Ernst Mach (1838–1916) was an Austrian physicist and philosopher. He made important contributions to physics (notably the study of shock waves, leading to the Mach number for speed relative to sound) and to the philosophy of science. Mach championed a radical empiricist approach (sometimes called empirio-criticism) that regarded scientific knowledge as an economical organization of sensory experiences, not as an account of a mind-independent reality. His arguments against concepts like absolute space and time influenced Albert Einstein, who credited Mach’s ideas (often paraphrased as Mach’s principle) as an inspiration in developing relativity. At the same time Mach’s philosophical strictures (for example, his rejection of atoms as unobservable) attracted sharp criticism from contemporaries, including the Marxist philosopher Vladimir Lenin, who saw Mach’s views as a form of idealism. Despite controversies, Mach’s legacy endures widely: his name lives on in terms like the Mach number in aerodynamics and the Mach band visual illusion, and his insistence on grounding science in experience helped shape 20th-century logical positivism and pragmatism.

Early Life and Education

Ernst Mach
Institutions	University of Prague; University of Vienna
Nationality	Austrian
Known for	Empirio-criticism; Mach principle; inertial frames
Fields	Physics; philosophy of science
Influence	Logical positivism; Albert Einstein
Occupation	Physicist; philosopher
Notable works	The Science of Mechanics; The Analysis of Sensations
Eponym	Mach number; Mach bands
Wikidata	Q93996

Ernst Mach was born on February 18, 1838, in the Moravian village of Chirlitz (now Chrlice in Brno, Czech Republic) in the Austrian Empire. His father, Jan Mach, was a well-educated tutor, and Ernst was mostly educated at home by his parents until age 14. He then attended a gymnasium (secondary school) in Kroměříž and in 1855 entered the University of Vienna. There Mach studied physics and mathematics (and briefly medical physiology), receiving his doctorate in 1860. Even as a student and young teacher in Vienna, he showed an interest in both scientific experiment and the theory of knowledge: for example, he combined physical research with studies of human perception.

After earning his doctorate, Mach remained in Vienna as a lecturer in mechanics and physics. In 1864 he accepted a professorship in mathematics (later physics) at the University of Graz. In 1867 he moved to Charles-Ferdinand University in Prague as professor of experimental physics, a position he held for 28 years. In the mid-1890s Mach returned to Vienna: in 1895 he was appointed professor of the history and philosophy of science at the University of Vienna. (After a stroke in 1897 he retired from active research in 1901 and served in the Austrian parliament.) Mach died on February 19, 1916, at his son’s home near Munich, Germany, one day after his 78th birthday.

Scientific Work and Contributions

Mach conducted pioneering experiments and theoretical studies in several fields of physics. One of his most famous contributions was to shock-wave physics: using optical methods (schlieren photography), he studied how objects moving faster than sound create compressed air waves. In 1887 Mach and his son Ludwig Mach photographed the cone-shaped shock waves produced by supersonic projectiles. From this work arose the now-standard concept of the Mach number, defined as the ratio of an object’s speed to the speed of sound in the medium. (Thus an object at Mach 2 moves twice as fast as the local sound speed.) The Mach number remains a fundamental parameter in aerodynamics; for example, it governs how air behaves around high-speed aircraft or rockets. Mach’s experiments also revealed the patterns now called Mach diamonds in the exhaust of supersonic rockets and bullets.

Beyond fluid dynamics, Mach made important discoveries in human perception. He independently identified in 1865 the Mach bands: the illusory bright and dark stripes people see at the edges between differing shades of gray. This optical effect is now understood in terms of how the visual system enhances contrast at edges. Mach also studied the human sense of balance (vestibular system). Working with physician Josef Breuer in 1873, he showed how the fluid motion in the inner ear’s semicircular canals conveys a sensation of orientation. In related work Mach devised a rotating chair to study motion perception, an apparatus that also indirectly influenced his thinking on inertial motion (see below).

Mach’s written works include treatises on optics, heat, and mechanics. His book The Science of Mechanics (first published in German in 1883) surveys the history of mechanics while also exercising philosophical critique of its foundations. In it Mach argued against Newton’s assumption of an absolute space; he criticized Newton’s famous spinning bucket experiment by pointing out that we can only judge rotation relative to other bodies. Throughout his career, Mach promoted experimental and historical investigations as essential to science. He believed in describing phenomena simply and closely tied to observation – an outlook summarized in his remark that scientific laws are “economical summaries” of sensory data.

Philosophy and Empirio-Criticism

Mach’s philosophical views were dominated by empiricism and phenomenalism. He held that all knowledge comes from immediate experience (sensations), not from some unknowable reality behind those sensations. Accordingly, he avoided talk of unobservable entities. For Mach, scientific terminology is a useful language for summarizing and predicting sensory impressions, not statements about an independent world of atoms or forces. This stance is known as empirio-criticism (or Machism), a rigorous form of positivism. In this view, even concepts like “time,” “space,” or “atoms” have meaning only insofar as they help organize what we observe. Mach insisted on what he called an economy of thought: theories should be as simple and minimal as possible, sticking to observables and discarding any extra metaphysical baggage.

For example, Mach famously doubted the reality of atoms (in his time, atoms could not be directly seen or measured). He quipped after hearing Ludwig Boltzmann’s atomic theory lectures that he did not believe atoms existed. Similarly, Mach rejected Newton’s idea of empty absolute space. He argued that if no distant stars or matter existed to provide a frame of reference, the notion of motion itself would lose meaning. Thus he said that laws of physics should use only directly measurable quantities and relative motion. Mach expressed this by noting that scientists should attribute “inertia” not to some empty space but to the influence of all other masses in the universe (a core idea behind Mach’s principle below).

Mach elaborated his philosophical position in works like Beiträge zur Analyse der Empfindungen (“Contributions to the Analysis of Sensations,” 1886). In this work he and other essays Mach argued that physical science must umash “sensation” as the primary data. He claimed scientific laws were conventions that summarize experience: the goal of science is simply to render our sensory experiences more comprehensible and predictable, not to uncover an underlying essence. This approach influenced thinkers of the early 20th century, particularly the logical positivists (e.g. members of the Vienna Circle like Rudolf Carnap) and the pragmatists in America (such as William James). They admired Mach’s emphasis on empirical verification and his suspicion of metaphysics.

Mach’s radical empiricism also drew sharp criticism. Critics said that if only sensations are real, one could drift into solipsism (the idea that only one’s own mind is sure to exist). Vladimir Lenin, in his 1909 book Materialism and Empirio-criticism, attacked Mach’s philosophy as a reactionary form of idealism. Lenin and other Marxists held that there is an objective material world independent of perception, and they argued that Mach’s views undermined scientific realism. In particular, they viewed Mach’s rejection of atoms as harmful to physics. (Ironically, soon after Mach’s death experimental evidence for atoms and molecules became overwhelming.) Scientists motivated by atomic theory often saw Mach as overly conservative. Nonetheless, Mach’s insistence on basing theories strictly on observation became a central pillar of 20th-century philosophy of science.

Mach’s Principle and Inertial Frames

One of Mach’s most famous ideas (though he never formulated it as a strict law) concerns the origin of inertia and the nature of inertial frames. An inertial frame is a reference frame in which Newton’s laws hold in their simplest form: for example, an object moving without any forces will travel in a straight line at constant speed. In Newtonian mechanics, inertial frames were thought to exist absolutely (defined by absolute space) – a controversial point Newton tried to justify with his bucket experiment. Mach challenged this notion: he argued that what we call inertia (an object’s resistance to changes in motion) actually arises from the object’s interaction with the rest of the mass in the universe. In modern terms, Mach proposed that “mass out there influences inertia here.”

Put concretely: imagine you are standing still and the sky is full of stars. If an elevator or train suddenly accelerates, you feel yourself pushed back. Mach suggested this force is not due to “empty space” but to your acceleration relative to the fixed stars (or the distant galaxies that define the inertial frame). He illustrated the idea with a colorful saying, reportedly phrased by his friend philosopher Philipp Frank: “When a subway train starts, it is the fixed stars that push you back.” In other words, Mach meant that centrifugal forces in a rotating frame emerge from relative motion against the distant stars. If the entire universe were rotating, even an object at rest locally would feel those same forces. Thus there would be nothing “absolute” about rotation or straight-line motion.

Mach’s principle (as later called by Einstein) can be stated loosely as: the local inertial frame is determined by the overall distribution of matter in the universe. In practice, this means that if all distant matter were removed, inertia would disappear. Mach uses this reasoning to reject absolute space or time: concepts like “true rest” or “absolute motion” have no meaning except in relation to other bodies. (For example, he argued that Newton’s bucket experiment only shows water rotating relative to the bucket; if the bucket were as large as the universe, the same curved surface would appear. Any effect Newton attributed to absolute rotation actually depends on the frame of reference defined by all masses.)

Although Mach put forth these ideas in his writings (especially in The Science of Mechanics, 1883), the name “Mach’s principle” was coined by Einstein around 1918. Einstein was influenced by Mach’s critique of Newtonian space and time when he developed general relativity. Early on, Einstein hoped that his theory would embody Mach’s principle – that is, that the gravitational field (which determines inertia) would be completely sourced by matter. He even cited Mach’s arguments as one of his guiding principles. In later years, Einstein admitted that general relativity did not fully satisfy Mach’s ideas (for example, there are solutions to Einstein’s equations with matter rotating independently of spacetime, like the Gӧdel universe). Nevertheless, Mach’s intuition that physics should not rely on a fixed background space had a lasting effect on Einstein’s worldview.

In summary, under Mach’s view inertial motion (and hence inertial frames) has meaning only in relation to the rest of the universe. An inertial frame of reference is a coordinate system in which an object unacted upon moves uniformly. Mach held that such a frame should ultimately be defined by the average motion of all mass. This relational view of motion challenged the Newtonian idea of an absolute empty space as the arbiter of inertia.

Method and Philosophy of Science

Beyond any single principle, Mach’s overall method in science was distinctive. He treated scientific concepts as convenient fictions or shorthand, not literal truths. For Mach, a theory’s value lay in its economy of thought and predictive power. He famously advised that scientists should strive to describe phenomena in the most “economical” way, avoiding unnecessary entities. (This attitude was a forerunner of the logical positivist dictum that we should not multiply entities beyond what is needed for explanation.) His historical studies of mechanics, optics, and thermodynamics reflect this mindset: he was as much interested in why certain ideas came to dominate as in the technical details. He often rewrote his histories of science to reflect the current state of understanding, emphasizing the role of observation.

An example of Mach’s methodological influence is in his treatment of space and geometry. He distinguished between physiological space (how we perceive the world with our senses) and geometrical space (the abstract mathematical space used in physics). He argued that concepts like straight lines or Cartesian coordinates are not inherent truths but useful descriptions that fit our senses. This view anticipated later arguments in the philosophy of science about theories being models or instruments.

Mach’s philosophical approach was holistic and operational: he insisted that quantities should be defined by how they are measured. This stance echoes the later operational definitions of time and length in relativity and quantum mechanics. Indeed, his insistence that physical concepts be grounded in observation paved the way for 20th-century thinkers. The Vienna Circle philosophers saw Mach as a precursor, emphasizing that scientific statements must tie to experience. In the United States, the pragmatist John Dewey praised Mach’s idea that ideas are tools for coping with experience.

Influence and Legacy

Ernst Mach’s ideas left a strong imprint on both science and philosophy. In physics, although his own stance was sometimes conservative, his name became attached to essential concepts. The Mach number is now standard terminology in aerodynamics, and educators teach the Mach band illusion in vision science. There are also O´ symbols like the Mach disc and Mach diamonds in supersonic exhaust patterns. Several astronomical and geographical features have been named after Mach (for example, a lunar crater named Mach), reflecting his prominence.

In philosophy and the foundations of science, Mach’s influence was especially pronounced in the early 20th century. Albert Einstein acknowledged Mach as a major influence on his thinking. He once wrote that he had eagerly studied Mach’s work before formulating relativity, and he called Mach the “precursor” to general relativity. (Of course, Einstein also had reservations later.) Beyond Einstein, Mach’s insistence on empiricism fed into the logical positivist tradition: philosophers like Moritz Schlick and Rudolf Carnap cited Mach as a forerunner who showed how science could dispense with metaphysics. Machian ideas were also popular in psychological theory: the Gestalt psychologists regarded Mach’s analysis of perception as a step toward understanding how we organize sensory data into structured experience. In cognitive science, Mach’s bands are still discussed as a basic example of edge detection in vision.

Not all influence was positive, however. Mach’s followers included some Marxist thinkers, such as Alexander Bogdanov, who tried to merge Mach’s empiricism with socialist thought (though Lenin disagreed strongly). In the political and philosophical debates of the time, Machianism became a catchword in the controversy between materialism and idealism. Even today, Mach is often cited in historical discussions of how philosophical ideas shaped physics.

Critiques

Despite his stature, Mach drew criticism on several fronts. The most famous critique came from the philosopher-politician Vladimir Lenin, who argued that Mach’s rejection of atoms and his focus on subjective experience undermined the objectivity of science. Lenin claimed that Mach’s “empirio-criticism” essentially collapsed the material world into impressions in the mind, an unacceptable idealism from the Marxist perspective. Contemporary scientists who embraced the atomic theory (like Boltzmann and later physicists) also pointed out that Mach’s strict anti-metaphysical line could lead to diametrically wrong conclusions: for example, Mach predicted in 1897 that deeper evidence of atoms might never be found or prove them “undemonstrable.” In fact, within a few years experiments in Brownian motion and other phenomena gave convincing proof of atoms and molecules, which Mach reluctantly accepted only late in life.

Mach’s stance on relativity also generated discussion. He was skeptical of the concept of an all-pervading ether (the presumed medium for light) and thus in some ways anticipated Einstein’s quips. But he also at times criticized aspects of the new physics. Near the end of his life he was not enthusiastic about Einstein’s theory, perhaps because it still used fields and geometry he thought unnecessary. In any case, modern physicists rarely commit to Mach’s principle as a hard law; many formulations of Mach’s principle exist, and general relativity itself does not fully enforce it. So one could say that Mach’s specific ideas about inertia did not become part of mainstream physics, even if philosophically inspiring.

More broadly, philosophers have noted that Mach’s empirio-criticism, taken to extremes, can be self-contradictory. If all we know are sensations, then the claim “all knowledge is sensations” is itself a claim not directly verifiable by sense. Defenders respond that Mach viewed his own statements as models or conventions as well. In practical terms, most scientists saw Mach as too positivist: they argued that science needed theoretical entities (like atoms, or even the field concept) as useful models, even if unobservable.

Legacy

Today, Ernst Mach is remembered as a brilliant experimenter and a provocative thinker. His name lives on in multiple ways. In physics and engineering, Mach’s legacy is embodied by the Mach number – it is hard to work on supersonic jets or shock waves without encountering his name. In anatomy and visual science, “Mach bands” are a classic classroom demonstration. In philosophy, Mach gradually became emblematic of a certain scientific stance: skepticism toward unseen entities, and emphasis on experience. Beginning in the 1930s, philosophers of science like Henri Poincaré, Albert Einstein, and later Karl Popper referenced Mach as championing an empirical, operational view of concepts. Even if the details of Mach’s own proposals (like his particular version of Mach’s principle) are superseded, the spirit of his views – that our theories should rest on careful control of observation – permeates modern science.

Selected memorials to Mach include an asteroid (3949 Mach) and a crater on the Moon, acknowledging his contributions to physics. Culturally, people sometimes quote him for his aphorisms (for instance his sardonic question “Did Wilhelm tell you that?” in response to hearsay in a lecture, meaning “note of doubt”). His influence on younger scientists was personal as well: he was a friend and mentor to many, and he was even the godfather of Nobel physicist Wolfgang Pauli.

Selected Works

Beiträge zur Analyse der Empfindungen und das Verhältnis des Physischen zum Psychischen (Contributions to the Analysis of Sensations, 1886) – Mach’s influential philosophical work on perception and scientific concepts.
Die Mechanik in ihrer Entwicklung historisch-kritisch dargestellt (The Science of Mechanics: A Critical and Historical Account of Its Development, 1883) – A classic history and critique of mechanics, where Mach discussed inertia and absolute space.
Principles of the Theory of Heat (1877) – Foundational lectures on thermodynamics.
Popular Scientific Lectures (1895, 1898) – Collections of essays on topics in physics and philosophy for general readers.
Erkenntnis und Irrtum: Skizzen zur Psychologie der Forschung (Knowledge and Error: Sketches on the Psychology of Research, 1905) – Philosophical reflections on the nature of scientific inquiry.
Die Analyse der Empfindungen und das Verhältnis des Physischen zum Psychischen (1905) – A later edition expanding on his theory of sensations.

Other notable publications include numerous scientific papers (for example the 1887 paper with P. Salcher on photography of shock waves) and Mach’s autobiography (Erlebtes und Erkanntes, 1910). These works span his dual interests in experimental physics and the philosophical foundations of science.

Overall, Ernst Mach stands out in the history of science as a thinker who bridged empirical research and philosophical analysis. He mapped out how science evolves in step with experience, and challenged his contemporaries to rethink the concepts of space, time, and inertia. Modern physics still grapples with questions he raised, and his name remains attached to key ideas in aerodynamics, perception, and the philosophy of science.