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Mathematics may be defined as the economy of counting. There is no problem in the whole of mathematics which cannot be solved by direct counting.

Ernst Mach

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Ernst Mach
Age: 78 †
Born: 1838
Born: February 18
Died: 1916
Died: February 19
Pedagogue
Philosopher
Photographer
Physicist
Politician
Professor
Ernst Mach Jr
Whole
Math
Defined
Mathematics
Direct
Economy
Cannot
Solved
Problem
Counting
May
Mathematical

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Strange as it may sound, the power of mathematics rests on its evasion of all unnecessary thought and on its wonderful saving of mental operations.
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The fact is, every thinker, every philosopher, the moment he is forced to abandon his one-sided intellectual occupation by practical necessity, immediately returns to the general point of view of mankind.
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The history of the development of mechanics is quite indispensable to a full comprehension of the science in its present condition. It also affords a simple and instructive example or the processes by which natural science generally is developed.
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When I recall today my early youth, I should take the boy that I then was, with the exception of a few individual features, for a different person, were it not for the existence of the chain of memories.
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The acquisition of the most elementary truth does not devolve upon the individual alone: it is pre-effected in the development of the race.
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A movement that we will to execute is never more than a represented movement, and appears in a different domain from that of the executed movement, which always takes place when the image is vivid enough.
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Archimedes constructing his circle pays with his life for his defective biological adaptation to immediate circumstances.
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I can accept the theory of relativity as little as I can accept the existence of atoms and other such dogmas.
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A colour is a physical object as soon as we consider its dependence, for instance, upon its luminous source, upon other colours, upon temperatures, upon spaces, and so forth.
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Science is the most complete presentment of facts with the least expenditure of thought
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Ordinarily pleasure and pain are regarded as different from sensations.
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What Mach calls a thought experiment is of course not an experiment at all. At bottom it is a grammatical investigation.
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Not bodies produce sensations, but element-complexes (sensation-complexes) constitute the bodies. When the physicist considers the bodies as the permanent reality, the 'elements' as the transient appearance, he does not realise that all 'bodies' are only mental symbols for element-complexes (sensation-complexes).
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To us investigators, the concept 'soul' is irrelevant and a matter for laughter. But matter is an abstraction of exactly the same kind, just as good and just as bad as it is. We know as much about the soul as we do of matter.
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Many an article that I myself penned twenty years ago impresses me now as something quite foreign to myself.
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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.
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Scientists must use the simplest means of arriving at their results and exclude everything not perceived by the senses.
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I once knew an otherwise excellent teacher who compelled his students to perform all their demonstrations with incorrect figures, on the theory that it was the logical connection of the concepts, not the figure, that was essential.
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Thing, body, matter, are nothing apart from the combinations of the elements, - the colours, sounds, and so forth - nothing apart from their so-called attributes.
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All this, the positive and physical essence of mechanics, which makes its chief and highest interest for a student of nature, is in existing treatises completely buried and concealed beneath a mass of technical considerations.
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