"The aim of natural science is to obtain connections among phenomena. Theories, however, are like withered leaves, which drop off after having enabled the organism of science to breathe for a time." (Ernst Mach, "Die Geschichte und die Wurzel des Satzes von der Erhaltung der Arbeit", 1871)
"Historical investigation not only promotes the understanding
of that which now is, but also brings new possibilities before us, by showing
that which exists to be in great measure conventional and accidental. From the higher
point of view at which different paths of thought converge we may look about us
with freer vision and discover routes before unknown."
"[…] not only a knowledge of the ideas that have been accepted and cultivated by subsequent teachers is necessary for the historical understanding of a science, but also that the rejected and transient thoughts of the inquirers, nay even apparently erroneous notions, may be very important and very instructive. The historical investigation of the development of a science is most needful, lest the principles treasured up in it become a system of half-understood prescripts, or worse, a system of prejudices." (Ernst Mach, "The Science of Mechanics", 1883)
"Purely mechanical phenomena do not exist […] are abstractions, made, either intentionally or from necessity, for facilitating our comprehension of things. The science of mechanics does not comprise the foundations, no, nor even a part of the world, but only an aspect of it." (Ernst Mach, "The Science of Mechanics", 1883)
"A person who knew the world only through the theatre, if brought behind the scenes and permitted to view the mechanism of the stage’s action, might possibly believe that the real world also was in need of a machine-room, and that if this were once thoroughly explored, we should know all. Similarly, we, too, should beware lest the intellectual machinery, employed in the representation of the world on the stage of thought, be regarded as the basis of the real world." (Ernst Mach, "The Science of Mechanics; a Critical and Historical Account of Its Development", 1893)
"In every symmetrical system every deformation that tends to destroy the symmetry is complemented by an equal and opposite deformation that tends to restore it. […] One condition, therefore, though not an absolutely sufficient one, that a maximum or minimum of work corresponds to the form of equilibrium, is thus applied by symmetry." (Ernst Mach, "The Science of Mechanics: A Critical and Historical Account of Its Development", 1893)
"Properly speaking the world is not composed of 'things' as its elements, but colors, tones, pressures, spaces, times, in short what we ordinarily call individual sensations." (Ernst Mach, "The Science of Mechanics", 1893)
"That branch of physics which is at once the oldest and the simplest and which is therefore treated as introductory to other departments of this science, is concerned with the motions and equilibrium of masses. It bears the name of mechanics." (Ernst Mach, "The Science of Mechanics: A Critical and Historical Account of Its Development", 1893)
"The atomic theory plays a part in physics similar to that of certain auxiliary concepts in mathematics: it is a mathematical model for facilitating the mental reproduction of facts. Although we represent vibrations by the harmonic formula, the phenomena of cooling by exponentials, falls by squares of time, etc, no one would fancy that vibrations in themselves have anything to do with circular functions, or the motion of falling bodies with squares." (Ernst Mach, "The Science of Mechanic", 1893)
"In algebra we perform, as far as possible, all numerical operations which are identical in form once for all, so that only a remnant of work is left for the individual case. The use of the signs of algebra and analysis, which are merely symbols of operations to be performed, is due to the observation that we can materially disburden the mind in this way and spare its powers for more important and more difficult duties, by imposing all mechanical operations upon the hand." (Ernst Mach, "The Economical Nature of Physical Enquiry", Popular Scientific Lectures, 1895)
"Strange as it may sound, the power of mathematics rests upon its evasion of all unnecessary thought and on its wonderful saving of mental operation. Even those arrangement-signs which we call numbers are a system of marvelous simplicity and economy. When we employ the multiplication-table in multiplying numbers of several places, and so use the results of old operations of counting instead of performing the whole of each operation anew; when we consult our table of logarithms, replacing and saving thus new calculations by old ones already performed; when we employ determinants instead of always beginning afresh the solution of a system of equations; when we resolve new integral expressions into familiar old integrals; we see in this simply a feeble reflexion of the intellectual activity of a Lagrange or a Cauchy, who, with the keen discernment of a great military commander, substituted new operations for whole hosts of old ones. No one will dispute me when I say that the most elementary as well as the highest mathematics are economically-ordered experiences of counting, put in forms ready for use." (Ernst Mach, "Popular Scientific Lectures", 1895)
"The aim of research is the discovery of the equations which subsist between the elements of phenomena." (Mach Ernst, 1898)
"[…] scientific research is somewhat like unraveling complicated tangles of strings, in which luck is almost as vital as skill and accurate observation." (Ernst Mach, "Knowledge and Error: Sketches on the Psychology of Enquiry", 1905)
"A symbolical representation of a method of calculation has the same significance for a mathematician as a model or a visualisable working hypothesis has for a physicist. The symbol, the model, the hypothesis runs parallel with the thing to be represented. But the parallelism may extend farther, or be extended farther, than was originally intended on the adoption of the symbol. Since the thing represented and the device representing are after all different, what would be concealed in the one is apparent in the other." (Ernst Mach, "Space and Geometry: In the Light of physiological, phycological and physical inquiry", 1906)
"Geometry, accordingly, consists of the application of mathematics to experiences concerning space. Like mathematical physics, it can become an exact deductive science only on the condition of its representing the objects of experience by means of schematizing and idealizing concepts." (Ernst Mach, "Space and Geometry: In the Light of physiological, phycological and physical inquiry", 1906)
"Physiological, and particularly visual, space appears as a distortion of 'geometrical space when derived from the metrical data of geometrical space. But the properties of continuity and threefold manifoldness are preserved in such a transformation, and all the consequences of these properties may be derived without recourse to physical experience, by our representative powers solely." (Ernst Mach, "Space and Geometry: In the Light of physiological, phycological and physical inquiry", 1906)
"Physics shares with mathematics the advantages of succinct description and of brief, compendious definition, which precludes confusion, even in ideas where, with no apparent burdening of the brain, hosts of others are contained." (Ernst Mach)
"Scientists must use the simplest means of arriving at their results and exclude everything not perceived by the senses." (Ernst Mach)
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