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To those who inquire as to the purpose of mathematics, the usual answer will be that it facilitates the making of machines, the travelling from place to place, and the victory over foreign nations, whether in war or commerce. ... The reasoning faculty itself is generally conceived, by those who urge its cultivation, as merely a means for the avoidance of pitfalls and a help in the discovery of rules for the guidance of practical life.

 

Bertrand Russell

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Since reasoning, or inference, the principal subject of logic, is an operation which usually takes place by means of words, and in complicated cases can take place in no other way: those who have not a thorough insight into both the signification and purpose of words, will be under chances, amounting almost to certainty, of reasoning or inferring incorrectly.

 

John Stuart Mill

 

The principal lever relied on by the insurgents for exciting foreign nations to hostility against us, as already intimated, is the embarrassment of commerce. Those nations, however, not improbably saw from the first that it was the Union which made as well our foreign as our domestic commerce. They can scarcely have failed to perceive that the effort for disunion produces the existing difficulty, and that one strong nation promises more durable peace and a more extensive, valuable, and reliable commerce than can the same nation broken into hostile fragments.

 

Abraham Lincoln

 

When, as a result of what was called Enlightenment in the eighteenth century, the priests had in fact almost entirely lost this function of guidance. Their place was taken by writers and scientists. In both cases it is equally absurd. Mathematics, physics, and biology are as remote from spiritual guidance as the art of arranging words. When that function is usurped by literature and science it proves there is no longer any spiritual life.

 

Simone Weil

 

Let us never forget that the cultivation of the earth is the most important labor of man. Man may be civilized, in some degree, without great progress in manufactures and with little commerce with his distant neighbors. But without the cultivation of the earth, he is, in all countries, a savage. Until he gives up the chase, and fixes himself in some place and seeks a living from the earth, he is a roaming barbarian. When tillage begins, other arts follow. The farmers, therefore, are the founders of human civilization.

 

Daniel Webster

 

Before entering into these details, however, it may be well to reply to the very natural question: What would be the use of such extreme refinement in the science of measurement? Very briefly and in general terms the answer would be that in this direction the greater part of all future discovery must lie. The more important fundamental laws and facts of physical science have all been discovered, and these are so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote. Nevertheless, it has been found that there are apparent exceptions to most of these laws, and this is particularly true when the observations are pushed to a limit, i.e., whenever the circumstances of experiment are such that extreme cases can be examined. Such examination almost surely leads, not to the overthrow of the law, but to the discovery of other facts and laws whose action produces the apparent exceptions.

As instances of such discoveries, which are in most cases due to the increasing order of accuracy made possible by improvements in measuring instruments, may be mentioned: first, the departure of actual gases from the simple laws of the so-called perfect gas, one of the practical results being the liquefaction of air and all known gases; second, the discovery of the velocity of light by astronomical means, depending on the accuracy of telescopes and of astronomical clocks; third, the determination of distances of stars and the orbits of double stars, which depend on measurements of the order of accuracy of one-tenth of a second—an angle which may be represented as that which a pin's head subtends at a distance of a mile. But perhaps the most striking of such instances are the discovery of a new planet by observations of the small irregularities noticed by Leverier in the motions of the planet Uranus, and the more recent brilliant discovery by Lord Rayleigh of a new element in the atmosphere through the minute but unexplained anomalies found in weighing a given volume of nitrogen. Many instances might be cited, but these will suffice to justify the statement that "our future discoveries must be looked for in the sixth place of decimals." It follows that every means which facilitates accuracy in measurement is a possible factor in a future discovery, and this will, I trust, be a sufficient excuse for bringing to your notice the various methods and results which form the subject matter of these lectures.

 

Albert Abraham Michelson