I guessed that this would be an article about how mathematicians are starting to become newly important today, with the rise of ML. Most use of mathematics we rely on, up to the present, is performed by non-mathematicians.
Mathematics is used in statistics, physics, biology, finance, engineering, medicine, social science and more.
Even in the 15th century, most mathematics was done by non-mathematicans, it's not a new thing.
To a physicist like me, the assumption that mathematicians are the source of mathematical revelations is the same as assuming that the best novels are produced by people who studied english.
That is by no means an apt metaphor. Mathematicians are in the business of creating new mathematics, not in studying the "language" in which mathematics happens to be written. Of course the history of mathematics is replete with contributions from physicists and others, particularly during the period when the educated could turn their hand to nearly any such discipline and find a problem on which progress could be made.
Yes. In fact it has often been noted by physicists that when they make a new discovery they sometimes find that mathematicians have been there before them. Like Einstein and non-euclidean geometry etc.
It more often goes the other way, and more profoundly.
Group theory.
Laplace transforms were considered an irrelevant curiosity until somebody figured out they were isomorphic to the wildly effective Heaviside D operator. Then everyone did their best to bury Heaviside's demonstrations of their utility, and pretend that Laplace had done all the heavy lifting.
There are, of course, many exceptions to this "rule". From the preface to my favorite calculus book:
My aim is to exhibit the close connexion between analysis and its applications and, without loss of rigour and precision, to give due credit to intuition as the source of mathematical truth. The presentation of analysis as a closed system of truths without reference to their origin and purpose has, it is true, an aesthetic charm and satisfies a deep philosophical need. But the attitude of those who consider analysis solely as an abstractly logical, introverted science is not only highly unsuitable for beginners but endangers the future of the subject; for to pursue mathematical analysis while at the same time turning one's back on its applications and on intuition is to condemn it to hopeless atrophy. To me it seems extremely important that the student should be warned from the very beginning against a smug and presumptuous purism; this is not the least of my purposes in writing this book.
- Richard Courant, Differential and Integral Calculus [1]
Courant, who trained — under Hilbert, no less — as a pure mathematician, never tired of pointing out the importance of applications to pure mathematics and vice versa. In addition to the calculus books, see, e.g.,
At my workplace, any math that can't be done by spreadsheet or with the math built into software tools (such as CAD) is delegated to a small handful of "math people" who happen to have backgrounds in math or physical science, and who retained an interest in math after graduating.
Granted, what can be done with spreadsheet or CAD is pretty impressive.
How important were math books in English really? English mathematicians were reading and writing latin at the time. And would have had access to Euclid well before that book was written.
Somewhere is another HN article about all the navigation schools that sprung up during the time this article discusses, how eventually everything req you to build technical skills even a contract sailor hundreds of years ago.
Mathematics is the source of Western civilization. Modern science and technology is descended from philosophy and theology, which started with the mathematical presocratics and Plato's use of math to argue for unchanging forms, justifying the pursuit of philosophy in itself.
I kind of like thinkibg of mathematics and language as technology for the mind - in the sense that both enable you to think thoughts that you otherwise may not have been able to think .. and to teach others how to do it, thereby transferring the technology.
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[ 2.7 ms ] story [ 24.8 ms ] threadOff by only five centuries...
That's like saying: Most of the physical products we buy are assembled by non-inventors. ;)
Group theory.
Laplace transforms were considered an irrelevant curiosity until somebody figured out they were isomorphic to the wildly effective Heaviside D operator. Then everyone did their best to bury Heaviside's demonstrations of their utility, and pretend that Laplace had done all the heavy lifting.
My aim is to exhibit the close connexion between analysis and its applications and, without loss of rigour and precision, to give due credit to intuition as the source of mathematical truth. The presentation of analysis as a closed system of truths without reference to their origin and purpose has, it is true, an aesthetic charm and satisfies a deep philosophical need. But the attitude of those who consider analysis solely as an abstractly logical, introverted science is not only highly unsuitable for beginners but endangers the future of the subject; for to pursue mathematical analysis while at the same time turning one's back on its applications and on intuition is to condemn it to hopeless atrophy. To me it seems extremely important that the student should be warned from the very beginning against a smug and presumptuous purism; this is not the least of my purposes in writing this book.
- Richard Courant, Differential and Integral Calculus [1]
Courant, who trained — under Hilbert, no less — as a pure mathematician, never tired of pointing out the importance of applications to pure mathematics and vice versa. In addition to the calculus books, see, e.g.,
https://www.ams.org/journals/bull/1943-49-01/S0002-9904-1943...
[1] https://archive.org/details/DifferentialIntegralCalculusVolI...
Granted, what can be done with spreadsheet or CAD is pretty impressive.
https://en.m.wikipedia.org/wiki/History_of_coffee#England