Ernst Mach

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Science Of Mechanics


Ernst Mach

Ernst Mach.
Ernst Mach.

Ernst Mach (pronounced [max], see IPA) (February 18, 1838 – February 19, 1916) was an Austrian physicist and philosopher and is the namesake for the "Mach number" (also known as Mach speed) and the optical illusion known as Mach bands.



Ernst Mach was born in Chrlice (now part of Brno), Czech Republic. Up to the age of 15 he was educated at home by his parents. He then joined the local Gymnasium and in 1855 the University of Vienna. There he studied mathematics, physics and philosophy, and received a doctorate in physics in 1860. His early work was focused on the Doppler effect in optics and acoustics. In 1864 he took a job as Professor of Mathematics in Graz, in 1866 he was also appointed as Professor of Physics. During that period Mach became interested also in the physiology of sensory perception. In 1867 he took the chair of Professor of Experimental Physics at Charles University, Prague, where he stayed for 28 years.

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

In 1897 he suffered a stroke and in 1901 retired from the University and was appointed to the upper chamber of the Austrian parliament. On leaving that post in 1913 he moved to his son's home in Vatterstetten, near Munich where he continued writing books until his death.


Most of his studies in the field of experimental physics were devoted to interference, diffraction, polarization and refraction of light in different media under external influences. These studies were soon followed by his important explorations in the field of supersonic velocity. Mach's paper on this subject was published in 1877 and correctly describes the sound effects observed during the supersonic motion of a projectile. Mach deduced and experimentally confirmed the existence of a shock wave which has the form of a cone with the projectile at the apex. The ratio of the speed of projectile to the speed of sound vp/vs is now called the Mach number. It plays a crucial role in aerodynamics and hydrodynamics. He also contributed to cosmology the hypothesis known as Mach's principle.

Sensory perception

In the area of sensory perception, he is best known for an optical illusion called the Mach band.

Philosophy of science

Mach developed a philosophy of science which was influential in the 19th and 20th centuries. Mach held that scientific laws are summaries of experimental events, constructed for the purpose of human comprehension of complex data. Thus scientific laws have more to do with the mind than with reality as it exists apart from the mind. Some quotations from Mach's writings will illustrate his philosophy. These selections are taken from his essay The Economical Nature of Physical Inquiry, excerpted by Kockelmans (citation below).

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.

In accordance with this philosophy, Mach opposed Ludwig Boltzmann and others who proposed an atomic theory of physics. Since atoms are too small to observe directly, and 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. Albert Einstein called him the "forerunner of [the] Theory of relativity", though Mach would later, to Einstein's disappointment, reject Einstein's theory.

Mach's positivism was also influential on many Russian Marxists, such as Alexander Bogdanov. In 1908, Lenin wrote a philosophical work Materialism and Empirio-Criticism in which he criticized the views of "Russian Machists".

The question is, did Mach deny the relevance of the a priori apropos of some of its possible significances or did he deny the a priori all together? If we are able to conlcude the former (rather than the positivist position: the latter), then we will be in a good position to suggest that a great mind such as Mach's would never have denied the significance of Kantian doctrine for an understanding of the scientific endeavor (including the significance of the a priori).

Mach's central text, as far as this debate is concerned, is his quasi-textbook entitled, "The Science of Mechanics." Thankfully, we can cut to the chase here as there are only a couple times that a priori or a posteriori are mentioned and, to complicate things further, it is always in the same vain; Mach denies the use of any a priori in arriving at the concepts that make up law-based propositions. In one of his only explicit mentions of the a priori in the Mechanics, Mach writes, "All this[, the derivation of the valid application of certain laws within certain circumstances,] has often led men to attribute knowledge of this kind to an entirely different source, namely, to view it as existing a priori in us (previous to all experience). That this opinion is untenable was fully explained in our discussion of the achievements of Stevinus. (p.83)" The specific 'achievements of Stevinus,' though they are indispensable so far as Mach's narrative of mechanics is concerned, is dispensable as far as his treatment of the a priori is concerned and so the reader's time will not be wasted with such explanation. With our goal in mind, it is only necessary to understand here that Stevinus arrived at a special application (static) application of a special law (a law of equilibrium, retrospectively derivative from the law of the composition of forces) that he could only have arrived at through experiment in the physical world. He could not pull a 'Bernoullian' trick, as Mach would have it; that is, he could prove the applicability of the law of the parallelogram of forces to the real world just because such a law is mathematically demonstrable prior to experience in the world; i.e. just because the law is a priori.

A HA! exclaim the positivists, so Mach does throw the a priori to the wolves. Well, no, unfortunately for the positivists, that is not true. A more careful reading or, might one even go so far as to say, any fair reading of Mach's text at all, would reveal that Mach only denies the a priori in a specific sense: in knowing how to validly or, in his terminology, 'economically' apply laws to the physical world. Only experience can tell us that, Mach insists, so don't go believing you can fly because you can dream about it--Mach warns the Bernoullians of the world. This, however, seems to be the advice of every great mind of the period, from Poincare to Einstein. Not-a-one of them would advocate for such a pre-experiential application of the concepts of the a priori and, surprise-surprise, neither would Kant.

Some, such as Hans Reichenbach, suggest, however, that there is a more subtle strain in Mach's conception of the way that laws arise and are validly applied. Such academics insist that Mach's conception of the inception of physical law is based upon empirical circumstance, is empirically given, and therefore contingent only upon circumstance. This tendency towards the given, which is so central to logical atomism, is nowhere to be found in Mach!

Mach is all about the 'imagination': the Kantian constructor of concept. Therefore it is no surprise that Mach acclaims Newton and his universal law of gravity. This law is, for Mach, forever true and applicable (universal). Yet this law is of a synthetic a priori nature: the very consideration that lead Kant to embark upon the Critique of Pure Reason. Furthermore, Mach's text is rich with suggestions as to the timelessness of laws, be it those of Gallileo or those of Newton (which, as he later points out, are to be conflated/subsumed one in the other). What is contingent for Mach is not what Newton called a 'law', but what he called a 'proposition'; namely that the law of gravity take on the form of the inverse square. Yet experience has proved it to be so, and therefor, though it may be a synthetic statement that's application to the world would have been arbitrary and audacious otherwise, because experience has validated it, it has taken on a synthetic a priori nature (which it could not have done without experience). This is the place that Mach reserves for the a priori for, at the same time that it is implied by his text, he never suggests or says otherwise. Indeed, it is worth considering Mach's one explicit mention of Kant within his entire text.

Mach writes, "It is scarcely necessary to remark that in the reflections here presented Newton has again acted contrary to his expressed intention only to investigate actual facts. (p.229)" He is here speaking of Newton's conceptions of Absolute Space and Absolute Time. Newton's example of the rotating bucket is invoked to delineate between the cetrifugal forces acting on the water and the forces of the rotating bucket as not indicating anything about Absolute Space, as he imagines Newton thought it did, but merely of the relative action of complex forces upon the water.

Newton's point, however, could be read to mean that, while we are repeatedly confronted by the action of complex forces, the fact that, as scientists, we are made to consider the potential of relative motions in an infinite regress (the bucket acting upon the water at the same time as the relatively stationary earth actus upon it, etc.---why stop at the earth?) and so we are faced with a humbling fact: as far as our finite minds might consider relative motions, there seems to be the infinite possibility of further considerations of relativity. "But if we take our stand on the basis of facts," Mach admits, "we shall find we have knowledge only of relative spaces and motions. (p.232)" How then, is the scientist to decipher the outline of a law-like proposition, such as that of the inverse square, in the face of this daunting relativity?

Newton thus concludes that we must assume that the relativity stops somewhere, that there is such a thing as an absolute space and an absolute time, for by such a conclusion the scientist is once again liberated to act as if the propositions that he constructs as he observes the relative motions of physical phenomena have absolute/universal significance: the universal gravitation that Mach so enthusiastically applauds. "No one is competent to predicate things about absolute space and absolute motion; they are pure things of thought, pure mental constructs, that cannot be produced in experience. (p.229)" Precisely, Mr. Mach, that is precisely what they are, however, he writes of the Absolutes as, "idle metaphysical conception[s]. (p.224)" Mach understands the necessity of relinquishing man's grip on attempting to nail down relativity from its infinite regress, as he writes, "It is certainly fortunate for us, that we can, from time to time, turn aside our eyes from the overpowering unity of the All, and allow them to rest on individual details, (p.235)" and so one has to wonder exactly why Mach took such an aversion to such utterances of the absolutes: poetic utterances, metaphysical concepts and practical ones at that.

The answer stares us in the face: Mach, for whatever reason(s), was hellbent on keeping metaphysics separate from science. Is this symptomatic of a poor understanding of the relation between metaphysics and science in the long-time established Kantian doctrine, or is this a symptom of a time period in which ghosts like the ether and the atom haunted physics? I am inclined, at this point, to suggest the latter, for (while Mach seems to have supported the ether) we have Mach's rabid anti-atomism as at least one salient symptom of what may fairly be portrayed as a desire to keep physics pure of the kinds of dialectical illusions that Kant illustrates in his antinomies (atoms versus plenum). Mach concludes, "No one is warranted in extending these principles beyond the boundaries of experience. In fact, such an extension is meaningless, as no one possesses the requisite knowledge to make use of it. (p.229)" Admittedly, this is an alarmingly dismissive tone when one considers the subtlety of the Newtonian conception... Mach explains, "No one is competent to say how the experiment would turn out if the sides of the vessel increased in thickness and mass till they were ultimately several leagues thick. The one experiment only lies before us, and our business [as physicists] is, to bring it into accord with the other facts known to us, and not with the arbitrary fictions of our imagination. (p.232)" But what of our business as metaphysicians? While Mach is thankful for our ability to act as if we observe absolute motions (though we know epistemologically that our knowledge reflects only relative motions), he warns, "But we should not omit, ultimately to complete and correct our views by a thorough consideration of the things which for the time being we left out of account (in our practical discursions into the realm of the absolues).

In an appended remark, Mach reveals, "I regard... Newton's distinction as an illusion. (p.543)" This remark is revealing because, while he shows that he understands the necessity of acting as if motions that we observe occur in absolute space, he is here demonstrating that he does not understand Newton to be alluding to this practical necessity. Perhaps his admiration for Newton as a 'first rate philospher' falls short of affording Newton the subtle reading that his words demand (for such an interpretation of his words is possible). Or perhaps Mach has resigned such a subtle reading to the persistent co-opting of Newton by his contemporaries in the name of cruder reading (Streintz's criticism of Mach seems to suggest this latter position). In any case, Mach does portray Newton as having been sloppy in form as he writes, "Like the commander of an army, a great discoverer cannot stop to institute petty inquiries regarding the right by which he holds each post of vantage he has won... Newton might well have expected of the two centuries to follow that they should further examine and confirm the foundations of his work. (p.245)"--i.e. clean it up, not in substance but in presentation, for, as Mach writes, "[Newton] was, as it is possible to prove, not perfectly clear himself (in his writings) concerning the import and especially concerning the source of his principles. (p.244)"

Mach says what Newton ought to have said (and, of course, can be read as having said). However, there is one last point in treating of the philosophy underlying Mach's scientific endeavor, and that is whether he was actually aware that his well grounded treatment of the a priori (negative) kept the a priori within the bounds that Kant intended it to be: for that he certainly did. That is, was Mach aware of the positive place of the a priori within his own exposition or was he unwittingly re-inventing the wheel? Signs point to no, that he was not aware of his inherited philosophical foundation, as he writes, "Newton might well have expected... that, when times of greater scientific tranquillity should come (Mach is here implying the future after even his own time), the principles of the subject might acquire an even higher philosophical interest than all that is deducible from them. (p245)" What is there left to imagine in terms of deducing from principles, if all phenomenal deductions are excluded, than a 'transcendental' deduction of the Kantian variety? The evidence seems to suggest that Mach didn't so much deny the significance of the Kantian doctrine, but merely that he wasn't aware of the full implications of that doctrine.


  • J. Kockelmans. Philosophy of science: the historical background. New York: The Free Press, 1968.
  • B. Bliumis. "Mach and the A Priori". Bard College: Home Computer, 2007

This article might use material from a Wikipedia article, which is released under the Creative Commons Attribution-Share-Alike License 3.0.

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