Ernst Mach

Ernst Waldfried Josef Wenzel Mach (/ˈmɑːx/; German: [ˈɛɐ̯nst max]; 18 February 1838 – 19 February 1916) was an Austrian[7] physicist and philosopher, noted for his contributions to physics such as study of shock waves. The ratio of one's speed to that of sound is named the Mach number in his honor. As a philosopher of science, he was a major influence on logical positivism and American pragmatism.[8] Through his criticism of Newton's theories of space and time, he foreshadowed Einstein's theory of relativity.

Ernst Mach
Ernst Mach 01
Ernst Mach (1838–1916)
Ernst Waldfried Josef Wenzel Mach

18 February 1838
Brno, Moravia, Austrian Empire (present day Czech republic)
Died19 February 1916 (aged 78)
ResidenceAustrian Empire, German Empire
Alma materUniversity of Vienna
Known forMach number
Mach's principle
Shock waves
Mach waves
Mach reflection effects
Mach band
Criticism of Isaac Newton's bucket argument[1]
Mach diamonds
Scientific career
InstitutionsUniversity of Graz
Charles University (Prague)
University of Vienna
Doctoral advisorAndreas von Ettingshausen
Doctoral studentsHeinrich Gomperz
Ottokar Tumlirz
Other notable studentsAndrija Mohorovičić
InfluencesAndreas von Ettingshausen[2]
Gustav Fechner[3]
Carl Ludwig[4]
InfluencedVienna Circle
Russian Machism
Ludwig Boltzmann
Albert Einstein
Wolfgang Pauli
William James
Wilhelm Kienzl[5]
Pierre Duhem[6]
Ernst Mach Signature
He was the godfather of Wolfgang Pauli. The Mach–Zehnder interferometer is named after his son Ludwig Mach, who was also a physicist.


Ernst Waldfried Josef Wenzel Mach was born in Chrlice (German: Chirlitz), Moravia (then in the Austrian Empire, now part of Brno in the Czech Republic). His father, who had graduated from Charles University in Prague, acted as tutor to the noble Brethon family in Zlín in eastern Moravia. His grandfather, Wenzl Lanhaus, an administrator of the Chirlitz estate, was also master builder of the streets there. His activities in that field later influenced the theoretical work of Ernst Mach. Some sources give Mach's birthplace as Tuřany (German: Turas, now also part of Brno), the site of the Chirlitz registry-office. It was there that Ernst Mach was baptized by Peregrin Weiss. Mach later became a socialist and an atheist.[9] His theory and life, though, was sometimes compared with Buddhism, namely by Heinrich Gomperz who addressed Mach as the "Buddha of Science" due to the phenomenalist approach of the "Ego" in his Analysis of Sensations.[10][11]

Up to the age of 14, Mach received his education at home from his parents. He then entered a Gymnasium in Kroměříž (German: Kremsier), where he studied for three years. In 1855 he became a student at the University of Vienna. There he studied physics and for one semester medical physiology, receiving his doctorate in physics in 1860 under Andreas von Ettingshausen with a thesis titled "Über elektrische Ladungen und Induktion", and his habilitation the following year. His early work focused on the Doppler effect in optics and acoustics. In 1864 he took a job as Professor of Mathematics at the University of Graz, having turned down the position of a chair in surgery at the University of Salzburg to do so, and in 1866 he was appointed as Professor of Physics. During that period, Mach continued his work in psycho-physics and in sensory perception. In 1867, he took the chair of Experimental Physics at the Charles University, Prague, where he stayed for 28 years before returning to Vienna.

Mach's main contribution to physics involved his description and photographs of spark shock-waves and then ballistic shock-waves. He described how when a bullet or shell moved faster than the speed of sound, it created a compression of air in front of it. Using schlieren photography, he and his son Ludwig were able to photograph the shadows of the invisible shock waves. During the early 1890s Ludwig was able to invent an interferometer which allowed for much clearer photographs. But Mach also made many contributions to psychology and physiology, including his anticipation of gestalt phenomena, his discovery of the oblique effect and of Mach bands, an inhibition-influenced type of visual illusion, and especially his discovery of a non-acoustic function of the inner ear which helps control human balance.

One of the best-known of Mach's ideas is the so-called "Mach principle," concerning the physical origin of inertia. This was never written down by Mach, but was given a graphic verbal form, attributed by Philipp Frank to Mach himself, as, "When the subway jerks, it's the fixed stars that throw you down."

Photography of bow shock waves around a brass bullet, 1888
Ernst Mach's historic 1887 photograph (shadowgraph) of a bow shockwave around a supersonic bullet[12]

Mach also became well known for his philosophy developed in close interplay with his science.[13] Mach defended a type of phenomenalism recognizing only sensations as real. This position seemed incompatible with the view of atoms and molecules as external, mind-independent things. He famously declared, after an 1897 lecture by Ludwig Boltzmann at the Imperial Academy of Science in Vienna: "I don't believe that atoms exist!"[14] From about 1908 to 1911 Mach's reluctance to acknowledge the reality of atoms was criticized by Max Planck as being incompatible with physics. Einstein's 1905 demonstration that the statistical fluctuations of atoms allowed measurement of their existence without direct individuated sensory evidence marked a turning point in the acceptance of atomic theory. Some of Mach's criticisms of Newton's position on space and time influenced Einstein, but later Einstein realized that Mach was basically opposed to Newton's philosophy and concluded that his physical criticism was not sound.

In 1898 Mach suffered from cardiac arrest and in 1901 retired from the University of Vienna and was appointed to the upper chamber of the Austrian parliament. On leaving Vienna in 1913 he moved to his son's home in Vaterstetten, near Munich, where he continued writing and corresponding until his death in 1916, only one day after his 78th birthday.


Most of Mach's initial studies in the field of experimental physics concentrated on the interference, diffraction, polarization and refraction of light in different media under external influences. From there followed important explorations in the field of supersonic fluid mechanics. Mach and physicist-photographer Peter Salcher presented their paper on this subject[15] in 1887; it correctly describes the sound effects observed during the supersonic motion of a projectile. They deduced and experimentally confirmed the existence of a shock wave of conical shape, with the projectile at the apex.[16] The ratio of the speed of a fluid to the local speed of sound vp/vs is now called the Mach number. It is a critical parameter in the description of high-speed fluid movement in aerodynamics and hydrodynamics. Mach also contributed to cosmology the hypothesis known as Mach's principle.

Philosophy of science

Ernst Mach - Rathauspark, Vienna
Bust of Mach in the Rathauspark (City Hall Park) in Vienna, Austria


From 1895 to 1901, Mach held a newly created chair for "the history and philosophy of the inductive sciences" at the University of Vienna.[17] In his historico-philosophical studies, Mach developed a phenomenalistic philosophy of science which became influential in the 19th and 20th centuries. He originally saw scientific laws as summaries of experimental events, constructed for the purpose of making complex data comprehensible, but later emphasized mathematical functions as a more useful way to describe sensory appearances. Thus scientific laws while somewhat idealized have more to do with describing sensations than with reality as it exists beyond sensations.[18]

The goal which it (physical science) has set itself is the simplest and most economical abstract expression of facts.

When the human mind, with its limited powers, attempts to mirror in itself the rich life of the world, of which it itself is only a small part, and which it can never hope to exhaust, it has every reason for proceeding economically.

In reality, the law always contains less than the fact itself, because it does not reproduce the fact as a whole but only in that aspect of it which is important for us, the rest being intentionally or from necessity omitted.

In mentally separating a body from the changeable environment in which it moves, what we really do is to extricate a group of sensations on which our thoughts are fastened and which is of relatively greater stability than the others, from the stream of all our sensations.

Suppose we were to attribute to nature the property of producing like effects in like circumstances; just these like circumstances we should not know how to find. Nature exists once only. Our schematic mental imitation alone produces like events.

Mach's positivism also influenced many Russian Marxists, such as Alexander Bogdanov (1873–1928). In 1908, Lenin wrote a philosophical work, Materialism and Empirio-criticism (published 1909), in which he criticized Machism and the views of "Russian Machists" (Lenin also cited in this work the concept of the 'ether', as the medium through which light waves propagated, and the concept of time as an absolute). Empirio-criticism is the term for the rigorously positivist and radically empirical philosophy established by the German philosopher Richard Avenarius and further developed by Mach, which claims that all we can know is our sensations and that knowledge should be confined to pure experience.[19]

In accordance with empirio-critical philosophy, Mach opposed Ludwig Boltzmann and others who proposed an atomic theory of physics. Since one cannot observe things as small as atoms directly, and since no atomic model at the time was consistent, the atomic hypothesis seemed to Mach to be unwarranted, and perhaps not sufficiently "economical". Mach had a direct influence on the Vienna Circle philosophers and the school of logical positivism in general.

To Mach are attributed a number of principles that distill his ideal of physical theorisation—what is now called "Machian physics":

  1. It should be based entirely on directly observable phenomena (in line with his positivistic leanings)[20]
  2. It should completely eschew absolute space and time in favor of relative motion[21]
  3. Any phenomena that would seem attributable to absolute space and time (e.g., inertia and centrifugal force) should instead be seen as emerging from the large scale distribution of matter in the universe.[22]

The last is singled out, particularly by Albert Einstein, as "the" Mach's principle. Einstein cited it as one of the three principles underlying general relativity. In 1930, he stated that "it is justified to consider Mach as the precursor of the general theory of relativity",[23] though Mach, before his death, would apparently reject Einstein's theory.[24] Einstein was aware that his theories did not fulfill all Mach's principles, and no subsequent theory has either, despite considerable effort.

Phenomenological constructivism

According to Alexander Riegler, Ernst Mach's work was a precursor to the influential perspective known as constructivism.[25] Constructivism holds that all knowledge is constructed rather than received by the learner. He took an exceptionally non-dualist, phenomenological position. The founder of radical constructivism, von Glasersfeld, gave a nod to Mach as an ally.

Mach's spinning chair
Spinning chair devised by Mach to investigate the experience of motion


In 1873, independently of each other[26] Mach and the physiologist and physician Josef Breuer discovered how the sense of balance (i.e., the perception of the head's imbalance) functions, tracing its management by information which the brain receives from the movement of a fluid in the semicircular canals of the inner ear. That the sense of balance depended on the three semicircular canals was discovered in 1870 by the physiologist Friedrich Goltz, but Goltz did not discover how the balance-sensing apparatus functioned. Mach devised a swivel chair to enable him to test his theories, and Floyd Ratliff has suggested that this experiment may have paved the way to Mach's critique of a physical conception of absolute space and motion.[27]


Mach bands - animation
Exaggerated contrast between edges of the slightly differing shades of gray, appears as soon as they touch

In the area of sensory perception, psychologists remember Mach for the optical illusion called Mach bands. The effect exaggerates the contrast between edges of the slightly differing shades of gray, as soon as they contact one another, by triggering edge-detection in the human visual system.[28]

More clearly than anyone before or since Mach made the distinction between what he called physiological (specifically visual) and geometrical spaces.[29]

Mach's views on mediating structures inspired B. F. Skinner's strongly inductive position, which paralleled Mach's in the field of psychology.[30]


In homage his name was given to:

Mach's principal works in English

  • The Science of Mechanics (1883)
  • The Analysis of Sensations (1897)[31]
  • Popular Scientific Lectures (1895)
  • Space and Geometry from the Point of View of Physical Inquiry (October 1903) In The Monist, Vol. XIV, No. I
  • History and Root of the Principle of the Conservation of Energy (1911)
  • The Principles of Physical Optics (1926)
  • Knowledge and Error (1976)
  • Principles of the Theory of Heat (1986)
  • Fundamentals of the Theory of Movement Perception (2001)

See also


  1. ^ Mach, E. (1960 [1883]), The Science of Mechanics, LaSalle, IL: Open Court Publishing, p. 284.
  2. ^, Ernst Waldfried Josef Wenzel Mach
  3. ^ Jagdish Mehra, Helmut Rechenberg, The Historical Development of Quantum Theory, page 47
  4. ^, Ernst Mach First published Wed May 21, 2008; substantive revision Tue Apr 28, 2009, Mach interest in physiology, Johannes Peter Müller and his students, Ernst Brüke and Carl Ludwig, started a new school of physiology in 1840s.
  5. ^ John T. Blackmore, Ernst Mach: His Work, Life, and Influence, 1972, p. 44.
  6. ^ John T. Blackmore, Ernst Mach: His Work, Life, and Influence, 1972, p. 196.
  7. ^ "Ernst Mach". Encyclopædia Britannica. 2016. Retrieved January 6, 2016.
  8. ^ John T. Blackmore (1972), Ernst Mach; his work, life, and influence, Berkeley: University of California Press, ISBN 978-0520018495, OCLC 534406, 0520018494
  9. ^ R. S. Cohen; Raymond J. Seeger (1975). Ernst Mach, Physicist and Philosopher. Springer. p. 158. ISBN 978-90-277-0016-2. And Mach, in personal conviction, was a socialist and an atheist.
  10. ^ Cf. Ursula Baatz: "Ernst Mach – The Scientist as a Buddhist?" In: John Blackmore (ed.): Ernst Mach — A Deeper Look. Documents and New Perspectives (Boston Studies in the Philosophy of Science, vol. 143). Springer, Dordrecht 1992, pp. 183–199.
  11. ^ John T. Blackmore (1972). "Chapter 18 - Mach and Buddhism". Ernst Mach, His Work, Life, and Influence]. University of California Press. p. 293. ISBN 978-0520018495. Mach was logically a Buddhist and illogically a believer in science.
  12. ^ John D. Anderson, Jr. "Research in Supersonic Flight and the Breaking of the Sound Barrier -- Chapter 3". p. 65. Retrieved 5 September 2016.
  13. ^ On this interdependency of Mach's physics, physiology, history and philosophy of science see Blackmore (1972), Blackmore (ed.) 1992 and Hentschel 1985 against Paul Feyerabend's efforts to decouple these three strands.
  14. ^ Yourgrau, P. (2005). A World Without Time: The Forgotten Legacy of Gödel and Einstein. Allen Lane
  15. ^ Mach, Ernst; Salcher, Peter (1887). "Photographische Fixirung der durch Projectile in der Luft eingeleiteten Vorgänge". Sitzungsber. Kaiserl. Akad. Wiss., Wien, Math.-Naturwiss. Cl. (in German). 95 (Abt. II): 764–780. Retrieved 24 October 2015.
  16. ^ Scott, Jeff (9 November 2003). "Ernst Mach and Mach Number". Retrieved 24 October 2015.
  17. ^ On Mach's historiography, cf., e.g., Hentschel (1988); on his impact in Vienna, see Stadler et al. (1988), and Blackmore et al. (2001).
  18. ^ Selections are taken from his essay The Economical Nature of Physical Inquiry, excerpted by Kockelmans and slightly corrected by Blackmore. (citation below).
  19. ^ "empirio-criticism": entry in The Blackwell Dictionary of Western Philosophy.).
  20. ^ Barbour, J. The End of Time, p. 220: "In the Machian view, the properties of the system are exhausted by the masses of the particles and their separations, but the separations are mutual properties. Apart from the masses, the particles have no attributes that are exclusively their own. They — in the form of a triangle — are a single thing. In the Newtonian view, the particles exist in absolute space and time. These external elements lend the particles attributes — position, momentum, angular momentum — denied in the Machian view. The particles become three things. Absolute space and time are an essential part of atomism."
  21. ^ Penrose, R., The Road to Reality, p. 753: "Mach’s principle asserts that physics should be defined entirely in terms of the relation of one body to another, and that the very notion of a background space should be abandoned"
  22. ^ Mach, E. The Science of Mechanics. "> [The] investigator must feel the need of ... knowledge of the immediate connections, say, of the masses of the universe. There will hover before him as an ideal insight into the principles of the whole matter, from which accelerated and inertial motions will result in the same way.
  23. ^ Quoted in Pais, Subtle is the Lord, 2005, OUP
  24. ^ The preface of the posthumously published Principles of Physical Optics explicitly rejects Einstein's relativistic views but it has been argued that the text is inauthentic; see Gereon Wolters, Mach and Einstein, or Clearing Troubled Waters in the History of Science. "Einstein and the Changing Worldviews of Physics". Birkhäuser, Boston, 2012. 39-57.
  25. ^ Riegler, A. (2011) "Constructivism". In: L'Abete, L. (Ed.) Paradigms in Theory Construction, pp. 235–255 (doi:10.1007/978-1-4614-0914-4_13).
  26. ^ Hawkins, J.E. and Schacht, J. "The Emergence of Vestibular Science" (Part 8 of "Sketches of Otohistory") in Audiology and Neurotology, April 2005.
  27. ^ Ratliff, Floyd (1975). "On Mach's Contributions to the Analysis of Sensations". In Seeger, Raymond J.; Cohen, Robert S. (eds.). Ernst Mach, Physicist and Philosopher.
  28. ^ Ratliff, Floyd (1965). Mach bands: quantitative studies on neural networks in the retina. Holden-Day.
  29. ^ Mach, E. (1906) Space and Geometry. Chicago: Open Court Publishing.
  30. ^ Mecca Chiesa (1994). Radical Behaviorism: The Philosophy and the Science. Authors Cooperative. ISBN 978-0-9623311-4-5.
  31. ^ See Mach, Ernst (1897). Williams, C.W. (ed.). Contributions to the Analysis of Sensation (1 ed.). Chicago: Open Court Publishing Company. Retrieved 13 July 2014. via

Further reading

  • Erik C. Banks: Ernst Mach's World Elements. A Study in Natural Philosophy. Dordrecht: Kluwer (now Springer), 2013.
  • John T. Blackmore: Ernst Mach. His Life, Work, and Influence. Berkeley and Los Angeles: University of California Press, 1972.
  • John Blackmore and Klaus Hentschel (eds.): Ernst Mach als Außenseiter. Vienna: Braumüller, 1985 (with select correspondence).
  • John T. Blackmore (ed.): Ernst Mach – A Deeper Look. Documents and New Perspectives. Dordrecht: Springer, 1992.
  • John T. Blackmore, Ryoichi Itagaki and Setsuko Tanaka (eds.): Ernst Mach's Vienna 1895–1930. Or Phenomenalism as Philosophy of Science. Dordrecht: Springer, 2001.
  • John T. Blackmore, Ryoichi Itagaki and Setsuko Tanaka (eds.): Ernst Mach's Science. Kanagawa: Tokai University Press, 2006.
  • John T. Blackmore, Ryoichi Itagaki and Setsuko Tanaka: Ernst Mach's Influence Spreads. Bethesda: Sentinel Open Press, 2009.
  • John T. Blackmore, Ryoichi Itagaki and Setsuko Tanaka: Ernst Mach's Graz (1864–1867), where much science and philosophy were developed. Bethesda: Sentinel Open Press, 2010.
  • John T. Blackmore: Ernst Mach's Prague 1867–1895 as a human adventure, Bethesda: Sentinel Open Press, 2010.
  • William Everdell: The First Moderns. Profiles in the Origins of Twentieth-Century Thought. Chicago: University of Chicago Press, 1997.
  • Rudolf Haller and Friedrich Stadler (eds.): Ernst Mach – Werk und Wirkung. Vienna: Hoelder-Pichler-Tempsky, 1988.
  • Klaus Hentschel: "On Paul Feyerabend's version of 'Mach's theory of research and its relation to Albert Einstein'", Studies in History and Philosophy of Science 16 (1985): 387-394.
  • Klaus Hentschel: "Die Korrespondenz Duhem-Mach: Zur 'Modellbeladenheit' von Wissenschaftsgeschichte'", Annals of Science 45 (1988): 73-91 (with their complete correspondence).
  • Klaus Hentschel: "Ernst Mach", in Arne Hessenbruch (ed.): Reader's Guide to the History of Science. London: Routledge, 2013, p. 427f.
  • D. Hoffmann and H. Laitko (eds.): Ernst Mach – Studien und Dokumente. Berlin, 1991.
  • Joseph J. Kockelmans: Philosophy of science. The historical background. New York: The Free Press, 1968.
  • Jiří Procházka: Ernst Mach /1838–1916/ Genealogie, 3 vols. Brno, 2007–2010. ISBN 80-903476-3-0, 80-903476-7-3, 978-80-903476-0-1.
  • V. Prosser and J. Folta (eds.): Ernst Mach and the development of Physics – Conference Papers, Prague: Universitas Carolina Pragensis, 1991.
  • Joachim Thiele: Wissenschaftliche Kommunikation – Die Korrespondenz Ernst Machs", Kastellaun: Hain, 1978 (with select correspondence).

External links


Energeticism is the physical view that energy is the fundamental element in all physical change. It concerns posits an ontology, or philosophy of being, which holds that all things are ultimately composed of energy and is opposed to ontological idealism. Energeticism might be associated with the physicist and philosopher Ernst Mach, though his attitude to it is ambiguous. It was also propounded by the chemist Wilhelm Ostwald.Energeticism is largely rejected today, in part due to its Aristotelian and metaphysical leanings and its rejection of the existence of a micro-world (such as the one that chemists or physicists have discovered). Ludwig Boltzmann and Max Planck posited that matter and energy are distinct from each other and, hence, that energy cannot itself be the fundamental unit of nature upon which all other units are based.

Epistemological idealism

Epistemological idealism is a subjectivist position in epistemology that holds that what one knows about an object exists only in one's mind. It is opposed to epistemological realism.

Fraunhofer Institute for High-Speed Dynamics

The Fraunhofer Institute for High-Speed Dynamics (German: Fraunhofer-Institut für Kurzzeitdynamik), commonly known as the Ernst Mach Institute and also by the abbreviation Fraunhofer EMI, is a facility of the Fraunhofer Society in Germany. The Institute is based in Freiburg im Breisgau. Its activities are applied research and development in the fields of materials science and high-speed measurement techniques. The Institute also has offices in Efringen-Kirchen and Kandern.

The name "Ernst Mach Institute" is named for the physicist Ernst Mach (1838-1916), who first used high-speed photography to visualize ballistic and gas-dynamic processes.

Gustav Jaumann

Gustav Jaumann (1863–1924) was an Austrian physicist.

An assistant to the physicist Ernst Mach, he had a talent for mathematics, but disbelieved the existence of small particles like electrons and atoms. Between 1901 and 1924 he taught physics

at the German Technical University in Brno. He won the Haitinger Prize of the Austrian Academy of Sciences in 1911.

Hadamard's dynamical system

In physics and mathematics, the Hadamard dynamical system (also called Hadamard's billiard or the Hadamard–Gutzwiller model) is a chaotic dynamical system, a type of dynamical billiards. Introduced by Jacques Hadamard in 1898, and studied by Martin Gutzwiller in the 1980s, it is the first dynamical system to be proven chaotic.

The system considers the motion of a free (frictionless) particle on the Bolza surface, i.e, a two-dimensional surface of genus two (a donut with two holes) and constant negative curvature; this is the simplest compact Riemann surface. Hadamard was able to show that every particle trajectory moves away from every other: that all trajectories have a positive Lyapunov exponent.

Frank Steiner argues that Hadamard's study should be considered to be the first-ever examination of a chaotic dynamical system, and that Hadamard should be considered the first discoverer of chaos. He points out that the study was widely disseminated, and considers the impact of the ideas on the thinking of Albert Einstein and Ernst Mach.

The system is particularly important in that in 1963, Yakov Sinai, in studying Sinai's billiards as a model of the classical ensemble of a Boltzmann–Gibbs gas, was able to show that the motion of the atoms in the gas follow the trajectories in the Hadamard dynamical system.

Hubert Schardin

Hubert Hermann Reinhold Schardin (17 June 1902 Plassow – 27 September 1965 in Freiburg im Breisgau) was a German ballistics expert, engineer and academic who studied in the field of high-speed photography and cinematography.

He also was the director of the German-French Research Institute (ISL) in Saint-Louis (France) and founder and director of the Fraunhofer Society Institute for High-Speed Dynamics - Ernst-Mach-Institut (EMI) - in Freiburg im Breisgau.

Mach's principle

In theoretical physics, particularly in discussions of gravitation theories, Mach's principle (or Mach's conjecture) is the name given by Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach. The idea is that the existence of absolute rotation (the distinction of local inertial frames vs. rotating reference frames) is determined by the large-scale distribution of matter, as exemplified by this anecdote:

You are standing in a field looking at the stars. Your arms are resting freely at your side, and you see that the distant stars are not moving. Now start spinning. The stars are whirling around you and your arms are pulled away from your body. Why should your arms be pulled away when the stars are whirling? Why should they be dangling freely when the stars don't move?

Mach's principle says that this is not a coincidence—that there is a physical law that relates the motion of the distant stars to the local inertial frame. If you see all the stars whirling around you, Mach suggests that there is some physical law which would make it so you would feel a centrifugal force. There are a number of rival formulations of the principle. It is often stated in vague ways, like "mass out there influences inertia here". A very general statement of Mach's principle is "local physical laws are determined by the large-scale structure of the universe".This concept was a guiding factor in Einstein's development of the general theory of relativity. Einstein realized that the overall distribution of matter would determine the metric tensor, which tells you which frame is rotationally stationary. Frame-dragging and conservation of gravitational angular momentum makes this into a true statement in the general theory in certain solutions. But because the principle is so vague, many distinct statements can be (and have been) made that would qualify as a Mach principle, and some of these are false. The Gödel rotating universe is a solution of the field equations that is designed to disobey Mach's principle in the worst possible way. In this example, the distant stars seem to be revolving faster and faster as one moves further away. This example doesn't completely settle the question, because it has closed timelike curves.

Mach bands

Mach bands is an optical illusion named after the physicist Ernst Mach. It exaggerates the contrast between edges of the slightly differing shades of gray, as soon as they contact one another, by triggering edge-detection in the human visual system.

Mach number

In fluid dynamics, the Mach number (M or Ma) (/mɑːk/; German: [max]) is a dimensionless quantity representing the ratio of flow velocity past a boundary to the local speed of sound.


M is the Mach number,
u is the local flow velocity with respect to the boundaries (either internal, such as an object immersed in the flow, or external, like a channel), and
c is the speed of sound in the medium.

By definition, at Mach 1, the local flow velocity u is equal to the speed of sound. At Mach 0.65, u is 65% of the speed of sound (subsonic), and, at Mach 1.35, u is 35% faster than the speed of sound (supersonic).

The local speed of sound, and thereby the Mach number, depends on the condition of the surrounding medium, in particular the temperature. The Mach number is primarily used to determine the approximation with which a flow can be treated as an incompressible flow. The medium can be a gas or a liquid. The boundary can be traveling in the medium, or it can be stationary while the medium flows along it, or they can both be moving, with different velocities: what matters is their relative velocity with respect to each other. The boundary can be the boundary of an object immersed in the medium, or of a channel such as a nozzle, diffusers or wind tunnels channeling the medium. As the Mach number is defined as the ratio of two speeds, it is a dimensionless number. If M < 0.2–0.3 and the flow is quasi-steady and isothermal, compressibility effects will be small and simplified incompressible flow equations can be used.

The Mach number is named after Austrian physicist and philosopher Ernst Mach, and is a designation proposed by aeronautical engineer Jakob Ackeret. As the Mach number is a dimensionless quantity rather than a unit of measure, with Mach, the number comes after the unit; the second Mach number is Mach 2 instead of 2 Mach (or Machs). This is somewhat reminiscent of the early modern ocean sounding unit mark (a synonym for fathom), which was also unit-first, and may have influenced the use of the term Mach. In the decade preceding faster-than-sound human flight, aeronautical engineers referred to the speed of sound as Mach's number, never Mach 1.

Mach reflection

Mach reflection is a supersonic fluid dynamics effect, named for Ernst Mach, and is a shock wave reflection pattern involving three shocks.

Mach–Zehnder interferometer

In physics, the Mach–Zehnder interferometer is a device used to determine the relative phase shift variations between two collimated beams derived by splitting light from a single source. The interferometer has been used, among other things, to measure phase shifts between the two beams caused by a sample or a change in length of one of the paths. The apparatus is named after the physicists Ludwig Mach (the son of Ernst Mach) and Ludwig Zehnder: Zehnder's proposal in an 1891 article was refined by Mach in an 1892 article.

Materialism and Empirio-criticism

Materialism and Empirio-criticism (Russian: Материализм и эмпириокритицизм, Materializm i empiriokrititsizm) is a philosophical work by Vladimir Lenin, published in 1909. It was an obligatory subject of study in all institutions of higher education in the Soviet Union, as a seminal work of dialectical materialism, a part of the curriculum called "Marxist–Leninist Philosophy". Lenin argued that human perceptions correctly and accurately reflect the objective external world.

Lenin formulates the fundamental philosophical contradiction between idealism and materialism as follows:

"Materialism is the recognition of 'objects in themselves' or objects outside the mind; the ideas and sensations are copies or images of these objects. The opposite doctrine (idealism) says: the objects do not exist, outside the mind '; they are 'connections of sensations'."

Michele Besso

Michele Angelo Besso (Riesbach, 25 May 1873 – Geneva, 15 March 1955) was a Swiss/Italian engineer.Besso was born in Riesbach from a family of Italian Jewish (Sephardi) descent. He was a close friend of Albert Einstein during his years at the Federal Polytechnic Institute in Zurich, today the ETH Zurich, and then at the patent office in Bern, where Einstein helped him to get a job. Besso is credited with introducing Einstein to the works of Ernst Mach, the sceptical critic of physics who influenced Einstein's approach to the discipline. Einstein called Besso "the best sounding board in Europe" for scientific ideas. In Einstein's original paper on Special Relativity, he ended the paper stating, "In conclusion I wish to say that in working at the problem here dealt with I have had the loyal assistance of my friend and colleague M. Besso, and that I am indebted to him for several valuable suggestions."Besso died in Geneva, aged 81. In a letter of condolence to the Besso family, Albert Einstein wrote "Now he has departed from this strange world a little ahead of me. That means nothing. People like us, who believe in physics, know that the distinction between past, present and future is only a stubbornly persistent illusion." Einstein died one month and 3 days after his friend, on 18 April 1955.

Mécanique analytique

Mécanique analytique (1788–89) is a two volume French treatise on analytical mechanics, written by Joseph-Louis Lagrange, and published 101 years following Isaac Newton's Philosophiæ Naturalis Principia Mathematica. It consolidated into one unified and harmonious system, the scattered developments of contributors such as Alexis Clairaut, Jean le Rond d'Alembert, Leonhard Euler, and Johann and Jacob Bernoulli in the historical transition from geometrical methods, as presented in Newton's Principia, to the methods of mathematical analysis. The treatise expounds a great labor-saving and thought-saving general analytical method by which every mechanical question may be stated in a single differential equation.

Lagrange wrote that this work was entirely new and that his intent was to reduce the theory and the art of solving mechanics problems to general formulae, providing all the equations necessary for the solution of each problem. He stated that...No diagrams will be found in this work. The methods that I explain require neither geometrical, nor mechanical, constructions or reasoning, but only algebraical operations in accordance with regular and uniform procedure. Those who love Analysis will see with pleasure that Mechanics has become a branch of it, and will be grateful to me for having thus extended its domain.

Ernst Mach describes the work as follows:

Analytic mechanics... was brought to the highest degree of perfection... Lagrange's aim is... to dispose, once and for all, of the reasoning necessary to resolve mechanical problems, by embodying as much as possible of it in a single formula. This he did. Every case... can now be dealt with by a very simple... schema; and whatever reasoning is left is performed by purely mechanical methods. The mechanics of Lagrange is a stupendous contribution to the economy of thought.

Paul Pojman

Paul Theodore Pojman (October 11, 1966 – September 20, 2012) was a philosopher, activist, and gardener whose interdisciplinary work involved fields such as religion, economics, and ecology. He was a professor of philosophy at Towson University in Maryland from 2002 until his death; he lived in Baltimore city at the Baltimore Free Farm in Hampden.

Pojman was known for his community activism as well as his scholarship. He was involved in the Baltimore Green Currency Association, the Baltimore Free School, and the Baltimore Free Farm, and worked with Occupy Baltimore after it began in October 2011.Paul Pojman is the son of Louis Pojman, also a philosopher. Paul edited the fifth and sixth editions of his father's popular anthology textbook, Environmental Ethics.

Robert Bouvier

Robert Bouvier (1886–1978) was a Swiss philosopher noted for popularising the work of Ernst Mach in French.Bouvier spent much his career teaching evening classes and doing translation work before becoming a privatdocent at the University of Geneva.

Russian Machism

Russian Machism is a political/philosophical viewpoint which emerged in Imperial Russia in the beginning of the twentieth century before the Russian Revolution. They upheld the scientific and philosophical insights of Ernst Mach to be of great interest. Many of the Russian Machists were Marxists, and indeed viewed Machism as an essential ingredient of a materialist outlook on the world.

Shock diamond

Shock diamonds (also known as Mach diamonds, Mach disks, Mach rings, donut tails or thrust diamonds) are a formation of standing wave patterns that appear in the supersonic exhaust plume of an aerospace propulsion system, such as a supersonic jet engine, rocket, ramjet, or scramjet, when it is operated in an atmosphere. The diamonds are formed from a complex flow field and are visible due to the abrupt changes in local density and pressure caused by standing shock waves. Mach diamonds (or disks) are named after Ernst Mach, the physicist who first described them.

Vienna Circle

The Vienna Circle (German: Wiener Kreis) of Logical Empiricism was a group of philosophers and scientists drawn from the natural and social sciences, logic and mathematics who met regularly from 1924 to 1936 at the University of Vienna, chaired by Moritz Schlick.

The Vienna Circle's influence on 20th-century philosophy, especially philosophy of science and analytic philosophy, is immense up to the present day.

Among the members of the inner circle were Moritz Schlick, Hans Hahn, Philipp Frank, Otto Neurath, Rudolf Carnap, Herbert Feigl, Richard von Mises, Karl Menger, Kurt Gödel, Friedrich Waismann, Felix Kaufmann, Viktor Kraft and Edgar Zilsel. In addition, the Vienna Circle was occasionally visited by Alfred Tarski, Hans Reichenbach, Carl Gustav Hempel, Willard Van Orman Quine, Ernest Nagel, Alfred Jules Ayer, Oskar Morgenstern and Frank P. Ramsey. Ludwig Wittgenstein and Karl Popper were in close contact to the Vienna Circle, but never participated in the meetings of the Schlick Circle.The philosophical position of the Vienna Circle was called logical empiricism (German: logischer Empirismus), logical positivism or neopositivism. It was influenced by Ernst Mach, David Hilbert, French conventionalism (Henri Poincaré and Pierre Duhem), Gottlob Frege, Bertrand Russell, Ludwig Wittgenstein and Albert Einstein. The Vienna Circle was pluralistic and committed to the ideals of the Enlightenment. It was unified by the aim of making philosophy scientific with the help of modern logic. Main topics were foundational debates in the natural and social sciences, logic and mathematics; the modernization of empiricism by modern logic; the search for an empiricist criterion of meaning; the critique of metaphysics and the unification of the sciences in the unity of science.The Vienna Circle appeared in public with the publication of various book series – Schriften zur wissenschaftlichen Weltauffassung (Monographs on the Scientific World-Conception), Einheitswissenschaft (Unified Science) and the journal Erkenntnis – and the organization of international conferences in Prague; Königsberg (today known as Kaliningrad); Paris; Copenhagen; Cambridge, UK, and Cambridge, Massachusetts. Its public profile was provided by the Ernst Mach Society (German: Verein Ernst Mach) through which members of the Vienna Circle sought to popularize their ideas in the context of programmes for national education in Vienna.

During the era of Austrofascism and after the annexation of Austria by Nazi Germany most members of the Vienna Circle were forced to emigrate. The murder of Schlick in 1936 by a former student put an end to the Vienna Circle in Austria.

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