similar point was made previously by Helmholtz (von
Helmholtz 1847). For Du Bois-Reymond, this means
that the aim of science can merely be to describe the
motions of matter and not to investigate their causes.
But despite the broadness of ‘descriptionism,’ and the
fact that it can be found throughout the nineteenth cen-
tury and across many authors who otherwise had signif-
13 Schiemann (2008) describes a transition in Helmholtz’s
thought in the late 1860s and early 1870s toward the idea
that scientific knowledge is based on hypotheses and has no
claim to absolute truth, and interprets this as the develop-
ment of a modern understanding of knowledge: ‘Knowledge
is only modern when it is marked by a relativeness of its
claim to validity’ (Schiemann 2008,3).
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icantly different views on natural science, it does point
to a general tendency which was increasingly empha-
sized toward the end of the century: a rejection of meta-
physical explanation and of claims to absolute truth,
and a rhetoric of modesty in the aims of science.14
This descriptionist rhetoric is often found in lec-
tures for a general audience; in this period, scientists
were expected to regularly give public lectures, and
physicists often used these occasions for philosophical
reflections on their field (St¨oltzner 2011). Heilbron has
argued that descriptionism should primarily be under-
stood in the context of the engagement of scientists with
a general public, and was largely a reaction to a nega-
tive public image of natural science. Natural science was
held responsible for uncontrolled technological devel-
opment and was seen as materialistic, atheistic, crude
and arrogant, removing all mystery, emotion and value
from the world (Heilbron 1982, 57–58; and see MacLeod
1982). Heilbron argues that in order to counteract this
public image of science, many physicists in the late nine-
teenth century distanced themselves from broad scien-
tistic and materialistic world views and argued that
their aim was merely to describe the phenomena. These
public aspects are indeed significant, but do not give
the full picture, since as we have seen, there were also
strong internal causes for descriptionism. Physicists in
this period partly used the means of public lectures to
debate the aims and methods of physics, and the shift
in the ultimate aims and explanatory ideals which can
be seen in these lectures also (in different ways) shaped
scientific practice.
The modesty of descriptionism seems far removed
from the Laplacian ideal of predicting all occurrences
in nature by tracing the motion of atoms and goes
against the confidence and optimism often associated
with classical physics. Indeed, in the late nineteenth
and early twentieth century, it is rare to find physi-
cists making broad claims in the style of Laplace or Du
Bois-Reymond. However, despite the rhetoric of mod-
esty, descriptionism could also convey a greater degree
of certainty to physics: ‘By giving up metaphysics and
relying instead on mathematical description, physics
could eliminate whatever was doubtful and attain to
almost perfect rigor and certainty’ (Porter 1994, 135;
see also Staley 2008b). This point was already made by
Duhem, who argued in 1906 that by restricting the aim
of physics to describing and classifying the phenomena,
rather than explaining the underlying causes, physics
can be made independent of metaphysics, and thereby
a higher degree of certainty can be reached (Duhem
1991).
If descriptionism implies a focus on mathematical
description rather than metaphysical explanation, this
14 Mach and Boltzmann are often supposed to be directly
opposed to each other, as Boltzmann was a main advocate
of atomism while Mach was opposed to atomism. That they
nevertheless had something in common is supported by a
remark from Philipp Frank: ‘strange as it was, in Vienna
the physicists were all followers of Mach and Boltzmann.’
(Frank, quoted in St¨oltzner 1999, 86).
does not have to stand in the way of determinism.
However, there are other reasons why descriptionism
could undermine commitments to scientific determin-
ism. Among physicists in the late nineteenth century,
there was a widely shared idea that scientific theories
should be understood as models which do not corre-
spond with nature in every way, and that laws of nature
always involve some degree of idealization and abstrac-
tion. For example, Mach, Boltzmann and Poincar´e all
argued that laws of nature can only hold with good
approximation and not absolutely, and that we can-
not formulate laws of nature without idealizing. Mach
argued that in order to formulate laws of nature, we
have to look for regularities within the observed phe-
nomena, and in order to do this we must always abstract
away from the details and single out that which seems
important to us; therefore, there is always a certain loss
of information and a lack of complete accuracy (Mach
1903, 222). According to Boltzmann, ‘No equation rep-
resents any process with absolute precision; each ideal-
izes them, emphasizes common features and disregards
differences, and thus goes beyond experience’ (Boltz-
mann 1905, 222; see also Poincar´e1921, 340ff ).15
Careful consideration of the relation between theory
and reality can also be seen in Poincar´e’s attitude to the
problem of the stability of the solar system. Poincar´eis
commonly regarded as the grandfather of modern chaos
theory, and this is mostly because of his work on the
n-body problem: the problem of the motion of a small
number of mass points attracting each other with gravi-
tational force. In the late nineteenth and early twentieth
century, this problem drew attention because of its rel-
evance for the stability of the solar system: if the sun,
planets, and moons are modeled as mass points, can one
prove mathematically that the solar system is stable,
and will keep holding together even in the far future?
It turns out that mathematically, this is a very compli-
cated problem, and as a mathematician, Poincar´e made
important contributions to it (Parker 1998). Poincar´e
noted, however, that the problem was of limited physi-
cal interest. He argued that, as successful as the models
based on mass points and gravitational force were in
astronomy, they could never yield an exact representa-
tion of the solar system: the sun, planets, and moons are
of course not actual mass points, Newton’s law of grav-
ity may not be rigorously exact, and in our solar system,
there are other forces at play than gravitational forces
15 For Boltzmann, a specific reason for considering laws
of nature as approximations comes from the philosophy of
mathematics. Concerns about the notion of infinity led him
to the conviction that we cannot assume natural quanti-
ties to vary continuously. Boltzmann therefore thought that
while with proper care, differential calculus can be used as
a calculation tool in physics, laws of nature formulated in
terms of differential equations cannot be taken to directly
correspond to nature (Van Strien 2015; see also De Courte-
nay 2002).
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Eur. Phys. J. H (2021) 46:8 Page 13 of 20 8
between the centers of mass (Poincar´e1898).16 These
issues became all the more pertinent when it turned
out that in the mathematical model of the solar system,
the stability of the system depends sensitively on initial
conditions: even small differences in initial states could
lead to very different evolutions. As the actual state of
the solar system cannot be determined with mathemat-
ical precision and the mathematical model can never be
an exact representation of the actual system, this meant
that the question of the stability of our solar system had
to remain unanswered.17
For the issue of determinism, the claim that scientific
theories never exactly correspond with reality implies
that determinism on the level of scientific theories does
not necessarily have to correspond with determinism on
the ontological level. If the laws of nature are taken to
be rigorously valid, then the fact that these laws always
yield unique solutions for given initial conditions can be
used to demonstrate that nature itself is deterministic—
even if in practice, we can never make absolutely certain
predictions, because we can never know initial condi-
tions with absolute accuracy. But if the laws of nature
only hold approximately (even with a very good approx-
imation), the fact that they yield exact and unique pre-
dictions does not necessarily have to imply that nature
itself is deterministic.
Romizi (2019, 259) has recently argued that the ten-
dency of late nineteenth century scientists to prob-
lematize the relation between theory and reality, and
to regard scientific theories as only providing (possi-
bly very good) models of scientific reality, led to a
weakening of determinism in this period. She notes,
however, that this ‘gap’ between theory and reality
could in certain cases also be used to defend determin-
ism against possible counterexamples. In fact, Joseph
Bertrand appealed to the idea of a discrepancy between
theory and reality in his criticism of Boussinesq’s argu-
ments for indeterminism (Bertrand 1878, 520; Van
Strien 2014b). But although in this case, a problema-
tization of the relation between theory and reality pro-
vided a way to defend determinism, in general it prob-
ably had primarily the effect of undermining determin-
ism, through undermining the inference from theories
to reality. Although most physicists at the time did not
explicitly reflect on the issue, the fact that scientific
theories were generally conceived of as highly reliable
but still idealized models, which could not be taken to
directly correspond to reality, could be taken to imply
that even if there were a unified theory of physics which
always yielded unique solutions for given initial condi-
tions, one would still not be justified in inferring that
nature itself is deterministic.
16 Poincar´e argues that the mass point model does not
account for dissipation of energy, which will have an effect
on planetary motions in the long term.
17 Similar ideas were expressed by Duhem, who remarked
for this reason that the problem of stability of the solar
system had a clear mathematical sense, but may be devoid
of physical sense (Duhem 1991, 142).
3.2 Determinism as a presupposition of science
By the late nineteenth century, broad, sweeping claims
about universal determinism had gone out of fashion.
However, the ideal of determinism did not disappear
from physics altogether, but more and more, it took the
shape of a necessary presupposition of science, rather
than an established result within physics or a broad
metaphysical principle.18 This section focuses on the
conceptions of determinism of Mach, Boltzmann and
Poincar´e. They are three of the ma jor physicists of this
period, and all three formulated general ideas on phi-
losophy of science; this section argues that despite the
fact that their philosophies of science are quite different,
there is a striking similarity in their ideas on the sta-
tus of determinism in physics. In particular, all three
thought of determinism as an assumption which one
must make when doing scientific research.
In his book on the history of mechanics, Mach argues
that within mechanics, any process is uniquely deter-
mined by Newton’s laws of motion (Mach 1897 [1883],
257). Yet, the same book contains a sharp criticism of
Laplacian determinism:
If the French encyclopedists of the eighteenth cen-
tury believed they were close to the goal of explain-
ing all of nature physico-mechanically, if Laplace
imagined a spirit who could indicate the course of
the world in the whole future, if only it knew all
the masses with their positions and initial veloc-
ities, then this joyful overestimation of the scope
of the physical-mechanical insights gained in the
eighteenth century is forgivable, even a charming,
noble, uplifting spectacle, and we can vividly sym-
pathize with this intellectual joy, which is unique
in history.
But after a century, after we have become more
level-headed, the projected world view of the ency-
clopedists appears to us as a mechanical mythology
in contrast to the animistic mythology of the old
religions. Both views contain improper and fantas-
tic exaggerations of a one-sided knowledge. (Mach
1897 [1883], 455; see also Mach 1903, 217).
There are several reasons why, for Mach, Lapla-
cian determinism is not tenable. Although according
to Mach mechanics itself, as a scientific theory, is
deterministic in the sense that it describes processes
which are uniquely determined by the laws of motion,
Mach rejects the idea that all of science is reducible
to mechanics: mechanics only describes certain aspects
18 Throughout the twentieth century, determinism contin-
ued to have its proponents among physicists. But even for
a strong defender of determinism as Planck, determinism
was more about the presuppositions of science than about
ontology: according to Planck, determinism is a ‘heuristic
principle, a signpost and in my opinion the most valuable
signpost we possess, to guide us through the motley disor-
der of events and to indicate the direction in which scientific
inquiry should proceed in order to attain fruitful results’
(Planck 1932; translation: Earman 2007, 1372).
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8 Page 14 of 20 Eur. Phys. J. H (2021) 46:8
of nature (Mach 1897 [1883], 499). He particularly
objects to the idea that physiology and psychology
are reducible to mechanics, arguing that mechanics
is ultimately based on sensory perception which is to
be understood physiologically, and therefore mechan-
ics cannot be an ultimate foundation. As we have seen
in Sect. 2.4, Mach also objected to the idea that all
domains of physics should be reducible to mechanics.
Thus, even if mechanics is deterministic, this says little
about whether nature as a whole is deterministic.
Mach furthermore objected to the universal scope
and the temporality of Laplace’s determinism, accord-
ing to which the current state of the universe, together
with the laws of nature, determines all future and past
states. According to Mach, this depends on a notion
of absolute time which is not warranted. Mach argues
that time is relational, which means that to describe
how something changes in time, one has to describe
how it changes relative to a part of the world func-
tioning as a clock; e.g., describing the motion of the
heavenly bodies in time is equivalent to describing the
motion of the heavenly bodies relative to the rotation
of the earth. But for the universe as a whole, there is
no clock relative to which we can describe its changes.
Therefore, Laplace’s statement of determinism depends
on the false assumption that we have a notion of time
which can be applied to the universe as a whole (Mach
1872, 36-37). More generally, in Mach’s account, laws
of nature are essentially summaries of experience, and
this entails that they cannot be extrapolated too far
beyond experience. For these reasons, it would accord-
ing to Mach be overly confident to assume that our laws
of nature hold absolutely and are applicable to the uni-
verse as a whole. Laplacian determinism oversteps the
boundaries of science.
Nevertheless, there are no indications that Mach
would accept a role for fundamental chance in physics.
In fact, he still adhered to determinism in a weaker
sense. According to Mach, the aim of science is to find
dependencies between observable phenomena, and he
argues that the ‘law of causality’ should be interpreted
as the statement that there are indeed dependencies
between the phenomena to be found, or in other words,
that we can find laws of nature (Mach 1872, 34; Mach
1897 [1883], 492). According to Mach, this is an assump-
tion we need to make when doing science: to do sci-
entific research is by definition to look for dependen-
cies between phenomena, and thus, the assumption that
there are such dependencies to be found is the assump-
tion that science is possible. However, this assump-
tion has to do with our scientific activity, rather than
with nature: Mach stresses ‘that all forms of the law of
causality arise from subjective instincts, to which there
is no need for nature to correspond’ (Mach 1897 [1883],
495).
In later years, Mach argued that it is better to
avoid the terms ‘cause’ and ‘causality’ in the context of
physics and to speak only about functional dependence.
He argued that the notion of cause is imprecise, and
that to specify functional dependencies between phe-
nomena is both more precise and more informative than
to state causes of phenomena; therefore, an advanced
science like physics deals (or at least should deal) with
functions rather than with causes. In line with this, in
his book Erkenntnis und Irrtum (1905), he no longer
speaks about the ‘law of causality’ but rather uses
the term ‘determinism’ to express the assumption that
there are functional dependencies between phenomena.
Mach presents this as a postulate which is needed when
doing research, even though it cannot be proven:
The correctness of the position of ‘determinism’ or
‘indeterminism’ cannot be proven. It could only
be decided if science were completed or demon-
strably impossible. We are dealing here with pre-
suppositions that one brings to the consideration
of things, depending on whether one attaches a
greater subjective weight to the successes or to the
failures that research has attained so far. But dur-
ing research, every thinker is necessarily a theoret-
ical determinist (Mach 1905, 277).
However, scientists must always be aware of the pos-
sibility that their theories fail:
The researcher must (...) always be ready for disap-
pointment. He never knows whether he has already
taken into account all the dependencies that may
be considered in a given case. His experience is
limited in space and time and only offers him a
small section of world events. No fact of experience
is exactly repeated. Every new discovery reveals
flaws in our understanding and reveals a previ-
ously neglected remainder of dependencies. So also
he who advocates an extreme determinism in the-
ory must in practice remain an indeterminist, espe-
cially if he does not want to speculate away the
most important discoveries. (Mach 1905, 278).
Thus, determinism is an assumption which is needed
when doing scientific research, but is to be avoided
when reflecting on the scope of current scientific the-
ories. Science is about finding dependencies between
phenomena, and this is an ongoing process. When doing
science, one should look for dependencies between phe-
nomena and assume that such dependencies are to be
found. At the same time, scientists should take account
of the possibility that the dependencies that have been
found do not hold absolutely at every scale and should
be aware of the limits of their theories and open to the
possibility to have missed something. Ultimately, there
is no guarantee that we can find laws of nature deter-
mining every natural process in full detail. The issue of
determinism versus indeterminism is therefore for Mach
not a metaphysical issue. It is not about whether nature
is fully determined or whether there is pure chance—
this is a question which remains unanswerable. Rather,
the issue is about assumptions made by scientists and
the attitude of scientists toward their work.
A similar conception of determinism can be found in
the work of Boltzmann. Boltzmann is mostly known for
his work in statistical mechanics, which is usually seen
as the main source of indeterminism in the early twenti-
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Eur. Phys. J. H (2021) 46:8 Page 15 of 20 8
eth century; therefore, he is often seen as a forerunner of
later indeterminism in physics (St¨oltzner 1999). In his
work on statistical mechanics, Boltzmann introduced
probabilistic assumptions which could not be derived
from the dynamical laws and worked with a notion
of ‘molecular disorder’ (see Uffink 2007). Moreover, he
argued for the idea that the second law of thermody-
namics is a law of nature which holds statistically rather
than absolutely, thus supporting the idea that laws of
nature can be statistical.
Nevertheless, in a lecture in 1899, Boltzmann argued
that the assumption that natural processes are uniquely
determined is a precondition for science:
A precondition for all scientific knowledge is the
principle of unique determination of natural pro-
cesses; in the case of mechanics, the unique deter-
mination of all motions. This means that bodies
do not move purely by chance, now this way and
now that way, but that the motions are uniquely
determined by the circumstances under which the
body is located. If each body moved however it
wanted, if under the same circumstances now this,
now that motion would follow by chance, we could
only curiously observe the course of the phenom-
ena, not investigate it (Boltzmann 1905, 276–277).
Boltzmann notes that this principle becomes unus-
able if the ‘circumstances under which the body is
located’ include the universe as a whole. If the move-
ment of an object on earth could just as well be deter-
mined by processes taking place in another solar sys-
tem, this would make scientific investigation into the
movement impossible. It is therefore needed to make
a restriction to local circumstances: we should assume
‘that the same movement always occurs when the
direct environment is in the same state’ (Boltzmann
1905, 276–277). Boltzmann’s principle of determinism
is thereby stronger than Mach, who did not impose such
a locality restriction.
Boltzmann argues for the Kantian idea that we
have laws of thought, without which knowledge would
be impossible and our observations would be without
any connection. However, he argues against Kant that
these laws of thought have no a priori certainty (see
Boltzmann 1905, 398–399; Fasol-Boltzmann 1990, 160).
Rather, our laws of thought are evolutionary in ori-
gin. As such, they are usually very effective in helping
us to make sense of the world, but because they have
become such strong habits, they can in certain cases
overshoot the mark (‘¨uber das Ziel hinausschießen’—
Boltzmann 1905, 399). Boltzmann compares this with
a baby’s instinct to suck, which keeps the baby alive
but is not effective in every instance. One of these laws
of thought is the law of causality, which according to
Boltzmann is the ‘foundation of all knowledge’ (Boltz-
mann 1905, 321). However, also our tendency to look for
causes can overshoot the mark: it can lead to supersti-
tion when we look for causes of chance events and leads
to useless philosophical puzzles, for example when we
ask why we exist, why the world exists, or why the law
of causality holds (Boltzmann 1905, 321, 354, 398–399).
Also Boltzmann’s principle of unique determination
may be interpreted as a law of thought: it is required for
doing science, but at the same time, there is no a priori
guarantee that it holds. A problematic issue, according
to Boltzmann, is how to determine what counts as local
circumstances. If we really assume that the motion of
bodies is always determined by their immediate sur-
roundings, this excludes action at a distance, which
makes it problematic to account for gravity. Alterna-
tively, one could consider a larger area around the body:
in the case of an object falling toward the earth, the
earth can be included in its local surroundings, while
distant stars are excluded. However, there is no strict
criterion deciding which circumstances should be taken
into account, and there is no guarantee that we can
always find a set of local circumstances through which
occurrences are determined (Boltzmann 1905, 278).
Furthermore, we may ask how Boltzmann’s claim
that scientists need to assume that the motion of all
bodies is locally determined relates to Boltzmann’s sta-
tistical physics, in which molecules move around ran-
domly. In fact, Boltzmann suggested the possibility that
the laws of motion according to which molecules move
around in a gas may be statistical regularities which
hold on average for large amounts of molecules, and
that the motion of individual molecules is undetermined
(Boltzmann 1898, 260). Boltzmann does not explicitly
address the question of how this relates to the principle
of unique determination, but a plausible answer is that
it depends on what you want to investigate. In statisti-
cal physics, the behavior of a gas as a whole is studied,
and then it is permissible to make the assumption that
the individual molecules move around randomly; but if
the goal is to investigate the movement of an individ-
ual molecule, this is only possible under the assumption
that it does not move around by chance but that the
movement depends on circumstances.
Boltzmann’s principle of local determination is thus
neither a priori certain nor derivable from experience.
When doing science, we have to look for local determin-
ing circumstances of processes, but whether we can in
fact establish a theory in which all processes are fully
locally determined is an empirical question which Boltz-
mann leaves open.
In contrast to Boltzmann and Mach, Poincar´edoes
seem to argue explicitly for a strong, universal Lapla-
cian determinism. For example, in Poincar´e(1917
[1913], 7), he writes: ‘Knowing the present state of each
part of the universe, the ideal scientist who knew all the
laws of nature would possess fixed rules to deduce the
state that these same parts will have the next day; it
is conceivable that this process can be continued indef-
initely.’ Like Laplace, Poincar´e especially emphasizes
determinism in the context of astronomy: he writes that
the accurate predictions that are possible within astron-
omy provide a model for other sciences (Poincar´e1921,
289ff).
Nevertheless, in my view, also Poincar´e ultimately
regards determinism as an assumption we have to make
when doing scientific research, rather than as an onto-
logical claim about nature or an established feature of
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our current scientific theories. According to Poincar´e,
the idea that our universe is deterministic depends
on certain assumptions which scientists (in particular
physicists) make, and which are needed either to make
science possible or to make specific scientific problems
much easier to deal with. In his writings, Poincar´edis-
cusses a number of these assumptions.
First, as we have seen in Sect. 3.1, Poincar´e argues
that laws of nature are always approximate and incom-
plete. However, in order to do science, scientists need
to work with the assumption that it is possible to find
ever more accurate laws of nature, which will enable
them to make better predictions:
[N]o particular law will ever be more than approx-
imate and probable. Scientists have never failed
to recognize this truth; only they believe, right or
wrong, that every law may be replaced by another
closer and more probable, that this new law will
itself be only provisional, but that the same move-
ment can continue indefinitely, so that science in
progressing will possess laws more and more proba-
ble, that the approximation will end by differing as
little as you choose from exactitude and the prob-
ability from certitude. (Poincar´e1921, 341)
If we find that a certain law fails, that it fails beyond
a certain limit or that it leads to contradictions when
extrapolated, we assume that it can be replaced by
some other, more accurate law.
Secondly, Poincar´e argues like Boltzmann that physi-
cists generally assume that only the local environ-
ment of the phenomena they are studying has an effect
(Poincar´e1921, 346). In addition to this assumption
of spatial locality, physicists generally assume that
an event is determined by the situation immediately
before, and that one does not need to know what hap-
pened in the distant past in order to predict future
occurrences. The assumption that the state of a sys-
tem at an instant can be derived from the state at the
preceding instant makes it possible to work with differ-
ential equations (Poincar´e1921, 136).
Furthermore, physicists generally work with the
assumption that from similar antecedents follow sim-
ilar consequents; Poincar´e states that this can be made
mathematically exact through the assumption that the
consequent is a continuous function of the antecedent
(Poincar´e1921, 346). More generally, he claims that we
can always express the relations between physical quan-
tities by mathematical functions which are continuous,
differentiable and analytic. According to Poincar´e, this
assumption cannot be falsified, since any function can
be approximated closely enough by an analytic func-
tion to be observationally indistinguishable (Poincar´e
1905, 296–297; Poincar´e1921, 288; Van Strien 2015).
As Poincar´e points out, these assumptions automati-
cally lead to a deterministic conception of the world
(Poincar´e1905, 295). The assumption that the relations
between physical quantities can always be expressed
by analytic functions guarantees that any process can
be described by differential equations, and, although
Poincar´e does not make the point explicit, because of
the property of analyticity these equations are guaran-
teed to have a unique solution for given initial condi-
tions and thus to uniquely determine what will happen.
To summarize, according to Poincar´e, physicists work
among others with the following assumptions: (1) We
can always find more accurate laws of nature; (2) Phe-
nomena are determined by circumstances which are
close to them in both space and time; (3) Physical quan-
tities take continuous values, and the relations between
them can be expressed by continuous functions. These
assumptions are not empirically verifiable or falsifiable,
and neither of them can be taken as an a priori truth,
although we do have to make such assumptions in order
to make progress in science.
Generally, according to Poincar´e, scientific theories
are based on experience as well as on various types of
hypotheses. Among the types of hypotheses he distin-
guishes is that of ‘natural hypothesis’:
There are first those [hypotheses] which are per-
fectly natural and from which one can scarcely
escape. It is difficult not to suppose that the influ-
ence of bodies very remote is quite negligible, that
small movements follow a linear law, that the effect
is a continuous function of its cause. I will say
as much of the conditions imposed by symmetry.
All these hypotheses form, as it were, the common
basis of all the theories of mathematical physics.
They are the last that ought to be abandoned.
(Poincar´e1921, 135)
The examples which Poincar´e here gives of natu-
ral hypotheses are very similar to, and in part iden-
tical with, the above-mentioned assumptions. Natural
hypotheses are hypotheses which we cannot test empir-
ically, and of which we have no a priori guarantee that
they hold; nevertheless, it would be very difficult or
even impossible to do science without such hypotheses.
Poincar´e is known for his conventionalism and neo-
Kantianism in philosophy of science. De Paz (2014)has
argued that natural hypotheses can be understood as
a type of convention: despite the fact that they can-
not be verified, the scientist can decide to adopt these
hypotheses because they are useful for the constitu-
tion of scientific theories. It is clear that to label these
assumptions as ‘conventions’ does not mean that they
are arbitrary. In general, Poincar´e argues that although
conventions can be freely chosen, they are not arbi-
trary: ‘Experiment leaves us our freedom of choice, but
it guides us by aiding us to discern the easiest way. Our
decrees are therefore like those of a prince, absolute
but wise, who consults his council of state’ (Poincar´e
1921, 28). The choice for conventions is thus guided
by practical considerations and experience (Poincar´e
1921, 352; on Poincar´e’s conventionalism, see, e.g., Psil-
los 2014; Ivanova 2015). It can also be argued that nat-
ural hypotheses are stronger than conventions, as they
are necessary preconditions for science, and that they
123

Eur. Phys. J. H (2021) 46:8 Page 17 of 20 8
should rather be seen as playing the role of synthetic a
priori principles (Heinzmann and Stump 2017).19
Now we can come back to Poincar´e’s conception of
determinism. In Science and Value (first published in
1905), Poincar´e reacts to the views of the philosopher
´
Edouard Le Roy, who had used Poincar´e’s ideas in order
to argue for a more extreme version of conventional-
ism, according to which science is merely a human con-
struct (see Psillos 2014). Le Roy argues that determin-
ism depends on the assumption that our laws hold abso-
lutely, but since this is an assumption which we make
freely, it is of our own free will that we can end up
with a deterministic world view—thus, determinism is
inherently contradictory. Poincar´e is critical of Le Roy’s
anti-intellectualism and skepticism about science, but
admits that, in a sense, ‘we are determinists voluntar-
ily’ (Poincar´e1921, 347). In his final work, Derni`eres
Pens´ees, we find the following remark:
Science is deterministic; it is deterministic a pri-
ori; it postulates determinism, because without it,
science could not exist. It is also deterministic a
posteriori; if it started out by postulating deter-
minism, as an indispensable condition for its exis-
tence, it then demonstrates determinism precisely
by existing, and each of its conquests is a victory
for determinism. (Poincar´e1917 [1913], 244).
For Poincar´e, science is about finding laws of nature
through which events are determined. Scientific theo-
ries are thus necessarily deterministic in the sense that
they specify the conditions under which occurrences
take place. However, this does not mean that there is a
guarantee that determinism holds absolutely and that
we can be sure that every event that takes place in
nature is uniquely determined: although determinism
must be presupposed in order for science to be possi-
ble, the possibility of science is not guaranteed, and we
have to find out empirically how far we can get with the
development of scientific theories. Thus, we can only
confirm determinism insofar as science is successful. In
Poincar´e’s words, ‘science, rightly or wrongly, is deter-
ministic; wherever it enters, it brings in determinism’
(Poincar´e1917 [1913], 245).
Despite significant differences in the philosophies of
science of Mach, Boltzmann and Poincar´e, there is thus
a striking similarity in their positions on determinism.
All three argue that science is essentially about find-
ing the conditions under which phenomena occur (or,
with Mach, to find functional dependencies between the
phenomena), and therefore, in order to do scientific
research, one has to assume that such conditions can
be found, and that phenomena do not just take place
19 However, at least as regards the assumption of continu-
ity, Poincar´e does argue that the physicist has a choice in
whether or not to adopt it: ‘The physicist may, therefore, at
will suppose that the function studied is continuous, or that
it is discontinuous; that it has or has not a derivative; and
may do so without fear of ever being contradicted, either
by present experience or by any other future experiment’
(Poincar´e1921, 288).
by chance. In this sense, determinism is a necessary
feature of scientific theories. But all three of them ulti-
mately remain agnostic about whether nature is deter-
ministic at the ontological level. For these authors, the
issue of determinism is not about ontology, but rather
about the presuppositions of science, the methods and
assumptions that scientists use in their research, and
the attitude of scientists toward their theories.
Conclusion: the image of classical physics
The use of the term ‘classical physics’ may have col-
ored our perception of physics and mechanics in the
eighteenth and nineteenth century: this term suggests
a finished theoretical framework and may have con-
tributed to a perception of physics in this period as a
static and finished whole, rather than an active research
area which was changing and developing, and in which
there were a diversity of approaches and lively debates
about the foundations of the field. The same holds for
‘classical mechanics.’ I think that it is only through
the conception of classical physics as a unified and
essentially complete theoretical framework that the idea
that ‘nineteenth century physics was deterministic’ can
appear as straightforwardly true.20
Retrospectively, we can say that the aim of account-
ing for all phenomena within physics through a basic
set of laws of motion was never reached. By the late
nineteenth century, the idea that all processes within
physics are uniquely determined by the laws of mechan-
ics had lost a significant part of its plausibility: new
developments in physics showed a complex and diverse
set of phenomena, which were described through a
range of mathematical techniques, and it was increas-
ingly unfeasible to reduce all of physics to a basic
set of equations and to establish that these equations
uniquely determine the course any process. Moreover, it
was often necessary to work with idealizations, which in
the late nineteenth century led to the common accep-
tance of the idea that scientific theories could merely
offer idealized models, with which it may be possible to
formulate highly reliable predictions but which could
not be taken to exactly correspond to reality.
20 This has also been argued by Beller, who moreover argues
that the image of classical physics was created partly as
a rhetorical strategy to defend a certain interpretation of
quantum mechanics: ‘As this new paradigm [of quantum
mechanics] emerged, its founders constructed a profile of the
opposition and a description of past science simultaneously.
The ideas of the opposition were projected as most char-
acteristic of the overthrown past—in this way opponents
were automatically presented as conservatives; disposing of
the old and discrediting the opposition went hand in hand.
The opposition became simple-minded and reactionary; the
past became monolithic. The diversity, ingenuity, fluidity,
and epistemological resilience of past science was thus forced
into a few rigid, simplistic categories. In this way classical
physics became uniformly deterministic.’ (Beller 1999, 281).
123

8 Page 18 of 20 Eur. Phys. J. H (2021) 46:8
Nevertheless, the claim that physics was never deter-
ministic may be an overstatement. Among nineteenth
century physicists, one finds some who explicitly argued
for or against determinism, and quite a few who were
silent on the issue. Overall, the majority of physicists
in this period seems to have taken for granted that nat-
ural processes are uniquely determined and do not take
place by chance; even though they would not be able to
demonstrate that this was indeed the case, there were
also no compelling reasons to abandon this belief.21 If
determinism could not be demonstrated, neither could
it be shown that there was fundamental chance. With
the ontological humility that was characteristic of the
late nineteenth century, the question whether nature is
deterministic became undecidable. But even if physi-
cists remained agnostic about determinism at the onto-
logical level, it was still possible to hold on to determin-
ism as an assumption that has to be made when doing
scientific research. Examples can be found in the work
of Mach, Boltzmann and Poincar´e, who each conceived
of determinism as a presupposition of science: if science
is about finding out what happens under which con-
ditions, then the working scientist, when investigating
some phenomenon, should assume that its determining
conditions can be found.
Acknowledgements This paper is a result of a long
project for which I have many people to thank. I would
like to thank in particular Gabriel Finkelstein, Iulia Mihai,
Gregor Schiemann, and two anonymous referees for their
helpful comments on earlier versions of this paper.
Funding Open Access funding enabled and organized by
Projekt DEAL.
Open Access This article is licensed under a Creative Com-
mons Attribution 4.0 International License, which permits
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to the original author(s) and the source, provide a link to
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licence and your intended use is not permitted by statu-
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To view a copy of this licence, visit http://creativecomm
ons.org/licenses/by/4.0/.
21 For a contemporary perspective on the issue: Wilson
(2009,2013) argues that within classical physics, it is gener-
ally reasonable to assume that, if the equations describing a
process do not seem to yield unique solutions, the situation
can be fixed by attending to some details of the physical
situation which had not been taken into account before. He
argues that therefore, in cases in which equations do not
seem to yield unique solutions, it is better to speak about
‘descriptive holes’ rather than about ‘failures of determin-
ism.’
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