A few notes from - Men of Mathematics (Touchstone Book) (E. T. Bell):
It must not be imagined that the sole function of mathematics—“the handmaiden of the sciences”—is to serve science. Mathematics has also been called “the Queen of the Sciences.”
Mathematics has a light and wisdom of its own, above any possible application to science, and it will richly reward any intelligent human being to catch a glimpse of what mathematics means to itself.
Lagrange (whom we shall meet later) believed that a mathematician has not thoroughly understood his own work till he has made it so clear that he can go out and explain it effectively to the first man he meets on the street.
But it may be recalled that only a few years before Lagrange said this the Newtonian “law” of gravitation was an incomprehensible mystery to even highly educated persons.
Students of mathematics are familiar with the phenomenon of “slow development,” or subconscious assimilation: the first time something new is studied the details seem too numerous and hopelessly confused, and no coherent impression of the whole is left on the mind.
Skipping is not a vice, as some of us were told by our puritan teachers, but a virtue of common sense.
It must not be imagined that the sole function of mathematics—“the handmaiden of the sciences”—is to serve science. Mathematics has also been called “the Queen of the Sciences.”
Mathematics has a light and wisdom of its own, above any possible application to science, and it will richly reward any intelligent human being to catch a glimpse of what mathematics means to itself.
Lagrange (whom we shall meet later) believed that a mathematician has not thoroughly understood his own work till he has made it so clear that he can go out and explain it effectively to the first man he meets on the street.
But it may be recalled that only a few years before Lagrange said this the Newtonian “law” of gravitation was an incomprehensible mystery to even highly educated persons.
Students of mathematics are familiar with the phenomenon of “slow development,” or subconscious assimilation: the first time something new is studied the details seem too numerous and hopelessly confused, and no coherent impression of the whole is left on the mind.
Skipping is not a vice, as some of us were told by our puritan teachers, but a virtue of common sense.
In saying that Descartes was responsible for the creation of analytic geometry we do not mean to imply that the new method sprang fullarmed from his mind alone.
Relativity, for example, is sometimes said to have been the great invention reserved by time for the genius of Minkowski.
Archimedes had the fundamental notion of limiting sums from which the integral calculus springs, and he not only had the notion but showed that he could apply it. Archimedes also used the method of the differential calculus in one of his problems. As we approach Newton and Leibniz in the seventeenth century the history of the calculus becomes extremely involved. The new method was more than merely “in the air” before Newton and Leibniz brought it down to earth; Fermat actually had documented significant parts of it ...
Only by seeing in detail what manner of men some of the great mathematicians were and what kind of lives they lived, can we recognize the ludicrous untruth of the traditional portrait of a mathematician.
Returning for a moment to the movie ideal of a mathematician, we note that sloppy clothes have not been the invariable attire of great mathematicians.
Men who should have been above such brawls seem to have gone out of their way to court battles over priority in discovery and to accuse their competitors of plagiarism.
Both inventors and perfectors are necessary to the progress of any science. Every explorer must have, in addition to his scouts, his followers to inform the world as to what he has discovered.
Things that now seem as simple as common sense—our way of writing numbers, for instance, with its “place system” of value and the introduction of a symbol for zero, which put the essential finishing touch to the place system—cost incredible labor to invent. Even simpler things, containing the very essence of mathematical thought —abstractness and generality
The mystical “not-being” of the seventeenth century Leibniz is seen to have a “being” as simple as ABC.
On the life of Gauss:
Nevertheless Gauss crowned the higher arithmetic, in his day the least practical of mathematical studies, the Queen of all.
As a child he was respectful and obedient, and although he never criticized his poor father in later life, he made it plain that he had never felt any real affection for him. Gerhard died in 1806. By that time the son he had done his best to discourage had accomplished immortal work.
Dorothea Gauss took her boy’s part and defeated her obstinate husband in his campaign to keep his son as ignorant as himself.
Gauss himself cared little if anything for fame; his triumphs were his mother’s life.
Before this the boy had teased the pronunciations of the letters of the alphabet out of his parents and their friends and had taught himself to read.
Shortly after his seventh birthday Gauss entered his first school, a squalid relic of the Middle Ages run by a virile brute, one Büttner, whose idea of teaching the hundred or so boys in his charge was to thrash them into such a state of terrified stupidity that they forgot their own names.
Büttner was so astonished at what the boy of ten had done without instruction that he promptly redeemed himself and to at least one of his pupils became a humane teacher. Out of his own pocket he paid for the best textbook on arithmetic obtainable and presented it to Gauss. The boy flashed through the book. “He is beyond me,” Büttner said; “I can teach him nothing more.”
Before entering the Caroline College at the age of fifteen, Gauss had made great headway in the classical languages by private study and help from older friends, thus precipitating a crisis in his career. To his crassly practical father the study of ancient languages was the height of folly.
When Gauss left the Caroline College in October, 1795 at the age of eighteen to enter the University of Göttingen he was still undecided whether to follow mathematics or philology as his life work. He had already invented (when he was eighteen) the method of “least squares,”
Gauss shares this honor with Legendre who published the method independently in 1806. This work was the beginning of Gauss’ interest in the theory of errors of observation.
A cathedral is not a cathedral, he said, till the last scaffolding is down and out of sight. Working with this ideal before him, Gauss preferred to polish one masterpiece several times rather than to publish the broad outlines of many as he might easily have done.
His own contemporaries begged him to relax his frigid perfection so that mathematics might actually digest some of it...
Consequently some of his works had to wait for highly gifted interpreters before mathematicians in general could understand them, see their significance for unsolved problems, and go ahead. His own contemporaries begged him to relax his frigid perfection so that mathematics might advance more rapidly, but Gauss never relaxed.
the Duke came to his rescue, paid for the printing of his doctoral dissertation (University of Helmstedt, 1799), and granted him a modest pension which would enable him to continue his scientific work unhampered by poverty. “Your kindness,” Gauss says in his dedication, “freed me from all other responsibilities and enabled me to assume this exclusively.”
The word “imaginary” is the great algebraical calamity, but it is too well established for mathematicians to eradicate. It should never have been used.
But arithmetic was his first love, and he regretted in later life that he had never found the time to write the second volume he had planned as a young man.
