Aleksandrov, Kolmogorov, Lavrent'ev; Mathematics: Its Content, Methods and Meaning (ISBN 0486409163 (amazon.com, search))
Perhaps the most remarkable mathematics book I've read so far. Similar to FeynmanLecturesOnPhysics?, as it starts out with an introduction to the field (touching on its relation to other fields), then plunges into a survey of topics from algebra to topology, written by 18 mathematicians.
The introduction explains mathematics as an abstraction of some of human thought, with two main ideas: numbers and three-dimensional objects. (Or as they say, arithmetic and geometry.) Like FeynmanLecturesOnPhysics?, it argues against an overly idealistic philosophical view of mathematics. The authors claim, "Idealists and metaphysicists not only fall into confusion in their attempts to answer these basic questions, but they go so far as to distort mathematics completely, turning it literally inside out. Thus, seeing the extreme abstractness and cogency of mathematical results, the idealist imagines that mathematics issues from pure thought. In reality, mathematics offers not the slightest support for idealism or metaphysics."
This mirror's Feynman's claim that, "Philosophers, incidentally, say a great deal about what is absolutely necessary for science, and it is always, so far as one can see, rather naive, and probably wrong."
One example is of straight lines: in modern times, we understand them intuitively because we've grown up around many manufactured objects, which feature straight lines and smooth curves. But before manufacturing, straight lines occurred very infrequently in nature. Maybe a tree or still waterline would be particularly straight; or a moon be very circular. Eventually, we'd stretch a line taut to fit a bow, and sculpt smooth pottery. When farmers were very interested in partitioning their land, mathematicians would consciously develop the abstraction of straight lines... an abstraction which is impossible to realize in the real world, because things such as density and color are abstracted away (not to mention a couple of dimensions are missing, being 1-D). The authors explain that countless humans had to manufacture and observe countless straight lines, before the notion was seriously abstracted. And yet this unrealizable abstraction has large real-world applicability because those ties to a specific reality are severed.
A similar process occurred with numbers. Ancient people might have had different words for "three" when referring to three people vs. three goats. Through time, we generalized numbers considerably. Eventually, we modified our numeric notations for ease in representing them and for simplifying many common calculations.
These processes of abstracting things go on with time. The authors mentioned that currently (in the 1950's), the notion of "function" still underwent modifications, as the rest of our mathematics grew and stretched its foundational concepts thin.
One note though... it seems a couple sections on "material dialecticism" were censored from the book (and the cited Russian books, which weren't yet available in English, were not listed). While they give a little note saying that the dialecticism parts were already well explained by previous sections and therefore unnecessary, it's suspicious. (I probably wouldn't have cared for this über-philosophical Marxist stuff, but I would've liked to have made the decision not to read it on my own.)
(Keep in mind I'm still reading the book and haven't yet finished it.)
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