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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
Cannot
Solved
Problem
Counting
May
Mathematical
Whole
Math
Defined
Mathematics
Direct
Economy

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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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Man is pre-eminently endowed with the power of voluntarily and consciously determining his own point of view.
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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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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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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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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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Many an article that I myself penned twenty years ago impresses me now as something quite foreign to myself.
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The plain man is familiar with blindness and deafness, and knows from his everyday experience that the look of things is influenced by his senses but it never occurs to him to regard the whole world as the creation of his senses.
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The presentations and conceptions of the average man of the world are formed and dominated, not by the full and pure desire for knowledge as an end in itself, but by the struggle to adapt himself favourably to the conditions of life.
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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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Archimedes constructing his circle pays with his life for his defective biological adaptation to immediate circumstances.
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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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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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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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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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Without renouncing the support of physics, it is possible for the physiology of the senses, not only to pursue its own course of development, but also to afford to physical science itself powerful assistance.
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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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My table is now brightly, now dimly lighted. Its temperature varies. It may receive an ink stain. One of its legs may be broken. It may be repaired, polished, and replaced part by part. But, for me, it remains the table at which I daily write.
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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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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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