This is an important book that develops some fundamental ideas about mathematics and the philosophy of mathematics. It compares and contrasts Hilbert's conception of axiomatic methods with Euclid's and with Lawvere's more recent work using category theory. By this light, Hilbert's attempt to reorient mathematics completely away from intuitive understanding to rule-based manipulation of symbols based on the rock of pre-established logic was unsuccessful. Indeed, Rodin implies it was doomed to fail since it was divorced from mathematical practice and had a flawed sense of logic’s independence.
Using a historical and philosophical perspective, Rodin argues that both mathematics and logic are human-derived tools that can and should be tailored to suit the tasks at hand. He makes a convincing case for using construction in the Euclidian sense: i.e., a more dynamic mathematics that posits the ability to create things rather than the assumption of their pre-existence. One example he cites is Euclid’s assertion that one can construct a line joining two points in contrast to Hilbert’s stating that such a line (already) exists between any two points.
The author stakes a claim for a new axiomatic method based on the work of Lawvere and others. The final chapter presents a sketch of what this new method could look like.
This is by no means an easy read but it stimulates and rewards the reader who sticks with it.
It is disappointing that the book is so poorly edited. While the message comes through, the author's use of English is non-standard. The grammar is tortured and choice of vocabulary is occasionally jarring. He consistently uses "remind" when "recall" or "remember" would work better; he uses "boarders" when he means "borders". There are many similar example on every page. A major publishing house should have done a better job.
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他の国からのトップレビュー
Got'em All
Important book, needs a good edit
2015年3月9日にカナダでレビュー済みAmazonで購入
Hunter Washburne
excellent overview of some complex ideas suitable for non-mathematicians
2014年3月19日にアメリカ合衆国でレビュー済みAmazonで購入
