## 1. Biographical note and selective timeline



The definitive biography of Hermann von Helmholtz (1821-1894) is
by his friend and associate, the mathematician Leo Konigsberger.
Konigsberger's biography is available in toto from Google
Books, since it is in the public domain. While
Konigsberger's treatment of scientific subjects can be
dated somewhat, for biographical information his account is
unrivaled.



Selective Timeline




|  |  |
| --- | --- |
| 1845 | Joins Berlin Physical Society |
| 1847 | "On the Conservation of Force," pamphlet |
| 1849-1855 | Professor of Physiology at Konigsberg |
| 1850 | *Description of an Opthalmoscope for the Investigation of the
Retina in the Living Eye*, Berlin: Verlag von A. Forster |
| 1855-1858 | Professor of Physiology and Anatomy at Bonn |
| 1856-1867 | *Handbook of Physiological Optics*, Leipzig: Leopold
Voss |
| 1858 | "On Integrals of the Hydrodynamic Equations which Express
Vortex-Motions," Crelle's *Journal fur die reine
und angewandte Mathematik*, Vol. 55 |
| 1858-1871 | Professor of Physiology at Heidelberg |
| 1863 | *On the Sensations of Tone as a Physiological Basis for the
Theory of Music*, Braunschweig: Verlag von Fr. Vieweg und
Sohn |
| 1867 | Riemann's *Habilitationsrede*, given 10 June 1854,
"On the Hypotheses Underlying Geometry," published
posthumously by Dedekind, *Abhandlungen der Koniglichen
Gesellschaft der Wissenschaften zu Gottingen*, volume 13 |
| 1868 | "On the Factual Foundations of Geometry," Lecture in
Heidelberg, published in the Verhandlungen des
naturhistorisch-medicinischen Vereins zu Heidelberg. |
| 1868 | "On the Facts Underlying Geometry," 15th volume of
the *Abhandlungen der Koniglichen Gesellschaft der
Wissenschaften zu Gottingen* |
| 1869 | Correspondence with Beltrami |
| 1870-1874 | "On the Theory of Electrodynamics," Crelle's
*Journal fur die reine und angewandte Mathematik*, Part I,
Vol. 72 (1870); Part II, Vol. 75 (1873); Part III, Vol. 78 (1874) |
| 1871-1877 | Professor of Physics, University of Berlin |
| 1871 | "The Velocity of Propagation of Electrodynamic
Effects," Berliner Akademie |
| 1877-1887 | Professor of Physics, Military Institute for Medicine and
Surgery, Berlin |
| 1878 | "The Facts in Perception," address delivered on the
Foundation Day of the University of Berlin |
| 1886 | "On the Physical Significance of the Principle of Least
Action," Crelle's *Journal fur die reine und
angewandte Mathematik*, Vol. 100 |
| 1887-1894 | Founding president of the Physicalisch-Technische Reichsanstalt
in Berlin |
| 1893 | "Consequences of Maxwell's Theory Concerning the
Motions of the Pure Ether," Berliner Akademie |


## 2. Theory of Perception



Helmholtz's earliest study of physiology was with Johannes
Muller. Muller (1801-1858) was a keen experimenter and
naturalist along the lines of Ernst Haeckel and Alexander von
Humboldt, sailing to the tropics to find specimens for his studies
(Otis 2007, 6-14). Early in his career, Muller was also a
proponent of Schelling's *Naturphilosophie*.
*Naturphilosophie* took its cue from what Schelling took to be
Kant's assertion that, while each part of an organism supported
the whole, the only purpose of a living organism was itself.
"Muller's enormously influential *Handbook of
Human Physiology*, written and rewritten between 1833 and 1844,
shows his simultaneous commitments to vitalism, philosophy, and
rigorous science" (Otis 2007, 21).



According to the "projection" theory of the physiology of
perception, a stimulus causes its response directly. That is, our
nerves are malleable, like wax, and the objects "project"
signals onto the nerves directly, like a key pressed into the wax.
Based on experiments in his laboratory, and on the phenomena of
binocular vision revealed by the stereoscope Charles Wheatstone
invented in the 1830s, Muller observed that several phenomena of
the physiology of perception contradict the projection theory. One
such phenomenon is stereoscopic binocular vision, in which the two
images on our retinas are resolved into one image, the image we see.
If images are projected directly onto the sense nerves, Muller
asked, then how is stereoscopic vision possible? Muller also
cited the fact that images from objects are projected onto the retina
upside down, but we see them as right side up. The projection theory
has no explanation of these effects.



In response to experimental data, Muller constructed the
"law of specific sense energies"
[*Sinnesenergien*]. Muller argued that each nerve is
configured to receive a specific range of signals, as a radio is tuned
to receive sound on a certain wavelength (of course, Muller did
not use this analogy). Muller called the configuration a
"specific sense energy." Muller explained
stereoscopic vision by arguing that each retina is able to perceive
itself, that is, to perceive a priori the manifold, or grid, of points
that can be projected onto the retina. Muller hypothesized that
there was an organic correlation between the left and the right
retinas of the eye (Muller 1837-1840, 2: 351-385, see
also Turner 1993, 159-160). Each point on the left retina is
indexed to one and only one point on the right retina. Muller
argues that for each point seen in stereoscopic vision, the two
signals from the corresponding points on the two retinas are projected
onto a single point.



Muller's explanation appeals to an a priori perception of
the retina by itself, and to the activity of projecting two points
onto a theoretical projection surface a priori. Muller tries to
use Kantian philosophy to support his theory, arguing that by the
"manifold accessible to pure intuition" Kant meant, or
could have meant, the a priori possible manifold of physical points
that could be perceived by the retina. Robert DiSalle presents
evidence that this view is not Kantian (DiSalle 1993, 502ff).



From 1838 to 1842, Helmholtz studied medicine under Muller. From
1842 to 1848, while working as an army surgeon in Potsdam, Helmholtz
made many trips to Berlin to work in Gustav Magnus's library.
During these trips, Helmholtz stayed in contact with fellow students
of Muller's he had met while working in Muller's
laboratory, Emil du Bois-Reymond and Ernst Brucke. In 1845,
Helmholtz joined the Berlin Physical Society, which was founded by du
Bois-Reymond and also counted among its members Brucke and Werner
von Siemens. Du Bois-Reymond founded the Physical Society to support
experiment and to banish vitalism from scientific research. For more
on Helmholtz's early career and relationship with the Physical
Society, see Sulloway 1992, 14ff and 65ff, and Cahan 2012.



While Helmholtz did not contradict Muller by name in his writings
on physiology, Helmholtz's research, in Muller's and
Magnus's labs and later in his own labs in Bonn and
Konigsberg, is in conflict with the doctrine of specific sense
energies (see Otis 2007, 129-130). Helmholtz proposes a
"sign" theory, according to which sensations symbolize
their stimuli, but are not direct copies of those stimuli. While
Muller explains the correspondence between sensation and object
by means of an innate configuration of sense nerves, Helmholtz argues
that we construct that correspondence by means of a series of learned,
"unconscious inferences."



In his sign theory of perception as expressed in his early career
(1848-1868), Helmholtz argues that the mind makes a series of
mental adjustments, "unconscious inferences," to construct
a coherent picture of its experiences. Helmholtz argues that spatial
position, often used as a criterion to individuate objects, is an
interpretation of our sensations, and not their immediate result.
Again, stereoscopic vision shows that what may appear, to us, as a
single image is in fact two images resolved into one. Perspective can
distort size, as when one puts a finger in front of the moon.
Helmholtz believes that we learn how to interpret spatial concepts
through experience, which means that he has what he calls an empirical
theory of spatial perception. This theory coexists, in
Helmholtz's epistemology, with his above commitment to the sign
theory, according to which spatial properties are only properties of
representations. Helmholtz's invention of the ophthalmoscope in
1850-1851 contributed to his understanding of the physiology of
perception (Schett 1999; for images of the original Helmholtz
ophthalmoscope see De Schweinitz and Randall 1899, 172ff).
Philosophically, Helmholtz's epistemology commits him to the
view that representations arise in a physical process, but are signs
and not copies of their objects.



Helmholtz was inspired by the theory of
 Hermann Lotze
 (1817-1881) in his explanation of these phenomena. For Lotze,
Helmholtz observes, "to the sensations from spatially distinct
nerve endings correspond various determinate *Localzeichen*
[literally: place signs], whose spatial meaning is learned"
(Helmholtz 1968 [1869], 57). My various sensations of my finger are
originally unrelated, but I can relate them to each other by means of
the concept "my finger," which serves as a mental
*Localzeichen* that contains the data of all the sensations.
The space of perception is a general *Localzeichen* that
relates all possible sensations to each other. The usefulness of
Lotze's theory is that all psychological sensations are mapped
directly onto mental concepts, and even space becomes a tool for
constructing an interpretation of sense-data, akin to a language.



According to Helmholtz's explanations of the physiology of
perception, the qualities of sensations "belong only to our
nervous system," and we acquire our knowledge of spatial
ordering through perceiving an unchanging sequence of sense
impressions of the same object:



> 
> It is easy to see that by moving our fingers over an object, we can
> learn the sequences in which impressions of it present themselves and
> that these sequences are unchanging, regardless which finger we use.
> It is thus that our knowledge of the spatial arrangement of objects is
> attained. Judgments concerning their size result from observations of
> the congruence of our hand with parts or points of an object's
> surface, or from the congruence of the retina with parts or points of
> the retinal image. A strange consequence, characteristic of the ideas
> in the minds of individuals with at least some experience, follows
> from the fact that the perceived spatial ordering of things originates
> in the sequences in which the qualities of sensations are presented by
> our moving sense organs: the objects in the space around us appear to
> possess the qualities of our sensations. They appear to be red or
> green, cold or warm, to have an odour or a taste, and so on. Yet these
> qualities of sensations belong only to our nervous system and do not
> extend at all into the space around us. Even when we know this,
> however, the illusion does not cease, for it is the primary and
> fundamental truth. The illusion is quite simply the sensations which
> are given to us in spatial order to begin with (Helmholtz 1971 [1878],
> 376-7).
> 



As a concrete example of a judgment of relative spatial position,
Helmholtz gives the case of someone grasping a pen in her fingers. She
cannot infer directly from the sensation of the pen that it is in one
place, because each finger feels only the position of the pen relative
to the finger itself. She would have exactly the same sensations if
her fingers were touching two or three different pens, separated in
space. The belief that the pen is in one place only is based on her
knowledge that his fingers are close enough together that only one pen
will fit between them. As Helmholtz remarks,



> 
> When two different parts of the skin are touched at the same time, two
> different sensitive nerves are excited, but the local separation
> between these two nerves is not a sufficient ground for our
> recognition of the two parts which have been touched as distinct, and
> for the conception of two different external objects which follows.
> Indeed, this conception will vary according to circumstances. If we
> touch the table with two fingers, and feel under each a grain of sand,
> we suppose that there are two separate grains of sand; but if we place
> two fingers one against the other, and a grain of sand between them,
> we may have the same sensations of touch in the same two nerves as
> before, and yet, under these circumstances, we suppose that there is
> only a single grain. In this case, our consciousness of the position
> of the fingers has obviously an influence upon the result at which the
> mind arrives... What, then, is it which comes to help the
> anatomical distinction in locality between the different sensitive
> nerves, and, in cases like those I have mentioned, produces the notion
> of separation in space? (Helmholtz 1995 [1868], 175-6)
> 



Helmholtz argues that perceived properties such as separation in space
are well-founded inferences from two sources of knowledge: our
experience, and the properties of our sense organs. For Helmholtz,
knowledge of the way our physiology works in perception is essential
to any epistemological account of spatial properties. Helmholtz argues
that the more we know about the physiology of perception, the more
accurate our inferences about our experience will be. In the case of a
person grasping a pen or touching a grain of sand, Helmholtz argues
that we become aware that the object touched is a single object by
studying the position of our sense organs: the nerve endings in our
fingers, in this case. Our awareness of distinction of spatial
position, necessary for depth perception and perception of distance,
is learned and not innate. Lenoir (2006) is a rich history of
Helmholtz's influences on this score, including Johann Herbart,
Wilhelm Wundt, Carl Friedrich Gauss, and an analysis of
Helmholtz's theory of perception as
"operationalizing" the Kantian view.



In Helmholtz's early work, and in some later essays, perceptual
space is a mental generalization of our orientation with respect to
objects in space. We learn the general properties of space by
discovering what spatial properties do not change when objects move,
and when we move relative to the objects. The insight that the
spatially relevant properties of objects are the properties that
remain invariant when objects change position, or when we change
position relative to the objects, is fundamental to Helmholtz's
later work on topology.



In the third part of Helmholtz's *Handbook of Physiological
Optics*, Helmholtz draws a distinction between nativism and
empiricism in the physiology of perception. In drawing up the
*Handbook*, Helmholtz's "task was to survey the
field of physiological optics for the purpose of writing a work that
would appear as volume nine of Karsten's *Encyklopadie
der Physik*. The format of the Karsten series dictated that
Helmholtz provide a thorough review of both past and present work, a
task he performed diligently. Wishing to provide a systematic
theoretical framework for his survey of the field, Helmholtz divided
the major theoretical positions with respect to spatial perception
into two groups, which he characterized with the adjectives
*empiristisch* and *nativistisch*" (Hatfield 1991,
274-275. I follow Hatfield in translating the term
"empirismus" as "empirism."). Turner (1993)
has argued that Helmholtz's empirism was pragmatic - that
Helmholtz adopted an empirist stance to meet the challenges of
"imposing order on a chaotic new field of investigation"
and of defending his stance in his lifelong debate with Ewald Hering
(Turner 1993,155).



Significant questions in the empirism - nativism debate include
how to explain perception of depth and relief (Turner 1993, 156). The
"projection" theory explained relief perception by
claiming that the mind interprets visual cues to construct a combined,
relief picture out of two flat pictures provided by the eyes (Turner
1993, 158). By 1867, when Helmholtz published his *Handbook of
Physiological Optics*, Helmholtz had developed his
"sign" theory, according to which the brain's
construction of phenomena such as depth and separation in space is
learned. This account brought Helmholtz into conflict with Hering.



> 
> Nativism [Hering's account] is the view that at birth (or
> sometime thereafter, depending upon the ontogenetic timetable) the
> organism, upon visual stimulation, will experience a spatially
> organized array of colored areas. The key notion is that the
> disposition to have spatially organized visual experience is inborn
> and does not depend upon a process of learning. A nativist need not
> assert that an innately endowed organism will be visually competent at
> birth, for the visual system might become functional only after a
> further period of maturation. Empirism [Helmholtz's account] is
> the view that at least some of the spatial organization found in the
> visual experience of the adult is the result of learning; it asserts
> that some or all of the ability to perceive a spatially organized
> visual world is acquired (Hatfield 1991, 276).
> 



The debate between Helmholtz and Hering focused on several specific
cases, in particular, how to account for the so-called
"horopter" problem. The horopter is the set of points that
the eye perceives as equidistant from the perceiving subject when the
eye is focusing on a single point. The eye perceives points on the
horopter as if they were lying on a straight line at a fixed distance
from the eye. However, in reality the horopter line is a curve, that
is, in reality the points are not equally distant from the eye.



When looking at a single object, some points on the object stimulate
both eyes, but some points stimulate only the right or the left eye.
The brain resolves these common and distinct stimuli into a single
image (Hershenson 1999, 29-30). The fact that the brain must
resolve two distinct images into one accounts for the horopter effect.
When the eyes are focusing on a single point, the left and the right
eyes give different inputs to the brain about objects to the left or
right of the point of focus. The brain corrects for the different
inputs coming from the left and the right eyes by representing them as
equally distant from the eyes as the object of focus.



Hering and Helmholtz disagree, not about the empirical results or the
concept of the horopter itself, but about how to explain the
brain's resolution of two images into one. Helmholtz argues that
the brain adjusts the retinal images by a process of
"unconscious inferences." Helmholtz contends that a
child's brain learns to respond to stimuli as the child
develops, and that the brain unconsciously adjusts itself to produce a
coherent experience -- for instance, to resolve retinal
disparities. For recent work on unconscious inference see De Kock
2014a, De Kock 2014b, Patton 2018, Patton 2019.



Hering argues that the ability to experience objects as a single,
spatially ordered image is a disposition inborn in human children, and
not acquired. While children may not be born with the ability to
resolve two images into one, Hering claims that the ability develops
when a child grows to maturity, and is not learned. Hering argues that
depth perception and stereoscopic vision are inherent physical
abilities, like running or even breathing, that can be honed but are
not learned wholesale from experience. In fact, to Hering, the
adjustments the brain and eyes make to different inputs are automatic
and involuntary, like a heartbeat. Helmholtz contends that depth
perception and stereoscopic vision require reciprocal adjustment to
objects and thus are skills that must be learned through experience,
like shaving in front of a mirror (an example he takes from
Muller, see Otis 2007, 129). When shaving in front of a mirror,
one must learn to distinguish right from left in the mirror, and to
turn the right or left side of one's face to the mirror
correctly, even though the apparent motion in the mirror is the
opposite of the felt motion in one's body. Helmholtz argues that
the ability to shave in front of a mirror cannot be innate, because it
requires learned adjustments to the properties of one's
environment, that reflections in a mirror are inverted, for
instance.



Patton (2022) argues that the Helmholtz-Hering debate "revealed
deep fissures in nineteenth century accounts of scientific
explanation, as well as in the conception of how physiology,
psychology, physics, and philosophy are related": "Hering
was a pioneer of Lamarckian explanations, arguing for an early version
of the biogenetic law. Hering explains physical processes, including
perception, in terms of 'organic memory' that is supported
by 'vital forces' located throughout the body. Helmholtz,
on the other hand, argues that vital forces are in direct conflict
with the results he and others proved in the 1840s and 50s on the
conservation of force." 



Otis (2007) and Turner (1993) argue that the most significant
difference between Hering and Helmholtz is not in their distinct
explanations of the empirical evidence. Rather, the difference is in
the epistemological consequences of their higher-order commitments. If
Helmholtz is right, then our access to objective properties is not
direct, but constructed. In his early "sign" theory,
Helmholtz argues that perceptions of objects are not impressions like
the imprint of a key on wax, but are symbols or signs of their
objects, as a name is a symbol of a person. For Helmholtz, the degree
of resemblance between perception and object may be as remote as the
degree of resemblance between a written name and the physical person
to whom the name refers (Helmholtz 1882, 2:608, Helmholtz 1903,
1:41ff; cited in Schiemann 1998, 26). Hering objects strenuously to
these epistemological consequences of Helmholtz's sign theory,
and argues that we have direct access to real objects in
perception.



There is significant debate over the question of how far
Helmholtz's theory was influenced by Kantianism. David Galaty,
Nicholas Pastore, and David Leary have argued that Helmholtz was
Kantian in his approach (Galaty 1971, 159-66; Pastore 1978,
355-76; and Leary 1982, 36; cited in Hatfield 1991, 325). As
Michael Heidelberger has observed recently, the Neo-Kantian Alois
Riehl refers to Kant as a "man of the physiologists," and
Riehl refers to Helmholtz's "Kantian understanding of
sensory perception," although Heidelberger himself may not read
Helmholtz as a Kantian (Riehl 1876, v and 5; Heidelberger 2007, 30).
Gary Hatfield and Edwin Boring have argued that Helmholtz's
empiricism outweighs his Kantianism. Boring's classic history of
psychology inaugurated the contemporary reading of Helmholtz as an
empiricist or empirist (Boring 1942, chapter 15). Hatfield argues
that, while Helmholtz maintained some Kantian doctrines for a time,
his mature view departs from Kantianism (Hatfield 1991,
325-326), and Lenoir concurs (2006, 200-204). In a 1918
article, the Marburg Neo-Kantian Ernst Cassirer supports this mixed
reading of Helmholtz. According to Cassirer, although
Helmholtz's view was "linked deliberately to Kant,"
and Helmholtz was very influential in the origins of Neo-Kantianism,
Helmholtz made the a priori dependent on the results of natural
science, a significant departure from Kantianism (Cassirer 2005
[1918], 96). Edgar (2018) addresses the broader question of the place
of Helmholtz in the neo-Kantian tradition.



One area of debate centers on the influence of Johann Gottlieb Fichte
on Helmholtz's work. Heidelberger (1995) and De Kock (2018)
emphasize the influence of Fichtean idealist philosophy on Helmholtz,
noting, for instance, that Fichte's distinction between the
"I" and the "not-I" is employed in
Helmholtz's analysis of perceptual experience and in the
distinction between subjectivity and objectivity. Hatfield (2018)
responds that Helmholtz "rejected Fichtean idealism" and
was influenced more profoundly by empirical physiologists of
perception, including Muller and Wundt but also Johann Georg
Steinbuch and Caspar Theobald Tourtual. Hatfield argues that
Helmholtz's view in 1867 is better understood as "a form
of modest realism, a kind of structural realism" avant la
lettre, "and that, in 1878, he consolidated this position in a
way that renders it metaphysical" (2018, 33).



Swanson (2016) has argued that the 'predictive processing'
paradigm in contemporary cognitive and computational neuroscience has
roots in Kant and Helmholtz. Predictive processing means
"perception involves the use of a unified body of acquired
knowledge (a multi-level 'generative model') to predict
the incoming sensory barrage" (Clark 2015). As Swanson notes,
"The current PP paradigm emerged from early work on generative
models, and this work explicitly identifies itself as being directly
inspired by ... Hermann von Helmholtz ... For example, the
seminal article on the use of generative models in machine perception,
titled 'The Helmholtz Machine,' states that
'Following Helmholtz, we view the human perceptual system as a
statistical inference engine whose function is to infer the probable
causes of sensory input'" (Swanson 2016, 10; Dayan et al.,
1995, 89).



Tracz (2018) defends a reading that Helmholtz is a relationalist about
perceptual properties, drawing on a related argument from Allais
(2015) that transcendental idealism is a form of relationalism.
"Helmholtz's relationalism should be of interest to
readers of Kant, for according to one of the main strands of
interpretation, Kant's transcendental idealism treats all
properties that appear in perception as relational properties"
(2018, 65). Moreover, "relationalism, particularly color
relationalism, is alive in contemporary philosophy of perception
(e.g., Chirimuuta 2015, Cohen 2009). Despite this, the history of
relationalism is rarely thematized, so Helmholtz's views provide
some historical context of interest to present work in philosophy of
perception" (Tracz 2018, 65). Patton (2019, 73) adds that



> 
> While the relationalist reading of Helmholtz is certainly defensible,
> it is not the end of the story: restricting ourselves to an argument
> according to which Helmholtz *only* is defending contemporary
> relationalism would risk losing some of the force and complexity of
> Helmholtz's view... for Helmholtz, *habit* and
> *inference* - and even deliberate interference -
> can change the relational properties involved, and thus change my
> perceptual experience... for Helmholtz, Fechner, and Weber,
> perceptual experience of complex phenomena like music and colored
> objects is not reducible to the ground-level sensory response to
> physical stimuli. (Patton 2019, 73)
> 



Giulio Peruzzi and Valentina Roberti (2023) nominate Helmholtz as
one of the "chief architects of the new color theory", who
defined "the first non-Euclidean line element in color space,
i.e., a three-dimensional mathematical model used to describe color
differences in terms of color distances". Hyder (2009), Turner
(1996), and Kremer (1993) trace Helmholtz's engagement with the
work of James Clerk Maxwell and Hermann Grassmann on color. 


## 3. The geometry of physical space and topology



Even when he was writing about physiology, Helmholtz's vocation
as a mathematical physicist was apparent. Helmholtz used mathematical
reasoning to support his arguments for the sign theory, rather than
exclusively philosophical or empirical evidence. Throughout his
career, Helmholtz's work is marked by two preoccupations:
concrete examples and mathematical reasoning. Helmholtz's early
work in physiology of perception gave him concrete examples of how
humans perceive spatial relations between objects. These examples
would prove useful to illustrate the relationship between metric
geometry and the spatial relations between objects of perception.
Later, Helmholtz used his experience with the concrete science of
human perception to pose a problem for the Riemannian approach to
geometry.



Helmholtz's work on geometry was made public from 1868, the year
he gave a lecture, "On the Actual Foundations of
Geometry," in Heidelberg, and published his "On the Facts
Underlying Geometry." The latter "astonished" the
academic public, who had considered Helmholtz primarily an
experimental scientist and physiologist (Konigsberger 1906, 254).
However, Helmholtz had initially wanted to be a physicist, and had
always been alive to the mathematical consequences of his work in
physiology. Riemann's work revolutionized mathematics and
physics, and philosophy by extension, and Helmholtz was perhaps the
first to recognize the extent of the revolution. Indeed, Helmholtz
claimed that he had come to results similar to Riemann's, though
perhaps slightly later. Although the publication of Riemann's
*Habilitationsrede*, "On the Hypotheses Underlying
Geometry," meant that Helmholtz's own results would not be
seen as original, Helmholtz says in his own essay that he is content
that "so distinguished a mathematician [as Riemann] should have
considered these questions as worthy of his attention"
(Helmholtz 1883 [1868], cited in Konigsberger 261). With his 1868
essay, Helmholtz saw himself as contributing to the Riemannian
approach to the foundations of geometry (Konigsberger 1906, 254).
Helmholtz's title, "On the Facts Underlying
Geometry," is a deliberate echo of Riemann's "On the
Hypotheses Underlying Geometry." The difference in the titles,
facts versus hypotheses, underscores the fact that while
Helmholtz's approach is very similar to Riemann's, there
is at least one significant difference.



In "On the Hypotheses Underlying Geometry," Riemann
focuses on how to determine the overall topological properties of
space. Topological properties are those spatial properties of figures
that are invariant under transformation, that is, that stay the same
when the figures move. Riemann proves that a space of n dimensions,
that is, a space determined by n continuously and independently
varying magnitudes, has constant curvature, but only under the
hypothesis that all spatial figures can be moved or rotated anywhere
in space without changing their form, the so-called "Axiom of
Free Mobility" (cf. Konigsberger 1906, 260-261). For
the case of astronomy, Riemann observes:



> 
> If one assumes that bodies exist independently of place, then the
> measure of curvature is constant overall, and then it follows from
> astronomical measurements that the measure of curvature cannot be
> distinguished from null... However, if bodies are not independent
> of place, then one cannot deduce infinitely small mass relationships
> from the relationships between large masses. In that case, between any
> given points in three dimensions the measure of curvature can have any
> random value, provided that the entire curvature of any measurable
> part of space is not distinguishable from null (Riemann 1892 [1854],
> 285, my translation).
> 



The assumption "that bodies exist independently of place"
is valid if and only if the properties of bodies remain invariant when
moved. That is, if a body changes place, that is a movement, and if
the body continues to have the same properties in any different place,
then its properties are invariant under transformation. Riemann goes
on to say that, without this assumption, the units of comparison that
are the basis for astronomical and other measurements -- rays of
light as the shortest path between two points, rigid bodies like meter
sticks as the basis of distance measurement -- no longer have the
invariant properties on which one can base valid measurements.



In "On the Facts Underlying Geometry," Helmholtz
investigates a related, but distinct question. He agrees with Riemann
that geometry is not possible unless we can compare figures to each
other and measure them, and that measurement is not possible unless at
least some properties of the figures we are measuring do not change
when the figures move. Helmholtz asks, what are the most general
axioms of geometry that must hold for such motions, motions preserving
the spatial properties of figures, to be possible?



Helmholtz presents the idea of "rigid motions" to account
for these invariant properties. "Rigid motions" are those
motions that preserve a set of properties of the objects. For
instance, when a sphere rotates around its central vertical axis, that
motion preserves the sphere's symmetries about the x and y axes,
and thus these symmetries are invariant under that particular
transformation (rotation). A given object may have a group of
properties that remain invariant under transformation and a group that
change, that is, when objects move in a certain way they may stretch
or lose symmetry about an axis. After the publication of
Helmholtz's work, Sophus Lie argued that rigid motions form
groups, and described the set of rigid motions in the mathematical
terms of group theory.



While Riemann tries to describe the general properties of space,
Helmholtz asks what the most general geometrical axioms are that can
account for our empirical measurement of objects. These will be the
axioms that preserve observed rigid motions, given an underlying
system of geometry. Helmholtz investigates the question of which
systems of geometry (Euclidean, Lobachevskian, and Riemannian) are
possible under which assumptions. Here Helmholtz makes an initial
mistake, and argues that only Euclidean geometry can account for our
actual physical measurements. In April 1869, Eugenio Beltrami pointed
out in a letter to Helmholtz that Lobachevskian geometry could be
adequate to the task under the assumption that we live in a
"pseudosphere" (see Other Internet Resources). Helmholtz
concedes the point immediately (Konigsberger 1906, 263). In 1870,
in "On the Origin and Significance of the Axioms of
Geometry," Helmholtz investigates in much closer detail the
question of whether non-Euclidean geometries can be visualized. This
essay, perhaps even more than "On the Facts Underlying
Geometry," was widely influential. Moritz Schlick and Hans
Reichenbach engaged with Helmholtz's views in the 1870 essay
(see, e.g., Reichenbach 1920, Schlick's notes to Helmholtz
1921). A related article, "The Origin and Meaning of Geometrical
Axioms," was published in *Mind* in 1878, and Russell
responds to it in his doctoral dissertation.



In *Space, Number, and Geometry from Helmholtz to Cassirer*,
Francesca Biagioli (2016) analyzes Helmholtz's theories of
geometry and of measurement, including their origins in the work of
Kant, Riemann, and Gauss, and the reception, development, and
extension of Helmholtz's work through engagement with Felix
Klein's Erlanger Programm, Sophus Lie's theory of groups,
the conventionalism of Henri Poincare, and neo-Kantian
reception in Alois Riehl, Hermann Cohen, Ernst Cassirer, and Bruno
Bauch. Biagioli (2016), Biagioli (2014), and Biagioli (2018) deal with
the question of how Helmholtz and neo-Kantians, especially Ernst
Cassirer, could claim that space is transcendental without also
arguing for the universality of Euclidean geometry. Matthias Neuber
(2012, 163) argues that, while "Helmholtz's theory of
space had significant impact on Schlick's early 'critical
realist' point of view", Schlick ends up radically
transforming Helmholtz's account.



Marco Giovanelli (2017) argues that the mathematical core of the
development of special relativity is found in the Riemannian tradition
and especially in the work of Christoffel, Lipschitz, Ricci, and,
later, Levi-Civita (Giovanelli 2017, 328). Giovanelli argues
that the claims for the Helmholtzian tradition as the sole progenitor
of special relativity are exaggerated, and that the real development
was a 'collision' of traditions, in the sense well known
from John Norton's paper on the Kleinian and Riemannian
traditions in general relativity.



Biagioli (2023 and 2016), Darrigol (2003), Paula Cantu (2018),
and Matthias Neuber (Neuber 2018a, Neuber 2018b) delve into
Helmholtz's work on the theory of measurement, including his
essay "Uber Zahlen und Messen," as a precursor
of more recent treatments of the topic. Cantu focuses on the
question of the applicability of mathematics in general, a question
also found in Frege, Hale, Wright, and Batitsky, while Neuber and
Biagioli (2016) focus on Helmholtz's account of the a priori and
empirical conditions for measurement in physics and mathematics.
Biagioli (2023) focuses on Helmholtz's contributions to the
psychophysical problem of the measurement of sensation.



For more detailed explanations of the notions in this section, see the
entry on
 19th century geometry.



See Section 7 of this entry for references to work on the
philosophical implications of Helmholtz's theory of
geometry.


## 4. Conservation laws, electrodynamics, and acoustics


### 4.1 Conservation of energy: 1842-1854



From 1841 to 1842, Helmholtz finished his studies with Johannes
Muller in Berlin. As seen above in the section on physiology,
Muller's endorsement of experimental science was in
pragmatic conflict with his commitment to *Naturphilosophie*,
and in particular, to vitalism. According to vitalist theories, in
addition to the mechanical and physical forces present in a living
body, there is a "vital force" that makes the parts of the
body work together as an organism. The term for momentum, vis viva,
illustrates the belief. In the early part of the 19th
century, Ernst Heinrich Weber argued that the so-called "vital
forces" are in fact physical forces, and he argued that the most
pressing problem for 19th century physiology was to explain
the so-called vital forces in physical terms, and thus to banish
vitalism. According to at least one account, Muller took his cue
from Weber in setting problems for the students working in his lab,
including Helmholtz, Emil du Bois-Reymond, Rudolf Virchow, and Ernst
Brucke, although Muller knew that Weber's view was
inconsistent with his own commitments (Konigsberger 1906, 25). It
is evidence in favor of this view that in 1843, Muller published
Helmholtz's first essay, "On the Nature of Fermentation
and Putrefaction," written in support of Justus von
Liebig's anti-vitalist arguments against spontaneous generation,
in his own journal, *Muller's Archiv*.



After receiving his medical degree in 1842, Helmholtz worked as a
military surgeon in Potsdam from 1843 to 1848. However, he took
frequent trips to Berlin to work in Gustav Magnus's lab, and to
talk with Muller's other former students and with Carl
Ludwig. As mentioned above, in 1845 Helmholtz joined the Berlin
Physical Society, which du Bois-Reymond and Brucke founded with
the express intention of banishing vitalism. Also in 1845, Helmholtz
took a five month leave from his work as a surgeon to take his
qualifying exams. He used the time to work in the lab in Berlin, and
continued to focus on the consequences of Weber's and
Liebig's attacks on vitalism.



Liebig had focused attention on the question of whether an
organism's metabolism produces all its heat and mechanical
energy. If so, then appealing to a vital force to explain these
phenomena would be superfluous. Through experiments on frogs'
muscles using electrical currents, Helmholtz showed that the heat the
frogs' muscles produced was accounted for by metabolism and
muscular action. He published his results in 1845, again in
Muller's journal, in "On Metabolism during Muscular
Activity" (see Bevilacqua 1993, 297ff). Helmholtz realized,
again in 1845, that the question of whether forces could be accounted
for by mechanical means had a much more general application. The
presupposition of vitalism, that there is an inexhaustible
"vital force" that powers living bodies, had led some
investigators to posit that there is an inexhaustible force, whether
mechanical or not, that can power a machine indefinitely: a perpetual
motion machine, *perpetuum mobile*. The realization was
prompted by Helmholtz's study of the 18th century classics of
mathematical analysis: Euler, Bernoulli, D'Alembert, and
Lagrange. As Helmholtz remarked years later, by the end of the 18th
century these mathematicians had established the impossibility of a
perpetual motion machine powered by mechanical forces:



> 
> for all purely mechanical, that is to say, for moving
> forces,...all our machinery and apparatus generate no force, but
> simply yield up the power communicated to them by natural forces,
> falling water, moving wind, or by the muscles of men and animals.
> After this law had been established by the great mathematicians of the
> last century, a perpetual motion, which should make use only of pure
> mechanical forces...could only be sought after by bewildered and
> ill-instructed people (Helmholtz 1995 [1854], 24).
> 



But what about the non-mechanical forces: heat, light, electricity and
magnetism? When Helmholtz started to look into the question of the
character of non-mechanical forces, instead of looking for perpetual
motion, he asked "if perpetual motion is impossible, what is the
relationship between natural forces that must hold?"



On the 23rd of July in 1847, Helmholtz gave an address, "The
Conservation of Force," at the Physical Society.
"Force" [Kraft], as Helmholtz uses it, is equivalent to
the modern term "energy." Helmholtz's address was
very well received by the Society, but Helmholtz was forced to publish
it as a pamphlet after Poggendorff rejected it for his
*Annalen* as too speculative.



Helmholtz summarizes his conclusions in the essay as follows:



> 
> The deduction of the propositions contained in the memoir may be based
> on either of two maxims; either on the maxim that it is not possible
> by any combination whatever of natural bodies to derive an unlimited
> amount of mechanical force, or on the assumption that all actions in
> nature can be ultimately referred to attractive or repulsive forces,
> the intensity of which depends solely on the distances between the
> points at which the forces are exerted. That both these propositions
> are identical is shown at the commencement of the memoir itself
> (Helmholtz 1853 [1847], 114-115; cited in Konigsberger
> 1906, 39).
> 



Helmholtz argues that a proof that all natural actions can be
accounted for by universal action at a distance is equivalent to a
refutation of perpetual motion.



In 1842-1843, Robert Mayer and James Joule had formulated
principles of the conservation of energy. Their work asserted the
conservation of mv2, the mass of a particle times the
square of its velocity. Further, Joule and Mayer argued that heat and
mechanical work are interchangeable (see Kuhn 1969, Mach 1911).
Helmholtz called on Joule's work (though he claimed to be
unaware of Meyer's) to achieve three results. As Bevilaqua
(1993) sums it up, Helmholtz concluded that:



> 
> The principle of conservation of force implies that the maximum
> quantity of work available from a system is a determined, finite
> quantity if the acting forces do not depend on time and velocity; if
> they do so depend, or if the forces act in directions other than that
> joining the active material points, the "force" can be
> gained or lost ad infinitum; and under non-central forces, a system of
> bodies at rest could be set in motion by the effect of its own
> internal forces (Bevilaqua 1993, 315).
> 



Kenneth Caneva (2019) argues that Helmholtz's principle is a
"creative refashioning" of the tradition of rational
mechanics, especially the principles of vis viva and its
conservation.  Bevilacqua argues that the main innovation of
Helmholtz's work was to unite two fields, and to anticipate the
integration of potential energy into mechanics:



> 
> In the tradition of analytical mechanics, the stress had been on the
> conservation of *vis viva*; in the tradition of mechanical
> engineering on the conservation of work. Helmholtz, by contrast,
> stressed the equivalence of the two. It was the introduction of the
> term *Spannkraft* that brought the real shift in meaning: with
> tension forces we are very far from the concept of work and very close
> to that of potential energy (Bevilaqua 1993, 315).
> 



While Helmholtz's work was published in pamphlet form in
Germany, it was picked up quickly by English scientists and published
in English translation almost immediately. Helmholtz's work
became known quickly in English scientific and philosophical circles,
which contributed to his lasting influence in both fields (see Cahan
2012).



Helmholtz acknowledged in his essay that earlier scientists, including
Joule, Newton, Bernoulli, and Rumford, had arrived at various forms of
the principle of energy conservation. In his history of mechanics,
Ernst Mach argues that some form of Helmholtz's principle was
known to "almost all eminent investigators" in history
(Mach 1911, 20). In his landmark study of energy conservation, Thomas
Kuhn presents the work of Joule, Mayer, and others from the 1830s,
which argues that heat and work can be substituted for each other
quantitatively, which is a significant component of Helmholtz's
principle (Kuhn 1969, 321). Kuhn argues that the discovery of the
principle of energy conservation is a case in which several scientists
lay the experimental and conceptual framework for a principle
together.



Kuhn does insist that Helmholtz formulated the principle. Robert
Purrington observes, following Kuhn, that "There was a widely
held, intuitive view that the total quantity of force (or motion)
might be somehow constant, but it did not necessarily imply an ability
to distinguish the vector nature of force or momentum from the scalar
energy" (Purrington 1997, 105). In his "Philosophical
Introduction" to the essay, Helmholtz himself argues that his
task in the essay is not to find new experimental evidence or a
completely new principle, but rather to explain the "unknown
causes" of the phenomena from their "visible
actions" according to the "laws of causality":



> 
> The problem of the sciences...is, in the first place, to seek the
> laws by which the particular processes of nature may be referred to,
> and deduced from, general rules. These rules--for example, the
> law of the reflection and refraction of light, the law of Mariotte and
> Gay-Lussac regarding the volumes of gases--are evidently nothing
> more than general ideas by which the various phenomena which belong to
> them are connected together. The finding out of these is the office of
> the experimental portion of our science. The theoretic portion seeks,
> on the contrary, to evolve the unknown causes of the processes from
> the visible actions which they present; it seeks to comprehend these
> processes according to the laws of causality (Helmholtz 1853 [1847],
> 114-115; cited in Konigsberger 1906, 39).
> 



For the philosophical implications of Helmholtz's work on the
conservation of energy, see Bevilacqua (1993) and Hyder (2006 and
2009). Bevilacqua investigates two matters: Hemholtz's synthesis
of earlier work into a single principle, and his methodological
distinction, evident in the above citation, between theoretical and
experimental physics. Hyder investigates Helmholtz's argument in
the essay that "force functions must be definable with regard
only to the relative positions of the mass-points comprising a
physical system" (Hyder 2006, 1). Marshalling the Kantian
antecedents of the argument, Hyder maintains that Helmholtz's
response to criticisms of "On the Conservation of Force"
by Clausius in 1854 influenced Helmholtz's later work on
geometry. Hyder 2009 is an analysis of Helmholtz's
"radical critique" of Kantian geometry in the context of
Helmholtz's retention of a "transcendental element"
in physics, especially with respect to the relationship between
geometry and empirical determinacy (2009, 19). In contrast, Edward
Jurkowitz (2010) argues that "already in his 1847
'Uber die Erhaltung der Kraft' one may find important
aspects of [Helmholtz's] later empiricism" (p. 39).


### 4.2 Acoustics, electrodynamics, and fluid dynamics: 1855-1881



In 1857, Helmholtz published "On the Integrals of the
Hydrodynamic Equations which Express Vortex-Motion" in
Crelle's *Journal fur die reine und angewandte
Mathematik.* Konigsberger observes that this essay was a
"work of genius that proved him to be a mathematician of first
rank" (Konigsberger 1906, 167).



Helmholtz knew of Euler's and Lagrange's previous
mathematical formulas describing fluid motion. Euler's equation
treated fluids as continua or fields, whereas Lagrange's
formulation treats them as discrete particles (see Emanuel 2000, 8).
Helmholtz took up Euler's equation, which gives the equations of
the flow of fluids in which friction does not alter the fluid's
motion significantly (the "inviscid" fluids). Lagrange had
remarked that Euler's equations did not work for
"viscous" fluids, in which the fluid's friction
alters its movement, because Euler's equations work only on the
assumption of a general conservation law, and viscous forces are not
conservative (Farge 2004, cf. Konigsberger 1906, 167).



Helmholtz, who had formulated a conservation law ten years previously
in "On the Conservation of Force," entered the field at
this point. Helmholtz's own work had concentrated on applying
the notion of a potential to fluid movement. But viscous forces do not
have potentials. Helmholtz decided to disregard the viscous forces
themselves, but also to redefine the problem without appealing to
potentials. He defined the "vorticity" of a fluid in a
small region as its average rotation or circulation in that region:
the rotation at each point on the boundary of the region divided by
the area of the region, which is described mathematically as the curl
of the velocity of the fluid. (In technical terms, Helmholtz evaluated
this as the curl of Euler's equation.) The vorticity of a fluid
at a single point is a vector quantity. Helmholtz introduced two
further idealizations. A "vortex line" is the tangent to
the vorticity vector of a point in the fluid. If you draw vortex lines
on the boundary of a region of the fluid, and then make the region
around each line indefinitely small, the lines will converge to
"vortex filaments," now also called
"vortices."



Helmholtz was able to prove three theorems in fluid dynamics using
these notions. In their modern expression, they are:


1. "Fluid particles originally free of vorticity [rotation]
remain free of vorticity.
2. Fluid particles on a vortex line remain on a vortex line, so that
vortex lines move with the fluid.
3. The strength of the vorticity is proportional to the length of the
vortex line" (Fuhs and Shetz 1999, 736).



These laws are used still in fluid dynamics, though they are modified
slightly from Helmholtz's original version (they are cited above
in the contemporary, modified form).



Helmholtz's work on fluid dynamics is significant
philosophically because Helmholtz's equations requires
*ideal* fluids, that is, fluids that are free from viscosity
and perfect continua. Helmholtz's equations are a paradigm case
of mathematical idealizations in physics. For a discussion of
Helmholtz's work in this context, including his discussions with
Gustav Kirchhoff and the notion, very significant in contemporary
fluid dynamics, of Helmholtz-Kelvin instability (or Kelvin-Helmholtz
instability), see Eckert 2006, 19ff. For accounts of Helmholtz's
use of idealizations in physics and of the development and
significance of his work in fluid dynamics, see Patton 2009 and
2012.



In 1863, a professor of physiology at Bonn, Helmholtz published his
pioneering work on sound waves, acoustics, and musical theory in
*On the Sensations of Tone as a Physiological Basis for the Theory
of Music*. One of the most significant discoveries in *On the
Sensations of Tone* is the precise mathematical description of
"sound vibrations in an open cylindrical tube"
(Konigsberger 1906, 206). Helmholtz begins *On the Sensations
of Tone* with a description of noise as we experience it in
ordinary life:



> 
> We perceive that generally, a noise is accompanied by a rapid
> alternation of different kinds of sensations of sound. Think, for
> example, of the rattling of a carriage over granite paving stones, the
> splashing or seething of a waterfall or of the waves of the sea, the
> rustling of leaves in a wood (Helmholtz 1954 [1863], 7).
> 



To analyze the "rapid alternations" of these noises into
their component vibrations, Helmholtz constructed the "Helmholtz
resonator." Helmholtz began by removing the bottom of an empty,
uncorked wine bottle, stretching a membrane over the bottom, and
fastening the membrane with a band. Then he suspended a single thread,
with a bit of wax on the end, from the band so that it hung halfway
down the membrane. If the thread is agitated, the bit of wax hits the
membrane as a drumstick hits a drum. Based on experimental evidence
and mathematical reasoning, Helmholtz argued that there will be a
single tone, the "prime tone," that is the tone at which
the membrane vibrates at the highest frequency, based on the thickness
and size of the membrane. By performing experiments, Helmholtz was
able to distinguish the prime tones of various membranes. Then if he
played music or made noise, the vibration of the membranes would
reveal the component vibrations of the sound.



Then Helmholtz had the idea of constructing more sophisticated
resonators, made of spheres or cylinders of glass or metal. The
resonators narrow to a small, hollow point at one end, and have a
circular opening at the other. Helmholtz placed a small piece of hot
wax at the small end, let it cool, and then inserted it into his ear
or the ear of a subject. The prime tone of such a resonator is
determined by its own composition and by the sympathetic vibrations of
the ear. If a sound other than the prime tone of the resonator plus
ear system is played, the listener hears only muffled noises, but
"if the proper tone of the resonator is sounded, it brays into
the ear most powerfully" (Helmholtz 1954 [1863], 43). Again,
using resonators allows an experimenter to distinguish the component
vibrations of music or ordinary noises, by first determining the prime
or proper tone of the resonator, and then determining, using the
resonator, whether the music or noise contains that tone.



Heller (2012) considers Helmholtz's most enduring contributions
to be his theories of dissonance and musical harmony (p. 441). In the
theory of pitch, Heller analyzes a debate between "Helmholtz and
Georg Ohm on one side and Rudolf Koenig and August Seebeck on the
other", which engaged physicists and instrument makers alike for
decades (441ff.). Heller's work provides an appreciation of the
strengths and weaknesses of Helmholtz's work from the
perspective of modern physics and acoustical theory. Deutsch 1984
provides a complementary perspective on Helmholtz's place in the
history of the physiology and psychology of music perception.



Hui 2011 and especially 2012 are detailed histories of research in
psychophysics and in acoustics in 19th century Germany, which
illuminate not only the social and cultural context, but also the
debates over scientific advances and experimental results.



It is typical of Helmholtz that, even as he was working on acoustics,
he realized that his work on sound waves could apply to related wave
phenomena, if supplemented with mathematical reasoning and
experimentation. Recent work on electrodynamics supported the
conclusion that electricity was a wave phenomenon. In 1845, Faraday
had demonstrated the connection between electric and magnetic
phenomena. In 1856, Wilhelm Weber and Rudolf Kohlrausch discovered the
ratio between electromagnetic and electrostatic units of charge (Assis
1994, 18). If he could describe sound vibrations in a cylindrical
tube, Helmholtz reasoned, similar equations might be successful for
describing the motion of electrical waves around a circular
boundary.



In 1861, Helmholtz gave a lecture, "A Universal Method of the
Transformation of Problems of Electrical Distribution," to the
Society for Natural History and Medicine at Heidelberg. Helmholtz was
unaware that, in letters to Liouville, William Thomson (Lord Kelvin)
had achieved the same results that he had, but he was told of this
immediately afterward. Helmholtz, who had met Kelvin in 1855 and had
been corresponding with him since 1856, acknowledged Kelvin's
priority in the Transactions of the Heidelberg society in 1862, and
wrote to Kelvin to ask whether he would be willing to publish his
results about the "distribution of electricity at a circular
edge" (Konigsberger 1906, 206). Kelvin responded with his
own results. Between 1861 and 1864, James Clerk Maxwell postulated
that light was an electromagnetic wave in the ether (Assis 1994, 18).
In 1864, Helmholtz traveled to England. During the trip, Helmholtz met
with Kelvin, John Tyndall, George Stokes, James Joule, Michael
Faraday, Thomas Huxley, Thomas Graham, Max Muller, and Maxwell.
He visited Graham's, Maxwell's, and Kelvin's
laboratories. By the end of the 1860s, Helmholtz was well informed on
the experimental and theoretical basis of the new electrodynamic
theory in England.



Alisa Bokulich (2015) explores the analogies that Maxwell and
Helmholtz draw between hydrodynamics and electrodynamics, arguing that
their methods give "insight into debates about the
representational and explanatory power of mathematical models"
(p. 28). Susan Sterrett (2017) "provides a critical history of
the concept of physically similar systems", including the use of
the concept in Helmholtz's theoretical hydrodynamics.



In 1870, Helmholtz published the first Part of "On the Theory of
Electrodynamics," "Equations of Motion of Electricity in
Conductors at Rest," in Crelle's *Journal fur die
reine und angewandte Mathematik*. In the essay, Helmholtz
supported Maxwell's work, but criticized Wilhelm Weber's
electrodynamic equations, charging that Weber's equations posit
an infinite kinetic energy, which contradicts Helmholtz's
conservation law of 1847. Weber and Helmholtz disputed the question
throughout the 1870s. Over the next several years, Helmholtz published
two more Parts of "On the Theory of Electrodynamics," in
which he responded to Weber and continued to support Maxwell's
assertion that light is an electromagnetic wave in the ether. The
debate between Helmholtz and Weber did not reach a conclusion until
the end of the 1880s, when the explanation of electromagnetic force in
terms of action at a distance between particles in the ether gave way
to the field theory.


## 5. Thermodynamics, the least action principle, and free energies: 1881-1887



In 1880, Helmholtz became the director of the Institute of Physics in
Berlin. Between 1881 and 1884, Helmholtz attacked the question of how
to integrate energy conservation and Maupertuis's principle of
least action to describe thermodynamic and chemical processes.



Over the 19th century, the application of the least action principle
dominated the approach of analytical dynamics to physical problems.
Lagrangian mechanics seeks to determine the trajectory of a system of
particles over time by solving Lagrangian equations for the system.
These equations are formulated using the least action principle. In
the Lagrangian formalism, a system of particles follows a path that
minimizes the action over time. Lagrange's equations can be
applied easily to systems of polar coordinates and of Cartesian
coordinates alike, a reason that the Lagrangian formalism was adopted
preferentially over the 19th century.



While the Lagrangian formalism is well adapted to evaluating
mechanical systems of particles, it is not as well suited to
evaluating some transfers of energy, which involve calculations over a
vast multitude of molecules. The Hamiltonian formalism is better
suited to this task. The Hamiltonian equations also evaluate the
action in a system, but use an integral sum of the momenta of the
elements of the system. In the simplest case, the Lagrangian deals
with the velocity of a particle and the Hamiltonian with the momenta
of the particle. The Hamiltonian gives a minimum value for a function
over any path given the initial and end states of the action. Thus,
the Hamiltonian formalism yields the same result, namely the
mathematical determination of the path taken by a system.



The Hamiltonian expression of the momenta of the system can be derived
from the Lagrangian, and vice versa, using a Legendre transform. In
fact, the Hamiltonian just *is* the Legendre transform of the
Lagrangian for any given system. The formalisms are equivalent, but
there are some situations, including evaluating heat transfer, in
which the Hamiltonian is preferred.



In 1882, Helmholtz gave an address, "The Thermodynamics of
Chemical Processes," at the Berlin Academy. Up until
Helmholtz's address, chemical reactions had been explained by
"chemical forces" or "affinities" between
chemical substances, measured quantitatively by the heat developed
during a chemical reaction. Gustave Coriolis had clarified the notion
of work as the product of force over distance in 1821, and this notion
was in common use by the late 19th century. In his address, Helmholtz
"proved that affinity was not given by the heat evolved in a
chemical reaction but rather by the maximum work produced when the
reaction was carried out reversibly" (Kragh 1993, 405). However,
while kinetic and mechanical energy can be converted into heat in
every case, only in restricted cases can heat be converted into
kinetic and mechanical energy. Hence, the equations describing
chemical processes involving heat could not always be reversed. These
are the conditions under which the Legendre transform could not be
applied, and thus the Hamiltonian could not be determined for that
system.



In particular, the equations for a system containing heat as a
variable contain entropy as a variable quantity. Entropy is an
inconvenient variable, difficult to control for and hold constant as
one can hold temperature, pressure, and volume constant. The Legendre
transform allows a researcher to convert equations containing entropy
into equations expressed only in terms of temperature, pressure, and
volume. The Legendre transform can be applied correctly only under
certain conditions, which must be specified.



Helmholtz proposed the notion of a "free energy" to
account for cases involving heat and entropy. The Helmholtz free
energy is defined as *F* = *E* [?] *TS*,
where *E* is energy, *T* is temperature, and *S*
is entropy. The free energy equation yields a quantity, *F*,
that is independent of heat and entropy. Many equations involving
*F* and not *T* or *S* are fully reversible, and
so Helmholtz's work allowed for the application of the
Hamiltonian to many chemical processes. Hence while "Helmholtz
was neither the sole nor the most important contributor" to
theoretical chemistry, "his thermodynamic theory of
1882-1883 was the pioneering work on which much of the new
theoretical chemistry rested" (Kragh 1993, 406).



For discussions of the philosophical and scientific implications of
Helmholtz's work in thermodynamics, see Kragh 1993 and Campisi
2005.


## 6. The Berlin School of Physics: 1878-1894



By the end of the 1870s, the experimental basis of electrodynamics had
been established, but mathematics lagged behind. In memoirs of his
training, Arnold Sommerfeld recalls that until the 1880s, the laws of
electromagnetic phenomena were constructed using unwieldy extensions
of Coulomb's, Biot-Savart's, and Weber's
generalizations of Newton's laws (Sommerfeld 1952, 1-2).
Further, although the relationship between electric and magnetic
phenomena had been established, there was no direct empirical evidence
for the existence of electromagnetic waves.



Between 1870 and 1887, Helmholtz was Director of the Institute of
Physics at Berlin University. Between 1880 and 1933, the following
scientists worked in Berlin: "Max Planck [Helmholtz's
student] and Albert Einstein above all, also Gustav Robert Kirchhoff,
Friedrich Kohlrausch, Emil Warburg, Walther Nernst, Max von Laue,
James Franck, Gustav Hertz, Erwin Schrodinger, Peter Debye"
and others (Hoffmann 1998, 1).



Perhaps the most significant relationship Helmholtz had with a student
or colleague was with Heinrich Hertz. Beginning in 1878, Hertz did
research as a graduate student in Helmholtz's and
Kirchhoff's laboratories. In 1879, Helmholtz set a prize problem
for the Prussian Academy of Science: "To establish
experimentally any relation between electromagnetic forces and the
dielectric polarization of insulators" (von Harnack 1900, 617,
cited Hoffmann 1998, 6). While Hertz was very interested in the
problem, he put it aside in 1879 to pursue other work. In 1880, Hertz
graduated and took a position as Helmholtz's assistant. From
1886 on in his own laboratory in Karlsruhe, Hertz performed his
now-famous experiments, which Hertz himself acknowledged were prompted
by Helmholtz's prize question, demonstrating the existence of
radio waves (which are a kind of electromagnetic wave). Hertz
"discussed his progress and results immediately and in great
detail with Helmholtz" (Hoffmann 1998, 6).



Hertz's experiments and their consequences finally laid to rest
the disputes between Helmholtz and Wilhelm Weber. Among those
consequences was the death of Helmholtz's view that
electromagnetism is a kind of action at a distance, and the
replacement of the action at a distance view with a field theory,
"one of the major turning points in the history of
physics" (Heidelberger 1998, 9). Since Hertz revealed empirical
evidence for the existence of electromagnetic waves, "the
sources of the electrodynamic effect are not anymore to be sought in
agents that are hidden from our view but in the surrounding
medium" (Heidelberger 1998, 17). While Helmholtz supported
Maxwell's view that electromagnetism was a wave in the ether,
Helmholtz explained wave propagation by means of action at a distance.
Hertz demonstrated that electromagnetic waves were perturbations in a
medium, which at the time was thought to be the ether, and which later
was re-conceived as a field. More importantly, Hertz explained the
propagation of electromagnetic waves directly "from the
interactions themselves," based on his empirical data, and thus
did not need to appeal to actions at a distance (Buchwald 1994,
265ff).



Although Hertz's results were in conflict with his own theory,
Helmholtz supported Hertz's career with unflagging enthusiasm
until Hertz's death at the age of 37. Recent scholarship by Jed
Buchwald emphasizes the influence of Helmholtz's methods and
theory on Hertz's discoveries (Buchwald 1994).



In 1888 Helmholtz assumed the founding directorship of the
Physikalisch-Technische Reichsanstalt, an institute for the study of
physics and industrial technology in Berlin founded in part by
Helmholtz's lifelong associate Werner von Siemens. For a
comprehensive history of the institute that illuminates the history of
Helmholtz's later career, see Cahan 1989. Helmholtz retained the
post until the end of his life.



For philosophical treatments of the shift from action at a distance to
field theory, and from Helmholtz to Hertz, see Heidelberger 1998 and
Leroux 2001. See also Purrington 1997.


## 7. Epistemology



In 1921, 100 years after Helmholtz was born, Paul Hertz and Moritz
Schlick brought out an edition of Helmholtz's epistemological
writings: *Schriften zur Erkenntnistheorie*. This edition made
available "On the Origin and Significance of the Axioms of
Geometry," "On the Facts Underlying Geometry,"
"Numbering and Measuring from an Epistemological
Viewpoint," and "The Facts in Perception."
Schlick's part in the selection of work having to do with
geometry and empirical measurement is significant. Schlick, a close
associate of Einstein's, saw the relevance Helmholtz's
work had for Einstein's theory. For a discussion of the
influence of Helmholtz on the early career of Einstein, see Cahan
2000, 70ff.



Schlick and Paul Hertz assess the relevance of Helmholtz's work
as follows: Helmholtz's "classic works on physics --
one may for example recall the energy principle or his work on vortex
motions -- stand at the *end* of lines of
development... his work seems for science at its present stage to
be something completed and left behind. There is only one field, but a
very important one, to which this does not apply: it does not hold for
Helmholtz's epistemological studies" (Hertz and Schlick
1977 [1921], xxxiv). In his Notes on "On the Origin and
Significance of the Axioms of Geometry," Schlick sees the
"chief epistemological result" of Helmholtz's work
in the replacement of the Kantian necessary a priori with a judgment
that "Euclidean space is *not* an inescapable form of our
faculty of intuition, but a product of experience"
(Schlick's note to Helmholtz 1921 [1868], 35).



For a thorough discussion of the anti-Kantian implications of
"On the Origin and Significance of the Axioms of
Geometry," see DiSalle 1993, 516ff. DiSalle discusses
Helmholtz's famous debate with the Kantian J. P. N. Land, as
does David Hyder (Hyder 2006, Land 1877). DiSalle also mentions that
in 1972, Jurgen Ehlers, Felix Pirani, and Alfred Schild
"derived the metric of general-relativistic spacetime from
assumptions about falling bodies and light rays" (DiSalle 1993,
520n). Ehlers, Pirani, and Schild note that their method "has
some similarity to Helmholtz's derivation of the metrics of
spaces of constant curvature" (Ehlers et al. 1972, 65).



Despite Schlick's conviction that Helmholtz's view is
antithetical to Kantianism, Helmholtz's notion that the a priori
in space consists of the manifold of possible orientations in space
has inspired new readings of Kant's a priori. This influence
comes partly through the mediation of Hans Reichenbach, whose reading
of Helmholtz was significantly different from Schlick's
(Reichenbach 1920). Reichenbach and Schlick both pointed out the
relationship between the views of Helmholtz and those of
Poincare, though Schlick realizes that Helmholtz is not a
conventionalist and Reichenbach may not have done (see Schlick's
note 40 to Helmholtz 1921 [1868], 34). For a discussion of
Helmholtz's influence on Michael Friedman's reading of
Kant, also influenced by Robert DiSalle, see Friedman 2000. Also see
Friedman 2001, Chapter One.



In *The Mechanics of Meaning*, David Hyder traces discussions
of Helmholtz's theory regarding "sensation-spaces and
their relation to the philosophy of space and perception" by
Boltzmann, Poincare, Carnap, Russell, and Weyl (Hyder 2002,
19n). De Kock 2013 is an analysis of the problem of externality in
Helmholtz's theory of perception. Biagioli (2014) analyzes
Helmholtz's debate with Albrecht Krause over the status of
geometrical axioms; more generally, Biagioli's work illuminates
Helmholtz's relationship to Poincare and to Holder.
Darrigol 2003 also places Helmholtz's work in context, including
with respect to related work by Grassmann, Fechner, and Wundt.
Darrigol's focus is Helmholtz's theory of number and
measure in physics and mathematics.



In *Hermann von Helmholtz's Mechanism*, Gregor Schiemann
sees Helmholtz's work in the sciences as characteristic of a
shift from the early modern to the modern tradition, as emblematic of
the "relativization of claims to validity" of scientific
knowledge, the "loss of certainty" characteristic of the
modern attitude toward science.



The influence of Hertz, and indirectly Helmholtz, on Wittgenstein is
argued by many. Brian McGuinness, Wittgenstein's biographer, and
Peter Hacker, in *Insight and Illusion*, are perhaps the first
to have so argued (see, e.g., Grasshoff 1993, 243, Hacker 1986,
2-4). Gerd Grasshoff traces the development in Grasshoff 1993,
arguing that Hertz and Boltzmann were the first to develop a
"picture theory," according to which physical and
linguistic theories give us pictures [*Bilder* in the original
German] of the world. Hertz's and Boltzmann's picture
(*Bild*) theories go well beyond Helmholtz's view.
Helmholtz argues that the constraints on using *Bilder* to
construct scientific explanations are given by physical experience and
natural science, and Hertz and Boltzmann argue that these constraints
are determined mathematically. However, according to this
interpretation, the historical roots of the picture theory can be
found in Helmholtz's substitution of the "sign
theory" for nativistic explanations (see, e.g., Schiemann 1998,
Patton 2009). According to Helmholtz's "sign"
theory, our physical perceptions are images or symbols that do not
resemble their objects directly. Michael Heidelberger and Jean Leroux
argue as well for the influence of Helmholtz's sign theory on
Hertz's picture theory in Heidelberger 1998 and Leroux 2001. See
section 2 for a detailed discussion of the sign theory.



In his early writings, Helmholtz defends a broadly Kantian methodology
against a more far-reaching idealism. In "On the Relation of
Natural Science to Science in General", Helmholtz opposes Kant
to Hegel. Helmholtz argues that for Kant "a principle discovered
a priori was a rule applicable to the method of pure thought, and
nothing further; it could contain no real, positive knowledge"
(Helmholtz 1995 [1862], 79). For Hegel, on the other hand, "not
only the physical phenomena, but even the actual world...were the
result of an act of thought on the part of the creative mind"
(*ibid.*). As a result, according to Helmholtz, Hegel attempted
to unify the human and the natural sciences (the Geistes- and the
Naturwissenschaften).



The question of the independence of the methods of the human sciences,
prominently including history, from the physical sciences became the
theme of heated debate over the 19th century. Wilhelm Dilthey (1883
[1989]) defended the independence of the methods of the human
sciences, while Wilhelm Windelband (1980/1894) argued that only the
natural, physical sciences are "nomothetic," or
law-governed, and thus genuinely scientific (see Beiser 2012 and
Patton 2019 for more detail on this debate and its history).



In an essay on the relation of the natural sciences to science in
general, Helmholtz observes that it is true that the human sciences
became increasingly scientific over the first half of the 19th
century. As a result, "the opposition between them and the
physical sciences has become less and less marked" (Helmholtz
1995 [1862], 81). Helmholtz argues that nonetheless there is a
profound difference, and that,



> 
> In short, there is no denying that, while the moral sciences deal
> directly with the nearest and dearest interests of the human mind, and
> with the institutions it has brought into being, the natural sciences
> are concerned with dead, indifferent matter, obviously indispensable
> for the sake of its practical utility, but apparently without any
> immediate bearing on the cultivation of the intellect (Helmholtz 1995
> [1862], 81).
> 



Helmholtz then addresses the old question of the "dispute of the
faculties" -- is it better to separate the study of the
natural sciences from all others? He concludes that the *logical
induction* characteristic of natural sciences must always be
supplemented with the *aesthetic induction* of the human
sciences, which draws conclusions about human character and behavior
(Helmholtz 1995 [1862], 85). Helmholtz concludes that the best outcome
for human knowledge is to encourage the strengths of aesthetic and
logical induction, to maintain the presence of both in the same
academy, while keeping their foundations and methods strictly
separate. The Hegelian program of methodological and conceptual
unification must give way to a separation between the
*Geisteswissenschaften* and the
*Naturwissenschaften*.



Helmholtz observes that what he sees as the attempt by the Hegelian
philosophy to legislate for natural science resulted in the following
impasse:



> 
> That in the moral sciences traces of the activity of the human
> intellect and of the several stages of its development should present
> themselves, was a matter of course; but surely, if nature really
> reflected the result of the thought of a creative mind, the system
> ought, without difficulty, to find a place for her comparatively
> simple phenomena and processes. It was at this point that
> Hegel's philosophy, we venture to say, utterly broke down. His
> system of nature seemed, at least to natural philosophers, absolutely
> crazy.... The philosophers accused the scientific men of
> narrowness; the scientific men retorted that the philosophers were
> crazy. And so it came about that men of science began to lay some
> stress on the banishment of all philosophical influences from their
> work... Thus, it must be confessed, not only were the
> illegitimate pretensions of the Hegelian system to subordinate to
> itself all other studies rejected, but no regard was paid to the
> rightful claims of philosophy, that is, the criticism of the sources
> of cognition, and the definition of the functions of the intellect.
> (Helmholtz 1995 [1862], 79-80)
> 



Helmholtz concludes that once natural scientists, physiologists of
perception such as himself, entered the philosophical fray, "the
path of future investigation was basically prescribed by the inductive
methods of the natural sciences" (*ibid.* 394).
Helmholtz's work in epistemology epitomizes the return to the
methods of the natural sciences in the mid-19th century. See Patton
(2019) and Schiemann (2013) for more on the background of changes to
the approach to the sciences over the 19th century.